Asymptotically adjusted self-consistent multiplicative parameter exchange-energy-functional method: Application to diatomic molecules

PHYSICAL REVIEW A, VOLUME 65, 032515

Asymptotically adjusted self-consistent multiplicative parameter exchange-energy-functional method: Application to diatomic molecules ˜a Valentin V. Karasiev and Eduardo V. Luden Centro de Quı´mica, Instituto Venezolano de Investigaciones Cientı´ficas, IVIC Apartado 21827, Caracas 1020-A, Venezuela ~Received 30 August 2001; published 27 February 2002! An asymptotically adjusted self-consistent a (AASCa ) method is advanced for the purpose of constructing an accurate orbital-dependent local exchange potential with correct asymptotic behavior. This local potential is made up of the Slater potential plus an additional term containing a multiplicative parameter a x ~a selfconsistently determined orbital functional! times a local response potential that is approximated using standard exchange-energy functionals. Applications of the AASCa functionals to diatomic molecules yield significantly improved total, exchange, and atomization energies that compare quite well, but at a much lower computational cost, with those obtained by the exact orbital-dependent exchange energy treatment @S. Ivanov, S. Hirata, and R. J. Bartlett, Phys. Rev. Lett. 83, 5455 ~1999!; A. Go¨rling, Phys. Rev. Lett. 83, 5459 ~1999!# ~in fact, the present results are very close to the Hartree-Fock ones!. Moreover, because in the AASCa method the exchange potential tends toward the correct (21/r) asymptotic behavior, the ionization potentials approximated by the negative of the highest-occupied-orbital energy have a closer agreement with experimental values than those resulting from current approximate density functionals. Finally, we show that in the context of the present method it is possible to introduce some generalizations to the Gritsenko-van Leeuwen-van LentheBaerends model @O. Gritsenko, R. van Leeuwen, E. van Lenthe, and E. J. Baerends, Phys. Rev. A 51, 1944 ~1995!#. DOI: 10.1103/PhysRevA.65.032515

PACS number~s!: 31.15.Ew, 31.10.1z, 31.15.Ne, 71.45.Gm

I. INTRODUCTION

The Kohn-Sham ~KS! approach to density-functional theory ~DFT! @1,2# is a standard tool in condensed matter physics and theoretical-chemistry calculations. Because the exact density functional for the exchange-correlation energy is unknown, it becomes necessary, however, to rely on approximate functionals. But, currently available approximate exchange functionals do not reproduce the correct asymptotic form of the electrostatic potential (21/r) and fail to eliminate the self-interaction correction ~SIC!. In fact, the apparent success of most of the generalized gradient approximation ~GGA! and hybrid functionals is actually due to error cancellation between the exchange and correlation contributions as well as due to the fact that these functionals do not satisfy the variational principle and allow, therefore, for the occurrence of energies lower than the exact one. In order to avoid the shortcomings of GGA-based exchange functionals, several methods involving an exact treatment of the DFT exchange energy have been developed. Among these methods we can mention the optimizedpotential model ~OPM! @3#, ~with the approximation introduced by Krieger, Li, and Iaffrate ~KLI-OPM! @4#!, the localscaling transformation version of DFT ~LST-DFT! @5#, and the exact orbital-dependent exchange energy ~EXX! @6,7#. The common trait of all these methods is the elaboration of more or less complicated prescriptions for the construction of optimal ~in some sense! local exchange potentials. A different and quite simple method for obtaining densitydependent local exchange potentials, was recently introduced @8#. This method, which was denoted as the SCa method, contains as a basic ingredient the multiplicative parameter a , which is a functional that depends on the orbitals ~or on the 1050-2947/2002/65~3!/032515~8!/$20.00

density! and which is determined self-consistently through iterations of the Kohn-Sham equations. Results of the application of various approximations to the SCa potentials @8# to diatomic molecules were reported in Ref. @9#. In Ref. @10# a generalization of this method was introduced, for the purpose of extending the use of the multiplicative parameter even to those cases where the standard exchange functional is of the hybrid type. This new approach, labeled as the selfconsistent multiplicative constant ~SCMC! method permitted us to correct any approximate exchange-energy functional and to improve it from a fundamental point of view by completely removing the SIC and by restoring the variational property to the functional. It failed, however, to yield a local exchange potential having the correct asymptotic behavior (21/r) for large r. In the present work we further develop the SCa and the SCMC methods @8,10# having in mind adjusting the local exchange potential so that it attains the correct (21/r) behavior. This approach, which we denote as the asymptotically-adjusted self-consistent a (AASCa ) method, partakes by construction of the same advantages characterizing the orbital-dependent exchange-energy treatment EXX @6,7#, namely: ~i! it is free from the self-interaction error; ~ii! it leads to an exchange potential with the correct (21/r) asymptotic behavior; ~iii! for the x-only calculations, its total energy satisfies the variational principle, i.e., this energy is the expectation value of the N particle Hamiltonian with reˆ u F & . It spect to a one-determinantal wave function E5 ^ F u H remains, however, a low-cost method, as only slight changes in the handling of the Kohn-Sham equations are necessary for its application. The AASCa method is implemented here using both existing exchange-energy functionals ~LDA and GGA! as well

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PHYSICAL REVIEW A 65 032515

as an approximate orbital-dependent model expression. Comparison of atomization energies D 0 and exchangeenergy contribution to D 0 with Hartree-Fock ~HF! and EXX values shows that the present AASCa results are much closer to the exact EXX values than those resulting from the original approximate functionals. Also, the HOMO energies coming from self-consistent calculations with AASCa exchange potential are very close to the EXX ones. II. METHOD A. Asymptotically adjusted self-consistent multiplicative parameter exchange potential

The exact exchange-energy of N occupied KS orbitals N can be defined as the electrostatic interaction energy $ c i % i51 between the KS density r and the Slater potential v S @11#

~viz, the electrostatic potential of the Fermi hole!, E x@ $ c i% # 5

1 2

E

r ~ r!v S ~ r! dr,

~1!

where 1 v S ~ r! 52 r ~ r! 3

E

~5! The first term of Eq. ~5! ensures the correct asymptotic behavior of 21/r for large r. The second term does not have a contribution in the asymptotic region. Hence, this term may be adequately approximated by resorting to existing exchange-energy functionals or else, it may be modeled. For any approximate exchange-energy functional E 0x @ r # 5 * r (r) e 0x (r)dr5 * e 0x (r)dr, where e 0x is the exchangeenergy density per electron and e 0x is the exchange-energy density, the second term in Eq. ~5! may be approximated by 1 0 0 ˜v resp ~ r! '˜v resp ~ r! 2 v 0S ~ r!# 5 v 0x ~ r! 22 e 0x ~ r! , ~ r! 5 @v resp 2 ~6! 0 52 v 0x 2 v 0S are where v 0x 5 d E 0x / d r , v 0S 52 e 0x 52e 0x / r , v resp the exchange potential and its components arising from a particular choice of E 0x . For example, in the LDA approxi52C x r 4/3, and thus Eq. ~6! yields mation e LDA x

2 1 LDA LDA ˜v resp ~ r! 2 v LDA ~ r!# 5 C x r 1/3. ~ r! 5 @v resp S 2 3

N

( d ~ s i , s j ! f *i ~ r! f j ~ r! i, j51

f *j ~ r8 ! f i ~ r8 ! u r8 2ru

dr8 .

~2!

The exchange potential v x defined as the functional derivative of the exchange-energy functional given by Eq. ~1! has two components 1 1 v x ~ r! 5 v S ~ r! 1 v resp ~ r! . 2 2

Er

~ r8 !

d v S ~ r8 ! dr8 . d r ~ r!

~4!

For the case of one- or two-electron systems ~one occupied atomic or molecular orbital!, the response potential is equal to the Slater potential and the exchange potential is just v S . In general, however, the response potential v resp is described by an expression involving the inverse of the static KS linear response function @13# and for this reason its handling is somewhat cumbersome. The LDA and GGA approaches show deficiencies in approximating v resp @12#. Although is has been possible to device a family of GGA exchange functionals that satisfy the large r asymptotic conditions, a proper treatment of the intermediate region has not yet been attained. To avoid these difficulties we have rearranged terms in Eq. ~3! in the following way:

~7!

Of course, other suitable approximations, not necessarily coming from the exchange-energy density functional, may be 0 . In the present work, we approximate ˜v resp (r) used for ˜v resp by multiplying the left-hand side of Eq. ~6! by the functional a x so that the local exchange potential is given by 1 0 a v AASC ~ r! 5 v S ~ r! 1 a x @ $ c i % #@v resp ~ r! 2 v 0S ~ r!# x 2 0 5 v S ~ r! 1 a x˜v resp ~ r! .

~3!

The first one is the Slater potential, Eq. ~2!; the second, the response potential @12# v resp ~ r! 5

1 v x ~ r! 5 v S ~ r! 1 @v resp ~ r! 2 v S ~ r!# [ v S ~ r! 1˜v resp ~ r! . 2

~8!

To ensure the satisfaction of the variational principle ~for x-only calculations! the parameter a x ~i.e., the functional a x @ $ c i % # ) is defined self-consistently, from the condition that the exchange-energy corresponding to the exchange potential, Eq. ~8!, be equal to the exchange-energy calculated through the exact expression defined by Eqs. ~1! and ~2!,

a x@ $ c i% # 5

E x @ $ c i % # 2E xL P @v S # , 0 ˜ resp E xL P @v #

~9!

where the exchange-energy E xL P @v# corresponding to exchange potential v is calculated through the Levy-Perdew virial expression @14,15# E xL P @v# 5

E

@ 3 r ~ r! 1r•“ r ~ r!#v~ r! dr.

~10!

It follows from Eq. ~9! that the exchange-energy expression corresponding to the approximate potential of Eq. ~8! is equal to the exact one

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0 a ˜ resp 5E xL P @v S # 1 a x E xL P @v E AASC # 5E x @ $ c i % # . ~11! x

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Let us consider now the theoretical basis underlying the AASCa method defined through Eqs. ~8!–~11!. Taking the a given variational derivative of the exchange-energy E AASC x by Eq. ~11! with respect to the one-particle orbital c i* and taking into account that a x @ $ c i % # is a functional of oneelectron orbitals, we obtain a d E AASC @ $ c i% # x

d c *i ~ r!

0 5 v S ~ r! c i ~ r! 1 a x˜v resp ~ r! c i ~ r!

0 ˜ resp 1E xL P @v #

d a x@ $ c i% # d c i* ~ r!

Eq. ~16! cancels out the local part of D vˆ x and Eq. ~16! is transformed into the usual HF integrodifferential equation. 0 and its modification B. GLLB model for ˜v resp

Recently, Gritsenko, van Leeuwen, van Lenthe, and Baerends ~GLLB! proposed a self-consistent approximation to the KS exchange potential in the form of Eq. ~8! @16# where the 0 expression adopted by these authors for the potential ˜v resp is based on the form taken by the OPM exchange potential in the KLI approximation @4# N 0 ˜v resp ~ r! 5

a [ v AASC ~ r! c i ~ r! 1D vˆ x ~ r! c i ~ r! , ~12! x

where D vˆ x is an additional exchange operator term arising in the KS equations as the result of varying a x @ $ c i % # with respect to the orbital c * i ,

d a x@ $ c i% # 0 ˜ resp D vˆ x ~ r! c i ~ r! 5E xL P @v 5 @vˆ xi ~ r! 2 v S ~ r! # d c *i ~ r! 0 2 a x˜v resp ~ r!# c i ~ r! ,

~13!

and vˆ xi is the usual HF exchange operator

v i 5 Am 2 e i ,

E c* j

~ r8 ! c i ~ r8 !

u r2r8 u

v i5

~14!

dr8 .

The contribution of D vˆ xi to the exchange-energy is 0 ˜ resp DE x @ $ c i % # 5E x @ $ c i % # 2E xL P @v S # 2 a x E xL P @v #.

H

i

ˆ ~ r!@v KLI x ~ r ! 2 v xi ~ r !# c i ~ r ! dr.

v N 50.

~19!

~20!

In the context of the AASCa method, in order to estimate appearing in the integral of the values of v i we replace v KLI x Eq. ~19! by the exchange potential of Eq. ~5!, and in addi0 tion, to simplify matters, we use the approximate form v resp of the response potential,

v 0i 5

J

E c* H i

1 0 ~ r! 2 v S ~ r!# ~ r! v S ~ r! 1 @v resp 2

J

2 vˆ xi ~ r! c i ~ r! dr.

~16!

As the corresponding exchange-energy expression does not depend on the value of the constant C, it is still equal to the exact one given by Eqs. ~1! and ~2!. Of course, in order to fulfil the requirement that the exchange operator appearing in the KS equations Eq. ~16! be local ~i.e., that it be multiplicative exchange potential! the value of C must be fixed at 0. a in On the other hand, for C51, the local potential v AASC x

~18!

To determine the values of v i the system of linear equations must be solved. For the highest occupied orbital c N the expotential and of the vˆ xi exchange pectation values of the v KLI x operator are equal @4# so that the corresponding parameter v N vanishes

1 a 2 ¹ 2 1 v ext ~ r! 1 v H ~ r! 1 v AASC ~ r! 1CD vˆ x ~ r! c i ~ r! x 2 5 e i c i ~ r! .

E c*

~15!

Taking into account Eq. ~9! we conclude that DE x 50, i.e., that D vˆ xi does not contribute to the exchange-energy expression. Consequently, the D vˆ xi c i term in the KS equations can be multiplied by any constant C without affecting the exchange-energy expression. In that case, the x-only KS equations take the following form:

~17!

and where m is the one-electron energy of the highest occupied orbital m 5 e N . In the KLI approximation, the values of v i are defined as the differences between the expectation values of the KLI exchange potential and that of the HF exchange operator Eq. ~14! with respect to KS orbitals

d E x@ $ c i% # 52 ( d ~ s i , s j ! c j ~ r! j51 d c *i ~ r! 3

u c i ~ r! u 2 , r ~ r!

where v i is a semiempirical parameter whose form is assumed to be

N

vˆ xi ~ r! c i ~ r! 5

(

i51

vi

~21!

Upon calculation of the expectation values of the Slater potential Eq. ~2! and the HF exchange operator Eq. ~14!, 8 and ^ c i u vˆ xi u c i & 52E xi 9 , we obtain ^ c i u v S u c i & 52E xi

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v 0i 5

1 2

Ec u

i ~ r! u

2 0 8 2E xi 9 ! 2E xi 8, v resp ~ r! dr12 ~ E xi

~22!

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PHYSICAL REVIEW A 65 032515

where the following different definitions of the ‘‘contribution N E 8xi of orbital i’’ to the exchange-energy, E x @ $ c i % # 5 ( i51 N 9 , are introduced: 5 ( i51 E xi

85 E xi 9 52 E xi

1 2

E

where r l (r)5l 3 r (lr). From Eqs. ~8!, ~17!, ~24!, and ~25! it follows that the scaling property of the asymptotically adjusted exchange potential given by Eq. ~8! is a ~@ r l # ;r! 5l v S ~@ r # ;lr! v AASC x

u c i ~ r! u 2 v S ~ r! dr,

1 2

( d ~ s i , s j ! E c *i ~ r! c j ~ r! j51

3

E c*

1

N

j

~ r8 ! c i ~ r8 !

u r2r8 u

dr8 dr.

v i5~ e N2 e i !

v 0i ,

~23!

~24!

where a is an empirical parameter. Note that the scaling property of v i Eq. ~24! depends on the value of the parameter a. This fact, however, does not alter the scaling property of the exchange potential @17,18# given by Eq. ~8!. This can be readily shown by writing down the scaling properties of 0 of Eq. ~17! and v 0i and v i of Eq. ~24!, of the potential ˜v resp of a x of Eq. ~9!,

v 0i ~@ r l # ! 5l v 0i ~@ r # ! , v i ~@ r l # ! 5l 2a11 v i ~@ r # ! , 0 0 ˜v resp ~@ r l # ;r! 5l 2a11˜v resp ~@ r # ;lr! ,

a x ~@ r l # ! 5

l l

2a11

a x ~@ r # ! ,

l

0 a x l 2a11˜v resp ~@ r # ;lr!

a 5l v AASC ~@ r # ;lr! . x

8 'E xi 9 , it follows that the second As one can show that E xi term of Eq. ~22! is very small in comparison with the last one. Unfortunately, Eq. ~22! with the approximate potential 0 does not satisfy the condition given by Eq. ~20!. This v resp may be remedied by multiplying v 0i times a function of the difference ( e N 2 e i ), a

l 2a11

~25!

Thus, in view of the fact that the choice of v i given by Eq. ~24! does not alter the scaling properties of a ( @ r l # ;r), it is clear that the choice adopted by Gritv AASC x senko et al., Eq. ~16! corresponds to a particular case of Eq. ~24!. To obtain the simplest approximation for the v i parameters, we neglect the first term in Eq. ~22!. In that case, Eq. ~24! gives the following approximate expression for v i :

8. v i 52 ~ e N 2 e i ! a E xi

~27!

It is not important which E xi is used in Eq. ~27!, because the 9 are very close to each other. Due 8 and E xi two quantities E xi to the fact that Eq. ~27! satisfies the condition stated by Eq. 0 of the AASCa exchange potential ~20!, the component ˜v resp has a short-range behavior and does not contribute to the asymptotic region. III. RESULTS

In order to test the proposed AASCa method, comparative numerical x-only calculations have been performed for first row atoms, and for selected diatomic molecules. AASCa results are compared with EXX x-only calculations. The modified version of the spin-restricted finite-difference full-numerical program developed by Laaksonen and coworkers @19,20# was used. The calculations employed the standard @ 1693193;25/28 bohr# grids for atoms and molecules ~see details of the program in Refs. @19,20#!. The LDA

TABLE I. Total absolute energies for atoms from various x-only calculations. All values are in hartree.

Li Be Bb Cc Nd Oe Ff Ne

~26!

AA-LDA

AA-PW91

GLLB

AA-m1

AA-m2

OPM a

7.4271 14.5658 24.5169 37.6735 54.3828 74.7906 99.3885 128.5243

7.4283 14.5674 24.5200 37.6773 54.3870 74.7960 99.3945 128.5305

7.4323 14.5723 24.5248 37.6834 54.3960 74.8076 99.4069 128.5444

7.4323 14.5723 24.5275 37.6863 54.3977 74.8076 99.4072 128.5446

7.4323 14.5723 24.5273 37.6861 54.3976 74.8077 99.4073 128.5446

7.4325 14.5724 24.5283 37.6889 54.4034 74.8121 99.4092 128.5454

a

Values are taken from Ref. @27#. 1 s 2 2 s 2 1 p 1 , S52. c 1 s 2 2 s 2 1 p 2 , S53. d 1 s 2 2 s 2 1 p 2 3 s 1 , S54. e 1 s 2 2 s 2 1 p 2 3 s 2 , S53. f 1 s 2 2 s 2 1 p 3 3 s 2 , S52.

b

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TABLE II. Spin-restricted finite differences HF total energies ~in hartree! and differences between KS-x2only and HF total energies ~in mhartree!. The last column is the ‘‘exact’’ results obtained from basis-set calculations.

H2 FH OH N2 O2 F2 NH CO a

HF@ FD #

D(LDA)

D(PW91)

D(AA-LDA)

D(AA-PW91)

D(GLLB)

D(AA-m1)

D(AA-m2)

D(EXX) @ BS # a

21.1336 2100.0708 275.4213 2108.9931 2149.6674 2198.7722 254.9782 2112.7909

89.9 919.6 833.9 1237.0 1479.0 1715.2 779.9 1260.2

7.7 239.0 6.6 265.4 278.9 2133.4 76.3 254.6

0.0 24.1 23.0 32.3 32.6 55.5 24.9 45.8

0.0 17.0 16.1 30.4 32.1 39.4 18.7 31.1

0.0 5.9 6.6 11.4 13.4 16.8 9.9 12.2

0.0 3.7 3.8 8.4 9.0 13.4 6.8 12.4

0.0 3.7 3.9 8.3 9.3 13.6 6.8 9.1

0.0 2.0 2.4 5.2 6.6 8.6 2.0 5.1

Reference @6#.

and Perdew-Wang 91 ~PW91! exchange functional @21,22# x-only results are compared with those obtained by the 0 in Eq. ~8! is apAASCa scheme when the potential ˜v resp proximated by the LDA or PW91 exchange potential components appearing in Eq. ~6!. These AASCa calculations are denoted as AA-LDA and AA-PW91, respectively. Results of Gritsenko and collaborators @16# with v i defined by Eq. ~18! are denoted as GLLB. The AASCa scheme based on the 0 ˜v resp potential of Eq. ~17! with v i defined by Eq. ~27! with a51,

8, v i 52 ~ e N 2 e i ! E xi

new parameter v i8 according to Eq. ~17! leads to a new response potential, which is effectively the old one times the constant C. On the other hand, this new response potential introduces the parameter C in the denominator of the righthand side of Eq. ~9! so that the new a x is equal to the old one divided by the constant C. Since in Eq. ~8! the last term is the product of a x times the response potential, C cancels out. Thus, the new parameters v 8i 5C v i , i51, . . . N, yield exactly the same results, except for the final a x value. Bearing this fact in mind we introduce a new set of parameters

~28!

is labeled as AA-m1. For this case, to check the sensitivity of the method to the semiempirical parameters v i , we have also performed calculations with

v i5 e N2 e i .

~29!

This scheme is denoted as AA-m2. The parameters v i of Eq. ~29! satisfy the condition given by Eq. ~20! and showing a similar behavior as those parameters defined by Eqs. ~18! and ~28! they tend to become bigger for inner KS orbitals. Let us remark that if we had chosen a new set of parameters $ v i8 % as the product v 8i 5C v i @where v i is defined by Eq. ~28! or by Eq. ~29! and C is a constant#, the AA-m1 and AA-m2 methods remain unmodified. The reason is that the

V i5

vi N

~30!

,

( vi

i51

where v i is defined by Eq. ~28! for AA-m1 or by Eq. ~29! for AA-m2. The parameters of Eq. ~30! are dimensionless and normalized N

( V i 51,

~31!

i51

and the AASCa schemes based on these parameters remain unmodified. The different AASCa atomic total energies are compared with OPM values in Table I. The error of the AA-LDA and

TABLE III. Atomization energies of molecules ~kcal/mol!.

H2 LiH OH FH Li2 LiF CO N2 O2 F2

LDA

PW91

AA-LDA

AA-PW91

GLLB

AA-m1

AA-m2

EXXa

145 77 168 183 30 157 357 389 227 95

147 79 155 170 27 149 323 355 189 75

84 34 68 99 2 92 176 122 34 238

84 33 69 100 1 92 180 118 27 235

84 33 67 99 3 94 181 119 24 237

84 33 69 100 3 93 179 119 27 235

84 33 69 100 3 94 181 119 27 235

84 34 69 98 4 92 172 112 30 242

a

Reference @7#. 032515-5

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PHYSICAL REVIEW A 65 032515

TABLE IV. Exchange-energy contribution to the atomization energies of molecules ~kcal/mol!.

H2 LiH OH FH Li2 LiF CO N2 O2 F2

LDA

PW91

AA-LDA

AA-PW91

GLLB

AA-m1

AA-m2

HF

110 83 142 168 33 180 262 295 191 104

109 86 123 150 30 172 226 258 153 90

21 34 14 65 15 105 32 241 244 251

21 35 13 63 16 102 38 242 250 244

21 37 21 76 11 136 55 223 247 236

21 37 19 72 11 138 40 248 264 257

21 37 20 73 11 138 43 245 261 254

21 33 13 66 2 132 41 256 271 243

AA-PW91 values is in the second decimal digit. The GLLB and AA-m1, AA-m2 schemes reproduce the OPM atomic total energies much better with an error in the third decimal digit. In Table II, we present the spin-restricted finite-difference HF energies calculated at the experimental geometries @24# and the differences between KS-x-only and HF total energies: D5E KS2x 2E HF . The ‘‘exact’’ EXX values of D, evaluated from results presented in @6#, change in magnitude between 0 ~for H2 molecule! and 8.6 mhartrees ~for F2 molecule!. The LDA values for D turn out to be larger ~by two orders of magnitude! than the EXX ones. The PW91 functional yields erratic results that change sign and differ by one order of magnitude from the exact values. The AA-LDA and AA-PW91 schemes reproduce D values that are much closer to the exact ones than those obtained from the original LDA and PW91 functionals; these values, however, are still be far away from the EXX ones. The GLLB and AA-m1, AA-m2 approximations yield D values in the range between 0 ~for H2 molecule! and 16.8 mhartree ~for F2 molecule! showing good agreement with the EXX values. In Table III, the atomization energies generated within the LDA, PW91, AA-LDA, AA-PW91, GLLB, AA-m1, and AA-m2 approximations are compared with the EXX results. The LDA and PW91 exchange functionals overbind mol-

ecules with respect to the EXX method. All GLLB and AASCa results are in excellent agreement with the exact exchange; the maximum error is of 10 kcal/mol for the N2 molecule in the AA-LDA scheme. The values of the different AASCa schemes are observed to be very close to each other. The exchange-energy contributions to the atomization energy, D x 5E x @ A # 1E x @ B # 2E x @ AB # , calculated within the LDA, PW91, AASCa , and HF approaches are presented in Table IV. As for the atomization energies, the AASCa D x values are in excellent agreement with the HF results. The AASCa potentials have the correct asymptotic behavior and Table V shows that HOMO energies obtained from these potentials are very close to the EXX orbital energies with a maximum error of 1 eV for all molecules except for the EXX HOMO energy of the F2 molecule taken from Ref. @7#. The negatives of the HOMO energies calculated within the AA-LDA and AA-PW91 approximations are also much closer to the experimental ionization potentials than the corresponding LDA and PW91 values. In Fig. 1, we compare the negative v x and the positive ˜v resp potentials for the ground state of the CN 2 molecule along the interatomic axis for the PW91, AA-PW91, GLLB, and AA-m2 approximations with the corresponding ‘‘exact’’ Kohn-Sham potentials obtained by the iterative procedure described in Ref. @23#.

TABLE V. Ionization potential ~in eV! approximated by the negative of the highest-occupied-orbital energies. Comparison with experimental values. LDA PW91 AA-LDA AA-PW91 GLLB AA-m1 AA-m2 HF H2 9.0 OH 7.1 FH 8.5 Li2 2.5 CO 7.8 2 9.1 N 2 →N 1 ( ( ) g 2 2 N 2 →N 1 2 ( P u ) 10.6 4.7 O2 F2 8.2

9.7 7.4 8.8 2.7 8.3 9.5 10.7 5.0 8.6

16.2 13.5 15.5 5.2 12.9 12.4 13.8 8.5 15.8

16.2 11.4 13.3 5.0 11.1 12.7 13.9 9.0 13.4

16.2 13.9 16.5 5.0 13.7 15.5 16.2 9.6 15.9

a

Reference @7#. From Ref. @24#. c Reference @6#. b

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16.2 15.9 18.4 5.0 15.2 17.1 18.3 13.7 19.0

16.2 15.7 18.2 5.0 15.0 17.0 18.0 13.2 18.7

EXXa

16.2 16.2 15.6 15.0 17.7 17.4 5.0 5.1 15.1 14.1 17.3 14.1a ~17.2!c 16.7 18.1c 14.5 13.6 18.2 14.5

Expt.b 15.5 13.0 16.0 5.0 14.0 15.6 16.7 12.1 15.7

ASYMPTOTICALLY ADJUSTED SELF-CONSISTENT . . .

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FIG. 1. Comparison of KS exchange potential and its component ˜v resp with various approximate potentials for the ground state of the CN2 molecule.

The AA-PW91 v x and ˜v resp potentials are closer to the exact ones than those potentials generated within the PW91 approximation. This is true for all distances, and specially for the interatomic and asymptotic regions. Due to the step form of ˜v resp , the GLLB and AA-m2 potentials reproduce well the shell structure of the exact exchange potential in the intermediate region. All AASCa exchange potentials have the correct asymptotic behavior due to the presence of the v S term. IV. CONCLUSIONS

In this paper, the SCMC method is applied to define a parameter a x that is included in the exchange potential constructed from the Slater potential plus an additional term. The Slater potential provides the correct (21/r) asymptotic behavior, and the additional short-range term reproduces the shell structure associated with the exchange potential. Clearly, by relying on the SCMC a x scheme we guarantee: first, that our method is free from the self-interaction error,

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W. Kohn and L.J. Sham, Phys. Rev. A 140, 1133 ~1965!. P. Hohenbegr and W. Kohn, Phys. Rev. 136, B864 ~1964!. J.D. Talman and W.F. Shadwick, Phys. Rev. A 14, 36 ~1976!. J.B. Krieger, Yan Li, and G.J. Iafrate, Phys. Rev. A 45, 101 ~1992!. ˜ a, V. Karasiev, R. Lo´pez-Boada, E. Valderrama, and E.V. Luden J. Maldonado, J. Comput. Chem. 20, 155 ~1999!, and references therein. S. Ivanov, S. Hirata, and R.J. Bartlett, Phys. Rev. Lett. 83, 5455 ~1999!. A. Go¨rling, Phys. Rev. Lett. 83, 5459 ~1999!. ˜ a, and R. Lo´pez-Boada, Int. J. QuanV. Karasiev, E.V. Luden tum Chem. 70, 591 ~1998!.

and second, that for x-only calculations the total energy is the expectation value of the N-particle Hamiltonian with respect to a one-determinantal wave function and, hence, that it satisfies the variational principle. To test the quality of the AASCa method advanced here, we have performed x-only atomic and molecular calculations. From the comparison of the AASCa results with the exact exchange ones, we conclude, that the AASCa scheme provides an excellent approximation to the total, atomization, and HOMO EXX energies. Thus, the AASCa method may be viewed as a less-costly alternative to the EXX method. Moreover, since essentially the purpose of the EXX method is to obtain the optimal approximation to the Hartree-Fock exchange through a local potential, and bearing in mind that the AASCa method mimicks quite well the EXX method, one may use the AASCa results to replace the Hartree-Fock terms in the hybrid functionals generated from the adiabatic connection model @25,26#. Thus, for example, starting from the hybrid functional LSD ACM , 1DE GGA 1DE GGA 50.25E HF E xc ! 1E LSD c c x x 10.75~ E x ~32!

we may obtain a new and totally local ‘‘hybrid’’ functional by replacing the Hartree-Fock exchange terms by the corresponding AASCa ones, a 1DE GGA . 10.75~ E LSD 1DE GGA E xc 50.25E AASC ! 1E LSD c c x x x ~33! These changes would certainly speed up applications involving hybrid functionals, although they do not address the question of error cancellation. Finally, we would expect this functional to have the same performance in all respects as the original nonlocal functional. Some preliminary numerical results confirm this assertion.

ACKNOWLEDGMENTS

The authors would like to gratefully acknowledge support of this work by CONICIT of Venezuela through Group Project No. G-97000741. The authors would also like to thank Dr. Carlos Gonzalez for help in the implementation of the SCMC methods in Gaussian code.

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