Dispersion, an Important Weak Force:
Case Studies and Development of an Anisotropic Parameter-free
Procedure for the Evaluation of Dispersion
Ph.D. dissertation
András Olasz
Supervisor:
Prof. Tamás Veszprémi
BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS
Department of Inorganic and Analytical Chemistry
H-1521 Budapest, Hungary
2016
NYILATKOZAT
Alulírott Olasz András kijelentem, hogy ezt a doktori értekezést magam
készítettem és abban csak a megadott forrásokat használtam fel. Minden olyan
részt, amelyet szó szerint, vagy azonos tartalomban, de átfogalmazva más
forrásból átvettem, egyértelműen, a forrás megadásával megjelöltem.
Budapest, 2016.05.30.
…...………………
Olasz András
Motto:
"John, when people thought the earth was flat, they were wrong. When
people thought the earth was spherical, they were wrong. But if you think
that thinking the earth is spherical is just as wrong as thinking the earth is
flat, then your view is wronger than both of them put together."*
- Isaac Asimov
*
The Skeptical Inquirer, Fall 1989, Vol. 14, No. 1, 35-44
ii
Köszönetnyilvánítás
Köszönöm Dr. Veszprémi Tamásnak, amiért tizenöt éve folyamatosan hozzájárul, hogy a
kvantumkémia egyre nagyobb mélységeit megismerhessem. Köszönöm, hogy türelmes
magyarázataira, tanácsaira és megértésére minden körülmények között szakmailag és emberileg is
számíthattam. Hálás vagyok, amiért akkor is bízott bennem, amikor én már feladtam volna.
Köszönöm Dr. Paul Geerlingsnek, hogy számításaimat az ő kutatócsoportjában végezhettem.
Köszönöm Dr. Pierre Mignonnak közös cikkünk megírásához nyújtott segítséget, és hogy a
gyengén kölcsönható π-aromás rendszereket megismertette velem.
Köszönöm Dr. Kenno Vanommeslaeghenek az izgalmas együttgondolkodásokat, Fortran kódírással
töltött órákat és a magas szintű szakmai és emberi hozzállását.
Köszönöm Dr. Mu-Hyun Baiknak és Dr. Dongwhan Leenek, hogy számításaimat az ő
kutatócsoportjaikban végezhettem.
Köszönöm Dr. Pintér Balázsnak a közös kutatással töltött éveket, a jóhangulatú beszélgetéseket és
a felejthetetlen edzéseket. Hálás vagyok neki és feleségének, Skara Gabriellának az ábrák és
táblázatok szerkesztéséhez nyújtott segítségéért.
Hálás vagyok Dr. Zsombok Györgynek a sok jótanácsért, a kérdéseimre adott kimerítő válaszokért
és a gyakran könnyfakasztó faviccekért, mely tekintetben behozhatatlan lemaradásban vagyok.
Hálás vagyok erkölcsi példamutatásáért.
Köszönöm Dr. Oláh Juliannának a konceptuális DFT elméleti összefoglalásában nyújtott
felbecsülhetetlen segítségéért (2.1.4-es szekció).
Köszönöm Dr. Szilvási Tibornak a modern DFT módszerek tárgyalásához nyújtott kiváló szakmai
segítségét (3-as fejezet) és a gondolatébresztő szakmai beszélgetéseket.
Köszönöm édesapámnak, Olasz Miklósnak a hivatkozások és rövidítések szerkesztéséhez nyújtott
segítségét.
Köszönöm Dr. Höltzl Tibornak az izgalmas együttgondolkodásokat, és hogy a Python
programozási nyelvet megismertette velem.
Hálás szívvel gondolok Dr. Oláh Juliannára, Dr. Szieberth Dénesre, Dr. Hajgató Balázsra, Dr.
Kárpáti Tamásra és Dr. Bajor Gáborra amiért tudásukat önzetlenül és türelmesen megosztották
velem. Meghálálni csak úgy tudom, ha én is hasonlóképpen bánok az új generációval.
A tudományosan és emberileg inspiráló közösség a „Veszprémi Csoport” létrehozásához külön
köszönettel tartozom még Dr. Veszprémi Tamásnak, Dr. Zsombok Györgynek, Dr. Pintér
Balázsnak, Dr. Petrov Klárának, Dr. Höltzl Tibornak, Dr. Németh Balázsnak, Dr. Oláh
Juliannának, Dr. Szieberth Dénesnek, Dr. Hajgató Balázsnak, Dr. Kárpáti Tamásnak és Dr. Bajor
Gábornak.
Nagyon hálás vagyok Varga Noéminek, aki segített az utolsó néhány száz formázási hiba
javításában.
Hálás köszönöttel tartozom szüleimnek, akik nélkül sem a jelen disszertáció sem a szerző nem
jöhetett volna létre. Feltétlen bizalmuk és támogatásuk sok nehézségen átsegített.
iii
Table of Contents
2
1.1
Introduction to the Author .................................................................................................. 4
1.2
Introduction to Weak Interactions in Chemistry ................................................................. 5
Quantum Chemical Methods ...................................................................................................... 8
2.1
A Hierarchy of Approximations ......................................................................................... 9
2.1.1 The Hamiltonian ........................................................................................................... 12
2.1.2 The Hartree-Fock (HF) and Hartree-Fock-Roothaan (HFR) Methods ......................... 14
2.1.3 Density Functional Theory (DFT) ................................................................................ 20
2.1.4 DFT-derived chemical concepts (Conceptual Density Functional Theory) ................. 23
3
Performance of Kohn-Sham Density-Functional Theory in Dispersion Interactions ............... 29
4
π–π Stacking ............................................................................................................................. 36
5
Results ....................................................................................................................................... 39
5.1
Effect of the π-π stacking interaction on the acidity of phenol ......................................... 39
5.1.1 Motivation ..................................................................................................................... 39
5.1.2 Discussion ..................................................................................................................... 40
5.1.3 Stacking Interaction ...................................................................................................... 42
5.1.4 Estimation of pKa ......................................................................................................... 45
5.1.5 Isolated substituted benzenes properties ....................................................................... 47
5.1.6 Discussion of π-stacking energy decomposition........................................................... 49
5.1.7 Conclusion .................................................................................................................... 53
5.2
The Art of Stacking ........................................................................................................... 54
5.2.1 Motivation ..................................................................................................................... 54
5.2.2 Discussion ..................................................................................................................... 56
5.2.3 Computational Details .................................................................................................. 64
5.2.4 Conclusion .................................................................................................................... 65
5.3
Building New Dispersion Terms ...................................................................................... 67
1
5.3.1 Motivation ..................................................................................................................... 67
5.3.2 The Electron and Its Exchange Hole ............................................................................ 70
5.3.3 Stockholder Atoms in Molecule (AIM) – the Hirshfeld-C and Hirshfeld-I methods ... 75
5.4
Implementation and Benchmark Calculation Details of the New Dispersion Term ........ 80
5.4.1 Results from a New Post-DFT Method ........................................................................ 83
5.4.2 Conclusion .................................................................................................................... 92
6
Concluding Thoughts ................................................................................................................ 94
7
Outlook ..................................................................................................................................... 95
8
References ................................................................................................................................. 96
2
Acronyms
AIM:
Atoms in molecules; a conceptual
partitioning of the molecule
B3LYP: Becke, 3 parameter, Lee–Yang–Parr
exchange-correlation functional
BOA: Born–Oppenheimer approximation
BSSE: Basis set superposition error
CAM: Coulomb-attenuated method
CC:
Coupled cluster post-HF method
CCS:
Coupled cluster with single excitations
only, produced by Tˆ1
CCSD: Coupled cluster with single and double
excitations produced by Tˆ1 and T̂2
CCSDT: Coupled cluster with single, double
and triple excitations produced by Tˆ1 ,
T̂2 and T̂3
CBS:
Complete basis set
CI:
Configuration interaction
CISD: Configuration interaction with every
singly and doubly occupied
determinant
CISDT: Configuration interaction every singly
doubly and triple occupied
determinant
DCACP: Dispersion corrected atom-centered
potential
DF:
Density functional
DFT:
Density functional theory
DFT-D: Density functional theory with
empirical dispersion correction
DNA: Deoxyribonucleic acid
DZ:
Double zeta (ζ) quality basis set
FCI:
Full CI
FMO: Frontier Molecular Orbitals
GGA: Generalized gradient approximation
HOMO: Highest Occupied Molecular Orbital
IRC:
Intrinsic Reaction Coordinate
HF:
Hartree–Fock (theory)
HFR:
Hartree–Fock–Roothan–Hall
implementation of HF theory
HSAB: Hard Soft Acid Base (theory)
IPA:
Independent Particle Approximation
LC:
Long-range corrected (prefix)
LDA:
LUMO:
Local density approximation
Lowest Unoccupied Molecular
Orbital
MBPT:
Many Body Perturbation Theory
MCSCF: Multiconfigurational self-consistent
field
MEP:
Molecular electrostatic potential
(also minimum energy pathway)
MM:
Molecular mechanics
MO:
Molecular orbital
MOS-MP2: Modified opposite spin MP2 (as
opposed to SCS-MP2)
MP:
Møller-Plesset (perturbation theory)
MP2:
Møller-Plesset perturbation theory
based method of the 2nd order
PA:
Proton affinity
PBE:
Perdew–Burke–Ernzerhof exchange
model (DFT functional)
PCM:
Polarizable continuum model
PES:
Potential Energy Surface
PVDZ:
Polarized valence double zeta (ζ)
quality basis set
PVQZ:
Polarized valence quadruple zeta
(ζ) quality basis set
RNA:
Ribonucleic acid
RPA:
Random phase approximation
S:
Softness (in the HSAB theory)
SAPT:
Symmetry adapted perturbation
theory
SCF:
Self-consistent field (also a
synonym for HF theory)
SCS-MP2: Spin-component scaled MP2
SOS-MP2: Scaled opposite spin MP2
TPSS:
Tao–Perdew–Staroverov–Scuseria
(meta-GGA functional)
TZ:
Triple zeta (ζ) quality basis set
VUB:
Vrije Universiteit Brussel –
Free University of Brussels
WF:
Wave function
TZ:
Triple Zeta (ζ) quality basis set
XDM:
Exchange-dipole moment model of
Becke and Johnson
XDM-I:
XDM improved
3
1.1 Introduction to the Author*
It all started with a Big Bang. Somewhat later the author was born.† He was blessed with a loving
family; among them a grandmother who was teaching chemistry at a grammar school. The
combination of this nurturing environment and the author’s – possibly hereditary and certainly
beyond recovery – curiosity led him inevitably to the Budapest University of Technology and
Economics. There he became a freshman chemical engineer. Mesmerized by the major secrets of
the universe – that is called thermodynamics – and obsessed with the minor secrets, also known as
quantum mechanics, he was honored by the offer from the teacher of both subjects to join his
research group. The author took the generous offer without hesitation and joined the group of
computational chemists supervised by Prof. Tamás Veszprémi.
The first investigation involved rearrangement reactions of novel unsaturated organosilicon
compounds that have old traditions at the Department of Inorganic and Analytical Chemistry. Two
papers have been coauthored based on this work.1,2 Most of this research was done in tandem with
Balázs Pintér, a good friend, making good use of work synergy.‡ Unfortunately, cooperation is
strongly discouraged when writing a thesis. Therefore one of them had to abandon the joint
investigation. The author had the opportunity to join ongoing research in Belgium in the group of
Prof. Paul Geerlings at the Free University of Brussels (VUB). The main focus was on π-stacking
aromatic rings, which have great significance in e.g. the study of the genetic material. Dr. Pierre
*
The present thesis is in the form of what Aldous Huxley would call “objective and factual essay”. In such the authors
“do not speak directly of themselves, but turn their attention outward to some literary or scientific or political theme”
and the use of the first person is often discouraged. The latter does not come naturally to native Hungarian speakers as
the passive voice is rarely used in Hungarian.
†
This time interval is approximately 13.6(3) billion years:
Menegoni, E.; Galli, S.; Bartlett, J. G.; Martins, C. J. A. P.; Melchiorri, A. Physical Review D 2009, 80, 087302.
‡
He should be addressed as Dr. Balázs Pintér more recently.
4
Mignon supervised the investigation and introduced the author to the intricate world of weak
interactions, and the results were published in Chemical Physics Letters.3 While studying the
intriguing interplay between stacking aromatic rings, a lack of efficient tools for calculating
dispersion forces became evident.
At that time in 2005 new corrections of density functional theory (DFT) to treat dispersion
effectively had just been starting to emerge. One such method in particular caught the attention of
Dr. Kenno Vanommeslaeghe, a postdoc in Dr. Geerling’s group, who invited the author to
implement and improve the new method of Becke and Johnson.4-6 Dr. Vanommeslaeghe and the
author admired Becke’s approach because it was free of any empirical parameters and was based
on ab initio theory. Two papers have been published in The Journal of Chemical Physics describing
the augmentations of the new method, coined as XDM.7,8
Later the author was honored to join the groups of Drs Dongwhan Lee and Mu-Hyun Baik at the
Inorganic Chemistry Department of Indiana University Bloomington. Where Dr. Ho Yong Lee was
already studying structural folding and self-assembly of branched π-conjugation assisted by
O--H···O and C--H···F hydrogen bonds of weak interactions and self-assembling structures.9 The
author joined happily and 5 more publications followed in the next couple of years.10-14
1.2 Introduction to Weak Interactions in Chemistry
Weak interactions in chemistry are orders of magnitudes weaker than the usual chemical bonds.
Looking at the values of selected interaction energies in Table 1.2.1, one can deduce that van der
Waals forces are the weakest by at least 2 orders of magnitudes compared to covalent or ionic
bonds. Ionic bonds are formed by attractive electrostatic forces. These forces can only act between
moieties with opposite charges. The attraction is inversely proportional to the square of the distance
5
and directly proportional to the charge difference. There must be some repulsion present, however,
otherwise charged particles would fall into each other.
Table 1.2.1
Non-Covalent Forces
van der Waals
Hydrogen Bonding (OH…O)
Selected Covalent Bonds
C–H
C–C
C=C
C=O
C–F
Selected Ionic Bonds
NaCl
LiF
Comparison of Covalent and Non-covalent Bond Energies
Typical Energy (kJ/mol)
2-10
20-40
413
348
614
799
488
787
1036
How does a very weak interaction become an important force? Planets, stars and even galaxies are
held together by gravity, another very weak force. Our Earth, the Moon and the Sun were created
from interstellar dust by the concerted gravitational attraction between every single dust particle.
But on the scale of elementary particles, the force is extremely small, only roughly 10-38 times that
of the strong force. The power of gravity for organizing galaxies into even larger structures lies in
its ability to act between any two particles regardless of charge. Opposite to nuclear forces (weak,
strong) and similar to electromagnetic force it has a long range effect and its effect does not vanish
at long range due to cancellation of opposite charges, as is the usual case with electrostatic
interactions. As an analogy to gravity, the London dispersion force is always attractive at
intermolecular range and its strength is proportional to the area of touching surfaces. This property
makes it a very important assembling force between molecules.
Every freshman chemist learns of dispersion, one kind of the van der Waals interactions but the
cause remains somewhat mystical. Becke and Johnson suggested a very interesting approach to
dispersion. The novel conceptual approach towards dispersion forces involves the interaction
between instantaneous dipoles and higher order multipoles of the exchange holes of the interacting
6
atoms. Becke and Johnson published proof of concept calculations that have been successfully
reproduced and later he stopped publishing on the subject. Dr. Vanommeslaeghe and the author
found this idea very interesting and decided to dig deeper in the subject. The author had to realize
soon that more programming was needed than expected before. The project’s arc stretches from
understanding fundamental theoretical equations, coding in Fortran to finally validating results.
Besides programming challenges the author had to broaden his skills in mathematics, physics and
data parsing.
The aim of this thesis is to illustrate the importance of van der Waals interactions and to show how
a post-DFT procedure for calculating interaction energies between larger molecules was made and
how it works. Four published papers form the base of this thesis, which are co-authored by the
author of this thesis: Refs. 3,7,8,9.
7
2 Quantum Chemical Methods
Accurate models of chemical systems need only a working model of the electromagnetic force, the
remaining three forces (gravity, weak and strong nuclear forces) are neglected. Most theories of
submicroscopic dimensions can be interpreted as approximations of the Dirac equation. Even
classical physical approaches can be treated as rough approximations. Every single model has to
balance accuracy and computation cost. Computation cost for results in a given paper is rarely
expressed in any currency but it is very real. It is not attempted to calculate such costs here but one
should consider hardware and software, maintenance (including software updates), electricity,
personnel, and time. Time being the mission critical resource most of the time. One can easily
design a task that would take centuries for a high end computer to finish. Such a result would lose
its significance by the time it is ready. A theory is considered to be efficient if the accuracy/cost
ratio is high enough – in a very subjective way. Sometimes like in the case of interaction energies,
accuracy below a certain level is practically useless. It is relatively cheap to calculate some values
like geometry but plenty of orders of magnitudes more expensive to calculate some others that
come as small differences of large values. A handful of methods are introduced below to show the
basic toolbox of a computational chemist accentuated towards capability of describing dispersion
interaction.
Often it comes down to the most accurate calculation that can be carried out in a reasonable amount
of time. Qualitative assessment for computation time will be given at each presented method. For
the sake of simplicity and following general textbook tradition all wave functions will be treated as
real in contrast to complex-valued wave functions.
8
2.1 A Hierarchy of Approximations
In order to describe properties of molecules it is often enough to describe a microsystem of
electrons and nuclei in equilibrium and independent of time. Still, one needs quite an amount of
mathematical tools. The practically solvable approximated formalism for computing energies of a
microsystem is the Schrödinger equation*. In its simplest form it can be written as
   , where
2.1.1
̂ is the Hamilton operator or Hamiltonian – the operator assigned to the physical quantity of the
energy of the system.†
 is the wave function describing the state of the system containing all the information to
be known.
 is the energy that belongs to the state of the system (given by  ).
Besides the simplest systems the Schrödinger equation cannot be solved in a closed form and
approximations are necessary to solve it numerically. A series of approximations are given below
which are implemented in a typical calculation at the current level of theory and computers.
First: Non-relativistic approximation. It means neglecting relativistic effects altogether. Therefore
the spin property of electrons must be attached as a correction because it occurs naturally from the
relativistic model‡. Other errors caused by this approximation are insignificant in the first three
*
More to read about the axioms of quantum mechanics and the Schrödinger equation: Frank Jensen, Introduction to
Computational Chemistry (John Wiley & Sons Ltd, England, 2001).
†
Every measurable physical quantity is represented by a mathematical object called operator. The actual measurement
is then represented by a transformation that involves the operator acting on the wave function of the system.
‡
”Although electronic spin is often said to arise from relativistic effects, […] spin arises naturally in the non-
relativistic limit of the Dirac equation. It may also be argued that electron spin is actually present in the non-relativistic
case, as the kinetic energy operator p 2 2m is mathematically equivalent to σ  p 2 2m . If the kinetic energy is written
as σ  p 2 2m in the Schrödinger Hamiltonian, then electron spin is present in the non-relativistic case, although this
9
rows of the periodic table*. Calculation for heavier elements are usually corrected by pseudo
potentials or rarely by other simplified models instead of using explicit relativistic formulae. It
should be mentioned here that even the time-dependent form of the Schrödinger equation fails to
remain invariant under Lorentz transformations so further mathematical modification is necessary
to give a relativistic quantum mechanical description of microsystems.
Second: The Born-Oppenheimer approximation (BOA). Further simplification is possible
considering that the electron’s mass is approximately 1/1836 that of the proton†. The BOA (also
known as adiabatic approximation) assumes that motion of the electrons is separable from that of
the nuclei being three orders of magnitudes heavier. So, the wave function can be written as the
product of two functions, one containing just the electron coordinates and the other containing just
the nucleus coordinates:
Ψ(1, 2, … , 𝑛, 1,2, … , 𝑁) = Ψe (1, 2, … , 𝑛)|1,2,…,𝑁 ΨN (1,2, … , 𝑁)
2.1.2
where 1,2,…n is for electrons, 1,2 …N is for nuclei (three spatial coordinates and one spin
coordinate). Please note that Ψe still depends parametrically on the nucleus coordinates.
Making nucleus and electron motion independent is an efficient approximation for most systems. It
may cause a significant error if mobile hydrogen atoms are present e.g. in H-bonds. Writing the
wave function as a product means that the Hamiltonian can be written as a sum of terms. This
allows the calculation to be carried out in much simpler separate steps greatly reducing calculation
cost. The theoretical impact is also honorable. The BOA allows plotting the electron energy as a
would only have consequences in the presence of a magnetic field.” Frank Jensen, Introduction to Computational
Chemistry (John Wiley & Sons Ltd, England, 2001), p. 208.
*
The author’s favorite online periodic table can be found here: http://ptable.com
†
More precisely 1836.15267245(75) – according to 2010 CODATA recommended values
(http://physics.nist.gov/cgi-bin/cuu/Value?mpsme)
10
function of different nucleus configurations and this is called the potential energy surface (PES). It
is a multi-dimensional surface having 3N-6 dimensions.* Minima on this surface match stable
configurations,† first order saddle points match transition states. A minimum energy path
connecting two minima is the elementary step of a reaction. Such paths can be found via intrinsic
reaction coordinate (IRC) calculations furnishing the toolbox of the computational chemist. These
reaction paths are ideally adiabatic therefore dissipative reactions can take different paths.
Third: IPA – Independent Particle Approximation. IPA is at the heart of quite a few methods such
as Hartree-Fock theory and density functional theory which are very popular methods to solve the
electronic Schrodinger Equation. Within the IPA each particle is independent, ie each particle is in
a different orbital, so that we can write the wavefunction in a product form:
|Ψ〉 =
1
√𝑛!
𝓐[Ψ1 (1)Ψ2 (2) … Ψn (n)] ,
2.1.3
where 𝓐 is the antisymmetrizer operator. This way each particle can be computed independently of
each other in such a way that any given particle experiences the mean field of the remaining
particles. Therefore the alternate name for this approximation: mean field approximation.
Also, every single determinant description is necessarily an IPA: The energy eigenfunction (ψ) is
assumed to be in a form of a single determinant consisting of one-electron functions, i.e. orbitals.
Therefore the conceptually attractive picture of electrons occupying given orbitals is a result of an
approximation. Reality is even more complex.
A more realistic approach to the 3rd approximation, IPA, is at the core of post-HF methods.
*
3N-5 in case of linear systems but who would not want to see how the system behaves when bending?
†
Provided that a full vibration level fits at the ”bottom” of the minimum.
11
Fifth: Finite basis. Quantum chemical calculations are performed in finite basis, despite axiomatic
requirement of any solution of the Schrödinger equation being an element of Hilbert space.
Therefore the wavefunction expansion is inherently infinite. A practical implementation has to use
a finite basis due to the finite nature of computational resources. The basis then becomes a
collection of a finite number of basis functions. This means that the wavefunction is represented as
an n-dimensional vector over complex numbers, where the components correspond to coefficients
of a linear combination of basis functions. In most implementations the complex vectors are rotated
into real space. It can be done because physical quantities have a relationship with the
wavefunction multiplied by its complex conjugate, which eliminates the imaginary part. So as long
as only the wavefunction multiplied by its complex conjugate has physical relevance such rotation
to real space is allowed. This is why the coefficients of an actual wavefunction calculation are real
in most of the computational chemistry program suites.
2.1.1 The Hamiltonian
In this section subjects introduced in section 2.1 will be expanded and therefore partially repeated.
The author’s aim is to increase readability rather than to annoy the Reader.
The Hamiltonian contains kinetic and potential energy terms:
 n n Z Z  N N 1 N n Z 
2 N 2 n 2
2
2
Hˆ  




e
  
 i 1 2M i 

2m i 1
i i  j rij
i  ri 
     R


2.1.4
where m and e are the mass and charge of the electron, Ma and Za are the mass and charge of 
nucleus, Ris the distance between the  and  nuclei, ri between electron i and the nucleus ,
and rij between electrons i and j.  is the Laplacian and the summation goes over N electrons and n
nuclei. The first two terms are the kinetic energy operators of the electrons and nuclei, Tˆe and T̂N ,
respectively. The potential energy operator is built up from three terms: nucleus-nucleus repulsion (
VˆNN ) and electron-electron repulsion ( Vˆee ), and the nucleus-electron attraction ( VˆNe ).
12
The wave function has 4(n+N) variables, because in non-relativistic quantum mechanics the spin
must be taken into account phenomenologically:
x,y,z,msx,y,z(ms),
2.1.5
where is the one-electron spin function of a given electron, and ms is the spin projection quantum
number (usually denoted as sz if the projection axis is z). Sadly, the Schrödinger equation cannot be
solved analytically for systems with more than one particle. Although, a numerical solution would
be theoretically possible, further approximations are introduced on practice. The main strategy is to
approximate the Hamiltonian as the sum of several operators, because if it is possible, the wave
function can be written as the product of several functions with fewer variables. Due to the BornOppenheimer approximation (Eq. 2.1.2), the electronic Schrödinger equation (Eq. 2.1.6) and an
equation containing only the nuclear coordinates (Eq. 2.1.7) can be solved independently.

He e  Ee e

H n n  En n
2.1.6
2.1.7
where Ee is the electronic energy, and En is the energy of the nuclei. Ee is also a function of the
nuclear coordinates as parameters, representing the potential energy of the nuclei. The Ee function
varies with the nuclear positions, and it is often referred to as the potential energy (hyper)surface
(PES). In the critical points of the potential energy surface the gradient is zero and the eigenvalues
of second derivatives matrix (Hessian) can be used to characterize the critical points. Of these,
from the chemist’s point of view, the most important ones are the minima corresponding to stable
structures and the first-order saddle points corresponding to transition states. Following the routes
with the steepest descent from a first-order saddle point one can reach at least two minima. This
path is called the intrinsic reaction coordinate (IRC), and it gives some indication on the changes
during the reaction. In reality, the reaction follows a different path above the minimum depending
on the kinetic energy of the system, which is completely ignored in the above picture. As a
13
consequence of the Born-Oppenheimer approximation, the electronic wave function only contains
the coordinates of the electrons. Further simplifications are introduced by using the independent
particle model, in which the interaction between the electrons is approximated by an average oneelectron potential Vi eff i  . The Hamiltonian is written as:
𝑁
𝑁
̂ ≈ ∑[ℎ̂𝑖 + 𝑉𝑖eff ] = ∑ 𝐹̂ (𝑖)
𝐻
𝑖=1
2.1.8
𝑖=1
where ℎ̂𝑖 , is a one electron operator, containing the kinetic energy operator of the electrons and the
electron-nuclear attraction potential energy term, and 𝐹̂ (𝑖) is the Fock-operator. In this model each
electron moves in the potential field generated by every other electron. The electronic wave
function now can be written in terms of one-electron functions, each depending only on the
coordinates of one electron. Due to Pauli’s exclusion principle, the wave function must be
antisymmetric for the interchange of two electrons, and therefore an appropriate linear combination
of the products of the above mentioned one-electron functions have to be used. This linear
combination can be represented by a determinant (, the Slater-determinant). For N electrons,
including the normalization constant:
𝜑1 (1) ⋯
⋱
Φ=
| ⋮
√𝑁! 𝜑𝑁 (1) ⋯
1
𝜑1 (𝑁)
⋮ |
𝜑𝑁 (𝑁)
2.1.9
where i are the one-electron wave functions, which are the product of the one-electron orbital (i)
and one-electron spin functions (i).
2.1.2 The Hartree-Fock (HF) and Hartree-Fock-Roothaan (HFR) Methods
In the Hartree-Fock method, the variational principle is used to find the minimum of the energy and
the corresponding one-electron wave functions by solving the Hartree-Fock (HF) equations:
𝐹̂ (𝑖)𝜑𝑖 = 𝜀𝑖 𝜑𝑖
𝑖 = 1, … , 𝑁
2.1.10
14
As the Fock operators contain the one-electron wave functions an iterative procedure is used, in
which a starting (0) function series is used to construct the Fock operators, than the HF equations
are solved, resulting in a new (1) series of functions and the energy. With these new functions the
process is repeated until the change of the energy between two cycles becomes smaller than a given
threshold. In this case it is said that the self-consistent field (SCF) threshold is reached.
The original HF equations are integro-differential equations, which can be solved for spherical
atoms. However, a similar solution is impossible for molecules, * therefore the Hartree-FockRoothan (HFR, or more precisely Hartree-Fock-Roothan-Hall) method is used. In the HFR method,
the one-electron wave functions are constructed as the linear combinations of pre-defined basis
functions   , and in the solution of the HF equations only the coefficients are varied.
m
 i   ci  
 1
i  1, 2, 3, ...
2.1.11
Using an infinite basis set an infinite number of one-electron wave functions are obtained from the
solution of the HFR equation together with infinite number of eigenvalues. The Molecular Orbital
(MO) theory uses these one-electron wave functions and eigenvalues to demonstrate the bonding in
the molecule. The one-electron wave functions are associated to molecular orbitals and the
corresponding eigenvalues to orbital energies. From these orbitals the most important ones are the
HOMO (highest occupied molecular orbital) and the LUMO (lowest unoccupied molecular orbital),
as these determine primarily the reactivity of the molecule and they play a key role in the FMO
(Frontier Molecular Orbital) theory.
The use of finite basis sets instead of infinite is a further necessary approximation, therefore its size
and quality is a very important factor. Nowadays, due to their computational efficiency the
*
With a very few very simple exceptions.
15
Gaussian-type orbitals are used as basis functions. The number of basis functions is bounded by the
number of electrons and the spherical symmetry of the atom (minimal or single zeta basis set) from
below, and by the computational resources from above. In the double zeta (DZ) and triple zeta (TZ)
basis sets the number of basis functions are twice or three times larger than in the minimum basis
set. In the so-called split-valence basis set, the inner shells are single zeta and the valence shell is of
double or triple zeta quality. The basis sets often contain polarization and diffuse functions.
Polarization functions give information on the surroundings around the atom making the basis set
more flexible. In the case of hydrogen atom, they are of p or d type, while for heavier elements d or
f type functions are used. Diffuse functions (denoted by + or ++) are especially important in the
description of weak interactions and anions. Recently, the correlation consistent (cc) basis sets,
proposed by Dunning have become very popular. They are known by their acronyms as cc-PVDZ,
cc-pVTZ, cc-PVQZ, .., (correlation consistent polarized valence double/triple/quadruple/…. Zeta).
These basis sets include by construction the polarization functions. Diffuse functions can be added,
which is assigned by an “aug” in front of the acronym. An important advantage of the Dunning
basis sets is the ability to generate a sequence of basis sets, which converge toward the basis set
limit (the best result that can be achieved within the approximations of a certain methodology). It
seems, however, that the Karlsruhe basis sets15 are the most effective ones as a BSSE-corrected*
def2-SVPD basis set energies are as good as uncorrected energies calculated at 3-4 times bigger
basis sets.16 Development of these basis sets were not known to the author until 2010.
The Hartree-Fock method can be used to calculate the energy of the system, but with increasing
basis set, the energy converges to the so-called Hartree-Fock limit, which is asymptotically always
higher than the exact solution of the non-relativistic Schrödinger equation. The correlation energy
(the energy difference between the HF-limit and the exact solution) is a consequence of the oneelectron model in the HF-method. The absolute value of the correlation energy is very small
*
See Eq. 5.1.3 and the surrounding text for the definition of BSSE.
16
compared to the total energy of the system, but it may have a very important influence on
quantities, which arise as energy differences e.g. activation energies and relative energies.
As the correlation energy is a direct consequence of the use of the effective potential in the
Hamiltonian and the Slater determinant, modern quantum chemical methods try to take into
account the correlation between the electrons by improving the wave function. The main strategy is
to use the one-electron wave functions obtained in the HF-method to develop the exact wave
function of the system. Via exciting electrons from the occupied orbitals to the virtual ones it is
possible to create formally excited determinant wave functions, which are also eigenfunctions of
the Hamiltonian and can be used for the development of the exact wave function.

   Ci  i
2.1.12
i 1
Although, this wave function is the superposition of different determinants, it is generally called the
configuration interaction (CI) method (Mulliken calls the method “configurational mixing”.) The
finite size of the basis set limits the number of determinants, so the wave function obtained from a
finite basis set is called FCI (full CI). This calculation includes every singly, doubly etc. excited
determinants. The variational principle is used to determine the coefficients of the determinants.
Due to the large number of determinants, FCI calculations can only be performed for molecules
with only a few atoms. Therefore, it is necessary to select a subset of those determinants, which are
important for the expansion of the wave function.
1. There are systems whose wave function cannot be described with one single determinant,
and a linear combination of determinants must be used. This is called the static electron
correlation (sometimes also called as non-dynamic electron correlation). Semi-dissociated
systems usually require the treatment of static electron correlation. In the general case if one
finds that the CI development of the system contains several determinants with non-
17
negligible coefficients, then these determinants have to be taken into account. This can be
the case if the studied electronic state is near in energy to another state.
2. The number of excited determinants, whose contribution to the wave function is small, is
very large. Therefore, their cumulated effect is very important and in many cases it gives
the largest part of the correlation energy (a.k.a. dynamic electron correlation).
If the first two types of correlation energy are important, a multiconfigurational self-consistent field
method (MCSCF) can be applied to describe the system. In this calculation a selected set of
determinants are used as reference instead of the single determinant and the orbital coefficients are
usually optimized simultaneously with the CI coefficients in a variational procedure. The more
detailed description of these and other important methods can be found in various textbooks.17-19
As very often the dynamic electron correlation turns out to be the most important, various
approximating approaches have been developed.
1. The CI-series. As the number of determinants is very large in the FCI approach, a
systematic series of calculations were developed based in the CI-method. In the CIS method
every singly occupied determinant, in the CISD every singly and doubly occupied
determinant are taken into account, while the CISDT method includes all single, double and
triple excitations. The serious drawback of these methods is that they are not sizeconsistent.
Even though the CIS method is mentioned here for consistency reasons, the CIS method is
unable to take the dynamic correlation into account. The reason is that the singly excited
determinants have zero interaction with the stationary energy single determinant.
2. Coupled Cluster (CC) calculations. The problem of size-consistency is solved by the
coupled-cluster methods, which use an exponential operator to produce the excited
determinants:
18
Ψ  eT̂ Φ0
2.1.13
The Tˆ operator can be written in the “ T̂  T̂1  T̂2  T̂3  ... ” form, where the subscript denotes the
order of the produced excitation. Substituting this into Eq. 2.1.13 the following equation can be
obtained:
1
1


Ψ   1  T̂1  T̂2  T̂12  T̂3  T̂1T̂2  T̂13  ...Φ0
2
3!


2.1.14
3. This shows that obtaining the Tˆ1 operator allows one to take into account not only the single
excitations but a part of the doubly and higher order excitations. Depending on the order of
the Tˆ operator, CCS, CCSD, CCSDT and CCSD(T) etc. methods have been developed so
far. In the abbreviations S refers to the inclusion of the excitations produced by Tˆ1 , D by T̂2
and T by T̂3 . In the CCSD(T) method only the excitations by Tˆ1 and Tˆ2 are taken into
account explicitly, and a perturbation method is used to estimate the effect of the triples. As
a result, this method is more accurate than CCSD, but computationally cheaper than
CCSDT.
One of the most frequently used method is the Many Body Perturbation Theory (MBPT, or MøllerPlesset, MP methods) to include the dynamic electron correlation in the calculation. According to
this theory, if the Hamiltonian of the system S1 only slightly differs from that of a known system
S0, it can be expected that the solution will be similar, too. The Hamiltonian of S 0 is taken as the
non-perturbed Hamiltonian and the Hamiltonian of S1 is:
Ĥ  Ĥ 0  V
2.1.15
where V is the difference between the Hamiltonians of S1 and the non-perturbed system, and  is
the perturbation parameter. The wave function of the non-perturbed system is used to develop the
wave function of S1. In quantum chemical calculations the Schrödinger-Rayleigh perturbation
method is often used in the development of the wave function and the energy, and the HF wave
19
function is taken as the wave function of the non-perturbed system. (This was first described by
Møller and Plesset). Including higher order of perturbation, the correlation energy can be estimated
better and better. The disadvantage of the MP-methods is that they are not variational methods, and
the obtained energy may be lower or higher than the exact solution. However, it is often true, that
using higher order, one gets closer to the exact solution. The two, most often used methods are
MP2 and MP4, in which the energy is calculated up to the second and the fourth term of the
expansion series. An important advantage of the MP2 method is that it accounts for about 90% of
the correlation energy, it is size-consistent, and its computation demand is only 20-50% higher than
that of the HF-method for a few dozen atoms.
The accuracy of quantum chemical methods is essentially determined by the sophistication of the
applied correlation method and the size and quality of the basis set. As the theoretical level and the
basis set increase, one obtains more and more accurate results, while the computational demands
increase extensively. In the past decade two series of methods (the Gaussian (G) and the Complete
Basis Set (CBS) series, respectively) have been developed to obtain very accurate energies at
moderate computational demand. The basic idea is to estimate the correlation energy and/or the
effect of the finite basis set by extrapolation from the results of several, lower level single point
calculations at an acceptable geometry (it depends on the method which geometry is used as
reference).
2.1.3 Density Functional Theory (DFT)
DFT allows one to replace the complicated N-electron wave function by a much simpler quantity,
the electron density:  ( r). The electron density gives the probability of finding an electron in a
given volume of space:
20
 

  

 r   N   ....  Ψ 1,2,...N  Ψ1,2,...N dr2 ...drN
2.1.16
Since the electrons in the molecules cannot be distinguished from one another, the above definition
is generally true for the electron density. The first model was proposed by Thomas and Fermi in the
1920s, but the Hohenberg-Kohn theorem in 1964 was the first to legitimize the use of the electron
density instead of the wave function of the system. The theorem states: “The external potential v(r)
is determined, within a trivial additive constant, by the electron density.” The external potential is
due to the nuclei and (r) determines the number of electrons:
  r dr  N
2.1.17
It follows from the theorem, that (r) also determines the ground state wave function and all other
electronic properties of the system, among others the total energy, the kinetic energy and the
potential energy of the system.
Ev    T    Vne    Vee    FHK      r vr dr
2.1.18
where T[] is the kinetic energy functional, V[] are the potential energy functional (the constant
nuclear-nuclear repulsion term is not included in the equation). FHK[] is the Hohenberg-Kohn
functional which contains the electron-electron repulsion term and the kinetic energy functional.
The second Hohenberg-Kohn theorem provides the possibility to determine the electron density:
“For a trial density ~ r  such that ~ r   0 and
  r   N ,
~
~  E
Ev 
0
2.1.19
where Ev ~  is the energy functional” of Eq.2.1.18. This is analogous to the variational principle
for wave functions; one has to search for the (r) minimizing E. For the optimal (r) the energy
does not change upon variation of (r) provided that (r) integrates to n, the number of electrons:
 E   r   0
2.1.20
where  is the undetermined Lagrange multiplier. This leads to the (so-called Euler) equation:
21
vr  
FHK

 r 
2.1.21
The difficulty of the solution of Eq.2.1.21 is that the components of the Hohenberg-Kohn
functional, FHK are not known. Kohn and Sham have introduced a procedure, which makes possible
the calculation based on Eq. 2.1.21. They started from an n-electron non-interacting system
showing the same electron density as the exact electron density of the real interacting system. The
Schrödinger equation of this non-interacting system can be easily separated into one-electron
equations and its wave function is properly described by the determinant wave function. The
electron density can be expressed as:
n
   i
2
2.1.22
i 1
The exact kinetic energy functional of this system Ts[] can be used as an approximation to the
kinetic energy operator of the real system. Using Ts[], FHK[] can be written as:
FHK    Ts    J    E xc  
2.1.23
where J[] is the classical electron-electron repulsion term and Exc is the exchange correlation
functional which contains the difference between the T[] and Ts[] and the non-classical part of
Vee[]. The solution of the variational problem can be found by solving the following Kohn-Sham
equation:
N
 2 2






v
r


 ij j

eff

 i
j 1
 2m

2.1.24
where veff(r) is an effective potential, i a one-electron function (Kohn-Sham orbitals). A similarity
transformation can be used to obtain the canonical Kohn-Sham equation, which can be solved in an
iterative manner similarly to the one used in the case of the HF equations.
The accuracy of DFT calculation strongly depends on the applied Exc functional, which is in
practice divided to an exchange and a correlation part. The available functionals today differ in
22
their exchange and/or correlation part. As the HF method contains an exchange part, some recent
“hybrid” functionals contain a mixture of HF and DFT exchange functionals. The very popular
B3LYP functional contains Becke’s three-parameter functional in conjunction with the Lee-YangParr functional.20-23 It contains three parameters optimized to atomization energies, ionization
energies etc, therefore it can also be regarded as a semi-empirical method. Due to its accuracy, ab
initio fundamentals, and despite DFT being a semi-empirical method, DFT can be as useful as
some other ab initio method.
2.1.4 DFT-derived
chemical
concepts
(Conceptual
Density
Functional
Theory)*
The basic idea of conceptual DFT is to use the electron density to define commonly accepted but
poorly defined chemical properties. In the earliest work in the field it was recognized that
Lagrangian multiplier  in the Euler equation (Eq. 2.1.21) could be written as the partial derivative
of the system’s energy with respect to the number of electrons at fixed external potential:
 E 
  
 N  v r 
2.1.25
This electronic “chemical potential” has been identified as the negative of electronegativity.24
 E 

 N  v r 
  
2.1.26
Since then various other derivatives have been identified; Fig. 2.1.1 shows the hierarchy of the
different energy derivatives. The global indices (derivatives of the energy at constant external
potential (i.e. at fixed nuclear positions)) are found in the left hand side of the diagram, while the
right side of the diagram contains the local indices, whose value changes from point to point. The
*
The author thanks Dr. Julianna Oláh for the invaluable help provided on the subject of conceptual DFT presented in
this section.
23
evaluation of the N-derivatives of the energy raises a fundamental question: whether the energy is
differentiable with respect to N, as the number of electrons must be an integer for isolated atoms
and molecules. Although, there have been various solutions proposed to overcome this problem,
the most common one up to date is the finite difference approach, in which the derivative is
approximated as the average of left and right hand derivatives.
E  EN , (vr 
energy of the system
 E 

    
 N  v r 
electronic chemical potential
 2 E 
  
 2   
 
 N v r   N v r 
hardness
 E 

   r 
 vr  N
electron density
  2 E    
  r 

  
  
  f r 
 Nvr   vr  N  N v r 
fukui function
Fig. 2.1.1
 2 E 
  r 

  
   r, r

 vr vr  N  vr N
linear response function
Derivatives of the energy
Electronegativity
The definition of electronegativity according to Linus Pauling is: “The power of an atom in a
molecule to attract electrons to itself”.25 During the last century several electronegativity scales
have been proposed, all showing the same tendencies but differing in their philosophy and
numerical values. Conceptual DFT offers a way to calculate the electronegativies not only of
atoms, but functional groups, ions etc. In conceptual DFT the electronegativity of a system is
defined according to Eq. 2.1.26. This derivative is approximated with the finite differences
approach:
   E ( N  N0  1)  E ( N  N0 )  I
2.1.27
   E ( N  N0 )  E ( N  N0  1)  A
2.1.28
24

  
2

I  A  HOMO  LUMO

2
2
2.1.29
where I and A are the ionization energy and the electron affinity of the neutral atom. This formula
is the same as proposed by Mulliken to calculate the electronegativity.26 An approximation is
introduced using Koopmans’ theorem so finally the electronegativity becomes proportional to the
HOMO-LUMO gap.27
Global hardness and softness
These terms originate from Pearson’s Hard and Soft Acids and Bases (HSAB) principle28. Pearson
studied the reactions of Lewis acids and bases and tried to find an explanation for the observed
reaction entalphies of the neutralization reactions. He classified the investigated compounds as soft
or hard. In his system acids and bases with low charge and greater volume were classified as soft,
while acids and bases having a smaller volume and a greater charge were classified as hard. This
classification turns out to be essentially polarizability based: low polarizability = hard bases: NH3,
F-, high polarizability = soft bases: H-, R-, R2S. Pearson concluded that hard acids preferably
interact with hard bases and soft acids with soft bases. Although, the HSAB principle could explain
the observed interaction energies of acids and bases, the definition of hard and soft was quite hazy
and the classification of a new acid or base was ambiguous. This problem was solved when Parr
and Pearson identified the hardness as:29
1  2E 
1
   2  
2  N  2S
2.1.30
Softness, S is considered as the reciprocal.
The qualitative formulation of the HSAB principle is expressed by the following equation:30,31
1  A   B 
1 

S ASB 
2 S A  SB
2 S A  SB
2
E AB
2.1.31
where the energy change is estimated as the sum of two terms.
25
The first term expresses the energy gain upon equalizing chemical potentials at fixed external
potential and the second term is the energy change due to rearrangement at fixed chemical
potential.  is a constant related to the effective number of valence electrons in the interaction and
to the ratio of the softnesses of the reacting partners. Chandrakumar and Pal32 defined the value
as the difference of electron densities of the interacting system A (or alternately B) before and after
the interaction:
M
M
j 1
j 1
 A    Ajeq    Aj0
2.1.32
There exists a semilocal and local form of Eq. 2.1.31 in which the local softness (sA and/or sB) are
used instead of the global softness SA or SB.
Electronic Fukui function
The electronic Fukui function f(r) is the mixed partial second derivative of the energy with respect
to the number of electrons and the external potential. As a result, the Fukui function changes from
point to point in space. The Fukui function links frontier MO theory and the HSAB principle. It can
be interpreted either as the change of the electron density at each point in space when the total
number of electrons is changed or as the sensitivity of the system’s chemical potential to an
external perturbation at a particular point r:
  r     

f r   
  
 N  v  vr   N
2.1.33
As the evaluation of the Fukui function encounters the problem of the integer number of electrons,
left- and right-hand side derivatives had to be introduced, both considered at a given number of
electrons N=N0. In 1986, a condensed form of Fukui function was introduced based on the idea of
integrating the Fukui function over atomic regions. This technique combined with the finite
differences approximation lead to the following working equations:
26

  r  

f r   
  N0 1 r    N0 r   q A N 0  1  q A N 0 
  N   v  r 

2.1.34

  r  

f r   
  N0 r    N0 1 r   q A N 0   q A N 0  1
  N   v  r 

2.1.35
where qA is the electronic population of atom A in the reference system. For a given system the
sum of the Fukui functions of the atoms is always equal to 1. As these equations make use of a
population analysis, the obtained results will depend on the method, basis set and the used
partitioning scheme. A detailed description can be found e. g. in the papers of De Proft
Arulmozhiraja and Gilardoni.33-35
Local softness and local hardness
The local softness is defined as:
sr   Sf r 
2.1.36
This can be interpreted as if the Fukui function redistributed the global softness among the different
parts of the molecule. The local softness contains the same information as the Fukui function, if
one examines intramolecular properties, but as it contains more information on the softness of the
molecule, it can be used more easily for the comparison of different molecules.
Contrary to the local softness, the definition of local hardness is ambiguous, and so far no
consensus has been reached.36 However, the local hardness has, among others, been shown to be
proportional to both the electronic part of the electrostatic potential and the total potential in each
point in the valence region of the molecule (e.g. at a distance of 4 Bohr from the atom, very good
results have been obtained).
Electrophilicity
The electrophilicity index has been recently proposed by Parr et al.37 The model was first proposed
by Maynard et al. to interpret ligand-binding phenomena in biochemical systems.38 The idea was to
27
“submerge” a ligand in an ideal, zero temperature free electron sea of zero chemical potential and
calculate the stabilization energy of the ligand when it becomes saturated by electrons. At the point
of saturation the electronic chemical potential of the ligand becomes zero:
 E 
 0
 N v
 
2.1.37
The Taylor expansion of the energy change of the ligand ( E ) up to second order due to electron
flow ( N ) from the free electron sea is:
1
2
E  N  N 
2
2.1.38
From the combination of Eq. 2.1.37 and Eq 2.1.38 the maximal electron flow can be obtained as:


2.1.39
2
2
2.1.40
N  
and the energy change is:
E  
The negative of this energy change was considered as the electrophilicity of the ligand

2
2
2.1.41
The authors pointed out that electron affinity (A) and the electrophilicity index () are similar
quantities, but electron affinity expresses the energy change at the uptake of exactly one electron
while  is related to maximal electron flow. Indeed, in the case of compounds with small electron
affinities, the deviation between A and  is relatively large, but as A increases the correlation
becomes increasingly better.
28
3 Performance of Kohn-Sham Density-Functional Theory in
Dispersion Interactions*
Discussion of this section contains exact and near exact quotations from the excellent review article
by Stefan Grimme: Density functional theory with London dispersion corrections.39 These
paragraphs are enclosed within quotation marks. Every quoted paragraph in section 3 is therefore
from Grimme’s review paper. Extensive quotation from Grimme is intentionally not avoided
because the author could not have summarized the quoted parts any better.
“Development of approximate a Kohn-Sham density-functional theory (DFT) that can give an
accurate model of both the physically and chemically important London dispersion interactions 40,41
is a very active field of research.42-46” Treatment of the dispersion interaction in the history of everevolving DFT functionals will be shortly summarized in this section. The significance of the
subject is highlighted in another excellent review paper by Weitao Yang and coworkers:
Challenges for Density Functional Theory.47,†
“It has now become very clear especially for the chemistry and physics of large systems, e.g., in
bio- or nanoarchitectures, that inclusion of dispersion interactions in theoretical simulations is
indispensable in order to reach so-called chemical accuracy. Because the van der Waals-type
effects are the result of omnipresent electron correlations, they also influence the accuracy of
theoretical (reaction) thermodynamics.” A recent paper also highlights the importance of dispersion
correction based on water studies.48 Another paper shows that the main reason for the artificially
*
Dr. Tibor Szilvási – member of the Veszprémi Group – provided invaluable help to the author on how to organize the
very rich literature of novel DFs discussed in this section.
†
This paper was suggested for reading by Ágnes Szabados.
29
low density of (dispersionless) DFT computed water is the lack of dispersion interaction.49 A
surface chemistry paper also highlights that accurate description of surface-adsorbates requires
dispersion (or rather vdW) correction.50
Cost-to-performance ratio of DFT is better than or equal to post HF methods.51,52 DFT has become
the most popular electronic structure method for large scale ground state systems.53 It would be
favorable to include the description of dispersion interactions in DFT methods without an increase
in the computation cost or at least without increased scaling with the system size.
An essential ingredient of DFT, the exchange-correlational functional (Exc) remains unknown and
has to be approximated. Widely used hybrid functionals, like B3LYP,20,22 have several
shortcomings around asymptotic regions, such as unphysical self-interaction54-57 effects upon
dissociation and they fail to describe long-range charge-transfer excitations.58-60 Also, very poor
description of dispersion forces is a disadvantage of the otherwise well-performing classical hybrid
functionals. That is to say hybrid functionals often give qualitatively wrong results under
asymptotic circumstances.
A qualitative solution to the self-interaction error comes in the form of long-range corrected (LC)
hybrid density functionals,61-73 which exhibit 100% HF exchange for long-range electron
interactions. This is accomplished by the inclusion of increasingly higher contribution of HF
exchange for long range and, consequently, the contribution of HF exchange diminishes in shortrange interactions. Over the past several years the LC hybrid scheme has been attracting attention
because of its low computational cost comparable to standard hybrid functionals. 20 Notable
examples for LC functionals are LC-ωPBE,74-76 CAM-B3LYP,77 or the meta-GGA based ωB97X.62
LC functionals, however, still do not give satisfactory results for dispersion interactions (London
forces).78,79 The main reason is that the community’s focus was on the exchange part of Exc and
30
enhancement of the correlational part is essential in order to increase the performance of DFT to be
practically acceptable for dispersion interactions.
One may ask whether all these novel DFT implementations are trustworthy both in terms of
precision and reproducibility between software packages. A recent thorough work published in
Science shows that it is indeed the case.80
The simplest method to treat this shortcoming of DFT is to augment the given functional with an
empirical correction term. The methodology has well-established roots in the field of molecular
mechanics (MM) simulations. There are plenty of examples of such functionals, we only mention a
few possible corrections for B3LYP: B3LYP-D, B3LYP-D2,81 B3LYP-D3,82 B3LYP+XDM,83
B3LYP+dDXDM,84 B3LYP-lg.85 A very promising empirically corrected functional for modeling
weak interactions is the ωB97X-D.62,86,87
From the practical point of view, where the focus is on robustness and computational speed,
empirical −C6·R−6 corrections to standard density functionals seem promising. The idea to treat the
(quantum mechanically) difficult dispersion interactions semiclassically and to combine the
resulting potential with a quantum chemical approach (a kind of quantum mechanical–molecular
mechanical hybrid scheme) goes back to the 1970s in the context of Hartree–Fock theory88,89. The
method has been forgotten for almost 30 years and was rediscovered about 10 years ago as
problems with DFT became more evident.90,91 It is usually termed DFT-D (or sometimes DFT +
disp). Although this name is quite unspecific, it has been accepted meanwhile generally. Probably,
the first published paper in which a standard DFT calculation has been combined with a damped
dispersion energy is the work of Gianturco et al.92 (for the special case of Ar–CO). An even earlier
approach along the same lines, however, not employing a standard DF but LDA-based expressions
31
for the repulsive short-range part combined with a damped interatomic pairwise treatment can be
found in the paper of Cohen and Pack, which likely represents the first true DFT-D work.93
Meanwhile, many modifications of the DFT-D approach have been published. All are based on an
atom pairwise additive treatment of the dispersion energy (for extensions to include three-body
nonadditive dispersion effects, see Refs 82,94). The general form for the dispersion energy (which is
simply added to the Kohn–Sham DFT energy) is
DFT−D
𝐸disp
= −∑
∑
AB 𝑛=6,8,10,…
𝑠𝑛
𝐶𝑛AB
𝑛 𝑓damp (𝑅AB ),
𝑅AB
2.1.1
where sum is over all atom pairs, 𝐶𝑛AB is the isotropic dispersion coefficient of order n for atom pair
AB, and 𝑅AB is their internuclear distance. DF-dependent scaling factors sn are typically used to
adjust the correction to the repulsive behavior of the chosen DF.95 If this is done only for n > 6 as in
DFT-D382, asymptotic exactness is fulfilled when the 𝐶6AB are exact. Note that the contribution of
the higher-ranked multipole terms n > 6 is more short ranged and rather strongly interferes with the
(short-ranged) DF description of electron correlation. The higher order Cn terms can be used to
adapt the potential specifically to the chosen DF in this mid-range region. The question how many
higher-order terms are necessary is not completely clear at present. Although C8 and C10 contribute
significantly in the equilibrium regions (roughly 50% of Edisp for heavier atoms), owing to the huge
values of these coefficients, their corresponding errors are grossly amplified and make the
correction somewhat unstable.82 Some kind of consensus has been reached in that C6 alone is not
sufficient to describe medium/short-range dispersion.82,84,96
In order to avoid near singularities for small R and double-counting effects of correlation at
intermediate distances, damping functions fdamp are used, which determine the range of the
dispersion correction.97 If only noncovalent interactions are considered, the results are only weakly
dependent on the specific choice of the function. Typical expressions are given in Ref 98 and Ref 81.
32
𝑓damp (R AB ) =
1
1+
𝑓damp (𝑅AB ) =
2.1.2
−𝛾
6(𝑅AB ⁄(𝑠𝑟,𝑛 𝑅0AB ))
1
1+𝑒
−𝛾(𝑅AB ⁄(𝑠𝑟,𝑛 𝑅0AB −1))
,
2.1.3
where 𝑅0AB is a cutoff radius for atom pair AB, sr,n is a DF-dependent (global) scaling factor and γ is
a global constant that determines the steepness of the functions for small R. For the cutoff radii,
often (averaged) empirical atomic vdW radii are used. An ab initio based approach by Grimme to
determine pair-specific values is described in Ref 82. A fundamental difference between the existing
damping approaches is their behavior for small R. Although in most methods (e.g., for the above
given formulas) the damping function (and thus Edisp) approaches zero for R→0, Becke and
coworkers4-6,96 use rational damping in the form
𝐸𝑑𝑖𝑠𝑝 = ∑
𝑅6
AB AB
𝐶6
+ const.
2.1.4
This leads to a constant contribution of Edisp to the total correlation energy from each bonded atom
pair. Although this seems theoretically justified,99 it basically requires special adjustment of the
standard correlation DF used, whereas the zero-damping method works very well with standard
functionals and also for thermochemistry.81,82
All these empirical corrections are based on optimized (scalar) coefficients for different atom types.
Such corrections are fundamentally incapable of describing any anisotropy or orientational effect
between atoms of molecules. Also, these empirical models are completely insensitive to the change
in the chemical environment of individual atoms, i.e. only given atom types are included in the
collection of coefficients. There is treatment to this situation in the form of the exchange-hole
dipole moment (XDM) method. The XDM method developed by Becke and Johnson is not an
empirical approach but employs computed coefficients based on theoretical principles.4-6,100 The
XDM method is discussed in detail in section 5.3. The XDM could be applied as an additive
33
correction term to existing local DFT functionals. According to the author’s knowledge, the only
XDM implementation is present in the Q-Chem suite of quantum chemical programs.
The next theoretical step to treat dispersion in the DFT framework is just evolving (in 2014). One
of the most attracting features of dispersion-corrected functionals is their superior performance
compared to their uncorrected equivalents – still, without notable increase in computational costs.
The root of the problem, however, is not fixed; the correlational functional is far from being
complete. An attempt to fix the correlational functional comes in the form of double-hybrid
functionals, like the B2PLYP101 and later the B2GP-PLYP,102 MC3BB,103 mPW2PLYP,104 and
XYG3105 functionals. These double hybrid functionals lack the usual fourth order scaling of
computational time, instead they exhibit fifth order scaling like MP2, this way their computational
cost becomes prohibitive for bigger systems which are still treatable with dispersion-corrected
functionals. Another problem is their inability to fix the description of dispersion interactions – one
of the reasons they have been developed. In defense of MP2 and double hybrid methods, it has to
be mentioned that MP2 algorithms that scale linearly with system size have been shown to exist
long ago.106,*
The third problem with double-hybrid functionals is that they introduce unnecessary error in short
distance interactions, despite that the density functional description is almost exact for short
distance interactions. In order to unravel this problem, several different versions of range-separated
double hybrid functionals were developed.65,69,71,72,107-115 The price of the increased chemical
accuracy for the double hybrid functionals, is their computational cost. Some functionals like the
range-separated double hybrid B2-OS3LYP,116 however, successfully incorporate the advantages
of double hybrid functionals and retain the fourth order scaling of classical KS methods through
*
The author thanks Ágnes Szabados for suggesting this paper for reading.
34
tricks similarly implemented in the MOS-MP2 method. It is not clear yet, whether MOS-MP2 or
SOS-MP2-like long range corrections suit better with double hybrid functionals.
There are two new branches of DFT functional development, which may prove to be superior to
current functionals once they reach a mature stage in development. Virtual orbital dependent, also
called as random phase approximation (RPA); or fragment-based methods, e.g. DFT-symmetry
adapted perturbation theory (SAPT); are not evaluated here because either they are not completely
generalized, in some kind of preliminary development stage, or currently not applicable to large
systems. Mihály Kállay, however, has already published a linear-scaling implementation of the
dRPA method.117,*
We, however, mention an RPA theory related promising approach for future double hybrid
functionals, the JMP2 method, which is similar to SOS-MP2 with a different scaling factor for like
spins.118 The JMP2 method can be an essential building block for computationally available DFT
functionals capable of quantitatively dealing with dispersion in the future. In the meantime the
XDM-based corrections may be able to save “classical” hybrid functionals; or, at least, may
become a viable alternative.
*
The author thanks Ágnes Szabados for suggesting this paper for reading.
35
4 π–π Stacking
Fig. 2.1.1
Simplified schemes of π-stacking conformations
Aromatic π-π interactions are now known to play a key role in a wide range of important problems,
including the stereochemistry of organic reactions,119 organic host-guest chemistry and crystal
packing,120-122 protein folding and structure,123-126 DNA and RNA base stacking,127 protein-nucleic
acid recognition,128 drug design,129,130 and asphaltene (heavy crude oil) aggregation and fouling.131
There has been intensive effort to establish the potential energy surfaces of aromatic molecular
dimers. Detailed knowledge of the intermolecular structures adapted by simple model aromatic
liquids is currently lacking, however, they are essential to our fundamental understanding of π-π
interactions in condensed matter. Benzene and toluene are the archetype of aromatic liquids and the
simplest molecules, which can allow an attempt to understand the structures resulting from
intermolecular π-orbital interactions. Furthermore, these liquids are extremely important nonpolar
organic solvents in their own right and used in a very wide range of laboratory and industrial
processes. Toluene is also recognized as one of the molecularly most simple van der Waals glass
formers, exhibiting particularly pronounced temperature dependence of the mean reorientation
times, a property known as fragility.132 The delocalized π electrons impart a quadrupolar moment to
benzene, toluene, and other aromatic molecules. Favorable electrostatic interactions occur when
36
molecules are either in a parallel (PD) or perpendicular (T or Y) geometry (Fig. 2.1.1).133 These
electrostatic interactions compete with the attractive dispersion forces, which favor maximum
molecular contact through a parallel linear “sandwich” (S) orientation.134 For the benzene dimer,
extensive experimental135-139 and theoretical140-144 studies have shown that the T and PD
configurations are almost isoenergetic but the former (T) is now viewed as the global energy
minimum. This T interaction is sometimes referred to as an “anti-hydrogen bond” due to the fact
that the donating >C–H bond is shortened.141 What is termed a “Y-shaped” configuration by
Headen,145 in which two H atoms are directed towards the acceptor aromatic ring, is also found
close to the global energy minimum.144 In the case of toluene, the increased dispersion interactions
stabilize the stacked PD geometry146,147 and the molecular dipole moment favors a staggered
(antiparallel) arrangement of the methyl groups over the eclipsed geometry.128 The crystal
structures of benzene148 and toluene149-151 are well defined and dominated by the lowest energy
dimer motifs, resulting in perpendicular and parallel contacts, respectively. In contrast, there is only
very limited experimental diffraction data for the structure of liquid benzene and none in the case
of liquid toluene. Parallel displaced and T-shaped stacking give the most favorable electrostatic
interactions, while the sandwich geometry provides the maximum orbital overlap. In the Y-shaped
geometry, two H atoms are directed toward the acceptor aromatic ring. For toluene, the lowest
energy arrangement for PD places the methyl groups antiparallel (staggered) with respect to each
other and for T the –CH3 is in an uppermost position.133,134,142,144 X-ray and neutron
diffraction152,153 along with optical Kerr effect spectroscopy studies154 of liquid benzene point
toward local ordering that is mostly perpendicular.128,155 Without the use of isotopic substitution,
however, the neutron diffraction data can only give limited insight into the orientational structure.
In response to this lack of conclusive experimental data, a large number of computational studies
have been directed toward simple aromatic molecules such as benzene and toluene. Simulations of
liquid benzene using classical atom-centered force fields have shown random orientations or a
slight preference for perpendicular arrangements of nearest neighbor molecules.156-158 The use of
37
atom-centered force fields, however, has a drawback in that they do not accurately describe the
charge distribution of the aromatic molecule.133,159 Simulations using partial charges above and
below the aromatic ring have attempted to address this issue and for benzene yield a higher
preference for perpendicular arrangements of nearest neighbor molecules. 154,160 Liquid toluene has
received rather less attention, in spite of the fact that it is viewed as a better model for π-π
interactions in proteins.161 Recent simulations, however, indicate a prevalence of parallel stacking
of the aromatic planes, with staggered disposition of –CH3 groups.128,155,162 Since these predictions
are highly sensitive to the model force field, high resolution experimental data are required to guide
the theoretical chemist.
38
5 Results
The present dissertation starts with analyzing the surprisingly strong intermolecular effect between
two non-bonding aromatic rings resulting in an increased acidity on the phenol part. The author
shows that dispersion plays an important role in stacking and how this affects acidity. Next he
describes an intimate interplay of O--H···O and C--H···F hydrogen bonds and π-π stacking, which
allows a phenyleneethynylene-based dendritic molecule to fold and self-assemble into two
distinctively different molecular crystals as pseudopolymorphs. Later he argues that accurate
dispersion figures are unavailable for bigger systems and offers a new, attractive parameter-free
procedure for the evaluation of dispersion energies. Description of this new procedure is a
significant part of this chapter.
5.1 Effect of the π-π stacking interaction on the acidity of phenol
5.1.1 Motivation
As the connection between stacking and protein side chain acidity has been established in section 4
it seemed natural to investigate the influence of the stacking interaction on the acidity of tyrosine
side chains. The ultimate goal was to find a computationally cheap property of the stacking (guest)
system that should be in good correlation with the acidity of the tyrosine side chain (host). Ideally
the property can be measured on the isolated guest system and the property is local. If such a
property is found a consistent place for the “measurement” has to be determined and an acceptable
correlation must exist between said property and acidity. In order to evaluate correlation, a
procedure has been built to calculate acidity change of the host molecule with several different
guest molecules. The stacking interaction had to be studied in detail to guide the search for a key
effect. The host system was chosen as the simplest tyrosine ligand model that retains aromatic
properties, namely phenol. A series of monosubstituted benzenes were chosen as guests. Guest
39
molecules are complexed in T-shaped and parallel displaced arrangements with phenol and are
optimized fully at the MP2 level of theory. The aqueous phase proton affinity of the stacked phenol
is computed at the same level and used to estimate the ΔpKa of phenol upon stacking. Intrinsic
properties of the stacking substituted benzenes such as electronegativity and local hardness as well
as interaction energy components between stacked rings are used to analyze the influence of
stacking on the pKas.
5.1.2 Discussion
T
PD
Fig. 5.1.1
S
Typical relative positions of π-stacking rings
Experimentally observed acidity change of tyrosine sidechain supported by crystallographic data
suggested that π–π interactions should have an important role in affecting acidity of aromatic
hydroxy groups.163,164 Three main orientations had been known for stacking aromatic rings (Fig.
5.1.1): the parallel displaced (PD) configuration, the T-shaped (T) orientation, and the face to face
parallel or sandwich (S) system which is not a minimum of the PES.
40
Phenol was chosen as a model molecule of tyrosine, being part of its side chain yet hopefully
retaining its acidic-aromatic character. The stacking partner could not be much bigger either, due to
prohibitive computational costs at the MP2 level. Therefore benzene derivatives were chosen with
a single substituent on each covering a wide chemical variety. So, the following stacking partners
were chosen: Ph–X (X = NO2, CN, CHO, F, H, CH3, OH, NH2). Geometry optimizations for the
dimers were carried out at the MP2/6-31G* level for configurations PD and T. A separate series of
optimizations were carried out with the phenolate ion to calculate proton affinity values. During
these calculations no geometric restrictions were applied which is quite unique. This was done to
model the intermolecular interaction in the most realistic way. This treatment caused some
optimizations to converge in several hundred steps. Step sizes had been reduced to deal with the
extremely flat PES around the optima. For the same reason Newton-Raphson algorithm was
replaced by steepest descent at the cost of less efficient steps. Even so, dimers containing the
phenolate ion would rather converge to the more stable H-bonded form. In order to validate
optimized structures frequency calculations were carried out at the same level of theory for dimers
with the unsubstituted benzene (X = H). Structure A was found to be free of imaginary frequencies
validating the optimum. Not in case of T and the anions though, as 2-3 imaginary frequencies were
found against all efforts to find true minima. These frequencies correspond to twisting and swaying
motions of the rings relative to each other meaning that their relative orientations have surprising
flexibility. Not even highest level calculations found by the author exhibit optimized benzene
dimers without similar imginary frequencies. Interaction energies were calculated at the MP2/631G*(0.25) level.165 Where *(0.25) indicates polarization functions more diffuse than usual on
heavy atoms. This modified basis is validated by CCSD(T) calculations with a basis extrapolated to
the complete basis limit giving 0.1-0.2 kcal mol–1 energy difference for benzene dimers.
41
5.1.3 Stacking Interaction
Interaction energies and other properties of the dimers can be found in Table 5.1.1. Besides
interaction energy binding energy is calculated for each dimer. Both values are the energy
differences between the sum of the dimers and the dimer itself. In the case of interaction energy
monomer geometries remain unchanged when being calculated separately. In the case of binding
energy monomer energies are calculated using separately optimized geometries, therefore binding
energy contains geometry relaxation energy.
When comparing systems of different sizes, it becomes important that a finite number of basis
functions are employed. In case of e.g. dimers, the electrons of the monomer occupy partially filled
orbitals of other monomer and vice versa. This improves the description of the dimer. This will,
however, give a more accurate description of the dimer than the isolated monomers, for which the
electrons are restricted into the monomer’s finite basis set.
42
Table 5.1.1 Properties of the optimized T and PD complexes: ΔEi (kcal/mol) binding energies at the MP2 and
interaction energies at the MP2, Hartree–Fock (HF) levels and its correlation (ΔEcorr) and electrostatic (ΔEelec)
components (kcal/mol), charge transfer to the pyrimidine, Δq (a.u.), Molecular Electrostatic Potential (MEP)
minimum around the acidic hydrogen atom (a.u.) and estimated pKa of the stacking phenol.
ΔEMP2 bind
T - phenol
NH2
OH
CH3
H
F
CHO
CN
NO2
T – phenolate ion
NH2
OH
CH3
H
F
CHO
CN
NO2
PD – phenol
NH2
OH
CH3
H
F
CHO
CN
NO2
PD – phenolate ion
NH2
OH
CH3
H
F
CHO
CN
NO2
ΔEMP2 int
ΔEHF
ΔEcorr
ΔEelec
Δq
-3.77
-2.32
-2.24
-2.28
-2.58
-2.73
-2.94
-2.97
-2.41
-2.56
-2.51
-2.56
-2.81
-3.06
-3.31
-3.34
1.94
1.72
1.85
1.73
1.39
1.10
0.79
0.61
-4.35
-4.27
-4.36
-4.29
-4.20
-4.15
-4.10
-3.96
-0.70
-0.84
-0.81
-0.88
-1.15
-1.34
-1.25
-1.61
-6.24
-5.61
-5.05
-5.38
-7.73
-9.25
-10.74
-11.29
-5.26
-6.22
-5.77
-6.17
-8.45
-10.68
-12.41
-12.85
0.04
-0.73
0.11
-0.40
-3.25
-5.09
-6.70
-8.24
-5.30
-5.49
-5.88
-5.77
-5.21
-5.59
-5.71
-4.60
-2.01
-2.69
-2.67
-2.62
-3.00
-3.47
-3.23
-3.87
-2.29
-2.63
-2.64
-2.59
-2.90
-3.43
-3.68
-3.80
5.93
5.47
5.73
5.57
5.08
4.83
4.50
4.21
-2.76
-1.51
-1.78
-1.77
-4.30
-6.98
-8.24
-8.56
-2.28
-1.83
-2.35
-2.22
-4.77
-8.36
-9.70
-9.80
5.80
6.18
6.26
6.13
2.99
0.71
-1.28
-2.86
MEP
pKa
0.033
0.034
0.033
0.038
0.040
0.045
0.049
0.045
0.1030
0.1052
0.1047
0.1053
0.1078
0.1096
0.1120
0.1114
8.9
8.7
8.8
8.7
8.2
7.6
7.1
7.0
-5.20
-4.41
-3.93
-4.15
-6.42
-8.35
-10.34
-10.58
0.077
0.084
0.082
0.086
0.089
0.107
0.114
0.121
-0.1950
-0.1938
-0.1945
-0.1943
-0.1916
-0.1878
-0.1857
-0.1850
-8.22
-8.09
-8.37
-8.16
-7.98
-8.26
-8.17
-8.01
0.74
2.20
2.70
2.33
1.55
0.63
0.03
-0.32
-0.006
0.001
-0.002
0.000
0.008
0.012
0.015
0.019
0.0993
0.1018
0.1021
0.1024
0.1052
0.1069
0.1096
0.1099
-8.08
-8.01
-8.61
-8.35
-7.76
-9.07
-8.42
-6.94
0.24
-0.35
0.67
0.98
-1.17
-3.49
-4.74
-4.03
0.041
0.067
0.061
0.068
0.075
0.096
0.096
0.099
-0.1980
-0.1991
-0.1982
-0.1985
-0.1957
-0.1904
-0.1893
-0.1895

9.4
9.7
9.6
9.6
9.0
8.1
7.8
7.9
An error would only arise if we compare the energy of monomers with that of the dimer. Doing so
would be similar to compare energies calculated with different basis sets. This effect is the basis set
superposition error (BSSE). The interaction and binding energy calculations were corrected by the
counterpoise166,167 procedure. This is to calculate the energy of the dimer so that one monomer
(electrons and nuclei) is removed from the system, but all basis functions are kept unchanged
(ghost atoms). This process is then repeated for the other monomer. The energy differences
43
between the monomers and the respective energy calculated with ghost atoms are summed up and
called the counterpoise BSSE energy correction term. The corrected interaction energy term is
expressed as
Eint  E AB  E A( AB )  E B ( AB )
5.1.1
BSSE  E A( AB )  E B( AB )  E A  E B
5.1.2
while the BSSE correction term is
where the left hand side of Eq. 2.1.2 is the corrected interaction energy, EAB is the energy of the
dimer, E A( AB) and E B( AB) are energies of the isolated monomers and the ghost atoms of the other
monomer. The BSSE can be applied during the optimization or simply just after the optimization as
a single point calculation correction. The author and coworkers chose the latter option because it is
simpler to carry out, the differences are small and none of the two ways is clearly superior to the
other.
The interaction energy of structure T is higher than the binding energy with the exception of
aniline. The same is true for the deprotonated form of T. The trend is reversed for PD, although the
difference is much smaller. Benzonitrile and aniline are excluded from the trend. Interaction energy
is greater again for the deprotonated complexes. This means that (except for the non-deprotonated
parallel displaced form) geometry changes of the monomers are energetically unfavorable.
The electrostatic interaction was computed from a distributed multipole analysis using the O RIENT
program version 3.2.168,169 Multipoles were computed at the nuclear positions with GDMA version
1.3 from the MP2/6-31G* (0.25) wave function170.
The correlation energy term (ΔEcorr in Table 5.1.1) has been calculated as the difference between the
HF and the MP2 energy in order to show the explicit correlation and its assumed stabilization
44
effect. Stabilization effect of ΔEcorr on the neutral complex is about 1 to 1.5 kcal mol–1 and that
exceeds the total interaction energy. However, for the deprotonated forms of the same structures
ΔEcorr
is smaller than the overall interaction energy. This more pronounced in case of electron
withdrawing substituents. This can be explained by the significantly elevated electrostatic
interaction due to the extra electron on the phenolate ion. The groups with higher electron
withdrawing character enhance charge separation. This explanation is supported by the electrostatic
energy terms (ΔEelec in Table 5.1.1) and charge separation values (Δq) as well. Relative stability of
T-shaped conformations is slightly lower by 0.2 kcal mol–1 on average, which is probably within
systematic error of the calculations. Energy decomposition, however, shows that the nature of the
interaction is different between T and PD. The stabilizing effect of ΔEcorr is about twice as big for
PD than for T.
5.1.4 Estimation of pKa
A theoretical approach was followed in which the gas-phase proton affinity of an acid and the
solvation energy of the acid and its conjugate base are calculated and combined to estimate pKas.
Such methods have been employed by several groups with considerable success,124,171-173 for
isolated molecules as well as for molecules undergoing chemical reactions, thus involved in strong
intermolecular interactions.174 In order to estimate the change in the pKa of phenol upon stacking, a
calibration curve is constructed by plotting experimental pKa values of a series of substituted
phenols (with pKas ranging from 0.38 to 10.35)175 versus their computed proton affinity in aqueous
phase (PA(aq)). Linear extrapolation thereof yields estimated pKas for the stacked phenols, using
their computed PA(aq) in the stacked complexes. PA(aq) was computed according to:
PAaq   PAgas   ΔΔGsolv  PAgas   ΔΔEsolv
5.1.3
where PA(gas) is the gas-phase proton affinity and ΔΔGsolv is the solvation energy difference
between the acid and conjugate base, calculated at the MP2/6-31G*(0.25) level.
45
12
10
8
6
pKa
y = -87.09x - 47.13
R2 = 0.88
4
2
-0.6700
-0.6300
-0.5900
0
-0.5500
PA(aq)
Fig. 5.1.2
Aqueous-phase proton affinities (a.u.) computed for substituted phenols (see) vs. experimental pKas
The solvation energy was estimated as the energy difference of the gas-phase optimized structure
and the same structure in the aqueous phase. This method has been successfully used to compute
the pKa of acetic acid during α-proton abstraction along the reaction coordinate,174 and yields a
reasonable correlation between experimental pKas and PA(aq) (Fig. 5.1.2 and Table 5.1.2). In order
to assess the change in acidity of phenol upon stacking, we computed PA(aq) values, as well as the
molecular electrostatic potential (MEP) values 2 Ǻ from the oxygen atom of the stacked phenol in
the direction of the O–H bond, as was done in a previous study on the acidity of alcohols. 176 The
charge transfer Δq was calculated as the sum of the ChelpG charges on phenol in the complex.
Solvated species have been calculated by the polarizable continuum model (PCM) as implemented
in Gaussian03.177 The PCM approach is not without flaws for bigger systems,178 however, small
aqueous systems are well tested. The ChelpG partial charges were computed as implemented in
Gaussian03.179
46
Table 5.1.2
Experimental pKa-s for substituted phenols, measured in water 175 and computed aqueous-phase proton
affinities according to Eq 5.1.3 (a.u.).
pKa
PA(aq)
Hydroquinone
10.35
-0.6618
Pyrocatechol
9.85
-0.6325
Resorcinol
9.81
-0.6575
o-cresol
10.2
-0.6565
m-cresol
10.01
-0.6618
p-cresol
10.17
-0.6615
Phenol
9.89
-0.6615
2-chlorophenol
8.49
-0.6383
3-chlorophenol
8.85
-0.6387
4-chlorophenol
9.18
-0.6403
2,3-dichlorophenol
7.44
-0.6331
2-nitrophenol
7.17
-0.6405
3-nitrophenol
8.28
-0.6203
4-nitrophenol
7.15
-0.6038
2,4-dinitrophenol
3.96
-0.5941
3,6-dinitrophenol
5.15
-0.6042
Picric acid
0.38
-0.5589
5.1.5 Isolated substituted benzenes properties
In order to represent the influence of the stacking substituted benzene on the pKa of the stacked
phenol, we tested a global and a local DFT based reactivity index.180 First, the electronegativity χ,
as a global descriptor, is used to quantify the substituted benzene’s ability to receive electrons and
thus to modify the electron distribution of the stacked phenol. χ is defined as the first partial
derivative of the energy with respect to the number of electrons. Using a finite difference
approximation, one gets
PAaq   PAgas   ΔΔGsolv  PAgas   ΔΔEsolv
5.1.4
47
with I as the vertical ionization energy, and A as the vertical electron affinity. For the substituted
benzenes considered in this study, the calculated electron affinity values were found to be negative;
hence the electronegativity was taken as half the ionization energy. In previous studies on stacked
complexes, the local hardness η(r) has been successfully used in the estimation of the electrostatic
interaction between the stacked rings. Indeed η(r) reflects the accumulation of negative charge at a
defined point independently of the number of electrons of the system and was useful to explain the
inverse correlation between electrostatic repulsion and hydrogen bonding ability of stacked
aromatic nitrogen bases.166,181,182 η(r) can be approximately written as in Eq.2.1.30:147,183-186
 (r )  
V el
2N
5.1.5
Here N is the number of electrons of the system and Vel(r) the electronic part of the electrostatic
potential. For the parallel-displaced conformations we used η1(r), for which Vel(r) was evaluated at
a distance of 1.7 Ǻ above the isolated benzene rings (Fig. 5.1.3). η2(r) was used for the T-shaped
geometries, Vel(r) was computed at a distance of 2 Å from the carbon atom in para position with the
benzene substituent.
Fig. 5.1.3
Local hardnesses evaluation for isolated substituted benzenes
48
5.1.6 Discussion of π-stacking energy decomposition
12.00
10.00
8.00
pKa
6.00
4.00
2.00
0.00
Aniline (NH2) Phenol (OH)
Toluol (CH3) Benzene (H)
Fluorobenzene (F)
Benzaldehyde Nitrobenzene
(CHO)
(NO2)
Benzonitrile
(CN)
Isolated
phenol
T-shaped
8.90
8.66
8.81
8.73
8.15
7.58
7.00
7.11
10.26
parallel displaced
9.41
9.71
9.59
9.62
9.01
8.10
7.93
7.84
10.26
substituents
Fig. 5.1.4
Acidity on the phenol part plotted against increasing substituent electron withdrawing effect. Isolated
phenol is given for comparison. All values are calculated, not the experimental values.
10,0
9,0
pKa
R2 = 0,86
8,0
T-shaped
7,0
R2 = 0,89
Parallel-displaced
6,0
0,32
0,34
0,36
0,38
0,4

Fig. 5.1.5
Electronegativity of the stacking benzenes (a.u.) vs. the corresponding (computed) pKas for the stacked
phenol
49
All isolated benzene properties as well as complex properties can be found in Table 5.1.1 and
Table 5.1.2, respectively. For complexes with phenol the dispersion energy (corresponding to the
correlation part of the interaction (DEcorr)) is found to be the major contribution to the stabilization,
with values about twice larger for parallel displaced arrangements than for T-shaped (Table 5.1.2).
In contrast, the inverse is found for the electrostatic interaction which turns out to be repulsive for
parallel conformations. Nevertheless, due to the Hartree–Fock contribution to the interaction, the
total and Table 5.1.2 interaction energies are almost the same for the two types of stacking
arrangements. Binding energies which show the parallel arrangements to be the more stable,
display the lower values for the unsubstituted stacking benzene, in agreement with previous
studies.134,147,171,172,185,186 This seems to contradict the Hunter–Sanders rules133 stating that
electrostatics governs substituents effects on interaction energy and that electron withdrawing
substituents should decrease the π electron density above the ring and thus should decrease the
repulsion (while electron-donating substituents should behave oppositely). However, these rules are
valid considering only electron-withdrawing substituents and methyl for parallel-displaced
arrangements: the electrostatic repulsion decreases with the increasing electron withdrawing
character of the substituents. For phenolate/benzene complexes the elctrostatic portion of the
interection energy (DEelec) increases the magnitude of the binding and interaction energies, and it is
clearly the major source of stabilization of T-shaped complexes, as could be expected from the
negative charge on the phenolate ring. Both stacking arrangements lead to lower pKa values
compared to the unstacked phenol (pKa = 9.89). The largest effect is due to the T-shaped
conformations decreasing the pKa by almost 2 units. This is in good agreement with an
experimental study on the low pKa of tyrosine 9 involved in stacking interaction with a phenyl ring
in Glutathione S-Transferase A1-1; there the effect of stacking is found to decrease the pKa by 2
units as well. Interestingly, the change in the pKa of phenol upon stacking appears to be related to
the character of the benzene substituent: the pKa decreases with increasing electron-withdrawing
character of the substituent and electronegativity of the substituted benzene. This relationship
50
between pKa and electronegativity is quite intuitive, the more electronegative is the stacking
benzene, the more charge is pulled out from phenol, the more polarized the O–H bond will be and
thus the more acidic the phenol will be (see Fig. 5.1.5). Concomitantly with the effect of the
stacking conformation on the change in pKa, the charge transfer is larger for T-shaped structures.
Likewise, this interpretation is applicable using the MEP computed around the acidic hydrogen
atom. MEP values are higher around more acidic hydrogen atoms. One may notice however that
the charge transfer is not the only cause for the decrease in pKa. Indeed for parallel-displaced
complexes with toluene and aniline the charge transfer is inverse (from the substituted benzene to
phenol) while the pKa of the stacked phenol is still lower than that of the unstacked phenol.
The local hardness η(r) reflects the accumulation of negative charge, above the substituted benzene
ring η1(r), in para position with the substituent η2(r) (see Fig. 5.1.3). One may notice from Table
5.1.1 that the substituent effect is not reflected in the η(r) values. According to Hunter–Sanders
rules stating that the negatively charged π-electron density is responsible for the repulsion between
stacked rings, η1(r) and η2(r) can be used to estimate the electrostatic interaction for the parallel
displaced and T-shaped arrangements, respectively. In agreement with a previous study,172 Fig.
5.1.7 shows a better correlation for parallel geometries compared to T-shaped. η(r) as a local
descriptor is also related to the charge transfer between the rings. This relationship is less intuitive,
but we observe that the larger the η(r), the larger the charge transfer. Considering the above
interpretation concerning the charge transfer and the pKa of phenol, η(r) may be used to estimate the
change in pKa. Indeed, the correlation coefficients for η(r) vs. pKa curves are R2 = 0.88 and R2 =
0.76 for parallel-displaced and Tshaped arrangements, respectively. Moreover, it indicates that the
pKa is related to the electrostatic interaction between the rings. Fig. 5.1.6 shows that the larger the
electrostatic repulsion, the larger the pKa of phenol.
51
Fig. 5.1.6
Local hardness η(r) (a. u.) vs. electrostatic interaction energy between the rings ΔEelec (kcal mol-1).
10,0
9,5
y = 114,27x - 2,58
R2 = 0,89
9,0
8,5
pKa
y = 106,64x - 7,88
R2 = 0,76
8,0
7,5
7,0
6,5
0,0800
T
merőleges
PD
párhzamosan
eltolt
0,1000
0,1200
0,1400
0,1600
 (r)
Fig. 5.1.7
pKa values for the stacked phenol vs. the local hardness of the stacking benzenes (a.u.)
Both global and local descriptors are successfully used to analyze the effect of stacking on the
acidity of phenol. On the one hand, the electronegativity helps us to investigate the intrinsic role of
the stacking partner. On the other hand, the local hardness is of great utility to understand the
52
influence of the stacking interaction. Although the latter is characterized by a large contribution of
the dispersion forces, the electrostatic component of the interaction reflects relatively well the
effect of electron withdrawing substituents.
5.1.7 Conclusion
The influence of the stacking interaction on the pKa of phenol is not negligible. In agreement with
experimental findings,164 it is found that the stacking interaction decreases the pKa of the phenol
ring up to 2 pKa units, with the largest for T-shaped stacking. Two major causes of this decrease
have been found, the electronegativity of the stacking benzene, via charge transfer from phenol,
and the electrostatic repulsion between the rings which increases with increasing pKa.
53
5.2 The Art of Stacking
Pseudopolymorphism of a phenyleneethynylene-based C3-symmetric π-conjugated molecule is
described in this section.
5.2.1 Motivation
Starting 2010 the author was fortunate to visit the Inorganic Chemistry Division of Indiana
University Bloomington. Dr. Ho Yong Lee, a postdoctoral researcher in Dr. Dongwhan Lee’s
group had ongoing synthesis of an intriguing new phenyleneethynylene-based C3-symmetric
molecule 1 having a fluorinated molecular core (Fig. 5.2.1).
Fig. 5.2.1
Synthetic route to 1
The author was surprised to see the sketch of the molecule, which somewhat reminded him of the
nucleic bases he studied earlier: planar π-conjugated skeleton prone to H-bonding through
apparently electron deficient hydroxyl groups, and a possibility for cofacial (PD) π-π stacking.
Later it was explained to him that 1 was designed to have a solvent dependent conformation switch,
resulting in different conformers. An ability of a molecule to have different structures in the solid
state is called polymorphism.187-193 When the different polymorphs depend on solvent choice it is
pseudopolymorphism.194 A practical importance of this phenomenon has been recognized in the
54
field of pharmaceuticals since (pseudo)polymorphs often display different physicochemical
properties.187,189,195,196
A delicate balance of van der Waals interactions “decide” the fate of structural folding and
stacking, similar to protein folding. Concerted effort from the fields of synthetic, analytical (X-ray
crystallography and NMR) and theoretical chemistry was necessary to reveal every aspect of the
causes of the pseudopolymorphism behavior of 1.
Structure analysis on two distinctively different molecular crystals of 1 revealed an intimate
interplay of strong O–H···O and weak C–H···F hydrogen bonds in structural folding and solidstate self-assembly (Fig. 5.2.2). Intermolecular stacking and especially docking was not by design,
and came as surprise.
Fig. 5.2.2
Conformational switching to achieve shape complementarity for self-association.
Hydrogen bonding is one prominent type of non-covalent interaction, the directional nature of
which is proven to play a critical role in the spontaneous structural ordering of many chemical
architectures.197-199 While strong hydrogen bonds are typically constructed using N–H···O, O–
H···O, or O–H···N motifs, weak hydrogen bonds involving C–H···O, C–H···N, or C–H···π
contacts have recently been recognized as an important class of non-covalent interactions.200-202
55
Recent examples of linear foldamers and heterodimers that are stabilized by N–H···F hydrogen
bonding support the notion that organo-fluorine fragments can also be utilized as a hydrogen
bonding acceptor (HBA) for the assembly of higher-order structures.203-205 Undoubtedly, the
charged F- anion is a strong HBA,206 but the ability of covalently bound fluorine atom, as present in
the C–F group, to function as an effective HBA still remains controversial.*192,207-210 Statistical
analysis on crystallographically determined structures suggests, however, that C–H···F interactions
could be as important as C–H···O or C–H···N hydrogen bonds.211-215
5.2.2 Discussion
Quantum chemistry-based insight was requested to help understanding the relative importance of
intramolecular interactions, which was carried out by the author. All the synthetic and analytical
chemistry results, however, belong to coauthors of The art of stacking paper.9 The complete work
will be presented in this section, however, as the theoretical part would seem aimless without
context.
The final round of cross-coupling carried out by Dr. Ho Yong Le between 4 and 1,3,5-trifluoro2,4,6-triiodobenzene resulted in a mixture of the desired product 1, along with partially coupled
byproducts and the homocoupling product of 4. A simple layering of hexanes over a CHCl3
solution of this material, however, induced selective crystallization of 1, which was isolated and
fully characterized (Fig. 5.2.3). Self-association of 1 in solution was subsequently investigated by
concentration-dependent 1H NMR studies in CDCl3 at T = 298 K. A systematic downfield shift of
the propargylic O–H resonance from 2.52 to 2.73 ppm with increasing concentration within the
range of 10 to 40 mM could be fitted with K = 4.1 (±0.5) M-1 using the EK model.216 This
observation implicated the involvement of O–H···O hydrogen bonding in the self-association
*
Understanding the supramolecular chemistry of the C–F group is thus considered as one of the last frontiers in
intermolecular interactions.
56
process, which should become more favorable with decreasing polarity in the CHCl 3/hexane binary
solvent mixture used in crystallization.
57
Fig. 5.2.3 (a) 1H NMR spectrum (400 MHz, T = 298 K), and (b) 13C NMR spectrum (100 MHz, T = 298 K) of 1
in CDCl3. (c) Partial 13C NMR spectrum for the spectral window, which is indicated with a box in (b). 1H NMR
and 13C NMR spectra were recorded on a 300 MHz Varian Gemini 2000 or a 400 MHz Varian Inova NMR
Spectrometer. Chemical shifts were reported versus tetramethylsilane and referenced to the residual solvent
peaks.
58
Fig. 5.2.4 Top view of X-Ray structures of 1U (a) and 1F (b) with thermal ellipsoids at 50% probability, where
O is red and F is blue. The O–H···O and C–H···F interactions are represented by red and blue dotted lines,
respectively.
Fig. 5.2.5
Side view of X-Ray structures of 1U (a) and 1F (b); cf. Fig. 5.2.4
Intriguingly, two different kinds of single crystals of 1 were obtained depending on the method of
crystallization. Specifically, colorless blocks of the completely ‘‘unfolded’’ conformer 1U (Fig.
5.2.4a) were obtained by slow diffusion of hexane into a CHCl3 solution of 1.* On the other hand,
an instant mixing of a CHCl3 solution of 1 with hexane or benzene produced colorless needles of
the ‘‘folded’’ conformer 1F (Fig. 5.2.4b),* which was also characterized by single crystal X-ray
crystallography. As shown in Fig. 5.2.4 and Fig. 5.2.5, the fluorinated benzene core and peripheral
aryl groups in 1U are disposed in an essentially perpendicular manner with dihedral angles of τ =
79.3(2) and 83.4(7)°. The dihedral angle here is the angular relationship between the central C 6F3
fragment and the peripheral aryl rings, all treated as least squares planes calculated from
crystallographic data. In this analysis, τ = 90° corresponds to a perfectly orthogonal arrangement,
*
In analogy to protein secondary structures, the term ‘‘folding’’ here describes conformational restrictions imposed by
non-covalentbonds.
59
whereas τ = 0° corresponds to a perfectly coplanar arrangement. For the other conformer 1F,
however, two of the three ethynyl-extended ‘‘wingtip’’ O–H groups engage in an intramolecular
O–H···O hydrogen bond (O···O = 2.731(4) Å), which, along with C–H···F contact to the
fluorinated core (C···F = 3.209(5) Å), effectively flattens the entire structure with relatively small τ
values of 9.0(2), 14.1(2), and 35.5(1)°. A close inspection of the intermolecular packing in the
lattice provided an initial clue to the origin of conformational isomerism of 1. Instead of making
intramolecular O–H···O bond as in 1F (Fig. 5.2.4b), 1U uses strong intermolecular O–H···O
hydrogen bonds (O···O = 2.689(5) Å) to form a discrete dimeric structure (1U)2. As shown in Fig.
5.2.6 and Fig. 5.2.7, the two 1U units comprising the dimer maximize their shape complementarity
by adopting an essentially ‘‘staggered’’ orientation, in which two C3-symmetric molecules are
rotated by 60 degrees with respect to each other to achieve a tight docking.
Fig. 5.2.6 Side view of capped stick (a) and space-filling (b) representations of the solid-state structure of
dimeric (1U)2 generated using X-ray coordinates. The O–H···O and C–H···F interactions are represented by
dotted lines, the close-up views of which are provided in partial structures (c) and (d).
60
Fig. 5.2.7
Top view of (a) capped stick and (b) space-filling representations (blue, top molecule; red, bottom
molecule) of the solid-state structure of dimeric (1U)2; cf. Fig. 5.2.6.
An interdigitated arrangement of phenyleneethynylene fragments in the dimeric (1U)2 thus
provides a well-shielded hydrophobic cavity that encapsulates a cyclic hydrogen bonding network
constructed by six ‘‘wing-tip’’ OH groups (Fig. 5.2.6c and d). A more fascinating feature of the
dimeric (1U)2 is additional stabilization provided by six C–H···F contacts between the propargyl
C–H groups and the fluorinated benzene core (Fig. 5.2.6a and c). The relatively short C···F
distance of 3.228(8) Å observed here lies toward the shorter end of the values determined for
previously reported structures.211,213-215,217-219
The functional relevance of such C–H···F contacts in the self-assembly process was evaluated
further by density functional theory (DFT) computational studies on the simplified model
compounds 1ʹ and 5 (at B3LYP/6-31G** theory level). The pseudo D3d-symmetric optimized
geometry (Fig. 5.2.8) of (1ʹ)2 provided an average value of d(F···H) = 2.45 Å along the C–H···F
contacts, which is ca. 0.17 Å shorter than the sum of van der Waals radii of F and H.
61
Fig. 5.2.8
1ʹ
5
(1ʹ)2
(5)2
Top, scheme of 1ʹ and 5; bottom, capped-stick representation of the DFT (B3LYP/6-31G**)
geometry optimized model (1ʹ)2 and (5)2.
Energy evaluation at the B3LYP/cc-pVTZ(-f) level further supported the notion that such C–H···F
interactions contribute to the thermodynamic stability of the dimer. Specifically, (1ʹ)2 is stabilized
by 7.4 kcal mol−1 relative to the essentially isostructural (5)2 lacking the C–F units (Fig. 5.2.8),
when the energy difference between the individual monomers and the corresponding dimers was
compared (Fig. 5.2.9). Divided by the total number of contacts within the dimer, each C–H···F
interaction contributes to the overall stability by ca. 1.2 kcal mol−1.
Similarly to the self-association of 1U, its conformational isomer 1F also forms a dimeric structure
in the solid state, but the topology of (1F)2 is determined by a quite different set of non-covalent
interactions. As shown in Fig. 5.2.4b, 1F adopts a planar conformation, which is anticipated to
promote cofacial (PD) stacking. Indeed, the assembly of dimeric (1F)2 is assisted by π– π stacking
(ca. 3.38 Å) between electron-deficient fluorinated benzene core and electron-rich tert-butyl-
62
substituted phenyl ring, the complementarity of which is further strengthened by the existence of
two such contacts within the dimer (Fig. 5.2.10a–c).
Fig. 5.2.9
Interaction energies of (1ʹ)2 and (5)2 referenced to the corresponding monomers 1ʹ and 5,
respectively.
In addition, the ‘‘dangling’’ OH group in the monomeric 1F engages in an intermolecular hydrogen
bond (O···O = 2.654(4); 2.670(4) Å) to the O–H···O–H unit of the stacking partner (Fig. 5.2.10a
and c) to complete an extended O–H···O–H···O–H array. Intriguingly, this crescent-shaped O–
H···O–H···O–H motif on each side of (1F)2 can further associate with the complementary O–
H···O–H···O–H unit of the neighboring dimer to close up the six-membered cyclic hydrogen bond
(Fig. 5.2.10d). The crystal structure of 1F also revealed an intermolecular C–H···F contact (C···F =
3.207(5) Å).
63
Fig. 5.2.10 Capped stick and space-filling representations of the solid-state structure of dimeric (1F)2 generated
using X-ray coordinates: (a) and (b) top view; (c) side view. In (d) is shown the extended structure of hydrogenbonded (1F)2 with hexameric OH cluster motif highlighted within a red box. The O–H···O and C–H···F
interactions are represented by dotted lines.
5.2.3 Computational Details
Geometry optimizations were performed using the Jaguar 7.7 ab initio quantum chemistry
program*220 at the B3LYP/6-31G** level of theory.20,21 The energies of the optimized structures
were re-evaluated by additional single-point calculations on optimized geometry using Dunning’s
correlation-consistent triple-ζ cc-pVTZ(-f) basis set.221 Optimizations were carried out using the
simplified model compounds 1ʹ and 5, in which the tert-butyl groups on the phenyl rings of the
parent system 1 were replaced by methyl groups to render the calculations computationally
*
Jaguar, v7.7; Schrödinger, LLC: New York, 2010. A review paper is given as reference from the year 2013.
64
tractable. Absolute energies of monomers, 1ʹ and 5, were calculated on frozen geometries obtained
from the optimized dimers, (1ʹ)2 and (5)2, respectively. Interaction energies were calculated as the
difference of refined energies between twice the monomer and the dimer. No further refinement of
the interaction energies was carried out (such as BSSE or solvent corrections) since the relevant
information is the difference of interaction energies where the effect of refinements are anticipated
to cancel out.
Optimized geometries maintain the pseudo-D3d symmetry of the dimer. In (1ʹ)2, the C–F···H
contacts are 2.45 Å apart from each other on average, which is 0.174 Å shorter than the sum of van
der Waals radii of F and H. In (5)2, the C–H···H contacts have an average distance of 2.69 Å,
which is 0.29 Å longer than the sum of van der Waals radii. Energy evaluation further supports the
hypothesis that C–F···H contacts contribute to the thermodynamic stability of the dimer. While (5)2
is stabilized by an interaction energy of 34.6 kcal mol−1 (referenced to the monomeric 5), (1ʹ)2 is
stabilized by 42.0 kcal mol−1 (referenced to monomeric 1ʹ). Divided by the number of C–F···H
contacts, each contact contributes to the overall dimer stability by 1.2 kcal mol−1.
5.2.4 Conclusion
An intimate interplay of O–H···O/C–H···F hydrogen bonds and π–π stacking collectively gives rise
to conformational pseudopolymorphs of a dendritic π–conjugated molecule (Fig. 5.2.2). For the
highly symmetric and non-planar conformer 1U, a maximum number of O–H···O and C–H···F
contacts is achieved by shape complementarity in monomer–monomer docking. For the less
symmetric and largely planar conformer, 1F, the molecule takes a different strategy to maximize
monomer–monomer contacts by using predominantly a combination of π–π stacking and O–H···O
interactions. One may think of the conjugated π system as conductive wires and the conformation
65
change as some kind switching action, which brings the idea of molecular circuits one tiny step
closer.
This section illustrates that theoretical, synthetic, and analytical chemistry are inseparably
intertwined; and investigative thoroughness and creativity is much more important than the specific
field of study. Van der Waals interactions drive the protein-like folding of 1: electrostatic
interaction, hydrogen bonds, C–H···F contacts, π–π stacking, and to some unexplored extent
dispersion forces. Among all van der Waals forces dispersion is the worst reproduced by current
DFT functionals, which necessitated MP2 calculations in section 5.1, and motivated section 5.3.
66
5.3 Building New Dispersion Terms
5.3.1 Motivation
Dispersion interactions play a fundamental role in physics, chemistry and biology, where they
appear for example in π-π stacking interactions contributing to the structural,124 catalytic163,222,223
and binding properties224-226 of proteins as well as nucleic acids. A theoretical description of these
interactions would therefore be desirable in order to improve our understanding of molecular
recognition processes and protein folding. This is not easily accomplished as dispersion
interactions can only be described at a level of theory that includes electron-correlation.227 Since
Density Functional Theory (DFT) methods do not correctly reproduce the dispersion component of
stacking interactions, at least 2nd order Møller-Plesset (MP2) theory must be used,227 which has
only in recent years become applicable to systems of a size that is relevant for the study of aromatearomate stacking interaction.163,171-173,227 Although the exponential increase in computer power will
bring ever larger systems within reach, the unfavorable scaling of MP2 methods makes this
progress go slowly. Therefore, substantial efforts are being made to find DFT methods with a
correct description of dispersion interactions through the development of functionals which take
long-range dispersion interactions into account,228 as well as by the inclusion of empirical
dispersion corrections, analogous to Molecular Mechanics (MM) force fields.229 Once such a
method is established, it becomes possible to study stacking phenomena in larger systems fully at
quantum chemical level.
In several studies adjustments were made to existing density functionals to improve their
performance for non-covalent interactions. Xu et al.228 designed an X3LYP extended functional,
based on the well-known B3LYP functional, that improves the accuracy for Van der Waals
complexes. (However, its greatest advantage lies in describing H–bonds) Zhao and Truhlar229-231
developed functionals based on simultaneously optimized exchange and correlation functionals.
67
Lin and Rothlisberger developed a dispersion-corrected DFT, where they augment the B3LYP
functional with dispersion corrected atom-centered potentials (DCACPs).232,233 Also, several
studies by Hirao and Lundqvist have been performed for developing special correlation functionals
which take long-range dispersion interactions into account.64,234-236 The inclusion of empirical
dispersion coefficients, as advocated by Grimme, Hobza and Head-Gordon, has booked some
success as a cost-efficient method for examination of stacking phenomena in larger systems, since
such applications are today not yet possible with the more elaborated methods mentioned
above.45,81,95,237-241 A different approach consists of calculating dispersion energies from dispersion
coefficients. For instance, Van Gisbergen et al. derivated van der Waals dispersion coefficients
from frequency dependent polarizabilities using time dependent DFT.98,242 On the other hand,
Becke and Johnson4-6 developed an approximation for the calculation of dispersion coefficients
from exchange hole dipole moments, that allows to obtain dispersion energies at the DFT level in
an easy and efficient fashion. However, the method of Becke and Johnson has two major
drawbacks. First, Becke and Johnson use inappropriate values for the polarizabilities of atoms in
the molecules, thus undermining the theoretical foundation of the method significantly. In their first
publication they used empirical values for polarizabilities, while in their later works they
approximated atomic polarizabilities by scaling free-atom polarizabilities by the Hirshfeld effective
volume of the atom in the molecule. Such rough approach for obtaining atom-in-molecule
polarizabilities ignores many relevant effects, such as electron density reorganization due to the
applied electric field. Second, Becke and Johnson make use of a damping function that strongly
reduces the values of dispersion energy even at equilibrium geometries by an approximate factor of
2.
In this section, the use of intrinsic polarizabilities will be presented, obtained from the Hirshfeld
method7,243-245, which is the joint work of the author and Dr. Vanommeslaeghe. As a result, not
only the dubious character of the atomic polarizabilities is eliminated, but also realistic dispersion
68
energies are obtained at equilibrium geometry without applying a damping function. In addition,
the presented all-Hirshfeld approach will be enhanced further in two separate ways. First,
anisotropy for the derivation of the dispersion coefficients will be introduced, in order to improve
the accuracy for complexes of reduced symmetry. Second, the iterative Hirshfeld method
(Hirshfeld-I) will be introduced into the calculation of the dispersion coefficients. The Hirshfeld-I
method, recently developed by Bultinck et al.,
245
brings several fundamental improvements to the
classic Hirshfeld weight function and the resulting partitioned properties such as charges, dipole
moments and polarizabilities. By its iterative nature the method eliminates the somewhat arbitrary
origin of the weight function and can also be applied to charged systems. It has been recently
shown by Van Alsenoy that not only the charges obtained with Hirshfeld-I are more in line with the
oxidation state of the atoms in the molecules, but also the intrinsic polarizabilities are more
adequate.20 These results are a significant step towards a meta-GGA density functional model of
intermolecular dispersion interactions that eliminates several weaknesses of dispersion corrected
DFT methods based on the London formula.
The current model has been developed from the idea based on the instantaneous dipole moment of
the electron and its exchange hole presented by Becke and Johnson.4-6,100,243 It has evolved through
several steps before reaching its present form – which is still just a foundation of an augmented,
reoptimized density functional envisioned by the author et al. A short introduction will be given
below to show the basic concepts before combining them into one method for the sake of clarity.
Sometimes discussing the author’s results necessitates the detailed explanation of the work of
others in tandem, which can blur the line between original results and previous results by others.
The author tries to draw the line with explicit statements showing whether he is talking about
original results or not. Also, the results the author describes as original are indeed the fruit of labor
of the coworkers and the author.
69
5.3.2 The Electron and Its Exchange Hole
As an electron moves through a multielectron system, it repels other electrons from its vicinity
through a combination of Pauli and Coulomb correlations. Pauli correlation arises from
multielectron wave-function antisymmetry and acts between electrons of parallel spin. It has the
consequence that the probability of finding another electron of parallel spin at the position of a
given electron is zero. This electron density depletion around the electron is called the exchange
hole. Coulomb correlation arises from the 1 r repulsive interaction between electron pairs and acts
between electrons of both opposite and parallel spins*. Coulomb correlation is much less important
than Pauli correlation. In, e.g., a single Slater determinant Hartree-Fock theory, Pauli correlation
contributes a significant exchange energy component to the total energy. Coulomb correlation, on
the other hand, contributes nothing in Hartree-Fock theory.243 Therefore only Pauli correlations are
considered in this work and it is assumed that each monomer in an intermolecular interaction is
described by a single Slater determinant.
The exchange hole or Fermi hole ( hX r1 ,r2  ) is the conditional probability density depletion for
finding an arbitrary electron of σ spin at point r2 if an electron of the same spin takes place at
position r1.
hX r1 , r2   
1
 r  r  r  r 
 r1  ij i 1 j 1 i 2 j 2
5.3.1
where i r  is the (real) value of the i-th orbital at position r and the summa goes over the
occupied orbitals of a restricted system.  is the total electron density of σ spin. For unrestricted
systems the expression lacks symmetry in terms of σ.
*
The Pauli and Coulomb repulsion is responsible for the space occupying nature of macroscopic objects and e.g. the
stability of stellar bodies like white dwarfs.
70
The relationship between the exchange hole and the exchange energy for Kohn-Sham or HartreeFock orbitals is shown below:
E X r1 , r2  
h r , r 
1
 r1  X 1 2 d 3r2 d 3r1

2
r12
5.3.2
The above equation holds a great conceptual power. It enables us to visualize self-interaction
correction and exchange. As the electron at r1 moves through the system the hole gives the
probability depletion of finding another electron of the same spin at r2. The probability density of
finding another electron of the same spin at the same place r1 = r2 is zero. It can be shown by
writing r1 in place r2 in Eq. 5.3.1.
hX r1 , r1     r1 
5.3.3
Therefore if the electron is localized at r1, the probability of finding another electron of the same
spin at r1 vanishes completely. This result agrees with the requirement of the Pauli exclusion
principle. Another property of the hole that it is always negative meaning that same spin densities
are always diminished around the electron.

1 
hX r1 , r2   
i r1 i r2 


 r1   i

2
5.3.4
It can be shown that the hole has a total charge of -1 integrated over the whole space.
 h  r , r d r
3
X
1
2
2
 1
5.3.5
As it can be derived using the orthonormality of the orbitals. This equation guaranties that the
negative charge depletion caused by the hole plus the charge of the electron give exactly zero.
The exchange hole can be used to conceptualize the first of Hund’s rules. Namely, electrons would
occupy near degenerate energy levels on a subshell with the most possible spins being the same
(Fig. 5.3.1) because this way their exchange hole prevents them to come too close to each other. By
avoiding each other in most of the time a good amount of Coulomb repulsion can be avoided. This
71
energy gain (avoidance of energy loss) greatly exceeds the slight repulsion originating from the
paramagnetic alignment.
Fig. 5.3.1
Two electron configurations of p2 electrons: 3P and 1D, from left
to right
Right here, an example of 2p electrons of carbon will be given, borrowed from Dan Dill (Fig.
5.3.2.)
Fig. 5.3.2 The red dot is the position of the reference electron, 2py, with alpha spin. The surface is the
conditional electron density of the 2px electron in the xy-plane. There is no magnetic field, so both orbitals can
rotate around the z axis. To the left: More overlap. The surface (2px) electron has beta spin (opposite), therefore
it is possible (very rarely) that the 2px density maximum coincides with the reference electron. (Fermi heap) To
the right: Avoided overlap. The surface (2px) electron has alpha spin (same), therefore the 2px density maximum
always avoids the reference electron via the Pauli repulsion. (Fermi hole)
The hole is almost never spherical around r1. It is only spherical under special circumstances. Such
an example is the uniform electron gas. Even in a perfectly spherical system the hole is
nonspherical unless it is right at the center of symmetry. So, while the charge of the electron and its
Fermi hole is zero, the dipole moment is different from zero. Since the exchange energy (Eq. 5.3.2)
senses only the spherical average of the hole around each r1, a nonzero dipole moment has no
effect on the energy of the system containing the electron.
72
The model presented by Becke and Johnson4-6,100,243 is based on the instantaneous dipole moment
of the electron (with spin σ) and its exchange hole dXσ at reference point r1 as :
 1

d X r1   
i r1  j r1    r2i r2  j r2 d 3r2   r1

  r1  ij

Fig. 5.3.3
5.3.6
The electron e- and its exchange hole h+ in a spherical atom
Fig. 5.3.3 shows when electron e– is at distance r from the nucleus of a spherical atom at solid angle
Ω. Notice that the mean position of the hole h+ does not coincide with position of the electron. Its
distance from the nucleus is r  d X  . Higher multipoles M  can be approximated using the
nucleus as origin:

M σ   r   r  d Xσ 


5.3.7
so the expectation value takes the form of:


M 2     r  r   r  d X  d 3r

 2
5.3.8
Based on Eq. 5.3.8, explicit expressions were derived for the calculation of the familiar C6, C8 and
C10 coefficients, characterizing the dispersion energy between two non-overlapping systems A and
B.
73
Edisp  
C6, AB C8, AB C10, AB
 8  10 .
6
RAB
RAB
RAB
5.3.9
Moreover, as Becke and Johnson pointed out, the dispersion coefficients of Eq. 5.3.9 could be
expressed, to a high level of accuracy, in terms of atom-atom interaction between the atoms
constituting both systems A and B. Explicitly for C6,AB:
A
B
a
b
C6, AB   C6,ab
5.3.10
With
C6,ab 
 a b M12
a M
2
1 a
a
M12
 b M
b
2
1 b
5.3.11
in which α and <M12> denote the polarizability and the ground state expectation value of the square
of the  =1 multipole moment, (i.e. dipole moment) respectively.
For C8 and C10 analogous expression were presented in terms of the higher order multipoles:243
M 2
a


2
    r1  r   r  d X  d 3r1


5.3.12
leading to
C8,ab
C10,ab  2
3  a b

2

 a b M 12
M
M 32
a
2
1 a

M12
M
b
a
2
1 a
 M 32
b  M
2
1 b
M 22
b
 M 22
b  M
a
a
M 12
b
2
1 b
a
M12
b

5.3.13
a
  21
 a b M 22
a
M 22
b
5 M 12  b  M 12  a
a
5.3.14
b
74
5.3.3 Stockholder Atoms in Molecule (AIM) – the Hirshfeld-C and Hirshfeld-I
methods
This section is based on the paper by Carbó-Dorca et al. and the author took some statements
directly from their paper.246 It is basically a shortened version with a few simplifications by the
author. A short introduction is necessary, because the Hirshfeld partitioning allows the use of ab
initio atomic polarizabilities in the sense of atoms in molecules (AIM) concept.
It has been shown so far how one can estimate dispersion interaction between two systems if
atomic properties are known in a molecule. However, it is not trivial to decompose molecular
properties into atomic ones. The recently developed Hirshfeld-I method246 has been used for
calculating all atomic properties based on purely molecular properties. The decomposition of the
multipoles of the exchange hole is quite straight forward (Eq. 5.3.12 ) using the Hirshfeld weight
factor when integrating over the whole space. Decomposing molecular polarizability on the other
hand is not so obvius. Before introducing the latter, a basic description of the Hirshfeld scheme is
discussed. Detailed discussion of the Hirshfeld-I partitioning is not attempted here, however. The
author kindly suggests the paper by Bultinck et al. for anyone interested in the details of the
Hirshfeld-I partitioning.
The Hirshfeld (or stockholder) decomposition is one of many AIM methods. The AIM concept
lacks true physical meaning, however, it is hard to argue with the conceptual power of the image of
atoms held together by chemical bonds. Within the large set of different definitions of the AIM, the
Hirshfeld scheme follows the Hilbert space (WF) approach as opposed to the Cartesian space
approach. The former deals with atom centered basis functions while the latter identifies AIM by
dissecting the Cartesian space in so-called atomic basins.
The Hirshfeld scheme is very popular within so-called conceptual DFT180,247 and the weighting
function, which allows identifying the AIM as that which is most similar to the isolated atom,248
has been shown to be directly derivable from information entropy.249-254 The Hirshfeld approach
75
results in overlapping atoms. Therefore it is also similar to other techniques such as the Voronoi
approach of Baerends and co-workers,255,256 the fuzzy atoms of Mayer and Salvador,257 or Becke’s
scheme.258
Despite the fact that the Hirshfeld AIM is quite popular, clearly there are shortcomings of the
model. These shortcomings, however, can be alleviated by a new scheme that has been
implemented by Bultinck and coworkers.246 They call it the iterative Hirshfeld scheme (HirshfeldI) while they suggest Hirshfeld-C as an abbriveation of the classical method.
The Hirshfeld method allows to partition properties into atomic contributions by means of a weight
function. The weight of each atom is determined by the density of the corresponding free spherical
atom, normalized by the sum of all the free atomic densities of the atoms in the molecule,
wA r  
 Afree r 
  Bfree r 
5.3.15
B
The shortcomings of the Hirshfeld-C method are summarized briefly without thorough explanation:
 Hirshfeld atomic charges tend to be virtually zero.259
 The Hirshfeld AIM populations depend on the choice of the promolecular density.260
 Hirshfeld charges are only available for neutral molecules.259
 Care needs to be taken in connecting Hirshfeld charges to information entropy.249
The Hirshfeld-I scheme answers every point of the above criticism, providing an iterative approach
that converges to a unique nonarbitrary solution. All the previous weaknesses have been solved by
introducing the constraint of no net charge transfer taking place in the molecule at the final iteration
step. This way the weight factors are iterated until this requirement is fulfilled and the AIM
properties only depend on the molecular density. Whereas in Hirshfeld-C the weight function is
predefined by the atomic densities of the free spherical atoms, in Hirshfeld-I the weight function is
76
iterated until self-consistency and therefore loses its somewhat arbitrary character. This means that
the starting atomic densities lose their significance because the result of the iterations is
independent of the starting atomic densities. The Hirshfeld-C weight function is used as a first
guess, and in each consecutive iteration the new weight function is constructed from the atomic
densities obtained from the weight function of the previous iteration,
 An1 r 
w r  
  Bn1 r 
n
A
5.3.16
B
The process is repeated until consecutive results differ only by a reasonably small value.
5.3.3.1 Intrinsic Atomic Polarizabilities
Krishtal et al.7 proposed a Hirshfeld scheme to partition molecular polarizabilities in terms of
atomic contributions as a sequel to previous work on electron density partitioning.261 It was proved
that the molecular polarizability can be written as
 xyA    xya  xa qa y  
5.3.17
a A
Thus, the xy element of the molecular polarizability tensor can be decomposed as the sum of the
intrinsic atomic polarizability (  xya ) the so called and charge delocalization term ( qa y  ) with xa
being the Cartesian coordinate of atom a in the x direction. The atomic intrinsic polarizability is
obtained using the following numerical integration:
 xya   wa r1 xa   y  r1 dr1
5.3.18
where wa(r1) is the Hirshfeld weight factor defined in Eq. 5.3.16 and   y  r1  is the first order
perturbed molecular density in the y direction, which can be calculated by methods such as
coupled-perturbed Hartree-Fock or coupled-perturbed Kohn-Sham.262 qa y  in Eq. 5.3.17 is the first
order perturbed atomic charge on atom a, given by
qa y    wa r1   y  r1 dr1
5.3.19
77
The use of the intrinsic atomic polarizabilities allows the calculation of highly accurate dispersion
coefficients. This way the use of damping functions can be circumvented. This is how the author
and Dr. Vanommeslaeghe implemented atomic polarizabilities in their dispersion correction term.
5.3.3.2 Incorporating Anisotropic Contributions via the Polarizability Tensor
This section shows how the author and Dr. Vanommeslaeghe implemented anisotropy. in their
dispersion correction term.
The use of atomic intrinsic polarizabilites, which are obtained in the form of an atomic
polarizability tensor, allows to introduce anisotropy into the model. Going back to the classical
paper by Buckingham,263,264 standard second order Rayleigh–Schrödinger perturbation theory for
long-range intermolecular forces was shown to yield the following general expression for the R −6
contribution to the dispersion energy for two molecules or two atoms a and b:
qa y    wa r1   y  r1 dr1
5.3.20
where  a  and  b  are the (dipole) polarizability tensors of the interacting systems and the
elements of the T2 tensor are defined as
1
T2 ,ij  i  j Rab
5.3.21
where Rab is the intermolecular distance and i and j stand for the Cartesian coordinates x, y, z. This
expression was obtained using an Unsöld- or London-type of approximation265,266 by simplifying
the energy denominator in the second order perturbation theory expression by introducing an
average excitation energy (or ionization energy) for a and b, namely, Ua and Ub. Eq. 5.3.20 can
then be rewritten as267
 
Eab,disp R 6  
   
U aU b
Tr Π α a  Π α b  
4U a  U b  

5.3.22
78


where the Π and α tensors involve the elements T2,ij and α ij . It is easily shown that in the case of
isotropic tensors Eq 5.3.22 reduces to
 
Eab,disp R
6

1
U aU b
 a  b   

α
α
Tr
Π


6
U a  U b 
4 Rab
 
2

5.3.23

where α a  and α b  are now equal to the diagonal elements of α a  and α b  (or one-third of their
2

traces), respectively. As it can be shown that Tr Π   6 , Eq. 5.3.22 finally reduces to
 
 
3
U aU b
α a α b 
6
2 Rab U a  U b 
Eab,disp R 6  
5.3.24
which is the standard London dispersion formula. Concentrating now on a pairwise atom-atom
interaction scheme, where in Eq. 5.3.22 a and b refer to isolated atoms or atoms in molecules, and
replacing in Eq.5.3.22, in the spirit of Becke and co-worker’s treatment4, the average excitation
energies Ua and Ub by expressions of the type 2 M12 3α , where α is the isotropic polarizability
of the atom in the molecule, we arrive at
aniso
ab,disp
E
R 
6
M 12
1

6 α b M 12
M 12
    a    b  
 Π α Π α 
Tr
 α a M 12


a 
b 
a 
b 
5.3.25
In the isotropic case the expression reduces to
  
Eab,disp R
6
M 12
αb M
2
1
a 
a 
M 12
b 
 αa M
2
1
α a α b 
b 
1
6
Rab
5.3.26
Anisotropy corrections to the R−8 and R−10 terms could be treated in a similar way necessitating,
however, the evaluation of terms involving quadrupole and mixed dipole-quadrupole
polarizabilities. As these contributions are expected to be smaller and lend themselves less easily to
an interpolation into the framework of Eq. 5.3.25, only the expression of an anisotropy corrected C6
term, optionally combined with isotropic C8 and C10 terms will be considered in the present thesis.
79
On the whole, anisotropy corrections to dispersion coefficients were relatively seldom studied in
the literature.268
5.4 Implementation and Benchmark Calculation Details of the New
Dispersion Term
It will be shown in this section that the use of intrinsic atomic polarizabilities in Eqs.5.3.11, 5.3.13
and 5.3.14 leads to highly accurate values for the dispersion energy, when evaluated at the
equilibrium geometry of a series of Van der Waals complexes. This method allows avoiding the
use of a damping function and opens a way to the idea of anisotropic corrections. Without a
damping function the current implementation does not contain any changeable parameters
eliminating any risk of over fitting.
The main goal of this work is to reproduce accurate interaction energies obtained from high level
calculations by adding dispersion energy corrections to interaction energies obtained at the DFT
level. For this purpose, a set of 18 different complexes was examined. In order to ensure that the
benchmark set is of good and consistent quality, individual geometries of the different complexes
have been optimized as follows. First the geometries of the monomers have been fully optimized at
the CCSD(T)/aug-cc-pVTZ level with tight convergence criteria. Subsequently, the geometries of
the complexes were optimized at the same level keeping the internal geometries of the monomers
frozen. For most of the complexes this meant optimizing only one parameter, namely, the distance
between the two monomers. For a few others the lateral displacement of the two monomers has
been taken into account. Symmetries of the complexes with the lowest reported energies were
taken from the literature. Once the equilibrium geometries were obtained, the interaction energies
at the CCSD(T) level and at the DFT level using the B3LYP, PBE, and TPSS functionals were
80
calculated taking into account the counterpoise BSSE correction.* Since the geometry of the
monomers was kept unchanged in the dimers, the interaction energies are given by

 



CCSDT 
CCSDT 
Einter
 EAB
 EACCSDT   EBCCSDT  ,

 
5.4.1



DFT
DFT
Einter
 EAB
 EADFT  EBDFT .
5.4.2
The asterisks in Eqs. 5.4.1 and 5.4.2 denote that the energies were obtained using all the basis
functions of the dimer. To ensure high quality reference data the values for the interaction energy at
the CCSD(T) level were extrapolated to the complete basis set (CBS) limit using the following
focal point analysis. To obtain the BSSE-corrected interaction energy one needs to extrapolate the
energies of the dimers and the complexes,

 



CCSDT  CBS
CCSDT  CBS
Einter
 EAB
 EACCSDT  CBS  EBCCSDT  CBS ,
5.4.3
where the total energy of each entity is a sum of the HF energy and the correlation energy,
CCSDT  CBS
HF CBS
CCSDT  CBS
Etot
 Etot
 Ecorr
.
5.4.4
Since it has been shown by Sinnokrot and Sherrill269 that the correlation energies at the MP2 and
CCSD(T) levels converge very similarly with the size of the basis set, the difference between the
two remaining constants for the set of the aug-cc-pVXZ basis sets upon the increase in X (ζ), it is
sufficient to extrapolate the correlation energy at the MP2 level to the basis set limit and add the
correlation energy value between the two methods obtained for a smaller basis set:


CCSDT  CBS
MP2 CBS
CCSDT  aug-cc-pVTZ
MP2 aug-cc-pVTZ
Ecorr
 Ecorr
 Ecorr
 Ecorr
.
5.4.5
The MP2 correlation energy at the CBS limit was obtained using Helgaker and co-worker’s linear
extrapolation formula,270,271
MP2 CBS
Ecorr

*
MP2 aug-cc-pVXZ
MP2 aug-cc-pVYZ
X 3 Ecorr
 Y 3 Ecorr
,
X 3 Y3
5.4.6
The counterpoise correction is defined in Eq.5.1.2.
81
where X=5 and Y =4 were chosen for all the complexes. Finally, since the HF total energy
converges toward the CBS limit fast and monotonously,271 the energies at the HF/aug-cc-pV6Z
HF CBS
level were used to estimate Etot
.
In the following step the interaction energy at the DFT level, obtained with the B3LYP, PBE, and
TPSS functionals, was corrected for dispersion,
corr.DFT
DFT
DFT
Einter
 Einter
 Edisp
,
5.4.7
where the dispersion interaction correction was calculated in four ways of increasing complexity.
The first two approaches, which neglect anisotropy, yield the following equations:

Isotropic C6 term only,
A
B
a
b
DFT
Edisp,
 
C iso
6

5.4.8
Three isotropic terms,
 C6iso,ab C8iso,ab C10iso,ab 
   6  8  10 .
Rab
Rab 
a b  Rab
A
DFT
disp,full
E

C6iso,ab
.
6
Rab
B
5.4.9
In a subsequent step anisotropy is introduced. In order to reach an optimal quality cost ratio
only the C6 term was corrected for anisotropy as it may be expected that the main
contribution to anisotropy will essentially be due to the C6 term. The counterpart of Eq.
5.4.8 may be written as follows:

Anisotropic C6 term only,
A
B
a
b
 
DFT
aniso
6
Edisp,
  Edisp
.
,ab R
C aniso
6

5.4.10
A compilation of the anisotropic C6 term and isotropic C8 and C10 terms finally gives
 aniso 6 C8iso,ab C10iso,ab 
   Edisp,ab R  8  10 .
Rab
Rab 
a b 
A
DFT
disp,mixed
E
B
 
5.4.11
82

The coefficients used in Eqs. 5.4.8-5.4.11 were obtained by the Hirshfeld-I method. Note
that no damping function has been used in these equations, since we are interested here in
dispersion energies of complexes at their equilibrium geometry, where the interatomic
distances are relatively large.
The geometries and polarizabilities of the different molecules were calculated using the
GAUSSIAN03272 program. The values of atomic intrinsic polarizabilities and expectation values of
the squared atomic multipole moments were obtained using the STOCK244 program. Finally the
different dispersion coefficients and the corresponding dispersion energy corrections were
calculated using the ATDISP7 program.
5.4.1 Results from a New Post-DFT Method
Table 5.4.1 lists the interaction energies obtained with the CCSD(T)/CBS, B3LYP/aug-cc-pVTZ,
PBE/aug-cc-pVTZ, and TPSS/aug-cc-pVTZ methods. The high level interaction energies vary
from –0.02 to –1.76 kcal mol–1, whereas the interaction energies obtained at the DFT level seem to
be dependent on the choice of functional.
83
Table 5.4.1
Interaction energies calculated with CCSD(T)/CBS, B3LYP/aug-cc-pVTZ, PBE/aug-cc-pVTZ, and
TPSS/aug-cc-pVTZ methods. All values are in kcal mol–1.
Complex
CCSD(T)
B3LYP
PBE
TPSS
He–He
−0.020
0.041
−0.056
−0.043
He–Ne
−0.037
0.040
−0.083
−0.058
He–Ar
−0.056
0.072
−0.078
−0.047
Ne–Ne
−0.071
0.045
−0.108
−0.066
Ne–Ar
−0.119
0.021
−0.123
−0.056
Ar–Ar
−0.272
0.177
−0.120
0.017
L–He–N2
−0.043
0.088
−0.073
−0.033
T–He–N2
−0.061
0.109
−0.075
−0.027
He–FCl
−0.096
0.072
−0.116
−0.046
FCl–He
−0.126
0.048
−0.220
−0.079
Ne–CH4
−0.175
0.066
−0.123
0.022
CH4–C2H4
−0.449
0.428
−0.101
0.158
CH4–CH4
−0.537
0.490
−0.025
0.273
SiH4–CH4
−0.824
0.549
−0.130
0.312
C2H2–C2H2
−1.403
0.153
−0.928
−0.548
CO2–CO2
−1.476
0.757
−0.388
0.133
OCS–OCS
−1.761
0.761
−0.083
0.572
C2H4–C2H4
−1.493
0.544
−0.284
0.431
For the B3LYP functional all interaction energy values are repulsive, as expected at the DFT level
where nonlocal dispersion interactions are not included in the exchange-correlation functional.
However, PBE produces negative interaction energies for all of the examined complexes while the
values obtained with the TPSS functional are partially negative. Since the attraction for most of the
examined complexes can be attributed only to dispersion energy, the negative interaction values at
the DFT level are not physically justified, B3LYP thus being in this aspect the most reliable of the
three functionals examined.
84
Table 5.4.2
Four different types of post-DFT interaction energies calculated with Hirshfeld-I method and
B3LYP functional. All values are in kcal mol–1.
Complex
C6iso
C6aniso
Full
He–He
0.01
0.01
−0.01
−0.01
−0.020
He–Ne
−0.01
−0.01
−0.04
−0.04
−0.037
He–Ar
0.01
0.01
−0.04
−0.04
−0.056
Ne–Ne
−0.03
−0.03
−0.08
−0.08
−0.071
Ne–Ar
−0.09
−0.09
−0.19
−0.19
−0.119
Ar–Ar
−0.08
−0.08
−0.34
−0.34
−0.272
L–He–N2
0.02
0.03
−0.02
−0.01
−0.043
T–He–N2
0.02
0.01
−0.05
−0.05
−0.061
He–FCl
−0.04
−0.04
−0.13
−0.13
−0.096
FCl–He
−0.04
−0.04
−0.09
−0.09
−0.126
Ne–CH4
−0.15
−0.15
−0.28
−0.28
−0.175
CH4–C2H4
0.06
0.03
−0.11
−0.14
−0.449
CH4–CH4
−0.06
−0.06
−0.30
−0.30
−0.537
SiH4–CH4
−0.21
−0.19
−0.62
−0.61
−0.824
C2H2–C2H2
−0.50
−0.48
−1.16
−1.15
−1.403
CO2–CO2
0.03
−0.04
−0.75
−0.81
−1.476
OCS–OCS
−0.33
−0.47
−1.56
−1.71
−1.761
C2H4–C2H4
−0.69
−0.55
−1.64
−1.50
−1.493
R
0.7064
0.8098
0.9412
0.9544
−
σ
0.44
0.37
0.18
0.16
−
Mixed CCSD(T)
Tables 5.4.2 – 5.4.4 give the dispersion-corrected post-DFT interaction energies obtained by the C6
isotropic (Eq.5.4.8), C6 anisotropic (Eq. 5.4.10), full isotropic (Eq.5.4.9), and mixed (Eq.5.4.11)
models for the B3LYP, PBE, and TPSS functionals, respectively. Dispersion corrections are
obtained with the Hirshfeld-I method. It has been shown in a study by Krishtal et al.273 that atomic
polarizabilities obtained with the Hirshfeld-I method are of better quality, therefore also leading to
more reliable dispersion coefficients than those obtained by the Hirshfeld-C scheme. The two last
lines in Tables 5.4.2 – 5.4.4 also show the correlation coefficient between the post-DFT interaction
energies and the high level interaction energies listed in Table 5.4.1 and the standard error of the
85
linear regression. Also the benchmark CCSD(T)/CBS energies are listed in the last column in each
table for convenience.
Table 5.4.3 The four different types of post-DFT interaction energies calculated with the Hirshfeld-I method and the
PBE functional. All values are in kcal mol–1. L–He–N2 refers to a colinear geometry of the van der Waals complex of
He–N2, while T–He–N2 refers to a T-shaped geometry of the He–N2 complex, where He is above the middle point of
the N-N bond.
C6iso
C6aniso
Full
He–He
−0.09
−0.09
−0.11
−0.11
−0.020
He–Ne
−0.13
−0.13
−0.17
−0.17
−0.037
He–Ar
−0.15
−0.15
−0.20
−0.20
−0.056
Ne–Ne
−0.19
−0.19
−0.25
−0.25
−0.071
Ne–Ar
−0.24
−0.24
−0.35
−0.35
−0.119
Ar–Ar
−0.38
−0.38
−0.64
−0.64
−0.272
L–He–N2
−0.14
−0.13
−0.18
−0.18
−0.043
T–He–N2
−0.17
−0.17
−0.24
−0.24
−0.061
He–FCl
−0.23
−0.24
−0.33
−0.33
−0.096
FCl–He
−0.31
−0.31
−0.37
−0.37
−0.126
Ne–CH4
−0.35
−0.35
−0.49
−0.49
−0.175
CH4–C2H4
−0.48
−0.54
−0.66
−0.71
−0.449
CH4–CH4
−0.59
−0.57
−0.85
−0.82
−0.537
SiH4–CH4
−0.92
−0.90
−1.35
−1.34
−0.824
C2H2–C2H2
−1.58
−1.57
−2.25
−2.24
−1.403
CO2–CO2
−1.15
−1.21
−1.96
−2.02
−1.476
OCS–OCS
−1.19
−1.33
−2.44
−2.59
−1.761
C2H4–C2H4
−1.54
−1.40
−2.50
−2.36
−1.493
R
0.9595
0.9777
0.9902
0.9953
−
σ
0.18
0.13
0.12
0.09
−
Complex
Mixed CCSD(T)
From Table 5.4.2 it appears that for the B3LYP functional, the addition of a dispersion interaction
correction based on the C6 dispersion coefficient reduces most of the interaction energies to
negative values, although for five complexes the interaction energies remain positive. Taking into
account the remaining C8 and C10 coefficients further reduces all post-B3LYP interaction energies
to values similar to the high level interaction energies Fig. 5.4.1 depicts the correlation between the
post-B3LYP interaction energy values, obtained using the isotropic and anisotropic C6 dispersion
86
coefficients, and the high level values. Although the correlation is poor, being only R=0.7064 and
R=0.8098 for the isotropic and anisotropic models, respectively, the effect of the addition of
anisotropy can be seen very clearly in this figure. Obviously, the anisotropic contributions are only
of significance for the complexes where the monomers are not spherically symmetric, so the values
for the smaller complexes remain the same. While for the isotropic model the values for the CO 2–
CO2 and C2H4–C2H4 complexes are situated far from the trend line, introduction of the anisotropy to
the C6 coefficient lowers the value for the C2H4–C2H4, although the CO2–CO2 complex remains an
outlier. One can conclude that the large standard error (listed in Table 5.4.2) and the very low value
of the slope (shown in Fig. 5.4.1) indicate that the C6-based models are insufficient by far for
reproducing accurate post-DFT interaction energies and that the higher coefficients are
indispensable.
Fig. 5.4.2 depicts the correlation between the high level interaction energy values and the postB3LYP interaction energy values for the full isotropic and mixed models. As was mentioned in
Sec. II, the derivation of anisotropic dispersion coefficients higher than C6 becomes complicated as
extra terms appear in the equations and the model loses its simplicity, which is one of its major
advantages. Therefore a mixed model is examined here, which on the one hand combines the
improvement achieved in the C6 coefficient by introducing anisotropy and on the other hand uses
isotropic C8 and C10 coefficients, which are vital for reproduction of accurate interaction energies.
The mixed model performs better than the full isotropic method, both of them performing
significantly better than the C6-based models, having correlation coefficients of 0.94 and 0.95 for
the former and the latter, respectively. Also the value for the CO2–CO2 complex improves
considerably by the addition of the higher dispersion coefficients. The difference between the two
models is the largest for the four complexes with the highest interaction energy values, namely
CO2–CO2, OCS–OCS, C2H2–C2H2, and C2H4–C2H4. As can be seen in Fig. 5.4.2, when only
isotropic contributions are taken into account, the order of interaction energies for those three
87
complexes is not correctly reproduced: while CCSD(T) interaction energies follow the order (in
absolute values) OCS–OCS > C2H4–C2H4 > C2H2–C2H2 > CO2–CO2, the interaction energies
obtained with the full isotropic model follow the order C2H4–C2H4 > OCS–OCS > C2H2–C2H2 >
CO2–CO2.
Table 5.4.4
The four different types of post-DFT interaction energies calculated with the Hirshfeld-I method
and the TPSS functional. All values are in kcal mol–1.
C6iso
C6aniso
Full
He–He
−0.07
−0.07
−0.09
−0.09
−0.020
He–Ne
−0.11
−0.11
−0.14
−0.14
−0.037
He–Ar
−0.11
−0.11
−0.16
−0.16
−0.056
Ne–Ne
−0.14
−0.14
−0.20
−0.20
−0.071
Ne–Ar
−0.17
−0.17
−0.28
−0.28
−0.119
Ar–Ar
−0.24
−0.24
−0.49
−0.49
−0.272
L–He–N2
−0.10
−0.09
−0.14
−0.13
−0.043
T–He–N2
−0.12
−0.12
−0.18
−0.18
−0.061
He–FCl
−0.16
−0.16
−0.25
−0.25
−0.096
FCl–He
−0.16
−0.16
−0.22
−0.22
−0.126
Ne–CH4
−0.20
−0.19
−0.33
−0.33
−0.175
CH4–C2H4
−0.21
−0.27
−0.37
−0.43
−0.449
CH4–CH4
−0.27
−0.25
−0.51
−0.48
−0.537
SiH4–CH4
−0.43
−0.42
−0.83
−0.82
−0.824
C2H2–C2H2
−1.19
−1.18
−1.83
−1.82
−1.403
CO2–CO2
−0.61
−0.67
−1.38
−1.45
−1.476
OCS–OCS
−0.50
−0.65
−1.70
−1.85
−1.761
C2H4–C2H4
−0.79
−0.65
−1.72
−1.58
−1.493
R
0.8572
0.8934
0.9791
0.9853
−
σ
0.32
0.28
0.13
0.11
−
Complex
Mixed CCSD(T)
Once anisotropy is introduced, however, the correct order is restored in the mixed model. The
reason for this difference is the lower symmetry of the complexes, where the monomers are
laterally shifted with respect to each other and anisotropy becomes more important. Therefore one
88
can expect the linear regression parameters to improve further for the mixed model when more
complexes of lower symmetry are taken into account.
Fig. 5.4.1
Correlation between high level interaction energies and post-B3LYP interaction energies obtained
with the isotropic and anisotropic C6 models. All values are in kcal mol–1.
Although the correlation achieved for the full isotropic and mixed models is satisfying, the linear
regression parameters are not optimal yet. With a slope of 0.82 in the full isotropic model and 0.83
in the mixed model, both models underestimate the interaction energy by an approximate 20%. The
PBE functional seems to suffer from an opposite problem, as can be seen from the values listed in
Table 5.4.3. The interaction energies obtained by the full and mixed models are overestimated by
no less than 40%, as can also be seen from the linear regression coefficients in Fig. 5.4.3. The
source of the overestimation of the values must be sought in the pure DFT interaction energies
obtained with this functional. As can be seen in Table 5.4.1, the PBE functional produces all
negative interaction energies even though dispersion is not included in the functional. Since the
89
dispersion energy correction is added to the original pure DFT interaction value, the spurious
potential well produced by the PBE functional for the examined complexes causes a serious
overestimation of the interaction energies. On the other hand, the correlation coefficients for the
full isotropic and mixed models are very high, being more than 0.99, while the standard error is
reduced by almost half in size in comparison to the B3LYP functional. One can therefore conclude
that our method for the calculation of dispersion energy corrections performs surprisingly well for
the PBE functional, but for accurate interaction energies the values must be scaled by a factor of
0.71.
Fig. 5.4.2
Correlation between high level interaction energies and post-B3LYP interaction energies obtained
with the full isotropic and mixed models. All values are in kcal mol–1.
It must also be mentioned that the effect of anisotropy on the interaction energy values is analogous
for the PBE functional: the replacement of the isotropic C6 coefficient by an anisotropic one in the
mixed model restores the correct order in interaction energy values for the four largest complexes.
The addition of the higher order coefficients C8 and C10 is here also of importance. Although the
correlation coefficients are quite high for the C6-based models, the values are less reliable. For
90
example, the interaction value for the C2H2–C2H2 is an evident outlier in Table 5.4.3 for those two
models.
Fig. 5.4.3
Correlation between high level interaction energies and post-PBE interaction energies obtained with
the full isotropic and mixed models. All values are in kcal mol–1.
The post-TPSS interaction energies are listed in Table 5.4.4 for the four different models and
depicted in Fig. 5.4.1 for the full isotropic and mixed models. This functional seems to perform
very well, the correlation for the full isotropic and mixed models being 0.98 and the standard error
on the linear regression being almost as low as for the PBE functional. The main strength of this
functional is the perfect slope of 1.00 for the mixed model, enabling us to reproduce accurate
interaction energies without the need for up- or downscaling, as is the case for the B3LYP and PBE
functionals. Among the larger complexes only the interaction energy value of the ethene dimer
appears to be problematic, being overestimated by 0.4 kcal mol–1. As was also the case with the
PBE functional, the source for this overestimation may lie in the negative pure DFT interaction
energy value of this complex (–0.548 kcal mol–1). It seems that our method is performing very well
91
for obtaining dispersion energies, but care must be taken if the potential energy surface produced
by the functional is incorrect.
Fig. 5.4.4
Correlation between high level interaction energies and post-TPSS interaction energies obtained
with the full isotropic and mixed models. All values are in kcal mol–1.
5.4.2 Conclusion
Becke and Johnson’s exchange dipole moment approach was combined with Hirshfeld-type
partitioning scheme for molecular polarizabilities, multipoles of the exchange hole and AIM to
reproduce chemically interesting interaction energies at the CCSD(T) level with simple dispersion
energy corrected DFT interaction energies. This approach leads to a simple, accurate, and
inexpensive method for evaluation of interaction energies of weakly interacting dimers. Three
functionals different by nature were examined here to test the robustness of our proposed method,
namely, B3LYP, PBE, and TPSS. Two major conclusions can be drawn from the present results.
First, the inclusion of anisotropy in the evaluation of the C6 coefficient leads to an improvement of
the corresponding dispersion energies. This improvement can be seen especially for the dispersion
92
energy values of the four complexes with lower symmetry, namely, CO2–CO2, OCS–OCS, C2H2–
C2H2, and C2H4–C2H4, where the mixed model produced significant improvements. Second, our
method performs well for different functionals regardless of their nature and might therefore be
universally applicable to different DFT functionals. However, since the final result for the
interaction energy is not only dependent on our dispersion energy correction but also on the pure
DFT interaction energy, care must be taken in the choice of the functional. From the three
functionals examined here TPSS seems to perform the best, producing accurate interaction energies
for most of the complexes. However, for complexes where the TPSS functional produces
questionable pure DFT interaction energies, such as was the case for the C2H2–C2H2 complex, the
B3LYP functional is a better choice.
93
6 Conclusions
Based on the already published papers (Refs. 3,7,8,9) the author has shown the significance of the
weak forces of chemistry, the van der Waals forces, as the intramolecular organizing forces:
1. The effect of π-stacking on the acidity of phenol is substantial – acidity can change two
orders of magnitude (zero to -2 pKa units). The T conformer shows stronger effect than the
PD.3
2. The increase in acidity, which has been shown experimentally,164 has two main sources.
The electronegativity, whose effect acts through intermolecular charge transfer, and the
repulsive electrostatic interaction, which decreases with acidity.3
3. With the help of very simple DFT calculations it has been shown that phenyleneethynylenebased dimers are indeed stabilized by H-bond-like F…H contacts.9
4. Based on concerted efforts with Dr. Vanommeslaeghe an exchange-hole based dispersion
correction method has been presented, which is an improvement over the method of Becke
and Johnson.7,8
5. The inclusion of the iterative Hirshfeld atomic partition procedure, and anisotropic
treatment of polarizability increases the quality of the dispersion correction, while the
computational cost is still insignificant compared the respective DFT calculation.7,8
So far the author presented the main steps towards a new ab initio tool for the computational
chemist and maybe for the material scientist. With the method faster or more accurate energy and
geometry computations become viable. This tool, let us call it XDM-I as in XDM improved,
however, is not ready for users yet; it is in an experimental stage. On the one hand, XDM-I is
indeed an improved implementation over the XDM method. On the other hand, the XDM method
itself is only part of a single computational program suite. It will take significant efforts before a
matured state of XDM-I will be reached.
94
7 Outlook
When the dispersion term presented here is combined with a properly chosen DF - especially in
larger molecules - there can be a significant portion of electron correlation at intermediate
interelectronic distances that are “double counted”. To fix this, empirical parameters of the DF
should be refitted to gain consistent thermochemical behavior for the total energy. One promising
candidate is the dispersionless DFT published in Phys. Rev. Lett.274
There are other subjects to be investigated, as well. E.g. condensed state computations: Whether
there is a significant “shielding effect” in closely packed systems? We have computed small
systems including dispersion terms C6, C8, C10. Do the odd terms have practical benefits? How
about higher order terms? Do the same rules apply for large systems? Is it possible to calculate
other molecular properties with XDM-I? The number of questions will only increase once we have
answers for some of them. The next step, however, seems to be obvious. Combine XDM-I with a
properly chosen – and re-optimized – DF and come up with analytical first and second derivatives
of the energy so geometry optimizations and frequency calculations become a possibility. Only
then can one hope that the new tool will be used by the community.
95
8 References
(1)
Pinter, B.; Olasz, A.; Petrov, K.; Veszpremi, T. Organometallics 2007, 26, 3677.
(2)
Veszprémi, T.; Olasz, A.; Pintér, B. Silicon Chem. 2007, 3, 187.
(3)
Olasz, A.; Mignon, P.; De Proft, F.; Veszprémi, T.; Geerlings, P. Chem. Phys. Lett. 2005, 407, 504.
(4)
Becke, A. D.; Johnson, E. R. J. Chem. Phys. 2005, 122, 154104.
(5)
Becke, A. D.; Johnson, E. R. J. Chem. Phys. 2005, 123, 154101.
(6)
Becke, A. D.; Johnson, E. R. J. Chem. Phys. 2006, 124, 014104.
(7)
Olasz, A.; Vanommeslaeghe, K.; Krishtal, A.; Veszpremi, T.; Van Alsenoy, C.; Geerlings, P. J.
Chem. Phys. 2007, 127.
(8)
Krishtal, A.; Vanommeslaeghe, K.; Olasz, A.; Veszpremi, T.; Van Alsenoy, C.; Geerlings, P. J.
Chem. Phys. 2009, 130.
(9)
Lee, H. Y.; Olasz, A.; Pink, M.; Park, H.; Lee, D. Chem. Commun. 2011, 47, 481.
(10)
Jo, J.; Olasz, A.; Chen, C. H.; Lee, D. J. Am. Chem. Soc. 2013, 135, 3620.
(11)
Crestani, M. G.; Olasz, A.; Pinter, B.; Bailey, B. C.; Fortier, S.; Gao, X. F.; Chen, C. H.; Baik, M. H.;
Mindiola, D. J. Chem. Sci. 2013, 4, 2543.
(12)
Lee, H. Y.; Olasz, A.; Chen, C. H.; Lee, D. Org. Lett. 2012, 14, 6286.
(13)
Jo, J.; Lee, H. Y.; Liu, W. J.; Olasz, A.; Chen, C. H.; Lee, D. J. Am. Chem. Soc. 2012, 134, 16000.
(14)
Hutt, J. T.; Jo, J.; Olasz, A.; Chen, C. H.; Lee, D.; Aron, Z. D. Org. Lett. 2012, 14, 3162.
(15)
Rappoport, D.; Furche, F. J. Chem. Phys. 2010, 133, 134105.
(16)
Witte, J.; Neaton, J. B.; Head-Gordon, M. J Chem Phys 2016, 144, 194306.
(17)
Levine, I. N. Quantum Chemistry; 5th ed.; Prentice Hall, 1999.
(18)
Veszprémi, T.; Fehér, M. Quantum chemistry: fundamentals to applications; 1st ed.; Springer, 1999.
(19)
Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Introduction to Advanced Electronic
Structure Theory; Dover Publications, 1996.
(20)
Becke, A. J. Chem. Phys. 1993, 98, 5648.
(21)
Lee, C. T.; Yang, W. T.; Parr, R. G. Phys Rev B 1988, 37, 785.
(22)
Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623.
(23)
Vosko, S. H.; Wilk, L.; Nusair, M. Can J Phys 1980, 58, 1200.
(24)
Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68, 3801.
(25)
Pauling, L. The nature of the chemical bond and the structure of molecules and crystals; an
introduction to modern structural chemistry; 3rd ed.; Cornell University Press: Ithaca, N.Y.,, 1960.
(26)
Mulliken, R. S. J. Chem. Phys. 1934, 2, 782.
(27)
Koopmans, T. A. Physica 1934, 1, 104.
(28)
Pearson, R. G. J. Am. Chem. Soc. 1963, 85, 3533.
(29)
Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512.
(30)
Mendez, F.; Gazquez, J. L. J. Am. Chem. Soc. 1994, 116, 9298.
(31)
Gazquez, J. L.; Mendez, F. J. Phys. Chem. 1994, 98, 4591.
(32)
Chandrakumar, K. R. S.; Pal, S. J. Phys. Chem. A 2002, 106, 11775.
(33)
Arulmozhiraja, S.; Kolandaivel, P. Mol. Phys. 1997, 90, 55.
(34)
DeProft, F.; Martin, J. M. L.; Geerlings, P. Chem. Phys. Lett. 1996, 256, 400.
(35)
Gilardoni, F.; Weber, J.; Chermette, H.; Ward, T. R. J. Phys. Chem. A 1998, 102, 3607.
(36)
Langenaeker, W.; Deproft, F.; Geerlings, P. J. Phys. Chem. 1995, 99, 6424.
96
(37)
(38)
Parr, R. G.; Von Szentpaly, L.; Liu, S. B. J. Am. Chem. Soc. 1999, 121, 1922.
Maynard, A. T.; Huang, M.; Rice, W. G.; Covell, D. G. Proc. Natl. Acad. Sci. U.S.A. 1998, 95,
11578.
(39)
Grimme, S. Wiley Interdisciplinary Reviews: Computational Molecular Science 2011, 1, 211.
(40)
Kaplan, I. G. Intermolecular Interactions; John Wiley & Sons, Ltd: Chichester, 2006.
(41)
Stone, A. J. The theory of intermolecular forces; Oxford University Press: Oxford, 1997.
(42)
Sato, T.; Nakai, H. J. Chem. Phys. 2009, 131, 224104.
(43)
Johnson, E. R.; Mackie, I. D.; DiLabio, G. A. J. Phys. Org. Chem. 2009, 22, 1127.
(44)
Grafenstein, J.; Cremer, D. J. Chem. Phys. 2009, 130, 124105.
(45)
Grimme, S.; Antony, J.; Schwabe, T.; Muck-Lichtenfeld, C. Org. Biomol. Chem. 2007, 5, 741.
(46)
Kong, J.; Gan, Z.; Proynov, E.; Freindorf, M.; Furlani, T. Phys. Rev. B 2009, 79, 042510.
(47)
Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Chem. Rev. 2012, 112, 289.
(48)
Gillan, M. J.; Alfe, D.; Michaelides, A. J Chem Phys 2016, 144, 130901.
(49)
Willow, S. Y.; Zeng, X. C.; Xantheas, S. S.; Kim, K. S.; Hirata, S. The journal of physical chemistry
letters 2016, 7, 680.
(50)
Wellendorff, J.; Silbaugh, T. L.; Garcia-Pintos, D.; Nørskov, J. K.; Bligaard, T.; Studt, F.; Campbell,
C. T. Surf. Sci. 2015, 640, 36.
(51)
Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864.
(52)
Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133.
(53)
Kohn, W.; Becke, A. D.; Parr, R. G. J. Phys. Chem. 1996, 100, 12974.
(54)
Bally, T.; Sastry, G. N. J. Phys. Chem. A. 1997, 101, 7923.
(55)
Dutoi, A. D.; Head-Gordon, M. Chem. Phys. Lett. 2006, 422, 230.
(56)
Braïda, B.; Hiberty, P. C.; Savin, A. J. Phys. Chem. A. 1998, 102, 7872.
(57)
Mori-Sánchez, P.; Cohen, A. J.; Yang, W. J. Chem. Phys. 2006, 125, 201102.
(58)
Dreuw, A.; Head-Gordon, M. J. Am. Chem. Soc. 2004, 126, 4007.
(59)
Dreuw, A.; Head-Gordon, M. Chem. Rev. 2005, 105, 4009.
(60)
Dreuw, A.; Weisman, J.; Head-Gordon, M. J. Chem. Phys. 2003, 119, 2943.
(61)
Savin, A. In Recent developments and applications of modern density functional theory; Seminario, J.
M., Ed.; Elsevier: Amsterdam, 1996, p 327.
(62)
Chai, J.; Head-Gordon, M. J. Chem. Phys. 2008, 128, 084106.
(63)
Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. J. Chem. Phys. 2004, 120, 8425.
(64)
Iikura, H.; Tsuneda, T.; Yanai, T.; Hirao, K. J. Chem. Phys. 2001, 115, 3540.
(65)
Toulouse, J.; Colonna, F.; Savin, A. J. Chem. Phys. 2005, 122, 014110.
(66)
Vydrov, O. A.; Heyd, J.; Krukau, A. V.; Scuseria, G. E. J. Chem. Phys. 2006, 125, 074106.
(67)
Vydrov, O. A.; Scuseria, G. E. J. Chem. Phys. 2006, 125, 234109.
(68)
Gerber, I. C.; Ángyán, J. G.; Marsman, M.; Kresse, G. J. Chem. Phys. 2007, 127, 054101.
(69)
Ángyán, J. G.; Gerber, I. C.; Savin, A.; Toulouse, J. Phys. Rev. B 2005, 72, 012510.
(70)
Goll, E.; Werner, H.-J.; Stoll, H. Phys. Chem. Chem. Phys. 2005, 7, 3917.
(71)
Goll, E.; Werner, H.-J.; Stoll, H.; Leininger, T.; Gori-Giorgi, P.; Savin, A. Chem. Phys. 2006, 329,
276.
(72)
Leininger, T.; Stoll, H.; Werner, H.-J.; Savin, A. Chem. Phys. Lett. 1997, 275, 151.
(73)
Gill, P. M. W.; Adamson, R. D.; Pople, J. A. Mol. Phys. 1996, 88, 1005.
(74)
Rohrdanz, M. A.; Martins, K. M.; Herbert, J. M. A long-range-corrected density functional that
performs well for both ground-state properties and time-dependent density functional theory excitation energies,
including charge-transfer excited states; AIP, 2009; Vol. 130.
(75)
Henderson, T. M.; Janesko, B. G.; Scuseria, G. E. Generalized gradient approximation model
exchange holes for range-separated hybrids; AIP, 2008; Vol. 128.
(76)
Tawada, Y.; Tsuneda, T.; Yanagisawa, S.; Yanai, T.; Hirao, K. A long-range-corrected timedependent density functional theory; AIP, 2004; Vol. 120.
(77)
Yanai, T.; Tew, D. P.; Handy, N. C. Chem. Phys. Lett. 2004, 393, 51.
(78)
Kristyán, S.; Pulay, P. Chem. Phys. Lett. 1994, 229, 175.
(79)
Dobson, J. F.; McLennan, K.; Rubio, A.; Wang, J.; Gould, T.; Le, H. M.; Dinte, B. P. Aust. J. Chem.
2001, 54, 513.
(80)
Lejaeghere, K.; Bihlmayer, G.; Bjorkman, T.; Blaha, P.; Blugel, S.; Blum, V.; Caliste, D.; Castelli, I.
E.; Clark, S. J.; Dal Corso, A.; de Gironcoli, S.; Deutsch, T.; Dewhurst, J. K.; Di Marco, I.; Draxl, C.; Dulak, M.;
Eriksson, O.; Flores-Livas, J. A.; Garrity, K. F.; Genovese, L.; Giannozzi, P.; Giantomassi, M.; Goedecker, S.; Gonze,
X.; Granas, O.; Gross, E. K.; Gulans, A.; Gygi, F.; Hamann, D. R.; Hasnip, P. J.; Holzwarth, N. A.; Iusan, D.; Jochym,
D. B.; Jollet, F.; Jones, D.; Kresse, G.; Koepernik, K.; Kucukbenli, E.; Kvashnin, Y. O.; Locht, I. L.; Lubeck, S.;
Marsman, M.; Marzari, N.; Nitzsche, U.; Nordstrom, L.; Ozaki, T.; Paulatto, L.; Pickard, C. J.; Poelmans, W.; Probert,
M. I.; Refson, K.; Richter, M.; Rignanese, G. M.; Saha, S.; Scheffler, M.; Schlipf, M.; Schwarz, K.; Sharma, S.;
Tavazza, F.; Thunstrom, P.; Tkatchenko, A.; Torrent, M.; Vanderbilt, D.; van Setten, M. J.; Van Speybroeck, V.; Wills,
J. M.; Yates, J. R.; Zhang, G. X.; Cottenier, S. Science 2016, 351, aad3000.
97
(81)
Grimme, S. J. Comput. Chem. 2006, 27, 1787.
(82)
Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. J. Chem. Phys. 2010, 132, 154104.
(83)
Kong, J.; Gan, Z.; Proynov, E.; Freindorf, M.; Furlani, T. R. Phys. Rev. B 2009, 79, 042510.
(84)
Steinmann, S. N.; Corminboeuf, C. J. Chem. Theory Comput. 2010, 6, 1990.
(85)
Liu, Y.; Goddard, W. A. J. Phys. Chem. Lett. 2010, 1, 2550.
(86)
Zhao, Y.; Truhlar, D. G. J. Chem. Theory Comput. 2011, 7, 669.
(87)
Goerigk, L.; Grimme, S. Phys. Chem. Chem. Phys. 2011, 13, 6670.
(88)
Hepburn, J.; Scoles, G.; Penco, R. Chem. Phys. Lett. 1975, 36, 451.
(89)
Ahlrichs, R.; Penco, R.; Scoles, G. Chem. Phys. 1977, 19, 119.
(90)
Wu, Q.; Yang, W. J. Chem. Phys. 2002, 116, 515.
(91)
Elstner, M.; Hobza, P.; Frauenheim, T.; Suhai, S.; Kaxiras, E. J. Chem. Phys. 2001, 114, 5149.
(92)
Gianturco, F. A.; Paesani, F.; Laranjeira, M. F.; Vassilenko, V.; Cunha, M. A. J. Chem. Phys. 1999,
110, 7832.
(93)
Cohen, J. S.; Pack, R. T. J. Chem. Phys. 1974, 61, 2372.
(94)
von Lilienfeld, O. A.; Tkatchenko, A. J. Chem. Phys. 2010, 132, 234109.
(95)
Grimme, S. J. Comput. Chem. 2004, 25, 1463.
(96)
Kannemann, F. O.; Becke, A. D. J. Chem. Theory Comput. 2010, 6, 1081.
(97)
Liu, Y.; Goddard, W. A., III Mater. Trans. 2009, 50, 1664.
(98)
Chai, J.-D.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2008, 10, 6615.
(99)
Koide, A. J. Phys. B 1976, 9, 3173.
(100) Johnson, E. R.; Becke, A. D. J. Chem. Phys. 2005, 123, 024101.
(101) Karton, A.; Tarnopolsky, A.; Lamère, J.-F. o.; Schatz, G. C.; Martin, J. M. L. J. Phys. Chem. A. 2008,
112, 12868.
(102) Zhao, Y.; Lynch, B. J.; Truhlar, D. G. J. Phys. Chem. A. 2004, 108, 4786.
(103) Schwabe, T.; Grimme, S. Phys. Chem. Chem. Phys. 2006, 8, 4398.
(104) Zhang, Y.; Xu, X.; Goddard, W. A. Proc. Natl. Acad. Sci. U.S.A. 2009, 106, 4963.
(105) Goerigk, L.; Grimme, S. J. Chem. Theory Comput. 2010, 7, 291.
(106) Ayala, P. Y.; Scuseria, G. E. J. Chem. Phys. 1999, 110, 3660.
(107) Fromager, E.; Réal, F.; Wåhlin, P.; Wahlgren, U.; Jensen, H. J. J. Chem. Phys. 2009, 131, 054107.
(108) Friedrich, J.; Hanrath, M.; Dolg, M. J. Chem. Phys. 2007, 126, 154110.
(109) Pollet, R.; Savin, A.; Leininger, T.; Stoll, H. J. Chem. Phys. 2002, 116, 1250.
(110) Gerber, I. C.; Ángyán, J. G. Chem. Phys. Lett. 2005, 416, 370.
(111) Toulouse, J.; Colonna, F.; Savin, A. Phys. Rev. B 2004, 70, 062505.
(112) Toulouse, J.; Colonna, F.; Savin, A. Mol. Phys. 2005, 103, 2725.
(113) Toulouse, J.; Savin, A.; Flad, H.-J. Int. J. Quantum Chem 2004, 100, 1047.
(114) Kohn, W.; Meir, Y.; Makarov, D. E. Phys. Rev. Lett. 1998, 80, 4153.
(115) Nozières, P.; Pines, D. Phys. Rev. 1958, 111, 442.
(116) Benighaus, T.; DiStasio, R. A.; Lochan, R. C.; Chai, J.-D.; Head-Gordon, M. J. Phys. Chem. A. 2008,
112, 2702.
(117) Kállay, M. J. Chem. Phys. 2015, 142, 204105.
(118) Janesko, B. G.; Scuseria, G. E. Phys. Chem. Chem. Phys. 2009, 11, 9677.
(119) Evans, D. A.; Chapman, K. T.; Hung, D. T.; Kawaguchi, A. T. Angew. Chem., Int. Ed. Engl. 1987,
26, 1184.
(120) Muehldorf, A. V.; van Engen, D.; Warner, J. C.; Hamilton, A. D. J. Am. Chem. Soc. 1988, 110, 6561.
(121) Vallée, R.; Damman, P.; Dosière, M.; Toussaere, E.; Zyss, J. J. Am. Chem. Soc. 2000, 122, 6701.
(122) Ferguson, S. B.; Sanford, E. M.; Seward, E. M.; Diederich, F. J. Am. Chem. Soc. 1991, 113, 5410.
(123) Burley, S. K.; Petsko, G. A. Science 1985, 229, 23.
(124) McGaughey, G. B.; Gagne, M.; Rappe, A. K. J. Biol. Chem. 1998, 273, 15458.
(125) Ranganathan, D.; Haridas, V.; Gilardi, R.; Karle, I. L. J. Am. Chem. Soc. 1998, 120, 10793.
(126) Vondrášek, J.; Bendová, L.; Klusák, V.; Hobza, P. J. Am. Chem. Soc. 2005, 127, 2615.
(127) Hunter, C. A. Philos. Trans. R. Soc. London, Ser. A 1993, 345, 77.
(128) Baker, C. M.; Grant, G. H. Biopolymers 2007, 85, 456.
(129) Frederick, C. A.; Williams, L. D.; Ughetto, G.; van der Marel, G. A.; van Boom, J. H.; Rich, A.;
Wang, A. H. J. Struct. Biochem. 1990, 29, 2538.
(130) Meyer, E. A.; Castellano, R. K.; Diederich, F. Angew. Chem., Int. Ed. 2003, 42, 1210.
(131) Marshall, A. G.; Rodgers, R. P. Acc. Chem. Res. 2004, 37, 53.
(132) Martinez, L. M.; Angell, C. A. Nature 2001, 410, 663.
(133) Hunter, C. A.; Sanders, J. K. M. J. Am. Chem. Soc. 1990, 112, 5525.
(134) Tsuzuki, S.; Honda, K.; Uchimaru, T.; Mikami, M.; Tanabe, K. J. Am. Chem. Soc. 2002, 124, 104.
(135) Ebata, T.; Hamakado, M.; Moriyama, S.; Morioka, Y.; Ito, M. Chem. Phys. Lett. 1992, 199, 33.
(136) Janda, K. C.; Hemminger, J. C.; Winn, J. S.; Novick, S. E.; Harris, S. J.; Klemperer, W. J. Chem.
Phys. 1975, 63, 1419.
98
(137) Law, K. S.; Schauer, M.; Bernstein, E. R. J. Chem. Phys. 1984, 81, 4871.
(138) Spirko, V.; Engkvist, O.; Soldan, P.; Selzle, H. L.; Schlag, E. W.; Hobza, P. J. Chem. Phys. 1999,
111, 572.
(139) Steed, J. M.; Dixon, T. A.; Klemperer, W. J. Chem. Phys. 1979, 70, 4940.
(140) Hobza, P.; Selzle, H. L.; Schlag, E. W. J. Phys. Chem. 1996, 100, 18790.
(141) Hobza, P.; Spirko, V.; Selzle, H. L.; Schlag, E. W. J. Phys. Chem. A 1998, 102, 2501.
(142) Sherrill, C. D.; Takatani, T.; Hohenstein, G. H. J. Phys. Chem. A 2009, 113, 10146.
(143) Sinnokrot, M. O.; Sherrill, C. D. J. Phys. Chem. A 2006, 110, 10656.
(144) Tsuzuki, S.; Honda, K.; Uchimaru, T.; Mikami, M. J. Chem. Phys. 2005, 122, 144323.
(145) Headen, T. F.; Howard, C. A.; Skipper, N. T.; Wilkinson, M. A.; Bowron, D. T.; Soper, A. K. J. Am.
Chem. Soc. 2010, 132, 5735.
(146) Gervasio, F. L.; Chelli, R.; Procacci, P.; Schettino, V. J. Phys. Chem. A 2002, 106, 2945.
(147) Sinnokrot, M. O.; Sherrill, C. D. J. Am. Chem. Soc. 2004, 126, 7690.
(148) Jeffery, G. A.; Ruble, J. R.; McCullan, R. K.; Pople, J. A. Proc. R. Soc. London A 1987, 41, 447.
(149) Anderson, M.; Bosio, L.; Bruneaux-Poulle, J.; Fourme, R. J. Chim. Phys. Phys. Chim. Biol. 1977, 74,
68.
(150) Andre, D.; Fourme, R. J. Mol. Struct. 1982, 81, 253.
(151) Ibberson, R. M.; David, W. I. F.; Prager, M. J. Chem. Soc., Chem. Commun. 1992, null, 1438.
(152) Misawa, M.; Fukunaga, T. J. Chem. Phys. 1990, 93, 3495.
(153) Narten, A. H. J. Chem. Phys. 1977, 67, 2102.
(154) Righini, R. Science 1993, 262, 1386.
(155) Baker, C. M.; Grant, G. H. J. Chem. Theory Comput. 2006, 2, 947.
(156) Cabaco, M. I.; Danten, Y.; Besnard, M.; Guissani, Y.; Guillot, B. J. Phys. Chem. B 1997, 101, 6977.
(157) Jorgensen, W. L.; Laird, E. R.; Nguyen, T. B.; Tirado-Rives, J. J. Comput. Chem. 1993, 14, 206.
(158) Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. J. Am. Chem. Soc. 1996, 118, 11225.
(159) Rai, N.; Siepmann, J. I. J. Phys. Chem. B 2007, 111, 10790.
(160) Zorkil, P. M.; Lanshina, L. V.; Bogdan, T. V. J. Struct. Chem. 2008, 49, 524.
(161) Chipot, C.; Jaffe, R.; Maigret, B.; Pearlman, D. A.; Kollman, P. A. J. Am. Chem. Soc. 1996, 118,
11217.
(162) Fioroni, M.; Vogt, D. J. Phys. Chem. B 2004, 108, 11774.
(163) Versees, W.; Loverix, S.; Vandemeulebroucke, A.; Geerlings, P.; Steyaert, J. J. Mol. Biol. 2004, 338,
1.
(164) Ibarra, C.; Nieslanik, B. S.; Atkins, W. M. Biochemistry 2001, 40, 10614.
(165) Hobza, P.; Sponer, J. J. Am. Chem. Soc. 2002, 124, 11802.
(166) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553.
(167) Simon, S.; Duran, M.; Dannenberg, J. J. J. Chem. Phys. 1996, 105.
(168) Stone, A. J. Chem. Phys. Lett. 1981, 83, 233.
(169) Stone, A. J.; Dullweber, A.; Hodges, M. P.; Popelier, P. L. A.; Wales, D. J. In ORIENT, a Program
for Studying Interactions between Molecules Version 3.2 ed.; University of Cambridge: 1995.
(170) Stone, A. J.; Alderton, M. Mol. Phys. 1985, 56, 1047.
(171) Mignon, P.; Loverix, S.; De Proft, F.; Geerlings, P. J. Phys. Chem. A 2004, 108, 6038.
(172) Mignon, P.; Loverix, S.; Geerlings, P. Chem. Phys. Lett. 2005, 401, 40.
(173) Mignon, P.; Loverix, S.; Steyaert, J.; Geerlings, P. Nucleic Acids Res. 2005, 33, 1779.
(174) Perakyla, M. Chem. Commun. 1996, 361.
(175) Perakyla, M. Phys. Chem. Chem. Phys. 1999, 1, 5643.
(176) Potter, M. J.; Gilson, M. K.; McCammon, J. A. J. Am. Chem. Soc. 1994, 116, 10298.
(177) Tomasi, J.; Mennucci, B.; Cammi, R. Chem. Rev. 2005, 105, 2999.
(178) Jiao, H.; Junge, K.; Alberico, E.; Beller, M. J Comput Chem 2016, 37, 168.
(179) Breneman, C. M.; Wiberg, K. B. J. Comput. Chem. 1990, 11, 361.
(180) Geerlings, P.; De Proft, F.; Langenaeker, W. Chem. Rev. 2003, 103, 1793.
(181) Berkowitz, M. J. Am. Chem. Soc. 1987, 109, 4823.
(182) DeProft, F.; Liu, S. B.; Parr, R. G. J. Chem. Phys. 1997, 107, 3000.
(183) Roos, G.; Loverix, S.; De Proft, F.; Wyns, L.; Geerlings, P. J. Phys. Chem. A 2003, 107, 6828.
(184) Roos, G.; Messens, J.; Loverix, S.; Wyns, L.; Geerlings, P. J. Phys. Chem. B 2004, 108, 17216.
(185) Sinnokrot, M. O.; Valeev, E. F.; Sherrill, C. D. J. Am. Chem. Soc. 2002, 124, 10887.
(186) Sinnokrot, M. O.; Sherrill, C. D. J. Phys. Chem. A 2003, 107, 8377.
(187) Bernstein, J. Polymorphism in Molecular Crystals; Oxford University Press, New York, 2007.
(188) Bonnett, P. E.; Carpenter, K. J.; Dawson, S.; Davey, R. J. Chem. Commun. 2003, 698.
(189) Stahly, G. P. Cryst. Growth Des. 2007, 7, 1007.
(190) Bernstein, J.; Dunitz, J. D.; Gavezzotti, A. Cryst. Growth Des. 2008, 8, 2011.
(191) Stanger, A. Chem. Commun. 2009, 15, 1939.
(192) Desiraju, G. R. Angew. Chem. Int. Ed. 2007, 46, 8342.
99
(193) Yu, L. Acc. Chem. Res. 2010, 43, 1257.
(194) Nangia, A. Cryst. Growth Des. 2006, 6, 2.
(195) Polymorphism in the Pharmaceutical Industry; Wiley-VCH, Weinheim, 2006.
(196) Brittain, H. G. J. Pharm. Sci. 2009, 98, 1617.
(197) Jeffrey, G. A. An Introduction to Hydrogen Bonding; Oxford University Press, New York, 1997.
(198) Prins, L. J.; Reinhoudt, D. N.; Timmerman, P. Angew. Chem. Int. Ed. 2001, 40, 2382.
(199) Steiner, T. Angew. Chem. Int. Ed. 2002, 41, 48.
(200) Desiraju, G. R. Acc. Chem. Res. 1991, 24, 290.
(201) Desiraju, G. R. Acc. Chem. Res. 1996, 29, 441.
(202) Hobza, P.; Havlas, Z. Chem. Rev. 2000, 100, 4253.
(203) Zhao, X.; Wang, X.-Z.; Jiang, X.-K.; Chen, Y.-Q.; Li, Z.-T.; Chen, G.-J. J. Am. Chem. Soc. 2003,
125, 15128.
(204) Li, C.; Ren, S.-F.; Hou, J.-L.; Yi, H.-P.; Zhu, S.-Z.; Jiang, X.-K.; Li, Z.-T. Angew. Chem. Int. Ed.
2005, 44, 5725.
(205) Li, C.; Zhu, Y.-Y.; Yi, H.-P.; Li, C.-Z.; Jiang, X.-K.; Li, Z.-T.; Yu, Y.-H. Chem.–Eur. J. 2007, 13,
9990.
(206) Harrell, S. A.; McDaniel, D. H. J. Am. Chem. Soc. 1964, 86, 4497.
(207) Howard, J. A. K.; Hoy, V. J.; O'Hagan, D.; Smith, G. T. Tetrahedron 1996, 52, 12613.
(208) Dunitz, J. D.; Taylor, R. Chem.–Eur. J. 1997, 3, 89.
(209) Meyer, E. A.; Castellano, R. K.; Diederich, F. Angew. Chem. Int. Ed. 2003, 42, 1210.
(210) Hof, F.; Diederich, F. Chem. Commun. 2004, 477.
(211) Thalladi, V. R.; Weiss, H.-C.; Bläser, D.; Boese, R.; Nangia, A.; Desiraju, G. R. J. Am. Chem. Soc.
1998, 120, 8702.
(212) Desiraju, G. R. Acc. Chem. Res. 2002, 35, 565.
(213) Choudhury, A. R.; Guru Row, T. N. Cryst. Growth Des. 2004, 4, 47.
(214) Schwarzer, A.; Seichter, W.; Weber, E.; Stoeckli-Evans, H.; Losada, M.; Hulliger, J. CrystEngComm
2004, 6, 567.
(215) Hyacinth, M.; Chruszcz, M.; Lee, K. S.; Sabat, M.; Gao, G.; Pu, L. Angew. Chem. Int. Ed. 2006, 45,
5358.
(216) Martin, R. B. Chem. Rev. 1996, 96, 3043.
(217) Choudhury, A. R.; Nagarajan, K.; Guru Row, T. N. Acta Crystallogr. Sect. C: Cryst. Struct. Commun.
2004, 60, o644.
(218) Mountford, A. J.; Hughes, D. L.; Lancaster, S. J. Chem. Commun. 2003, 2148.
(219) Weiss, H.-C.; Boese, R.; Weiss, H.-C.; L. Smith, H.; M. Haley, M. Chem. Commun. 1997, 2403.
(220) Bochevarov, A. D.; Harder, E.; Hughes, T. F.; Greenwood, J. R.; Braden, D. A.; Philipp, D. M.;
Rinaldo, D.; Halls, M. D.; Zhang, J.; Friesner, R. A. Int. J. Quantum Chem 2013, 113, 2110.
(221) Dunning, J. T. H. J. Chem. Phys. 1989, 90, 1007.
(222) Steyaert, J. Eur. J. Biochem. 1997, 247, 1.
(223) Raines, R. T. Chem. Rev., 98, 1045.
(224) Finnin, M. S.; Donigian, J. R.; Cohen, A. J.; Richon, V. M.; Rifkind, R. A.; Marks, P. A.; Breslow,
R.; Pavletich, N. P. Nature 1999, 401.
(225) Nielsen, T. K.; Hildmann, C.; Dickmanns, A.; Schwienhorst, A.; Ficner, R. J. Mol. Biol., 354, 107.
(226) Wang, D. F.; Wiest, O.; Helquist, P.; Lan-Hargest, H.-Y.; Wiech, N. L. J. Med. Chem. 2004, 47,
3409.
(227) Hobza, P.; Šponer, J. Chem. Rev. 1999, 99, 3247.
(228) Xu, X.; Goddard, W. A. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 267.
(229) Zhao, Y.; Truhlar, D. G. J. Chem. Theory Comput. 2005, 1, 415.
(230) zhao, Y.; Truhlar, D. G. J. Chem. Theory Comput. 2007, 289.
(231) zhao, Y.; Truhlar, D. G. Acc. Chem. Res. 2008, 41, 157.
(232) Lilienfeld, v. O. A.; Tavernelli, I.; Rothlisberger, U.; Sebastiani, D. Phys. Rev. Lett. 2004, 93,
153004.
(233) Lilienfeld, v. O. A.; Tavernelli, I.; Rothlisberger, U.; Sebastiani, D. Phys Rev B 2005, 71, 195119.
(234) Lin, I.-C.; Rothlisberger, U. Phys. Chem. Chem. Phys. 2008, 10, 2730.
(235) Dion, M.; Rydberg, H.; Schröder, E.; Langreth, D. C.; Lundqvist, B. I. Phys. Rev. Lett. 2004, 92,
246401.
(236) Dion, M.; Rydberg, H.; Schröder, E.; Langreth, D. C.; Lundqvist, B. I. Phys. Rev. Lett. 2005, 95,
109902.
(237) Sato, T.; Tsuneda, T.; Hirao, K. J. Chem. Phys. 2007, 126, 234114.
(238) Kamiya M.; Tsuneda T.; Hirao, K. The Journal of Chemical Physics 2002, 117, 6010.
(239) Anthony J.; Grimme, S. Phys. Chem. Chem. Phys. 2006, 8, 5287.
(240) Grimme, S. Angew. Chem., Int. Ed. 2008, 47.
(241) Jurečka P.; Černý J.; Hobza, P.; Salahub D. R. J. Comput. Chem. 2007, 28, 555.
100
(242) Černý J.; Hobza, P. Phys. Chem. Chem. Phys. 2007, 9, 5291.
(243) Becke, A. D.; Johnson, E. R. J. Chem. Phys. 2007, 127, 154108.
(244) Rousseau, B.; Peeters, A.; Alsenoy, C. V. Chem. Phys. Lett. 2000, 324, 189.
(245) Krishtal, A.; Senet, P.; Yang, M.; Alsenoy, C. V. J. Chem. Phys. 2006, 125, 034312.
(246) Bultinck, P.; Van Alsenoy, C.; Ayers, P. W.; Carbo-Dorca, R. J. Chem. Phys. 2007, 126, 144111.
(247) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press:
New York, 1989.
(248) Ayers, P. W. J. Chem. Phys. 2000, 113, 10886.
(249) Parr, R. G.; Ayers, P. W.; Nalewajski, R. F. J. Phys. Chem. A 2005, 109, 3957.
(250) Nalewajski, R. F.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 2000, 97, 8879.
(251) Nalewajski, R. F.; Switka, E.; Michalak, A. Int. J. Quantum Chem 2002, 87, 198.
(252) Nalewajski, R. F. Chem. Phys. Lett. 2003, 372, 28.
(253) Nalewajski, R. F.; Broniatowska, E. Int. J. Quantum Chem 2005, 101, 349.
(254) Ayers, P. W. Theor. Chem. Acc. 2006, 115, 253.
(255) Bickelhaupt, F. M.; Hommes, N. J. R. V.; Guerra, C. F.; Baerends, E. J. Organometallics 1996, 15,
2923.
(256) Baerends, E. J.; Guerra, C. F.; Handgraaf, J. W.; Bickelhaupt, F. M. J. Comput. Chem. 2004, 25, 189.
(257) Mayer, I.; Salvador, P. Chem. Phys. Lett. 2004, 383, 368.
(258) Becke, A. D. J. Chem. Phys. 1988, 88, 2547.
(259) Davidson, E.; Chakravorty, S. Theoret. Chim. Acta 1992, 83, 319.
(260) Ayers, P. W.; Morrison, R. C.; Roy, R. K. J. Chem. Phys. 2002, 116, 8731.
(261) De Proft, F.; Vivas-Reyes, R.; Peeters, A.; Van Alsenoy, C.; Geerlings, P. J. Comput. Chem. 2003,
24, 463.
(262) McWeeny, R. Rev. Mod. Phys. 1960, 32, 335
(263) Buckingham, A. D. Adv. Chem. Phys. 1967, 12, 107.
(264) Buckingham, A. D. Intermolecular Interactions: From Diatomics to Biopolymers; Wiley: New York,
1978.
(265) London, F. Trans. Faraday Soc. 1937, 33, 8.
(266) Unsöld, A. Z. Phys. 1927, 43, 563.
(267) Pullman, B.; Claverie, P.; Caillet, J. Proc. Natl. Acad. Sci. U.S.A. 1966, 55, 904.
(268) Rijks, W.; Wormer, P. E. S. J. Chem. Phys. 1989, 90, 6507.
(269) Sinnokrot, M. O.; Sherrill, C. D. J. Phys. Chem. A. 2004, 108, 10200.
(270) Bak, K. L.; Jorgensen, P.; Olsen, J.; Helgaker, T.; Klopper, W. J. Chem. Phys. 2000, 112, 9229.
(271) Halkier, A.; Helgaker, T.; Jørgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K. Chem.
Phys. Lett. 1998, 286, 243.
(272) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.;
Montgomery, J., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.;
Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.;
Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.;
Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.;
Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.;
Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck,
A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B.
B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al-Laham, M. A.; Peng, C.
Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J.
A. In Gaussian 03, Revision C.02; Gaussian, Inc.: Wallingford CT, 2004.
(273) Krishtal, A.; Senet, P.; Alsenoy, C. V. J. Chem. Theory Comput. 2008, 4, 2122.
(274) Pernal, K.; Podeszwa, R.; Patkowski, K.; Szalewicz, K. Phys. Rev. Lett. 2009, 103, 263201.
101