Theoretical study of electronic properties of some aromatic rings
PhD Thesis Zino Boisdenghien
1. Faculty of Science & Bio-Engineering Sciences
Research Group General Chemistry
The Linear Response Function
in Conceptual Density Functional
Theory
Fundamental Aspects and Application to Atoms
Zino Boisdenghien
Thesis submitted in partial fulfillment of the requirements for the academic degree
of Doctor in Sciences
Academic Year 2015-2016
Promotors:
Em. Prof. Dr. Paul Geerlings
Prof. Dr. Frank De Proft
Dr. Stijn Fias
December 2015
3. Acknowledgements
This thesis is the culmination of four years of research conducted in the
General Chemistry Group (Eenheid Algemene Chemie, ALGC) at the Vrije
Universiteit 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 opportunity
to perform my research and their continued guidance and discussions that
have 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 my
PhD.
I would like to thank all of my colleagues at the VUB over the years for
creating a pleasant atmosphere to work in and for the interesting discussions
over coffee, kind words of encouragement and much more over the years.
Mercedes, Fran, Freija, Songül, Jan, Thijs, Ann-Sophie; I am certain that I
have 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 it
was letting of steam in the dojo (merci Sensei Jean en Sensei Oli en al mijn
vrienden van de jiu) or over a couple of beers afterwards. I especially want
to 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, bedankt
voor 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 leven
met jou door te brengen.
iii
5. Abstract
The research presented here is situated in the field of Conceptual Density
Functional Theory (Conceptual DFT), a chemical reactivity theory rooted
in Density Functional Theory (DFT). Density Functional Theory presents a
description of quantum mechanics that takes the electronic density as its cen-
tral object rather than the wavefunction. The principal idea of Conceptual
DFT is to define reactivity indices as (functional) derivatives of the energy
which 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 of
the 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 external
potential, which provides us with the intuitive interpretation of the linear
response kernel as the response of the density to changes in the external
potential.
By taking a step back and focussing our attention on atoms we were able to
study the linear response function in its own right whereas previous stud-
ies have obtained numerical data by employing an atom-atom condensation
scheme.
We evaluate and represent the (uncondensed) linear response function for
hydrogen through argon using both the Independent Particle Approxima-
tion as well as the Coupled Perturbed Kohn-Sham approach. The resulting
figures nicely illustrate the trends that the linear response function captures
throughout the periodic table, such as the periodicity. We also investigate
spin polarized versions of the linear response kernel, which provide insight
in how α or β electrons react differently to perturbations in the α or β parts
of 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 also
provides us with a straightforward definition of the local polarizability, its
v
6. 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 Dependent
DFT using the Sternheimer equations, which form the time dependent ana-
logue to the Coupled Perturbed Kohn-Sham equations. In the limit where
the frequency tends to zero, this provides a direct comparison between the
static and the frequency dependent linear response kernel. In practice we
compare static an dynamic linear response by calculating the static and dy-
namic local polarizability for atoms and a single molecular system.
vi
7. 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 plaats
van de golffunctie. Het basisprincipe van Conceptuele DFT is het definiëren
van reactiviteitsindices in termen van (functionele) afgeleiden van de energie
die inzicht kunnen verschaffen in de (inherente) reactiviteit van een systeem.
We focussen ons op één van deze reactiviteitsindices, namelijk de lineaire
respons 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 intuïtieve interpretatie van
de lineaire response kernel verschaft als het antwoord van de dichtheid op
veranderingen in de externe potentiaal.
Door een stap terug te nemen en ons te focussen op atomen waren we in staat
om de lineaire respons functie als dusdanig te bestuderen in tegenstelling tot
vorige studies waar numerieke data werden bekomen d.m.v. een atoom-atoom
condensatie procedure.
We evalueren en visualiseren de (ongecondenseerde) lineaire respons functie
voor Waterstof t.e.m. Argon, gebruik makend van zowel de Onafhankelijke
Deeltjes Benadering als de Gekoppeld Geperturbeerde Kohn-Sham benader-
ing. De resulterende figuren bieden een mooie illustratie van de tendensen
die door de lineaire respons functie kunnen worden blootgelegd doorheen
de periodieke tabel, zoals bijvoorbeeld de periodiciteit. We bestuderen ook
spingepolarizeerde versies van de lineaire response kernel, die ons inzicht ver-
schaffen in hoe α of β electronen anders reageren op perturbaties van het α
of β gedeelte van de externe potentiaal.
De lineaire respons kernel is nauw gerelateerd aan het concept van polariz-
vii
8. abiliteit. De relatie tussen de lineaire respons functie en de polarizabiliteit
verschaft ons ook met een eenduidige definitie van de lokale polarizabiliteit
tesamen met de evaluatie en evolutie van deze lokale polarizabiliteit doorheen
de periodieke tabel. Na integratie verkrijgen we terug de polarizabiliteit en
de 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. In
de 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 respons
door de statische en dynamische lokale polarizabiliteit te berekenen voor
atomen en een moleculair systeem.
viii
9. List of Abbreviations
B3LYP Becke three-parameter hybrid functional
with the Lee-Yang Parr-correlation functional
CPKS Coupled Perturbed Kohn-Sham
DFT Density Functional Theory
HF Hartree-Fock
HK Hohenberg-Kohn
KS Kohn-Sham
MO Molecular Orbital
PBE Perdew, Burke and Ernzerhof
pVTZ polarized valence triple-zeta basis set
SD Slater determinant
TDDFT Time Dependent Density Functional Theory
VWN Vosko, Wilk and Nusair
(LDA correlation functional)
xc exchange-correlation
ix
11. Publication List
The following is a list of publications containing the work presented in this
thesis 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. De
Proft, 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. De
Proft, 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 Time
Dependent Density Functional Theory, Z.Boisdenghien, S.Fias, F.Da
Pieve, F.De Proft, P.Geerlings, Mol. Phys., 2015, 113, 1890
xi
16. 5.5.2 A Perturbation Theoretical Expression for the Linear
Response 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
xvi
17. Chapter 1
Introduction
As soon as 1927, only one year after Erwin Schrödinger published the wave
equation that governs quantum systems, Walter Heitler and Fritz London
applied quantum mechanics to the diatomic hydrogen molecule [1]. This
idea, to apply quantum mechanics to chemical systems and concepts (in their
case the chemical bond), is based on the fact that at its core, the properties of
atoms and molecules should be based on the interaction of quantum objects
(specifically, electrons and nuclei). In other words, chemistry at its core is a
quantum 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 the
Bohr model for the hydrogen atom which was published in 1913 and was
developed to be consistent with experimental data available for the atomic
emission spectrum. This observation was one of the pillars of what is now
known 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 for
the self-consistent and more complete quantum theory that starts with the
Schrödinger equation.
To make sense of the huge amount of available experimental data, chemists
have been searching for easily understandable qualitative models, concepts
and principles that are capable of categorising and predicting properties
of molecules, in particular chemical reactivity. Examples of such concepts
and principles include the Lewis-dot structures [2], orbital hybridisation and
resonance [3], electronegativity [4, 5], frontier (highest occupied and lowest
unoccupied) molecular orbital (MO) concepts [6], the Woodward-Hoffmann
rules [7], etc. Due to the complicated nature of quantum mechanics though,
most concepts have been derived only in highly approximate contexts and
1
18. tend to lose their validity when moving to more quantitatively sound ap-
proaches [8, 9]. The question is thus: can we find chemical descriptors that
emerge from a theoretically rigorous basis.
In the case of chemical reactivity descriptors these efforts culminated in Con-
ceptual Density Functional Theory [8–15], a chemical reactivity theory that
stems 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 describe
larger 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 need
to describe 10 electrons which means that the wavefunction depends on 30
variables (40 if we include spin). In 1964 Pierre Hohenberg and Walter Kohn
proved that you don’t need to use the wavefunction with its monstrous num-
ber of variables, you can get all the information you need by just using the
electronic density function [16]. This marked the beginning of Density Fun-
tional Theory (DFT) [10, 17]. It is this reformulation of quantum mechanics
that forms the theoretically sound basis for Conceptual DFT.
Conceptual DFT is a branch of chemistry that developed alongside regular
DFT and primarily focussed on the formalization of several chemical concepts
and ideas that at the time lacked a strong theoretical foundation. It does
so by introducing reactivity indices, which are defined as (mixed) derivatives
of the electronic energy E[N, v] w.r.t. the number of electrons N and/or the
external potential v due to the nuclei. As chemical reactions can be thought
of as perturbations of the system in either the number of electrons and/or in
the external potential, it is natural to start from the quantity E[N, v]. The
reactivity indices defined in this way can be used to probe the (inherent)
chemical reactivity of a system. Because of the link to DFT however, we
avoid the increasing complexity that troubles wavefunction based theories
when aiming for increasing accuracy. Some examples of reactivity indices
that 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 the
electrophilicity [21]. Spin polarized versions of Conceptual DFT indices [22]
have also been introduced, which have been successfully used to describe
open shell systems, radical chemistry [23–26], 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 all
the reagents and products as well as any transition states along the reaction
path. To complicate matters even more, this must possibly be known in the
presence of a solvent. On the other hand, from observation one can see that
specific molecules will interact in similar ways for a range of reagents. This
2
19. presence of a systematic trend leads to the idea that it should be possible to
characterize the chemical behaviour of molecules in response to perturbations
without explicit reference to their partner reagents [8, 27]. Conceptual DFT
as 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 the
potential energy surface, but rather takes the second approach. It tries to
capture 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, refinement
and extension of reactivity descriptors and conceptual DFT in general as a
valid chemical reactivity theory. The work performed during my PhD and
presented in this thesis can be seen in this tradition. Previous work in our
group has focussed on the (numerical) calculation of functional derivatives
with respect to the external potential, specifically the linear response kernel
χ(r, r′), which is defined as the second order derivative of the electronic
energy w.r.t the external potential.
The importance of the linear response kernel lies in its ability to measure
how 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 the
first order derivative of the electronic density w.r.t. the external potential.
Its importance in Conceptual DFT is highlighted as it represents the answer
of the electronic density ρ(r) at position r to an external potential pertur-
bation at position r′ at constant number of electrons (i.e. (δρ(r)/δv(r′))N ),
a situation which is at the heart of understanding the course of a chemical
reaction. The linear response kernel is also in a very clear way related to
the polarizability through double integration. Intuitively, the link to the po-
larizability is clear: an external field will generate a dipole moment because
the electrons get shifted from their usual location. This moment is (as long
as the perturbing field is not too strong) approximately proportional to the
field itself, with the proportionality constant being the polarizability. As
the linear response kernel provides a measure of how the electronic density
responds to changes in the external potential, it is intuitively clear that the
polarizability and the linear response kernel are linked. The huge benefit of
this link however is that it also provides us with a straightforward definition
of the local polarizability, an object that is not so unambiguously defined as
the global polarizability. This link to the local and global polarizability will
be 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 other
indices. It has been discussed in certain more formal works, for example in
Senet [28, 29], which focussed on the exact relation between linear (and non-
3
20. linear) response functions and the ground state electronic density in terms
of the universal Hohenberg-Kohn functional Fhk[ρ] (vide infra). The work
done by Cohen et al. [30], Ayers and Parr [31], and Ayers [32] focussed on
the properties of these functions within a Kohn-Sham framework. For an
overview of some of the mathematical properties, we refer to Liu et al. [33].
Beside the more formal work mentioned above, relatively little attention has
been paid to the actual calculation and more importantly the interpretation
of the chemical and physical information contained in the linear response
function. For example, the previous work performed in our group focussed
on the calculation of the linear response kernel using numerical methods and
extracting qualitative and quantitative information about inductive, reso-
nance and hyperconjugation effects. In order to do so, a discretized version
of the linear response kernel, integrated over atomic domains in order to
create a linear response matrix χAB was used [34–37],
χAB =
∫
VA
∫
VB
drdr′
χ(r, r′
). (1.1)
Some works on the chemical information contained in these atom condensed
linear response matrices are Baekelandt et al. [38] and Wang et al. [39], which
use 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 successfully
applied to is the study of aromaticity [42–44].
The research presented here can be seen in this context but offers a fresh
perspective on the linear response kernel. Our aim was to focus on the
linear response function χ(r, r′) itself, without resorting to integration over
atomic domains, in order to extract the physically and chemically relevant
information contained in this function. To this end we decided to take a step
back and focus on single atoms while performing a systematic study of the
linear response kernel throughout the periodic table.
It is worth mentioning that the linear response kernel is also linked to some
interesting 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 is
that in systems with many electrons at constant electronic chemical poten-
tial, the change in the electron density at a point r induced by a perturbation
in the external potential at a point r′, |∆ρ(r0)|, where |r′ − r| > R, will al-
ways be smaller than a maximum value ∆ρ(r, R) and this is independent of
the size of the perturbation ∆v(r′). The language used here implies a close
connection between the concept of nearsightedness and the linear response
4
21. kernel. We can therefore introduce the quantity
(
δρ(r)
δv(r′)
)
µ
(1.2)
The only difference being the condition of constant electronic chemical po-
tential µ, which indicates that this is quantity is naturally defined when
working in an open system with a grand potential Ω[µ, v] performing the
same role as the energy functional E[N, v] in Conceptual DFT. Indeed, the
initial development of Conceptual DFT was very reminiscent of classical
chemical thermodynamics, where the canonical ensemble (at 0K) uses the
electron number N and the external potential v(r) as the basic variables.
The basic variables can be changed through Legendre transformations. The
grand canonical ensemble is then obtained by introducing the grand potential
Ω = Ω[µ, v] defined through
Ω = E − Nµ (1.3)
which exchanges the number of electrons for the chemical potential as a basic
variable.
It turns out that the quantity defined above in eq. (1.2) is minus the softness
kernel s(r, r′) which appears as the natural counterpart to the linear response
kernel in open systems described by a grand potential Ω[µ, v]. The link
between both quantities is given by the rules of differentiation with different
constraints:
(
δρ(r)
δv(r′)
)
N
=
(
δρ(r)
δv(r′)
)
µ
+
(
δρ
δµ
)
v
(
δµ
δv
)
N
(1.4)
or
χ(r, r′
) = −s(r, r′
) +
s(r)s(r′)
S
, (1.5)
where we introduced the total softness. This last equation is the famous
Parr-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 the
concept of alchemical derivatives [49], i.e. derivatives of the energy w.r.t. one
or more nuclear charges. We think of this as the linear response function
where the perturbation δv(r) results from a nuclear charge variation. The
first order derivative is given by
(
∂Eel
∂Zα
)
N
=
∫
dr
(
δEel
δv(r)
)
N
(
∂v(r)
∂Zα
)
N
=
∫
drρ(r)
−1
|r − Rα|
, (1.6)
5
22. which is nothing else than the electronic part of the Molecular Electrostatic
Potential [50] (MEP) at position of nucleus α. At second order, the mixed
derivative gives us
(
∂2Eel
∂Zα∂Zβ
)
N
=
∫
drdr′
(
δ2Eel
δv(r)δv(r′)
)
N
∂v(r)
∂Zα
∂v(r′)
∂Zβ
=
∫
drdr′
χ(r, r′
)
1
|r − Rα|
1
|r − Rβ|
, (1.7)
where we retrieved the linear response function. Recent work on the use of
alchemical 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 linear
response kernel in Conceptual DFT.
Everything we have mentioned up to now dealt with static systems, described
by the stationary Schrödinger equation. The natural question that arises is
how 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 the
Schrödinger equation can be extended to a time dependent variant, Density
Functional Theory can be extended to Time Dependent Density Functional
Theory (TDDFT). TDDFT has been succesfully applied to calculate and
predict excited-state properties in chemistry as well as solid state physics
and even biophysics. Examples of situations where TDDFT is used include
the 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 properties
of solids etc. [53, 54]
Contrary to DFT, linear response calculations are commonplace in TDDFT.
They are most frequently used however to calculate excitation energies [55–
57], which correspond to the poles of the frequency dependent linear response
function. The most commonly known method to perform these linear re-
sponse calculations is using the Casida equations. The linear response kernel
is rarely studied in its own right however, and the extension of our previous
work to the time dependent domain was much less straightforward as one
might expect. It was our goal to examine the linear response kernel itself
and focus on the chemical information that is contained in it.
6
23. 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 mainly
be interested in stationary systems with definite energy levels, which are
described by the stationary Schrödinger equation. We first describe some
fundamental 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 possess
too many degrees of freedom than are physically necessary. What follows is
an introduction to Density Functional Theory, starting with an overview of
the proof of the Hohenberg-Kohn theorem that provides the theoretical basis
for 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 is
called the linear response kernel and is the main focus of this thesis. We
give a brief overview of a numerical evaluation method that was used in
previous work from our group, followed by a method to (analytically) cal-
culate the linear response kernel using perturbation theory at various levels
of approximation, from the independent particle approximation to the full
coupled perturbed description. The theory is first introduced using Hartree-
Fock theory but later extended to Kohn-Sham theory. An important point
to note is that the derivation and structure of the equations are nearly the
same for Hartree-Fock and Kohn-Sham, only differing in two points. First
of all, the orbitals involved are obviously different, being either HF or KS
orbitals. The second difference occurs at the coupled perturbed level, where
Hartree-Fock theory incorporates exact exchange whereas Kohn-Sham the-
ory incorporates exchange-correlation. In the limit where we approximate
the xc-potential by exact exchange however, the equations become exactly
the same, only differing in the orbitals that are used. In order to trans-
late these equations to a more computationally friendly language, a matrix
formulation is introduced through the use of basis functions that are the
product of two orbitals.
In the following chapter, chapter 4, we discuss the results of my research
[58–61], which focusses on the systematic evaluation and visualization of
the linear response function throughout the periodic table. We introduce
both two dimensional contour plots and one dimensional plots for Hydrogen
through Argon and discuss the trends and the chemical information that is
contained in these plots. In this chapter we also introduce spin polarized
versions of the linear response kernel in two different representations and
7
24. discuss how to switch between them. Finally we make the link to both the
total and the local polarizability tensor.
The extension of DFT to TDDFT is introduced in chapter 5. We begin, as
we 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 a
density based time dependent theory. Once we have our theoretical basis for
a density based theory, we reintroduce orbitals through the time-dependent
Kohn-Sham equations [63]. After a quick detour to address some concerns
that might arise dealing with causality, we introduce the equations for the
linear 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 link
to the (local) polarizability, this time in the time-dependent case.
8
25. Chapter 2
Theoretical Background
2.1 Many Body Quantum Mechanics
The evolution of a general quantum state |Ψ⟩, represented here by a vector in
an abstract Hilbert space representation H, is governed by the Hamiltonian
ˆH of the system through the Schrödinger equation1
i
∂
∂t
|Ψ⟩ = ˆH|Ψ⟩. (2.1)
Note that since we will only be interested in the electronic problem, we will
use the Born-Oppenheimer [64] to separate the electronic problem from the
nuclear one. We will also strictly work in the non-relativistic regime.
For the most part, we will be interested in stationary states with definite
energies, for example the ground state (see chapter 5 for time-dependent
systems). These states are solutions to the eigenvalue equation
ˆH|Ψ⟩ = E|Ψ⟩ (2.2)
known as the stationary Schrödinger equation. Alternatively, stationary
states can be obtained by varying ⟨Ψ| ˆH|Ψ⟩ w.r.t. |Ψ⟩ subject to the con-
straint ⟨Ψ|Ψ⟩ = 1. The eigenvalue E then appears as a Lagrange multiplier
for the constraint.
We are interested in systems of N identical particles moving in a given
external field v. In such a case, the Hamiltonian takes on the form
ˆH = ˆT + ˆV + ˆW (2.3)
1
We are working here in atomic units, in which = m = e = 1.
9
26. where ˆT denotes the kinetic energy operator, ˆV describes the interaction of
the particles with the external field, and ˆW is the two-particle interaction
operator. We will usually work in the Schrödinger representation of the
Hilbert space, which uses eigenstates of the coordinate operator ˆr and of the
z-component of the spin operator ˆσ as basis vectors:
ˆr|r⟩ = r|r⟩, ˆσz|s⟩ = s|s⟩. (2.4)
We introduce a combined spin-position variable, x = (s, r). In this rep-
resentation, the N-particle quantum state |Ψ⟩ is represented by an L2-
wavefunction
Ψ(x1, . . . , xn) = ⟨x1 · · · xn|Ψ⟩. (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, |Ψ⟩ must be taken from the anti-symmetric sector
of the N-particle Hilbert space in question, i.e. it must satisfy
Ψ(x1, . . . , xi, . . . , xj, . . . , xn) = −Ψ(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 Hilbert
space, eq. (2.3), to the Schrödinger representation of the N-particle Hilbert
space of wave functions, we write
ˆHΨ(x1, . . . , xn) = ⟨x1 · · · xn| ˆH|Ψ⟩, s (2.8)
where the operator ˆH 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 Schrödinger
representation is given by (still using atomic units)
ˆH = −
1
2
N∑
i=1
∇2
i +
N∑
i=1
v(xi) +
N∑
i<j
w(|ri − rj|)
=
N∑
i=1
h(xi) +
N∑
i<j
w(|ri − rj|), (2.9)
where v(xi) is the potential of the external field acting on particle with spin-
position variable xi and w is the two-particle interaction potential.
10
27. In general, let {|L⟩} be a complete orthonormal set of N-particle states. A
general element of our Hilbert space of states can be expanded as
|Ψ⟩ =
∑
L
|L⟩⟨L|Ψ⟩ =
∑
L
|L⟩CL, (2.10)
where CL is the projection of the eigenstate onto a basis vector |L⟩. Inserting
the expansion of the eigenstate into the Schrödinger equation eq. (2.2) and
projecting into the chosen representation leaves us with
∑
K
(HLK − EδLK) CL = 0, (2.11)
where HLK = ⟨L| ˆH|K⟩ are the matrix elements of the Hamiltonian operator.
Switching back to the Schrödinger representation, suppose we have a com-
plete set of spin-orbitals (single particle wavefunctions) {φℓ}, i.e. a set which
satisfies ∑
k
φk(x)φ∗
k(x′
) = δ(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 Slater
determinant:
ΦL(x1, . . . , xn) =
1
√
N!
det(φℓi
(xk)). (2.13)
Here L is a now multi-index (ℓ1 · · · ℓn) denoting a specific orbital configura-
tion. We can write this determinant as
ΦL(x1, . . . , xn) =
1
√
N!
∑
σ∈Sn
N∏
i=1
sgnσϕℓi
(xσ(i)) (2.14)
=
1
√
N!
( N∧
i=1
ϕℓi
)
(x1, . . . , xn) (2.15)
where SN denotes the symmetric group of order N. In the last line we
have borrowed a notation common in differential geometry, where the wedge
product 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 tensor
product reduces to an ordinary product of functions.
In this case, we can express the matrix elements HLK = ⟨ΦL| ˆH|ΦK⟩ in
terms of the spin-orbitals. Note that the Hamiltonians we are interested in
will consist of (sums of) one- or two-particle operators. In general, we write
ˆO1 =
N∑
i=1
ˆo1(xi) (2.16)
11
28. for a one-electron operator and
ˆO2 =
N∑
i<j
ˆo2(xi, xj) (2.17)
for a two-electron operator.
For one-electron operators, note that
⟨ΦL| ˆO1|ΦK⟩ = N⟨ΦL|ˆo1(x1)|ΦK⟩, (2.18)
while for a two-electron operator one has
⟨ΦL| ˆO2|ΦK⟩ =
N(N − 1)
2
⟨ΦL|ˆo2(x1, x2)|ΦK⟩. (2.19)
We can systematically build up the matrix elements by starting from the
case where K = L, then the case where K and L differ in one index, and so
on.
If K = L, we find
⟨ΦL| ˆO1|ΦL⟩ =
N
N!
∫
dx1 · · · dxn
(
∑
σ∈Sn
N∏
i=1
φℓi (xσ(i))
)
ˆo1(x1)
(
∑
τ∈Sn
N∏
i=1
φℓi (xτ(i))
)
.
(2.20)
Since ˆo1 only acts on electron one, the integration over electrons 2 through
N will be zero unless σ = τ. Once electron one is put into a certain spin-
orbital, the remaining electrons can take on (N −1)! possible configurations.
Thus, the sum over all Sn reduces to a sum over the spin orbitals (selecting
in which orbital you put electron one) multiplied by a factor (N − 1)!:
⟨ΦL| ˆO1|ΦL⟩ =
N∑
i=1
∫
dx1 φℓi
(x1)ˆo1(x1)φℓi
(x1)
=
N∑
i=1
⟨ℓi|ˆo1|ℓi⟩, (2.21)
where we have introduced the shorthand |ℓi⟩ for |φℓi
⟩.
If K and L differ in only one index, say L = (ℓ, ℓ2, . . . , ℓn) and K =
(k, ℓ2, . . . , ℓn), then we find that
⟨ΦL| ˆO1|ΦK ⟩ =
N
N!
∫
dx1 · · · dxn
( ∑
σ∈Sn
φℓ(xσ(1))
N∏
i=2
φℓi (xσ(i))
)
ˆo1(x1)
( ∑
τ∈Sn
φk(xτ(1))
N∏
i=2
φℓi (xτ(i))
)
.
(2.22)
As before, the integration over electrons 2 through N forces σ = τ. How-
ever, now we have the additional constraint that if φℓ and φk don’t contain
12
29. electron one, the matrix element will be zero as well since the spin-orbitals
are orthonormal. Thus we are just left with
⟨ΦL| ˆO1|ΦK⟩ =
N
N!
∑
σ∈Sn−1
∫
dx1 · · · dxn φℓ(x1)
N∏
i=2
φℓi
(xσ(i))ˆo(x1)φk(x1)
N∏
i=2
φℓi
(xσ(i))
=
∫
dx1 φℓ(x1)ˆo1(x1)φk(x1)
= ⟨ℓ|ˆo1|k⟩. (2.23)
In the case that K and L differ in more than one index, we can immediately
see that the matrix elements will be zero. As in the previous cases, the
integration over electrons 2 through N will force σ = τ. Then in the previous
case, since the two orbitals which were different are also orthogonal, the
matrix element would be zero unless we force electron one to be in those
orbitals. Now, however, we have two pairs of orthogonal orbitals remaining,
and while we can force the integration over one pair of those to be nonzero
by putting electron one in it, the integral over the other pair will still yield
zero.
We can use similar reasoning to calculate matrix elements for two-electron
operators. In that case however, matrix elements will only be zero once
K and L differ in three or more indices, since - following the reasoning of
the last paragraph - we can avoid integration over two pairs of orthogonal
orbitals to give zero by putting electrons one and two in those pairs, but
introducing a new pair of orbitals will give zero yet again.
We summarize the results in table Table 2.1. Here, we introduce the notation
for two electron integrals:
⟨ij|ˆo2|kℓ⟩ =
∫
dxdx′
φ∗
i (x)φ∗
j (x′
)ˆo2(x, x′
)φk(x)φℓ(x′
). (2.24)
Note that in the specific case where ˆo2 is the Coulomb interaction, o2(xi, xj) =
r−1
ij , we will drop the operator from this expression and just write
⟨ij|kℓ⟩ =
∫
dxdx′
φ∗
i (x)φ∗
j (x′
)
1
|r − r′|
φk(x)φℓ(x′
). (2.25)
In particular, for the energy E = ⟨L| ˆH|L⟩ we have
E =
N∑
i=1
⟨ℓi|ˆh|ℓi⟩ +
1
2
N∑
i,j=1
(⟨ℓiℓj| ˆw|ℓiℓj⟩ − ⟨ℓiℓj| ˆw|ℓjℓi⟩) (2.26)
13
30. 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 operators
L = K ⟨ΦL| ˆO1|ΦL⟩ =
∑
ℓi∈L⟨ℓi|ˆo1|ℓi⟩
L = (ℓ, ℓ2, . . . , ℓn), K = (k, ℓ2, . . . , ℓn) ⟨ΦL| ˆO1|ΦK⟩ = ⟨ℓ|ˆo1|k⟩
K and L differ by more than one entry ⟨ΦL| ˆO1|ΦK⟩ = 0
Two-electron operators
L = K ⟨ΦL| ˆO2|ΦL⟩ = 1
2
∑
ℓi,ℓj ∈L
(
⟨ℓiℓj|ˆo2|ℓiℓj⟩ − ⟨ℓiℓj|ˆo2|ℓjℓi⟩
)
L = (ℓ, ℓ2, . . . , ℓn), K = (k, ℓ2, . . . , ℓn) ⟨ΦL| ˆO2|ΦK ⟩ =
∑
ℓi∈L
(
⟨ℓℓi|ˆo2|kℓi⟩ − ⟨ℓℓi|ˆo2|ℓik⟩
)
K = (ℓ, ℓ′, ℓ3, . . . , ℓn), K = (k, k′, ℓ2, . . . , ℓn) ⟨ΦL| ˆO2|ΦK⟩ = ⟨ℓℓ′|ˆo2|kk′⟩ − ⟨ℓℓ′|ˆo2|k′k⟩
K and L differ by more than two entries ⟨ΦL| ˆO2|ΦK⟩ = 0
2.2 Hartree-Fock Theory
For an interacting system, one cannot expect a single Slater determinant to
accurately describe the ground state. However, since the ground state can be
described using a variational principle, it is natural to ask, given a set of spin-
orbitals, which Slater determinant best approximates the true N-particle
ground state, i.e. which Slater determinant minimizes the expectation value
of ˆH. This minimum will give an upper limit to the exact ground state
energy. When the number of given spin-orbitals climbs to infinity, the upper
limit 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 a
definite total spin S, a Slater determinant does - in general - not have a
definite total spin. Assume for the moment that the ground state is a closed
shell state (which implies that N must be even). Given two spin one-half
particles, the only way to obtain a spin zero state is if the spin part of the
state is the antisymmetric combination
⟨s1s2|S = 0⟩ =
1
√
2
(
χ+
(s1)χ−
(s2) − χ−
(s1)χ+
(s2)
)
(2.27)
where χ± represent the spin part of a spin-up or -down spin-orbital respec-
tively. The two particles may then occupy the same spatial orbital without
violating the antisymmetry principle. A product of N/2 particle pairs with
spin-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 same
time symmetric w.r.t. pair exchange. Extending this to a Slater determinant
of spin-orbitals, with each orbital doubly occupied, will lead to a spin-zero
state satisfying the correct antisymmetry.
Taking such a Slater determinant |Φ⟩ and still using the shorthand |i⟩ for
14
31. the orbital ϕi(x), one can see that using eq. (2.26)
E = ⟨Φ| ˆH|Φ⟩ = 2
N/2
∑
i=1
⟨i|ˆh|i⟩ + 2
N/2
∑
i,j=1
⟨ij| ˆw|ij⟩ −
N/2
∑
i,j=1
⟨ij| ˆw|ji⟩, (2.28)
where we call the first term the one-particle energy, the second one the
Hartree energy and the last one the exchange energy. The Hartree term, if
w(rij) equals the Coulomb repulsion, r−1
ij , is the classical Coulomb repulsion
term. The exchange term on the other hand is a purely quantum mechanical
term which has no classical analogue.
The next step is to minimize this expression under variation of the orbitals
with the constraint that they must remain orthonormal. Introducing La-
grange multipliers εk leads to the condition2
(ˆhϕk)(r) + (ˆvHϕk)(r) + (ˆvxϕk)(r) = ϕk(r)εk, (2.29)
where the Hartree operator boils down to multiplication with the Hartree
potential,
vh(r) = 2
N/2
∑
j=1
∫
dr′
ϕ∗
j (r′
)w(|r − r′
|)ϕj(r′
), (2.30)
and the effect of the exchange potential operator is given by
(ˆvxϕk)(r) = −
N/2
∑
j=1
∫
dr′
ϕ∗
j (r′
)w(|r − r′
|)ϕk(r′
)ϕj(r). (2.31)
We can write the Hartree-Fock equations eq. (2.29) in short as
ˆFϕk = εkϕk, (2.32)
which takes the form of an effective one-particle Schrödinger equation with
the Fock operator ˆF taking the role of the Hamiltonian,
ˆF = −
1
2
∇2
+ ˆveff , (2.33)
with an effective potential ˆveff = ˆv + ˆvh + ˆvx called the mean field.
The Hartree-Fock method gives us a set of spin-orbitals {φk} with energies
εk. The Hartree-Fock ground state |Φ0⟩ is the determinant formed from the
N spin-orbitals with the lowest orbital energies (called the occupied orbitals).
2
as before we will interpret the Lagrange multiplier as an energy, specifically the orbital
energy.
15
32. The remaining spin-orbitals are called virtual orbitals. From eq. (2.29) we
see that
N/2
∑
i=1
εi =
N/2
∑
i=1
⟨i|ˆh|i⟩ + 2
N/2
∑
i,j=1
⟨ij| ˆw|ij⟩ −
N/2
∑
i,j=1
⟨ij| ˆw|ji⟩. (2.34)
This leads to
Ehf =
N/2
∑
i=1
(εi + ⟨i|ˆh|i⟩) = 2
N/2
∑
i=1
εi − ⟨ ˆW⟩, (2.35)
which is the sum over all occupied orbital energies εi minus the double-
counted interaction energy.
In theory, the set {φk} is infinite. In practice, one solves the HF equations by
introducing a set of spatial basis functions {ϕµ|µ = 1, . . . , K}, which leads
to a set of 2K spin-orbitals, N of which will be occupied and 2K − N of
which will be virtual. Of course, as K → ∞, the HF energy E0 = ⟨Φ0| ˆH|Φ0⟩
will 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, {φ}, the
HF ground state is only one of the possible
(2K
N
)
determinants that can
be formed. The other possible determinants can be described by how they
differ form the HF ground state. For example, if we relabel the HF spin-
orbitals in order of ascending energy, the ground state corresponds to the
specific configuration L = (1 · · · ij · · · N). A singly excited determinant then
corresponds to a configuration L′ = (1 · · · ai · · · N), i.e.
|ΦL′ ⟩ = |ϕ1 · · · ϕaϕi · · · ϕn⟩, (2.36)
a doubly excited determinant to a configuration L′′ = (1 · · · ab · · · N), et
cetera up to N-tuply excited determinants. These excited determinants serve
as N-electron basis functions to expand exact N-electron states in,
|Ψ⟩ = c0|Φ0⟩ +
∑
L′
cL′
|ΦL′ ⟩ +
∑
L′′
cL′′
|ΦL′′ ⟩ + . . . +
∑
LN
cLN
|ΨLN ⟩, (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 the
ground and excited state are then given by the eigenvalues of the Hamilto-
nian matrix ⟨ΦL| ˆH|ΦK⟩, the lowest eigenvalue corresponding to the exact
ground state energy E0
3. Since each excited determinant is specified by a
3
Or rather, exact within the Born-Oppenheimer approximation and without taking
into account relativistic effects.
16
33. certain configuration of spin-orbitals this technique is called configuration
interaction.
As in theory the number of spin-orbitals is infinite, the number of excited
determinants is also infinite. However, even if we only use a finite basis
to expand the spin-orbitals in, the number of excited determinants quickly
becomes to large to handle. In the case one does use all possible excited
determinants we call that procedure full CI. Note that for finite K, the
(2K
N
)
determinants don’t form a complete basis set, but diagonalizing the Hamil-
tonian matrix formed with these determinants leads to solutions that are
formally 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 form
⟨ΦL| ˆH|ΦK⟩. See the results summarized in table Table 2.1
2.2.2 Exchange and correlation energy
Assume for a moment that the Hamiltonian ˆH is the simple sum of one-
electron Hamiltonians4 and look at an N-electron wavefunction Φ that is
just the product of spin-orbitals, rather than the antisymmetric product.
This wavefunction is again an eigenfunction of ˆH with eigenvalue E =
∑
εi.
However, this wavefunction is uncorrelated, i.e. the probability density |Φ|2
is simply the product of the individual probability densities of each orbital:
|Φ(x1, . . . , xn)|2
= |φi(x1)|2
|φj(x2)|2
· · · |φk(xn)|2
. (2.38)
Aside from the obvious lack of antisymmetry, there is another reason why this
wavefunction is not appropriate to describe electrons. Since the wavefunction
is uncorrelated, the probability of finding electron 1 at any given point in
space is independent of the position of electron 2. Physically however, both
electrons 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, for
a two electron system where the electrons have parallel spin
∫
dω1dω2 |Φ|2
=
1
2
(
|ϕ1(r1)|2
|ϕ2(r2)|2
+ |ϕ1(r2)|2
|ϕ2(r1)|2
− (ϕ∗
1(r1)ϕ2(r1)ϕ∗
2(r2)ϕ1(r2) + ϕ1(r1)ϕ∗
2(r1)ϕ2(r2)ϕ∗
1(r2))
)
, (2.39)
4
Note that this is more general than setting ˆW = 0 since we could include two-particle
effects in an average way.
17
34. where the extra cross term introduces correlation. Setting r1 = r2 shows
that indeed the probability of finding two electrons at the same point in
space with parallel spins is zero. Note that it is the motion of electrons with
parallel spins that becomes correlated - electrons with anti-parallel spins
remain uncorrelated: for electrons with opposite spins we find
∫
dω1dω2 |Φ|2
=
1
2
(
|ϕ1(r1)|2
|ϕ2(r2)|2
+ |ϕ1(r2)|2
|ϕ2(r1)|2
)
. (2.40)
Setting r1 = r2 here yields a non-zero result, so the probability of finding
two electrons with opposite spins at the same point in space is non-zero.
In general we say that the determinantal wavefunction is an uncorrelated
wavefunction 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 system
are the eigenvalues of the Hamiltonian matrix, with the lowest one being the
exact (non-relativistic and within the Born-Oppenheimer approximation)
ground state energy E0. Since Hartree-Fock only incorporates exchange, we
define 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 of
freedom, it quickly becomes very difficult to handle. A natural question that
arises is thus if we really need all those degrees of freedom. We have seen that
in the calculation of matrix elements for one- and two-electron operators one
integrates 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 typically
sufficient 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 in
finding these reduces the number of physically relevant degrees of freedom.
In explaining the basics of DFT in the following paragraphs we will mainly
follow Eschrig’s [65] approach in which the physical but also the mathemat-
ical foundations of DFT are described with great rigour.
18
35. 2.3.1 The Particle Density
We will introduce the particle density operator through the density matrix
of the system. The benefit of this is that while in most cases one cannot
express the kinetic energy in terms of the particle density, you can express
it using density matrices.
The (spin-dependent) single-particle density matrix of a state |Ψ⟩ is defined
by
γ1(x, x′
) = N
∫
dx2 · · · dxn Ψ(x, x2, . . . , xn)Ψ∗
(x′
, x2, . . . , xn). (2.42)
From this definition it follows that (assuming |Ψ⟩ is normalized)
trγ1 = N. (2.43)
Note that for an (anti)-symmetric product of spin orbitals φi(x) the single-
particle density matrix is given by
γ1(x, x′
) =
∑
i
φi(x)φi(x′
). (2.44)
The probability density of measuring one of the particles at r is given by
ρ(r) = γ1(r, r) =
∫
ds γ1(x, x). (2.45)
Note that ∫
ρ = trγ1 = N. (2.46)
The particle density operator ˆρ is defined by (for an N-particle system)
ˆρ(x) =
N∑
i=1
δ(r − ˆri)δsˆσi
, (2.47)
where ˆri is the position operator for particle i and ˆσi is its spin operator. In
general, expressions like δ(r − ˆri) and δsˆσi
should be interpreted as follows:
first act with ˆri or ˆσi on the wavefunction that follows the expression and
use the result to evaluate the Dirac or Krönecker delta.
The spin-dependent number density in the state |Ψ⟩ is then the expectation
value
ρ(x) = ⟨Ψ|ˆρ(x)|Ψ⟩. (2.48)
19
36. To obtain the spatial density function, we integrate out the spin-dependence,
ˆρ(r) =
N∑
i=1
δ(r − ˆri), ρ(r) = ⟨Ψ|ˆρ(r)|Ψ⟩. (2.49)
Take a spin-independent one-particle operator ˆO1 =
∑
ˆo1(ri). Its expecta-
tion value is given by
⟨ ˆO1⟩ =
∫
dx1 . . . dxn Ψ∗
(x1, . . . , xn)
N∑
i=1
ˆo1(ri)Ψ(x1, . . . , xn)
=
∫
dr [ˆo1(r′
)γ1(r, r′
)]r=r′
= tr ˆo1γ1. (2.50)
Here the middle line was obtained because we set r = r′ after ˆo1 acts on
γ1(r, r′) to ensure it only acts on the x variable of Ψ, not of Ψ∗. As an
example, the expectation value of the kinetic energy is given by
⟨ ˆT⟩ = −
1
2
∫
dr [∇2
γ1(r, r′
)]r=r′ (2.51)
= −
1
2
tr ∇2
γ1. (2.52)
Note that if we have
ˆO1 =
N∑
i=1
ˆo1(xi), (2.53)
then
⟨ ˆO1⟩ =
⟨ ∫
dx ˆo1(x)
N∑
i=1
δ(r − r1)δssi
⟩
=
∫
dx o1(x)
⟨ N∑
i=1
δ(r − r1)δssi
⟩
=
∫
dx ˆo1(x)ρ(x). (2.54)
In the case of a two-body operator,
ˆO2 =
1
2
N∑
i̸=j
ˆo2(ri, rj), (2.55)
20
37. we find analogously that
⟨ ˆO2⟩ =
∫
drdr′
o2(r, r′
)γ2(r, r′
; r, r′
) (2.56)
where
γ2(x1, x2; x′
1, x′
2)
=
N(N − 1)
2!
∫
dx3 · · · dxn Ψ(x1, x2, x3, . . . , xnΨ∗
(x′
1, x′
2, x3, . . . , xn)
(2.57)
is the spin-dependent two-particle density matrix from which we obtain the
spin-independent version through summation over the relevant spin variables
as in the one-particle case,
ρ2(x1, x2) = 2γ2(x1, x2; x1, x2). (2.58)
In terms of the particle density operator, we can write
⟨ ˆO2⟩ =
⟨
1
2
∫
drdr′
ˆo2(r, r′
)
∑
ij
δ(r − ˆri)δ(r′
− ˆrj) −
∑
i
δ(r − ˆri)δ(r′
− ˆri)
⟩
=
1
2
∫
drdr′
o2(r, r′
)[⟨ˆρ(r)ˆρ(r′
)⟩ − ρ(r)δ(r − r′
)]. (2.59)
With this knowledge, we can write the energy E of a system governed by
the hamiltonian eq. (2.9) as
E = ⟨ ˆH⟩ = −
1
2
∫
dr [∇2
γ1(r, r′
)]r=r′ +
∫
dx v(x)ρ(x)
+
1
2
∫
drdr′
ρ(r′
)w(|r − r′
|)ρ(r) +
1
2
∫
drdr′
w(|r − r′
|)h(r, r′
)
= Ekin + Epot + Eh + Exc, (2.60)
where
h(r, r′
) = ρ2(r, r′
) − ρ(r)ρ(r′
). (2.61)
2.3.2 The Hohenberg-Kohn Theorems
From now on, we consider Hamiltonians of the form
ˆH[v] = ˆT + ˆV + ˆW
= −
1
2
N∑
i=2
∇2
i +
N∑
i=1
v(xi) +
1
2
N∑
i̸=j
w(|ri − rj|), (2.62)
21
38. with the independent particle Hamiltonian w = 0 used as a reference system,
H0
[v] = −
1
2
N∑
i=2
∇2
i +
N∑
i=1
v(xi). (2.63)
The potential v will be taken from the set U, which is the set of potentials
such that the energy is finite. We will later elaborate more on this set. The
Hamiltonian ˆH is assumed to be bounded from below, however there might
not be a ground state |Ψ0⟩ minimizing the expectation value of ˆH. Instead,
the ground state energy is defined as
E0[v] = inf{⟨Ψ| ˆH[v]|Ψ⟩|Ψ ∈ Wn}, (2.64)
where the Wn defined as
Wn = {|Ψ⟩ | |Ψ⟩ antisymmetric , ⟨Ψ|Ψ⟩ = 1, ∇iΨ ∈ 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 ’0’
to denote ground state quantities, i.e. we will write E for the ground state
energy E0, as long as it does not lead to confusion.
Note that
E[v(x) + c] = E[v(x)] + Nc. (2.66)
Since the reference level E = 0 can be chosen arbitrarily, this introduces a
gauge freedom in our system. From now on, we will consider potentials v1
and v2 to be different if they differ by more than a constant. We will also
only consider spin-independent potentials.
We define the class of potentials (or rather families of potentials in the sense
that 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 to
say that this choice encompasses all Coulomb type potentials.
Then, for v ∈ Vn the infimum in eq. (2.64) becomes a minimum and we can
write for the ground state energy
E[v] = ⟨Ψ0| ˆH[v]|Ψ0⟩ = ⟨Ψ0| ˆT + ˆW|Ψ0⟩ + ⟨Ψ0|
N∑
i=1
v(xi)|Ψ0⟩, (2.68)
22
39. where |Ψ0⟩ denotes the ground state or - in the case of degeneracy - one of
the ground states. We can rewrite the last term, which is the only system-
specific term, in terms of the ground state density ρ, after which the energy
becomes
E = ⟨Ψ0| ˆT + ˆW|Ψ0⟩ + (v|ρ), (2.69)
where
(v|ρ) =
∫
dx v(x)ρ(x). (2.70)
Obviously, ρ depends on v through the ground state Ψ0 which is uniquely
determined by v up to degeneracy. The tricky part however is finding how
ρ determines v. The (first) Hohenberg-Kohn theorem now states
Any v ∈ Vn is a unique function of the ground state density ρ(x).
Proof. Suppose we have two distinct vi ∈ Vn (i.e. differing by more than a
constant) with the same density ρ. Each of these has its own ground state
Ψvi defined through the respective Schrödinger equations
H[vi]Ψvi = E[vi]Ψvi (2.71)
with associated energy
E[vi] = ⟨Ψvi |H[vi]|Ψvi ⟩. (2.72)
The ground state energy is by definition the variational minimum of the
expectation value of the Hamiltonian, using any other wavefunction will
raise the expectation value. Specifically,
E[v1] < ⟨Ψv2 |H[v1]|Ψv2 ⟩ = ⟨Ψv2 |H[v2]|Ψv2 ⟩ + (v1 − v2|ρ),
E[v2] < ⟨Ψv1 |H[v2]|Ψv1 ⟩ = ⟨Ψv1 |H[v1]|Ψv1 ⟩ + (v2 − v1|ρ). (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 ρ(x), there is at most one potential v(x) (up
to a constant) for which ρ(x) is the ground state density. Of course, the
converse is always valid: if v1 and v2 are different potentials in Vn (i.e. they
differ by more than a constant), they lead to 2 different hamiltonians H[vi],
each with their own ground states Ψvi and densities ρvi .
23
40. Complementary to the class of potentials Vn, we define the class of densities
An as
An = {ρ(x)|ρ comes from an N-particle ground state} (2.75)
which is called the class of (pure-state) v-representable densities. Using this
notation, we can think of the first Hohenberg-Kohn theorem as defining a
mapping between An and Vn where each element of Vn (viewed as a family
of gauge equivalent potentials) is the image of at least one An (more than
one implies degeneracy) but two distinct (up to degeneracy) elements of An
always map to different elements of Vn.
The second Hohenberg-Kohn theorem introduces a variational principle for
the electronic energy. It uses a new functional, called the (universal) Hohenberg-
Kohn Functional, defined on the class of densities An as
Fhk[ρ] = E[vρ] − (vρ|ρ), (2.76)
where vρ is the image of ρ under the first Hohenberg-Kohn theorem.
For any given ρ, the first Hohenberg-Kohn theorem assures that we have an
associated potential vρ, which in turn defines a Hamiltonian ˆH[vρ] which has
a ground state Ψρ with ground state energy E[vρ] = ⟨Ψρ| ˆH[vρ]|Ψρ⟩. Taking
any other v ∈ Vn independently of ρ, (i.e. v ̸= vρ), we find
Fhk[ρ] + (v|ρ) = ⟨Ψρ| ˆH[v]|Ψρ⟩
≥ ⟨Ψ[v]| ˆH[v]|Ψ[v]⟩
= E[v]. (2.77)
Thus the second Hohenberg-Kohn theorem (also known as the Hohenberg-
Kohn variational principle) reads
E[v] = min
ρ∈An
{Fhk[ρ] + (v|ρ)} , v ∈ Vn. (2.78)
As a side note, the original paper of Hohenberg and Kohn defined the uni-
versal functional as
Fhk[ρ] = ⟨Ψ0| ˆT + ˆW|Ψ0⟩, (2.79)
which is only possible if we confine the study to the classes V′
n of potentials
having a non-degenerate ground state and A′
n of densities coming from a
non-degenerate ground state. In that case, the mapping from ρ to v is not
only single-valued and surjective (each v ∈ Vn is the image of at least one
ρ ∈ An), it is also injective (two distinct densities are never mapped to the
same potential) and thus bijective.
24
41. The variational principle can be rewritten as as a stationary principle for the
energy w.r.t variations in the density subject to the constraint
∫
dr ρ = N.
We introduce a Lagrange multiplier µ for this constraint:
δ
{
E[vρ] − µ
( ∫
dr ρ − N
)}
= 0, (2.80)
where we explicitly wrote vρ to indicate that ultimately the energy depends
on the density. From this we retrieve the Euler-Largrange equation for DFT:
δE[vρ]
δρ
− µ = 0. (2.81)
Since E = Fhk + (v|ρ), this is equivalent to
δFhk
δρ
= µ − v. (2.82)
The Lagrange multiplier µ is called the electronic chemical potential.
2.3.3 Constrained Search
The Hohenberg-Kohn functioal was defined as
Fhk[ρ] = E[vρ] − (vρ|ρ) (2.83)
for ρ ∈ An. The Hohenberg-Kohn variational principle in turn defines the
energy as
E[v] = min
ρ∈An
{
Fhk[ρ] + (v|ρ)
}
(2.84)
for v ∈ Vn. This presents us with a v-representability problem: the sets Vn
and An are unknown. Levy and Lieb independently found a way around this
problem by defining instead of Fhk the Levy-Lieb functional,
Fll[ρ] = inf
{
⟨Ψ| ˆT + ˆW|Ψ⟩ Ψ ∈ Wn, Ψ → ρ
}
, (2.85)
on the extended domain In, which is the set of N-representable densities,
In =
{
ρ|ρ(x) ≥ 0, ∇ρ1/2
∈ L2
,
∫
ρ = N
}
. (2.86)
Levy and Lieb then went on to prove that any non-negative density that
integrates to N and such that ∇ρ1/2 ∈ L2 comes from a Ψ ∈ Wn, implying
that Fll is well defined. Since E[v] = inf{⟨Ψ| ˆH|Ψ⟩|Ψ ∈ Wn} we find that
E[v] = inf
ρ∈In
{
Fll[ρ] + (v|ρ)
}
. (2.87)
25
42. In other words, for a given density we first search for the wavefunction (which
yields that density) which infinimizes the energy and subsequently we search
for the density that infinimizes the energy. The benefit here is that In is
explicitly known and one can show that An is dense in In. Furthermore, one
can prove that for ρ ∈ In one can replace the infimum in the definition of
Fll by a minimum. The minimizing Ψ does not have to be a ground state
but if it is, we see that Fll[ρ] = Fhk[ρ] on An. This means that Fll is a
continuation 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)+δf(x).
We can write the difference F[f + δf] − F[f] = δF[f] as
∫
dx A(x)δf(x) as
a linear approximation in δf to F. We can think of A(x) as the functional
derivative:
δF =
∫
dx
δF
δf(x)
δf(x). (2.88)
To more formally introduce functional derivatives, we introduce the con-
cepts of Gâteaux derivatives which generalize the notion of the directional
derivative 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 define
the directional derivative of f in a point a in a direction y ∈ Rn {0} as
Dyf(a) = lim
λ→0
f(a + λy) − f(a)
λ
(2.89)
(if the limit exists). Another notation is ∂f
∂y (a). In the specific case that y
is one of the basis vectors ei, we call this a partial derivative and we can
calculate it by taking the one-dimensional derivative w.r.t. the ith variable
while keeping the others fixed.
Our goal here is to extend the previous notions of differentiability to the case
of Hilbert spaces. We will specifically be interested in functionals, i.e. maps
F : X → R where X is a Hilbert space. The directional derivative of F at a
point f ∈ U an open subset of X, in the direction g ∈ X is defined as the
limit
DgF[f] = lim
λ→0
F[f + λg] − F[f]
λ
. (2.90)
For any given f ∈ U, F is said to be Gâteaux differentiable at f if the
directional derivative exists for all g ∈ X and they can be assembled into a
26
43. single map φ : X → R such that DgF[f] = φ[g] for all g ∈ X. We write the
functional φ as
φ[g] =
δF[f]
δf
[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, Thomas
and Fermi were able to derive an expression for the energy for one of the few
systems in which you can express the kinetic energy in terms of the density,
i.e. the homogeneous, independent fermion gas:
Etf = Ctf
∫
dr ρ5/3
(r) +
∫
dr v(r)ρ(r) +
1
2
∫
drdr′
ρ(r′
)w(|r − r′
|)ρ(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 original
system. In this case, the kinetic energy is expressible in terms of orbitals
while 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 particle
approximation. In this case, H = H0 = T + (v|ρ). In other words, the uni-
versal Hohenberg-Kohn functional is exactly the kinetic energy of the ground
state of the non-interacting N-particle system. We define the domain of the
kinetic energy functional T[ρ] to be
A0
n = {ρ(x)|ρ comes from a determinantal N-particle ground state}
(2.93)
and write
T[ρ] = E0
[v0
[ρ]] −
∫
dx v0
[ρ]ρ, ρ ∈ A0
n. (2.94)
Writing the interaction free ground state as a determinant formed from or-
thonormal orbitals ϕi, |Ψ0
0⟩ = 1√
N!
∧
σ ϕi(xσ(i)), as we have seen the density
27
44. of a determinantal state like this is
ρ(x) =
N∑
i=1
ϕi(x)ϕ∗
i (x) (2.95)
while its kinetic energy is
T = ⟨Ψ0
0| ˆT|Ψ0
0⟩ = −
1
2
N∑
i=1
⟨ϕi|∇2
|ϕi⟩. (2.96)
We can rewrite the definition of the kinetic energy functional T[ρ], eq. (2.94),
as the minimization
T[ρ] = min
ϕ∗
i ,ϕi
{
−
1
2
N∑
i=1
⟨ϕi|∇2
|ϕi⟩|⟨ϕi|ϕj⟩ = δij,
N∑
i=1
ϕiϕ∗
i = ρ
}
(2.97)
which extends the definition of T[ρ] beyond A0
n to cases where no v0[ρ] exists
for that ρ.
The Hohenberg-Kohn variational principle, eq. (2.78) then reads
E0
[ρ] = min
ρ
{
T[ρ] + (v|ρ)
}
(2.98)
= min
ϕ∗
i ,ϕi
{ N∑
i=1
(
−
1
2
⟨ϕi|∇2
|ϕi⟩ + ⟨ϕi|v|ϕi⟩
)
|⟨ϕi|ϕj⟩ = δij
}
. (2.99)
Introducing Lagrange multipliers εi for the constraints, we find the Kohn-
Sham equations, which are the one-particle Schrödinger equation for the N
lowest energy orbitals in the non-interacting case,
(
−
1
2
∇2
+ ˆvks(x)
)
ϕi(x) = εiϕi(x), (2.100)
where
ˆvks(x) = ˆv + ˆvJ + ˆvxc. (2.101)
In the interacting case, we write the kinetic energy functional from above as
Ts[ρ] and we split up the total kinetic energy functional in Ts[ρ] and the rest
which is put in an exchange-correlation term Exc[ρ].
28
45. Chapter 3
Linear Response
3.1 Conceptual Density Functional Theory
As we have seen in the previous chapter, the electronic energy is a functional
of the external potential through eq. (2.64). We have spent the last chapter
discussing how, through the Hohenberg-Kohn theorem, we can use the elec-
tronic density as the basic variable in our theory as it determines the external
potential and so the Hamiltonian. In light of chemical reactivity theory how-
ever, we turn back to a description in terms of the external potential and
change the previously fixed variable N to a variable as well. The reason
for this is that chemical reactions can be thought of as perturbations in the
external potential (due to a rearrangement of nuclei) and/or the number of
electrons of the reagents. In other words, we move to a study of the energy
functional E[v, N], the response of which should in theory give us insight into
the reactivity of the system. In order to avoid having to study the energy
response 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
(
∂2
E
∂N2
)
v(r)
(∆N)2
+ . . .
+
∫
dr
(
δE
δv(r)
)
N
∆v(r)
+ ∆N
∫
dr
(
∂
∂N
(
δE
δv(r)
)
N
)
v(r)
∆v(r) + . . .
+
1
2
∫
drdr′
(
δ2
E
δv(r)δv(r′)
)
N
∆v(r)∆v(r′
) + . . .
(3.1)
29
46. 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 a
mix of both, can be viewed as a reactivity index or a response function. For
example,
ρ(r) =
(
δE
δv(r)
)
N
, (3.2)
while the other first order derivative gives
µ =
(
∂E
∂N
)
v
. (3.3)
This links us back to the previous chapter as these two objects are central
to DFT: the density ρ is the titular fundamental object and the chemical
potential µ appears as the lagrange multiplier for the constraint
∫
ρ = 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 to
DFT, the energy functional E[v, N], even at first order, retrieves objects
that are central to DFT.
The study of chemical reactivity through the use of reactivity indices is called
Conceptual Density Functional Theory [12], a branch of theoretical chemistry
developed alongside DFT. The aim of this framework was to formalize certain
well 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, ρ(r) =
(
δE
δv(r)
)
N
and µ =
( ∂E
∂N
)
v(r)
are well
studied and provide a link to DFT as mentioned before. As mentioned in
the foundational work by Parr et al. [14], earlier work done by Iczkowski and
Margrave [66] defined the electronegativity χ of a system as
χ = −
( ∂E
∂N
)
v
(3.4)
leading to the close relation between a fundamental DFT quantity and a well
known but difficult to formally define chemical quantity:
χ = −µ. (3.5)
At second order, the N-derivative
(
∂2E
∂N2
)
v(r)
is identified as the chemical
hardness η whereas the mixed derivative f(r) =
(
∂
∂N
(
δE
δv(r)
)
N
)
v(r′)
is known
as the Fukui function, a concept that is closely related to the frontier MO
concept of Fukui et al. [67]. The second order functional derivative w.r.t. v
is known as the linear response kernel
χ(r, r′
) =
(
δ2E
δv(r)δv(r′)
)
N
. (3.6)
30
47. 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 Acids
and Bases (HSAB) principle. It wasn’t until Parr and Pearson [20] however
that a rigourous definition of the chemical hardness was given as the second
order derivative of the electronic energy w.r.t. the number of electrons. The
related concept of global softness is defined as the inverse of the hardness,
S =
1
η
. (3.7)
The Fukui function [18, 68] can be used to describe the regionselectivity for
soft or orbital-controlled reactions. It can also be used to define the local
softness 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 first
order derivatives mentioned above, this function can be rewritten as
χ(r, r′
) =
(
δ2E
δv(r)δv(r′)
)
N
=
(
δρ(r)
δv(r′)
)
N
, (3.8)
which (as already mentioned in the introduction) gives us the extremely
useful interpretation of the linear response kernel as the change in electron
density in a point r in response to a perturbation of the external potential
in a point r′.
Some third order derivatives have also been studied in the literature [15],
specifically the hyperhardness [15]
(
∂3E
∂N3
)
and the dual descriptor [70]
(
∂f(r)
∂N
)
v(r)
which provides a one shot picture of electrophilic and nucleophilic regions
around 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 partial
derivation w.r.t. particle number N whereas moving to the right represents
functional 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, some
indices 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
48. E[N, v(r)]
( ∂E
∂N
)
v(r)
= µ
(
δE
δv(r)
)
N
= ρ(r)
(
∂2
E
∂N2
)
v(r)
= η
(
∂2
E
∂Nδv(r)
)
= f(r)
(
δ2
E
δv(r)δv(r′)
)
N
= χ(r, r′
)
(
∂3
E
∂N3
)
v(r)
(
∂3
E
∂N2δv(r)
)
= f(2)
(r)
(
∂3
E
∂Nδv(r)δv(r′)
) (
δ3
E
δv(r)δv(r′)δv(r′′)
)
N
Scheme 3.1: Energy Derivatives and Response Functions in the Canonical En-
semble, δn+m
E/∂Nn
δvm
, (m + n ≤ 3)
As mentioned, we will focus on the diagonal second order response function
χ(r, r′), defined as
χ(r, r′
) =
(
δ2E
δv(r)δv(r′)
)
N
=
(
δρ(r)
δv(r′)
)
N
. (3.9)
As we can see from the second equality in eq. (3.9), the linear response kernel
gives a (first order) measure of the change in electronic density in response
to a change in the external potential.
The importance of the linear response kernel in conceptual DFT is evident
from the Berkowitz-Parr relationship [47]:
χ(r, r′
) = −s(r, r′
) +
s(r)s(r′)
S
, (3.10)
where s(r, r′) is the softness kernel,
s(r, r′
) = −
(
δρ(r)
δv(r′)
)
µ
, (3.11)
s(r) is the local softness and S is the global softness. The softness kernel is
the inverse of the hardness kernel η(r, r′) [73] in the sense that
∫
dr′
s(r, r′
)η(r′
, r′′
) = δ(r − r′′
), (3.12)
through which it is ultimately connected to the local hardness η(r) [74–
78]. Note that Senet [28] derived exact functional relations between both
32
49. the linear and non-linear response functions and the ground state density in
terms of the universal Hohenberg-Kohn functional Fhk[ρ].
Besides the more formal work mentioned above, some earlier work done in
our group focussed on the calculation of the linear response kernel using
numerical methods [34–37, 42] and extracting qualitative and quantitative
information about inductive, resonance and hyperconjugation effects. Due
to computational reasons, an atom-atom condensation scheme of χ(r, r′) was
chosen, resulting in
χAB =
∫
VA
∫
VB
drdr′
χ(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 condensed
linear response matrices are Baekelandt et al. [38] and Wang et al. [39], which
use 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:
χ(r, r′
) =
( δ2E
δv(r)δv(r′)
)
N
=
( δρ(r)
δv(r′)
)
N
. (3.14)
Note that this quantity is symmetric in r and r′. As we have noted in
chapter 2, these derivatives are to be understood in the sense of Gâteaux
derivatives (see section 2.4). In the next section, we will discuss a method to
approximate the linear response kernel in terms of molecular (Hartree-Fock
or Kohn-Sham) orbitals using perturbation theory.
The linear response kernel is real-valued and symmetric [33], implying that
its eigenvalues hi, defined by
∫
dr′
χ(r, r′
)ωi(r′
) = hiωi(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, zero
included: ∫
dr′
χ(r, r′
) = 0. (3.16)
33
50. Note that these properties can be interpreted physically: the fact that there
exists a zero eigenvalue indicates that shifting the potential by a constant
leaves the density unchanged,
0 =
∫
dr′
χ(r, r′
) =
∫
dr′
(
δρ(r)
δv(r′)
)
N
, (3.17)
whereas the arbitrarily small eigenvalues indicate that very large changes in
the external potential do not necessarily yield big changes in the density.
3.3 Evaluation of χ(r, r′
)
3.3.1 Numerical evaluation
To find an approach to numerically evaluate the linear response kernel χ(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 δQ/δv(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
(
δQ[v]
δv(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 {βj|j = 1, . . . , K}:
(
δQ[v]
δv(r)
)
N
=
K∑
j=1
qjβj(r) (3.19)
with expansion coefficients qj. Combining both equations we find a set of
linear equations
Q[v + wi] − Q[v] =
K∑
j=1
qj
∫
dr βj(r)wi(r). (3.20)
Being a set of linear equations, we can rewrite them in matrix form as
d = Bq, (3.21)
where
di = Q[v + wi] − Q[v] (3.22)
and
Bij =
∫
dr wi(r)βj(r). (3.23)
34
51. In practice P is chosen to be larger than K and eq. (3.21) is solved via least
squares fitting, where P is varied until the result converges. The perturba-
tions themselves are point charge perturbations (zi)
wi(r) =
zi
|r − Ri|
(3.24)
and the expansion functions are uncontracted s- and p-type Gaussians on
each center. This approach has been used successfully to study the Fukui
function f+(r) in previous work in our group [79].
Following Sablon et al. [34], we can extend this procedure to the second order
derivatives, specifically the linear response function χ(r, r′) being (δ2E/δv(r)δv(r′))N .
Similar to eq. (3.18) we can write
E[v + wi] − 2E[v] + E[v − wi] =
∫
drdr′
χ(r, r′
)wi(r)wi(r′
) (3.25)
and we can expand χ(r, r′) as
χ(r, r′
) =
K∑
k,l
qklβk(r)βl(r′
), (3.26)
which again leads to a set of linear equations which can be written as a
matrix equation
d = Bq (3.27)
where B is now a P × K2 matrix composed of the integrals over the various
basis functions and the external perturbations:
Bj,(k−1)K+l =
∫
dr βk(r)wj(r)
∫
dr′
βl(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(k−1)K+l = qkl (k,l = 1, . . . , K).
Initially, the six dimensional kernel χ(r, r′) was represented by an atom-atom
condensed linear response matrix with elements
χAB =
∫
VA
∫
VB
drdr′
χ(r, r′
), (3.29)
which can be expanded as
χAB =
∑
k∈A
∑
l∈B
qkl
∫
dr βk(r)
∫
dr′
βl(r′
). (3.30)
In this thesis, we focussed on the representation and interpretation of the
full, non-condensed linear response kernel.
35
52. 3.4 A perturbational approach to the linear response
kernel
We will see now how to analytically express the linear response function
using standard perturbational methods.
We assume a single Slater determinant ansantz, be it HF or KS. For the sake
of simplicity, we shall assume a closed shell type system and real orbitals.
Under these assumptions, the density becomes
ρ(r) = 2
N/2
∑
i
φ2
i (r). (3.31)
Assume a perturbation of the external potential δv(r). We can express the
orbitals in a perturbation expansion, which is a formal power series, a gen-
eralization of polynomials which can have an infinite amount of terms:
|φi⟩ = |φ
(0)
i ⟩ + λ|φ
(1)
i ⟩ + · · · (3.32)
where φ
(1)
i represents the first order correction to the unperturbed orbital
φ
(0)
i . For the density, the first order correction is given by
ρ(1)
(r) = 4
N/2
∑
i
φ
(0)
i (r)φ
(1)
i (r). (3.33)
The solutions φ
(0)
i of the unperturbed problem are assumed to form a com-
plete set. This means we can express |φ
(1)
i ⟩ as
|φ
(1)
i ⟩ =
∑
a
Cia|φ(0)
a ⟩. (3.34)
Note that one can prove that we can limit the summation to unoccupied
orbitals1 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-Fock
equation is (switching to bra-ket notation)
F|φi⟩ = εi|φi⟩ (3.35)
1
In the following, indices i, j, k, . . . will be used to denote occupied orbitals while indices
a, b, c, . . . will refer to unoccupied orbitals.
36
53. with
F = −
1
2
∇2
+ veff
= h + G, (3.36)
where h and G combine the one and two electron operators respectively:
h = −
1
2
∇2
+ v, (3.37)
G = vh + vx. (3.38)
We can write the the perturbation expansions
F = F(0)
+ λF(1)
+ . . . (3.39)
εi = ε
(0)
i + λε
(1)
i + . . . (3.40)
|φi⟩ = |φ
(0)
i ⟩ + λ|φ
(1)
i ⟩ + . . . (3.41)
where for F(0) it is assumed that we can explicitely solve the Hartree-Fock
equations and the higher order terms are considered small in comparison to
F(0). Plugging these expansions into the Hartree-Fock equation we retrieve
at zeroth order the Hartree-Fock equations for F(0),
F(0)
|φ
(0)
i ⟩ = ε
(0)
i |φ
(0)
i ⟩, (3.42)
and at first order
(
F(0)
− ε
(0)
i
)
|φ
(1)
i ⟩ +
(
F(1)
)
|φ
(0)
i ⟩ = ε
(1)
i φ
(0)
i . (3.43)
Note that h(1) = δv.
Inserting the expansion eq. (3.34) and taking the inner-product with |φ
(0)
b ⟩
(still assuming real orbitals).
∑
a
Cia⟨φ
(0)
b |F(0)
− ε
(0)
i |φ(0)
a ⟩ = −⟨φ
(0)
b |F(1)
|φ
(0)
i ⟩ + ε
(1)
i ⟨φ
(0)
b |φ
(0)
i ⟩. (3.44)
The last term equals zero since |φb⟩ is an unocccupied orbitals, which are
orthogonal to the occupied orbitals. Introducing the shorthand |i⟩ for |φ
(0)
i ⟩,
this leads to ∑
a
Cia⟨b|ε(0)
a − ε
(0)
i |a⟩ = −⟨b|F(1)
|i⟩ (3.45)
or ∑
a
Cia(ε(0)
a − ε
(0)
i )δab = −⟨b|F(1)
|i⟩. (3.46)
37
54. The independent particle approximation
As a first approximation, we will assume that a perturbation in the external
potential δv does not influence vh or vx through the perturbed orbitals, i.e.
F(1)
= h(1)
= δv. (3.47)
This is known as the independent particle approximation. With this, eq. (3.46)
becomes
Cia(ε(0)
a − ε
(0)
i ) = −⟨a|δv|i⟩, (3.48)
which becomes
Cia = −
⟨a|δv|i⟩
ε
(0)
a − ε
(0)
i
. (3.49)
Random Phase Approximation
The next step is to include the effect of the Coulombic part of G(1):
F(1)
= δv + v
(1)
h , (3.50)
which turns eq. (3.46) into
Cia(ε(0)
a − ε
(0)
i ) = −⟨a|δv|i⟩ − ⟨a|v
(1)
h |i⟩. (3.51)
From eq. (2.30) we see that in general
v
(1)
h (r) = 4
∑
j
∫
dr′
ϕ
(0)
j (r′
)w(|r − r′
|)ϕ
(1)
j (r′
), (3.52)
hence
⟨a|v
(1)
h |i⟩ = 4
∑
j,c
Cjc
∫
drdr′
φ(0)
a (r)φ
(0)
j (r′
)
1
|r − r′|
φ(0)
c (r′
)φ
(0)
i (r)
= 4
∑
j,c
Cjc⟨aj|ci⟩. (3.53)
Then eq. (3.51) becomes
Cia(ε(0)
a − ε
(0)
i ) = −⟨a|δv|i⟩ − 4
∑
j,c
Cjc⟨aj|ic⟩. (3.54)
38
55. 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(ε(0)
a − ε
(0)
i ) = −⟨a|δv|i⟩ − ⟨a|v
(1)
h |i⟩ − ⟨a|v(1)
x |i⟩. (3.56)
From eq. (2.31) we can see that
(v(1)
x ϕk)(r) = −2
∑
j
∫
dr′
ϕ
(0)
j (r′
)w(|r − r′
|)ϕk(r′
)ϕ
(1)
j (r)
= −2
∑
j,c
∫
dr′
ϕ
(0)
j (r′
)w(|r − r′
|)ϕk(r′
)Cjcϕ(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⟩ = −2
∑
j,c
Cjc
∫
drdr′
φ(0)
a φ
(0)
j (r′
)
1
|r − r′|
φ
(0)
i (r′
)φ(0)
c (r)
= −2
∑
j,c
Cjc⟨aj|ic⟩. (3.58)
This leaves us with
Cia(ε(0)
a −ε
(0)
i ) = −⟨a|δv|i⟩−4
∑
j,c
Cjc⟨aj|ic⟩+
∑
j,c
Cjc⟨aj|ci⟩+
∑
j,c
Cjc⟨ac|ji⟩
(3.59)
for the expansion coefficients.
3.4.2 Matrix Formulation
Looking back at eq. (3.59), we can rewrite this as
∑
j,c
Cjc
[
(ε(0)
c − ε
(0)
j )δacδij + 4⟨aj|ic⟩ − ⟨aj|ci⟩ − ⟨ac|ji⟩
]
= −⟨a|δv|i⟩.
(3.60)
Introducing the matrix M formed from elements
Mia,jc = (ε(0)
c − ε
(0)
j )δacδij + 4⟨bj|ic⟩ − ⟨aj|ci⟩ − ⟨ac|ji⟩, (3.61)
39
56. where the elements are labeled by pairs of indices, this can be written as
MC = −δV, (3.62)
where C and δV are column matrices with elements Cia and δVia = ⟨a|δv|i⟩
respectively. Solving for the expansion coefficients yields
C = −M−1
δV, (3.63)
or, for the components,
Cia = −
∑
jc
(
M−1
)
ia,jc
δVjc. (3.64)
From eq. (3.33),
ρ(1)
(r) = 4
N/2
∑
i
φ
(0)
i (r)φ
(1)
i (r)
= 4
∑
i,a
Ciaφ
(0)
i (r)φ(1)
a (r)
= −4
∑
i,a
∑
j,c
(
M−1
)
ia,jc
φ
(0)
i (r)φ(1)
a (r)δVjc. (3.65)
From this, remembering that δVjc are the elements of δv projected into a
basis, we get for the linear response kernel projected in a basis
χ(r, r′
) =
δρ(r)
δv(r′)
= −4
∑
i,a
∑
j,c
(
M−1
)
ib,jc
φi(r)φa(r)φj(r′
)φc(r′
) (3.66)
where the superscripts "(0)" have been dropped to ease the notation.
For example, in the non-interacting case, this reduces to
χ(r, r′
) = −4
∑
i,b
∑
j,c
1
εc − εj
φi(r)φb(r)φj(r′
)φc(r′
)δijδbc
= −4
∑
i,b
φi(r)φb(r)φi(r′)φb(r′)
εb − εi
. (3.67)
40
57. 3.4.3 Kohn-Sham
The results from Hartree-Fock theory can be readily converted to Kohn-Sham
theory through eq. (3.46), which becomes
∑
a
Cia(ε(0)
a − ε
(0)
i )δab = −⟨b|δvks|i⟩, (3.68)
where, from the definition of the Kohn-Sham potential
δvks = δv + δvh + δvxc. (3.69)
In the independent particle case, where δvks = δv, the matrix M again
consists of
Mia,jc = (εc − εj)δacδij, (3.70)
where the energies are now Kohn-Sham orbital energies. Including the effect
of δv on vh, this becomes
Mia,jc = (εc − εj)δacδij + 4⟨aj|ic⟩. (3.71)
Finally, for the full CPKS approach, i.e. where we include both vh and
vxc in δvks, 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|δvxc|i⟩ =
∫
drdr′
φ
(0)
b (r)
(
δvxc[ρ]
δρ(r′)
δρ(r′
)
)
φ
(0)
i (r)
=
∫
drdr′
φ
(0)
b (r)
(
δ2Exc[ρ]
δρ(r)δρ(r′)
δρ(r′
)
)
φ
(0)
i (r). (3.72)
Using δρ(r′) = ρ(1)(r′) this becomes
⟨b|vxc|i⟩ = 4
∑
j,a
Caj
∫
drdr′
φ
(0)
b (r)φ
(0)
i (r)
δ2Exc
δρ(r)δρ(r′)
φ(0)
a (r′
)φ
(0)
j (r′
).
(3.73)
In the full coupled-perturbed Kohn-Sham case, the matrix elements of M are
Mia,jc = (εc − εj)δacδij + 4⟨aj|ic⟩ + 4⟨ia|fxc(r, r′
)|jc⟩, (3.74)
where
fxc(r, r′
) =
δ2Exc
δρ(r)δρ(r′)
. (3.75)
41
58. These fxc integrals are calculated numerically (using the Becke integration
scheme [81]). In the case of a GGA functional for example, the exchange-
correlation energy takes the general form
Exc[ρ] =
∫
dr exc(ρ(r), σ(r)), (3.76)
where σ(r) is the generalized gradient. The derivative of this functional is
calculated as a Gâteaux derivative (see section 2.4). Note that in the case
of a GGA, changes in the density will also influence the gradient through
which in turn the exchange-correlation energy is affected. The second order
derivative of the energy will thus not only contain a term δ2exc/δρ(r)δρ(r′)
but also e.g. δexc/δσ(r), δ2exc/δρ(r)δσ(r′) et cetera. These derivatives can
be 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 main
focus of the research presented in this thesis. After a brief overview of the
mathematical 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 consists
of equating F(1) or δvks (in the HF or KS case respectively) to the change in
the external potential, δv. The next level consists of including the effect of
the perturbation in the external potential on the Coulombic part vh. Up to
this 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. either
they 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 theory
includes exact exchange whereas Kohn-Sham theory incorporates exchange-
correlation effects. If we were to take the limit case of Kohn-Sham with no
correlation and only exact exchange, both methods would fully agree (again,
up to the specific orbitals used).
In section 3.3.1, we introduced a way to expand a function of two variables
in a basis (eq. (3.26)), in which it is represented as a matrix B. In order
to connect the analytical expressions we introduced in section 3.4 to a more
computer friendly language, we reintroduce basis functions. In this case,
42
59. the basis functions that are used are products of orbitals, one occupied and
one unoccupied: φi(r)φa(r). The resulting expression for the linear response
function written in this "particle-hole" basis is given by the matrix M in-
troduced in section 3.4.2. Even though both matrix expressions (B vs. M)
are fundamentally very different, they perform the same role in their respec-
tive level of theory, namely that of the linear response kernel expressed in
a basis. In the limit of an infinite basis set both descriptions converge to
an exact description. This is where the comparison ends however, as in the
limit of infinite basis functions B becomes theoretically exact whereas M -
while analytical - remains only a first order result.
We finally mention that the difference between the approximated χKS and
the full χ can be written in a compact way, found in Solid State Physics
texts as a Dyson equation [83, 84] (see also Ayers [32]).
From the definition of the response function χ(r, r′) (Scheme 3.1), it follows
that
δρ(r) =
∫
dr′
χ(r, r′
)δvext(r′
). (3.77)
Considering an equivalent non-interacting system, this can also be written
terms of δvKS as
δρ(r) =
∫
dr′
χKS(r, r′
)δvKS(r′
). (3.78)
Using the definition of the Kohn-Sham potential,
δvks = δv + δvh + δvxc, (3.79)
and using (cfr. eq. (3.73))
δvJ + δvxc =
∫
dr′
(
1
|r − r′|
+ fxc(r, r′
)
)
δρ(r′
), (3.80)
we obtain
∫
dr′
χ(r, r′
)δvext(r′
) =
∫
dr′
χKS(r, r′
)
[
δvext(r′
)
+
∫
dr′′
(
1
|r′ − r′′|
+ fxc(r′
, r′′
)
)
δρ(r′′
)
]
(3.81)
43
60. or
∫
dr′
(
χ(r, r′
) − χKS(r, r′
)
)
δvext(r′
) =
∫
dr′
dr′′
dr′′′
χKS(r, r′
)
[
1
|r′ − r′′|
+ fxc(r′
, r′′
)
]
χ(r′′
, r′′′
)δvext(r′′′
). (3.82)
Upon interchanging r′′′ and r′ in the last integral, one obtains
∫
dr′
(
χ(r, r′
) − χKS(r, r′
)
)
δvext(r′
) =
∫
dr′
dr′′
dr′′′
χKS(r, r′′′
)
[
1
|r′′′ − r′′|
+ fxc(r′′′
, r′′
)
]
χ(r′′
, r′
)δvext(r′
), (3.83)
yielding an integral equation involving the two response functions:
χ(r, r′
) − χKS(r, r′
) =
∫
dr′′
dr′′′
χKS(r, r′′′
)
×
[
1
|r′′′ − r′′|
+ fxc(r′′′
, r′′
)
]
χ(r′′
, r′
)δvext(r′
). (3.84)
This expression can be written down more elegantly by combining the Coulomb
and the exchange-correlation contributions into fhxc and introducing the ⋆
operator defined as
F ⋆ G =
∫
dr′′
F(r, r′′
)G(r′′
, r′
). (3.85)
Using this notation, the Dyson equation becomes
χ − χKS = χKS ⋆ fhxc ⋆ χ, (3.86)
yielding
χ = χKS ⋆ (1 + fhxc ⋆ χ) (3.87)
44
61. or
χ = χKS ⋆ (1 − χKS ⋆ fhxc)−1
(3.88)
or
χ−1
= χ−1
KS − fhxc. (3.89)
In principle χ can be obtained from χks in the functional Exc[ρ] is known.
For an extension to the time dependent case we refer to Casida and Huix-
Rotllant [85], Van Leeuwen [86], Ghosh [87], . . . in which the time dependent
perturbation expressed in the frequency domain is δvext(r, ω). In the limit
of ω → 0 the DFT results are retrieved (vide infra chapter 5)
45
63. Chapter 4
Evaluation and Graphical
Representation of the Linear
Response Kernel
4.1 Introduction
The goal of the research presented here was to study the linear response
kernel in its non-condensed form, i.e. χ(r, r′). As we have mentioned before,
most of the work done on the linear response kernel before we started our
research focussed on an atom-atom condensed version. To be able to focus on
the function itself we decided to take a step back and look at atoms. Inspired
by the work of Savin et al. [88], we first calculated the linear response kernel
for light, closed shell elements in the independent particle approximation
with real orbitals, i.e. using
χ(r, r′
) = −4
N/2
∑
i=1
∞∑
a=(N/2)+1
φi(r)φa(r)φa(r′)φi(r′)
εa − εi
. (4.1)
Due to the fact that we work with closed shell elements, spherical symmetry
reduces this to a function χ(r, r′) which helps with the visualization of this
otherwise highly complicated function.
The obvious extension of these results is to calculate the linear response
function for other closed shell atoms throughout the periodic table, i.e. the
noble gasses and the IIA elements [58]. Our final goal however was to sys-
47
64. tematically calculate χ(r, r′) throughout the periodic table. Still working in
the independent particle case, we can extend the previous expression for the
linear response kernel to an expression that also holds for open shell atoms
by splitting up the summation over α and β electrons:
χ(r, r′
) = −2
∑
i,a
ψα
i (r)ψα
a (r)ψα
a (r′)ψα
i (r′)
ϵα
a − ϵα
i
− 2
∑
j,b
ψβ
j (r)ψβ
b (r)ψβ
b (r′)ψβ
j (r′)
ϵβ
b − ϵβ
j
. (4.2)
This allows us to calculate the linear response kernel for all elements from
Hydrogen through Argon [59]. This systematic walk through the periodic
table is discussed in section 4.3.
In section 4.4, we introduce spin polarized versions of the perturbation the-
oretical expressions we have encountered in chapter 3 in two different rep-
resentations and show how to switch between both representations while in
section 4.5 we discuss the graphical representations of these spin polarized
versions of χ(r, r′).
We conclude this chapter with a section on the connection between the linear
response kernel an the the (local) polarizability (section 4.6).
4.2 The Independent Particle Approximation
Our initial steps to investigate the linear response function consisted of cal-
culating said function for light, closed shell elements in the independent
particle approximation, i.e. using
χ(r, r′
) = −4
N/2
∑
i=1
∞∑
a=(N/2)+1
φi(r)φa(r)φa(r′)φi(r′)
εa − εi
. (4.3)
The orbitals contained in this sum over states were obtained using the Gaus-
sian 09 [89] software package at the PBE [90] level of theory and using both
a 6-311+G* [91] and an aug-cc-pVTZ [92–94] basisset. The summation for
these initial results was carried out through use of the Stock software [95]. In
order to ease the visual representation of the linear response kernel, we inte-
grate out the remaining angular dependencies to obtain a quantity χ(r, r′).
48
65. In practice, this means that we numerically integrate r and r′ over two
spheres with radii r and r′ respectively. We then end up with a quantitiy
χ(r, r′) which is easier to visualize through contour plots as the ones we have
introduced before. These spherical integrations were carried out on Lebedev
grids with 74 angular points and a radial spacing of 0.02 a.u. extending 5
a.u. away from the origin.
Figure 4.1: Contour plots for the radial distribution of the linear response kernel
r2
χ(r, r′
)r′2
for He and Be. The orbitals were calculated with both a
6-311+G* basis set and an aug-cc-pVTZ basis set.
The plots in Figure 4.1 (as well as similar plots that will be shown throughout
this thesis) show the ’radial distribution’ of the linear response kernel in the
case of a spherical potential perturbation, i.e. r2χ(r, r′)r′2, here specifically
for He and Be. Note that the quantities on the axes are the distance to the
origin/nucleus for r and r′ which in turn are the coordinates of the points
where we investigate the change in electron density δρ(r) for a perturbation
49
