Local and Gradient-Corrected Density Functionals - American

Chapter 30

Local and Gradient-Corrected Density Functionals John P. Perdew, Kieron Burke , and Matthias Ernzerhof Downloaded by NORTH CAROLINA STATE UNIV on October 8, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch030

1

Department of Physics and Quantum Theory Group, Tulane University, New Orleans, LA 70118

The generalized gradient approximation (GGA) corrects many of the shortcomings of the local spin-density (LSD) approximation. The accuracy of GGA for ground-state properties of molecules is comparable to or better than the accuracy of conventional quantum chemical methods such as second-order Møller-Plesset perturbation theory. By studying various decompositions of the exchange-correlation energy E , we show that the real-space decomposition of E facilitates the most detailed understanding of how the local spin-density approximation and the Perdew-Wang 1991 GGA work. The real-space decomposition shows that the near universality of the on-top value for the exchange-correlation hole connects the homogeneous electron gas to inhomogeneous systems such as atoms and molecules. The coupling-constant decomposition shows that the exchange-correlation energy at full coupling strengthE ,λ=1is ap proximated more accurately by local and semi-local functionals than is the coupling-constant average E . We use this insight both to critique popular hybrid functionals and to extract accurate energies from exact electron densities by using functionals for the exchange -correlation energy at full coupling strength. Finally, we show how a reinterpreted spin density functional theory can be applied to sys tems with static correlation. XC

XC

XC

XC

Density functionals in quantum chemistry T h e m a i n goal of quantum chemistry is the reliable prediction of molecular proper ties [1]. T h e development of generalized gradient approximations ( G G A ' s ) [2-10] has made density functional theory [11-13] a serious competitor to conventional quantum chemistry methods for ground-state properties. T h e latter methods include Configuration Interaction techniques, Coupled-Cluster methods, and the 1

Address after July 1,1996: Department of Chemistry, Rutgers University, Camden, NJ 08102 0097-6156/96/0629-0453$15.00/0 © 1996 American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

454

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

Downloaded by NORTH CAROLINA STATE UNIV on October 8, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch030

M0ller-Plesset perturbation expansion [14]. T h e y account for electron correlation, but suffer i n general from basis set problems. These basis set problems are much less severe i n current density functional methods, since (as we show below) the pair density is not expanded i n a basis of one-particle functions. T h e accuracy of the G G A is usually comparable to conventional quantum chemistry methods, at much lower computational cost. G G A ' s offer significant improvements i n the calculation of molecular properties compared to their an cestor, the local spin-density ( L S D ) approximation [15]. T h e local spin-density approximation has not been popular amongst chemists, m a i n l y because of its tendency to overestimate the binding energy of molecules. In practical electronic structure calculations based on density functional the ory [12], a set of independent-particle equations (the K o h n - S h a m equations [15]) is solved. These equations require as input an approximation to the exchangecorrelation energy E as a functional of the electron density. T h e G G A approx imations to E depend on both the local spin-density η (τ) and the gradient of the local spin-density. A m o n g the popular G G A ' s , the Perdew-Wang ( P W 9 1 ) [7-10] functional allows the most detailed understanding of how G G A ' s work and why they work, since this approximation contains no empirical parameter and is constructed from first principles. Results of calculations w i t h this form [9] show that it t y p i c a l l y reduces exchange energy errors from 10% i n L S D to 1%, and correlation energy errors from 100% to about 10%. ΡW91 corrects the L S D over estimate of atomization energies for molecules and solids i n almost a l l cases, it enlarges e q u i l i b r i u m bond lengths and lattice spacings, usually correctly, and re duces v i b r a t i o n a l frequencies, again usually correctly [10]. P W 9 1 also generally improves activation barriers [16]. For recent results w i t h P W 9 1 , see Refs. [17-27]. xc

σ

xc

A s indicated above, the exchange-correlation energy Exc as a functional of the electron density is the crucial quantity i n K o h n - S h a m calculations. In this article we discuss various decompositions of the exchange-correlation energy, and we show which of these decompositions is accurately approximated by L S D and by the P W 9 1 approximation. T h i s analysis makes it possible to understand how and why local and semilocal functionals work even for highly inhomogeneous electron systems, such as atoms and molecules.

Decompositions of E

x c

T h e basic formula which serves as the starting point for various of E

xc

decompositions

is [11-13] E

xc

= y

o

d\

du 2πυ, J

J d r n(r) n , ( r , r + u). 3

x c

A

(1)

The exchange-correlation hole η λ ( Γ , r -f u) at coupling strength A is given i n terms of the pair density Ρ\(τσ, (r + ιι)σ') by ΧΟ)

n(r) [n(r + u) + n , ( r , r + u)] = £ x c

A

Ρ ( Γ σ , (r + ιι)σ'). λ

σ,σ'

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

(2)

30. PERDEW ET AL.

455

Local and Gradient-Corrected Density Functionals

σ and σ' are the spin variables of the electrons. Ρ\(γσ,γ'σ')

(where r ' = r + u)

gives the probability density to find an electron w i t h spin σ at r and an electron w i t h spin σ' at r ' . T h e pair density is related to the many-electron wave function Φ (Γ,σ,Γ ,σ ,...,Γ ,σΛτ) /

Λ

/

by

Ν

Ρχ(τσ, r V )

N(N-l)

=

£

/d r 3

3

... / d

χ Φ (Γ,σ,Γ ,σ ,...,Γ ,σΛΓ) /


Λ

,

Λ Γ

3

·

(3)

ΦΑ is the ground-state wave function of a system i n which the electron-electron repulsion operator is multiplied by λ and the external potential is varied w i t h λ so that the electron density is equal to the physical ground-state density for all values of λ. ΨΑ=Ι is the interacting wavefunction found by traditional correlated methods, while ΨΑ=Ο is the exact exchange wavefunction (which is similar to that of Hartree-Fock [28]). Eqs. 1 and 2 suggest a number of possibilities to decompose ^occ by simply permuting the sequence of integrations and summations. T h e decomposed approximate exchange-correlation energy can then be compared w i t h the corresponding exact quantity. E n e r g y - d e n s i t y d e c o m p o s i t i o n o f E , A - We examine the integrand of the expression X C

£XC,A

d r n(r) e ,A(r),

= /

3

(4)

XC

where roo

Εχο,λ( ) = J

r

du 2πυ,

Γ

Π ,Α(Γ, r + u ) .

J

(5)

ΧΟ

(E is related to E \ by E = f d\E ,\.) In the local spin-density approxi m a t i o n the energy per particle e A(r) of an arbitrary inhomogeneous system is approximated by that of a homogeneous electron gas w i t h spin density n ( r ) , i.e., ^ x c ( r ) = e ^ (n|(r),n|(r)). T h e error i n Exc resulting from this approximation is t y p i c a l l y about 10%. T h e energy per particle close to a nucleus is usually over estimated, and that i n the t a i l region of the electron density is underestimated [29]. However, the region close to the nuclei has a small volume and the contribu tion from the t a i l region is very s m a l l , since the energy per particle gets weighted by the electron density i n E q . 4. In the valence region of atoms and molecules, χ ο * ' ( ΐ ( ) > ΐ ( ) ) shows a semiquantitative agreement w i t h the exact energy per particle. However, the P W 9 1 energy density is not so useful for understanding how the P W 9 1 functional works [30], since the energy density ηεζ^χ has been simplified by an integration by parts, which leaves E unchanged but leads to an ill-defined n ( r ) e ^ ( r ) . xc

XCt

l

xc

0

xc

XC>

a

x

€

η

Γ

if

η

Γ

1

xc

Real-space d e c o m p o s i t i o n of E E ,\, which is defined by

X C )

A«

T h e real-space decomposition of the

xc

roo

Exc,\ = Ν / Jo

du 2KU

(n x(u)) XCt


(6)

456


where the system- and spherically averaged hole is (n A )) xc

u

= J

^ J dr () 3

n r

n , ( r , r + u), x c

A

(?) offers the most detailed insight into the L S D and P W 9 1 functionals.

In fact

the P W 9 1 exchange-correlation functional is based on a model for the systemaveraged exchange-correlation hole / d r n(r) n , A ( r , r + u)/N. 3

Detailed studies

XC

[31] of (n ,A(w)) have shown that even the L S D approximation to this quantity


XC

is remarkably accurate. M a n y exact conditions on the exact (n ,x(u)), xc

the normalization condition / °° du 4TU

2

0

(n \(u)) XCf

such as

= —1, are satisfied by L S D ,

since the L S D exchange-correlation hole is the hole of a possible physical system. T h e normalization condition on the hole together w i t h the on-top (u = 0) value for n , A ( r , r + u) set the scale for the exchange-correlation hole and therefore the XC

scale for (n \(u)) XCi

and E \. XCi

Thus the on-top value of n , A ( r , r + u) plays a cru XC

cial role i n density functional theory. Investigations [32] on a number of systems show that the on-top value of the hole as a function of the local density is almost universal among C o u l o m b systems. Thus the L S D approximation to this quantity is very accurate, especially i n the valence and t a i l regions of the electron density [32]. A s a consequence, any approximate density functional should reproduce the correct L S D on-top value of the exchange-correlation hole i n the l i m i t of slowly varying electron densities. T h e P W 9 1 functional has the L S D on-top value built i n . It is based on a systematic expansion of the exchange-correlation hole i n terms of the local density and the gradient of the local density. T h i s gradient expansion approximation ( G E A ) to second order i n V n improves the description of the hole at intermediate u values, but its spurious large-u behavior [33] violates a number of important constraints on the exact hole, such as the normalization condition. B y restoring these conditions v i a the real-space cutoff procedure [8], we obtain the P W 9 1 model for the exchange-correlation hole. Since the r integration i n E q . 7 involves an integration by parts which changes the local hole rc ,A(r> r + u), only XC

the system-averaged hole is a well-defined quantity i n the P W 9 1 construction. Detailed studies [29,31] of the spherical- and system-averaged hole for molecules and atoms show that the P W 9 1 approximation to this quantity significantly i m proves the L S D model. Other popular G G A ' s [5,6] do not provide models of the exchange-correlation hole and thus do not allow a detailed analysis of correlation effects on molecular bond formation. Spin decomposition

of E , A -

Another decomposition of the exchange-

X C

correlation hole and therefore of the exchange-correlation energy distinguishes between electrons w i t h parallel and anti-parallel spins: E c,x = Ell X

x

+ E ^

+ Ell , x

where


(8)


30. PERDEW ET AL.


T h e P a u l i principle prevents two parallel-spin electrons from coming close to each other, i.e., P ( r a , τσ) = 0. Electron-electron repulsion cannot deepen the cor responding hole at u = 0, so the spatial extent of the hole is not significantly reduced. Note that the deeper the hole at u = 0, the shorter-ranged it must be to satisfy the normalization condition. O n the other h a n d , the on-top value of the exchange-correlation hole for two electrons w i t h antiparallel-spin orienta tion is significantly lowered by electron-electron repulsion, and the normalization therefore assures that the spatial extent of the hole gets reduced by correlation. Local and semilocal approximations work best if the exchange-correlation hole is confined to a small region of space around the reference electron. In this case the information about the local density and the gradient of the density is suffi cient to capture the important features of the exchange-correlation hole. Thus it is not surprising that local and semilocal functionals work better for the corre lation energy between antiparallel-spin electron than they do for the correlation energy between parallel-spin electrons [29]. However, attempts to construct a hy b r i d scheme which uses a G G A for antiparallel-spin correlation and wave function methods for parallel-spin correlation are of l i m i t e d use [29], since the correlation effects between parallel-spin electrons are as difficult to describe w i t h i n a finite basis set approach as are correlation effects between antiparallel-spin electrons. F i n a l l y we note that the approximate G G A for antiparallel-spin [34] predicts E^ + E^ to be 20% of the total correlation energy of ATe, i n good agreement w i t h sophisticated wave function calculations which give a value [29] of 24%. C o u p l i n g - c o n s t a n t d e c o m p o s i t i o n of E . T h e kinetic correlation energy contribution T to the total energy need not be explicitly approximated as a functional of the electron density. T is i m p l i c i t l y accounted for i n E of E q . 1 v i a the coupling-constant integration over the λ-dependent exchange-correlation hole [35]. T h i s coupling-constant integration leads to another decomposition of the exchange-correlation energy: x c

c

c

E

xc

=

where

xc

Jo

dX £ , A ,

(10)

X C

^ E ,x xc

= J

q

du 2τπζ J

J d r n(r) η , ( Γ , r + u ) . 3

ΧΑ

(11)

Λ

T h i s decomposition has become a popular tool i n density functional theory [36-38]. For λ = 0, the electrons are not Coulomb-correlated, so E ,\=o = E accounts for the self-interaction correction and for the P a u l i exclusion principle. Compared to the hole at finite values of λ, the hole at λ = 0 is shallower and therefore more long-ranged. A t full coupling-strength (λ = 1), E \-i = E — T . Electrons close to the reference electron get pushed away at small u-values and pile up at large u-values, m a k i n g the hole deeper at u = 0 and more short-ranged. L o c a l and semilocal approximations usually work best for small u. T h u s they are least suitable for λ = 0 (the exchange-only l i m i t ) . These expectations about the range of the hole are confirmed by a study [31] of the λ-dependent exchange-correlation hole, and are probably true for a l l systems. xc

XCf

x

xc

c


457

458


T h e popular h y b r i d schemes [36-38] can be viewed as attempts to exploit this observation. We consider only h y b r i d schemes which recover the slowly-varying electron gas l i m i t . In such a scheme, the coupling-constant integral is replaced by the weighted sum of the integrands at the endpoints of the coupling-constant inte gration and the density functional approximation to the exchange energy E ,x=o is replaced by the Hartree-Fock exchange energy [29,38]. In formulas, xc

£


+ (1 - a)E

= aE»

F

xc

GGA

(12)

+ E . GGA

The parameter a is usually adjusted to minimize the root-mean-square errors of various molecular properties. However, it has been demonstrated [32,29] that the parameter a is far from universal for molecular systems. T h i s can be seen by considering the stretched H2 molecule: The restricted Hartree-Fock hole is always distributed equally over both / / - a t o m s , whereas the exact hole and its G G A model are localized on the H atom at which the reference electron is located. Thus no finite amount of exact exchange should contribute to E in the l i m i t of infinite stretching. Exchange-correlation potential. It has been shown [39,40] that the P W 9 1 approximation to the exchange-correlation potential v = SE [n]/6n , which appears i n the K o h n - S h a m H a m i l t o n i a n , deviates considerably from the exact potential, especially in the core and tail regions of atoms. However, we have argued that approximate local and semilocal density functionals give good approximations only to system-averaged quantities such as the system-averaged exchange-correlation hole. It is therefore more appropriate to study quantities which involve system-averages of the exchange-correlation potential v and its components v and v , rather than the potential itself. T h e v i r i a l theorem shows that [41] xc

xc

xc

xc

x

c

E

x

(13)

= - Jd rn(r)vVv (r) 3

x

and E

+ T = - j d r n(r) r · V u ( r ) .

c

3

c

c

(14)

In this system average, the region close to the nuclei and the tail region of the electron density, where v is not well approximated by P W 9 1 , get little weight. Due to error cancellation within the integral of Eqs. 14, the system-averaged quantities are far better approximated by P W 9 1 than by L S D [31]. xc

E x c h a n g e - c o r r e l a t i o n e n e r g i e s f r o m e x a c t d e n s i t i e s . A n approach which makes use of the improvement i n accuracy of the L S D and G G A approximations at full coupling strength (λ = 1) relative to the coupling-constant average can be formulated based on knowledge of the exact density n(r) corresponding to a known external potential v(r) [42]. Several methods are known for calculating the exact K o h n - S h a m potential v (r) from a given electron density [43-47]. T h e exact exchange-correlation potential can then be obtained according to s

«xc(r) = « . ( r ) - v(r) - J d r' s


(15)

30. PERDEW ET AL.

459

Local and Gradient-Corrected Density Functional

The v i r i a l relations Eqs. 13 and 14 show that the virial of the exchange-correlation potential is given by the sum of T + E . Since E \=\ = E — T , we obtain the exact relation c

E

=

x c

xc

i[£xc,A=i -

/

XCy

dr 3

n(r)

r · Vt,

xc

x c

c

(r)].

(16)

E ,\=i on the right hand side of this equation w i l l now be replaced by its lo cal or semilocal approximation, a replacement typically involving less error than the functional approximation of E . A s shown i n Table 1, the resulting expres sions for E give a significant improvement compared to the local and semi-local approximations of E itself. xc


xc

xc

xc

Table 1: Exchange-correlation energies in Hartrees for several atoms [42] We com pare exact values w i t h those of L S D and P W 9 1 , and w i t h improvements thereof (marked by a prime) using E q . (16). Unless otherwise indicated, the exact val ues are from Ref. [48], while the approximate f u n c t i o n a l are evaluated on the Hartree-Fock densities of Ref. [49]. atom H H" > He Li Be++ Ne + Be Be Ν Ne a

a

6

6

8

6

a

c

α

exact

LSD

LSD'

-0.312 -0.423 -1.068 -1.827 -2.321 -6.073 -2.772 -2.772 -6.78 -12.48

-0.290 -0.409 -0.997 -1.689 -2.107 -5.376 -2.536 -2.545 -6.32 -11.78

-0.307 -0.420 -1.048 -1.786 -2.243 -5.776 -2.686 -2.691 -6.61 -12.20

PW91 -0.314 -0.425 -1.063 -1.821 -2.298 -5.989 -2.739 -2.748 -6.77 -12.50

PW91' -0.316 -0.422 -1.066 -1.829 -2.312 -6.036 -2.755 -2.760 -6.78 -12.47

A p p r o x i m a t e f u n c t i o n a l evaluated o n exact

densities. b

E x a c t r e s u l t s f r o m R e f . [39].

c

E x a c t r e s u l t s f r o m R e f . [43].

S t a t i c c o r r e l a t i o n i n d e n s i t y f u n c t i o n a l t h e o r y . In the context of den sity functional theory, systems which show only dynamical correlation effects are called normal systems [50,51]. For small values of the coupling-constant λ, normal systems are well described by a single Slater determinant, and the on-top value of the hole is well reproduced by L S D and P W 9 1 . Systems w i t h static correla tion, such as H2 stretched beyond the Coulson-Fisher point, are called abnormal systems. T h e exact wavefunction of stretched H (with bond length R —• 0 0 ) does not reduce to a single determinant as λ —• 0. Instead, it remains a HeitlerLondon type wavefunction, keeping one electron localized on each a t o m . B u t in a restricted single-determinant approximation, the electrons cannot localize on i n d i v i d u a l atoms. A s a consequence, the on-top value of the exchange-correlation 2


460


hole that results from a restricted calculation (Hartree-Fock, L S D , or G G A ) is quite incorrect. T o see this, note first that for a single Slater determinant, the on-top pair density is simply P ( r , r) = 2nj(r)ra|(r). In a restricted calculation, i n which n ( r ) = n ( r ) = n ( r ) / 2 , we find ΡΑ=Ι(Γ,Γ) = P j ^ / ( n ( r ) / 2 , n ( r ) / 2 ; u = 0), instead of the correct result, ΡΑ=Ι(Γ, r ) = 0 for all r. A cure for this problem is provided by an alternative interpretation of spin-density functional theory [50,51]. In this alternative interpretation, the quantities predicted are not the individual spin-densities n j ( r ) and n j ( r ) , but the total density n(r) = n j + rij and the full coupling strength on-top pair density ΡΑ=Ι(Γ, r ) = Ρχ^ί{η],nj;u = 0). In ab normal systems, the spin symmetry can then be broken w i t h impunity. In the stretched Hi molecule, an electron w i t h up-spin localizes on one hydrogen atom and an electron w i t h down-spin localizes on the other. T h e spin-density i n such an unrestricted calculation is obviously no longer accurately reproduced, but the total density is. Furthermore, ΡΑ=Ι(Γ, r) = ΡΑ=Ο(Γ, Γ) correctly vanishes for a l l values of r, since either n j ( r ) or n j ( r ) is zero everywhere. T h i s behavior of the on-top value of the pair density ensures that the unrestricted K o h n - S h a m calculation gives an accurate dissociation energy.


T

x

Acknowledgments This work has been supported i n part by N S F grant DMR95-21353 and i n part by the Deutsche Forschungsgemeinschaft.

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Local and Gradient-Corrected Density Functionals - American

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