The generalized SCF transition operator

-Chemical

Physics 29 (1978) 67-76

0 North-Holland

Publishing

Company

THE GENERALIZED

SCF TRANSITION

OPERATOR

D. FIRSHT *, B-T_ PICKUP and R. McWEENY Department of Chemistry, The University, SheffieId S3 7HF. UK Received 20 September

1977

The transition operator concept for the direct calculation of electronic energy diffcrenws is generalized to many-shell SCF theory. The correct stationary conditions are incorporated using a density matrix formalism. Trial calculations arc presented for the IP’s and KLL Auger energies of Ne using the configurational average RHF of McWeeny and compared to UHF results of Goscinski et al.

1. Introduction

The effects of relativity Of the direct

The shortcomings of the Koopmans’ theorem (KT) estimates of inner-shell ionization potentials (IP’s) are well-known. [l]. Greatly improved estimates of core IP’s are given by the A.&,, procedure [I], i.e. by subtracting the total energies obtained in two separate SCF calculations on the parent and ionized systems. Quite apart from the conceptual disadvantage associated with performing two calculations to obtain just one quantity, there are a number of drawbacks connected with the AEs,-, method: (a) Correlation effects are not included, so that for good agreement with the experimental IP the correlation energies of parent and ionized systems must be similar and, therefore, cancel. (b) The non-orthogonality problem in the calculation of generalized overlap amplitudes - quantities important in the theory of fast ionization processes [3,4] - or, more generally, of transition density matrices [5] when separately optimized wavefunctions are used [6]. (c) The difficulty of ensuring that the two basis sets used in the calculation describe evenly their respective states, so that one SCF solution is not closer to its Hartree-Fock (HF) limit than is the other.

are not considered available,

here.

KT. while

over-

coming obstacles (b) and(c), is quite inappropriate for the calculation of core ionization intensities, as well as the energies, although the “frozen” orbital estimate of an intensity, unlike that of the associated energy, is not necessarily much improved by the incorporatiofi of orbital relaxation [7]. In the extension of KT to correlated wavefunctions [8] (a) is overcome, but relaxation effects are stiil not included [9]. With the Green’s function method [3,10], which in principle yields exact IP’s (and electron affinities), all the above disadvantages are removed, but this approach is outside the scope of the present work. In order to retain the simplicity of KT but obtain core W’sof LIE,,, quahty, a method related to Slater’s transition state concept [l I ] was introduced. This transition operator (TO) method [ 121~ involving one calculation per orbital ionization, automatically overcomes the non-orthogonality difficulty of(b) and gives, as well as ionization energies that mimic AEs,F calculations very closely, good approximations to generalized overlap amplitudes obtained from separately optimized initial and final state HF wavefunctions [ 131. Of course, disadvantage (a) still remains, although for core ionizations (c)

* Resent address: Imperial College of Science and Technology, Dept. of Physics, Prince Consort Rd., London SW7 2% UK.

methods

this is not usually troublesome,

while

is to a large extent obviated. The question of basis

sets is considered in section 4. The development of the TO method has been chiefly in terms of unrestricted-HF (UHF) theory.

68

D. Firsht et a1.fThe generalized SCF transition opemror

For an ionization given by i;

= i +iFfI

from spin-orbital $ti the TO is

Wjll$j) +&-lirLj) ,

(1.0

where the operator h is the one-electron part of the hamiltonian and the “Coulomb-minus-exchange” operator, C~j;-II~j), for thejth spin-orbital is defined

by

(lL,(2jll$j(2))441) = s 1Lj(2)*ri~(l -PI21 rCj(2) 441)dT2_

(1.2)

Here, integration is over space and spin variables, ‘12 = Irt - r21 is the spatial distance between points 1 and 2, and PI2 is the interchange operator. Henceforth we shall reserve i, j for occupied and II, u for unoccupied spin-orbitals, while !>I,II will be used when no distinction is meant. Eq. (1.1) may be written alternatively as Q = $(P

f P)

,

O-3)

where

(I-4)

LB=I; +igi,

($jll$ji>

by use of the TO solutions [ 1511 The TO (1.1) or (1.3) may in fact be derived through a variational treatment [14,16] in which the transition energy functional ,I?= = $r+’ +@)

(1.7)

constructed with a common spin-orbital set for E* and EB, is made stationary subject to the restriction that the spin-orbitals remain orthonormal. This may be regarded as a compromise to the ideal of individual fully stationary E* and EB, resulting in TO solutions which are unsuited to the calculation of the separate state properties, but which are eminently suitable for transition quantities. Clearly from (1.7), it is the actual state energy function& that are interpolated whereas in Slater’s method [I l] the interpoiation is on the orbital occupancies [14,17]. It is also possible to define TOS for other electronic transitions [ 14,161 such as excitation from I$~ to rJL,,for which we have [ 141

~~jil~j~+~E~~,ill9~~~~~~ll~~~l (1.8) = :(iA + t;B),

‘L ‘i’+jgj)

where now I;* =I; + jf&

,

,

<~jll$jLi>-w~llkq 1

7

the UHF operators for initial and final states, respectively, expressed now in terms of a common spinorbital set. The transition spin-orbitals, I$:}, arise as the selfconsistcnt solutions of the (pseudo-) eigenvalue equation ,;TIG;-= eilJlj )

(1-5)

and the self-consistent eigenvalue e: was found to give the exceptionally good [ 12,14,15] approximation _+E;-+

E&

- El&

= A#,,

(1.6)

where E* and EB are the initial and final state UHF energy functionals, respectively, the subscript T denoting their construction with {$,?I. This result was explained using Rayleigh-Schrodinger perturbation theory, originally through second-order where all quantities were expressed in terms of the initial state UHF solutions [ 121, but later extended to third-order

(1.9) iB = L ’ jg.,

($jlltij)+ Cti,II ti,,).

Indeed, when such transitions involve the core orbit& directly (the Auger process, for example) a method of it least AE,,, quality is certainly needed [ 18]_ Recently, the variational and perturbational arguments of the TO method were found to be generally applicable to any atomic or molecular transition process calculated within any level of approximation [19]: for example, the calculation of the energy difference associated with a change in geometry using a correlated wavefunction. The present work, however. deals only with electronrc transitions, and in the next section extends the above UHF TO method to manyshell SCF theory by use of projection operator techniques [5,20], while in section-3 the connection between the two is established. Section 4 presents some simple illustrative atomic applications within the

restricted HF (RHF) average energy formulation of McWeeny [ZO] , and the questions of basis set choice and the control of convergence are’discussed.

energy functional for the state defined by the above electronic E({&)

2. The TO method

69

D. Firxhtet al./The generaiizedSCF transitionoperator

*

in many-shell

SCF theory

conflguration. = 5

VM t&

The assumed form is + :&)jM

I

(2.4)

where “tr” IS used in a sense analogous to its normal

matrix meadng, as an abbreviation for [21] Consider a system in which the orthonormal spinorbitals are grouped into a set of shells. For generality spin-orbitals are assumed for the present, although in RHF cases, where spins have been integrated out, the corresponding equations would contain only (space) orbitals; the structure of the following derivation would not be altered. The electronic configuration is, in turn, specified by a set of fractional occupancies for the shells: for shell M containing pM spin-orbitals and qM (
between states not

necessarily

having exactly coincident unoccupied shells. Associated with shell M is the density projection operator,

and its coordinate representative 1211. (2.2) the M-shell density matrix, while resolution of the identity requires

gjM=i.

(2.3)

the unit operator for the space spanned by the given spin-orbital set. Strictly speaking, for the space to be complete the summations in (2.1) and (2.2) may contain integrations over the continuum, with consequent implications for the calculation of quantities like ionization cross sections. Note, however, the work of Shull and Lijwdin [33] in this context. In the present work we limit the discussion to a discrete (and in practice truncated) basis set. We must make one further assumpiion which concerns the structure of the

trip=

c I’4

&l)p[l;l’)d~t

,

for some operator A. 6& has the general form eif = F [aMN j(p~)

+ ~MN &pnf)l

s

(2.5)

where the Coulomb and exchange operators are defined, respectively, by

Ail many-shell SCF theories rely on this structure for the energy functional [22] and the actual values of the coefficients &MN. bMN) will depend on the electronic state considered and the kind of approximate wavefunction used, with the latter not necessary a $ngle determinant. By virtue of the property [5] trJ(Pnl)PN = tr J(pN)pAw; tr &pM)pN = tr K@N)pnf, the firstorder variation in E is SE= CL!

M ”

tr

I;,,&M ,

(2.6)

where ^ __

hM=h+Gbf,

(2.7)

is a Fock-type hamiltoniah operator for shell hf. The structure of eM is similar to (2.5) and normally the {Use, bMN) are related in a way that gives GM = GhThe necessary and sufficient conditions on the shell density matrices that make .!?({pM}) stationary (6E = 0) whilst preserving condition (2.3) and the orthonormality conditions [5,20,22j &rt&, = S,$,

;

‘W N >

pare embodied in the effective hamiltonian-[22]

(2.8)

70

D. Firsl~ietal.fThegeneralizedSCFtransition

Here, (iM is an arbitrary hermitean operator and the arbitrary, non-z&o scalars {\v&~) are related by ~vMN = -w;M. Note that IvM~~ need not be specified since the corresponding term in the second &mmatibn of (2.9) is zero anyway. Furthermore, it is unnecessary to define a Fock-type operator (2.7) for an unoccupied shell, U, since then vu = 0. These arbitrary factors in

operator

--

I’.

1

-- -: ’ :_

as stated previously, ii unnec&sary. This’is p%cisely the circumstance for the 2s orbital in th&K Lo L 4 i Auger transition N1?[[ls*2s’2p~] -+ Ne’+[ls-2s02p6]. The constrained stationary requir&$nt &ET = 0 gives the transition effective hamiltonian,

principle do not affect the stationary conditions although in actual calculations they have proved critical in influencing convergence [23,24]. The koperators may also be defined in a manner that lends physical significance to the self-consistent eigenfunctions of (2.9) [22,X]. Consider, now, a transition between two such states A and B where the shell-structure is the same for both and which may involve any number of sheIIs, as well as a possible change in the total number of electrons. For instance, the K L, L3 3 Auger transition Nei[ls12s22p6] +Ne2+[ls?2si2i3] involves all three occupied shells rrrzd a change in the number of electrons. The initial and final state configurations are specified by the sets of fractional occupancies, {v&}, Z = A, B. For the same physical reasons outlined in the previous section we wish to optimize the transition energy functional E-VIP&

+ @wP~f3)1%

= ; [@((P&)

(1.7’)

subject to the conditions (2.3) and (2.8). Making the variation pi -+ phf + Qfif for each shell gives

and the optimum densities (~$3 are obtained through (2.2) from the self-consistent soluti&s of the eigenvalue equation for (2-i-2). We now require that 7iT be related to gA and ffB, the effective hamiltonians for the individual cdnfigurations, in the same manner as (1.3) for the UHF case. This may be achieved by the requirement that the arbitrary parameters {B&) (Z = A, B) occurring in gz, where from (2.9)

satisfy IVfilv = ‘“&$r = W& ;

‘dfil, N _

(l-13)

On defining i= = $A

+ I;B> ,

(2.14)

we then obtain the explicit form given in (3-l 2) with SET = i(fiE* + SEB) = $tr

i&o,,,

,

(IlO)

where I;nTI= f(V$lcq

+$&)

)

(2.1 I)

and I;$ (Z = A, B) is the corresponding operator (2.7) for configuration Z, each, of course, constructed from the same set of shell density matrices. We now see the value of only implicitly including the virtual subspace: if configuration A contains just one unoccupied shell, U, while configuration B contains an occupied subset of the spin-orbitals in U, the definition of the transition Fock-type operator (3.11) for this shell would be ambiguous. The division of the shell into U’ and V, where U’ is occupied in B alone whereas V is unoccupied in both, overcomes the

problem by making available two operators $ and !I:, although the actual definition of @j,, fi$ and RF,

i;f = $(@

+ iff)

_

(1.15)

and -A h and -B k in (2.14) constructed from a common set of densities. We see, therefore, that in the manyshell case it is possible to have a formally similar result to that in UHF TO theory where, now, the UHF operators in (1.3) are simply replaced by effective hamiltonians, providing (2.13) is imposed. (AB) for the transition The approximation to AEScF A+ B afforded by (~$1 is good through second order in self-consistent perturbation theory when all quantities are expanded about the transition solutions. Briefly, this follows from the introduction of the interpolating functional [19]

=ET(b49f3)+ ~ET({~,&).

(2.16)

71

D. Finht et ol./TJte generalized SCF transition operator

-. where

i(d)

AE(b,i& = E k (b,& - EB(bnil)-

(2.17)

We shall use {p&Q} to denote the self-consistent densities that make (2.16) stationary, with value E(X), subject to (2.3) and (2.8), so that PM(O) = p&, pIM($J = pi and pM(-k) = ,&, {&} (Z= A, B) being the self-consistent densities for the individual states. These optimum densities are formed from the selfconsistent solutions of the effective hamiltonian, analogous to (2.14) E(X) = ($ + $A

+(f - X)@ ,

= EA({&})

- EB({&})

&E=

c M=I,J,LT~,~tr(~+f~~)~~;

Z=A,B,

where the only Gtr requiring definition are

where, clearly, E(O) = iT. Expanding E(h) as a Taylor series in powers of X about ET( {p&}) for X = +-i and taking the difference we find E($) - E(-f)

onto their respective subspaces. The UHF energy functionals for initial and ionized states A and B, respectively, can then be recast as (noting that no prime is needed for the G-operators)

(~j;-[l$j~+(+~ll~~~

2

and

= A,$@)

h=O

i;JA =jg.,

’

since

(2.18) where the Hellman-Feynman

theorem [ 191,

has been used. The last term on the far right-hand side of (2.18) is, in fact, a third order correction to the transition approximation AE({p&]) and it may easily be verified that ~11even order corrections (powers of Xl) are absent in the extension of (2.18) to intinite order [ 191. The form of the third order correction may be derived by the methods of selfconsistent perturbation theory [26] although we do not take this up here, preferring for the present to test the efficacy of the many-shell TO method numericaily.

3. Connection with UHF TO theory To establish the relationship of the preceding treatment with the UHF method of section 1 we firstly need to express the familiar UHF energy functional in the form of (2.4). In the case of ionization from spin-orbital JJ~it is convenient to define the three shells, 1, J and CJwhich have the density projection operators

and 2w&, = ~7~ = w& = 2, and use of the relations

I;MC~illlL~~~=/;,<9ill~i)P,=O

; iw=r,J7u,

in (2.12), an effective hamiltonian zr? may be obtained such that flT=j&;(i,A+$$

I

I

7

in accord with (1.3). The two approaches are therefore entirely equivalent for an ionization within the UHF approximation. For the excitation ~i--f $,, the situation is somewhat different. It is now necessary to define the four shells 1,J, U and V which have the corresponding projection operators

D. Fink et &%egeneralized SCF transitionoperator

and by a procedure similar to that above the transition effective hamiltonian is found to differ from the TO of Goscinski et al. 1143, given in (1.8), by a correction term h’ where (3&l) and

tion, without~allowing transfer of electrons between shells- The shells are now defined by groups of (space) orbitals and the equations corresponding tb those in section 2 appear with all spin integmtipns completed. Since we solve the equations by the analytic linear expansion technique (the analogue of the LCAO method in MO theory), rather than by numerical integration, we further introduce a set of k basis functions, {Xl(‘), X2(‘)* **** X/J’)]9 common to each shell, and to each of the configurations in the transition, whereby (2.4) becomes [20,22], using standard notation for one- and two-electron integrals [S], E$( {RM.l) = c&

Here [h.c.] denotes the hermitean conjugate of the preceding square-bracketed terms. This discrepancy can be explained when it is realized that the equations of Coscinski et al.

+ Bit)*ItijLi)+ @lc,jlI$[,) i

Nz =il

ivIf

tr(h f ~G$)R~

Z=A,B, (4.1)

with hab = t&@

;

1x$,

G$=v~G(R~)+

c r$,G(R& N(+M)

7

where {or,,,, ) are La,orange multipliers, while satisfied by the correct self-consistent solutions, {Szr,), are not in themselves sufficient to determine them. This argument was more fully developed by Cook [27] section 2 when confronted by an analogous situation in his treatment of paired excitation mnlticontiguntion SCF theory. It is evident from equations (3.1) that diagonalization of ZrL alone. as suggested in ref. [14], will not yield spinorbitals that make the a propriate transition energy functional stationary P.

The matrices h, G, G& and R are all of dimensionality k X Ic, whereas T&f is the rectangular k X pnf matrix whose columns contain the expansion coefficients of the orbitals of shell AI. The quantity pM is the number of orbitals in shell Af, so that now 0 < r& = &/P~ < 2 and ZI,$= 2 (& - l)/(2pfif - l), a term preventing ‘self-interaction” when s& = 1. Application to (4.1) of a variational treatment formally identical to that in section 2 gives, instead of (2.14), the k X k transition effective hamiltonian matrix

4. Application to the Ne atom

T;= = +(
We have carried out trial calculations of the If% and KLL Auger energies of the Ne atom within the RHF many-shell theory of McfYeeny [20]. This particular scheme requires that the optimal orbitals are those which make stationary the averuge energy of all the variously vector coupled states of the configura-

where

* These points have been discussed privately with one of the authors (O.C.) in ref. [ 14 1.

(4.2)

hZ=CRMMMM ciZR

and h$= h + G& (Z = A, B); the cancellation properties exhibited in (2.16)-(2.18) still obtaining. The

D. Firshr et ai.f The generalized SCF transition operator

matrix forms of (4.1) and (4.2) are equivalent to the corresponding operator forms of section 2 in the limit of an infinite basis set. The particular reduction of the above equations for the case of atomic rotational symmetry is easily obtained and is given elsewhere [ 181, the averaging in this case being over L.Scoupled states with the shells specified by their principal and azimuthal quantum numbers. The basis functions used were Slatet-type otbitals (STO’s) but no exponent optimization was performed since we did not wish to employ non-linear procedures in our calculations. It was therefore iplportant to be reasonably certain that the exponents of the chosen basis sets here well suited to the transition under consideration. To this end the equivalent core (EC) concept guided us in our choice [ 181 from the basis sets tabulated by Clementi [2g]. It has been shown previously [ 181 that satisfactory results for all the hole states of hTeconsidered here, can be obtained when a basis set is used which is appropriate for the EC ion hTa+(3s*), where 3s* indicates a hole in the 3s shell (two asterisks a double hole, etc.). For transitions between two hole states there is no ambiguity in the choice of exponents, but for W’s the ground state (GS) is involved, and the GS SCF energy obtained by the use of an EC basis set is defmitely inferior to the optimum SCF energy. To overcome this difficulty we have adopted a compromise to an optimum “transition basis” by using exponents that are the average of corresponding exponents optimized for the GS and EC ion separately. In table 1 the results obtained with this zeroth order approximation to the transition basis are listed alongside those obtained by use of the Table 1 The IP’s (in eV) of Ne by the a&cF This

73

GS and EC bases. It should be noted that here AL&P =E$({R&]) - Efvv({R$]) and AET = E$(IR&)) E&({R&}). All three approximations to AEscF are very good although, with the exception of the 2s IP, the transition basis gives the closest values, the exceptional case being only marginally (0.02 eV) better represented by the GS basis. For comparison the RHF results of Goscinski et al. [14] are also included in table 1. These are not in exact agreement with ours since a numerical HF procedure rather than our analytic expansion method was used to obtain them. In addition the exact stationary conditions associated with the numerical HF are uncertain. Table 2 presents a summary of all the transitions. considered for this work along with an analysis of the errors. The transition approximations can be seen to mimic the “exact” AESCF values excellently, reinforcing the success of the original UHF work. An idea of the errors involved may be obtained by considering the quantities A+ = $ tE,&) + &,(-~)I - E,,(O) and A_ = AESCF - AET. The former equals the sum of all even order corrections to E,,(O) in the expansion of E,,(X) and the latter is twice the sum of all odd order corrections to AE, (cf. eqs. (2.16)-(2.18)) [19]. The following salient features arise from this analysis: (a) The absolute errors (]A_ 1) are very small and roughly the same within each group of transitions, the ls*2p** case being something of an exception, and the precentage errors in all cases are respectably small. (b) In all cases A+ dominates A_, for some transitions by as much as one or two orders of magnitude.

and TO methods

work

Goscinskiet al. b)

AET EC

GS

basis

basis

transition basis

‘%cF~’

‘=T

SSCF

IS

868.30

868.75

868.65

868.59

868.67

868.62

2S

48.89

49.37

49.27

49.33

49.34

49.33

7P

19.56

20.02

19.91

19.88

19.98

19.85

a) GS calculation with GS basis: hole state calculations with EC basis. b) From ref. [la]. C)From ref. [ 301.

Esperiment ‘)

870.2 48.42 21.59

74 Table 2 Data for

D. Fksht et &The

various

generaked

transitions in the Ne atom (energies in hartrees)

Transition

GS/ls* GS/2s* cst2p* 1s*/zs** 1s*/zs*2p*

lS*/2p**

might be expected.

*+

31.921 1.813 0.731

31.923 1.811 0.732

-0.212 -0.029 -CO32

-0.002 0.002 -0.001

0.006 0.133 0.148

27.451 28.661

27.460 28.649

-0.033 -0.030

-0.009 0.012

0.035 0.044

29.682 --

29.685

-0.029

-0.003

0.009

An important

factor

point on the HF energy surface, but with

the recognition that for excited states it will generally be a saddle point and not a local minimum the task of obtaining self-consistency becomes less formidable. We may account for this feature during iteration through the signs allocated to the various \w&.,,parameters (chosen real for the purposes of calculation): this matter is critical since use of just one wrong sign will cause divergence. The whole question of deciding on the signs, which is of wider application than to solely TO calculations, is discussed in detail elsewhere [24] _For the present we simply summarize the additional measures taken that ensured convergence in all but one of the cases, the exception proving somewhat L‘pathological“. The d-matrices were defined by d;=h;;

d$=~h;/N,,,; I

Percentage error

*ET

influencing convergence is the nature of the requisite stationary

A_

*&XX

This trend is significant since the terms cancelling in the approximation of AE,,, by AE, are those that constitute A+. In many-shell calculations such as these convergence difficulties

SCF transition operator

Z=A,B,

(4.3)

where U is unoccupied in A and B, Nocc is the number of occupied shells including those occupied in only one of the configurations, and I ranges over all N,,., occupied shells. Note that this choice gives an average KT [ZO] for occupied shells in the individual configurations while ensuring that the eigenvalues of the virtual subspace lie above those of the occupied subspaces. Whereas level shifters [22,23] of about 10 hartree were used to sarantee this relationship between occupied and virtual subspace eigenvalues, this measure was unnecessary for occupied shells since

in AET

the use of (4.3) alone gives 1s and 2s ievels well separated at about 30 hartree apart. In the case of the 2s** configuration h,, was conveniently still defined by h2s= h f G2s with 42s = 0 in z$,_ The number of cycles required for convergence was normally between 20 and 30 and after each diagonalization of (4.2) the shell allocation of the eigenvectors was carried out according to a principle of least change effected by a “maximum overlap” criterion [23,24]. For this work no attempt was made to speed convergence further although it was felt that this was within the capability of the computer program used. The pathological case was the ls*/2s*2p* calculation for which E,,,(O) =Z -110.9 hartree. Unfortunately it has a near degeneracy with 2s*/ls*2p*, a solution on the same HF surface where now E,,(O) 1: -110.6 hartree, and we believe that this is the source of trouble. Starting the calculation for ls*/2~*2p* with different sensible initial guesses of {TI} would give convergence to different solutions, implying that the energy surface for this case is complicated by the presence of several local stationary points close together. We found that by performing one cycle for each configuration individually and using the average of the resulting T-matrices as the initial guess for the TO run proper (a kind of zeroth order approximation to the transition Tmatrices), convergence to the desired solution could be obtained in about twice as many cycles as normal when level shifters of gross order lo2 hartree were used! Recently, Hehenberger [29] also encountered some severe convergence difficulties in a molecular UHF TO calculation. Table 3 displays data for the KLL Auger energies of Ne resolved in the LS limit. Using standard methods the symmetry states where constructed from the optimum orbit& of ls*/2s** in the case of KL, L,,

75

D. Firsht et a1.f The generalized SCF tramition operator Table 3 The KLL Auger energies (eV) of Ne

Experiment b)

RHF

UHFa)

(iO.6) TO

~ESCF

TO

“ESCF

KLILI&)

147.2

746.9

747.2

747.0

748.7

KLILz,&)

770.1

770.8

770.3

770.8

772.1

782.9

783.0

782.6

781.9

782.6

800.9

800.9

795.7

79.5.6 .

801.1

(‘D)

806.0

806.0

804.2

8O.t.l

804.8

(3P)

809.5

809.4

811.1

811.0

808 .O

(3P) KL&z,&)

a) Fromref.

[16].

b) Fromref.

[30].

ls*/2s*2p* in the case of KL, Lr,3, and Is*/Zp** in the case of KLZ,3L2,3. The agreement with the corresponding AE,,, values is excellent although, strictly speaking, the cancellation properties outlined at the end of section 2 apply only to the average ener,v and not the energies of the resolved symmetry states. For comparison, the results of UHF TO calculations [ 161 are also given and may be seen to mimic their associated AEscF values marginally more closely in one or two cases. There is, however no significant difference in the accuracies of the UHF and RHF TO methods.

5. Conclusion

The concept of a transition operator has been extended to many-shell SCF theory, effective hamiltonians replacing the UHF hamiltonians of the original scheme, and the resulting transition effective hamiltonian embodies the correct stationary conditions for the many-shell transition energy functional. Numerical testing of the many-shell method was extremely encouraging and some obvious further applications would be in the calculation of “shake-up” and “shake-off” energies [30] and, generally, the treatment of molecular systems. It might be said here that A,!&, quality calculations for inner shell processes might not always agree as well with experiment as do those for Ne [ 181. There is also a greater likelihood of encountering convergence difficulties in molecular examples particularly in cases where

there are a number of highest occupied and lowest unoccupied molecular orbitals lying close together in a KT sense. The use of level shifters and the correct signs for the {IV&~}, however, should largely overcome such problems. Another extension of the present work would be to the calculation of transition density matrices by the TO method. Preliminary investigations of this kind have already been made [3 l] for Slater’s transition state X, model [l l] _ As a final remark we note that the analysis of section 2 als? applies to certain types of multiconfiguration (MC) SCF schemes [22,27]. In view of the important role that correlation can play in determining core Ionization intensities [33] and, to a smaller degr !e, energies, the prospect of including correlation effects in transition properties by an MC SCF TO method becomes attractive, although it may well involve severe numerical problems. The recent work of Godrefroid et al. [34] is of interest in this respect.

References [II D.A. Shirley, in: Advances in chemical physics, Vol. 23, eds. I. Prigogine and S.A. Rice (Wiley, New York, 1973). PI P.S. Bagus, Phys. Rev. 139 (1965) A619. [31 B-T. Pickup and 0. Goscinski. Mol. Phys. 26 (1973) 1013. I41 B.T. Pickup, Chem. Phys. 19 (1977) 193. 151 R. McWeeny and B.T. Sutcliffe, Methods of molecular quantum mechanics (Academic Press, New York, 1969). t61 R.L. Martin and D.A. Shirby, J. Chem. Phys. 64 (1976) 3685. r71 I.H. Hillier and J. Kendrick, .I. Chem. Sot. Faraday II 71 (1975) 1654.

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D. Firsht et aLlThe genemlized

iSI D.W. Smith and 0-W. Day, J. Chem. Phys. 62 (1975) 113. [9] B.T. Pickup, Chem. Phys. Letters 33 (1975) 422. [lo] J. Linderbergand Y. Giun, Propagators in quantum chemistry (Academic Press, New York, 1973). 1111 J.C. Slater, in: Advances in quantum chemistry, Vol. 6, ed. P.~.LBwdht (Academic Press, New York, 1972). [12] 0. Goscinski, B-T. Pickup and G. Purvis, Chem. Phys. Letters 22 (1973) 167. [I?] 0. Goscinski and B.T. Pickup, Chem. Phys. Letters 33 (1975) 265. [14] 0. Goscinski, G. Howat and T. Aberg, J. Phys. B 8 (1975) 11. [15] 0. Goscinski, M. Hehenberger, B. Roos and P. Siegbahn, Chem. Phys. Letters 33 (1975) 427. [161 0. Goscinski, Intern. J. Quantum Chem. Symp. 9 (1975) 221. (171 D. Firsht and B.T. Pickup, to be published. [ISI D. Firsht and R. hlcWeeny, Mol. Phys. 32 (1976) 1637. [Is] B.T. Pickup and D. Firsht, Chem. Phys. 24 (1977) 407. [20) R. hlcWceny, Mol. Phys. 28 (1974) 1273. [21] R. McWeeny, Rev. Mod. Phys. 32 (1960) 33.5. [22] R. McWceny, Chem. Phys. Letters 35 (1975) 13. [23] V-R. Saunders and I.H. Hither, Intern. J. Quantum Chem. 7 (1973) 699; M.F. Guest and V.R. Saunders, Mol. Phys. 28 (1974) 819.

SCF transition opemtor [24] D. Firsht and B.T. Pickup, Intern. J. Quantum Chem., ..~ to be published. [25] K. Hiio, J. Chem. Phys. 6L (1974) ;247. [26] R. McWeeny, Phys. Rev. 126 (1962) 1028; R. McWeeny and G. Diercksen. J. Chem. Phys. 49 (1968) 4852. [27] D.B. Cook, Mol. Phys. 30 (1975) 733. 1281 E. Clementi, Tables of atomic functions (IBM Corporation, San Jose, 1965). [29] M. Hehenberger, Chem. Phys. Letters 46 (1977) 117. [30] K. Siegbahn, C. Nordling, G. lohansson, J. Hedman, P.F. Heden, K. Hamrin, U. Gelius, T. Bergmark, L.O. Werme, R. Manne and Y. Baer, ESCA applied to free molecules (North-HoUand, Amsterdam, 1969). 1311 P.G. Ellis and 0. Goscinski, Phys. Scripta 9 (1974) 104. r321 _ . T. Darko, I.H. Hillier and J. Kendrick, Chem. Phys. Letters45 (1977) 188. [33] H. Shull and P.G. Ltiwdin, Svensk Kern. Tidskr. 67 (1955) 373; S. Hagstrom and H. Shull, J. Chem. Phys. 30 (1959) 1314. [34] M. Godrefroid, J.J. Berger and G. Verbaegen, I. Phys. B9 (1976) 2181.