Definition of Molecular Structure
By Choice or by Appeal to Observation?
Paul Jerabek
Group Seminar
10.05.2011
1 / 35
The Author
Richard F. W. Bader, McMaster–University (Hamilton, Ontario)
2 / 35
A Philosophical Dilemma
R. Hoffmann
“I also have philosophical reservations about the reductionist
framework in which AIM (atoms in molecules) resides; I believe the
most interesting ideas of chemistry are not reducible to physics. I
think we have a fundamental difference of opinion between us on
this matter.”
3 / 35
A Philosophical Dilemma
R. Hoffmann
“I also have philosophical reservations about the reductionist
framework in which AIM (atoms in molecules) resides; I believe the
most interesting ideas of chemistry are not reducible to physics. I
think we have a fundamental difference of opinion between us on
this matter.”
R. Bader
“The time has come for entering students of chemistry to be
taught that the electron density can be seen, touched, and
measured and that the chemical structures they learn are in fact
the tracings of “bonds” onto lines of maximum density that link
bonded nuclei. Matter, as we perceive it, is bound by the
electrostatic force of attraction between the nuclei and the electron
density.”
3 / 35
Where do the classical models fail?
Hypervalent molecules
SF6 has octahedral symmetry (via Raman and infrared
spectroscopy)a b and can not be written without ’expanded
valence’.
a
b
Steffens, C. C.; Gross, S. T. J. Chem. Phys. 1934, 2, 311
Eucken, A.; Ahrens, H. Z. Phys., Chem. B 1934, 26, 297
4 / 35
Where do the classical models fail?
Hypervalent molecules
SF6 has octahedral symmetry (via Raman and infrared
spectroscopy)a b and can not be written without ’expanded
valence’.
a
b
Steffens, C. C.; Gross, S. T. J. Chem. Phys. 1934, 2, 311
Eucken, A.; Ahrens, H. Z. Phys., Chem. B 1934, 26, 297
Hydrogen Bridges
In the Lewis model no interactions between closed shell fragments
can happen. The discovery of hydrogen bonds a lead to a
broadening of this model.
a
Latimer, W. M.; Rodebush, W. A. J. Am. Chem. Soc. 1920, 42, 1419
4 / 35
Where do the classical models fail?
3c–2e–Bonds
It was proposed for B2 H6 to possess an ethane-like stucture.a
Through spectral data a D2h symmetry was observed b and
three–center two–electron bonds were proposed and accepted.
a
b
Schlesinger. Burg. Chem. Rev. 1942, 31, 1
Longuet-Higgins, H. C.; Bell, R. P. J. Chem. Soc. 1943, 250
5 / 35
Basics of QTAIM
The central quantity is the electron density ρ(r ) which can be
calculated or measured.
Atoms are defined within zero–flux surfaces which no
gradients cross.
6 / 35
Basics of QTAIM
The central quantity is the electron density ρ(r ) which can be
calculated or measured.
Atoms are defined within zero–flux surfaces which no
gradients cross.
6 / 35
Basics of QTAIM
The atoms in the molcules are well defined through the
intersecting surfaces and lead to proper open systems (as
quantum mechanics propose).
7 / 35
Basics of QTAIM
Voronoi diagram
8 / 35
Basics of QTAIM
Nuclei of interacting atoms are linked through atomic
interaction lines of maximum charge density.
When the interaction is attractive they are called bond paths.
Bond paths cross the zero–flux surfaces at the so called
critical points (cp).
9 / 35
Basics of QTAIM
cp can be distinguished with respect to their gradients.
Negative gradients in all 3 dimensions:
P
Nuclear attractor ( : −3).
One positive gradient (towards the nuclei) and two negative
P
gradients (leaving the bond): Bond critical point ( : −1).
Two positive gradients and one negative gradient:
P
Ring critical point ( : +1).
P
Three positive gradients: Cage critical point ( : +3).
10 / 35
Basics of QTAIM
Bond strengths can be estimated via electron density at the
cp.
The sign of the Laplacian gives an hint if a bond is more
covalent or ionic.
The itegration over the atomic basins lead to atomic
properties (charges, etc.) which strengthens the picture of
atoms as building blocks of matter.
Bond paths 6= bonds.
11 / 35
Advantages of QTAIM
Build on solid physical ground. No ’fuzzy concept’ of the
chemical bond.
Bonds as ’bridges of density’ between atoms.
Leads to a sensible topologic representation of the molecules
including every interaction (and treating them equally).
12 / 35
Advantages of QTAIM
13 / 35
A Little Physics
Two forces and a theorem
Chemical bonding can be understood with the Ehrenfest force, the
Feynman force and the virial theorem.
14 / 35
A Little Physics
Two forces and a theorem
Chemical bonding can be understood with the Ehrenfest force, the
Feynman force and the virial theorem.
The Virial Theorem
When the energy of a system consists of E = T + V then
2T = V at a stationary point.
14 / 35
The Feynman force
Acts on the nuclei.
Is defined as the negative gradient of the potential energy
dE
with respect to a nucleus (F (R) = − dR
).
Attractive for larger distances, zero for Re and repulsive for
close nuclei.
15 / 35
The Feynman force
Acts on the nuclei.
Is defined as the negative gradient of the potential energy
dE
with respect to a nucleus (F (R) = − dR
).
Attractive for larger distances, zero for Re and repulsive for
close nuclei.
15 / 35
The Feynman force
Solely responsible for the movements of the nuclei.
Vanishes at Re .
16 / 35
The Feynman force
Solely responsible for the movements of the nuclei.
Vanishes at Re .
T = –E + RF(R).
V = 2 E – RF(R).
16 / 35
The Feynman force
Solely responsible for the movements of the nuclei.
Vanishes at Re .
T = –E + RF(R).
V = 2 E – RF(R).
Kinetic energy can only decrease when the Feynman force is
attractive (RF(R) < 0).
16 / 35
The Feynman force
Beginning of the
bonding between
two H atoms (T is
maximally positive
and V maximally
negative):
Density is
removed from the
immediate vicinity
of the nuclie and
distributed in the
bonding region
which relaxes the
gradient of ρ.
This decreases T
and increases V.
17 / 35
The Feynman force
Accumulation of electron density in the bonding region leads
to attractive Feynman forces (until they vanish when Re is
reached).
Each nucleus is attracted by its own polarized charge density.
T can only be lowerd when the Feynman forces are attractive
Models which address bonding to a reduction of kinetic
energy due to orbital overlap must be questioned (cannot
happen at equilibrium geometry!).
18 / 35
The Ehrenfest Force
The Ehrenfest force F acts on atoms.
Defined as the negative gradient of the potential energy with
respect to the coordinates of an eletron averaged over the
whole charge density.
Like the Feynman force a result of repulsion (larger distances
– electrons repell each other) and attraction (close distances –
electron are attracted by the nuclei).
F is not zero at Re , in contrary: It is the reason for the
bonding.
Atoms which share a surface A—B have opposed values of F:
FA|B = −FB|A . [Atom A is attracted by atom B as B is
attracted by A.]
19 / 35
Correcting the VSEPR model
VSEPR predict molecular geometries based of the behavoir of
electron pairs.
Wrong prediction for CaF2 , BaH2 , TiO2 which should be
linear!
20 / 35
Correcting the VSEPR model
VSEPR predict molecular geometries based of the behavoir of
electron pairs.
Wrong prediction for CaF2 , BaH2 , TiO2 which should be
linear!
But: CaF2 has bent geometry. (Bytheway, I.; Gillespie, R. J.; Tang,
T.-H.; Bader, R. F. W. Inorg. Chem. 1995, 34, 2407-2414)
20 / 35
Correcting the VSEPR model
Charge concentraions (CC) in inner core lead to bending.
VSEPR doesn’t account for the higher polarizability of the
heavier atoms (deviation from a spherical shell).
21 / 35
Forces in Ionic Crystals
Lewis model not applicable in ionic crystals. One Possibility:
Use of frational bond orders (e.g. 61 in NaCl).
22 / 35
Forces in Ionic Crystals
Lewis model not applicable in ionic crystals. One Possibility:
Use of frational bond orders (e.g. 61 in NaCl).
Complicated for other but the simplest compounds.
For example: Perowskit AMX3 (A = Na+ , K+ ,... M = Mg2+ ,
Ca2+ ,... X = F− , Cl− ,...).
22 / 35
Forces in Ionic Crystals
AIM does find the anticipated anionic–cationic (a–c)
interactions together with a–a interactions for X = Cl− , I−
which were proposed by Pauling for LiCl, LiBr and LiI.
(Luaa, V.; Costales, A.; Pendas, A. M. Phys. Rev. B 1997, 55,
4285-4297)
23 / 35
Forces in Ionic Crystals
a–a interactions are be found in LiI! (Pendas, A. M.; Costales, A.;
Luaa, V. Phys. Rev. B 1997, 55, 4275-4284)
24 / 35
Forces in Ionic Crystals
Also c–c interactions are present (Cs+ ) together with small
anions.
Classical explanation fails: Same charges repell each other.
25 / 35
Forces in Ionic Crystals
Also c–c interactions are present (Cs+ ) together with small
anions.
Classical explanation fails: Same charges repell each other.
Responsible quantity is the Ehrenfest force (not the Feynman
force as the nuclei are in their equilibrium structures).
The repulsion between the eletrons is smaller than the
attraction between the electron density and the nuclei.
25 / 35
Pauli repulsion
Classical problem: What leads to the rotational barrier in
ethane and biphenyl? Traditional explanation: Pauli repulsion
between the o–hydrogen atoms, C–C bond lengthening.
26 / 35
Pauli repulsion
EDA: TS possesses lower Pauli repulsion in equilibrium
geometry than the minimum structure.
But when rings in the TS are moved closer together the Pauli
repulsion increases.
Thus: Rotational barrier is a result of avoided Pauli repulsion.
(J. Poater, M. Sola, F. M. Bickelhaupt, Chem. Eur. J. 2006, 12, 2889.)
27 / 35
Pauli repulsion
AIM: Bond paths between o–hydrogen atoms.
From minimum to TS: Energy decrease in the atomic basin of
the o–hydrogen atoms.
But also: Big energy increase in C1 and C7 because of bond
lengthening.
Energy increase of bond lengthening higher than decrease
because of o–hydrogen atom interaction.
28 / 35
Pauli repulsion
AIM: Bond paths between o–hydrogen atoms.
From minimum to TS: Energy decrease in the atomic basin of
the o–hydrogen atoms.
But also: Big energy increase in C1 and C7 because of bond
lengthening.
Energy increase of bond lengthening higher than decrease
because of o–hydrogen atom interaction.
H atoms are interacting in an attractive way, not an repulsive
way. There is no Pauli repulsion present!
Discussion of this problem in:
Krapp, A.; Frenking, G. Chem.—Eur. J. 2007, 13, 8256-8270.
28 / 35
Non–bonded or Van–der–Waals Interactions
Electron pair model was never extended to interacting closed
shell systems. VdW forces have been tradionally distinguished
from chemical bonds.
Projection of a six–dimensional problem (electron pairs!) onto
three dimensions leads to diffculties in the descriptions.
Origin of dispersion forces: Classical explanation: Fluctuating
dipoles lead to attaction between the atoms.
29 / 35
Non–bonded or Van–der–Waals Interactions
Electron pair model was never extended to interacting closed
shell systems. VdW forces have been tradionally distinguished
from chemical bonds.
Projection of a six–dimensional problem (electron pairs!) onto
three dimensions leads to diffculties in the descriptions.
Origin of dispersion forces: Classical explanation: Fluctuating
dipoles lead to attaction between the atoms.
According to Feynman: Each nucleus is attracted by its own
distorted charge distribution which gives an attractive R17
force. A Feynman force for opposed atomic dipoles would be
repulsive.
29 / 35
Non–bonded or Van–der–Waals Interactions
The weakest and longest chemical bond between two He
atoms (62 ± 10Å) was easily accepted as a bond (a BCP was
also found).
Luo, F.; Giese, C. F.; Gentry, W. R. J. Chem. Phys. 1996, 104,
1151-1154.
30 / 35
Non–bonded or Van–der–Waals Interactions
The weakest and longest chemical bond between two He
atoms (62 ± 10Å) was easily accepted as a bond (a BCP was
also found).
Luo, F.; Giese, C. F.; Gentry, W. R. J. Chem. Phys. 1996, 104,
1151-1154.
Therefore: Why distinguish between VdW and chemical
bonds?
30 / 35
Non–bonded or Van–der–Waals Interactions
Example: Ng2 @C60 .
Krapp, A.; Frenking, G. Chem.—Eur. J. 2007, 13, 8256-8270.
31 / 35
Non–bonded or Van–der–Waals Interactions
Example: Ng2 @C60 .
Krapp, A.; Frenking, G. Chem.—Eur. J. 2007, 13, 8256-8270.
For Ng = Xe: ’The results give clear evidence that the Xe–Xe
and the Xe2 –C60 interactions in Xe2 @C60 should be
considered as genuine chemical bonds.’ (When you read carefully:
Also Ar2 and Kr2 are considered to possess genuine chemical bonds..)
31 / 35
Non–bonded or Van–der–Waals Interactions
Example: Ng2 @C60 .
Krapp, A.; Frenking, G. Chem.—Eur. J. 2007, 13, 8256-8270.
For Ng = Xe: ’The results give clear evidence that the Xe–Xe
and the Xe2 –C60 interactions in Xe2 @C60 should be
considered as genuine chemical bonds.’ (When you read carefully:
Also Ar2 and Kr2 are considered to possess genuine chemical bonds..)
Why is a Ng–dimer not automatically considered an molecule?
31 / 35
Non–bonded or Van–der–Waals Interactions
Example: Ng2 @C60 .
Krapp, A.; Frenking, G. Chem.—Eur. J. 2007, 13, 8256-8270.
For Ng = Xe: ’The results give clear evidence that the Xe–Xe
and the Xe2 –C60 interactions in Xe2 @C60 should be
considered as genuine chemical bonds.’ (When you read carefully:
Also Ar2 and Kr2 are considered to possess genuine chemical bonds..)
Why is a Ng–dimer not automatically considered an molecule?
Physically there is no difference between a strong bonding and
a weak bonding.
31 / 35
Non–bonded or Van–der–Waals Interactions
Example: Ng2 @C60 .
Krapp, A.; Frenking, G. Chem.—Eur. J. 2007, 13, 8256-8270.
For Ng = Xe: ’The results give clear evidence that the Xe–Xe
and the Xe2 –C60 interactions in Xe2 @C60 should be
considered as genuine chemical bonds.’ (When you read carefully:
Also Ar2 and Kr2 are considered to possess genuine chemical bonds..)
Why is a Ng–dimer not automatically considered an molecule?
Physically there is no difference between a strong bonding and
a weak bonding.
Who decides when a chemical bond ceases to be a ’true’ bond
and becomes a weak van–der–Waals interaction?
31 / 35
Personal Opinion
Who decides when a chemical bond ceases to be a ’true’ bond
and becomes a weak van–der–Waals interaction?
IUPAC
According to the 1997 definition of IUPAC, “there is a chemical
bond between two atoms or groups of atoms in the case that the
forces acting between them are such as to lead to the formation of
an aggregation with sufficient stability to make it convenient for
the chemist to consider it as an independent ”molecular species”.
32 / 35
Personal Opinion
Our current bond models: Handy tools for the everyday (lab)
use. [(Nearly) nobody believes that these model are the reality
(e.g. orbitals). ]
33 / 35
Personal Opinion
Our current bond models: Handy tools for the everyday (lab)
use. [(Nearly) nobody believes that these model are the reality
(e.g. orbitals). ]
They help to get an grip into the principles in chemistry even
though the flaws of each model should be taught in more
detail.
33 / 35
Thank you
Thank you for your attention!
34 / 35