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AB INITIO DENSITY FUNCTIONAL THEORY
By
IGOR VITALYEVICH SCHWEIGERT
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2005
Copyright 2005
by
Igor Vitalyevich Schweigert
TABLE OF CONTENTSpage
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Ab-Initio Wavefunction-Based Methods . . . . . . . . . . . . . . . 31.2 Kohn-Sham Density Functional Theory . . . . . . . . . . . . . . . 91.3 Problems with Conventional Functionals . . . . . . . . . . . . . . 121.4 Orbital-Dependent Functionals . . . . . . . . . . . . . . . . . . . . 141.5 Ab initio Density Functional Theory . . . . . . . . . . . . . . . . 15
2 EXACT ORBITAL-DEPENDENT EXCHANGE FUNCTIONAL . . . . 17
2.1 Exact Exchange Functional . . . . . . . . . . . . . . . . . . . . . . 172.2 Optimized Effective Potential Method . . . . . . . . . . . . . . . . 192.3 Performance of the Auxiliary-Basis EXX Method . . . . . . . . . 24
3 CORRELATION FUNCTIONALS FROM SECOND-ORDERPERTURBATION THEORY . . . . . . . . . . . . . . . . . . . . . . . 31
3.1 Correlation Functional from Second-Order Perturbation Theory . 313.2 Correlation functional from Second-Order Perturbation Theory
with Partial Infinite-Order Resummation . . . . . . . . . . . . . 333.3 Implementation of the PT2 and PT2SC Functionals . . . . . . . . 393.4 Numerical Tests for Ab initio Functionals . . . . . . . . . . . . . . 40
4 OTHER THEORETICAL AND NUMERICAL RESULTS . . . . . . . . 48
4.1 Connection between Energy, Density, and Potential . . . . . . . . 484.2 Diagrammatic Derivation of the Optimized Effective Potential
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Mixing Exact Nonlocal and Local Exchange . . . . . . . . . . . . 554.4 Second-Order Potential within Common Energy Denominator
Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
iii
APPENDIX
A FUNCTIONAL DERIVATIVE VIA THE CHAIN RULE . . . . . . . . . 69
B SINGULAR VALUE DECOMPOSITION . . . . . . . . . . . . . . . . . 73
C DERIVATIVE OF THE SECOND-ORDER CORRELATION ENERGIES 75
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
iv
LIST OF TABLESTable page
21 Effect of basis set on the performance of the EXX method. . . . . . . 26
22 Effect of the explicit asymptotic term on the performance of the EXXmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
23 Effect of the Singular Value Decomposition threshold on theperformance of the EXX method . . . . . . . . . . . . . . . . . . . . 28
24 Performance of the EXX methods for the 35 closed-shell molecules ofthe G1 test set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
31 Performance of ab initio and conventional correlation functionals inthe high-density limit . . . . . . . . . . . . . . . . . . . . . . . . . . 42
32 Performance of ab initio correlation functionals for closed-shell atoms . 42
33 Density moments of Ne calculated with ab initio DFT, ab initiowavefunction and conventional DFT methods . . . . . . . . . . . . . 46
41 Performance of the hybrid ab initio functional EXX-PT2h withoptimized fraction of nonlocal exchange . . . . . . . . . . . . . . . . 57
v
LIST OF FIGURESFigure page
21 Explicit asymptotic terms for Ne and the corresponding EXXpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
31 Performance of ab initio DFT and ab initio wavefunction methods intotal energy calculations for the G1 test set. . . . . . . . . . . . . . 43
32 Performance of ab initio DFT, ab initio wavefunction, andconventional DFT methods in calculations of the total energy asa function of the bond lengths . . . . . . . . . . . . . . . . . . . . . 45
33 Performance of ab initio DFT, ab initio wavefunction, andconventional DFT methods in dipole moment calculations for theG1 test set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
41 The total energy and first density moment of Be calculated withthe EXX-PT2h functional with various fractions of the nonlocalexchange operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
vi
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
AB INITIO DENSITY FUNCTIONAL THEORY
By
Igor Vitalyevich Schweigert
August 2005
Chair: Rodney J. BartlettMajor Department: Chemistry
Ab initio Density Functional Theory (DFT) is a new approach to the
electronic structure problem that combines elements of both density-functional
and wavefunction-based approaches. It avoids the limitations of conventional
DFT methods by using orbital-dependent functionals based on the systematic
approximations of wavefunction theory.
The starting point of ab initio DFT is the exact exchange functional. This
functional was implemented with the auxiliary-basis Optimized Effective Potential
method. The effect of numerical parameters on the performance of the method was
also examined.
It has been suggested in the literature to use perturbation theory to construct
the correlation counterpart of the exact exchange functional. In this study,
an ab initio correlation functional from second-order perturbation theory was
implemented. However, numerical tests showed that this functional fails to provide
an adequate description of correlation effects in molecules. This problem was
attributed to the poor convergence of the perturbation series based on the Kohn-
Sham determinant and a partial infinite-order resummation of one-body terms was
vii
proposed as a solution. The new functional offers a more balanced description of
correlation effects, as was demonstrated in applications to a number of closed-shell
atoms and molecules. It resulted in energies and densities superior to conventional
(Mller-Plesset) second-order perturbation theory or DFT methods, accurately
reproduced potential energy surfaces, and led to qualitatively correct effective
potentials and single-electron spectra.
An extension of the method based on mixing exact local and nonlocal
exchange and an approximate second-order correlation potential were also
examined.
viii
CHAPTER 1INTRODUCTION
The underlying physical laws necessary for the mathematical theory ofa large part of physics and the whole of chemistry are thus completelyknown, and the difficulty is only that the exact application of these lawsleads to equations much too complicated to be soluble.
P. A. M. Dirac, Proc. Roy. Soc. London, p. 174, 1929
At the microscopic level, a chemical reaction is the transition from one stable
conglomerate of nuclei and electrons (reagent) to another one (product). Given
the initial configuration of the system, the transition properties and the final
state are determined by the interactions of the particles with each other and with
the environment. Since the nature of these interaction is known, it is then the
task of theoretical chemistry to predict the outcome of the reaction by solving
the fundamental equation describing these particles. Diracs famous words state
the ultimate goal of theoretical chemistry the complete substitution of the
experiment by a theoretical calculation and warn about the ultimate difficulty
the immense complexity of the problem. Even now, given all the computational
power at our disposal, the near-exact solutions of the electronic problem are still
limited to few-electron systems.
Facing the intractability of the exact solution, one must rely on
approximations. Although, for systems beyond several thousands particles one
has no choice but to rely on classical mechanics, most chemical phenomena require
a quantum-mechanical description to obtain at least qualitative resemblance
with reality. In quantum theory, a chemical system is described by the molecular
1
2
Hamiltonian (neglecting magnetic and relativistic effects for simplicity)
H = elec.i=1
1
22i
nucl.A=1
1
22A
elec.i
nucl.A
ZAri RA +
elec.i
3
1.1 Ab-Initio Wavefunction-Based Methods
Hartree-Fock Method.
The simplest approximate wavefunction that retains the correct fermion
symmetry is given by the antisymmetric product of single-electron wavefunctions
(x1, .., xN) = (N !)1/2A[1(x1)..N(xN)
], (1-5)
where
A =
P
(1)P P (1-6)
ensures that is antisymmetric with respect to a permutation of the labels of
any pair of electrons. This type of wavefunction can be conveniently written as a
determinant
HF =
1(r1) . . . 1(rN)
. . . . . .
N(r1) . . . N(rN)
(1-7)
and is often called a Slater determinant or single-determinant wavefunction.
In the Hartree-Fock method, the single-electron wavefunctions (or orbitals)
are determined by the condition that the corresponding determinant minimizes the
expectation value of the true many-electron Hamiltonian [1]
EHF =HF
HHF
= min
H, (1-8)
subject to the constraint that the orbitals remain orthonormal,p
q
= pq.
Inserting the expression for from Eq. 1-5 into this expectation value, one
obtains the expression for the Hartree-Fock energy in terms of the orbitals
EHF =HF
HHF
=elec.
i
i
122 + vext
p
+1
2
elec.ij
ij
ij (1-9)
4
whereij
ij is the Dirac notation for the two-electron integrals defined byEq. 1-10.
ij
ij = ijij ij
ji
=
drdr
i (r)j(r
)i(r)j(r)r r
drr
i (r)j(r
)j(r)i(r)r r (1-10)
Requiring that EHF be stationary with respect to an arbitrary variation of
{i} one obtains the Hartree-Fock equations
(122 + vext + vH + vnlx)
p
=
q
pqq
(1-11)
where
vext(r) =nucl.
A
ZAr RA (1-12)
is the external Coulomb field created by the nuclei and
vH(r) =
dr
(r)r r (1-13)
is the Hartree potential (i.e., the Coulomb field created by the total electron
density),
(r) =elec.
i
i (r)i(r), (1-14)
vnlx is the nonlocal exchange operator,
rvnlx
p
=elec.
i
i(r)
dr
i (r)p(r)r r
, (1-15)
and pq are the Lagrange multipliers that ensure the orthonormality of the Hartree-
Fock orbitals.
Note that the number of solutions of Eq. 1-11 is not limited to the number of
electrons. The lowest N solutions are referred to as occupied orbitals (N being the
number of electrons) and the remaining solutions are referred to as virtual orbitals.
5
Using the fact that the Fock operator
f = 122 + vext + vH + vnlx (1-16)
is invariant with respect to any unitary transformation of the occupied orbitals, one
can transform Eq. 1-11 to its canonical form,
fp
= p
p, (1-17)
which is the eigenvalue problem for the Fock operator.
Since the Fock operator depends on {i} through the vH and vnlx, Eq. 1-11 isan integro-differential equation that can be solved iteratively, until self-consistency
is reached. Therefore, the Hartree-Fock approximation belongs to the class of
Self-Consistent Field (SCF) approximations.
One can solve the Hartree-Fock equations numerically. However, a more
practical approach is to use a finite basis set (usually atom-centered Gaussian-type
functions) to expand the HF orbitals. As the result, the Hartree-Fock integro-
differential equations are transformed into a matrix problem.
Electron-Correlation Methods.
The Hartree-Fock method can recover as much as 99% of the total electronic
energy. Still, even the remaining error of 1% is too large on the chemical scale and
may lead to a qualitatively wrong theoretical prediction.
The difference between the SCF and exact solutions is due to electron-
correlation effects. In ab initio electron-correlation methods, one relies on elaborate
many-body techniques to go beyond the SCF approximation and account for the
simultaneous electron-electron interactions. These methods, in contrast to the
relatively simple Hartree-Fock approximation, can be quite challenging conceptually
and computationally.
6
The correlation limit (i.e., the exact solution of Eq. 1-3 in a given basis set)
can be obtained via the Full Configuration Interaction method. In this method, the
correlation correction to the Hartree-Fock determinant is expanded over all possible
excited determinants
FCI = HF +occ.
i
virt.a
Cai ai +
occ.
i6=j
virt.
a 6=bCijC
ababij + . . . (1-18)
where ai , abij , etc. are formed by by substituting several occupied orbitals in the
Hartree-Fock determinant by virtual orbitals, e.g.
ai = (N !)1/2A[1(x1)..a(xi)..N(xN)
]. (1-19)
The expansion coefficients are found from the variational condition on the
expectation value of the true Hamiltonian
EFCI = minCai ,C
abij ,...
FCI
HFCI
FCI
FCI (1-20)
However, the number of possible excited determinants grows exponentially
with the number of electrons and basis functions, therefore, the Full CI method
is computationally intractable for any but very small systems. Among the
approximate electron-correlation methods, the most common are the truncated
and multi-reference Configuration Interaction methods, Coupled-Cluster methods
[2], and Many-Body Perturbation Theory [3]. For example, for systems where
the multi-reference treatment is not necessary (i.e., when the Hartree-Fock
wavefunction dominates the Full CI expansion), the Coupled-Cluster methods
have proved to be the most systematic and computationally robust approach to the
many-electron problem.
7
Many-Body Perturbation Theory.
In some cases, perturbation theory can provide an accurate description of
electron-correlation effects at a significantly lower cost than required by Coupled-
Cluster or multireference methods. For example, second-order Rayleigh-Schrodinger
perturbation theory is the simplest and least expensive ab initio method for
electron correlation. That is why it was chosen as the basis for the ab initio
correlation functional (Chapter 3).
In such perturbation theory, one finds the solution of the many-body problem
(Eq.1-21) using an SCF model (Equations 1-22 and 1-23) as the reference.
H = E (1-21)
(122 + u)
p
= pp
. (1-22)
H0 =
elec.i
(122 + u)
= E0, (1-23)
where is the single-determinant wavefunction constructed from the N lowest
solutions to Eq. 1-22. The remaining eigenfunctions of H0 are obtained by
substituting the corresponding number of occupied orbitals in by the virtual
orbitals.
To do this, the true Hamiltonian is partitioned into the reference Hamiltonian
and perturbation
H = H0 + V (1-24)
where
V = H H0 =elec.
i
vext(ri) +elec
i6=j
1ri rj
elec.i
u (1-25)
The solution to Eq. 1-21 is then found by introducing the perturbation
parameter and expressing the corrections to the reference wavefunction and
8
energy as series of terms of increasing powers of
H = H0 + V (1-26)
= + (1) + 2
(2) + . . . (1-27)
E = E0 + E(1) + 2E(2) + . . . (1-28)
These order-by-order corrections can be found by neglecting all higher terms
from the Schrodinger equation
(E0 H0)(1) = (V E(1)) (1-29)
(E0 H0)(2) = (V E(1))
(1) E(2) (1-30)
and so forth.
Choosing the perturbative corrections to be orthogonal to the reference
wavefunction,(n)
= 0, one can readily obtain the expressions for the order-by-order contributions to the energy by projecting the Equations 1-29 and 1-30
onto the reference space
E(1) =
V (1-31)
E(2) =
V(1) (1-32)
The order-by-order contributions to the wavefunction can be written in terms
of the resolvent operator [4] (the inverse of integro-differential operator E0 H0 inthe Hilbert subspace)
(1) = R0V (1-33)
9
(2) = R0(V E1)(1) = R0(V E1)R0
, (1-34)
where
R0 = QE0 H0 , (1-35)
and Q = 1 is the projector onto the complementary space of (Hilbertspace with
excluded.)The actual expression for the resolvent operator can readily be found by
recognizing that E0 H0 is diagonal in terms of eigenfunctions of H0
R0 =all
n 6=0
n
n
E0 En (1-36)
Note that the Hartree-Fock SCF model presents a special case as the reference
for the perturbation expansion. First, the HF energy is correct through first order
E0 + E(1) =
H0 +
V =
H = EHF (1-37)
Second, the HF SCF Hamiltonian cancels all the effective one-body terms of the
true Hamiltonian, leaving only two-body terms in the perturbation, so that only
double-excited determinants contribute to the second-order energy
E(2)HF =
occ.i,j
virt.
a,b
ijab
2i + j a b (1-38)
1.2 Kohn-Sham Density Functional Theory
Density Functional Theory is an alternative approach to the electronic
structure problem of Eq. 1-3 that uses the electronic density rather than the
wavefunction as the basic variable. The formal basis of DFT is provided by two
theorems introduced by Hohenberg and Kohn [5]. The first theorem establishes
the one-to-one correspondence between the electronic ground-state density and
the external potential. Since it is the external potential that defines a particular
10
molecule, the existence of such a correspondence ensures that the ground-state
electronic density alone carries all the information about the system. In particular,
the ground-state energy can be written as a functional of the density. However,
there is no equation of motion for the electronic density. Instead, one must rely on
the second Hohenberg-Kohn theorem that states that the ground-state energy as a
functional of the density is minimized by the true ground-state density. Therefore,
given the energy functional, one can obtain the ground-state density and energy by
variational minimization of the functional.
However, the formal definition of Density Functional Theory does not tell
how to construct such functional. Several approximate forms have been suggested;
however, they are far from accurate. The kinetic energy of electrons is particularly
difficult to approximate as a functional of the density.
The idea of Kohn and Sham [6] was to use a SCF model (Eq. 1-39) to
transform the variational search over the density into a search over the SCF
orbitals that integrate to a given trial density.
[122 + vs(r)]p(r) = p(r) (1-39)
Such a transformation does not restrict the variational space, provided that every
physically meaningful density correspond to a unique set of SCF orbitals (the
v-representability condition). Not only does the use of the Kohn-Sham SCF model
ensure that the variational search be to fermionic densities, but also it provides
a good approximation for the kinetic energy. Indeed, provided that the orbitals
integrate to the exact density, the so-called noninteracting kinetic energy
Ts =s
elec.
i
1
22
s
=occ.
i
i
122
i
(1-40)
should account for a large part of the true kinetic energy.
11
The remaining unknown terms of the energy functional are grouped into the
exchange-correlation functional
Exc[] = E[] Ts Eext EH (1-41)
where EH is the Hartree energy, which (as well as Ts) can be readily calculated
for a given set of SCF orbitals. Since Ts should reproduce a large part of T ,
this procedure eliminates the necessity to model the entire kinetic energy as a
functional of the density. Thus, it is expected that Exc is easier to approximate as a
functional of the density than E.
Note that according to the definition of the exchange interaction in
wavefunction theory, the exchange component of the exchange-correlation
functional is defined as
Ex =s
Vees
EH (1-42)
and the correlation component is the remaining part
Ec = Exc Ex (1-43)
The Kohn-Sham SCF orbitals are defined by the effective potential vs.
Transforming the variational condition on the energy functional into the condition
for the constrained search over the orbitals, one can obtain
vs(r) =[E[] Ts
]
(r)=
[Eext + EH + Exc
]
(r)= vext + vH(r) + vxc(r) (1-44)
where the exchange-correlation potential is defined as the functional derivative of
the exchange-correlation functional
vxc(r) =Exc(r)
(1-45)
Thus, given an exchange-correlation functional, one defines the exchange-
correlation potential and then solves the Kohn-Sham SCF equations. Note that
12
since it is an SCF model, practical implementation of the KS procedure is very
similar to the Hartree-Fock method. Usually, the SCF orbitals are expanded in
a Gaussian-type atom-specific basis, which transforms the Kohn-Sham integro-
differential equation into a matrix SCF equation. After self-consistency is reached,
the SCF orbitals are guaranteed to reproduce the true density of the many-electron
system. Also, the true energy can be found by inserting this density into the energy
functional.
Virtually all modern implementations of DFT use the Kohn-Sham scheme.
However, the theory still leaves open the question of how to construct the
exchange-correlation functional. Therefore, the principal challenge for the
theoretical development of DFT remains the construction of accurate exchange-
correlation functionals.
1.3 Problems with Conventional Functionals
The conventional approach is to approximate the energy functional as an
analytical expression of the density and its gradients. The effective potential can
then be obtained in analytical form as well, and the KS equations can be solved
readily. This approach started with the simplest Local Density Approximation
(LDA) where the energy is given through an integral of a local functional
of the density. The next-level, Generalized Gradient Approximation (GGA)
functionals, improved on the LDA functional form by including the dependence
on the gradients of the density. This extension provided a certain freedom in
defining the form of the functional, and a number of different forms have been
suggested. Typically, the basic form of a GGA functional is chosen to satisfy
a set of conditions known to be satisfied by the exact functional. The basic
form is then either parameterized to reproduce experimental data (empirical
functionals) or further modified to satisfy an extended set of conditions (non-
empirical functionals).
13
With the conventional functionals, KS DFT surpasses the quality of the HF
method, and becomes comparable with the simplest ab initio correlation methods.
Nevertheless, restricting the functional form to analytical expressions of the density
imposes certain limitations on the energy functional. GGA exchange functionals
are not capable of complete elimination of the spurious self-interaction component
of the Hartree energy. Since the exchange part often dominates the exchange-
correlation energy, the self-interaction error can considerably reduce the accuracy
of the GGA functional. Similarly, semilocal correlation functionals cannot describe
pure nonlocal components of the correlation energy such as dispersion. This
omission greatly reduces the applicability of the conventional KS DFT methods to
weakly-interacting systems.
Another problem is that while the GGA functionals result in relatively
accurate energies, the functional derivatives (i.e., the corresponding KS potentials)
are not nearly as accurate, especially in the inter-shell and asymptotic regions.
Consequently, one should not expect the same level of accuracy for the density
as for the energy. Furthermore, the qualitatively incorrect potentials reduce
substantially the usefulness of the KS orbitals and orbital energies, which are often
used to calculate certain ground-state properties or as the basis for response and
time-dependent KS DFT calculations.
Some of these problems can be addressed without extending the functional
form. Several post-SCF corrections have been suggested to partially remove the
self-interaction error. For example, after the KS equations have been solved, one
can introduce corrections to the energy to include dispersion or ensure the correct
asymptotic behavior of the KS potential. However, these corrections are specific to
the particular functional and class of systems and they likely are incompatible with
each other. Clearly, one needs to go beyond the GGA functional form to resolve
these problems in a consistent and universal fashion.
14
1.4 Orbital-Dependent Functionals
It has now been fully recognized that KS orbitals can provide extra
information about the system that cannot be extracted easily from the density or
its gradients. The next-generation functionals (hybrid- and meta-GGA) augment
the GGA functional form with terms that depend explicitly on the orbital rather
than the density.
An alternative approach is to dismiss completely the conventional hierarchy of
approximations and construct the functional using solely the orbitals. In contrast
to the conventional functionals, orbital-dependent functionals are analytical
expressions of the orbitals (and orbital eigenvalues). They still are implicit
functionals of the density, however. Indeed, the central assumption of KS DFT
is that there exists one-to-one mapping between the exact density and some a local
potential. Therefore, a given density uniquely defines the potential, which, in turn,
uniquely defines the orbitals through the KS SCF equations. Therefore, the orbitals
and explicitly orbital-dependent functional are implicit functionals of the density.
One can think of the KS orbitals as the intermediate step in the mapping from the
density to the energy.
The most significant difference between the orbital-dependent and conventional
functionals is how the corresponding potential (i.e., the functional derivative with
respect to the density) is determined. The conventional functionals are given
as analytical expressions in terms of the density. Therefore, one can take the
functional derivative straightforwardly to obtain an analytical expression for the
potential. The orbital-dependent functionals are analytical expressions in terms
of the orbitals, whose dependence on the density is given through the effective
potential and KS integro-differential equation. Therefore, the analytical expression
for the functional derivative (hence, potential) cannot be obtained directly. Instead,
one must rely on the chain rule to obtain an integral equation for the potential
15
(Chapter 2). This integral equation is identical to the one used in the Optimized
Effective Potential (OEP) method. The OEP method is, therefore, the cornerstone
of DFT with orbital-dependent functionals.
The immediate advantage of the orbital-based approach is that the exact
exchange functional is known in term of orbitals. One can think of the EXX
method as an extension to the idea of Kohn and Sham, where SCF orbitals are
used to calculate both the larger part of the kinetic energy and a (presumably
larger) part of the exchange-correlation energy.
1.5 Ab initio Density Functional Theory
While the EXX functional provides the exact description of the exchange
interactions, it is just a first step towards the exact exchange-correlation functional.
It is the effective description of electron correlation effects that makes KS DFT
a powerful alternative to the ab initio wavefunction methods. Thus, one needs a
correlation functional that can be combined with the EXX functional.
Conventional (GGA or higher-level) correlation functionals are developed
in combination with the corresponding approximate exchange functionals and
often compensate the deficiencies of the latter. For example, the GGA correlation
functionals usually result in correlation potentials that have the opposite sign to
the exact one. The terms correcting the approximate exchange are hidden in the
correlation functionals and inseparable from the true correlation terms.
Thus, it is not surprising that substituting the approximate exchange by
its exact counterpart destroys the balance between the approximate exchange
and approximate correlation components and results in a functional inferior to
the exchange-only approximation. In other words, the conventional correlation
functionals are not compatible with the EXX functional. Thus, the primary
challenge in the orbital-based approach to exchange-correlation functional is
16
to develop an orbital-dependent correlation functional that can be combined
seamlessly with the exact exchange functional.
Ab initio DFT solves the problem of constructing an orbital-dependent
correlation functional by referring to ab initio wavefunction methods. The idea
is simple: the goal of ab initio methods is to calculate the correction to the exact
exchange approximation (i.e., correlation energy) in terms of the SCF orbitals.
Thus, such an energy expression treated as the orbital-dependent functional results
in a correlation functional that can be seamlessly added to the exact exchange
functional.
Ab initio DFT makes a plethora of wavefunction-based approximations
available as the correlation functionals. Unlike the conventional ones, ab initio
functionals are systematically improvable, since one can always use a higher-level
approximation to obtain a more accurate functional. They also have a well-defined
exact limit represented by the FCI method.
The next two chapters describe the formal development, implementation,
and some test applications for the exact exchange functional and the correlation
functional based on second-order perturbation theory. Chapter 4 discusses possible
extensions of the ab initio DFT approach. The results of the study are summarized
in Chapter 5.
CHAPTER 2EXACT ORBITAL-DEPENDENT EXCHANGE FUNCTIONAL
2.1 Exact Exchange Functional
The immediate advantage of constructing the energy functional in terms
of orbitals is that since the exchange energy is defined in terms of orbitals, the
exact orbital-dependent exchange functional is known. Indeed, since the exchange
component of the exchange-correlation energy is defined as
Ex =
Vee EH = 1
2
occ.i,j
ij
ji (2-1)
treating it as an implicit functional of the density results in the exact exchange
(EXX) functional.
The most important feature of the EXX functional is that, unlike any of the
conventional functionals, it completely eliminates the spurious self-interaction
component of the Hartree energy. Similarly, the corresponding EXX potential
cancels the self-interaction component of the Hartree potential. Thus, using the
EXX functional and potential will avoid many pathological problems caused by the
self-interaction error in conventional DFT approximations, both at the energy and
density levels.
The explicit dependence on the orbitals amounts to one complication, however.
Since the EXX functional does not depend on the density explicitly (i.e., it is not
an analytical expression of the density), the functional derivative cannot be taken
directly. Instead, one must rely on the chain rule, which accounts for the implicit
dependence expressing the derivative of interest through the product of known
derivatives.
17
18
The chain rule plays a central role in the orbital-based approach because it
allows to take the functional derivative of an expression in terms of orbitals with
respect to the density. To do that one needs to determine for which derivatives the
expressions are known, and then express the derivative of interest in terms of the
known derivatives.
First, one recognizes that the KS potential is the most convenient variable.
Indeed, the response of the orbitals and orbital energies to a infinitesimally
small change in the potential is readily available through the linear response
KS equations (Appendix A). And so is the response of the density. Thus, the
functional derivatives of the orbital, orbital energies, and density with respect
to the potential are known. Second, since the exchange energy is given as an
analytical expression in terms of orbitals, its derivative with respect to the orbitals
can be obtained directly.
Thus, starting with the definition for the exchange potential
vEXX(r) =EEXX(r)
=
dr
vs(r)
(r)
EEXXvs(r)
(2-2)
and recognizing that it is (r)/vs(r) that is known in analytical form, one
obtains drvEXX(r)
(r)vs(r)
=EEXXvs(r)
(2-3)
Thus, the EXX potential is given through an integral equation
drX(r, r)vexx(r) = w(r) (2-4)
where
X(r, r) =(r)vs(r)
=occ.
i
virt.a
i (r)a(r)a(r)i(r)
i a + c.c. (2-5)
19
and, as shown in Appendix A
w(r) =EEXXvs(r)
=occ.
i
dr
i(r)
vs(r)
EEXXi(r)
=occ.
i
virt.a
i (r)a(r)avnlx
i
i a + c.c. (2-6)
This integral equation was first written in the context of the Optimized
Effective Potential (OEP) method. The OEP method was originally introduced
by Sharp and Horton [7], long before the foundation of the DFT. Their goal was
to find a local approximation to the HF exchange operator. They defined the
optimized potential as the one that makes the single-determinant expectation value
of the true Hamiltonian stationary [8]. Much later was it realized that the OEP
method results in the exact exchange potential in the KS DFT context.
Because of this equivalence, the terms EXX method and OEP method
are often used interchangeably. However, the application of the chain rule is not
limited to the exchange functional. In the next chapter, the chain rule will be
applied to the orbital-dependent correlation functional. Thus, it is preferrably to
use the term OEP method to denote the way to determine the local potential for
a given orbital-dependent functional. Consequently, one refers to the EXX method
as the KS DFT method with the exact orbital-dependent exchange functional and
corresponding potential obtained via the OEP method.
2.2 Optimized Effective Potential Method
The OEP equation is a Fredholm integral equation of the first kind. Its
integral kernel as well as the right-hand side depend on the SCF orbitals and
orbital eigenvalues, therefore, it must be solved simultaneously with the SCF
equations.
Hirata et al.[9] analyzed the integral kernel and showed that it defines the
potential uniquely up to an irrelevant constant if the SCF orbitals form a complete
20
basis set. However, virtually all practical implementations of the SCF procedure
employ a finite basis to represent the orbitals. In this case, the SCF orbitals
do not form the complete set and the integral kernel, in general, has singular
eigenfunctions. This, means that the potential is defined up to a linear combination
of the singular eigenfunctions. Thus, in a practical implementation of the OEP
method, one must exclude the subspace spanned by the singular eigenfunctions of
the kernel from the solution.
The incompleteness of the orbital basis can also lead to the OEP integral
kernel that does not sample certain regions of the real space. For example,
Gaussian-type orbitals, which are typically used as the orbital basis, fall off too
rapidly with increasing r. They decay as er2, while the exact exchange potential
is known to have the 1/r asymptotic behavior. As a result, the OEP kerneldecays too rapidly and does not sample the solution in the asymptotic region.
Consequently, the potential obtained from the OEP equation in the finite orbital
basis can deviate arbitrarily from the exact solution in the asymptotic region .
The first implementation of the SCF procedure with the exchange potential
given by the OEP equation was reported by Talman and Shadwick [8]. They
used a expansion over a spatial grid to solve the integral equation. However, such
grid-based implementation is inevitably limited to atoms, for which the spherical
symmetry permits excluding the angular points from consideration. In the case
of a polyatomic molecule, the number of grid points necessary for an adequate
representation of the EXX potential is significantly larger and a grid-based OEP
method becomes computationally intractable.
Krieger et al.[10] suggested neglecting the orbital structure of the integral
kernel to avoid the solutions of the integral equation, the so-called KLI
approximation. The OEP integral equation in the KLI approximation reduces
to a very simple nonlinear equation for the approximate KLI exchange potential.
21
The KLI equation can easily be solved self-consistently even in the case of
polyatomic molecules. However, the resulting potential does not reproduce the
characteristic bumps of the EXX potential in the inter-shell region. Recently, a
more elaborated approximation to the OEP equation has been suggested, known
as Common Energy Denominator Approximation [11] or Localized Hartree-Fock
[12]. The exchange potential in this approximation more accurately reproduces
the structure of the EXX potential. However, the error introduced by this
approximation still cannot be measured a priori or controlled.
The auxiliary-basis approach [13, 14] presents an attractive alternative to grid-
based and approximate OEP methods. In this method, the potential is expanded
in a finite auxiliary basis set and the OEP integral equation is transformed into
a linear matrix problem. This is very similar to how the integro-differential SCF
equations are solved in the LCAO approximation. The auxiliary-basis approach to
the solution of the OEP equation does not require a fine spatial grid nor does it
introduce any approximation to the kernel. The error introduced by the finite-basis
expansion always can be reduced by increasing the size of the auxiliary basis set.
In this approach, the OEP integral equation
drX(r, r)vexx(r) = w(r) (2-7)
is transformed into a linear matrix problem
aux.
Xu = w (2-8)
by projecting the equation onto a finite basis set:
X(r, r) =aux.
,
X(r)(r) (2-9)
w(r) =aux.
w(r) (2-10)
22
u(r) =aux.
u(r), (2-11)
where the expansion coefficients over the orthonormal set of auxiliary basis
functions dr(r)(r) = (2-12)
are found by solving the linear matrix problem of Eq. 2-8.
As it has been already mentioned, in a finite orbital basis the occupied-
virtual orbital products do not span the entire space and, therefore, the integral
kernel may have nontrivial eigenfunctions with zero eigenvalue. Consequently, the
auxiliary-basis representation of the kernel, X, may have singular eigenvalues
and not be invertible. In this case, one can use the Singular Value Decomposition
(SVD) procedure.
The Singular Value Decomposition (Appendix B) with a given SVD threshold
will provide the approximate solution to Eq. 2-8
u = (XSV D)w (2-13)
that minimizes the error in a least-squares sense
SV D =
w aux.
Xu (2-14)
where the actual value of the residual error, SV D, depends both on the matrix X
and the SVD threshold. In the hypothetical case of the complete orbital basis, one
would have to set the threshold at the hardware-specific numerical precision. In the
case of a finite orbital (and finite auxiliary) basis set, retaining very small singular
values of X may introduce instabilities into the solution and ultimately lead to the
divergence of the iterative solution. Conversely, a large SVD threshold decreases
23
the quality of the approximate solution. Therefore, there is no a priori preferred
value for the threshold and its actual choice is subject to investigation.
Another problem mentioned in the beginning of this section concerns with
the asymptotic behavior of the EXX potential. The exact exchange potential must
decay as 1/r at large r. However, if the OEP kernel is obtained with Gaussian-type SCF orbitals it decays too rapidly and does not sample the potential in the
asymptotic region. The solution to this problem, considered by many authors
[15, 16, 17], is to use a numerical potential that has the asymptotic behavior of
the exact exchange potential. There are several choices for such a potential: the
Fermi-Amaldi scaled Coulomb potential
vfa(r) =N 1
N
dr
(r)r r , (2-15)
the local exchange energy density, defined so that
exx(r) =occ.i,j
i (r)j(r)(r)
dr
j(r)i(r)r r
+ c.c. (2-16)
and others. These potentials depend explicitly on the SCF orbitals and are
guaranteed to have the correct asymptotic behavior. One can then use them as the
explicit asymptotic term in the EXX potential
vexx(r) = veat(r) + vexx(r) (2-17)
and solve the OEP equation for vexx(r)
drX(r, r)vexx(r) = w(r)
drX(r, r)veat(r) (2-18)
Now choosing a larger SVD threshold will ensure that vexx decays rapidly and vexx
becomes veat at large r and thus has the correct asymptotic.
Thus, a practical implementation of the OEP method has several parameters
that will affect the quality of the corresponding EXX potential. The next
24
section discusses the typical choices for these parameters and their effect on
the performance of the EXX method.
2.3 Performance of the Auxiliary-Basis EXX Method
In the conventional DFT approximations, the quality of a given exchange-only
approximation is solely determined by the quality of the approximate functional.
In the EXX method the functional is known and the quality of the method is
determined by the implementation of the OEP method for the corresponding
potential. In this study, the auxiliary-basis implementation is chosen because it
allows applications to general polyatomic molecules and does not introduce any
simplifications to the integral kernel structure. The only error is introduced by the
incompleteness of the orbital and auxiliary bases, but this error can be controlled
by increasing the sizes of the basis sets. Also, given the inevitable incompleteness of
these bases, the quality of the potential is affected by the explicit asymptotic term
used to ensure the correct long range behavior and the SVD threshold.
The task of finding the optimal combination of the numerical parameters is
greatly facilitated by the fact that the reference for the EXX method is given by
the Hartree-Fock method and, thus, is readily available. Indeed, the only purpose
of the EXX method as the first step toward the exchange-correlation functional
is to accurately include the exchange interaction within the DFT framework.
Since the HF results represent the exact exchange limit in a given basis set, an
implementation of the EXX method must be assessed by how well it reproduces the
HF results.
One must understand, however, that these two methods are not identical.
First, both methods result in single-determinants that minimizes the expectation
value of the true Hamiltonian. However, in the Hartree-Fock case, the SCF
operator is not constrained to be local. Therefore, the Hartree-Fock energy is
always lower than the EXX one. Second, the two densities must be very close, but
25
not exactly equal, since the OEP condition ensures that the difference between
the two densities is zero only through first order. There also is a virial condition
on the exact local exchange potential that provide the equivalence between the
HOMO energies in the HF and EXX SCF models. But this equivalence is affected
by the incompleteness of the orbital basis set. Thus, there is no exact equivalence
on the energy, density, or HOMO energy; however, one expects these quantities to
be very close in the HF and EXX methods. Thus, in this work, all three quantities
energy, density, and HOMO energies were compared to provide an exhaustive
assessment of the EXX implementation.
There are exactly four parameters that affect the performance of the auxiliary-
basis OEP method: the size of the orbital basis set, the size of the auxiliary basis
set, the explicit asymptotic term, and the SVD threshold. The most fundamental
effect comes from the incompleteness of the basis set used to expand the molecular
orbitals. The second most important effect should come from the size of the
auxiliary basis. In the current implementation the same Gaussian-type atomic
basis was chosen as both the orbital and auxiliary bases. Conventional atomic
bases may not be the best choice to expand the potential because of the different
physical nature of the orbitals and potential. However, the use of conventional basis
sets significantly facilitates the implementation and also removes a necessity of
developing and testing auxiliary bases.
Table 21 reports the deviations of the EXX total energies, highest-occupied
orbital energies and densities from the Hartree-Fock values as a function of the
Gaussian-type bases. Since it is difficult to meaningfully compare the atomic or
molecular densities directly, the density moments were compared instead. In all
these calculations the exchange energy density was used as the explicit asymptotic
term and SVD threshold of 105 was used. As one can see from these results it is
difficult to define the optimal basis; however, the uncontracted Roos augmented
26
double zeta results in adequate errors for all the quantities. Unlike the single-zeta
bases (6-311G and 6-311G), it is extensive enough to describe the correlation
effects (which will become important in the next chapter) and not as large as
triple- and quadruple-zeta bases so the calculations remain affordable.
Table 21: Effect of basis set on the performance of the EXX method. Shown aredeviations of total energy (in milliHartree), HOMO energy (in eV) and densitymoments (in a.u.) from the HF values. (u) indicate uncontracted basis sets.
Basis E N Density momentsNe < r > < r2 > < r1 > < r2 >
DZP 0.6 0.54 0.000747 0.003584 0.002573 0.189502Roos-ADZP 0.2 0.50 0.000001 0.000151 0.000261 0.020306Roos-ATZP 0.9 1.19 0.000963 0.005382 0.001295 0.080455cc-pVTZ 0.5 7.35 0.000524 0.001968 0.001261 0.053778cc-pVQZ 0.9 1.18 0.000025 0.000270 0.000326 0.062478cc-pV5Z 1.8 0.65 0.000244 0.001795 0.001736 0.1984036-311G(u) 1.4 1.57 0.000032 0.000037 0.000173 0.0084716-311G**(u) 1.4 0.84 0.000030 0.000098 0.000186 0.007592DZP(u) 1.5 0.97 0.000031 0.000069 0.000118 0.005479Roos-ADZP(u) 1.6 0.38 0.000017 0.000117 0.000151 0.001379Roos-ATZP(u) 1.6 0.24 0.000007 0.000182 0.000141 0.001171cc-pVTZ(u) 1.5 0.76 0.000050 0.000040 0.000246 0.000281cc-pVQZ(u) 1.4 0.56 0.000010 0.000015 0.000109 0.008050
H2O < z > < x2 > < y2 > < z2 >
DZP 1.2 0.16 0.0017 0.0035 0.0011 0.0003Roos-ADZP 0.9 3.38 0.0193 0.0064 0.0008 0.0017Roos-ATZP 1.5 0.59 0.0170 0.0051 0.0038 0.0018cc-pVTZ 2.0 0.36 0.0027 0.0057 0.0127 0.0010cc-pVQZ 1.9 0.56 0.0164 0.0027 0.0073 0.00216-311G(u) 1.9 0.14 0.0073 0.0120 0.0144 0.01506-311G**(u) 2.0 3.58 0.0047 0.0006 0.0048 0.0040DZP(u) 2.0 1.47 0.0100 0.0005 0.0032 0.0010Roos-ADZP(u) 2.1 0.13 0.0161 0.0011 0.0043 0.0001Roos-ATZP(u) 2.3 0.16 0.0156 0.0040 0.0059 0.0022cc-pVTZ(u) 2.3 0.36 0.0139 0.0050 0.0111 0.0025cc-pVQZ(u) 2.2 0.42 0.0155 0.0032 0.0065 0.0025
Table 22 compares the performance of the EXX method with and without the
explicit asymptotic terms (EAT). As one can see, using the Fermi-Amaldi potential
as explicit asymptotic term only slightly improves the energy and density, but
27
dramatically changes the value of the HOMO energy. Using the exchange energy
density further improves the performance of the EXX method as this potential is
much better approximation to the EXX potential and, therefore, is better suited to
ensure the correct long range behavior.
Table 22: Effect of the explicit asymptotic term on the performance of the EXXmethod. Uncontracted Roos augmented double-zeta basis set and SVD threshold of105.
E N Density momentsNe < r > < r2 > < r1 > < r2 >None -128.545041 -7.85 18.52 7.891344 9.372690 31.112819 414.831247F-A -128.545041 -0.50 18.50 7.891344 9.372689 31.112820 414.831243exx -128.545041 0.37 18.49 7.891345 9.372692 31.112818 414.831234H2O < z > < x
2 > < y2 > < z2 >None -76.063395 -6.71 7.95 1.987344 5.639106 7.237257 6.509336F-A -76.063409 -0.74 8.37 1.986902 5.640095 7.236823 6.508710exx -76.063437 0.15 8.42 1.986697 5.640234 7.236424 6.508589
Figure 21 shows the Fermi-Amaldi potential, exchange energy density, and
EXX potentials with or without the explicit asymptotic terms. Note that the EXX
potential without the EAT was shifted to facilitate the comparison. As one can
see, the use of the EAT does not affect the shape of the potential but ensures the
correct asymptotic behavior.
Table 23 shows the results of the EXX calculations for Ne and H2O with
gradual increase of the SVD threshold. As can be seen up to about 105 106
atomic units, the SVD threshold had small effect the energy or density. However,
using the thresholds less than these values affects the quality of the HOMO energy.
Based on these results, we conclude that the optimal configuration for the
auxiliary-basis OEP implementation was achieved when we used uncontracted Roos
augmented double zeta, the exchange energy density as the explicit asymptotic
term, and the SVD threshold of 105 or 106.
As the final test for the EXX method, the HF and EXX calculations were
performed for the 35 closed-shell molecules with singlet ground state chosen from
28
-5
-4
-3
-2
-1
0
0 1 2 3 4 5 6 7 8
Exc
hang
e po
tent
ials
for
Ne
Distance from the nucleus, Angstrom
-1/rFermi-Amaldi
XEXX (shifted)EXX (FA) EXX (X)
-3
-2
-1
0.5 1
Figure 21: Explicit asymptotic terms for Ne and the corresponding EXXpotentials. Uncontracted Roos augmented double zeta ANO basis set.
Table 23: Effect of the Singular Value Decomposition threshold on theperformance of the EXX method. Uncontracted Roos augmented double-zeta basisset.
SV D E N Density momentsNe < r > < r2 > < r1 > < r2 >
101 0.0041 1.37 0.030887 0.121066 0.010612 0.217301102 0.0017 1.00 0.006959 0.034878 0.001004 0.004677103 0.0016 0.48 0.000239 0.001822 0.000122 0.001399104 0.0016 0.26 0.000045 0.000105 0.000152 0.001378105 0.0016 0.38 0.000017 0.000117 0.000151 0.001379106 0.0016 0.38 0.000017 0.000117 0.000151 0.001379
H2O < z > < x2 > < y2 > < z2 >
101 0.0062 1.12 0.0531 0.1112 0.0947 0.1061102 0.0023 0.81 0.0044 0.0336 0.0140 0.0225103 0.0022 0.61 0.0112 0.0068 0.0042 0.0071104 0.0021 0.46 0.0140 0.0017 0.0025 0.0029105 0.0021 0.13 0.0161 0.0011 0.0043 0.0001106 0.0021 0.59 0.0147 0.0025 0.0055 0.0016107 0.0021 1.28 0.0148 0.0026 0.0056 0.0017108 0.0021 1.15 0.0148 0.0026 0.0056 0.0017109 0.0021 1.15 0.0148 0.0026 0.0056 0.0017
29
the G1 test set. The G1 set was the first one from the series of sets [18] developed
to test standard electronic structure methods. Although not exhaustive, this set
aims at representing different types of molecules and chemical bonding. Also the
experimental (rather than computed) structures were used to avoid the ambiguity
in comparison. The experimental values for the bond length and angles are readily
available online (Computational Chemistry Comparison and Benchmark DataBase,
http://srdata.nist.gov/cccbdb).
Table 24 reports the HF energies, the absolute and relative differences
between HF and EXX energies, the HF dipole moments and the absolute and
relative difference between the HF and EXX dipole moments.
As one can see, the EXX energies are very close to the HF ones, with the
largest deviations of about 10 milliHartree for SO2, which is less than 2% of the
MP2 correlation energy of 631 milli-Hartree.
30
Table 24: Performance of the EXX methods for the 35 closed-shell molecules ofthe G1 test set. Uncontracted Roos augmented double zeta basis set. Energies arein milliHartree and dipole moments are in Debye.
System Energy DipoleLiH 0.2 0.0025% 0.006 0.10%CH4 1.5 0.0037%NH3 1.2 0.0021% 0.027 1.68%H2O 0.9 0.0011% 0.020 1.02%HF 0.6 0.0006% 0.009 0.47%SiH4 4.5 0.0016%PH3 3.9 0.0011% 0.039 5.78%H2S 3.3 0.0008% 0.030 2.79%HCl 2.3 0.0005% 0.031 2.64%Li2 0.7 0.0046%LiF 0.7 0.0006% 0.004 0.06%C2H2 1.7 0.0022%H2C=CH2 3.0 0.0038%H3C-CH3 3.9 0.0050%HCN 2.4 0.0026% 0.037 1.12%CO 3.0 0.0027%H2C=O 3.7 0.0032% 0.040 1.41%CH3-OH 3.7 0.0032% 0.037 2.05%N2 2.7 0.0025%HO-OH 4.0 0.0027% 0.022 1.31%F2 4.7 0.0024%CO2 5.5 0.0029%Na2 0.5 0.0002%P2 4.6 0.0007%Cl2 7.8 0.0008%NaCl 2.8 0.0004% 0.003 0.03%SiO 4.0 0.0011% 0.024 0.65%CS 6.0 0.0014% 0.045 2.77%ClF 5.8 0.0010% 0.011 0.98%H3Si-SiH3 8.4 0.0015%CH3Cl 5.6 0.0011% 0.034 1.65%H3C-SH 5.9 0.0013% 0.036 2.11%HOCl 5.8 0.0011% 0.012 0.74%SO2 9.2 0.0017% 0.042 2.14%Average 3.7 0.0019% 0.025 1.57%Maximum 9.2 0.0050% 0.045 5.78%
CHAPTER 3CORRELATION FUNCTIONALS FROM SECOND-ORDER
PERTURBATION THEORY
While the EXX functional provides the exact description of the exchange
interaction, one still needs a functional to account for electron-correlation
effects. Unlike the exchange case, there is no closed expression for the correlation
energy. Thus, the principal challenge is to find an adequate approximation for the
correlation functional.
Ab initio Density Functional Theory solves this problem by using the energy
expressions from ab initio wavefunction methods. Most ab initio methods calculate
the correlation energy in terms of the SCF orbitals. Treated as orbital-dependent
functionals, these expressions become the correlation functionals in the ab initio
DFT context.
The simplest ab initio approximation for the correlation energy comes from the
second-order perturbation theory. That is why the second-order energy expression
was chosen as the basis for the initial implementation of ab initio DFT.
3.1 Correlation functional from Second-Order Perturbation Theory
The idea of using perturbation theory to approach the exact correlation
functional belongs to Gorling and Levy [19]. They demonstrated how the exact
correlation functional can be formally constructed from the perturbation expansion
within the Adiabatic Connection formalism. Truncated in second order the
Gorling-Levy perturbation theory give the first approximation to the correlation
functional
E(2) =occ.
i
virt.a
ivnlx vxa2
i a +1
4
occ.i,j
virt.
a,b
ijab2i + j a b (3-1)
31
32
Similarly to the EXX functional, the corresponding functional derivative (i.e.,
the correlation potential) can be obtained using the chain rule
v(2)c (r) =E(2)
(r)=
dr
vs(r)
(r)
E(2)
vs(r)(3-2)
which results in the integral equation for the potential
drs(r, r)v(2)c (r
) =E(2)
vs(r)(3-3)
To derive the algebraic expression for this equation one must take the
functional derivative of the right-hand side. Engel et al. [20] were first to solve
this equation based on the grid-based OEP algorithm. However, they did not
include the potential into the SCF iterations and did not take into account the first
term of Eq. 3-1, which describes the contribution of the single excitations to the
second-order energy. Also, in their numerical solution of the OEP equation, certain
terms were treated separately which apparently lead to a numerical singularity.
Based on this singularity, they concluded that the correlation potential from
the second-order perturbation series does not vanish at large r, as the exact one
should. This raised the question whether the second-order perturbation theory can
lead to a meaningful correlation potential. Niquet et al.[21] disputed this conclusion
and argued that it is the numerical procedure used to calculate the potential that
is the source of this problem. Recently, they showed that the correlation potential
from the second-order perturbation theory has the correct C/r4 asymptote, atleast for closed-shell systems with spherical symmetry.
As it has been discussed in the previous chapter, any grid-based OEP
implementation is limited to small systems. Grabowski et al.[22] rederived the
expression for the second-order potential that included the contributions of single
excitations and implemented it based on the auxiliary-basis OEP method. They
have shown that the PT2 functional results in accurate correlation energies of
33
two-electron systems, quickly approaching the exact correlation energy in the
high-density limit and results in qualitatively correct correlation potentials for
atoms.
3.2 Correlation Functional from Second-Order Perturbation Theory withPartial Infinite-Order Resummation
More recently, the same authors [23] found that the iterative solution of
the OEP equation diverges for the Be atom and the PT2 functional significantly
overestimates the correlation energies for small molecules.
This poor performance of the second-order functional is not surprising. It is
known that the KS reference is usually a bad reference for perturbation expansions.
For example, Warken [24] analyzed the perturbation series based on the KS orbitals
and showed that it usually has radius of convergence smaller than 1. Therefore, the
KS-based perturbation series often diverges for molecular systems.
This presents an even bigger problem for the determination of the potential.
Indeed, truncating a divergent series at some finite order still results in a finite
energy. Moreover, it is known that some asymptotically divergent series may give
decent approximations in lower orders. However, the convergence of the series is
crucial to obtain even a lower-order potential. Because the OEP equation for the
potential is solved iteratively, the large terms will accumulate and the iterative
solution will diverge.
In our opinion, there are two primary cause for the divergence of the series.
First, the KS model features a local SCF potential, and as the result, the virtual
orbitals lie much lower then, for example, Hartree-Fock ones. As the result, the
occupied-virtual energy difference are smaller and the resulting series features
small denominators. Second, unlike the Hartree-Fock case, the KS reference
Hamiltonian is not equal to the one-body part of the true Hamiltonian. Thus, the
perturbation contains a one-body part that can be large and ultimately lead to the
34
divergence of the series. Although not immediately obvious, these two problems
are closely related. Indeed, as it will be shown below, the large one-body terms can
be removed from the series by resumming them to all orders. As the results, the
denominators are formed by diagonal matrix elements of the Fock operator, which
correspond to large differences.
To see where these terms arise, consider the true Hamiltonian in second-
quantized form
H =allp,q
hpqapaq +
1
4
p,q,r,s
pq
srapaqasar
=allp,q
[hpq +
occ.i
pi
qiapaq +1
4
p,q,r,s
pq
srapaqasar (3-4)
Thus, the effective one-body part of the Hamiltonian the Fock operator consists
of the core Hamiltonian and two-electron terms of one-body character
vH + vnlx =occ.
i
pi
qi. (3-5)
Note that the Hartree-Fock model uses the Fock operator as the SCF
Hamiltonian, therefore all the one-body terms are included in the reference
Hamiltonian. The KS Hamiltonian is based on a different potential, therefore the
one-body terms remain in the perturbation
V = H Hs =allp,q
[h+vH +vnlxhs
]pq
apaq +W =allp,q
[vnlxvxc
]pq
apaq +W (3-6)
where W stands for the two-body terms.
These one-body terms can be large and potentially lead to a divergent
perturbation series. In case of ab initio DFT, the situation is slightly improved,
because the use of the exact exchange potential reduces the size of certain one-body
35
terms. Indeed, the OEP equation for the EXX potential
occ.i
virt.a
i (r)a(r)avexx vnlx
i
i a + c.c. = 0 (3-7)
can be viewed as a fitting procedure for a local vexx that minimizes the difference
ofavnlx vexx
i with the given weights. Since the correlation contribution,avc
i, corresponds to the higher orders of perturbation theory (hence, is small),one should expect that the use of the EXX potential as the reference makes the
(a, i) part of the perturbation small.
However, this is not true for occupied-occupied or virtual-virtual elements
of the one-body part of the perturbation. Nothing in the definition of the vexx
indicates that termsivnlx vexx
j or avnlx vexx
b should be small. Thepresence of these terms in the perturbation may ultimately lead to the divergence
of the series.
A usual method to avoid the divergence of a perturbation series is to perform
an infinite-order resummation. In the case of the series under investigation, one is
particularly interested in resummations of the occupied-occupied and virtual-virtual
one-body terms. As we will see below, these terms are particularly easy to resum
because they do not mix the reference and complementary spaces. The infinite-
order resummation of these terms can be effectively performed by redefining the
reference Hamiltonian.
Let us demonstrate how the resummation of these terms can be performed.
First, recall that using the ab initio DFT Hamiltonian leads to the following
perturbation
V = V + V (3-8)
V =occ.
i
virt.a
ivnlx vxc
aai aa + c.c. + W (3-9)
36
V =occ.i,j
ivnlx vxc
jai aj (3-10)
where V indicates the problematic terms.
To find the solution of the Schrodinger equation using perturbation theory, one
partitions the equation(E H)
= 0 (3-11)
into(E0 H0
) (V E) = 0 (3-12)
Since is the ground-state eigenfunction of H0, the following is true
Q(E0 H0
) = 0 (3-13)
where Q = 1
is the projector on the complementary space spanned by allbut ground-state eigenfunctions of H0.
Projecting Eq. 3-12 on the complementary space and adding Eq. 3-13 one
obtains
Q(E0 H0
) Q(V E) = 0, (3-14)
or = Q
E0 H0(V E)
. (3-15)
Applying this relation iteratively, one obtains the series for the true
wavefunction
= +
= + Q
E0H0(V E)
= + Q
E0H0(V E)
+[
Q
E0H0(V E)
]2
=
n=0
[ QE0H0
(V E)]n
(3-16)
37
Note that in the expression, the terms of the sum above do not correspond to
any order of perturbation theory because E has terms of all orders. To obtain
the order-by-order expansion one would have to substitute the order-by-order
expression for E. This, however, is not necessary for the current discussion.
Instead, one proceeds by identifying the problematic terms V and reordering
them
=
+
n=0
[ QE0H0
(V E )]n
+
n=0
[ QE0H0
(V E )]n(V E )]n Q
E0H0(V E )
+ . . . (3-17)
However, since V is such that
Q[V E ]
= 0 (3-18)
one can perform the summation to obtain
=
n=0
{ m=0
[ QE0H0
(V E )]m Q
E0H0
](V E )
}n
=
n=0
{R(V E )
}n (3-19)
where
R =
m=0
[ QE0H0
(V E )]m Q
E0H0 =Q
E0 + E H0 V (3-20)
Thus, the new series where the occupied-occupied and virtual-virtual one-body
terms have been resummed is the perturbation series featuring a new resolvent
38
operator R0. It is based on the new reference Hamiltonian
H 0 = H0 + V
=all.p
phs + u
papap +occ.i,j
ivH + vnlx u
jai aj +virt.
a,b
avH + vnlx u
baaab
=occ.i,j
fij ai aj +
occ.
a,b
fabaaab (3-21)
where the fact thatphs + u
q = pqphs + u
q and
hs + u + vH + vnlx u = hs + vH + vnlx = f (3-22)
is the one-particle Fock operator were used.
Note that since H 0 is no longer diagonal in the basis of eigenfunctions of H0,
its inverse is no longer given by Eq. 1-36.
The case when the reference Hamiltonian is not diagonal in terms of the SCF
eigenfunctions is typical for the generalized Many-Body Perturbation Theory
[2]. Two possible solutions to this problem are to find the inverse (i.e., the
resolvent operator) iteratively or to find a unitary transformation that makes the
occupied-occupied and virtual-virtual blocks of the reference Hamiltonian diagonal.
While these two solutions are formally equivalent, the unitary transformation is
less computationally expensive since it involves only operations with two-index
quantities.
Thus, if at each SCF iteration, one transforms the occupied orbitals so that
fij = ijfii and the virtual orbitals so that fab = abfaa then H0 become diagonal in
this basis. Since this transformation does not mix the occupied and virtual orbitals,
all the physically relevant quantities such as energy and density are not affected
by this transformation. This new set of orbitals is one of the possible noncanonical
representations of the KS orbitals and is called semicanonical because it diagonalize
the occupied-occupied and virtual-virtual block of the Fock operator. This is
39
why the new perturbation series and corresponding functionals are referred to as
semicanonical (SC).
3.3 Implementation of the PT2 and PT2SC Functionals
As in the case of the EXX functional, the central element of implementation of
orbital-dependent correlation functionals is the OEP method for the corresponding
potential. As one can see from Eq. 3-3, the only difference between the integral
equations for the EXX and PT2 (or PT2SC) potentials is on the right-hand side.
Thus, if one has the OEP method implemented for the EXX potential, its extension
for the second-order potential is a tedious, but straightforward task.
As for the EXX potential, the implementation based on the auxiliary-basis
OEP method was used. In this method the integral OEP equation is transformed
into a linear matrix problem by projecting the real-space quantities onto an
auxiliary basis. The only difference is that the calculation of the right-hand side
requires a number of contractions of matrix elements of auxiliary basis functions
with the one- and two-electron integrals.
Thus, a typical SCF iteration with both the EXX and PT2 potentials proceeds
as follows Read in the one-electron integrals and construct the matrix elements of the
core Hamiltonian. If the PT2SC is used, read the two-electron integrals, construct the Fock
matrix, and diagonalize its occupied-occupied and virtual-virtual block toobtain the SCF coefficients in the semicanonical representation.
Read in the two-electron integrals and matrix elements of the auxiliarybasis functions. Transform them using the original or semicanonical SCFcoefficients.
Construct the OEP integral kernel in the auxiliary-basis representation andfind its inverse using the SVD procedure.
Construct the right-hand side of the OEP equation for the EXX potential.Contract it with the inverse of the kernel to obtain the EXX potential in theauxiliary-basis representation. Calculate its matrix elements with respect toatomic and molecular orbitals.
Construct the right-hand side of the OEP equation for the PT2 potentialusing the one- and two-electron integrals and matrix elements of the EXXpotential. Contract it with the inverse of the kernel to obtain the PT2
40
potential in the auxiliary-basis representation. Calculate its matrix elementswith respect to atomic orbitals.
Add the matrix elements of exchange and correlation potential to the coreHamiltonian. Diagonalize it to obtain new SCF coefficients, check theconvergence, and proceed to the next SCF iteration unless the convergence isreached.
The most expensive step in this procedure is the transformation of the two-
electron integrals. As in case of conventional MP2 energy calculation, the first
step in this transformation requires the loop over one occupied and four atomic
indeces. Therefore, the computational cost of the SCF iteration with the EXX
and PT2 potential is similar to the MP2 energy calculation and scales as NoccN4all,
where Nocc is the number of occupied orbitals and Nall is total number of orbitals.
Therefore, the overall cost of a EXX-PT2 calculation is the cost of MP2 times the
number of SCF iterations.
Since the incompleteness of the orbital basis set leads to the singularities
in the integral kernel, one has to use the SVD procedure to find an approximate
solution to the OEP matrix problem. In all the calculations reported below, the
SVD threshold was fixed at 106 atomic units. For example, in the case of Roos
augmented double zeta basis set, which was used as orbital and auxiliary bases in
all molecular calculations, the 106 threshold results in 0 or 1 singularities removed
in the majority of atoms and molecules considered below. The largest number of 3
singular eigenvalues (out of total 58 orbitals) was neglected for NaCl.
In the case of the exchange, the same basis set (contracted or uncontracted
Roos augmented double zeta) was used to expand the orbitals and potentials.
3.4 Numerical Tests for Ab initio Functionals
Correlation Energy in the High-Density Limit.
The first test was for the performance of the ab initio correlation functionals
in the high-density limit. It is known [19] that the contributions of order higher
than second scale as negative powers of the scaling parameter and, therefore,
41
vanish as the scaling parameter approaches infinity (i.e., in the high-density limit).
Therefore, the second-order energy expression is the exact limit for the correlation
functional at infinitely large scaling parameter and the combination of the exact
exchange and second-order correlation functionals becomes the exact exchange-
correlation functional in the high-density limit.
The PT2SC functional is based on the energy expression that does not scale
homogeneously due to the presence of the Fock operator in the denominators.
Nevertheless, it is equivalent to a infinite-order series, where the higher-order terms
again scale as the negative powers of the scaling parameter and vanish as in the
high-density limit. Therefore, the EXX-PT2SC functional must approach the exact
exchange-correlation functional as well.
To test the properties of ab initio functionals in the high-density limit,
we calculated the correlation energies of the series of two-electron atomic
ions with increasing nuclear charge Z (Table 31. Two-electron systems are
particularly convenient because the exact exchange potential is just half of the
Hartree potential, hence, there is no error associated with the auxiliary-basis
implementation of EXX potential. Moreover, the full CI energy is readily available
as only single and double excitations contribute to the correlated wavefunction (the
Coupled-Cluster method with single and double excitations [CCSD] was used to
obtain the full CI energy.)
The results demonstrate that the PT2 energy indeed rapidly approaches the
full CI value. This is in agreement with the results reported by Grabowski et al.
[22]. Note that the GGA correlation functionals such as PBE or LYP do not have
the correct scaling and result in nonvanishing error.
Correlation Energies of Closed-Shell Atoms.
The next test set consisted of the first six closed-shell atoms with singlet
ground states. Table 32 reports the deviation of the PT2 and PT2SC correlation
42
Table 31: Performance of ab initio and conventional correlation functionals in thehigh-density limit. The first column gives full CI correlation energies and theremaining columns give the differences between these values and correlation energycalculated with ab initio and conventional correlation functionals. UncontractedRoos augmented double zeta basis set. All values are in milliHartree.
Z Ion FCI PT2 PT2SC PBE BLYP2 He0+ 38.63 4.02(10%) 6.41(17%) 2.27(6%) 5.05(13%)4 Be2+ 39.78 1.84(5%) 3.42(9%) 5.75(14%) 9.48(24%)
10 Ne8+ 40.19 0.65(2%) 1.47(4%) 7.52(19%) 10.21(25%)12 Mg10+ 40.37 0.54(1%) 1.24(3%) 7.52(19%) 9.97(25%)18 Ar16+ 40.51 0.35(1%) 0.83(2%) 7.61(19%) 9.54(24%)20 Ca18+ 40.48 0.08(0%) 0.99(2%) 7.67(19%) 9.55(24%)
energies from CCSD(T) values. The Coupled Cluster method with single and
double excitations, and noniterative inclusion of triple excitations provides very
accurate energies for closed-shell atoms and molecules at the equilibrium geometries
and will be regarded as the correlation limit for the given basis set. The energies
obtained with second-order Mller-Plesset perturbation theory and CCSD method
are given for comparison.
Table 32: Performance of ab initio correlation functionals for closed-shell atoms.The first column give the CCSD(T) correlation energies and the remaining columnsgive absolute and relative deviations from these values. The MP2 and CCSD valuesare given for comparison. Roos augmented double zeta basis set. All values are inmilliHartree.
Atom CCSD(T) PT2 PT2SC MP2 CCSDHe 37.1 3.3 ( 9%) 6.9 (19%) 6.9 (19%) 0.0 (0%)Be 53.4 N/C 18.4 (34%) 18.5 (35%) 0.2 (0%)Ne 267.5 80.2 (30%) 3.7 ( 1%) 4.5 ( 2%) 4.1 (2%)Mg 51.2 24.5 (48%) 11.7 (23%) 11.9 (23%) 0.7 (1%)Ar 228.5 87.7 (38%) 15.6 ( 7%) 15.8 ( 7%) 4.1 (2%)Ca 84.9 42.8 (50%) 11.6 (14%) 12.0 (14%) 2.0 (2%)
As one can see, the PT2 functional results in accurate correlation energy for
He, but significantly overestimates the correlation energy for larger atoms. Also
the iterative solution for the PT2 potential did not converge in the case of Be. On
the contrary, the PT2SC energy is slightly worse for He, but gives much better
43
estimation of correlation energies for other atoms. Also, the iterative solutions for
the PT2SC potential converged in every case. Note that the PT2SC functional
performs slightly better than MP2.
Correlation Energies of Molecules.
Based on the results for atomic correlation energies one can conclude that
the PT2 functional significantly overestimates the correlation energies, while the
PT2SC functional offers a more adequate description of the correlation effects. To
further verify this conclusion a series of calculations was performed for the same set
of 35 closed-shell molecules that was used in Chapter 2.
Figure 31 reports the relative deviations of the correlation energies from the
CCSD(T) values averaged over 35 molecules
(method) =1
N
Nsystem
Ec[method] Ec[CCSD(T )]
Ec[CCSD(T )]
. (3-23)
0
20
40
60
80
100
Rel
ativ
e de
viat
ion
from
the
CC
SD(T
) co
rrel
atio
n en
ergy
, %
HF EXX
PT2
PT2SC MP2MP3
MP4 CCSD
0
2
4
6
8
10
12
14
PT2SCMP2
MP4
CCSD
Figure 31: Performance of ab initio DFT and ab initio wavefunction methods intotal energy calculations for the G1 test set. Shown are average relative deviationfrom the CCSD(T) values. Roos augmented double zeta basis set.
44
The results are very similar to those for atoms. First, the iterative solution
of OEP equation for the PT2 potential diverged for 19 molecules (including LiH,
NH3, N2, CO, and others). For the remaining 16 molecules, the PT2 functional
overestimated the correlation energy on average by 40%. On the contrary, the
iterative solution for the PT2SC potential converged for all 35 molecules. The
PT2SC functional led to an average error of 11.7%, slightly better than the MP2
value of 12.3%.
Note that for these systems, the HF-based perturbation theory indeed provides
a series that systematically converges to the exact answer. Including higher-order
corrections reduces the error from the MP2 value of 12.3%, to the MP3 value of
8.0% to the MP4 value of 3.8%. This supports the promise of ab initio DFT to
provide a series of systematically improving approximations to the correlation
functional. Of course, the Coupled Cluster method provides a more rapidly
converging series resulting in the average error of 6.1% already at the CCSD level.
It should be emphasized that it is the ab initio character of the PT2 and
PT2SC functionals that allows us to compare the absolute values of the correlation
energy. On the contrary, one cannot directly compare the GGA energies to ab
initio results. It is a well-known fact that the absolute values of the DFT energies
can be very different from the wavefunction correlation limit. Instead, one has
to compare relative quantities like atomization energies to assess the quality of
conventional functionals.
Total Energy as a Function of the Bond Length.
Next test (Figure 32) assessed the performance of ab initio DFT functionals
in description of the potential energy surfaces (i.e., the total energy of a molecule
as a function of the bond length.) Four molecules were chosen to represent different
types of chemical bonds: the ionic bond (HF), symmetric single covalent bond
(F2), double bond (H2O where the hydrogen atoms were simultaneously pulled
45
away from the oxygen atom), and a triple bond (N2). To remove the ambiguity
with respect to the curves absolute position, the curves were shifted vertically
so that all curves cross the CCSDT curve at the experimental bond lengths.
Such a shift not only facilitates the comparison of the shapes, but also allows a
direct comparison between the ab initio and conventional DFT (PBE in this case)
methods.
-100.35
-100.3
-100.25
-100.2
-100.15
0.6 0.8 1 1.2 1.4 1.6 1.8 2
Tot
al e
nerg
y, H
artr
ee
H-F bond length, Angstrom
HF
EXX-PT2 + 0.100HEXX+PT2SC - 0.005H
MP2 - 0.007HPBE + 0.067H
CCSDT
-199.28
-199.26
-199.24
-199.22
-199.2
1.2 1.4 1.6 1.8 2
Tot
al e
nerg
y, H
artr
ee
F-F bond length, Angstrom
F2
EXX-PT2 + 0.210HEXX-PT2SC - 0.007H
MP2 - 0.020HPBE + 0.147H
CCSDT
-76.35
-76.3
-76.25
-76.2
-76.15
-76.1
-76.05
-76
0.6 0.8 1 1.2 1.4 1.6 1.8 2
Tot
al e
nerg
y, H
artr
ee
H-O bond length, Angstrom
H2O
EXX-PT2 + 0.110HEXX-PT2SC - 0.010H
MP2 - 0.012HPBE + 0.062H
CCSDT-109.35
-109.3
-109.25
-109.2
-109.15
-109.1
0.8 1 1.2 1.4 1.6 1.8
Tot
al e
nerg
y, H
artr
ee
N-N bond length, Angstrom
N2
EXX-PT2 + 0.170HEXX-PT2SC - 0.009H
MP2 - 0.016HPBE + 0.095H
CCSDT
Figure 32: Performance of ab initio DFT, ab initio wavefunction, andconventional DFT methods in calculations of the total energy as a function of thebond lengths. Roos augmented double zeta basis set.
As one can see from the figures, the EXX-PT2 functional failed to reproduce
a meaningful energy curve for any of the four molecules. For all the systems except
N2 the EXX-PT2SC curves lies closer to the CCSDT one that MP2. The N2
plot clearly demonstrates that one should not apply a perturbative method to a
multiple-bond breaking problem.
46
Atomic and molecular densities.
The quality of a DFT functional must also be reflected in the converged
density, i.e., the density that minimizes the functional. Since it is difficult
to compare atomic or molecular densities directly, the density moments were
compared instead. Table 33 reports the density moments for Ne calculated
with ab initio DFT, ab initio wavefunction, and conventional DFT methods. The
density obtained with the CCSD(T) method was used as the reference. Also, unlike
in the case of total energies, the GGA density (obtained with PBE functional)
can be directly compared to the ab initio results. As evident from the table, the
overestimation of correlation effects by the PT2 functional leads to the situation
where the exchange-only (EXX) density is better than when the correlation effects
are included (EXX-PT2). However, the EXX-PT2SC functional results in a density
that is closer to CCSD(T) than any other method, in the regions both close to the
nucleus (as sampled by the average values of the negative powers of r) and away
from the nucleus (as sampled by the average values of the positive powers of r).
Table 33: Density moments of Ne calculated with ab initio DFT, ab initiowavefunction and conventional DFT methods. Uncontracted Roos augmenteddouble zeta basis set.
Methodr2
r1
r
r2
r3
r4
r5
PBE 415.180 31.089 8.004 9.833 15.972 32.714 82.036EXX 414.831 31.113 7.891 9.373 14.386 27.200 61.613MP2 414.819 31.104 7.913 9.447 14.599 27.811 63.472EXX-PT2 414.892 31.061 8.075 10.092 16.788 35.351 91.162EXX-PT2SC 414.878 31.097 7.972 9.686 15.398 30.486 72.924CCSD(T) 414.879 31.101 7.955 9.615 15.155 29.650 69.861
As the test for the electronic density of molecules, the molecular dipole
moments were calculated. Figure 33 shows the average (over 22 systems with
nonzero dipole moments) deviation of the computed dipole moments from the
experimental values. For 9 (out of 22) systems where the solution for the PT2
potential converged, the deviation from the experimental dipole moment is no less
47
than 0.13 Debye (HF) with the largest error of 0.56 Debye for H2O. This results in
an average error greater than that of exchange-only methods. Again, EXX-PT2SC
results in dipole moments that improves upon the MP2 values. In the case of EXX-
PT2SC, the largest errors in the computed dipole moment are for SiO: 0.45 Debye
(CCSD:0.14, PBE: 0.22) and SO2: 0.22 Debye (CCSD: 0.10, PBE: 0.16). In the
case of PBE, for NaCl: 0.40 Debye (OEP2(SC): 0.10, CCSD: 0.22) and LiH: 0.27
(OEP(SC): 0.04, CCSD: 0.01). In case of CCSD, the largest errors are for NaCl
and SiO.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Abs
olut
e de
viat
ion
from
exp
erim
enta
l dip
ole
mom
ents
, Deb
ye
HF
EX
X
EX
X-P
T2
EX
X-P
T2S
C
MP2
MP2
+ o
rbita
l rel
axat
ion
effe
cts
CC
SD
CC
SD(T
)
PBE
Figure 33: Performance of ab initio DFT, ab initio wavefunction, andconventional DFT methods in dipole moment calculations for the G1 test set.Shown are the average absolute deviation from the experimental values. Roosaugmented double zeta basis set.
CHAPTER 4OTHER THEORETICAL AND NUMERICAL RESULTS
4.1 Connection between Energy, Density, and Potential
The effective potential of the Kohn-Sham model is defined as the derivative of
the energy functional with respect to the density. This definition follows from the
variational condition on the energy functional. It ensures that if the Kohn-Sham
SCF model generates the density that minimizes the energy functional and, thus, is
equal to the exact ground-state density.
In the context of ab initio DFT, this should mean that the potential defined
through the functional derivative generates the SCF density