Information Carriers and Information Theory in Quantum ChemistryQuantum Chemistry

16-3-2009 1Herhaling titel van presentatie

Information Carriers and Information Theory in Quantum Chemistry

Paul GEERLINGS, Alex BORGOO, Sara JANSSENS, Pablo JAQUE

Department of Chemistry

Faculty of Sciences

Vrije Universiteit Brussel

Pag.

Vrije Universiteit Brussel

Pleinlaan 2, 1050 - Brussels

Belgium

16-3-2009 2

2009 Molecular Informatics and Bio-informaticsInstitute for Advanced Study, Collegium Budapest

Budapest, March 17-19, 2009

Quantum Chemistry

Information Carriers ()

Wave function

Electron Density

Ψ

↓

ρ

↓

How to “read” information content, or its difference between two systems A and B

↓Construction of a functional

1

2

Reading The Information

Pag.16-3-2009 3

Shape function

↓

σ

↓

The Holographic Shape Theorem

↓

Use of Information Theory

Case Studies

I Quantum Similarity of atomsII Dissimilarity of enantiomersIII Information Theory in reactions: information profiles

2

3

FAB= F[A,B]

1. 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

N-electron system

E

( ) ( ) ( )*

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

Pag.16-3-2009 4

Integration over 4N-3 variables!

• Density Functional Theory: ρ(r) contains all ground-stateinformation of the atom or molecule

• No need for an “overcomplicated” wave function containing too much information.

Fundamentals and "Computational" DFT

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 DFT

Pag.16-3-2009 5

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

ρ(r) as basic variable

•

•

• • •

•

•

•

••

•

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.16-3-2009 6

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

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

E0 ≤ E ˜ ρ [ ] ↑ exact ground

state energy

E ρ[ ] ?? : energy functional • what is

≥0 ∀∀∀∀r and ∫ dr=NSecond Theorem

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

• ρ should be so that lefthand site of the Euler equation is a contant

Pag.16-3-2009 7

• ρ should be so that lefthand site of the Euler equation is a contant

Practical Implementation: Kohn Sham equations

DFT as a computational method of ever increasing importance

(Chemical Abstracts: 2007: 13000 papers)

Conceptual DFT

Parr's landmark paper on the identification of µ as (the opposite of) the electronegativity

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

• 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.16-3-2009 8

As Ev = Ev[ρ] (Euler equation)

dEv = µ δρ(r)dr∫ = µdN

= - χ (Iczkowski - Margrave electronegativity)

( )( )

( )( )

v

v(r) v

δE δEdE = δρ r dr with µ=

δρ r δρr

r

∫

( ) ( )v rµ= E/ N∂ ∂

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

Pag.16-3-2009 9

∂2E

∂N2

v

= η Chemical Hardness S = 1/η Chemical Softness

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

∂ ∂

2. Information carriers: from ρ→σThe shape function : an even simpler carrier of information ?

Introduction

σ(r) =ρ(r)

Nshape function

• characterizes shape of the electron distribution• redistributes N in space ρ(r) = N σ(r)

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

Pag.16-3-2009 10

• redistributes N in space ρ(r) = N σ(r)cf. Electronic Fukui Function : redistributes S in space

s(r) = Sf(r)• σ(r)dr =1∫

σσσσ(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

Water :

Ethylene Cusps → Z , R →v(r)

Integration → N

ρ(r)

Pag.16-3-2009 11

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.16-3-2009 12

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

∂ ∂

What about N ?

• Convexity postulate

E1 - E2 > E2 - E3 > E3 - E4 ... until system becomes unbound

Successive ground state ionisation potentialsof a coulombic system increase•

•

E

E3

E1

E2

IN0 +1,N0< IN0,N0 −1 < IN0 −1,N0 −2

Pag.16-3-2009 13

ρ(r)r↑ → e

− 8I r(For a simple approach : see N.C. Handy in European Summer Schoolin Quantum Chemistry, Lund University, Chapter 10, 2000)

••

N

E4

E3

N0-2 N0-1N0-1N0-1 N0 N0+1

• Long range behaviour of ρρρρ(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 σ∑

Pag.16-3-2009 14

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)

[ ] [ ](2)

=1

IE σ = b µ σκ κκ∑

σσσσ(r) as carrier of information on DFT reactivity indices

Does the shape function also contain information about reactivity indicesη, S, f(r), s(r), ... containing N derivatives ?

ρ(r) = A(N)e− 8Ir

• The long range behaviour of the Fukui Function

f(r) =∂ρ(r)

∂N

→ lim

r→∞

∂∂r

lnf(r) = − 8Ir

large r

Pag.16-3-2009 15

f(r) =∂N

v

→ limr→∞ ∂r

lnf(r) 8Ir

f(r)

ρ(r)→ ∂

∂r

f(r)

ρ(r)

=

−2

2I

∂I

∂N

v

∂I

∂N

v→ η

r→∞:

Now∂∂r

f(r)

ρ(r)

=

∂∂r

∂σ ∂N[ ]vσ

= −

2

2I

∂I

∂N

v

Simultaneous knowledge of σ(r) and ∂σ(r)/∂N → η

• F. De Proft, P.W. Ayers, K.D. Sen, P. Geerlings, J. Chem. Phys., 120, 9969 (2004).

Pag.16-3-2009 16

• P.Geerlings, F.De Proft, P.W. Ayers in “Theoretical Aspects of Chemical Reactivity”,Theoretical and Computational Chemistry, Vol.19, A.Toro Labbé Editor, Elsevier, Amsterdam, 2006,p.1.

Case Study I: An Application of the σσσσ(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.16-3-2009 17 17

Normalized index

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

← 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

∫

∫ ∫

Carbo type Approach

Numerical Hartree Fock wave functions

The case of atoms

Noble Gases

( ) ( )1 1 1AB A BZ = ρ r ρ r dr∫AB

AA BB

ZSI=

Z Z

Pag.16-3-2009 18

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

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

Pag.16-3-2009 19

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 p(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∫

Pag.16-3-2009 20

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

(r)

400

600

800

∆SAρ =

ID( ) ( )

( )A

A

0

ρ rρ r log dr

ρ r∫

Pag.16-3-2009 21

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)

( ) ( )( )

A

A A

0

σ r∆S σ = σ r log dr

σ r∫

Pag.16-3-2009 22

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

( ) ( )( )

A

A A

0

σ r∆S σ = σ r log dr

σ r∫

Pag.16-3-2009 23

0

10

0 10 20 30 40 50 60

Z-1

He

XeLi

Periodicity and evolution in atomic properties throughout PT regained

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 Shape Based

( ) ( ) ( )

( )Aρ

A AA

0

0

ρ r∆S r =ρ r log

Nρ r

N

Pag.16-3-2009 24

• Cross Section of QSM(ZA,ZB)Completely Shape Based

ZB= 82

Information contained in the shape function of Nitrogen

to determine that of the elements of the next row

N

Al0.9865

Si0.9969

P1.0

S0.9973

Cl0.9903

Pag.16-3-2009 25

0.9865 0.9969 1.0 0.9973 0.9903

A different aspect of Periodicity Regained

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.16-3-2009 26

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

Case Study II: Information (Theory) in Molecules:

Similarity of enantiomers and the Holographic Density Theorem

Introduction

Enantiomers Different chemical behaviour towards chiral partners(e.g. receptors)

Medicinal Chemistry, Pharmacology

Pag.16-3-2009 27

Eudesmic Ratio (ER) = Potency ratio of eutomer and distomer

Relationship with • (dis)similarity of enantiomers ? (Richards)

Black-white or discrete property

?

• degree of chirality

Testing the Holographic Density Theorem through Hirshfeld partitioning of ρ(r)

The Holographic Density Theorem

Hohenberg - Kohn

→ E = Ev ρ[ ]••

••

• •

••

compatibility

with a single external potential v(r)

electrons

ρ(r) for a ground state

nuclei

••

Pag.16-3-2009 28

Holographic Density Theorem (P.G. Mezey, Mol.Phys. 96, 1691 (1990),J.Cem.Inf.Comp.Sci.,39, 324 (1999))

E = Ev ρΩ[ ]ρΩ(r)

••

••

• •

••

Ω ρ(r)

••

••

Hirshfeld partitioning : basics and information theory context

• Hirshfeld partitioning (Theoret. Chim. Acta 44, 129 (1977))

ρA(r) = wA(r)ρ(r)Contribution of atom A to electron density at position r

weight factorρA

0 (r)

ρX0 (r)

X

∑promolecular densities “stockholder" partitioning

• Nalewajski - Parr (Proc. Natl. Acad. Sci.USA, 97, 8879 (2000))

Pag.16-3-2009 29

P0(r) reference, well-behaved, probability distributionP(r) trial probability distribution whose information content we want

to make as close as possible to P0, subject to one or more constraints

Define ∆S P/Po[ ]= P(r)ln∫P(r)

Po(r)

dr

( ) ( )( )

( )0

A

A

0

ρ rρ r = ρ r

ρ rHirshfeld’s “stockholder” partitioning

Possibility to look at global and local similarity

Results

CHFClBr : the text-book example of a chiral molecule

• orientational problem : superimposing C* and two of its substitutents

6 orientations

• intuitively : superposition of C*, Cl, Br largest similarityadequate for local similarity analysis

Global similarity : B3PW91/6-311G* Carbo Index

superimposed

(Carbo index)

Pag.16-3-2009 30

• Numerical and analytical results quasi identical (10-3)• Effect of orientation

ClCBr 0,990

FCBr 0,915

HCBr 0,906

FCCl 0,098

HCCl 0,089

HCF 0,054

Analyticalsuperimposed

atoms

0,990

0,915

0,906

0,098

0,089

0,055

Numerical

Pag.16-3-2009 31

Relative orientation of the R and S enantiomers of CHFClBr: the asymmetric carbon atoms, chlorine and bromine are superimposed.

Local similarity via Hirshfeld partitioning

HCF 0,991 1,000

HCBr 0,980 0,996

∆ρ∆ρ∆ρ∆ρρρρρsuperimposed atoms

HCF 0,999 0,995

FCBr 0,998 0,623

superimposed

atomsρρρρ ∆ρ∆ρ∆ρ∆ρ

Asymmetric carbon atom (Carbo index) Hydrogen atom (Carbo index)

Calculation on the basis of

but importance of core electrons, e.g. C1S

- ∆ρ : deformation density : ∆ρ = ρ - ρ0 ρ0 = promolecular density

- ρ

Pag.16-3-2009 32

HCBr 0,980 0,996

HCCl 0,971 0,995

FCBr 0,998 0,623

FCCl 0,998 0,620

HCBr 0,996 0,951

ClCBr 0,996 0,446

HCCl 0,995 0,940

• For C : larger range with ∆ρ∆ρ gives complementary information (role of core electrons eliminated)

• For H : numerical evidence for Holographic Density Theorem. "Chirality is everywhere"

Conclusions

• As all ρ and ∆ρ results are identical to σ and ∆σ results ⇒(Dis)similarity of enantiomers = a shape function analysis

• Transition from global to local (dis)similarity

• First ab initio numerical testing of the Holographic Electron Density Theorem

G. Boon, G. Van Alsenoy, F. De Proft, P. Bultinck, P. Geerlings, J. Phys. Chem. A, 107, 11120 (2003).G. Boon, F. De Proft, C. Van Alsenoy, P. Geerlings, Theochem (R. Carbo Volume), 49,727 (2005)

Pag.16-3-2009 33

• Confirmed by studies on systems with two asymmetric centra(halogensubstituted ethanes)

S.Janssens, C.Van Alsenoy, P.Geerlings, J.Phys.Chem.A 111, 3143 (2007)

• An excursion into the relation with optical rotation

The case of substituted allenes: systems wtihout C*

C=C=C

Y

HH

X

Pag.16-3-2009 34

Optical Rotation Similarity Index

Sequences correspond with the sequence of the size of the substituents

→ As the substituents get larger, the optical rotation increases and the enantiomers become more and more dissimilar

S.Janssens, G.Boon, P.Geerlings, J.Phys.Chem.A110, 9267 (2006)

• An information theory based approach to (dis)similarity of Enantiomers

Local

( ) ( )( )

RRR

SS

σ rσ∆S = σ r ln dr

σ σ r

∫ Information in R distinguishing it from S

↓Local version for atom A

( )( )

0

A,R

A,R 0

X,R

x

ρ rw =

ρ r∑

( ) ( )( )

A,R RA,RA,R R

A,SA,S S

w σ rσ∆S = w σ r ln dr

σ w σ r

∫

Pag.16-3-2009 35

Return to CHFClBr(including also I as substituent)

S.Janssens, A.Borgoo, C.Van Aslenoy, P.Geerlings, J.Phys.Chem.A, 112, 10560 (2008)

Local entropy of C*

Conclusions

• Shape function contains basic information to look at dissimilarity of

enantiomers at both global and local level

• Various functionals from ρ and σ are conceivable to extract information from

it: Carbo Index,... local entropy based expressions.

• Dissimilarity of enantiomers permits chirality to pass from a black or white

property to a continuously varying property, the “grey” scale being not

evidently linked to optical rotation or constitution

Pag.16-3-2009 36

evidently linked to optical rotation or constitution

3. The Holographic Shape Theorem

ρΩ(r) determines ρ(r) all propertiesHK →

E = Ev[ρΩ]*

• Local similarity can never be perfectcf. The Chemist's idea of a molecule built up from functional

Holographic Density Theorem:

Pag.16-3-2009 37

cf. The Chemist's idea of a molecule built up from functionalgroups, transferable from one molecule to another

* → if ρΩ identical in two different molecules

ρ identical → molecules identical

Exact transferability is impossible, but the search for local similarity remains important because the theorem does not quantify the degree of resemblance of two ρρρρΩΩΩΩ(r) functions still allowing considerable variations of ρρρρ(r) outside ΩΩΩΩ.

The shape function σσσσ(r)

• determines v(r), N → all information on the system.

• cf. - experimental set up of X-ray diffraction

- ρ(r) plots ← complex transformation of refraction data whose intensities

are measured relative to each other

(cf. E.D. Stevens, P. Coppens, Acta Cryst. A 31, 5122 (1975))

⇒ X-ray diffraction experiments initially yield the

shape function

direct search for minimalization of σ A(r)− σB(r)∫2dr

Pag.16-3-2009 38

direct search for minimalization of σ A(r)− σB(r)∫ dr

in Similarity Analysis

direct obtention of ZAB expression

( )( )A B

AB 1/22 2

A B

σ (r)σ (r)drZ =

σ (r)dr σ (r)dr

∫

∫ ∫

Combination of Holographic Theorem and the fundamental importanceof the Shape Function

Knowledge of the shape function in a finite but otherwise arbitrarysubdomain of a Coulombic system determines all properties of the system.

Information content of the shape function

Pag.16-3-2009 39

ρ(r) = σN(r)N

From Hohenberg Kohn to a holographic shape theorem

a) Hohenberg - Kohn

→ E = Ev ρ[ ]••

••

• •

••

compatibility

with a single

external potential

v(r)ρ(r) for a ground

state

••

v(r),N

electrons nuclei

Pag.16-3-2009 40

b) Mezey : Holographic density theorem

→ E = Ev ρΩ[ ]

ρΩ(r)

••

••

• •

••

Ω (finite, arbitrary subdomain)

ρ(r)

•• •

•

v(r),N

c) Ayers : shape function

••

••

• •

••

σ(r)

→ E = Ev σ[ ]

d) Combining (b) and (c) → holographic shape theorem

••

v(r),N

Pag.16-3-2009 41

••

••

••

• •

•• → E = Ev σΩ[ ]

σΩ(r) σ(r)

••

v(r),N

Conclusion : Similarity in shape is a fundamental issue to be looked at, both globally and locally

P. Geerlings, G. Boon, C. Van Alsenoy, F. De Proft, Int. J. Quant. Chem., 101, 722 (2005)

Case Study III: Information Theory in Reactions:

Kullback Leibler Information Profiles

• Investigation of information profiles for processes involving geometrical and energetical changes along a coordinate: internal rotation, vibrations, chemical reactions

• Kullbacks Leibler information deficiency along ξ

↑

( ) ( ) ( ) ( )( )0

0

σ r∆S σ r σ r = σ r ln dr

σ r ∫ξ

ξ

ξ

Pag.16-3-2009 42

↑Shape function for reference geometry

• Can energetic relationships be retrieved/complemented from an information theory based analysis of the density or shape function

Cfr. Chemical ReactionE

ξ

∆s?

ξ

ξ=IRC

An example: SN2 reaction OH- +CH3F → CH3OH + F-

• Reaction Profile

Pag.16-3-2009 43

• Take TS as reference

• ∆S[σξ(r)/σTS (r)]= distance in information between shape function of TS (σTS) and system along the reacton path (σξ)

By convention: ∆S=0 at ξ=0 (TS)

∆S profile

Comparison between energetical data and shape function information measure

forward∆E =≠7.85 kcal mol-1

reverse∆E =≠

44.60 kcal mol-1

[ ]R TS∆S σ ,σ =1.83

[ ]P TS∆S σ ,σ =3.44

TS contains more information about reactants than about products

Much lower reaction barrier for forward than for reverse reaction

Hammond’s postulate (“exothermic reaction: early TS;

Pag.16-3-2009 44

Hammond’s postulate (“exothermic reaction: early TS; endothermic reaction: late TS”)

retrieved in the information profile related to the shape function

Other examples ( rotation, vibration, …)

A.Borgoo, P.Jaque, A.Toro-Labbé, C.Van Alsenoy, P.Geerlings, PCCP, 11, 476 (2009)

Conclusions

• The electron density function as a carrier of information is challenged by an even simpler function : the shape function.The latter contains enough information for describing chemical reactivity and permits an extension of the Holographic Density Theorem

• Information theory based analysis of the shape function of atoms reveals the periodicity in Mendeleev’s table.

Pag.16-3-2009 45

• Information theory based Hirshfeld partitioning of the electron density of moleculespaves the way to discuss local similarity in molecules, and in particular dissimiarity of enantiomers and to numerical studies on the Holographic Density Theorem

• Information theory based reading of the Shape function along a reaction coordinate yields a companion to the energy profile of a reaction.

Acknowledgements

Prof. P. Ayers (McMaster-Canada)

Dr. G. Boon (VUB)

Prof. F. De Proft (VUB)

Prof. M. Godefroid (ULB)

Pag.16-3-2009 46

Prof. C. Van Alsenoy (Antwerp)

Prof. P. Bultinck (Ghent)

Prof. K.D. Sen (Hyderabad)

Prof. A. Toro-Labbé (Santiago, Chile)