Chemical Applications of Density-Functional Theory - American

Chapter 3

Downloaded by UNIV MASSACHUSETTS AMHERST on October 10, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch003

Conventional Quantum Chemical Correlation Energy Versus Density-Functional Correlation Energy Ε. K. U . Gross, M . Petersilka, and T. Grabo Institut für Theoretische Physik, Universität Würzburg, A m Hubland, D-97074 Würzburg, Germany

We examine the difference between the correlation energy as defined w i t h i n the conventional quantum chemistry framework and its na mesake i n density-functional theory. B o t h correlation energies are rigorously defined concepts and satisfy the inequality E c ≥ E c. We give numerical and analytical arguments suggesting that the numerical difference between the two rigorous quantities is small. F i n a l l y , approximate density functional correlation energies resul ting from some popular correlation energy functionals are compared w i t h the conventional quantum chemistry values. QC

DFT

In quantum chemistry ( Q C ) , the exact correlation energy is traditionally defi ned as the difference between the exact total energy and the total selfconsistent Hartree-Fock ( H F ) energy:

W i t h i n the framework of density-functional theory ( D F T ) [1, 2], on the other hand, the correlation energy is a functional of the density [p]. T h e exact D F T correlation energy is then obtained by inserting the exact ground-state density of the system considered into the functional E^* - [p], i . e. 1

0097-6156/96/0629-0042$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.

3.

GROSS E T A L .

Density-Functional Correlation Energy

43

In practice, of course, neither the quantum chemical correlation energy (1) nor the D F T correlation energy (2) are known exactly. Nevertheless, both quantities are rigorously defined concepts. T h e a i m of the following section is to give a coherent overview of how the cor relation energy is defined i n the D F T literature [3-14] and how this quantity is related to the conventional Q C correlation energy. T h e two exact correlation energies E^ act and E^J^ are generally not identical. T h e y satisfy the inequali ty £ e xact > ^ e x L f Furthermore we w i l l give an analytical argument indicating that the difference between the two exact quantities is small. X


Q c

C

lCt

c

D

In the last section we compare the numerical values of approximate conventional Q C correlation energies w i t h approximate D F T correlation energies resulting from some popular D F T correlation energy functionals. It turns out that the difference between D F T correlation energies and Q C correlation energies is smallest for the correlation energy functional of Colle and Sal vet t i [15, 16] further indicating [17] that the results obtained with this functional are closest to the exact ones. Basic Formalism We are concerned w i t h Coulomb systems described by the H a m i l t o n i a n (3)

H = T + W \h + V C

where (atomic units are used throughout)

f

-

|(-5

V ?

(4)

)

(5) 1

N

1 (6)

V

=

£>( ) rj

.

i=l To keep the following derivation as simple as possible, we choose to work w i t h the traditional Hohenberg-Kohn [18] formulation rather than the constrained-search representation [4, 19, 20] of D F T . In particular, a l l ground-state wavefunctions (interacting as well as non-interacting) are assumed to be non-degenerate. B y virtue of the Hohenberg-Kohn theorem [18] the ground-state density p uniquely determines the external potential ν = ν [p] and the ground-state wave function Φ [p]. If v (r) is a given external potential characterizing a particular physical system, the Hohenberg-Kohn total-energy functional is defined as Ev [p) = (m\f+Wa + Vo\9\p)) . (7) A s an immediate consequence of the R a y l e i g h - R i t z principle, the total-energy functional (7) is m i n i m i z e d by the exact ground-state density p t corresponding 0

i

h

exac

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

44

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

to the potential v , the m i n i m u m value being the exact ground-state energy, i . e. 0

.exact — Ev [/>exact]

·

Q

(8)

In the context of the K o h n - S h a m ( K S ) scheme [21] the total-energy functional is usually written as E* [p] = T.[p] + Jp(r)v (r)d r+ -Jf^^d r^r' Downloaded by UNIV MASSACHUSETTS AMHERST on October 10, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch003

t

0

3

l

+ E [p]

3

xc

(9)

where T [p] is the kinetic-energy functional of non-interacting particles. B y virtue of the Hohenberg-Kohn theorem, applied to non-interacting systems, the density p uniquely determines the single-particle potential v [p] and the ground-state Slater-determinant s

s

K S

$

(10)

[p] = - ^ d e t { ^ H } s

and hence T [p] is given by s

m

=

(Φ *[p]\τ\Φ™[p\)

=

Σ

κ

Σ /Ψ™ Μ W ( " ^ V ) φ™ [p] (r) d r 2

We mention i n passing that for a "Hartree-Fock world" nes the external potential. functional of the density as

.

3

(11)

the Hohenberg-Kohn theorem can also be formulated [22], i m p l y i n g that the H F density uniquely determi Consequently the H F ground-state determinant is a well:

Φ™[p] = -±=ά*{ %[p]} ί

.

(12)

T h e resulting kinetic-energy functional T™[p]

(Φ [p]|Γ|Φ [p])

=

ΗΡ

ΗΓ

is different from T [p] because the orbitals i n (11) come from a local single-particle potential v [p] while the orbitals i n (13) come from the nonlocal H F potential v [/?]. However, the numerical difference between THF[/9] and T [p] has been found to be rather small [14]. s

s

HF

8

T h e remaining t e r m , E [/?], on the right hand side of equation (9) is termed the exchange-correlation (xc) energy. Comparison of equation (9) w i t h equation (7) shows that the xc-energy functional is formally given by xc

EM

= (n )\f P

+ W \m-TM-\jJ^ Z^")

Σ M O [

•

( 4 4 )

T h e constants a , 6, c and d are given by a = 0.04918, c = 0.2533,

6 = 0.132, d = 0.349.

In Table 2, the four approximate D F T correlation energy functionals are evaluated at the exact densities [13, 28] of H ~ , He, Be" " , N e , B e , N e and compared with the exact D F T correlation energies given by equation (22). O n average, the K L I - C S values are superior. 1

2

+ 8

In Table 3 selfconsistent D F T correlation energies are compared w i t h Q C values taken from [38]. In these selfconsistent calculations the approximate correlationenergy functionals ϋ £ , Ε™ , E^ are complemented with the approximate exchange-energy functionals E* [39], E™ [31] and £ £ , respectively. In the K L I - C S case, the D F T exchange-energy functional (17) is of course treated exactly. T h e numerical data show three main features: γ ρ

91

Ok

88

91

D A


50


Table 2: Non-relativistic absolute correlation energies resulting from various ap proximate DFT correlation energy functionals, evaluated at the exact groundstate densities [13, 28] of the respective atoms (in Hartree units). Exact values are from [13, 38]. \A\% denotes the mean value of \E - ^ ^ t l / l ^ i x L t l * percent. n


C

HHe Be+ Ne+ Be Ne

2

8

|Δ|%

es

LYP

0.0297 0.0416 0.0442 0.0406 0.0936 0.375 8.2

0.0299 0.0438 0.0491 0.0502 0.0955 0.383 9.5

PW91 0.0320 0.0457 0.0535 0.0617 0.0950 0.381 15.4

LDA

EXACT

0.0718 0.1128 0.1512 0.2030 0.2259 0.745

0.0420 0.0421 0.0443 0.0457 0.0962 0.394

175

Table 3: Non-relativistic absolute correlation energies of first and second row atoms from selfconsistent calculations with various DFT approximations. QC denotes the conventional quantum chemistry value [38]. |Δ|% denotes the mean value of | ( £ f - E? )/E$ \ in percent. All other numbers in Hartree units. F T

c

He Li Be Β C Ν 0 F Ne Na Mg Al Si Ρ S Cl Ar |Δ|%

c

KLI-CS 0.0416 0.0509 0.0934 0.1289 0.1608 0.1879 0.2605 0.3218 0.3757 0.4005 0.4523 0.4905 0.5265 0.5594 0.6287 0.6890 0.7435 3.13

BLYP 0.0437 0.0541 0.0954 0.1287 0.1614 0.1925 0.2640 0.3256 0.3831 0.4097 0.4611 0.4979 0.5334 0.5676 0.6358 0.6955 0.7515 4.52

LDA

PW91 0.0450 0.0571 0.0942 0.1270 0.1614 0.1968 0.2587 0.3193 0.3784 0.4040 0.4486 0.4891 0.5322 0.5762 0.6413 0.7055 0.7687

0.1115 0.1508 0.2244 0.2906 0.3587 0.4280 0.5363 0.6409 0.7434 0.8041 0.8914 0.9661 1.0418 1.1181 1.2259 1.3289 1.4296

5.10

120

QC 0.0420 0.0453 0.0943 0.1249 0.1564 0.1883 0.2579 0.3245 0.3905 0.3956 0.4383 0.4696 0.5050 0.5403 0.6048 0.6660 0.7223


3.


GROSS E T

51

Table 4: Non-relativistic absolute correlation energies of atoms from selfconsistent calculations with various DFT approximations. All numbers in Hartree units.


KLI-CS Κ Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

BLYP

0.8030 0.8269 0.8832 0.9371 0.9882 1.0073 1.0812 1.1597 1.2324 1.3009 1.3693 1.4273 1.4704 1.5101 1.5465 1.6177 1.6795 1.7355

PW91 0.7994 0.8467 0.9033 0.9613 1.0198 1.0736 1.1375 1.2158 1.2933 1.3700 1.4562 1.5212 1.5768 1.6343 1.6917 1.7662 1.8393 1.9112

0.7821 0.8329 0.8855 0.9374 0.9882 1.0086 1.0861 1.1620 1.2331 1.3010 1.3694 1.4303 1.4753 1.5174 1.5570 1.6288 1.6912 1.7493

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

KLI-CS 1.7688 1.8222 1.8763 1.9281 1.9475 1.9905 2.0796 2.1571 2.2278 2.3123 2.3561 2.4146 2.4600 2.5024 2.5419 2.6134 2.6763 2.7338

BLYP 1.7832 1.8355 1.8863 1.9363 1.9558 2.0003 2.0874 2.1637 2.2340 2.3154 2.3649 2.4247 2.4704 2.5135 2.5544 2.6252 2.6876 2.7456

PW91 1.9509 2.0056 2.0671 2.1307 2.1899 2.2551 2.3412 2.4254 2.5081 2.6074 2.6705 2.7373 2.7964 2.8577 2.9193 2.9965 3.0726 3.1475

1. For most atoms, the absolute value of E$° is smaller than the absolute cor relation energy obtained w i t h any D F T method, as it should be according to the relation (26). 2. T h e values of £ " , Ε™*, £ and E?° agree quite closely w i t h each other while the absolute value of Ε^ is too large roughly by a factor of two. We mention that due to the well known error cancellation between Ε^ and £ , the resulting L D A values for total xc energies are much better. C

K L I

C S

C

P W 9 1

ΌΑ

ΌΑ

C

L D A

3. T h e difference between £ and E?° is smallest for the £ " values, larger for £ and largest for £ . T h e difference between £ and E^^ has three sources: C

C

L Y P

D F T

c

c

(a) T h e values of E^

c

(b) T h e values of i Ç irDFT

K L I

P W 9 1

c s

C

Q C

are only approximate, i . e. not identical w i t h u ^ a c t F T

are only approximate, i . e. not identical w i t h

^c.exacf

(c) A s shown i n the last section, the exact values E^ not identical.

xact

and E^J^


are

52


Currently it is not known w i t h certainty which effect gives the largest con tribution. However, with the arguments given i n the last section, we expect the contribution of (c) to be small. Assuming that the quoted values of E^ are very close to E^ t we conclude that ^ is closest to E^J^. c

L

exac

I

C

S

Table 4 shows correlation energies of atoms Κ through X e obtained w i t h the various selfconsistent D F T approaches. In almost a l l cases, the absolute K L I - C S values for E are smallest and the ones from P W 9 1 are largest, while the L Y P values lie i n between. In most cases, J^KLI-CS j #BLYP g within less than 1 % while | ^ | i larger (by up to 10 %) as the atomic number Ζ increases. We emphasize that reliable values for E^ do not exist for these atoms. c


a n (

W 9 1

a

r e e

s

c

Acknowledgments We thank C . Umrigar for providing us with the exact densities and K S poten tials for H " , H e , B e , N e , B e and Ne. We gratefully appreciate the help of D r . E . Engel especially for providing us w i t h a K o h n - S h a m computer code and for some helpful discussions. We would also like to thank Professor J . Perdew for providing us w i t h the P W 9 1 xc subroutine. T h i s work was supported i n part by the Deutsche Forschungsgemeinschaft. + 2

+ 8

Literature cited

[1] R. M. Dreizler, E.K.U. Gross. Density Functional Theory; Springer-Verlag: Berlin Heidelberg, 1990 [2] R.G. Parr, W. Yang. Density-Functional Theory of Atoms and Molecules Oxford University Press: New York, 1989 [3] V. Sahni, M. Levy, Phys. Rev. Β 33, 3869 (1986) [4] M. Levy, Proc. Natl. Acad. Sci. USA 76, 6062 (1979) [5] S. Baroni, E. Tuncel, J. Chem. Phys. 79, 6140 (1983) [6] M. Levy, J. P. Perdew, V. Sahni, Phys. Rev. A 30, 2745 (1984) [7] H. Stoll, A. Savin. In Density Functional Methods in Physics; R. M. Dreiz ler, J. da Providencia, Eds.: NATO ASI Series B123; Plenum: New York London, 1985; p 177. [8] A. Savin, H. Stoll, H. Preuss, Theor. Chim. Acta 70, 407 (1986) [9] M. Levy, R. K. Pathak, J. P. Perdew, S. Wei, Phys. Rev. A 36, 2491 (1987) [10] M. Levy, Phys. Rev. A 43, 4637 (1991) [11] A. Görling, M. Levy, Phys. Rev. A 45, 1509 (1992) [12] A.Görling,M. Levy, Phys. Rev. A 47, 13105 (1993) [13] C.J. Umrigar, X. Gonze, Phys. Rev. A 50, 3827 (1994) [14] A. Görling, M. Ernzerhof, Phys. Rev. A, in press (1995) [15] R. Colle, D. Salvetti, Theor. Chim. Acta 37, 329 (1975) [16] R. Colle, D. Salvetti, Theor. Chim. Acta 53, 55 (1979) [17] T. Grabo, E.K.U. Gross, Chem. Phys. Lett., 240, 141 (1995); Erratum: ibid. 241, 635 (1995)



3. GROSS ET AL.


53

[18] P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964) [19] E.H. Lieb. In Physics as Natural Philosophy; A. Shimony, H. Feshbach, Eds.; MIT Press: Cambridge, 1982; p.111;a revised version appeared in Int. J. Quant. Chem. 24, 243 (1983) [20] M. Levy, Phys. Rev. A 26, 1200 (1982) [21] W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965) [22] P.W. Payne, J. Chem. Phys. 71, 490 (1979) [23] A.Görling,Phys. Rev. A 46, 3753 (1992) [24] R. van Leeuwen, E.J. Baerends, Phys. Rev. A 49, 2412 (1994) [25] Q. Zhao, R. C. Morrison, R. G. Parr, Phys. Rev. A 50, 2138 (1994) [26] E. Engel, J.A. Chevary, L.D. Macdonald, S.H. Vosko, Z. Phys. D 23, 7 (1992) [27] E. Engel, S.H. Vosko, Phys. Rev. A 47, 2800 (1993) [28] C. J. Umrigar, X. Gonze, unpublished; a preliminary version of the Ne data was published in High Performance Computing and its Applicatio to the Physical Sciences, proceedings of the Mardi Gras '93 Conference, D. A. Browne et al. Eds.; World Scientific, Singapore: 1993 [29] The QC values for Be and Ne are taken from [38]; the QC values for the two electron systems are the differences between the total energies taken from [13] and HF total energies obtained with our program. [30] C. Lee, W. Yang, R.G. Parr, Phys. Rev. Β 37, 785 (1988) [31] J.P. Perdew. In Electronic structure of solids '91, P. Ziesche and H. Eschrig Eds.; Akademie Verlag, Berlin: 1991; and J.P. Perdew and Y. Wang, Tulane University, unpublished. [32] S.J. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58, 1200 (1980) [33] R.T. Sharp, G.K. Horton, Phys. Rev. 90, 317, (1953) [34] J.D. Talman, W.F. Shadwick, Phys. Rev. A 14, 36 (1976) [35] J.B. Krieger, Y. Li, G.J. Iafrate, Phys. Rev. A 45, 101 (1992) [36] J.B. Krieger, Y. Li, G.J. Iafrate, Phys. Rev. A 46, 5453 (1992) [37] J.B. Krieger, Y. Li, G.J. Iafrate, Phys. Rev. A 47, 165 (1993) [38] S.J. Chakravorty, S.R. Gwaltney, E.R. Davidson, F.A. Parpia, C. Froese Fischer, Phys. Rev. A 47, 3649 (1993) [39] A.D. Becke, Phys. Rev. A 38, 3098 (1988)


Chemical Applications of Density-Functional Theory - American

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