The Periodic Table retrieved from the Electron Density and the

Shape function:

an Exercise in Information Theory.an Exercise in Information Theory.

6-8-2012 1Herhaling titel van presentatie

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

Pag.

Vrije Universiteit Brussel

Belgium

The Third International ConferenceThe Third International Conferenceon the Periodic Tableon the Periodic Table

1414ThTh--1616ThTh August, Cusco Peru, 2012August, Cusco Peru, 2012

6-8-2012 2

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

Pag.6-8-2012 3

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

• 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

Pag.6-8-2012 4

• 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!

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

Pag.6-8-2012 5

χχχχ

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)

*

Pag.6-8-2012 6

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 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)

→ ↑Electron density function: experimentally available

E

• 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 reduced density matrix ( Löwdin, Mc Weeny, Davidson, …)

( ) ( ) ( ) ( )N N-1

∫

2γ

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( ) ( ) ( ) ( )*

1 2 1 2 1 2 3 N 1 2 3 N 3 N2

N N-1γ x' , x' , x , x x' , x' , x ,..., x x , x , x ,..., x dx ...dx

2= Ψ Ψ∫

expectation values of one – and two electron operators

• Further integration leads to the first order reduced density matrix

( ) ( ) ( )*

1 1 1 2 N 1 2 N 2 N1γ x' , x N x' , x ,..., x x , x ,..., x dx ...dx= Ψ Ψ∫Still quite involved expressions, difficult to visualize or handle in an intuitive way

Integration over 4N-3 variables!

( ) ( ) ( )*

2 N 2 N 2 Nρ r =N x,x ,...x x,x ,...x ds...dx ...dxΨ Ψ∫ Electron density function

Diagonal element of the spinless first order reduced density matrix

function of only 3 variables

Pag.6-8-2012 8

• 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?

ρ(r) v(r) N

external potential (i.e. due to the nuclei)

Number of

electrons

H Ψ opE = E [ (r)]ρ

ZA RA ∀A: ,

The Hohenberg Kohn Theorems (P.Hohenberg, W.Kohn, Phys.Rev. B, 136, 864 (1964))

First Theorem

Fundamentals of Density Functional Theory

Pag.6-8-2012 9

“The external potential v(r) is determined, within a trivial additive contant, by the electron density ρ(r)"

ρ(r) as basic carrier of information

•

•

• • •

•

•

•

••

•

compatible with a single v( r)

ρ(r) for a given ground state

- nuclei - position/charge

electrons

v(r)

RA, ZA

No two different v(r) generate same ρ(r) for ground state

For a trial density , such that ˜ ρ (r) ˜ ρ (r) ˜ ρ (r)∀∀∀∀ ∫Second Theorem

Pag.6-8-2012 10

• how can we obtain ρ directly from the knowledge of E[ρ]?

� For a trial density , such that ˜ ρ (r) ˜ ρ (r) ˜ ρ (r)

• what is E[ρ]

≥0 ∀∀∀∀r and ∫ dr =NSecond Theorem

~

0E E ρ ≤

exact ground state energy

� Computational DFT

Variational procedure → Normalization of ρ(r) (Lagrangian multiplier(µ))

( )( )HKδF

v r µδρ r

+ = →→→→ Euler equation

• DFT analogue of Schrödinger’s time independent equation HΨ = EΨ

• FHK Hohenberg Kohn functional: universal but unknown

Pag.6-8-2012 11

• 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)

� Conceptual DFT

• Chemical reaction involves perturbation of a system in v(r) and/or N

Consider

dE =∂E

∂N

v(r)

dN +δE

δv(r)

∫

N

δv(r)dr

( )E=E N,v r

Pag.6-8-2012 12

R.G. Parr et al., J. Chem. Phys. 68, 3801 (1978)

= - χ (Iczkowski - Margrave electronegativity)( ) ( )v rµ= E/ N∂ ∂

Parr et al. proved that

From first order perturbation theory it can further be shown that ρ r( ) =δE

δv(r)

N

Identification of two first derivatives of E with respect to N and v(r) in aDFT context.

response functions

∂2E

∂N2

= η Chemical Hardness S = 1/η Chemical Softness

( ) ( ) ( )( ) ( )v r v rE / N , δE/δρ r ...∂ ∂

Pag.6-8-2012 13

∂N2

v

Fukui function Sf(r) = s(r) Local Softness

Review : P. Geerlings, F. De Proft, W. Langenaeker, Chem. Rev., 103, 1793 (2003)

vN

δµ ρ(r)= =f(r)

δv(r) N

∂ ∂

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)

Questions

• Is ρ(r) the simplest carrier of information: ( )σ r

• How does one read the information content of ρ(r) and for a given system, or as is often needed, its difference between two systems

( )σ r

Construction of a functional FAB= F[ʒA(r),ʒB(r)]

Information Theory

Pag.6-8-2012 14

• 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, …)

3. Information carriers: from ρ→σ

The shape function : an even simpler carrier of information ?

σ(r) =ρ(r)

Nshape function

• characterizes shape of the electron distribution• σ(r)dr =1∫

• Parr, Bartolotti (J. Phys. Chem. 87, 2810 (1983))

Pag.6-8-2012 15

• σ(r)dr =1∫

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

• σσσσ(r) as carrier of information

Water :

Ethylene Cusps → Z , R →v(r)

Integration → N

ρ(r)

Pag.6-8-2012 16

ρ(r) → N, v(r) → Hop → "Everything"

Ethylene Cusps → ZA, RA →v(r)

Ayers (Proc. Natl. Acad. Sci., 97, 1959 (2000)) :

σ(r) =1

Nρ(r) (2)

• It is easily seen that

• Cusps in ρ and σ occur at the same places as

1 1 σ(r) ∂

Pag.6-8-2012 17

{ }A Aσ(r) R ,Z v(r)→ →

σ(r) N→• - Convexity postulate

- Long range behaviour of ρ(r)

A

A

Ar=R

1 1 σ(r)Z =-

2 σ(r) r-R

∂ ∂

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

( ) [ ] ( )n-

n 2

0

µ σ = r σ r 4πr drκ

κκ

∞ ∫

[ ] [ ]N

(2)IE σ = b µ σκ κ∑

Pag.6-8-2012 18

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)

[ ] [ ]=1

κ κκ∑

4. The σ(r) Information Content : Quantum Similarity of Atoms

Quantum Similarity : Basics

largest value for ZAB = ρA (r)ρB(r)dr∫

smallest value ρA −ρB∫2dr⇒

?• Characterizing similarity of molecules A and B

↓ Electron density ρ(r)

Pag.6-8-2012 19

Normalized index

← σ (r)

( )( )A B

AB 1/22 2

A B

ρ (r)ρ (r)drZ =

ρ (r)dr ρ (r)dr

∫

∫ ∫

( )( )A B

1/22 2

A B

σ (r)σ (r)dr=

σ (r)dr σ (r)dr

∫

∫ ∫

M. Carbo et al., Int. J. Quant., 17, 1185 (1980)

Carbo type Approach

Numerical Hartree Fock wave functions

The case of atoms, nearly unexplored until 2003.

Noble Gases

( ) ( )1 1 1AB A BZ = ρ r ρ r dr∫AB

AA BB

ZSI=

Z Z

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Nearest neighbours alike!17

Noble Gases

Looking for periodicity: the introduction of Information Theory

• Kullback Leibler Information Deficiency ∆S for a continuous probability distribution

pk(x), p0(x): normalized probability distributions, p0(x): prior distribution.

( ) ( ) ( )( )

k

k 0 k

0

p x∆S p p p x log dx

p x≡ ∫ S. Kullback, R.A. Leibler, Ann.Math.Stat. 22, 79 (1951)

Pag.6-8-2012 21

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)

ρ0(r) Cf. Sanderson electronegativity scale

Take ρ(r) as p(x)

Choice of

( ) ( )( )

A

A

0

ρ rρ r log dr

ρ r∫

(Ratio of electron densities)

Pag.6-8-2012 22

Renormalized density of a hypothetical noble gas atom with equal number of electrons as atom A.

(r)

(Ratio of electron densities)

400

600

800

∆SAρ =

ID( ) ( )

( )A

A

0

ρ rρ r log dr

ρ r∫

Pag.6-8-2012 23

0

200

400

0 10 20 30 40 50 60

Z-1

ID

Periodicity reflected

Eliminating NA dependence by reformulation of the theory directly in terms of the shape functions (cf. role as information carrier)

( ) ( )( )

Aσ

A A

0

σ r∆S = σ r log dr

σ r∫ an example of F[ʒA,ʒB]

Pag.6-8-2012 24

Reference choice as before: the core of the atom i.e. the shape of the previous noble gas atom

20

30

40

50 Ne

Ar Kr

XeLi

( ) ( )( )

Aσ

A A

0

σ r∆S = σ r log dr

σ r∫

Pag.6-8-2012 25

0

10

0 10 20 30 40 50 60

Z-1

He

XeLi

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)

Similarity Measure based on a local version of Kullback’s Information Discrimination

• Consider

• ( ) ( )ρ ρ

AB A BZ = ∆S r ∆S r dr∫AB

AA BB

ZSI=

Z Z

• Cross Section of QSM(Z ,Z )

Quantum Similarity Measure (QSM)

↓Quantum Similarity Index

Completely Information Theory Based

( ) ( ) ( )

( )Aρ

A AA

0

0

ρ r∆S r =ρ r log

Nρ r

N

Pag.6-8-2012 26

• Cross Section of QSM(ZA,ZB)

ZB= 82 (Pb)

Dirac Fock Calculations

Relativistic Effects

Dirac Fock densities

• large and small components• qnκ : occupation number of relativistic subshell

Similarity Index between HF and DF densities

( ) ( ) ( )2 2

n n

n2n

P r +Q r1ρ r = q

4 r

κ κκ

κπ ∑

Pag.6-8-2012 27

A.Borgoo, M.Godefroid, P.Indelicato, P.Geerlings, J.Chem.Phys.126, 044102 (2007)

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 Si P S Cl

• 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, …

Pag.6-8-2012 28

Al0.9865

Si0.9969

P1.0

S0.9973

Cl0.9903

A different, “triangular”, aspect of Periodicity regained through the functional [ ]x N PF σ ,σ ,σ

Similar results when considering (P, As), (As, Sb), (Sb, Bi) but with lower sensitivity.

< < > >

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

Pag.6-8-2012 29

“reading” the information content of these information carriers.

P. Geerlings and A. Borgoo, Phys. Chem. Chem. Phys., 13, 911 (2011)

5. Electronegativity of the Group 14 Elements

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)

Electronegativity: one of the most pervading concepts in chemistry when discussing

structure, bonding, reactivity, electric properties… of molecules and the solid state.

Pag.6-8-2012 30

• 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

Has everything been said about electronegativity?

− on its definition: NO

cf. its natural appearance in conceptual DFT (Parr)

v

E= -

Nχ

∂ ∂

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

Pag.6-8-2012 31

cf. our work on Conceptual DFT

− on its behavior along the periodic table: NO

An example: electronegativity variation of Group 14 elements

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

C > Pb > Ge > Si > Sn

χAllen: C > Ge > Si > Sn

χMulliken: C > Si > Ge > Sn > Pb

Irregularities *

(Absent in Group

17-Halogens)

Relativistic

effects?

Pag.6-8-2012 32

χMulliken- Jaffe: C > Pb > Ge > Si > Sn

χGordy: C > Si > Ge > Sn

effects?

-XY3 Functional groups X: C, Si, Ge, Sn, Pb, Uuq (eka-Pb)

Y: F, Cl, Br, I, At, H, CH3

→ Evaluation of electronegativities in a Conceptual DFT context, including also

* Only C has fixed position * Si/Ge sequence * Pb position

Computational methods

� Geometry optimisation of H-XY3

Extract • XY3

� Computation of vertical I and A for • XY3

� Three levels of theory: Non Relativistic NR

Scalar Relativistic SR

Relativistic including spin orbit coupling SO

Pag.6-8-2012 33

� Group electronegativity:

� Group hardness:

( ) ( )3

• •

3 3

-XY

I XY + A XY=

2χ

( ) ( )3

• •

3 3

-XY

I XY - A XY=

2η

Ionisation energies & electron affinities

Pag.6-8-2012 34

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

Electronegativity

Atoms: C > Si > Ge > Sn > Uuq > Pb : Importance of Relativistic Effects (in accordance with “experimental” Mulliken scale)

Pag.6-8-2012 35

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.

Hardness

Pag.6-8-2012 36

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)

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

Pag.6-8-2012 37

“single document that captures the essence of chemistry

in an elegant pattern” (E. Scerri)

Thanks to the previous and present members of the ALGC group

in particular:

• Prof. F. De Proft

• Dr. A. Borgoo

• Dr. K.T. Giju

Pag.6-8-2012 38

• Dr. K.T. Giju

• Prof. P. Ayers (MC Master)

• Prof. V. D. Sen ( Hyderabad)

• and Prof. M. Godefroid ( Université Libre de Bruxelles)