Chapter 2
Effective OneElectron 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/bk19960629.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 KohnSham molecular orbital method, that is almost exclusively used, is more than a convenient tool to generate the required electron density. The effective oneelectron potential in the KohnSham 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 exchangecorrelation 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 KohnSham 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 densityfunctional methods have been around for some time, such as the ThomasFermi method and the Xα method, a rigorous foundation has only been given with the formulation of the HohenbergKolm theorems [1]. In the second place, the KohnSham oneelectron 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 socalled 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 localdensity
00976156/96/06290020$15.75/0 © 1996 American Chemical Society In Chemical Applications of DensityFunctional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
Downloaded by COLUMBIA UNIV on March 16, 2013  http://pubs.acs.org Publication Date: May 5, 1996  doi: 10.1021/bk19960629.ch002
2.
BAERENDS ET AL.
OneElectron Potential in KohnSham MO Theory
approximation (LDA) to the exchangeonly L D A or Χα method). In chemistry, appli cations almost invariably use the KohnSham molecular orbital model. The exchangecorrelation functional Εχο[p], which is defined in the context of the KohnSham model, has been the primary focus of theoretical work directed towards the immediate goal of obtaining highly accurate energies from KohnSham calculations. In this line of re search, the KohnSham oneelectron 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 KohnSham model has received comparatively little attention, as may be evident from the frequently voiced opinion that the KohnSham 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 KohnSham model. As we will demonstrate, the effective potential of the KohnSham model has an intimate connection with the physics of the exchange and correlation effects in atoms and molecules. The KohnSham orbitals thus represent electrons that move in a potential that is certainly as realistic as the HartreeFock "potential" and indeed has some advantages. There is no reason to believe that the KohnSham orbitals are any less "physical" or useful than the HartreeFock orbitals and they may and have indeed been used quite succesfully in qualitative M O explana tions that are so typical for presentday chemistry. The physics embodied in the effective oneelectron potential of the KohnSham 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 exchangecorrelation energy Exc[p], that the exchangecorrelation 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 KohnSham potential that belongs to a given density, in the sense that the occupied eigenfunctions of that potential produce the given density. Application of the HohenbergKohn theorem to the KohnSham system of noninteracting electrons proves that the KohnSham 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 KohnSham potentials as density functionals directly. Since KohnSham 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 DensityFunctional 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 DENSITYFUNCTIONAL T H E O R Y
Downloaded by COLUMBIA UNIV on March 16, 2013  http://pubs.acs.org Publication Date: May 5, 1996  doi: 10.1021/bk19960629.ch002
identical energies, and given their nonuniqueness it is hard to judge different proposals for energy densities. The KohnSham 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 KohnSham potential? This is indeed the case, when one is prepared to perform a lineintegral 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 KohnSham 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 KohnSham 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 exchangeonly 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 KohnSham potentials from a given density. The given density is the best available density, usually from an extensive configuration interaction calculation. If we multiply the KohnSham 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
In Chemical Applications of DensityFunctional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
2.
BAERENDS E T AL.
OneElectron Potential in KohnSham MO Theory
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 HohenbergKohn 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 noninteracting KohnSham 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 wellknown fact that Εχο is different from the traditional quantum chemical definition of the exchangecorrelation energy as the sum of the HartreeFock exchange energy plus the correlation energy, the latter being traditionally defined as the difference between the exact and HartreeFock 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 exchangecorrelation contribution to Τ (also often simply referred to as the correlation part of the kinetic energy, T ) . It is to be noted that the KohnSham exchangecorrelation energy consists of a kinetic part and a pure exchangecorrelation part of the electronelectron interaction energy. In contrast, the traditional definition E% + E contains  sometimes sizable  corrections to the electronnuclear and Hartree energies due to the difference A p between exact and HartreeFock 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 KohnSham definition has the advantage that it only consists of the exchangecorrelation corrections to the kinetic energy (T[p]T ) and electronelectron interaction energy (Wxc) and is not "cluttered" by other terms. These other terms such as the correlation correction to the electronnuclear 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 electronnuclear 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 electronelectron interaction energy. The traditional definition has s
2
In Chemical Applications of DensityFunctional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
2.
BAERENDS ET AL.
OneElectron Potential in KohnSham MO Theory
29
of course the operational advantage that the reference HartreeFock energy does not contain unknowns and can be calculated to virtually arbitrary accuracy. This evidently is not the case for the KohnSham system, Exc and its functional derivative v being only known approximately. Obtaining them exactly is equivalent to a full solution of the manyelectron problem. xc
We are now in a position to consider the physical interpretation of the KohnSham 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 paircorrelation factor has been redefined by the socalled couplingconstant 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
Downloaded by COLUMBIA UNIV on March 16, 2013  http://pubs.acs.org Publication Date: May 5, 1996  doi: 10.1021/bk19960629.ch002
xc
1 Ml)(ftl,2)l),(2)^ 2J ii2
=
(21)
I J (l)V (l)dl
=
P
scr
The screening (or the hole) potential v is now due to an average exchangecorrelation hole. It is referred to as the screening potential [7], since the exchangecorrelation ef fects embodied in g may be considered as screening effects on the full electronelectron interaction Ι / Γ 1 2 . The considerations concerning exchangecorrelation 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 exchangecorrelation 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 KohnSham potential Vs(r)
is related to Exc
=
since
V(r)
+
VHartree(r)
+
(22)
V
xc
x (23)
àExc Vxc = ΤΓΖΓ 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 KohnSham potential: the most important part of v is just the potential due to the averaged exchangecorrelation hole. It is interesting to note that this part of v xc
xc
xc
In Chemical Applications of DensityFunctional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
30
C H E M I C A L APPLICATIONS O F DENSITYFUNCTIONAL T H E O R Y
is directly related to the exchangecorrelation 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 exchangecorrelation energy and the screening potential part of v may be exploited when modelling exchangecorrelation 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 υrepresentable, then according to the HohenbergKohn theorem this changed density corresponds uniquely to an external potential ν + δν. For the system with external potential ν + δν we have the corresponding KohnSham system and the couplingconstant 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 KohnSham potential, the S C F density in the KohnSham 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
Downloaded by COLUMBIA UNIV on March 16, 2013  http://pubs.acs.org Publication Date: May 5, 1996  doi: 10.1021/bk19960629.ch002
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 leftright corre lation effect [10]. This particular form of correlation leads to a peak in υ ^ at the bond η
In Chemical Applications of DensityFunctional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.
2.
OneElectron Potential in KohnSham MO Theory
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 exchangeonly densityfunctional 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 screeningresponse parts of the KohnSham potential and to discuss the accuracy of the L D A and G G A approximations for these potentials. r
Downloaded by COLUMBIA UNIV on March 16, 2013  http://pubs.acs.org Publication Date: May 5, 1996  doi: 10.1021/bk19960629.ch002
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 singledeterminantal wavefunction. This problem, nowadays often referred to as the exchangeonly 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 singledeterminantal 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