Effective One-Electron Potential in the KohnâSham Molecular Orbital

Chapter 2

Effective One-Electron Potential in the Kohn—Sham Molecular Orbital Theory Evert Jan Baerends, Oleg V. Gritsenko, and Robert van Leeuwen

Downloaded by COLUMBIA UNIV on March 16, 2013 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch002

Afdeling Theoretische Chemie, Vrije Universiteit, De Boelelaan 1083, 1081 HV, Amsterdam, Netherlands

Density functional theoryhasreceived great interest mostly because of the accu rate bonding energies and related properties (geometries, force constants) it provides. However, the Kohn-Sham molecular orbital method, that is almost exclusively used, is more than a convenient tool to generate the required electron density. The effective one-electron potential in the Kohn-Sham equations is intimately related to the physics of electron correlation. We demonstrate that it is useful to break down the exchange -correlation part of the potential into a part that is directly related to the total energy (the hole potential or screening potential) and a socalled response part that is related to "response" of the exchange-correlation hole to density change. The latter part is poorly represented by the generalized gradient approximation, explaining why this ap proximation yields accurate total energies but fails for simple orbital related quantities such as the HOMO orbital energy. A simple modelling of the response potential is proposed. We stress the usefulness of Kohn-Sham orbitals in chemistry, both quantitatively (by providing in principle the exact electron density, to be used in density functionals for the energy) and qualitatively (for use in qualitative M O theory).

Although density-functional methods have been around for some time, such as the Thomas-Fermi method and the Xα method, a rigorous foundation has only been given with the formulation of the Hohenberg-Kolm theorems [1]. In the second place, the Kohn-Sham one-electron model [2] has endowed density functional theory with a very expedient and at the same time very successful method for practical applications. The use of both the old and the new density functional methods has been stimulated by their providing relatively high accuracy for relatively low cost. In particular the so-called generalized gradient approximations ( G G A ) [3, 4, 5] are clearly a major step forward (much more so in fact than adding electron gas correlation effects in the local-density

0097-6156/96/0629-0020$15.75/0 © 1996 American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.


2.

BAERENDS ET AL.

One-Electron Potential in Kohn-Sham MO Theory

approximation (LDA) to the exchange-only L D A or Χα method). In chemistry, appli cations almost invariably use the Kohn-Sham molecular orbital model. The exchangecorrelation functional Εχο[p], which is defined in the context of the Kohn-Sham model, has been the primary focus of theoretical work directed towards the immediate goal of obtaining highly accurate energies from Kohn-Sham calculations. In this line of re search, the Kohn-Sham one-electron model has been viewed primarily as a convenient method to generate accurate electron densities, which can be used in some approximate Exc[p] to obtain the total energy. The theoretical status of the Kohn-Sham model has received comparatively little attention, as may be evident from the frequently voiced opinion that the Kohn-Sham orbitals are just a means to generate electron densities but do not have any physical meaning themselves. However, this is a far too restricted view on the Kohn-Sham model. As we will demonstrate, the effective potential of the Kohn-Sham model has an intimate connection with the physics of the exchange and correlation effects in atoms and molecules. The Kohn-Sham orbitals thus represent electrons that move in a potential that is certainly as realistic as the Hartree-Fock "potential" and indeed has some advantages. There is no reason to believe that the Kohn-Sham orbitals are any less "physical" or useful than the Hartree-Fock orbitals and they may and have indeed been used quite succesfully in qualitative M O explana tions that are so typical for present-day chemistry. The physics embodied in the effective one-electron potential of the Kohn-Sham model leads to certain requirements that have to be fulfilled by these potentials. Wellknown ones are the — 1/r behaviour for r —> oo and the finiteness at the nucleus. Other properties, such as certain invariance properties [6], special behaviour at atomic shell boundaries [7, 8, 9] and at the bond midpoint [10, 11] have also been identified. It is a rather stringent test for approximations to the exchange-correlation energy Exc[p], that the exchange-correlation potential νχο that may be derived from it by functional differentiation, obeys these requirements. However, in order to obtain a complete assessment of the quality of trial Εχα[p] and their functional derivatives, it would be necessary to obtain the exact vxc at all points in space. Several procedures have been published [12, 6, 13] to generate the Kohn-Sham potential that belongs to a given density, in the sense that the occupied eigenfunctions of that potential produce the given density. Application of the Hohenberg-Kohn theorem to the Kohn-Sham system of non-interacting electrons proves that the Kohn-Sham potential so obtained is unique. A detailed study of the potentials derived from the current G G A ' s for Εχο[p] shows that, even if these G G A ' s are quite succesful for the energy, their potentials exhibit important deficiencies. This knowledge may be helpful when devising improved functionals for the energy.

One may also wonder if it is possible and useful to model Kohn-Sham potentials as density functionals directly. Since Kohn-Sham potentials can now be obtained for a variety of systems (the first applications to molecules are just appearing [14]), there will be more complete data to judge proposals for model potentials than there is for functionals for the energy. The latter are usually judged only by their performance for the energy, which is an integral over all space of the energy density, in which local deficiencies may have cancelled. Locally different energy densities may lead to (almost)

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

21

22

C H E M I C A L APPLICATIONS O F DENSITY-FUNCTIONAL T H E O R Y


identical energies, and given their non-uniqueness it is hard to judge different proposals for energy densities. The Kohn-Sham potential on the other hand is a unique, local function of r. These considerations naturally lead to the question: is it possible to determine the energy of a system, given its Kohn-Sham potential? This is indeed the case, when one is prepared to perform a line-integral in the space of densities on which Exc[p] and uxcW are defined [15]. This, however, requires knowledge of the K S potential for all p's along the line. It is maybe more practical to study and model Exc[p] and υχο[p] simultaneously, since it is possible to identify a part of the Kohn-Sham potential that can serve as energy density and thus leads directly to the energy, and a part that serves to add features to the potential that are required to generate the exact density, but that do not enter the energy calculation. It is the purpose of this contribution to summarize our present understanding of the Kohn-Sham potentials. The paper is structured as follows. In section 2 we introduce a method to calculate exact K S potentials from exact (in practice, highly accurate) elec tron densities. A comparison is made between such accurate potentials and the G G A potentials, highlighting important deficiencies of the G G A potentials. In section 3 the K S potential is analyzed and in particular its relationship established with traditional concepts in the theory of electron correlation such as density matrices and Fermi and Coulomb correlation holes [16]. In the last section we specialize to the exchange-only case and discuss the breaking up of the K S potential into a screening part, directly related to the potential of the exchange hole charge density, and a response part, re lated to the "response" of the hole to density change. Characteristic features of these components of the potential are used to model them accurately.

2

Exact and G G A potentials in atoms

We will use the procedure outlined in ref. [6] to generate Kohn-Sham potentials from a given density. The given density is the best available density, usually from an extensive configuration interaction calculation. If we multiply the Kohn-Sham equations (-^V

2

+ ».(f))*i(fO = iiA-(r-)

(1)

from which the density is obtained as: X > ( r - ) | = p(r-) t

(2)

a

(N is the number of electrons in the system), by φ* and sum over i , we obtain after dividing by p: Μ*1 =

Σ

\φ"^ Φί(τ) 2

+ U\m?

(3)

We now define an iterative scheme using this equation. We want to calculate the potential corresponding to the density p. Suppose that at some stage in the iteration


2.

BAERENDS E T AL.


23

we have calculated orbitals φ° with eigenvalues £? and density p° and potential v°. In the next step we define the new potential: • Ή = ^

Σ

i * r ( r > v * i ( f ) + W = Σ/*[p]W(-\v ).[p}(W t

(is)

2

This follows of course also immediately from the Hohenberg-Kohn theorem, since this theorem implies that the wavefunction of a nondegenerate system is a functional of the density, and therefore every expectation value is, including that of the kinetic energy. The theorem holds for systems with arbitrary electron interaction, therefore also for the non-interacting Kohn-Sham system. So the first three terms in the r.h.s. of eq.(17), as well as the l.h.s. are defined and this equation therefore defines the socalled exchangecorrelation energy Εχο· We emphasize the well-known fact that Εχο is different from the traditional quantum chemical definition of the exchange-correlation energy as the sum of the Hartree-Fock exchange energy plus the correlation energy, the latter being traditionally defined as the difference between the exact and Hartree-Fock energies. If we compare the equations (17) and (9) for the exact total energy, it is clear that Exc[p]

=

T[p] -T [p] +

=

Txc + Wxc

s

Wxc (19)

where we have used the fact that also the exact kinetic energy (T) is a functional of p, and have written the difference between the exact kinetic energy and the K S kinetic energy as the exchange-correlation contribution to Τ (also often simply referred to as the correlation part of the kinetic energy, T ) . It is to be noted that the Kohn-Sham exchange-correlation energy consists of a kinetic part and a pure exchange-correlation part of the electron-electron interaction energy. In contrast, the traditional definition E% + E contains - sometimes sizable - corrections to the electron-nuclear and Hartree energies due to the difference A p between exact and Hartree-Fock densities: c

F

corr

Δρ(1) fpHF , rp &X Τ &corr

=

P(1)-P (1)

_ —

T?HF , j? rpHF &X -T EJ — EL

=

T[p) -

HF

+ J

T

HF

Ap(l)v(l)dl Μ1)Ρ(2)

, 1 [

Λ Λ

Δρ(1)Δρ(2)^ r\2

(20)

+W

XC

The Kohn-Sham definition has the advantage that it only consists of the exchangecorrelation corrections to the kinetic energy (T[p]-T ) and electron-electron interaction energy (Wxc) and is not "cluttered" by other terms. These other terms such as the correlation correction to the electron-nuclear energy and the corrections to the Hartree energy are often quite large, see table 5.1 in ref. [16]. For instance, for the N molecule, the correlation correction / Δpvdv to the electron-nuclear energy is -13.8 eV, to be compared to the total correlation energy of -11.1 eV and -11.0 eV for the correlation correction to the electron-electron interaction energy. The traditional definition has s

2


2.

BAERENDS ET AL.


29

of course the operational advantage that the reference Hartree-Fock energy does not contain unknowns and can be calculated to virtually arbitrary accuracy. This evidently is not the case for the Kohn-Sham system, Exc and its functional derivative v being only known approximately. Obtaining them exactly is equivalent to a full solution of the many-electron problem. xc

We are now in a position to consider the physical interpretation of the Kohn-Sham potential v (r) = SExc/Sp(f). It is possible to take the above equation(19) as a starting point, but it is also possible to incorporate the kinetic energy part in an expression that formally is similar to eq.(16), but in which the pair-correlation factor has been redefined by the so-called coupling-constant integration [21, 22, 23, 24]. The coupling constant integrated hole is described by the "average" pair correlation factor (7(1,2), in terms of which Exc may be written as


xc

1 Ml)(ftl,2)-l),(2)^ 2J i-i2

=

(21)

I J (l)V (l)dl

=

P

scr

The screening (or the hole) potential v is now due to an average exchange-correlation hole. It is referred to as the screening potential [7], since the exchange-correlation ef fects embodied in g may be considered as screening effects on the full electron-electron interaction Ι / Γ 1 2 . The considerations concerning exchange-correlation holes, on which present day approximations for Εχα are based, use almost always the coupling con stant integrated form, either implicitly (e.g. by referring to electron gas calculations of exchange-correlation that employ g) or explicitly (cf. the explicit introduction of the λ = 0 limit ("exact exchange") by Becke [26]). We will use both expressions. scr

The Kohn-Sham potential Vs(r)

is related to Exc

=

since

V(r)

+

VHartree(r)

+

(22)

V

xc

x (23)

àExc Vxc = -Τ-ΓΖΓ 8p{r)

Using eq.(21) we may write v

xc

/ O Q

as

where « ^ ( 3 )

= \ Jp(l)^^p(2)dld2

(25)

Eq. 24 demonstrates the physical nature of the components of v and therefore of the Kohn-Sham potential: the most important part of v is just the potential due to the averaged exchange-correlation hole. It is interesting to note that this part of v xc

xc

xc


30

C H E M I C A L APPLICATIONS O F DENSITY-FUNCTIONAL T H E O R Y

is directly related to the exchange-correlation energy according to equation (21). In other words, the X C energy density is just onehalf times the screening potential: = / P(r)€ (r)rfr = ± J p(f)v (f)df

E

XC

(26)

scr

xc

This relation between the exchange-correlation energy and the screening potential part of v may be exploited when modelling exchange-correlation functionals for the energy and potential simultaneously. O f course v also contains an additional term, which we have called the response part [7]. It is a measure of the sensitivity of the paircorrelation factor to density variations. These density variations may be understood in the following way. If the density changes to p + δρ, also assumed υ-representable, then according to the Hohenberg-Kohn theorem this changed density corresponds uniquely to an external potential ν + δν. For the system with external potential ν + δν we have the corresponding Kohn-Sham system and the coupling-constant integrated paircorrelation factor g + 6g. So the derivative of g occurring in the response potential may be regarded as the response of g to density changes δρ caused by potential changes δυ. The response potential does not affect the energy directly, but it does so indirectly since it determines, as part of the Kohn-Sham potential, the S C F density in the Kohn-Sham calculation. A s we will see below (cf. also ref. [7]), the response part of the potential has less pronounced features than the hole potential but is certainly required to obtain accurate K S orbitals and densities. xc


xc

It is of course also possible to use eq.(19) when deriving v . It is convenient to use for the exact kinetic energy (T) = T[p] and for the kinetic energy of the noninteracting electrons T [p] the expressions derived in ref. [10]: xc

s

T\p] =

T

T [p]

T

a

=

+j

w

p{r)v (r)dr kin

(27)

+ j p(r)v (r)dr

w

9Mn

Here Tw is the Weiszacker kinetic energy for a density p, which is just Ν times the kinetic energy of the normalized density amplitude ("density orbital") \/p/N, (28)

Tw\p] = N J ^ ( - \ v ^ d f

The kinetic potential υ * can be related to the electron correlation by expressing it in terms of the conditional amplitude Φ/^/p or in terms of the derivative of the oneelectron density matrix [10]. Taking the functional derivative of Exc of eq.(19) to obtain v leads of course to kinetic potentials and their responses: ιη

xc

E

x

c

=

l _ J p(l)( (l 2)-lM2

fxc = ^

f l

;

= «W + C

) ( f 2 d l

l p o n s e

+

j

p

W

{

v

k

i

n

{

+ (»«. - ν*Μη) +

1

)

_

V a k i n { 1 ) ) d l

-

( 2 9 )

(30)

The kinetic potential plays an important role in the typical molecular left-right corre lation effect [10]. This particular form of correlation leads to a peak in υ ^ at the bond η


2.


BAERENDS ET AL.

midpoint (a ridge around the bond midplane) in molecules. Other features of these po tentials, such as step structure in v ^ in atoms, are discussed in ref. [7]. In the present paper however we will concentrate on atoms, where exchange is the dominating factor in the correlation between electron motions. We will for this case restrict ourselves to the exchange-only density-functional theory as made operational in the optimized potential method [27, 28]. This will enable us to highlight the relative importance of the screening and the screening-response parts of the Kohn-Sham potential and to discuss the accuracy of the L D A and G G A approximations for these potentials. r


4 Screening and response potentials in exchangeonly density functional theory Given an external potential, one may ask for the local potential that generates the or bitals that minimize the energy of a single-determinantal wavefunction. This problem, nowadays often referred to as the exchange-only D F T , has been addressed by Sharp and Horton [29] and Talman and Shad wick [27]. The local potential obtained within this "optimized potential model" (OPM) is plotted in fig. 2 for C d . The correlation hole of the single-determinantal wavefunction is just the Fermi hole and the screening part of the effective potential is just the potential due to this hole: Ρ

Xhole,-? . . \ \Τ σ\Γισί) = 2

r Γχ(Γ\σ, Γ σ) 6, --—

(31)

2

σσ

P( i

Effective One-Electron Potential in the KohnâSham Molecular Orbital

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Effective One-Electron Potential in the KohnâSham Molecular Orbital