Comments on the geometric approximation to the second-order perturbed energies

Voiume 58, number 4

cEiI%hticA.L PHYSICS LEITERS

15Oct&er1978

COMMENTS ON TDE GEOMETRIC APPROXIMATION TO TEIE SECOND-ORDER PERTURBED ENERGIES+ Andaej J. SADLEJ Inrritute of Organic Chemistry. ì’olkh Academy of Sciences. PL-OI-224

IVamw 42. Poland

Received 24 April 1978 Revïsed manuscript receîved 7 July1978 It is shown that the idezs SimïIarto these behïnd the ordinary gcometric approximation can be efficiently used to speed up the convergente of the coupled Hartree-Fock iterative procedure in the density matrix formalïsm. The formulation of the geometrïc approximation in the case of multiple perturbations is also considered and an alternative second-order energy formula is proposed. The validity of t&e upper bound formula for the error of the geometrïc approximation derived by Burrows, is discussed. 1 _ htrodution The so-calied geometrie approximation (GA) cl] has proved very successful in perturbation calculations of a variety of atomïc and molecular secondorder propertïes ]2-7]_ Though origindly deduced as a purely empirical scheme [l] , the GA was given then a fh-m theoretical justifìcation [S-101 and its computational advantages are frequently underlined [7,1 l] _ The GA has also been extended to deal with the second-order energïes arising from two simultaneous perturbations [4,5]. One of the purposes of the present note is to show that similar ideas can be efficiently employed to speed up the convergente of the coupled HartreeFock (CHF) [12,13] iteratïve calculations. Also some ambiguities of the definition of the GA in the case of multiple perturbations are discussed and an altemative solution is proposed. The GA formula derived in this paper does not suffer from singularities which are likely to occur within the ordinary approach [5] _ Finally, we would lilce to comment on Burrows’ derivation of bounds for the error of the GA [14]. 2. The GA and the convergente CRF ealeulations

of iterative

We shall closely fellow the deftitions and notation of ref_ [ 131. For a given 2nelectron closed-shell * This research was %upportedby_the Institute of Lnw Temperatures and Structure Researc h of rhe Poli& Academy of Sciences under contract No. MR-L9A.3.

system consïdered at the SCF HF ievel, the perturbation is represented by oneelectron operatorsfil). Then, according to eq. (30) of ref. [ 131, rhe secondorder energy functional can be written as $2)

= 2Tr[R:#,(0)R$;) +fq+')

_ R$;)h(O)R$;)

T $G(R(~))R(~)],

(1)

where

R(1) = Ri;) + R$;) , R:;)

= R$;)t

= R(o) R(1) (1 _ R(o)),

(2)

and ether symbols have their usual meaning [ 13]_ The variation of the functional(1) constrained by the density matrix idempotency condition leads to the Euler equation for the (12) projection of R(l) h(O) R:;)

-

Ry?h(‘i

-

[fit’

+ GIZ(R(l))]

= (3 _ (3)

Since tbe fïrst-order density matrix enters this equation through G,,(R(l)), the natural way of its solution is to use some iterative process. Suppose that the solution R(l)was already obtained in the nth step of a gïven iterative procedure. Following tbe ideas of the GA [lO] the fmal solution R(l) which mïniis the second-order energy functional(1) can be approximated as R<‘) = p; R
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Volume 55, mmber 4

Since Rfl)an follows from the solution of eq. (3) which bas the GIZ-term constructed using the density mat& of tbe previous iteratïon, RC1)snmX, the functiod (1) will have the ferm J’2’(irl)

=pf’(c

_p;)A(2)+pf2B(2)

,

(5)

where ~(2),TrffI.>R(~~.~, #?=TrfG(Rff>,n) =TrG(R(ï),n)

(6) _ G(Rfl)>=--f)] (R(‘),=

Rfï),=

_ R<%=-1)

_

_

(3)

(9)

is zdso preserved for any pair of two subsequent iterations. This feature can be used to speed up the convergente of the CHF iterative calculations. In this context one sbould mention that the calculation of # does nat require any additional computationaI effort in comparison with that already involved. Thus, a reasonable appro-ximation to the subsequent density matrix (4) can be easily obtained. SiQce each CHF iteration requires the processing of the whole file of two-eiectron integrals this maayresult in a substantial gain Uz the case of large basis sets and highIy accurate caIculations. The procedure described in this section bas recentIy btin mmlemented in our CHF progrm and some pilot calculations indicate its extremely high efficiency. For iQstance, in some cases it was possible to reduce the Qumber of required CHF iterations by more than four times. Let us ais0 point out that exacrly the same approach ca~ be utilized within the CI-IF scheme due to Stevens et al. [12]. However, the densìty matrix approach [í3,17] appears to be computa~oQ~y much faster aQd more convenient ]7,18]. It is 7.0 562

worth atteQtioQ that traditioQalIy the GA was employed m&ly as a one-step procedure [ 1&--IO J Readingfrom tbe uncoupled HF resuhs to some approxirnutíonto &t.eaccurate CHF second-order perturbed energy. In the present case a sir~ihu idea is used to speed up the convergente of the Chic iterations and the fmal value of Ec2> corresponds to the accuurte CHF rest&_ Some useful observations [2S] relating the ordinary GA to Aitken’s g2 traQsformation are aIso vahd for any pair of two subsequent iterations, 3. The GA in the case of muftiplepertdations

If one chooses the initial first-order density matrix approximation as R(1) po = 0, then for n = 1 the ordinary GA [1,8-101 is recovered and Bf2) bas the meaning of the so-cahed first-order correlation correction to the uncoupled HF result [15,16] _ However, it fohows from the present derivation that the geomettic structure of the second-order energy E(z) ECzt = @TA CL1

15 October 1978

17)

The miQimum of the functiond (5) is obtained for

&f-tfl = l/(l -@+X(2))

LETTERS

IR this section we shdl deal with the ordimuy GA [ 1JO] . Thus, the initial (n = 0) frrst-order density matrices are set equal to 0 and for the SEX iterative sohrtio~s (n = 1) the superscript n is replaced by the subscript u meaning that they folIow from the uncoupIed procedure. Let us consider two s~muhaneous perturbations re resented by oQe-eIe&ron operators jfl*O) and Pg ’ l)- For each of these perturbations one can easily form the correspoQding secondarder energy fnQctiona&Jf2**) aQdJ(Oy2) [lg]_ Then, if the CIE perturbed density matrices are approximated according to eq. (4), ene obtains J(W)

=p10(2

-tcl&4WI

+&3(2*0)

(10)

and $0*2> = pol (2 - pol) _.$UW) + &

RfW)

)

(11)

where, for instance /$f*,z> = fr fco.1) Ri%l’

,

REOJ) = Tr G (R$% 1))R$I)

(12)

and pio, %1 are the parameters to be determùted, If the same assumptions are used for the “tied” second-order energy fúnctional J(l J) the appropriate substitutions rest& in J
f&C1O1 -Plo~O1)A(lJ)

+ 2P10P01 B(lJ) where

,

(13)

is kear in both go1 and P:O and thus, does not provide any variation principle for tbe “mixed” second-order energy E(r*r). This inconvenience can be most easily remedied if ene assumes tbat pro = po1 [4,5,16] _ Stil& however, the corresponding fmrctionalwill suffer from the lack of bounding properties with respect to E
plo

&“.94

(23))

=

l/(l

_

=

l/(l

_B(0.2)/A(0s2)) _

(16)

and flol

15 October1978

CHEMICAL PHYSICS LETTERS

Volume 58, rwmber 4

(17)

As far as botb A(2*o) and A(o*2) are different from zero. the second-order energy formula E(l ll) = J(1.l) will not lead to any numerical problem. Using eq. (13) as tbe second-order energy formula with plo and po1 as determined from eqs. (16) and (17) appears also to be physically reasonable. It is obvious that the first-order correlation correction, and thus, the efficiency of the GA, must depend on the perturbation operator_ It is also worth attention tbat in the case of nuclear magnetic shielding constantseq. (13) will representthe so-called pararnagneticcontribution and will be linear in the gauge crigin coordinates only for fmed values of the GA parameters.Since the GA was found [2] to improve the degree of the gauge invariante of molecular magnetic properties calculated within the finite basis set approximation, eq. (13) can be also utilized for the same purpose. Its non-linear dependence on the gauge origin coordinates may prove another possibility for the Öptimization of tbe choice of the origin for the vector potential of the extemal magnetic field [21]. Finally, we would like to point out that using two distinct GA parametersin eq. (13) does not lead to any additional computational effort.

4. Rurrows’ bounds for the error of the GA Some years ago Burrows has derived the bounding formula for the error of the GA in comparison with the CHF result [ 14]_ His forrnula bas already been employed by several authors and appears to work properly 171. However, the lower bound formula for the CHF second-order energy (eq. (52) of ref. [14] ) derived by Burrows according to Robinson’s method 1221 follows from the assumption that the operator (Wo - E, + t) (eq. (31) of ref. [14]) or the operator i (eq. (25) in ref. [14] ) is positive deftite in the specified domain. The proof given by Burrows in appendix A to bis paper [ 141 contains, however, a misprint, i.e., the Ibs. of his eq. (6 1) is in fact negative [23] . TRus, the proof shows just the

opposite. It is interestingto investigatewhy the upper bound formula of Burrows did not fail thus far and when it should fail. What is required for tbis formula Po be valid is that the operator (Ho - E, + L) but not necessarilyg must be positive definite in a gïven domah. In fact one can take for El any vzlue larger than the ground state energy and lower than the

energy of the first excited state of Wo. It seems that until the stabïlity conditions of the SCF approximation 1241 are satisfied one can easily select tbe appropriate value ofEl. Tl& seems also to be the reason why Burrows’ formula, in spite of so;me inconsistencyin its derivation, was found useful and rather accurate [14,7].

References Cl1 J.M. Schulmanand J.I. Musher,J. Chem. Phya 49 (1968) 4845.

121A_T. Amos and H.G. Ff. Roberts, J. Chem.Phys. 50 (1969) 2375. 131 D. F.-T. Tuan and A. Davidz, J. Chem. Phys. 55

(1971) 1286. MOL Phys. 22 (1971) 141 A_J. Sadlej and M. Ja.szuiïskï, 761.

[SI DJ?.-T.Tuan and A. Davidz, J. Chem. Phys. 55 (1971) 1294;

161 G. Howaf M. Trsïc and 0. GosGnski, Intern. J. Quantum Chem. ll(1977)

283.

171 P. bzzzretti, R Zanasi and B. Cadio& J. Chem. Phys_67 (1977) 382.

181 0. Goscinskiand E. Brzindas,PRys.Rev. 182 (1969) 42.

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CHEhSCAL

[9] A_T. Amoq J. Chem_Phys_52 (1970) 603_ ClOt DF_-T_ Tuan. Chem_Phya Letters 7 (1970) 115. 1111P_Lazzaretti,B.CadioliandU.Pincelli,Intem_ J. Quaatum Chen 10 (1976) 771_ f121 RM_ Stevens,RM_ Pitier and WiN_ Lipscomb, J. Chem_Pbys 38 (1963) 550. [ 13: G.H_F_ Dieniksen and R bicWeeny, J. chem. Fhys. 44 (1966) 3554.

[ 141 BL. Bwraws, Iixtesa J. Quantumchem, 7 (1973) 345. fl5] D_F_-T_Tuan, S.T. Epstein and J.O. Hirscbfelïier, J. Chem. Phyr 44 (1966) 431. [IS] P_ hzzzetti, MOL phys 28 (1974) 1389. [ 171 R McWeeny. Pkys. Rcv_ 126 (1962) 1028. [ 18fJ-L. DaIda, R &lcWceny and AA !%adbj, hkk phyn 34 (1977) 1779_

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JL. Dodds, ti. McWeeny, W_T_Raynes and JS. Ritey, bfot Phys. 33 (1977) 611. 1201M. fcarplusand HJ. Kolker, J_Chaa phys 41(1964> 1259. 1211 S_S.Chzmand TP. Das, J. chem. phys. 37 (1962) 1527; R. Moccïa, Chem. Phys. Letters5 (197ö) 260,265; AJ. sadkj, chem_ Phys E&ters 36 (1975) 129; Acts F&ys_PoIon. A49 _ KT_ Jamiesonaad k Chafarian MOL Phys. 32 (X976) fW 1717, and referencesthexein

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