The Periodic Table retrieved from
the Electron Density and the
Shape function:
an Exercise in Information Theory.
6-8-2012
Herhaling titel van presentatie
1
The Periodic Table retrieved from the Electron Density
and the Shape function:
an Exercise in Information Theory.
Paul Geerlings, Alex Borgoo
Department of Chemistry
Faculty of Sciences
Vrije Universiteit Brussel
Belgium
The Third International Conference
on the Periodic Table
14Th-16Th August, Cusco Peru, 2012
6-8-2012
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1. Introduction
“ The periodic table of the elements is one of the most powerful icons in science: a
single document that captures the essence of chemistry in an elegant pattern.
Indeed, nothing quite like it exists in biology or physics, or any other branch of
science, for that matter.”
“ It is sometimes said that chemistry has no deep ideas, unlike physics, which can
boast quantum mechanics and relativity, and biology, which has produced the theory
of evolution. This view is mistaken, however, since there are in fact two big ideas in
chemistry. They are chemical periodicity and chemical bonding, they are deeply
interconnected.”
E.R Scerri, The Periodic Table,
Its story and Its Significance, Oxford UP, Oxford 2007
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Pag.3
• History of the Periodic Table
Underpinning the Periodic Table from “first principles”: reductionist approach
~ story of the 20th century
Quantum Mechanics
Quantum Chemistry
• Central role of the “orbital” concept → Electronic Configurations
↑
→ Periodicity of properties related to periodicity of Electronic Configurations
Wave Function Quantum Mechanics
• This approach has pervaded General Chemistry, Physical Chemistry and
Quantum Chemistry books
→ Students are educated along these lines
→ One sometimes needs to explain freshman students that the Periodic Table was originally
presented, merely based on experimental facts, before the advent of Quantum Mechanics, even
before the discovery of the electron!
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Pag.4
All in all wavefunction Quantum Mechanics/ Chemistry gives a convincing support to the
Periodic Table, though it should be realized that the Quantum Mechanical approach leads to
properties of neutral gas phase atoms. It supports the position of the elements and the
variation of their properties along rows and columns.
A typical example: electronegativity χ
χ
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Present talk
* Can one use a conceptually simpler QM approach, as compared to the horribly
complicated N-electron wave function Ψ, to retrieve periodicity from first
principles and in general the evolution of properties in the Periodic Table.
replace Ψ by other, simpler carriers of information?
which carriers?
how to extract the information from these carriers?
The Electron Density ρ(r) and the Shape Function σ(r)
Density Functional Theory (DFT)
Reading its information
content via
Information Theory
Periodicity?
*
Can DFT based concepts be used to explain/ interpret evolution of properties along
the Periodic Table.
CASE STUDY
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Pag.6
Electronegativity of the group 14 Elements
2. Information carriers: from Ψ to ρ.
From Conventional (= wave function) Quantum chemistry to Density Functional Theory
HΨ = EΨ → Ψ → properties, e.g. ρ(r)
→ E
↑
Electron density function: experimentally available
• In virtue of the nature of the operators involved in Quantum Chemistry (1- and 2- particle
operators), physical content of a system can be accurately described by the γ 2 reduced density matrix
( Löwdin, Mc Weeny, Davidson, …)
N ( N-1)
γ 2 ( x'1 , x'2 , x1 , x 2 ) =
Ψ ( x'1 , x'2 , x 3 ,..., x N ) Ψ * ( x1 , x 2 , x 3 ,..., x N ) dx 3 ...dx N
∫
2
expectation values of one – and two electron operators
• Further integration leads to the first order reduced density matrix
γ1 ( x'1 , x1 ) = N ∫ Ψ ( x'1 , x 2 ,..., x N ) Ψ * ( x1 , x 2 ,..., x N ) dx 2 ...dx N
Still quite involved expressions, difficult to visualize or handle in an intuitive way
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Diagonal element of the spinless first order reduced density matrix
ρ ( r ) =N ∫ Ψ * ( x,x 2 ,...x N )Ψ ( x,x 2 ,...x N ) ds...dx 2 ...dx N
Integration over 4N-3 variables!
Electron density function
function of only 3 variables
• Does ρ(r) still contains all ground-state information of the atom or
molecule
• If so no need for an “overcomplicated” wave function containing too
much information?
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Fundamentals of Density Functional Theory
The Hohenberg Kohn Theorems
First Theorem
(P.Hohenberg, W.Kohn, Phys.Rev. B, 136, 864 (1964))
external potential (i.e. due to the nuclei) : Z A , R A
ρ(r)
v(r)
N
H op
Ψ
∀A
E = E [ρ(r)]
Number of
electrons
ρ(r) as basic carrier of information
“The external potential v(r) is determined, within a trivial additive
contant, by the electron density ρ(r)"
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No two different v(r) generate same ρ(r) for ground state
electrons
•
•
•
•
•
•
•
•
•
•
•
ρ(r) for a given
ground state
compatible
with a
single v( r)
Second Theorem For a trial density
v(r)
- nuclei
- position/charge RA, ZA
ρ˜ (r) , such that ρ˜ (r) ≥0 ∀r and ∫ρ˜ (r) dr =N
~
E 0 ≤ E ρ 
 
exact ground state energy
• what is E[ρ]
• how can we obtain ρ directly from the knowledge of E[ρ]?
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Pag.10
Computational DFT
Variational procedure → Normalization of ρ(r) (Lagrangian multiplier(µ))
δFHK
+ v(r) = µ
δρ ( r )
→ Euler equation
• FHK Hohenberg Kohn functional: universal but unknown
• DFT analogue of Schrödinger’s time independent equation HΨ = EΨ
• ρ should be so that left hand side of the Euler equation is a contant
Practical Implementation: Kohn Sham equations
DFT as a computational method of ever increasing importance
(Chemical Abstracts: 2009: 15000 papers)
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Conceptual DFT
• Chemical reaction involves perturbation of a system in v(r) and/or N
Consider
E=E  N,v ( r ) 
 δE 
 ∂E 
dE =  
dN + ∫ 
 δv(r)d r
 ∂N  v(r)
 δv(r)  N
Parr et al. proved that
µ= ( ∂E/∂N ) v( r ) = - χ (Iczkowski - Margrave electronegativity)
R.G. Parr et al., J. Chem. Phys. 68, 3801 (1978)
6-8-2012
Pag.12
 δE 
ρ( r ) = 

 δv(r) N
From first order perturbation theory it can further be shown that
Identification of two first derivatives of E with respect to N and v(r) in a
DFT context.
( ∂E / ∂N )v( r ) , ( δE/δρ ( r ) )v( r ) ...
 ∂2E 


 2 = η
 ∂N  v
Chemical Hardness
 δµ   ∂ρ(r) 
 δv(r)  =  ∂N  =f(r)
v

N 
S = 1/η
Fukui function
response functions
Chemical Softness
Sf(r) = s(r) Local Softness
Review : P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev., 103, 1793 (2003)
P. Geerlings, F. De Proft, PhysChemChemPhys, 10, 3028 (2008)
P. Geerlings, P.W. Ayers, A. Toro-Labbé, P.K. Chattaraj, F. De Proft,
Acc. Chem. Res., 45, 683 (2012)
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Pag.13
Questions
• Is ρ(r) the simplest carrier of information:
σ (r)
()
• How does one read the information content of ρ(r) and σ r for a given
system, or as is often needed, its difference between two systems
Construction of a functional FAB= F[ʒA(r),ʒB(r)]
Information Theory
• Launched by Shannon in 1948
• Pioneered in Chemistry by Levine (Molecular Reaction Dynamics) and
in Quantum Chemistry by Sears, Parr, Dinur (1986)
• Breakthrough in QC in the last decade: focusing on the electronic
structure of atoms and molecules (Gadre, Sen, Parr, Nalewajski, …)
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Pag.14
3. Information carriers: from ρ→σ
The shape function : an even simpler carrier of information ?
• Parr, Bartolotti (J. Phys. Chem. 87, 2810 (1983))
σ(r) =
ρ(r)
N
shape function
• characterizes shape of the electron distribution
• ∫ σ(r)dr = 1
• σ(r) as carrier of information
P. W. Ayers
(Proc. Natl. Acad. Sci, 97, 1959 (2000))
σ(r) → v(r)
for a finite Coulombic system
Cfr. Bright Wilson’s arguments for the ρ(r) → v(r) relationship
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Pag.15
Water :
ρ(r)
Integration → N
Ethylene
Cusps → ZA, RA →v(r)
ρ(r) → N, v(r) → Hop → "Everything"
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Pag.16
Ayers
•
(Proc. Natl. Acad. Sci., 97, 1959 (2000)) :
Cusps in ρ and σ occur at the same places as σ(r) =
• It is easily seen that
1  1 ∂σ(r)
ZA =- 
2  σ(r) ∂ r-R A
• σ(r) → N


 r =R A
σ(r) → {R A ,ZA } → v(r)
- Convexity postulate
- Long range behaviour of ρ(r)
6-8-2012
Pag.17
1
ρ(r)
N
(2)
A first application:σ(r) contains enough information to predict atomic
Ionization Potentials.
Expressing the ionization energy in terms of
moments of the shape function
↓
∞ κ

(n)
2
µκ [ σ ] =  ∫ r σ ( r ) 4πr dr 
0

-n
κ
N
IE [ σ ] = ∑ bκ µκ(2) [ σ ]
κ =1
Neutral atoms and cations
H
→ W20+
(1081 species)
Already only two moments of σ give a fair fit with the data, whereas two moments of the
density are not adequate at all (σ better for periodic properties)
P.W. Ayers, F. De Proft, P. Geerlings, Phys. Rev. A 75, 012508 (2007)
6-8-2012
Pag.18
4. The σ(r) Information Content :
Quantum Similarity of Atoms
Quantum Similarity : Basics
• Characterizing similarity of molecules A and B
?
2
↓ Electron density ρ(r) ⇒ smallest value ∫ ρA − ρB dr
largest value for Z AB = ∫ ρ A (r)ρ B (r)dr
Normalized index
ZAB =
=
6-8-2012
∫ρ
A
(r)ρ B (r)dr
(
 ρ 2A (r)dr
 ∫
∫σ
(
A
Pag.19
2
B
(r)σ B (r)dr
 σ (r)dr
 ∫
2
A
)( ∫ ρ (r)dr )
1/2
)( ∫ σ (r)dr )
2
B
1/2
M. Carbo et al., Int. J. Quant., 17, 1185 (1980)
← σ (r)
The case of atoms, nearly unexplored until 2003.
Carbo type Approach
ZAB = ∫ ρ A ( r1 ) ρ B ( r1 ) dr1
SI=
ZAB
ZAA ZBB
Numerical Hartree Fock wave functions
Noble Gases
17
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Pag.20
Nearest neighbours alike!
Looking for periodicity: the introduction of Information Theory
• Kullback Leibler Information Deficiency ∆S for a continuous probability distribution
∆S( pk p0 ) ≡ ∫ pk ( x ) log
pk ( x )
dx
p0 ( x )
S. Kullback, R.A. Leibler, Ann.Math.Stat. 22, 79 (1951)
pk(x), p0(x): normalized probability distributions, p0(x): prior distribution.
∆S = distance in information between pk(x) and p0(x) or
as the information present in pk(x), distinguishing it from p0(x)
Choice of reference p0(x)
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Pag.21
Take ρ(r) as p(x)
∫ ρA ( r ) log
ρA ( r )
dr
ρ0 ( r )
Choice of
ρ0(r)
Cf. Sanderson electronegativity scale
(Ratio of electron densities)
(r)
Renormalized density of a hypothetical noble gas atom
with equal number of electrons as atom A.
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∆SA
ρ
= ∫ ρ A ( r ) log
ρA ( r )
dr
ρ0 ( r )
800
ID
600
400
200
0
0
10
20
30
Z-1
Periodicity reflected
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40
50
60
Eliminating NA dependence by reformulation of the theory directly in
terms of the shape functions (cf. role as information carrier)
∆S = ∫ σ A ( r ) log
σ
A
σA ( r )
σ0 ( r )
dr
an example of F[ʒA,ʒB]
Reference choice as before: the core of the atom i.e. the shape of the
previous noble gas atom
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∆SσA = ∫ σ A ( r ) log
σA ( r )
σ0 ( r )
dr
Ne
50
40
30
Kr
Ar
20
Xe
Li
10
He
0
0
10
20
30
40
50
60
Z-1
Periodicity and evolution in atomic properties throughout PT regained;
overall performance of σ(r) better than ρ(r)
A. Borgoo, M. Godefroid, K.D. Sen, F. De Proft, P. Geerlings, Chem.Phys.Lett., 399, 363 (2004)
6-8-2012
Pag.25
Similarity Measure based on a local version of Kullback’s Information Discrimination
ρ
• Consider ∆SA ( r ) =ρ A ( r ) log
ρA ( r )
N
ρ0 ( r ) A
N0
• ZAB = ∆SA ( r ) ∆SB ( r ) dr
∫
SI=
ρ
ρ
ZAB
ZAA ZBB
Quantum Similarity Measure (QSM)
↓
Quantum Similarity Index
Completely Information Theory Based
• Cross Section of QSM(ZA,ZB)
ZB= 82 (Pb)
Dirac Fock Calculations
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Relativistic Effects
Dirac Fock densities
1
ρ(r) =
4π
Pn2κ ( r ) +Q n2κ ( r )
q nκ
∑
2
r
nκ
• large and small components
• qnκ : occupation number of relativistic subshell
Similarity Index between HF and DF densities
A.Borgoo, M.Godefroid, P.Indelicato, P.Geerlings, J.Chem.Phys.126, 044102 (2007)
6-8-2012
Pag.27
• An alternative view: which atom in a given period belongs to a certain
column and how to order the “elements” in that period.
• Consider period ……Al, Si, P, S, Cl, …
Information contained in the shape function of Nitrogen to determine that
of the elements of the next row. Comparing information content in σN, on
σP with its information on σAl, σSi, σS, σCl.
N
Al
0.9865
<
Si
0.9969
<
P
1.0
>
S
Cl
0.9973 > 0.9903
A different, “triangular”, aspect of Periodicity regained through the
functional F [ σ x ,σ N ,σ P ]
Similar results when considering (P, As), (As, Sb), (Sb, Bi) but with
lower sensitivity.
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Conclusions
Periodicity can be regained through QM in an in principle orbital
free approach based on the properties of the electron density of
gas phase atoms, or an even simpler carrier of information, the
shape function. Information Theory plays a fundamental role in
“reading” the information content of these information carriers.
P. Geerlings and A. Borgoo, Phys. Chem. Chem. Phys., 13, 911 (2011)
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5. Electronegativity of the Group 14 Elements
Electronegativity: one of the most pervading concepts in chemistry when discussing
structure, bonding, reactivity, electric properties… of molecules and the solid state.
Plentiful methods for ( empirically/theoretically) quantifying this property and its
evolution along columns and rows of the Periodic Table have been presented.
• Pauling/ Allred (thermochemical) (1932-1961)
• Mulliken / Hinze/Jaffe, …(averaging ionization energy and electron affinity) (1934-1962)
• Gordy (electrostatic potential at covalent radius) (1946)
• Sanderson (electrondensity ratio) (1951)
• Allred/Rochow (attraction force for a valence electron) (1958)
• Iczkowski/Margrave: energy vs. number of electron curve (1961)
• …
Most of them show satisfactory mutual correlations.
• Electronegativity, K.D. Sen Ed., Structure and Bonding, 66, Springer, Berlin, 1987.
• M.R. Leach, Foundations of Chemistry, 14, xxx, 2012
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Has everything been said about electronegativity?
− on its definition: NO
cf. its natural appearance in conceptual DFT (Parr)
 ∂E 
χ= - 

 ∂N  v
with present day discussions on how to evaluate this
derivative and how to reconcile this global concept with
local characteristics( atom-in-molecule)
cf. our work on Conceptual DFT
− on its behavior along the periodic table: NO
An example: electronegativity variation of Group 14
elements
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Electronegativity variation of Group 14 elements :
χPauling:
C > Pb > Ge > Sn > Si
χSanderson:
C > Ge > Pb > Si > Sn
χAllred- Rochow:
C > Ge > Si > Sn > Pb
Irregularities *
(Absent in Group
17-Halogens)
C > Pb > Ge > Si > Sn
χAllen:
C > Ge > Si > Sn
χMulliken:
C > Si > Ge > Sn > Pb
χMulliken- Jaffe:
C > Pb > Ge > Si > Sn
χGordy:
C > Si > Ge > Sn
* Only C has fixed position
* Si/Ge sequence
Relativistic
effects?
* Pb position
→ Evaluation of electronegativities in a Conceptual DFT context, including also
-XY3 Functional groups
X: C, Si, Ge, Sn, Pb, Uuq (eka-Pb)
Y: F, Cl, Br, I, At, H, CH3
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Computational methods
Geometry optimisation of H-XY3
Extract
•
XY3
Computation of vertical I and A for
Three levels of theory:
•
XY3
Non Relativistic NR
Scalar Relativistic SR
Relativistic including spin orbit coupling SO
Group electronegativity:
χ -XY =
I ( • XY3 ) + A ( • XY3 )
3
Group hardness:
η-XY =
3
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2
I ( • XY3 ) - A ( • XY3 )
2
Ionisation energies & electron affinities
Atomic NR and SR values for these quantities:
Reasonable correlation with experimental values for the lighter elements
Heavier elements clearly need the inclusion of spin-orbit coupling which
correctly predicts experimental sequence. ( Uuq left out of consideration)
Order for X seldom transferred to XY3
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Electronegativity
Atoms: C > Si > Ge > Sn > Uuq > Pb : Importance of Relativistic Effects (in accordance
with “experimental” Mulliken scale)
XY3 with Y=H or CH3: electronegativity ordering:
C > Uuq > Pb > Sn > Ge > Si
Y= halogen:
Uuq > Pb > Sn > Ge > C > Si
• Passing from an atom to a functional group can dramatically change the
electronegativity sequences!
• In all cases Ge>Si as opposed to free atom case.
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Hardness
Atoms: C > Uuq > Pb > Si > Ge > Sn (in accordance with “experimental” Mulliken scale)
Opposed to the halogens: F > Cl > Br > I > At: monotonous decrease
Functional groups: all three methods following the ordening:
C > Si > Ge > Sn > Pb > Uuq
Calculations confirm that Group 14 shows less monotonicity in its
behavior as compared to Group 17
K.T. Giju, F. De Proft, P. Geerlings, J. Phys. Chem. A, 109, 2925 (2005)
6-8-2012
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Conclusions
The study of the Periodic Table is alive and well, also in a Quantum
mechanical context. Despite the “neutral gaseous atoms philosophy”
new approaches can shed further light on Periodicity and
(ir)regularities in the Periodic Table, the
“single document that captures the essence of chemistry
in an elegant pattern” (E. Scerri)
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Thanks to the previous and present members of the ALGC group
in particular:
• Prof. F. De Proft
• Dr. A. Borgoo
• Dr. K.T. Giju
• Prof. P. Ayers (MC Master)
• Prof. V. D. Sen ( Hyderabad)
• and Prof. M. Godefroid ( Université Libre de Bruxelles)
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