Fractional occupation in the hartree-fock method

Fractional occupation in the hartree-fock method

Volume 73, number 3 FRACTIONAL CHEMICAL PHYSICS LE’ITERS OCCUPATION IN THE HARTREE-FOCK 1 August 1980 METHOD * H S. BRANDI, M.M. DE MATOS Depa...

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Volume 73, number 3

FRACTIONAL

CHEMICAL PHYSICS LE’ITERS

OCCUPATION

IN THE HARTREE-FOCK

1 August

1980

METHOD *

H S. BRANDI, M.M. DE MATOS Departamento de Fhca, Pontzficia Univcrsidade Cotdlwa. 20000 Rio de Janeuo. RJ. Brasrl

and R. FERREIRA Departamento de Fisica. Inshtuto 13560 Sr70 Gzrlos. SP. Brad

de Fisrca e Quinma de Srio chrlos - USP,

Recewed 4 February 1980; III final form 6 May 1980

It is shown that cFF = cIx” if we aliow fractional occupation in the Hartree-Fock formahsm and substitute the local for the non-local exchange potential. Calculations of Ionization energies of closed-shell atoms show that the proposed method ISas accurate, and sunpler than the standard transltion*perator techniques.

XCY potenti

1. Introduction In the last few years a great deal of work has been done using Slater’s SCF Xar method [I], which greatly simphfies calculations since the non-local Hartree-Fock exchange potential is approximated by a local potential proportional to the l/3 power of the local charge dens@. Leite and Ferrelra [2] introduced withm the Xcu scheme the concept of the “transition state”, where the occupation numbers of the spm-orbitals are halfway between those cf the initial and final states. This concept has proved to be very successful for the study of electron bindmg energies and optical excitations [ 11, because it takes into account the possibility of relaxation of spin-orbitals, a fact which is neglected in the standard Hartree-Fock procedure and Koopmans’ theorem. Goscmski et al. [3] introduced the “transition Fock operator” method and demonstrated that it corresponds to the AEsCF techmque, in wluch two SCF calculations are made, separately optimized for the parent and hole states. Goscinsla et al. [3] have also shown by thetr perturbation approach how to just@ the concept of fractiona! occupation m Hartree-Fock schemes. In the present work a direct demonstration of the validity of the transltion state concept within the HartreeFock scheme is proposed. We have adopted a point of view related to the one proposed by Slater for the Xa method and wdl restrict our derivation to closed-shell systems. It is possible to show that ti procedure is convenient for the calculation of optical transitions and iomzation energies, and has intrinsic computational advantages over the usual usCF c&uh&ons since it involves only one self-consistent cdctiation. The assumption of fractional occupation in Hartree-Fock schemes leads to &fficulties since we necessarily deal with open-shell systems. For the purpose of calculation we obtained a simple expression relating the eigenenergies of the “transition Fock operator” and the hyper-Hartree-Fock (HHF) [l] one-electron energies. This expression was used to calculate the ionization energies of several closed-shells atoms. The results are basicalIy equivalent to those of Coscinski et al. 131, but the numerical calculations are simpler than their perturbation method.

* Work partially supp orted by the Brazilian Agencies CNPq and FINEP. 597

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energies

the average total Hartree-Fock

energy

for closed shells 111the standard

form

(1) where & =
Jl, = Qlglr/)

,

Kt, = (iilgl/r>

.

f and g are the one-electron

and two-electron operators appeanng in the hamrltonum. To obtam the Hartree-Fock equatrons we vary one of the occupred spm-orbitals preserving and orthogonality to the other spin-orbitals. The ergenvalues e,“’ are easily found to be [i ]

its normahzation

In the usual Hartree-Fock scheme the occupation numbers assume values 0 or 1, and it 1s clear that eiHF equals the sum of all terms of eq. (1) which are present when tz, = 1 and absent when n, = 0. If we use the notation L!THF(nl = 1)) and (EHF(Fz~ = 0)) for the total energy when n, = 1 and 12, = 0. respectively, we have = 1)) - (E,,(tt,

eHF = (EH&$ t

= 0)))

(3)

where rt was assumed that no relaxatron of the spin-orbitals takes place. function of the Now, let us make the same assumption of the X0 method, namely, that (EHF ) is a continuous occupatron number. In what follows the only basic difference wrth respect to the Xa method IS that the non-local potential IS strll present, Instead of the Xol local potential. Let us drfferentiate (E& [eq. (I)] wrth respect to the occupation number &?k_ a(&&iank

= fk + $ c

J

fir wk,

- &,I +
-

(4)

Obv-rously JzJ = JJ, and KIJ = K,, , therefore a(&F)lan, Companson

=

fk + c

J

n,(JkJ - KkJ) .

(5)

wrth eq. (2) leads to

a(EHF>/affk

HF = Ek .

(6)

eq. (6) was obtamed under the assumpthat efF depends exphcrtIy on all n,, except n, = nk_ Furthermore, tron that integrals JkJ and KkJ are not exphcrt functions of the occupatron numbers. In other words, eq. (6) is only vahd for unrelaxed orbrtals. It IS cIear that, under the above assumptions, expressions for the ergenvalues of the Hartree-Fock equatrons are obtamed as partral denvatrves of the Hartree-Fock energy wrth respect to the occupatron numbers, s~mrlarly exchange to the Xor method; evidently et?” = erxC’ rf the local Xar potential replaces the non-local Hartree-Fock potential. NotIce

3. Optical transitions and ionization

In the calculatron of excitatron and romzation energies the importance of obtainmg properly self-consistent wavefunctions both for the imtial and the final states is well known. We wrlI show that this can be approximately method as Slater has done for the Xcr achieved mtroducrng the concept of “transrtron state” m the Hartree-Fock method [1,2]. 598

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As the ergenvalue obtained m eq. (6) 1s the partial derivative of (EHF ) with respect to an occupation number. Slater and Wood [4,5] and expand (E& in a power series of the occupation numbers. Let us restnct our calculatron to a non-spm-polarized scheme and express UC) as a power series in ni - Q, where tr,, is the occupation number of a standard state about whrch the expansion is made; that is:

we will follow

(7) We calculate the energy difference between an initial state an a final state in whxh n, decreased by one and nj increased by one unit, that is, one electron has jumped from the ith to the ~th state [I] _ Assuming the initial state as the standard state around which we perform our series expansion, we have for this state n, = nio, ni = nlo, and we find that the third derivatives for the final state n, = nlo - 1, nI = nlo + 1. Using eq. (7) and making (E) = (E& are zero; the excitation energy is (E(fmal))

- cE(initial))

= nlo - 1, n, =nlo + 1)) -
= (E(n,

We see that a2(EHF)/&Zl an, IO 1s a correction to the Koopmans’ theorem. A better scheme than eq. (8) would be to have monoelectromc energies modified m such manner to cancel the second derivatives. This can be achieved If we use the “transrtion-state” concept, in which the expanstons of the initral and final states are expressed as a power senes in nk - Z,,. where ii,,o = rzlo - 5 and n,, = iiIO + i; in this case we find:

Agam, if(E)

= (E,F),

tEHF(fmal))

the third derivatrves

- (EHF(uutial))

=

are zero and we get:

e,HF -

EHF I

-

Notrce that we have neglected the explicit dependence of the spin-orbitals with respect to occupation numbers. processes, the ionization energy being simply The concept of “transitron state” can also be used for ionization the negative of the Hartree-Fock eigenvalue ezHF, wrth n, = tzto - a.

4 Results and discussion In order to calculate the ergenenergies of the “transition Fock operator’* codes. The HHJ! one-electron energies are gtven by the expression:

eHHF =I(f)+(q, I The quantities

- l)(&i)+,Gq,(i,l?

appeanng

-

III eq. (11) are defined

I(i) = I@,, 1,) , c

(i, i) = F”(nn, i, ; n, li) k=O (i. 1) = F”(n, I, ; n, II> - c

we used the available

HHF computer

(11) as in ref. [6] :

cQ,W,,O) 41i-+ 1

*Fk(nili;nl 1,)v

&r,,O;I/,O)

k [(41, + 2)(4$

Gk(nili;nj$). + 2)] 112

02) 599

Volume 73, number 3

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CHEhIICAL PHYSICS LETTERS

Table 1 Ionization encrges

of some closed-shell

atoms (m hartreel

Koopmans’ theorem a)

Present calculahon c)

Ref 131

Expenmentd

Atom

Hole orbIta

He

IS

0.91796

0.86168

0 85803

0 85751

0 90372

Be

Is 2S

4.73267 0.30927

4 5285 0 2957

452384 0 29464

4.52112 0 2929

4 45 e) 0 343

Ne

Is 2s 7-P

32 7727 1 93048 0 8504

319113 1.80816 0 7280

3190417 1 80432

Ar

IS 2s 3s 2P 3P

117.4189 11.9369 1.2170 9.1998 0 54247 __I__

117.4127 11.93157 12139

a) Ref. [S]

-

118 6106 12 3224 I.2775 9 571 0 5910

b) Cakuhted from SCF energies of ref. [Z].

In terms of these quantltles El

~=SCF

the Hartree-Fock

b,

117 4011 119303 12133 9.25 36

0.56859

C) Occupatton numbers = $

ergenenergres

319110 18040 0.7229

are expressed

d)

Refs [9,lOj

d)

3198 1 78 0.7926 117 81 1199 108

0 5792 e, Esttmated from ref i 11{

IR the form

HF=f(z)+CN,FO(z,l)-~(N,/Nr)ck(Zl,O,I/,O)Gk(i,~),

113)

3

where N,(q,) is the number of states (electrons) m the ith shell. For closed-she11 systems there IS a straghtforward relatlon between (12) and (13) under the assumptfon equivalence between the Slater mtegrals. Smce the degeneracy of the rth shell is (46 + 2):

of

Cl41 We present m table 1 the results of some representative calculations usmg the proposed scheme. We compare the present calculatrons wth the results of Koopmans’ theorem (column 3), the conventional AEsC, method (column 4). the transltlon operator techmque (column S), and the experrmental results (column 7) The calculatlons were performed m double preclslon, usmg a modlficatlon of the spm-polarized HF program of Froese-Fischer [7], adapted for HHF calculations The results of the present calculations are very sundar to those of Goscmski et al. [3], the differences ansmg basrcally from the &fferent wavefunctions (HHF mstead of HF spm-orbitals) used m the evaluation of the Slater Integrals The present approach has advantages over their perturbation method smce all quantifies appearmg in eq. (14) are automatIc~Iy obtamed from the output of the Froese-Fzscher numerical program 173. This srmplrfies the fiial calcufations of the “transition Fock operator” eigenvalues.

5. Conclusions that If we allow fractlonal occupation m a Hartree-Fock method and define a “transltion state” It is possible to obtam a very accurate appromation for the optical transltmns and lonizatron energres. The &fference of HF eigenvalues between the imtial and the fmal states calculated from a seIf_consistent transItion state are nearly equal to the difference between the totaJ self-consistent HF energres of the atom in the m~tial and foal states. (This IS confirmed by the calculations of ionization energzes.) Furthermore all the concepts mtroduced m the X~I method appear naturaUy from tks generalization of the &u-tree-Fock We have shown

as the one used in the Xa! method,

600

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L August L980

method, and for optical transitions and ioruzations of closed shells we do not have to worry about the powersenes convergence [ 11. Goscinski et al. [3] have previously shown that the AExF values obtained using a Rayleigh-Schrbdlnger perturbation theory agrees with that of the transition state calculations in the HartreeFock approxunation through third order. T~K work presents an alternative, but numerically simpler, scheme to the one suggested by Goscinski et al. [3] to justify the use of the “transition-state” concept within the closed-shell Hartree-Fock theory.

Acknowledgement The authors would hke to thank Professor B. Maffeo for heIpfu1 discusslons, and Professor T.M. Wilson for us wth the HHF computer codes.

prowling

References [l] [2] [3] [4] [S] ]6] [7]

J.C Slater, The self-consntent field for molecules and solids, Vol. 4 (McGraw-Hdl, New York, 1974). J.R. Leite and L G. Ferreira, Phys Rev. A3 (1971) 1224. 0. Goscmski. B.T. Plckup and G. Purvis,Chem Phys. Letters 22 (1973) 167. J.C. Slater and J.H Wood, Intern. J. Quantum Chem. 45 (1971) 3. J C. Slater. Advan. Quantum Chem. 6 (1972) 1. J C Slater, J B. Mann, T M. Wdson and J-W. Wood, Phys. Rev. 184 (1969) 672. C. Froese-Fischer, Dcscnption of a Hartree-Fock Program With Confiiuratron Mixmg, Computer Center, University of Bntlsh Columbia (June 1968). Can. J. Phys. 41(1963) 1895 [8] E ClementI, Tables of AtONC Functions, IBM Corporation, San JosC (1965). [9] C. Moore, Tables of Atomic Energy Levels, National Bureau of Standards, Washington, DC (1949). [lo] K. Slegbahn, C Nordlmg, G. Johanson, J. Hedman. P.F. Heden, K. Hanuin, U. Gebus, T. Bergmark, L 0. Werme, R. Manne and Y. Baer, ESCA apphed to free molecules (North-Holland, Amsterdam, 1969). [ll] J C. Slater, Phys Rev 98 (1955) 1039.