Faculty of Science & Bio-Engineering SciencesResearch Group General Chemistry

The Linear Response Functionin Conceptual Density FunctionalTheory

Fundamental Aspects and Application to Atoms

Zino BoisdenghienThesis submitted in partial fulfillment of the requirements for the academic degreeof Doctor in Sciences

Academic Year 2015-2016

Promotors:Em. Prof. Dr. Paul GeerlingsProf. Dr. Frank De ProftDr. Stijn Fias

December 2015

ii

Acknowledgements

This thesis is the culmination of four years of research conducted in theGeneral Chemistry Group (Eenheid Algemene Chemie, ALGC) at the VrijeUniversiteit Brussel (VUB) under the guidance of Prof. P. Geerlings, Prof.F. De Proft and Dr. S. Fias.

First of all, I would like to thank my promotors to give me the opportunityto perform my research and their continued guidance and discussions thathave shaped my research over the last years. I would also like the thank Prof.C. Van Alsenoy and Dr. F. Da Pieve for their guidance and help during myPhD.

I would like to thank all of my colleagues at the VUB over the years forcreating a pleasant atmosphere to work in and for the interesting discussionsover coffee, kind words of encouragement and much more over the years.Mercedes, Fran, Freija, Songl, Jan, Thijs, Ann-Sophie; I am certain that Ihave made some friends for life (even if they do move to Boston).

I want to thank my friends who have always been there for me. Whether itwas letting of steam in the dojo (merci Sensei Jean en Sensei Oli en al mijnvrienden van de jiu) or over a couple of beers afterwards. I especially wantto thank Jan; copain, bedankt om er altijd te zijn.

Ik wil ook mijn familie bedanken voor alle steun en vertrouwen in mij.

Last but not least, I want to thank Elisa, the love of my life. Liefje, bedanktvoor alles. Zonder je steun de afgelopen 9 jaar (en zeker de laatste maanden)zou ik nooit zover zijn geraakt. Ik kijk er naar uit om de rest van mijn levenmet jou door te brengen.

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iv

Abstract

The research presented here is situated in the field of Conceptual DensityFunctional Theory (Conceptual DFT), a chemical reactivity theory rootedin Density Functional Theory (DFT). Density Functional Theory presents adescription of quantum mechanics that takes the electronic density as its cen-tral object rather than the wavefunction. The principal idea of ConceptualDFT is to define reactivity indices as (functional) derivatives of the energywhich can provide insight into the (inherent) reactivity of a system.

We will focus on one of those reactivity indices, specifically the linear re-sponse kernel which is defined as the second order functional derivative ofthe electronic energy w.r.t. external potential. Alternatively, it can be writ-ten as the first order functional derivative of the density w.r.t. the externalpotential, which provides us with the intuitive interpretation of the linearresponse kernel as the response of the density to changes in the externalpotential.

By taking a step back and focussing our attention on atoms we were able tostudy the linear response function in its own right whereas previous stud-ies have obtained numerical data by employing an atom-atom condensationscheme.

We evaluate and represent the (uncondensed) linear response function forhydrogen through argon using both the Independent Particle Approxima-tion as well as the Coupled Perturbed Kohn-Sham approach. The resultingfigures nicely illustrate the trends that the linear response function capturesthroughout the periodic table, such as the periodicity. We also investigatespin polarized versions of the linear response kernel, which provide insightin how or electrons react differently to perturbations in the or partsof the external potential.

The linear response kernel is closely related to the concept of polarizability.The relation between the linear response function and the polarizabilty alsoprovides us with a straightforward definition of the local polarizability, its

v

evaluation and its evolution throughout the periodic table. Upon integration,the polarizability and its trends throughout the periodic table are retrieved.

A final research line is to study the linear response kernel in Time DependentDFT using the Sternheimer equations, which form the time dependent ana-logue to the Coupled Perturbed Kohn-Sham equations. In the limit wherethe frequency tends to zero, this provides a direct comparison between thestatic and the frequency dependent linear response kernel. In practice wecompare static an dynamic linear response by calculating the static and dy-namic local polarizability for atoms and a single molecular system.

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Samenvatting

Het onderzoek dat hier wordt gepresenteerd situeert zich in het veld van Con-ceptuele Dichtheidsfunctionaaltheorie (Conceptuele DFT), een chemische re-activiteitstheorie die zijn oorsprong vindt in Dichtheidsfunctionaaltheorie(DFT). Dichtheidsfunctionaaltheorie verstrekt een beschrijving van kwan-tummechanica waar de electronendichtheidsfunctie centraal staat in plaatsvan de golffunctie. Het basisprincipe van Conceptuele DFT is het definirenvan reactiviteitsindices in termen van (functionele) afgeleiden van de energiedie inzicht kunnen verschaffen in de (inherente) reactiviteit van een systeem.

We focussen ons op n van deze reactiviteitsindices, namelijk de lineairerespons kernel, gedefinieerd als de tweede orde functionele afgeleide van de(electronische) energie t.o.v. de externe potentiaal. Een alternatieve om-schrijving van deze kernel is als de eerste orde afgeleide van de dichtheids-functie t.o.v. de externe potentiaal, die ons de intutieve interpretatie vande lineaire response kernel verschaft als het antwoord van de dichtheid opveranderingen in de externe potentiaal.

Door een stap terug te nemen en ons te focussen op atomen waren we in staatom de lineaire respons functie als dusdanig te bestuderen in tegenstelling totvorige studies waar numerieke data werden bekomen d.m.v. een atoom-atoomcondensatie procedure.

We evalueren en visualiseren de (ongecondenseerde) lineaire respons functievoor Waterstof t.e.m. Argon, gebruik makend van zowel de OnafhankelijkeDeeltjes Benadering als de Gekoppeld Geperturbeerde Kohn-Sham benader-ing. De resulterende figuren bieden een mooie illustratie van de tendensendie door de lineaire respons functie kunnen worden blootgelegd doorheende periodieke tabel, zoals bijvoorbeeld de periodiciteit. We bestuderen ookspingepolarizeerde versies van de lineaire response kernel, die ons inzicht ver-schaffen in hoe of electronen anders reageren op perturbaties van het of gedeelte van de externe potentiaal.

De lineaire respons kernel is nauw gerelateerd aan het concept van polariz-

vii

abiliteit. De relatie tussen de lineaire respons functie en de polarizabiliteitverschaft ons ook met een eenduidige definitie van de lokale polarizabiliteittesamen met de evaluatie en evolutie van deze lokale polarizabiliteit doorheende periodieke tabel. Na integratie verkrijgen we terug de polarizabiliteit ende tendensen hiervan doorheen de periodieke tabel.

Een laatste onderzoekslijn is de studie van de lineaire response kernel in tijd-safhankelijke DFT d.m.v. de Sternheimer vergelijkingen, het tijdsafhankeli-jke analoog van de Gekoppeld Geperturbeerde Kohn-Sham vergelijkingen. Inde limiet waar de frequentie naar nul daalt geeft dit ons een directe vergelijk-ing tussen de statische en de frequentie afhankelijke lineaire response kernel.In de praktijk vergelijken we de statische en de dynamische lineaire responsdoor de statische en dynamische lokale polarizabiliteit te berekenen vooratomen en een moleculair systeem.

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List of Abbreviations

B3LYP Becke three-parameter hybrid functionalwith the Lee-Yang Parr-correlation functional

CPKS Coupled Perturbed Kohn-ShamDFT Density Functional TheoryHF Hartree-FockHK Hohenberg-KohnKS Kohn-ShamMO Molecular OrbitalPBE Perdew, Burke and ErnzerhofpVTZ polarized valence triple-zeta basis setSD Slater determinantTDDFT Time Dependent Density Functional TheoryVWN Vosko, Wilk and Nusair

(LDA correlation functional)xc exchange-correlation

ix

x

Publication List

The following is a list of publications containing the work presented in thisthesis as well as some additional results.

1. Evaluating and Interpreting the Chemical Relevance of the Linear Re-sponse Kernel for Atoms, Z. Boisdenghien, C. Van Alsenoy, F. DeProft, P. Geerlings, J. Chem. Theor. Comp., 2013, 9, 1007.

2. Analysis of aromaticity in planar metal systems using the linear re-sponse kernel, S. Fias, Z. Boisdenghien, T. Stuyver, M. Audiffred, G.Merino, P. Geerlings, F. De Proft, J. Phys. Chem. A, 2013, 117, 3556.

3. Conceptual DFT: Chemistry from the Linear Response Function, P.Geerlings, S. Fias, Z. Boisdenghien, F. De Proft, Chem. Soc. Rev.,2014, 43, 4989.

4. Evaluating and Interpreting the Chemical relevance of the Linear Re-sponse Function for Atoms II: Open Shell, Z. Boisdenghien, S. Fias, C.Van Alsenoy, F. De Proft, P. Geerlings, Phys. Chem. Chem. Phys.,2014, 16, 14614

5. The Spin Polarised Linear Response from Density Functional Theory:Theory and Application to Atoms, S. Fias, Z. Boisdenghien, F. DeProft, P. Geerlings, J. Chem. Phys., 2014, 141, 184107

6. The Local Polarizability of Atoms and Molecules: a Comparision Be-tween a Conceptual Density Functional Theory Approach and TimeDependent Density Functional Theory, Z.Boisdenghien, S.Fias, F.DaPieve, F.De Proft, P.Geerlings, Mol. Phys., 2015, 113, 1890

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xii

Contents

Acknowledgements iii

Abstract v

Samenvatting vii

List of Abbreviations ix

Publication List xi

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Theoretical Background 9

2.1 Many Body Quantum Mechanics . . . . . . . . . . . . . . . . 9

2.2 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Configuration Interaction . . . . . . . . . . . . . . . . 16

2.2.2 Exchange and correlation energy . . . . . . . . . . . . 17

2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 18

2.3.1 The Particle Density . . . . . . . . . . . . . . . . . . . 19

2.3.2 The Hohenberg-Kohn Theorems . . . . . . . . . . . . . 21

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2.3.3 Constrained Search . . . . . . . . . . . . . . . . . . . . 25

2.4 A note on functional derivatives . . . . . . . . . . . . . . . . . 26

2.5 Kohn-Sham Theory . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Linear Response 29

3.1 Conceptual Density Functional Theory . . . . . . . . . . . . . 29

3.2 Mathematical background . . . . . . . . . . . . . . . . . . . . 33

3.3 Evaluation of (r, r) . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.1 Numerical evaluation . . . . . . . . . . . . . . . . . . . 34

3.4 A perturbational approach to the linear response kernel . . . 36

3.4.1 Hartree-Fock . . . . . . . . . . . . . . . . . . . . . . . 36

3.4.2 Matrix Formulation . . . . . . . . . . . . . . . . . . . 39

3.4.3 Kohn-Sham . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Evaluation and Graphical Representation of the Linear Re-sponse Kernel 47

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 The Independent Particle Approximation . . . . . . . . . . . 48

4.2.1 General remarks . . . . . . . . . . . . . . . . . . . . . 51

4.3 Systematic Excursion throughout the periodic table . . . . . . 54

4.3.1 One dimensional plots . . . . . . . . . . . . . . . . . . 55

4.3.2 Functional and Basis set dependence . . . . . . . . . . 59

4.3.3 Two dimensional plots . . . . . . . . . . . . . . . . . . 60

4.3.4 Isoelectronic series . . . . . . . . . . . . . . . . . . . . 64

xiv

4.4 Spin polarized Linear Response using the Coupled PerturbedKohn-Sham approach . . . . . . . . . . . . . . . . . . . . . . . 68

4.4.1 General Theory . . . . . . . . . . . . . . . . . . . . . . 68

4.4.2 Analytical expressions for the spin polarized linear re-sponse functions in the [N, N] representation. . . . 71

4.4.3 Switching between both representations . . . . . . . . 74

4.5 Graphical representation of the linear response kernel in the[N, N] and [N,Ns] representation . . . . . . . . . . . . . . . 76

4.5.1 The noble gasses . . . . . . . . . . . . . . . . . . . . . 77

4.5.2 The [N, N] representation throughout the PeriodicTable . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5.3 The [N,Ns] representation throughout the PeriodicTable . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.6 Connection to polarizability . . . . . . . . . . . . . . . . . . . 94

4.6.1 Total polarizability . . . . . . . . . . . . . . . . . . . . 94

4.6.2 Local Polarizability using Coupled Perturbed Kohn-Sham . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.6.3 Spin polarized version . . . . . . . . . . . . . . . . . . 98

4.7 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5 TDDFT 105

5.1 Introduction and Reading Guide . . . . . . . . . . . . . . . . 105

5.2 The Runge-Gross theorem . . . . . . . . . . . . . . . . . . . . 106

5.3 Time Dependent Kohn-Sham Equations . . . . . . . . . . . . 109

5.4 Action Principle and the Causality Paradox . . . . . . . . . . 111

5.5 Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.5.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

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5.5.2 A Perturbation Theoretical Expression for the LinearResponse Kernel . . . . . . . . . . . . . . . . . . . . . 117

5.5.3 Switching to Frequency Space . . . . . . . . . . . . . . 121

5.5.4 The Sternheimer approach . . . . . . . . . . . . . . . . 122

5.6 Link to the Polarizability density . . . . . . . . . . . . . . . . 127

5.6.1 Local polarizability for atoms . . . . . . . . . . . . . . 128

5.6.2 Local Polarizability for Molecules . . . . . . . . . . . . 130

6 Conclusions 135

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Chapter 1

Introduction

As soon as 1927, only one year after Erwin Schrdinger published the waveequation that governs quantum systems, Walter Heitler and Fritz Londonapplied quantum mechanics to the diatomic hydrogen molecule [1]. Thisidea, to apply quantum mechanics to chemical systems and concepts (in theircase the chemical bond), is based on the fact that at its core, the properties ofatoms and molecules should be based on the interaction of quantum objects(specifically, electrons and nuclei). In other words, chemistry at its core is aquantum theory.

In fact, the history of the development of quantum mechanics is closely in-tertwined with the history of theoretical chemistry. A prime example is theBohr model for the hydrogen atom which was published in 1913 and wasdeveloped to be consistent with experimental data available for the atomicemission spectrum. This observation was one of the pillars of what is nowknown as the old quantum theory, a collection of models and results de-veloped in the first quarter of the 20th century that form a precursor forthe self-consistent and more complete quantum theory that starts with theSchrdinger equation.

To make sense of the huge amount of available experimental data, chemistshave been searching for easily understandable qualitative models, conceptsand principles that are capable of categorising and predicting propertiesof molecules, in particular chemical reactivity. Examples of such conceptsand principles include the Lewis-dot structures [2], orbital hybridisation andresonance [3], electronegativity [4, 5], frontier (highest occupied and lowestunoccupied) molecular orbital (MO) concepts [6], the Woodward-Hoffmannrules [7], etc. Due to the complicated nature of quantum mechanics though,most concepts have been derived only in highly approximate contexts and

1

tend to lose their validity when moving to more quantitatively sound ap-proaches [8, 9]. The question is thus: can we find chemical descriptors thatemerge from a theoretically rigorous basis.

In the case of chemical reactivity descriptors these efforts culminated in Con-ceptual Density Functional Theory [815], a chemical reactivity theory thatstems from a theoretically sound model while retaining qualitative intuition.After the introduction of quantum mechanics, it soon became clear that al-though it could nicely explain the hydrogen atom, once you try to describelarger systems it becomes nearly impossible to use. After all, even for rela-tively small systems such as the neon atom or the water molecule, we needto describe 10 electrons which means that the wavefunction depends on 30variables (40 if we include spin). In 1964 Pierre Hohenberg and Walter Kohnproved that you dont need to use the wavefunction with its monstrous num-ber of variables, you can get all the information you need by just using theelectronic density function [16]. This marked the beginning of Density Fun-tional Theory (DFT) [10, 17]. It is this reformulation of quantum mechanicsthat forms the theoretically sound basis for Conceptual DFT.

Conceptual DFT is a branch of chemistry that developed alongside regularDFT and primarily focussed on the formalization of several chemical conceptsand ideas that at the time lacked a strong theoretical foundation. It doesso by introducing reactivity indices, which are defined as (mixed) derivativesof the electronic energy E[N, v] w.r.t. the number of electrons N and/or theexternal potential v due to the nuclei. As chemical reactions can be thoughtof as perturbations of the system in either the number of electrons and/or inthe external potential, it is natural to start from the quantity E[N, v]. Thereactivity indices defined in this way can be used to probe the (inherent)chemical reactivity of a system. Because of the link to DFT however, weavoid the increasing complexity that troubles wavefunction based theorieswhen aiming for increasing accuracy. Some examples of reactivity indicesthat arise in Conceptual DFT include the electronic chemical potential [14](which turns out to be related to the electronegativity), the Fukui functions[18] (related to frontier MO indices), the chemical hardness [19, 20] and theelectrophilicity [21]. Spin polarized versions of Conceptual DFT indices [22]have also been introduced, which have been successfully used to describeopen shell systems, radical chemistry [2326], metal complexes etc.

In theory, in order to achieve a complete and accurate description of a chem-ical reaction, one must have knowledge of the (electronic) structure of allthe reagents and products as well as any transition states along the reactionpath. To complicate matters even more, this must possibly be known in thepresence of a solvent. On the other hand, from observation one can see thatspecific molecules will interact in similar ways for a range of reagents. This

2

presence of a systematic trend leads to the idea that it should be possible tocharacterize the chemical behaviour of molecules in response to perturbationswithout explicit reference to their partner reagents [8, 27]. Conceptual DFTas a reactivity theory does not presume to reach the same level of sophistica-tion as the first approach, which corresponds to a complete calculation of thepotential energy surface, but rather takes the second approach. It tries tocapture the essence of chemical processes by introducing a (hopefully small)number of relatively easily computable indices.

Much of the research in our group has focussed on the study, refinementand extension of reactivity descriptors and conceptual DFT in general as avalid chemical reactivity theory. The work performed during my PhD andpresented in this thesis can be seen in this tradition. Previous work in ourgroup has focussed on the (numerical) calculation of functional derivativeswith respect to the external potential, specifically the linear response kernel(r, r), which is defined as the second order derivative of the electronicenergy w.r.t the external potential.

The importance of the linear response kernel lies in its ability to measurehow the electron distribution reacts to a small change in the external poten-tial. Indeed, from its definition given above we can see it is also given by thefirst order derivative of the electronic density w.r.t. the external potential.Its importance in Conceptual DFT is highlighted as it represents the answerof the electronic density (r) at position r to an external potential pertur-bation at position r at constant number of electrons (i.e. ((r)/v(r))N ),a situation which is at the heart of understanding the course of a chemicalreaction. The linear response kernel is also in a very clear way related tothe polarizability through double integration. Intuitively, the link to the po-larizability is clear: an external field will generate a dipole moment becausethe electrons get shifted from their usual location. This moment is (as longas the perturbing field is not too strong) approximately proportional to thefield itself, with the proportionality constant being the polarizability. Asthe linear response kernel provides a measure of how the electronic densityresponds to changes in the external potential, it is intuitively clear that thepolarizability and the linear response kernel are linked. The huge benefit ofthis link however is that it also provides us with a straightforward definitionof the local polarizability, an object that is not so unambiguously defined asthe global polarizability. This link to the local and global polarizability willbe further explored in this thesis.

Even though we have highlighted the importance of the linear response ker-nel, it has received relatively little attention compared to some of the otherindices. It has been discussed in certain more formal works, for example inSenet [28, 29], which focussed on the exact relation between linear (and non-

3

linear) response functions and the ground state electronic density in termsof the universal Hohenberg-Kohn functional Fhk[] (vide infra). The workdone by Cohen et al. [30], Ayers and Parr [31], and Ayers [32] focussed onthe properties of these functions within a Kohn-Sham framework. For anoverview of some of the mathematical properties, we refer to Liu et al. [33].

Beside the more formal work mentioned above, relatively little attention hasbeen paid to the actual calculation and more importantly the interpretationof the chemical and physical information contained in the linear responsefunction. For example, the previous work performed in our group focussedon the calculation of the linear response kernel using numerical methods andextracting qualitative and quantitative information about inductive, reso-nance and hyperconjugation effects. In order to do so, a discretized versionof the linear response kernel, integrated over atomic domains in order tocreate a linear response matrix AB was used [3437],

AB =

VA

VB

drdr (r, r). (1.1)

Some works on the chemical information contained in these atom condensedlinear response matrices are Baekelandt et al. [38] and Wang et al. [39], whichuse highly approximate semi-empirical schemes and Morita and Kato [40, 41]using coupled perturbed Hartree-Fock theory.

Another area of research that the linear response kernel has been successfullyapplied to is the study of aromaticity [4244].

The research presented here can be seen in this context but offers a freshperspective on the linear response kernel. Our aim was to focus on thelinear response function (r, r) itself, without resorting to integration overatomic domains, in order to extract the physically and chemically relevantinformation contained in this function. To this end we decided to take a stepback and focus on single atoms while performing a systematic study of thelinear response kernel throughout the periodic table.

It is worth mentioning that the linear response kernel is also linked to someinteresting concepts, whose importance in quantum chemistry is still grow-ing. The first of those concepts is nearsightedness of electronic matter, in-troduced by Kohn [45], Prodan and Kohn [46]. What this concept tells us isthat in systems with many electrons at constant electronic chemical poten-tial, the change in the electron density at a point r induced by a perturbationin the external potential at a point r, |(r0)|, where |r r| > R, will al-ways be smaller than a maximum value (r, R) and this is independent ofthe size of the perturbation v(r). The language used here implies a closeconnection between the concept of nearsightedness and the linear response

4

kernel. We can therefore introduce the quantity((r)

v(r)

)

(1.2)

The only difference being the condition of constant electronic chemical po-tential , which indicates that this is quantity is naturally defined whenworking in an open system with a grand potential [, v] performing thesame role as the energy functional E[N, v] in Conceptual DFT. Indeed, theinitial development of Conceptual DFT was very reminiscent of classicalchemical thermodynamics, where the canonical ensemble (at 0K) uses theelectron number N and the external potential v(r) as the basic variables.The basic variables can be changed through Legendre transformations. Thegrand canonical ensemble is then obtained by introducing the grand potential = [, v] defined through

= E N (1.3)

which exchanges the number of electrons for the chemical potential as a basicvariable.

It turns out that the quantity defined above in eq. (1.2) is minus the softnesskernel s(r, r) which appears as the natural counterpart to the linear responsekernel in open systems described by a grand potential [, v]. The linkbetween both quantities is given by the rules of differentiation with differentconstraints: (

(r)

v(r)

)N

=

((r)

v(r)

)

+

(

)v

(

v

)N

(1.4)

or(r, r) = s(r, r) + s(r)s(r

)

S, (1.5)

where we introduced the total softness. This last equation is the famousParr-Berkowitz relation [47], whose role in quantifying the concept of near-sightedness is presently investigated in our group [48].

A second concept that is closely linked to the linear response kernel is theconcept of alchemical derivatives [49], i.e. derivatives of the energy w.r.t. oneor more nuclear charges. We think of this as the linear response functionwhere the perturbation v(r) results from a nuclear charge variation. Thefirst order derivative is given by(

EelZ

)N

=

dr

(Eelv(r)

)N

(v(r)

Z

)N

=

dr(r)

1|rR|

, (1.6)

5

which is nothing else than the electronic part of the Molecular ElectrostaticPotential [50] (MEP) at position of nucleus . At second order, the mixedderivative gives us

(2Eel

ZZ

)N

=

drdr

(2Eel

v(r)v(r)

)N

v(r)

Z

v(r)

Z

=

drdr(r, r)

1

|rR|1

|rR|, (1.7)

where we retrieved the linear response function. Recent work on the use ofalchemical derivatives have been delivered by ALGC members in collabora-tion with R. Balawender [51] and A. Von Lilienfeld [52].

Both research lines clearly illustrate the important position of the linearresponse kernel in Conceptual DFT.

Everything we have mentioned up to now dealt with static systems, describedby the stationary Schrdinger equation. The natural question that arises ishow one can extend this to the dynamic, time-dependent case. After all,dynamical processes are abundant in nature; there is nothing particulary in-teresting about a system that continues to sit in its groundstate. Just as theSchrdinger equation can be extended to a time dependent variant, DensityFunctional Theory can be extended to Time Dependent Density FunctionalTheory (TDDFT). TDDFT has been succesfully applied to calculate andpredict excited-state properties in chemistry as well as solid state physicsand even biophysics. Examples of situations where TDDFT is used includethe calculation of photo-absorption cross sections of molecules and nanos-tructures, the response of systems to either a weak or intense laser field,van der Waals interactions, chromophores in biophysics, optical propertiesof solids etc. [53, 54]

Contrary to DFT, linear response calculations are commonplace in TDDFT.They are most frequently used however to calculate excitation energies [5557], which correspond to the poles of the frequency dependent linear responsefunction. The most commonly known method to perform these linear re-sponse calculations is using the Casida equations. The linear response kernelis rarely studied in its own right however, and the extension of our previouswork to the time dependent domain was much less straightforward as onemight expect. It was our goal to examine the linear response kernel itselfand focus on the chemical information that is contained in it.

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1.1 Overview

In chapter 2 we provide a brief overview of many electron quantum mechan-ics, mainly focussing on a single determinantal description. We will mainlybe interested in stationary systems with definite energy levels, which aredescribed by the stationary Schrdinger equation. We first describe somefundamental many-body quantum mechanics techniques, such as Slater de-terminants and Hartree-Fock (HF) theory. We will then see how the com-plicated nature of the many body wavefunction indeed appears to possesstoo many degrees of freedom than are physically necessary. What follows isan introduction to Density Functional Theory, starting with an overview ofthe proof of the Hohenberg-Kohn theorem that provides the theoretical basisfor DFT. Also introduced in this chapter is the Kohn-Sham (KS) approach,which reintroduces orbitals in order to make the calculations easier.

Chapter 3 introduces Conceptual Density Functional Theory and the reac-tivity indices that appear in it. As describe above, one of these indices iscalled the linear response kernel and is the main focus of this thesis. Wegive a brief overview of a numerical evaluation method that was used inprevious work from our group, followed by a method to (analytically) cal-culate the linear response kernel using perturbation theory at various levelsof approximation, from the independent particle approximation to the fullcoupled perturbed description. The theory is first introduced using Hartree-Fock theory but later extended to Kohn-Sham theory. An important pointto note is that the derivation and structure of the equations are nearly thesame for Hartree-Fock and Kohn-Sham, only differing in two points. Firstof all, the orbitals involved are obviously different, being either HF or KSorbitals. The second difference occurs at the coupled perturbed level, whereHartree-Fock theory incorporates exact exchange whereas Kohn-Sham the-ory incorporates exchange-correlation. In the limit where we approximatethe xc-potential by exact exchange however, the equations become exactlythe same, only differing in the orbitals that are used. In order to trans-late these equations to a more computationally friendly language, a matrixformulation is introduced through the use of basis functions that are theproduct of two orbitals.

In the following chapter, chapter 4, we discuss the results of my research[5861], which focusses on the systematic evaluation and visualization ofthe linear response function throughout the periodic table. We introduceboth two dimensional contour plots and one dimensional plots for Hydrogenthrough Argon and discuss the trends and the chemical information that iscontained in these plots. In this chapter we also introduce spin polarizedversions of the linear response kernel in two different representations and

7

discuss how to switch between them. Finally we make the link to both thetotal and the local polarizability tensor.

The extension of DFT to TDDFT is introduced in chapter 5. We begin, aswe did for DFT, by giving a brief overview of the theory behind TDDFT,specifically the theorems by Runge and Gross [62] that allow us to define adensity based time dependent theory. Once we have our theoretical basis fora density based theory, we reintroduce orbitals through the time-dependentKohn-Sham equations [63]. After a quick detour to address some concernsthat might arise dealing with causality, we introduce the equations for thelinear response kernel that are parallel to the ones we introduced in chapter 3,both at the Independent Particle level as well as using the Sternheimer ap-proximation, the time-dependent equivalent of the coupled-perturbed Kohn-Sham expression we found in DFT. We conclude by again looking at the linkto the (local) polarizability, this time in the time-dependent case.

8

Chapter 2

Theoretical Background

2.1 Many Body Quantum Mechanics

The evolution of a general quantum state |, represented here by a vector inan abstract Hilbert space representation H, is governed by the HamiltonianH of the system through the Schrdinger equation1

i

t| = H|. (2.1)

Note that since we will only be interested in the electronic problem, we willuse the Born-Oppenheimer [64] to separate the electronic problem from thenuclear one. We will also strictly work in the non-relativistic regime.

For the most part, we will be interested in stationary states with definiteenergies, for example the ground state (see chapter 5 for time-dependentsystems). These states are solutions to the eigenvalue equation

H| = E| (2.2)

known as the stationary Schrdinger equation. Alternatively, stationarystates can be obtained by varying |H| w.r.t. | subject to the con-straint | = 1. The eigenvalue E then appears as a Lagrange multiplierfor the constraint.

We are interested in systems of N identical particles moving in a givenexternal field v. In such a case, the Hamiltonian takes on the form

H = T + V + W (2.3)1We are working here in atomic units, in which ~ = m = e = 1.

9

where T denotes the kinetic energy operator, V describes the interaction ofthe particles with the external field, and W is the two-particle interactionoperator. We will usually work in the Schrdinger representation of theHilbert space, which uses eigenstates of the coordinate operator r and of thez-component of the spin operator as basis vectors:

r|r = r|r, z|s = s|s. (2.4)

We introduce a combined spin-position variable, x = (s, r). In this rep-resentation, the N -particle quantum state | is represented by an L2-wavefunction

(x1, . . . , xn) = x1 xn|. (2.5)

Here, x1 xn| are the elements of the dual basis on H.

Remember that the L2 space is the collection of all functions f for which(dx |f(x)|2

)1/2< . (2.6)

We note that for fermions, | must be taken from the anti-symmetric sectorof the N -particle Hilbert space in question, i.e. it must satisfy

(x1, . . . , xi, . . . , xj , . . . , xn) = (x1, . . . , xj , . . . , xi, . . . , xn). (2.7)

Bosonic systems on the other hand are elements of the symmetric sector, i.e.they are invariant under the exchange of two particles.

To connect the formal Hamiltonian defined on the formal N -particle Hilbertspace, eq. (2.3), to the Schrdinger representation of the N -particle Hilbertspace of wave functions, we write

H(x1, . . . , xn) = x1 xn|H|, s (2.8)

where the operator H on the left hand side is the Hamiltonian acting on wave-functions and the right hand side contains the abstract Hilbert space opera-tor. Explicitly, the Hamiltonian acting on wavefunctions in the Schrdingerrepresentation is given by (still using atomic units)

H = 12

Ni=1

2i +Ni=1

v(xi) +Ni

In general, let {|L} be a complete orthonormal set of N -particle states. Ageneral element of our Hilbert space of states can be expanded as

| =L

|LL| =L

|LCL, (2.10)

where CL is the projection of the eigenstate onto a basis vector |L. Insertingthe expansion of the eigenstate into the Schrdinger equation eq. (2.2) andprojecting into the chosen representation leaves us with

K

(HLK ELK)CL = 0, (2.11)

whereHLK = L|H|K are the matrix elements of the Hamiltonian operator.

Switching back to the Schrdinger representation, suppose we have a com-plete set of spin-orbitals (single particle wavefunctions) {}, i.e. a set whichsatisfies

k

k(x)k(x

) = (x x). (2.12)

From any given selection of N spin-orbitals chosen from among this set,we can form an anti-symmetric N -electron wavefunction by taking a Slaterdeterminant:

L(x1, . . . , xn) =1N !

det(i(xk)). (2.13)

Here L is a now multi-index (1 n) denoting a specific orbital configura-tion. We can write this determinant as

L(x1, . . . , xn) =1N !

Sn

Ni=1

sgni(x(i)) (2.14)

=1N !

(Ni=1

i

)(x1, . . . , xn) (2.15)

where SN denotes the symmetric group of order N . In the last line wehave borrowed a notation common in differential geometry, where the wedgeproduct is the alternating tensor product of k-covectors on a vector space.Since we are dealing with linear functionals here, i.e. 1-covectors, the tensorproduct reduces to an ordinary product of functions.

In this case, we can express the matrix elements HLK = L|H|K interms of the spin-orbitals. Note that the Hamiltonians we are interested inwill consist of (sums of) one- or two-particle operators. In general, we write

O1 =

Ni=1

o1(xi) (2.16)

11

for a one-electron operator and

O2 =

Ni

electron one, the matrix element will be zero as well since the spin-orbitalsare orthonormal. Thus we are just left with

L|O1|K =N

N !

Sn1

dx1 dxn (x1)

Ni=2

i(x(i))o(x1)k(x1)

Ni=2

i(x(i))

=

dx1 (x1)o1(x1)k(x1)

= |o1|k. (2.23)

In the case that K and L differ in more than one index, we can immediatelysee that the matrix elements will be zero. As in the previous cases, theintegration over electrons 2 through N will force = . Then in the previouscase, since the two orbitals which were different are also orthogonal, thematrix element would be zero unless we force electron one to be in thoseorbitals. Now, however, we have two pairs of orthogonal orbitals remaining,and while we can force the integration over one pair of those to be nonzeroby putting electron one in it, the integral over the other pair will still yieldzero.

We can use similar reasoning to calculate matrix elements for two-electronoperators. In that case however, matrix elements will only be zero onceK and L differ in three or more indices, since - following the reasoning ofthe last paragraph - we can avoid integration over two pairs of orthogonalorbitals to give zero by putting electrons one and two in those pairs, butintroducing a new pair of orbitals will give zero yet again.

We summarize the results in table Table 2.1. Here, we introduce the notationfor two electron integrals:

ij|o2|k =

dxdx i (x)j (x

)o2(x, x)k(x)(x

). (2.24)

Note that in the specific case where o2 is the Coulomb interaction, o2(xi, xj) =r1ij , we will drop the operator from this expression and just write

ij|k =

dxdx i (x)j (x

)1

|r r|k(x)(x

). (2.25)

In particular, for the energy E = L|H|L we have

E =

Ni=1

i|h|i+1

2

Ni,j=1

(ij |w|ij ij |w|ji) (2.26)

13

Table 2.1: Matrix elements of one- and two-electron operators between Slater De-terminants expressed in terms of the constituent spin-orbitals.

One-electron operatorsL = K L|O1|L =

iLi|o1|i

L = (, 2, . . . , n), K = (k, 2, . . . , n) L|O1|K = |o1|kK and L differ by more than one entry L|O1|K = 0Two-electron operatorsL = K L|O2|L = 12

i,jL

(ij |o2|ij ij |o2|ji

)L = (, 2, . . . , n), K = (k, 2, . . . , n) L|O2|K =

iL

(i|o2|ki i|o2|ik

)K = (, , 3, . . . , n), K = (k, k, 2, . . . , n) L|O2|K = |o2|kk |o2|kkK and L differ by more than two entries L|O2|K = 0

2.2 Hartree-Fock Theory

For an interacting system, one cannot expect a single Slater determinant toaccurately describe the ground state. However, since the ground state can bedescribed using a variational principle, it is natural to ask, given a set of spin-orbitals, which Slater determinant best approximates the true N -particleground state, i.e. which Slater determinant minimizes the expectation valueof H. This minimum will give an upper limit to the exact ground stateenergy. When the number of given spin-orbitals climbs to infinity, the upperlimit will converge to what is known as the Hartree-Fock limit E0.

We should note that while in most cases the true ground state will have adefinite total spin S, a Slater determinant does - in general - not have adefinite total spin. Assume for the moment that the ground state is a closedshell state (which implies that N must be even). Given two spin one-halfparticles, the only way to obtain a spin zero state is if the spin part of thestate is the antisymmetric combination

s1s2|S = 0 =12

(+(s1)

(s2) (s1)+(s2))

(2.27)

where represent the spin part of a spin-up or -down spin-orbital respec-tively. The two particles may then occupy the same spatial orbital withoutviolating the antisymmetry principle. A product of N/2 particle pairs withspin-states as described here will therefore give a state with total spin S = 0,which is antisymmetric w.r.t. particle exchange within a pair and at the sametime symmetric w.r.t. pair exchange. Extending this to a Slater determinantof spin-orbitals, with each orbital doubly occupied, will lead to a spin-zerostate satisfying the correct antisymmetry.

Taking such a Slater determinant | and still using the shorthand |i for

14

the orbital i(x), one can see that using eq. (2.26)

E = |H| = 2N/2i=1

i|h|i+ 2N/2i,j=1

ij|w|ij N/2i,j=1

ij|w|ji, (2.28)

where we call the first term the one-particle energy, the second one theHartree energy and the last one the exchange energy. The Hartree term, ifw(rij) equals the Coulomb repulsion, r1ij , is the classical Coulomb repulsionterm. The exchange term on the other hand is a purely quantum mechanicalterm which has no classical analogue.

The next step is to minimize this expression under variation of the orbitalswith the constraint that they must remain orthonormal. Introducing La-grange multipliers k leads to the condition2

(hk)(r) + (vHk)(r) + (vxk)(r) = k(r)k, (2.29)

where the Hartree operator boils down to multiplication with the Hartreepotential,

vh(r) = 2

N/2j=1

dr j (r

)w(|r r|)j(r), (2.30)

and the effect of the exchange potential operator is given by

(vxk)(r) = N/2j=1

dr j (r

)w(|r r|)k(r)j(r). (2.31)

We can write the Hartree-Fock equations eq. (2.29) in short as

F k = kk, (2.32)

which takes the form of an effective one-particle Schrdinger equation withthe Fock operator F taking the role of the Hamiltonian,

F = 122 + veff , (2.33)

with an effective potential veff = v + vh + vx called the mean field.

The Hartree-Fock method gives us a set of spin-orbitals {k} with energiesk. The Hartree-Fock ground state |0 is the determinant formed from theN spin-orbitals with the lowest orbital energies (called the occupied orbitals).

2as before we will interpret the Lagrange multiplier as an energy, specifically the orbitalenergy.

15

The remaining spin-orbitals are called virtual orbitals. From eq. (2.29) wesee that

N/2i=1

i =

N/2i=1

i|h|i+ 2N/2i,j=1

ij|w|ij N/2i,j=1

ij|w|ji. (2.34)

This leads to

Ehf =

N/2i=1

(i + i|h|i) = 2N/2i=1

i W , (2.35)

which is the sum over all occupied orbital energies i minus the double-counted interaction energy.

In theory, the set {k} is infinite. In practice, one solves the HF equations byintroducing a set of spatial basis functions {| = 1, . . . ,K}, which leadsto a set of 2K spin-orbitals, N of which will be occupied and 2K N ofwhich will be virtual. Of course, as K , the HF energy E0 = 0|H|0will converge to a lowest bound called the Hartree-Fock limit.

2.2.1 Configuration Interaction

Given a set of 2K spin-orbitals obtained from the HF procedure, {}, theHF ground state is only one of the possible

(2KN

)determinants that can

be formed. The other possible determinants can be described by how theydiffer form the HF ground state. For example, if we relabel the HF spin-orbitals in order of ascending energy, the ground state corresponds to thespecific configuration L = (1 ij N). A singly excited determinant thencorresponds to a configuration L = (1 ai N), i.e.

|L = |1 ai n, (2.36)

a doubly excited determinant to a configuration L = (1 ab N), etcetera up toN -tuply excited determinants. These excited determinants serveas N -electron basis functions to expand exact N -electron states in,

| = c0|0+L

cL |L+

L

cL |L+ . . .+

LN

cLN |LN , (2.37)

where the summations run over all unique excitations. In this expression,LN refers to an N -tuply excited configuration. The exact energies of theground and excited state are then given by the eigenvalues of the Hamilto-nian matrix L|H|K, the lowest eigenvalue corresponding to the exactground state energy E03. Since each excited determinant is specified by a

3Or rather, exact within the Born-Oppenheimer approximation and without takinginto account relativistic effects.

16

certain configuration of spin-orbitals this technique is called configurationinteraction.

As in theory the number of spin-orbitals is infinite, the number of exciteddeterminants is also infinite. However, even if we only use a finite basisto expand the spin-orbitals in, the number of excited determinants quicklybecomes to large to handle. In the case one does use all possible exciteddeterminants we call that procedure full CI. Note that for finite K, the

(2KN

)determinants dont form a complete basis set, but diagonalizing the Hamil-tonian matrix formed with these determinants leads to solutions that areformally exact within the subspace spanned by these determinants (or alter-natively, within the one-electron subspace spanned by the 2K spin-orbitals).

To calculate exact energies using CI, we need to diagonalize the Hamilto-nian matrix, which means we have to calculate matrix elements of the formL|H|K. See the results summarized in table Table 2.1

2.2.2 Exchange and correlation energy

Assume for a moment that the Hamiltonian H is the simple sum of one-electron Hamiltonians4 and look at an N -electron wavefunction that isjust the product of spin-orbitals, rather than the antisymmetric product.This wavefunction is again an eigenfunction of H with eigenvalue E =

i.

However, this wavefunction is uncorrelated, i.e. the probability density ||2is simply the product of the individual probability densities of each orbital:

|(x1, . . . , xn)|2 = |i(x1)|2|j(x2)|2 |k(xn)|2. (2.38)

Aside from the obvious lack of antisymmetry, there is another reason why thiswavefunction is not appropriate to describe electrons. Since the wavefunctionis uncorrelated, the probability of finding electron 1 at any given point inspace is independent of the position of electron 2. Physically however, bothelectrons will repel each other and electrons will spatially avoid each other.This electron-electron interaction makes the motion of electrons correlated.

Introducing Slater determinants introduces exchange effects. Specifically, fora two electron system where the electrons have parallel spin

d1d2 ||2 =1

2

(|1(r1)|2|2(r2)|2 + |1(r2)|2|2(r1)|2

(1(r1)2(r1)2(r2)1(r2) + 1(r1)2(r1)2(r2)1(r2))), (2.39)

4Note that this is more general than setting W = 0 since we could include two-particleeffects in an average way.

17

where the extra cross term introduces correlation. Setting r1 = r2 showsthat indeed the probability of finding two electrons at the same point inspace with parallel spins is zero. Note that it is the motion of electrons withparallel spins that becomes correlated - electrons with anti-parallel spinsremain uncorrelated: for electrons with opposite spins we find

d1d2 ||2 =

1

2

(|1(r1)|2|2(r2)|2 + |1(r2)|2|2(r1)|2

). (2.40)

Setting r1 = r2 here yields a non-zero result, so the probability of findingtwo electrons with opposite spins at the same point in space is non-zero.In general we say that the determinantal wavefunction is an uncorrelatedwavefunction if only the motion of electrons with parallel spin is correlated.

In section 2.2.1 we have seen that the exact energies of the states of a systemare the eigenvalues of the Hamiltonian matrix, with the lowest one being theexact (non-relativistic and within the Born-Oppenheimer approximation)ground state energy E0. Since Hartree-Fock only incorporates exchange, wedefine the correlation energy of the system, Ecorr as

Ecorr = E0 E0. (2.41)

2.3 Density Functional Theory

The electronic wavefunction is a monstrous object. With its 4N degrees offreedom, it quickly becomes very difficult to handle. A natural question thatarises is thus if we really need all those degrees of freedom. We have seen thatin the calculation of matrix elements for one- and two-electron operators oneintegrates out almost all coordinates up to a few; when two determinants dif-fer by three or more orbitals the resulting matrix element will always be zero.As for our level of description one- and two-electron operators are typicallysufficient this already indicates a number of redundant degrees of freedom.What is more, physically relevant values (numbers we can actually measure)are given by expectation values of operators. The integration involved infinding these reduces the number of physically relevant degrees of freedom.

In explaining the basics of DFT in the following paragraphs we will mainlyfollow Eschrigs [65] approach in which the physical but also the mathemat-ical foundations of DFT are described with great rigour.

18

2.3.1 The Particle Density

We will introduce the particle density operator through the density matrixof the system. The benefit of this is that while in most cases one cannotexpress the kinetic energy in terms of the particle density, you can expressit using density matrices.

The (spin-dependent) single-particle density matrix of a state | is definedby

1(x, x) = N

dx2 dxn (x, x2, . . . , xn)(x, x2, . . . , xn). (2.42)

From this definition it follows that (assuming | is normalized)

tr1 = N. (2.43)

Note that for an (anti)-symmetric product of spin orbitals i(x) the single-particle density matrix is given by

1(x, x) =

i

i(x)i(x). (2.44)

The probability density of measuring one of the particles at r is given by

(r) = 1(r, r) =

ds 1(x, x). (2.45)

Note that = tr1 = N. (2.46)

The particle density operator is defined by (for an N -particle system)

(x) =

Ni=1

(r ri)si , (2.47)

where ri is the position operator for particle i and i is its spin operator. Ingeneral, expressions like (r ri) and si should be interpreted as follows:first act with ri or i on the wavefunction that follows the expression anduse the result to evaluate the Dirac or Krnecker delta.

The spin-dependent number density in the state | is then the expectationvalue

(x) = |(x)|. (2.48)

19

To obtain the spatial density function, we integrate out the spin-dependence,

(r) =Ni=1

(r ri), (r) = |(r)|. (2.49)

Take a spin-independent one-particle operator O1 =

o1(ri). Its expecta-tion value is given by

O1 =

dx1 . . . dxn (x1, . . . , xn)Ni=1

o1(ri)(x1, . . . , xn)

=

dr [o1(r)1(r, r)]r=r

= tr o11. (2.50)

Here the middle line was obtained because we set r = r after o1 acts on1(r, r

) to ensure it only acts on the x variable of , not of . As anexample, the expectation value of the kinetic energy is given by

T = 12

dr [21(r, r)]r=r (2.51)

= 12tr 21. (2.52)

Note that if we have

O1 =Ni=1

o1(xi), (2.53)

then

O1 =

dx o1(x)

Ni=1

(r r1)ssi

=

dx o1(x)

Ni=1

(r r1)ssi

=

dx o1(x)(x). (2.54)

In the case of a two-body operator,

O2 =1

2

Ni =j

o2(ri, rj), (2.55)

20

we find analogously that

O2 =

drdr o2(r, r)2(r, r; r, r) (2.56)

where

2(x1, x2;x1, x

2)

=N(N 1)

2!

dx3 dxn (x1, x2, x3, . . . , xn(x1, x2, x3, . . . , xn)

(2.57)

is the spin-dependent two-particle density matrix from which we obtain thespin-independent version through summation over the relevant spin variablesas in the one-particle case,

2(x1, x2) = 22(x1, x2;x1, x2). (2.58)

In terms of the particle density operator, we can write

O2 =

1

2

drdr o2(r, r)

ij

(r ri)(r rj)i

(r ri)(r ri)

=1

2

drdr o2(r, r)[(r)(r) (r)(r r)]. (2.59)

With this knowledge, we can write the energy E of a system governed bythe hamiltonian eq. (2.9) as

E = H = 12

dr [21(r, r)]r=r +

dx v(x)(x)

+1

2

drdr(r)w(|r r|)(r) + 1

2

drdr w(|r r|)h(r, r)

= Ekin + Epot + Eh + Exc, (2.60)

whereh(r, r) = 2(r, r

) (r)(r). (2.61)

2.3.2 The Hohenberg-Kohn Theorems

From now on, we consider Hamiltonians of the form

H[v] = T + V + W

= 12

Ni=2

2i +Ni=1

v(xi) +1

2

Ni=j

w(|ri rj |), (2.62)

21

with the independent particle Hamiltonian w = 0 used as a reference system,

H0[v] = 12

Ni=2

2i +Ni=1

v(xi). (2.63)

The potential v will be taken from the set U , which is the set of potentialssuch that the energy is finite. We will later elaborate more on this set. TheHamiltonian H is assumed to be bounded from below, however there mightnot be a ground state |0 minimizing the expectation value of H. Instead,the ground state energy is defined as

E0[v] = inf{|H[v]|| Wn}, (2.64)

where the Wn defined as

Wn = {| | | antisymmetric , | = 1,i L2 for i = 1, . . . , N}.(2.65)

This is essentially the set of properly normalized N -particle wavefunctions.

As we are talking about DFT, in what follows we will drop the subscript 0to denote ground state quantities, i.e. we will write E for the ground stateenergy E0, as long as it does not lead to confusion.

Note thatE[v(x) + c] = E[v(x)] +Nc. (2.66)

Since the reference level E = 0 can be chosen arbitrarily, this introduces agauge freedom in our system. From now on, we will consider potentials v1and v2 to be different if they differ by more than a constant. We will alsoonly consider spin-independent potentials.

We define the class of potentials (or rather families of potentials in the sensethat was described above)

Vn = {v| v admits an N -particle ground state}. (2.67)

We will not go into detail about the specific nature of the set, suffice it tosay that this choice encompasses all Coulomb type potentials.

Then, for v Vn the infimum in eq. (2.64) becomes a minimum and we canwrite for the ground state energy

E[v] = 0|H[v]|0 = 0|T + W |0+ 0|Ni=1

v(xi)|0, (2.68)

22

where |0 denotes the ground state or - in the case of degeneracy - one ofthe ground states. We can rewrite the last term, which is the only system-specific term, in terms of the ground state density , after which the energybecomes

E = 0|T + W |0+ (v|), (2.69)

where(v|) =

dx v(x)(x). (2.70)

Obviously, depends on v through the ground state 0 which is uniquelydetermined by v up to degeneracy. The tricky part however is finding how determines v. The (first) Hohenberg-Kohn theorem now states

Any v Vn is a unique function of the ground state density (x).

Proof. Suppose we have two distinct vi Vn (i.e. differing by more than aconstant) with the same density . Each of these has its own ground statevi defined through the respective Schrdinger equations

H[vi]vi = E[vi]vi (2.71)

with associated energy

E[vi] = vi |H[vi]|vi. (2.72)

The ground state energy is by definition the variational minimum of theexpectation value of the Hamiltonian, using any other wavefunction willraise the expectation value. Specifically,

E[v1] < v2 |H[v1]|v2 = v2 |H[v2]|v2+ (v1 v2|),E[v2] < v1 |H[v2]|v1 = v1 |H[v1]|v1+ (v2 v1|). (2.73)

Combining these equations, we find

E[v1] + E[v2] < E[v1] + E[v2] (2.74)

which is not possible.

In other words, for any given (x), there is at most one potential v(x) (upto a constant) for which (x) is the ground state density. Of course, theconverse is always valid: if v1 and v2 are different potentials in Vn (i.e. theydiffer by more than a constant), they lead to 2 different hamiltonians H[vi],each with their own ground states vi and densities vi .

23

Complementary to the class of potentials Vn, we define the class of densitiesAn as

An = {(x)| comes from an N -particle ground state} (2.75)

which is called the class of (pure-state) v-representable densities. Using thisnotation, we can think of the first Hohenberg-Kohn theorem as defining amapping between An and Vn where each element of Vn (viewed as a familyof gauge equivalent potentials) is the image of at least one An (more thanone implies degeneracy) but two distinct (up to degeneracy) elements of Analways map to different elements of Vn.

The second Hohenberg-Kohn theorem introduces a variational principle forthe electronic energy. It uses a new functional, called the (universal) Hohenberg-Kohn Functional, defined on the class of densities An as

Fhk[] = E[v] (v|), (2.76)

where v is the image of under the first Hohenberg-Kohn theorem.

For any given , the first Hohenberg-Kohn theorem assures that we have anassociated potential v, which in turn defines a Hamiltonian H[v] which hasa ground state with ground state energy E[v] = |H[v]|. Takingany other v Vn independently of , (i.e. v = v), we find

Fhk[] + (v|) = |H[v]| [v]|H[v]|[v]= E[v]. (2.77)

Thus the second Hohenberg-Kohn theorem (also known as the Hohenberg-Kohn variational principle) reads

E[v] = minAn

{Fhk[] + (v|)} , v Vn. (2.78)

As a side note, the original paper of Hohenberg and Kohn defined the uni-versal functional as

Fhk[] = 0|T + W |0, (2.79)

which is only possible if we confine the study to the classes V n of potentialshaving a non-degenerate ground state and An of densities coming from anon-degenerate ground state. In that case, the mapping from to v is notonly single-valued and surjective (each v Vn is the image of at least one An), it is also injective (two distinct densities are never mapped to thesame potential) and thus bijective.

24

The variational principle can be rewritten as as a stationary principle for theenergy w.r.t variations in the density subject to the constraint

dr = N .

We introduce a Lagrange multiplier for this constraint:

{E[v]

(dr N

)}= 0, (2.80)

where we explicitly wrote v to indicate that ultimately the energy dependson the density. From this we retrieve the Euler-Largrange equation for DFT:

E[v]

= 0. (2.81)

Since E = Fhk + (v|), this is equivalent to

Fhk

= v. (2.82)

The Lagrange multiplier is called the electronic chemical potential.

2.3.3 Constrained Search

The Hohenberg-Kohn functioal was defined as

Fhk[] = E[v] (v|) (2.83)

for An. The Hohenberg-Kohn variational principle in turn defines theenergy as

E[v] = minAn

{Fhk[] + (v|)

}(2.84)

for v Vn. This presents us with a v-representability problem: the sets Vnand An are unknown. Levy and Lieb independently found a way around thisproblem by defining instead of Fhk the Levy-Lieb functional,

Fll[] = inf{|T + W |

Wn, }, (2.85)on the extended domain In, which is the set of N -representable densities,

In =

{|(x) 0,1/2 L2,

= N

}. (2.86)

Levy and Lieb then went on to prove that any non-negative density thatintegrates to N and such that 1/2 L2 comes from a Wn, implyingthat Fll is well defined. Since E[v] = inf{|H|| Wn} we find that

E[v] = infIn

{Fll[] + (v|)

}. (2.87)

25

In other words, for a given density we first search for the wavefunction (whichyields that density) which infinimizes the energy and subsequently we searchfor the density that infinimizes the energy. The benefit here is that In isexplicitly known and one can show that An is dense in In. Furthermore, onecan prove that for In one can replace the infimum in the definition ofFll by a minimum. The minimizing does not have to be a ground statebut if it is, we see that Fll[] = Fhk[] on An. This means that Fll is acontinuation of Fhk on an explicitly known and convex domain.

2.4 A note on functional derivatives

Intuitively, we can think of functional derivatives as follows. Consider a func-tional F = F [f ] and consider an infinitesimal variation f(x) f(x)+f(x).We can write the difference F [f + f ] F [f ] = F [f ] as

dx A(x)f(x) as

a linear approximation in f to F . We can think of A(x) as the functionalderivative:

F =

dx

F

f(x)f(x). (2.88)

To more formally introduce functional derivatives, we introduce the con-cepts of Gteaux derivatives which generalize the notion of the directionalderivative to (locally convex) topological vector spaces, e.g. Hilbert spaces.

Recall that for f : G Rp, where G is an open subset of Rn, we can definethe directional derivative of f in a point a in a direction y Rn \ {0} as

Dyf(a) = lim0

f(a+ y) f(a)

(2.89)

(if the limit exists). Another notation is fy (a). In the specific case that yis one of the basis vectors ei, we call this a partial derivative and we cancalculate it by taking the one-dimensional derivative w.r.t. the ith variablewhile keeping the others fixed.

Our goal here is to extend the previous notions of differentiability to the caseof Hilbert spaces. We will specifically be interested in functionals, i.e. mapsF : X R where X is a Hilbert space. The directional derivative of F at apoint f U an open subset of X, in the direction g X is defined as thelimit

DgF [f ] = lim0

F [f + g] F [f ]

. (2.90)

For any given f U , F is said to be Gteaux differentiable at f if thedirectional derivative exists for all g X and they can be assembled into a

26

single map : X R such that DgF [f ] = [g] for all g X. We write thefunctional as

[g] =F [f ]

f[g]. (2.91)

The expression in eq. (2.90) opens the door for an implementation of func-tional derivatives through finite difference approximations.

2.5 Kohn-Sham Theory

Thinking back to eq. (2.60) and neglecting for a moment the exchange-correlation term, the only term that is not easily written in terms of the par-ticle density is the kinetic energy term. Before the advent of DFT, Thomasand Fermi were able to derive an expression for the energy for one of the fewsystems in which you can express the kinetic energy in terms of the density,i.e. the homogeneous, independent fermion gas:

Etf = Ctf

dr 5/3(r) +

dr v(r)(r) +

1

2

drdr (r)w(|r r|)(r).

(2.92)

The idea of Kohn-Sham is to introduce an auxiliary non-interacting system,i.e. governed by H0, which exactly reproduces the density of the originalsystem. In this case, the kinetic energy is expressible in terms of orbitalswhile the density (which by construction equals the density of the original,interacting system) is expressible as the sum of the square of orbitals.

The Hohenberg-Kohn theorem is valid for any w that keeps the Hamiltoni-ans bounded from below, in particular for w = 0, the independent particleapproximation. In this case, H = H0 = T + (v|). In other words, the uni-versal Hohenberg-Kohn functional is exactly the kinetic energy of the groundstate of the non-interacting N -particle system. We define the domain of thekinetic energy functional T [] to be

A0n = {(x)| comes from a determinantal N -particle ground state}(2.93)

and write

T [] = E0[v0[]]

dx v0[], A0n. (2.94)

Writing the interaction free ground state as a determinant formed from or-thonormal orbitals i, |00 = 1N !

i(x(i)), as we have seen the density

27

of a determinantal state like this is

(x) =

Ni=1

i(x)i (x) (2.95)

while its kinetic energy is

T = 00|T |00 = 1

2

Ni=1

i|2|i. (2.96)

We can rewrite the definition of the kinetic energy functional T [], eq. (2.94),as the minimization

T [] = mini ,i

{ 1

2

Ni=1

i|2|i|i|j = ij ,Ni=1

ii =

}(2.97)

which extends the definition of T [] beyond A0n to cases where no v0[] existsfor that .

The Hohenberg-Kohn variational principle, eq. (2.78) then reads

E0[] = min

{T [] + (v|)

}(2.98)

= mini ,i

{Ni=1

(12i|2|i+ i|v|i

)|i|j = ij

}. (2.99)

Introducing Lagrange multipliers i for the constraints, we find the Kohn-Sham equations, which are the one-particle Schrdinger equation for the Nlowest energy orbitals in the non-interacting case,(

122 + vks(x)

)i(x) = ii(x), (2.100)

wherevks(x) = v + vJ + vxc. (2.101)

In the interacting case, we write the kinetic energy functional from above asTs[] and we split up the total kinetic energy functional in Ts[] and the restwhich is put in an exchange-correlation term Exc[].

28

Chapter 3

Linear Response

3.1 Conceptual Density Functional Theory

As we have seen in the previous chapter, the electronic energy is a functionalof the external potential through eq. (2.64). We have spent the last chapterdiscussing how, through the Hohenberg-Kohn theorem, we can use the elec-tronic density as the basic variable in our theory as it determines the externalpotential and so the Hamiltonian. In light of chemical reactivity theory how-ever, we turn back to a description in terms of the external potential andchange the previously fixed variable N to a variable as well. The reasonfor this is that chemical reactions can be thought of as perturbations in theexternal potential (due to a rearrangement of nuclei) and/or the number ofelectrons of the reagents. In other words, we move to a study of the energyfunctional E[v,N ], the response of which should in theory give us insight intothe reactivity of the system. In order to avoid having to study the energyresponse in its entirety, it is customary to look at its Taylor expansion:

E[N0 +N, v0(r) + v(r)] E[N0, v0(r)] =(E

N

)v(r)

N +1

2

(2E

N2

)v(r)

(N)2 + . . .

+

dr

(E

v(r)

)N

v(r)

+ N

dr

(

N

(E

v(r)

)N

)v(r)

v(r) + . . .

+1

2

drdr

(2E

v(r)v(r)

)N

v(r)v(r) + . . .

(3.1)

29

For an analysis of the convergence and formal properties, see Ayers et al.[8]. Each of the derivatives of the energy, either with respect to N , v or amix of both, can be viewed as a reactivity index or a response function. Forexample,

(r) =

(E

v(r)

)N

, (3.2)

while the other first order derivative gives

=

(E

N

)v

. (3.3)

This links us back to the previous chapter as these two objects are centralto DFT: the density is the titular fundamental object and the chemicalpotential appears as the lagrange multiplier for the constraint

= N

when deriving the Euler-Lagrange equations for DFT [10]. In other words,even though we started with an object that seemingly had no relation toDFT, the energy functional E[v,N ], even at first order, retrieves objectsthat are central to DFT.

The study of chemical reactivity through the use of reactivity indices is calledConceptual Density Functional Theory [12], a branch of theoretical chemistrydeveloped alongside DFT. The aim of this framework was to formalize certainwell known chemical concepts that previously were defined rather vaguely.For this end, reactivity indices were used to describe these chemical concepts.The first order derivatives, (r) =

(E

v(r)

)N

and =(EN

)v(r)

are wellstudied and provide a link to DFT as mentioned before. As mentioned inthe foundational work by Parr et al. [14], earlier work done by Iczkowski andMargrave [66] defined the electronegativity of a system as

= ( EN

)v

(3.4)

leading to the close relation between a fundamental DFT quantity and a wellknown but difficult to formally define chemical quantity:

= . (3.5)

At second order, the N -derivative(

2EN2

)v(r)

is identified as the chemical

hardness whereas the mixed derivative f(r) =(

N

(E

v(r)

)N

)v(r)

is known

as the Fukui function, a concept that is closely related to the frontier MOconcept of Fukui et al. [67]. The second order functional derivative w.r.t. vis known as the linear response kernel

(r, r) =

(2E

v(r)v(r)

)N

. (3.6)

30

This function is the main focus of this thesis.

The concept of chemical hardness (and related to it the softness) was intro-duced by Pearson [19] in the 1960s in connection to the Hard and Soft Acidsand Bases (HSAB) principle. It wasnt until Parr and Pearson [20] howeverthat a rigourous definition of the chemical hardness was given as the secondorder derivative of the electronic energy w.r.t. the number of electrons. Therelated concept of global softness is defined as the inverse of the hardness,

S =1

. (3.7)

The Fukui function [18, 68] can be used to describe the regionselectivity forsoft or orbital-controlled reactions. It can also be used to define the localsoftness s(r) = f(r)S [69].

In contrast to the chemical hardness and the Fukui function, the linear re-sponse kernel has hitherto received relatively little attention. Using the firstorder derivatives mentioned above, this function can be rewritten as

(r, r) =

(2E

v(r)v(r)

)N

=

((r)

v(r)

)N

, (3.8)

which (as already mentioned in the introduction) gives us the extremelyuseful interpretation of the linear response kernel as the change in electrondensity in a point r in response to a perturbation of the external potentialin a point r.

Some third order derivatives have also been studied in the literature [15],specifically the hyperhardness [15]

(3EN3

)and the dual descriptor [70]

(f(r)N

)v(r)

which provides a one shot picture of electrophilic and nucleophilic regionsaround a molecule.

The diagram in Scheme 3.1 gives a graphical representation of the construc-tion of reactivity indices, where moving down and to the left denotes partialderivation w.r.t. particle number N whereas moving to the right representsfunctional derivation w.r.t. the external potential v(r).

We should note that not all reactivity indices are defined as derivatives.Examples include the description of the steric effect by Liu [71] and the de-scription of non-covalent interactions by Johnson et al. [72]. Similarly, someindices are defined as a combination of other indices (that can in turn be de-fined as derivatives) but cannot be written as a simple derivative themselves,e.g. the electrophilicity [21].

31

E[N, v(r)]

(EN

)v(r)

= (

Ev(r)

)N

= (r)

(2EN2

)v(r)

= (

2ENv(r)

)= f(r)

(2E

v(r)v(r)

)N

= (r, r)

(3EN3

)v(r)

(3E

N2v(r)

)= f (2)(r)

(3E

Nv(r)v(r)

) (3E

v(r)v(r)v(r)

)N

Scheme 3.1: Energy Derivatives and Response Functions in the Canonical En-semble, n+mE/Nnvm, (m+ n 3)

As mentioned, we will focus on the diagonal second order response function(r, r), defined as

(r, r) =

(2E

v(r)v(r)

)N

=

((r)

v(r)

)N

. (3.9)

As we can see from the second equality in eq. (3.9), the linear response kernelgives a (first order) measure of the change in electronic density in responseto a change in the external potential.

The importance of the linear response kernel in conceptual DFT is evidentfrom the Berkowitz-Parr relationship [47]:

(r, r) = s(r, r) + s(r)s(r)

S, (3.10)

where s(r, r) is the softness kernel,

s(r, r) = ((r)

v(r)

)

, (3.11)

s(r) is the local softness and S is the global softness. The softness kernel isthe inverse of the hardness kernel (r, r) [73] in the sense that

dr s(r, r)(r, r) = (r r), (3.12)

through which it is ultimately connected to the local hardness (r) [7478]. Note that Senet [28] derived exact functional relations between both

32

the linear and non-linear response functions and the ground state density interms of the universal Hohenberg-Kohn functional Fhk[].

Besides the more formal work mentioned above, some earlier work done inour group focussed on the calculation of the linear response kernel usingnumerical methods [3437, 42] and extracting qualitative and quantitativeinformation about inductive, resonance and hyperconjugation effects. Dueto computational reasons, an atom-atom condensation scheme of (r, r) waschosen, resulting in

AB =

VA

VB

drdr (r, r). (3.13)

It has also been used to extract information about aromaticity [43, 44]

Some works on the chemical information contained in these atom condensedlinear response matrices are Baekelandt et al. [38] and Wang et al. [39], whichuse highly approximate semi-empirical schemes and Morita and Kato [40, 41]using coupled perturbed Hartree-Fock theory.

3.2 Mathematical background

The linear response kernel is defined as a functional derivative:

(r, r) =( 2Ev(r)v(r)

)N

=( (r)v(r)

)N. (3.14)

Note that this quantity is symmetric in r and r. As we have noted inchapter 2, these derivatives are to be understood in the sense of Gteauxderivatives (see section 2.4). In the next section, we will discuss a method toapproximate the linear response kernel in terms of molecular (Hartree-Fockor Kohn-Sham) orbitals using perturbation theory.

The linear response kernel is real-valued and symmetric [33], implying thatits eigenvalues hi, defined by

dr (r, r)i(r) = hii(r), (3.15)

are also real. More specifically, the point-spectrum of the response func-tion contains an infinite number of eigenvalues arbitrarily close to zero, zeroincluded:

dr (r, r) = 0. (3.16)

33

Note that these properties can be interpreted physically: the fact that thereexists a zero eigenvalue indicates that shifting the potential by a constantleaves the density unchanged,

0 =

dr (r, r) =

dr

((r)

v(r)

)N

, (3.17)

whereas the arbitrarily small eigenvalues indicate that very large changes inthe external potential do not necessarily yield big changes in the density.

3.3 Evaluation of (r, r)

3.3.1 Numerical evaluation

To find an approach to numerically evaluate the linear response kernel (r, r),we can start by taking a look at an arbitrary functional of the external po-tential, Q = Q[v] and the evaluation of the functional derivative Q/v(r).Suppose we perturb the external potential by a set of P perturbations{wi(r)|i = 1, . . . , P}. Up to first order we then have

Q[v + wi]Q[v] =

dr(Q[v]

v(r)

)wi(r). (3.18)

This expression can be seen as a finite approximation of the limit in eq. (2.90).We can expand the functional derivative in a basis set {j |j = 1, . . . ,K}:(

Q[v]

v(r)

)N

=

Kj=1

qjj(r) (3.19)

with expansion coefficients qj . Combining both equations we find a set oflinear equations

Q[v + wi]Q[v] =Kj=1

qj

dr j(r)wi(r). (3.20)

Being a set of linear equations, we can rewrite them in matrix form as

d = Bq, (3.21)

wheredi = Q[v + wi]Q[v] (3.22)

andBij =

dr wi(r)j(r). (3.23)

34

In practice P is chosen to be larger than K and eq. (3.21) is solved via leastsquares fitting, where P is varied until the result converges. The perturba-tions themselves are point charge perturbations (zi)

wi(r) =zi

|rRi|(3.24)

and the expansion functions are uncontracted s- and p-type Gaussians oneach center. This approach has been used successfully to study the Fukuifunction f+(r) in previous work in our group [79].

Following Sablon et al. [34], we can extend this procedure to the second orderderivatives, specifically the linear response function (r, r) being (2E/v(r)v(r))N .Similar to eq. (3.18) we can write

E[v + wi] 2E[v] + E[v wi] =

drdr (r, r)wi(r)wi(r) (3.25)

and we can expand (r, r) as

(r, r) =Kk,l

qklk(r)l(r), (3.26)

which again leads to a set of linear equations which can be written as amatrix equation

d = Bq (3.27)

where B is now a P K2 matrix composed of the integrals over the variousbasis functions and the external perturbations:

Bj,(k1)K+l =

dr k(r)wj(r)

dr l(r)wj(r) (3.28)

with j = 1, . . . , P and k, l = 1, . . . ,K. The column matrix q is a K2 di-mensional column matrix with elements q(k1)K+l = qkl (k,l = 1, . . . , K).Initially, the six dimensional kernel (r, r) was represented by an atom-atomcondensed linear response matrix with elements

AB =

VA

VB

drdr (r, r), (3.29)

which can be expanded as

AB =kA

lB

qkl

dr k(r)

dr l(r). (3.30)

In this thesis, we focussed on the representation and interpretation of thefull, non-condensed linear response kernel.

35

3.4 A perturbational approach to the linear responsekernel

We will see now how to analytically express the linear response functionusing standard perturbational methods.

We assume a single Slater determinant ansantz, be it HF or KS. For the sakeof simplicity, we shall assume a closed shell type system and real orbitals.Under these assumptions, the density becomes

(r) = 2

N/2i

2i (r). (3.31)

Assume a perturbation of the external potential v(r). We can express theorbitals in a perturbation expansion, which is a formal power series, a gen-eralization of polynomials which can have an infinite amount of terms:

|i = |(0)i + |(1)i + (3.32)

where (1)i represents the first order correction to the unperturbed orbital(0)i . For the density, the first order correction is given by

(1)(r) = 4

N/2i

(0)i (r)

(1)i (r). (3.33)

The solutions (0)i of the unperturbed problem are assumed to form a com-plete set. This means we can express |(1)i as

|(1)i =a

Cia|(0)a . (3.34)

Note that one can prove that we can limit the summation to unoccupiedorbitals1 only [80].

Our goal is then to find the expansion coefficients Cia.

3.4.1 Hartree-Fock

We will start by applying this to Hartree-Fock theory. The Hartree-Fockequation is (switching to bra-ket notation)

F |i = i|i (3.35)1In the following, indices i, j, k, . . . will be used to denote occupied orbitals while indices

a, b, c, . . . will refer to unoccupied orbitals.

36

with

F = 122 + veff

= h+G, (3.36)

where h and G combine the one and two electron operators respectively:

h = 122 + v, (3.37)

G = vh + vx. (3.38)

We can write the the perturbation expansions

F = F (0) + F (1) + . . . (3.39)

i = (0)i +

(1)i + . . . (3.40)

|i = |(0)i + |(1)i + . . . (3.41)

where for F (0) it is assumed that we can explicitely solve the Hartree-Fockequations and the higher order terms are considered small in comparison toF (0). Plugging these expansions into the Hartree-Fock equation we retrieveat zeroth order the Hartree-Fock equations for F (0),

F (0)|(0)i = (0)i |

(0)i , (3.42)

and at first order(F (0) (0)i

)|(1)i +

(F (1)

)|(0)i =

(1)i

(0)i . (3.43)

Note that h(1) = v.

Inserting the expansion eq. (3.34) and taking the inner-product with |(0)b (still assuming real orbitals).

a

Cia(0)b |F(0) (0)i |

(0)a =

(0)b |F

(1)|(0)i + (1)i

(0)b |

(0)i . (3.44)

The last term equals zero since |b is an unocccupied orbitals, which areorthogonal to the occupied orbitals. Introducing the shorthand |i for |(0)i ,this leads to

a

Ciab|(0)a (0)i |a = b|F

(1)|i (3.45)

or a

Cia((0)a

(0)i )ab = b|F

(1)|i. (3.46)

37

The independent particle approximation

As a first approximation, we will assume that a perturbation in the externalpotential v does not influence vh or vx through the perturbed orbitals, i.e.

F (1) = h(1) = v. (3.47)

This is known as the independent particle approximation. With this, eq. (3.46)becomes

Cia((0)a

(0)i ) = a|v|i, (3.48)

which becomes

Cia = a|v|i

(0)a (0)i

. (3.49)

Random Phase Approximation

The next step is to include the effect of the Coulombic part of G(1):

F (1) = v + v(1)h , (3.50)

which turns eq. (3.46) into

Cia((0)a

(0)i ) = a|v|i a|v

(1)h |i. (3.51)

From eq. (2.30) we see that in general

v(1)h (r) = 4

j

dr (0)j (r

)w(|r r|)(1)j (r), (3.52)

hence

a|v(1)h |i = 4j,c

Cjc

drdr (0)a (r)

(0)j (r

)1

|r r|(0)c (r

)(0)i (r)

= 4j,c

Cjcaj|ci. (3.53)

Then eq. (3.51) becomes

Cia((0)a

(0)i ) = a|v|i 4

j,c

Cjcaj|ic. (3.54)

38

Coupled Perturbed Hartree-Fock

Finally, including the full effect of G(1), i.e.

F (1) = h(1) + v(1)h + v

(1)x , (3.55)

eq. (3.46) becomes

Cia((0)a

(0)i ) = a|v|i a|v

(1)h |i a|v(1)x |i. (3.56)

From eq. (2.31) we can see that

(v(1)x k)(r) = 2j

dr (0)j (r

)w(|r r|)k(r)(1)j (r)

= 2j,c

dr (0)j (r

)w(|r r|)k(r)Cjc(0)c (r). (3.57)

Applied to the problem at hand, the matrix elements of v(1)x are given by

a|v(1)x |i = 2j,c

Cjc

drdr (0)a

(0)j (r

)1

|r r|(0)i (r

)(0)c (r)

= 2j,c

Cjcaj|ic. (3.58)

This leaves us with

Cia((0)a

(0)i ) = a|v|i4

j,c

Cjcaj|ic+j,c

Cjcaj|ci+j,c

Cjcac|ji

(3.59)for the expansion coefficients.

3.4.2 Matrix Formulation

Looking back at eq. (3.59), we can rewrite this asj,c

Cjc

[((0)c

(0)j )acij + 4aj|ic aj|ci ac|ji

]= a|v|i.

(3.60)

Introducing the matrix M formed from elements

Mia,jc = ((0)c

(0)j )acij + 4bj|ic aj|ci ac|ji, (3.61)

39

where the elements are labeled by pairs of indices, this can be written as

MC = V, (3.62)

where C and V are column matrices with elements Cia and Via = a|v|irespectively. Solving for the expansion coefficients yields

C = M1V, (3.63)

or, for the components,

Cia = jc

(M1

)ia,jc

Vjc. (3.64)

From eq. (3.33),

(1)(r) = 4

N/2i

(0)i (r)

(1)i (r)

= 4i,a

Cia(0)i (r)

(1)a (r)

= 4i,a

j,c

(M1

)ia,jc

(0)i (r)

(1)a (r)Vjc. (3.65)

From this, remembering that Vjc are the elements of v projected into abasis, we get for the linear response kernel projected in a basis

(r, r) =(r)

v(r)

= 4i,a

j,c

(M1

)ib,jc

i(r)a(r)j(r)c(r

) (3.66)

where the superscripts "(0)" have been dropped to ease the notation.

For example, in the non-interacting case, this reduces to

(r, r) = 4i,b

j,c

1

c ji(r)b(r)j(r

)c(r)ijbc

= 4i,b

i(r)b(r)i(r)b(r

)

b i. (3.67)

40

3.4.3 Kohn-Sham

The results from Hartree-Fock theory can be readily converted to Kohn-Shamtheory through eq. (3.46), which becomes

a

Cia((0)a

(0)i )ab = b|vks|i, (3.68)

where, from the definition of the Kohn-Sham potential

vks = v + vh + vxc. (3.69)

In the independent particle case, where vks = v, the matrix M againconsists of

Mia,jc = (c j)acij , (3.70)

where the energies are now Kohn-Sham orbital energies. Including the effectof v on vh, this becomes

Mia,jc = (c j)acij + 4aj|ic. (3.71)

Finally, for the full CPKS approach, i.e. where we include both vh andvxc in vks, remember that Kohn-Sham theory incorporates an exchange-correlation term rather than only an exchange term as in Hartree-Fock the-ory. In general, one can write

b|vxc|i =

drdr (0)b (r)

(vxc[]

(r)(r)

)(0)i (r)

=

drdr (0)b (r)

(2Exc[]

(r)(r)(r)

)(0)i (r). (3.72)

Using (r) = (1)(r) this becomes

b|vxc|i = 4j,a

Caj

drdr(0)b (r)

(0)i (r)

2Exc(r)(r)

(0)a (r)

(0)j (r

).

(3.73)

In the full coupled-perturbed Kohn-Sham case, the matrix elements of M are

Mia,jc = (c j)acij + 4aj|ic+ 4ia|fxc(r, r)|jc, (3.74)

where

fxc(r, r) =

2Exc(r)(r)

. (3.75)

41

These fxc integrals are calculated numerically (using the Becke integrationscheme [81]). In the case of a GGA functional for example, the exchange-correlation energy takes the general form

Exc[] =

dr exc((r), (r)), (3.76)

where (r) is the generalized gradient. The derivative of this functional iscalculated as a Gteaux derivative (see section 2.4). Note that in the caseof a GGA, changes in the density will also influence the gradient throughwhich in turn the exchange-correlation energy is affected. The second orderderivative of the energy will thus not only contain a term 2exc/(r)(r)but also e.g. exc/(r), 2exc/(r)(r) et cetera. These derivatives canbe calculated using for example the xcfun library [82].

3.5 Closing Remarks

In this chapter, we introduced conceptual DFT and specifically the linear re-sponse kernel which appears as one of the reactivity indices and is the mainfocus of the research presented in this thesis. After a brief overview of themathematical properties of this function (section 3.2), we discuss two meth-ods to evaluate the linear response function, one numerical (section 3.3.1)and one based on perturbation theory (section 3.4).

We have derived an expression for the linear response kernel using pertur-bation theory for Hartree-Fock theory and Kohn-Sham theory. In each case,there are three levels of approximation. The crudest approximation consistsof equating F (1) or vks (in the HF or KS case respectively) to the change inthe external potential, v. The next level consists of including the effect ofthe perturbation in the external potential on the Coulombic part vh. Up tothis point, HF and KS yield the same expressions for the linear response ker-nel, the only difference being the orbitals used in said expressions (i.e. eitherthey are HF orbitals or they are KS orbitals). For the final level, the differ-ence between CPHF and CPKS is due to the fact that Hartree-Fock theoryincludes exact exchange whereas Kohn-Sham theory incorporates exchange-correlation effects. If we were to take the limit case of Kohn-Sham with nocorrelation and only exact exchange, both methods would fully agree (again,up to the specific orb