Theoretical basis for semi-empirical pseudopotentials

CHEMICAL PHYSICS LETTERS

Volume 29, number 1

I November

1974

:

Karl

F. FREED,*

i%el@?nes Fran& Institute and 771~Departmrnt of Chemistry, The Chiversity of Gricago.. Ciricago, Illinois 6063 7, USA Received 1 July 1974 It is shown that additional,“correIation” effects are incorporated into semi-empirical model pseudopoicntiais bccause the traditional equations for the theoretical pseudopotentids arc generally based upon a frozen core appro.xirnation. We derive the exact effective valence electron eqktions that are being hick& by the model potentials. It is shown how the energy dependence of model pseudopotent~a~s is partly a reflection‘of core polarization effects ffiat are absent in the usual theoretical psaudopotentials, and we comment on possible nonadditivity effects that correct for the assumed decomposition of many-electron pseudopotentials into those for oneelectron uses.

i. Introduction Pseudopotontials

were

first introduced

into

solid

state physics in order to provide equations for only the valence electrons which do not suffer from the problem of the “variational collapse” of the valence electrons into‘ the core [I]. Currently the method of pseudopotentials is being extensively applied to molecules [2-71, From the outset, two qu,~tative~y different types ‘, of pseudopotential theories should be distinguished. In the first approach the orthogonality constraint on the vaience orbit& is lifted to provide equations for the pseudowavefunctions to make the latter as smooth as possible. The orthogonality of the actual valence orbitals to a given set of core orbitals is translated into a repulsive pseudopotential [ 1, 2,6]. This applicatio,n of the pseudopotential method is a pureiy theoretical rewriting.of the orig&d equations for the valence wavefunction, just another repre~ntation of the fued core approbation. (The iatter is not necessarily restricted to a single determinanta! core wavefunction.)’ In the method of model pseudopotentials; on the otherhand, a, semi-empirical model potential is introduced to mimic the theoretic& pseudopotential of the frozen. core approximation [l-S, 7, 81, and the latter is fit to experimental data. With a relatively small number of pnrametersand usingvery simple mathematical forms, * Alfred’P. Sloan FoundationFellow. .. ‘,.

the model pseudopotentials are able to accurateIy reproduce observed Rydberg leve!s [2-S, 6,7]. The original theoretical pseudopotential equations are generally not equivalent to the full atomic or molecular Schriidinger equation - they are often expressed as being the frozen core approximation to the Hartree-Fock equations for the valence orbit&. Thus the theoretical pseudopotenti~ equations Ieave out some corre!ation effects.Mhen these equations are thzn modelled and fit to experinrental data, they must Perforce introduce these omitted correlation effects unIess they are truly negligible. The formal pseudo potentials can often be introduced in 2 seemingly general fashion in terms of abstract projection operators onto the core electron functions [2]. However, any concrete realization of these core projectors reverts these equations back to the valence equations in the frozen core approximation. This Lrnplies that some important correlation effects, such as core polarization, are absent from the formal p~udopotential equations. It is therefore clear that *he model psetniopotential ‘equations must follow from something more general than the formal or theoretical pseudopotential equations that have previously been given in a manner Cmilar to the ca&of semi-&mpirid the&es, such as the Paris&-ParrGople method which is based on a firmer theoretical foundation f9, IO] ,than is apparent from its customary derivation from minimum basis set self-consistent field equations. Thus, just as in the case ;‘.’

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of the& serrii-empirical valerice theories, _it’isof irr- : ~~‘po-ri&ce to fti* the +opei theoretical framework~for : the model p;eudopotentials in order to test the app& :, ~a~~ns:~vo~~d by this met&d akd to extend its ,’ : range of validity. For instance, given the pseudowave; function it’shduld be possible to calculate properties of the state,.other than the that depend only _’ ‘on the’valence electrons. But in this case it is necessary to orthogonalize ‘the pseudowavefunction to the : . core; A givenchoice of co& ~vavef~ction again reduces this cal&lation tq one that appears to be a fro zen core ~p~ro~ation, in spite of the obvious fact that the rkdelliitg and semi-empiricism has introduced. ‘-correlatior~e’ffects. More ~port~t,qu~stions arise, however;in the development of manyeIectron model, pseudopotentials f2,4, S,.ll]. Tire ~~~~“ofmodel ‘one~i~ctron.pseud~potenti~s is easily generated from :’ approximate equations g&erning the motion of 6ne eIe&on outside the core without the explicit knowtedge’of how correlation effects are incorporated. The .:

efE?K~j

situation

in the

two-electron case is quite a bit more

cempficated 12). Here the two-ekktron model pseudo-’ potential is t&en to be the sum 6f one-electron model pseudopotentiaIs.(with projectors’for the ot,her electron to +ep it. out of the core) along with the c,ore-removed bare e~ec~ro~~Iec~r~n repulsion. The forrnal’diffkulties involved in deriving the theoretical pseudopoten~aI equations for several valence electrons are ve’iy great. [Z, 4,‘5, 111, so’the rrk.ny-electron model pseudopotenti& are derived from ad~itte~y .inadequa:e equationa, The inadequacies of the latter foIIow f&m the fact that the core projection operators do not corn-. mute with the valence electron-other eledtron rep& sions. Sitiple app~o~mation~ have been introduced that often appear to be katisfktory, but improvements can,be expected to follow from the form of the exact effective equation for the valence electrons. It may be argued ‘&at the model p~udopotenti~~ need not introduce correlation effects, such as core polarization, becaui these are negligible for the Ryd; : krg states of interest. However; since the parameters of the model p~eudopot~nti~s are gene~~~y fit to ex-. I. perimental data invhin~ the ground and loweit, valence, electronic state, where conelati& effects are decidedly not negligibk, .t.heassertion that the rnodef:pseudopo: t+ial equations are equivalent to,*e biiginal thee.: re tical on& is;unteaabie. ye show, however, that the semizmpl~~isn‘dfthe.model potejntial fnethod in- : ” :.

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chides the eifects‘of core polar&&ion even thbugh they Gere adsent fro-m the on’&nal theoretical [email protected]

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In section ‘2, the notation is introduced’ by briefly ‘r&_ll.ing some basic facts from ~~eudo~oten~~ theory: In stidtion 3 thti exact effective valence eIectron .Schr&Xnger equatiohis obtained and shown to differ from rhat cur$or&&ly taken as the theore tical precursor of the model ps~~dopotenti~ theories. The case of. rn~~e~e~tron pseudopotentials is explicitly d&ussed.,

It k: convenient to begin by restating the basic equations of the P~ips-~e~~~ pseudopotenti~ equations as they serve both to introduce the notation and the necessary concepts El, 21. They begin with the eigenvnlue equation for the single vdence electron 3cl;ll”s CT-+v>& =E,9,

(2.1)

t

where $Cis the appropriate one~~ec~ron ~a~l~ani~~ and the valence electron wavefunc~ion is constgained to be orthogonal to the.core. This ar~ogo~~~ty eon&zint is written by them as (tfi”lQc: = 0,.

c = 1, .**,N,

where the set of core orbit& the ~ne~lectron equations

@;,J is obta.ine$’ from

_

X& =i&9,‘_ The

(2.2)

nonorthogonal

Gw

pseudowavefun&ion

A,,

‘~

(2.4) c=l

containkg arbitrary coefficients Q, s is then introduced as the solution ta the new equation

(2.51,

o-f+~~~~h,=&EL,,

wlrere’the valence eigenvakes in (2.1): a.& (2.5) are identic$. Here flRK is a reptilsive non-local pseudopotential

whkh repIac&the~ o~ho~o~~~, conkini (2.2). In . tkis cas& the use 0F a oneeI&rop~h+ltonian in “. (2.1) anii (is) &Ok,s (2.5) to be based on a further .“, ._ ::,: ,.’ ., .: . . .~ ,.

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CHE!vW.ZAL PHYSICSLE’ITERS

Volume 29, number 1

: frozen core approbation to the Hartree-Fqck approximation. Thus, the replacement of 1/E by a model form Vhf, whose parameters are fit to experiment, involves the introduction of correlation effects. The formal pseudopotential equations summarized by Weeks et al_ [23 appear to alleviate this ~I%rtrek-Fock fou~d~tio~ by use of the Fuli barn~t~~~ H and an abstract projector,ponto the core. Unfortunately, because H involves both core and valence electrons; while 3 involves only the core electrons, it has not been possible to implement these formalequations until an explicit form is ~tro~uced’for 9 which involves the assumption that the core ~vavefunction is either a single determmant or some other simple form. It then becomes apparent that core polarization effects have been omitted. In the two-electron case there is the added possibility that one valence electron interacts with the polarization of the core that is induced by the other valence electron, so the situation becomes somewhat more complex, and the basis for an exact pseudopotential theory is correspondingly less clear.

3. Exact eq&tions for the valence electrons in this section we derive the exact equations for the vaience electrons that include all correlation effects. The simplest case of a single valence electron is fist discussed as the general case then follows quite directly. In order to derive these exact equations, it is convenient to rewrite the conventional theoretical pseudopotential equations. Aa the theories discussed in section 2 ultimately n&t introduce a core wavefunction or its projector, we let %c represent theN-electron core ~vavefunction. QiEmay be quite general, but in practical appli&tions it is often represented as a sum of one or a few Slater determinants. introducing the valence electron wavefunction I),,, the theoretical pseudopotential equations are imp~ci~y based upon the use of the wavefun~tion. 0.1) where g&, is the antisymmetrizer between the core and valence electrons and ‘k, is taken to-be normalized to unity. The effective valence electron equations follow from the crayon of . ._ I ._

._

.

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:,

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”

EA = (_I, IHI

@A)/J*A /‘I!,> ,

(3.21

by varying $, withfixed Qc. However, it is well known that (3.1) is inadequate to represent,the exact wavefunction; it omits conflgutations where core electrons are excited into ‘rhe valence shell, etc. Thus, the tp-le wavefunction must be written as (3.3) where x represents all the cont
,

(3.4)

with respect to 9, and x with cDcJT.wJ”. The aigebra is somewhat involved, but it has ai1been presented before in other contexts [9, IO]. Although the exact equations for +e could be presented for arbitrary choice of @e, the algebra is straightforward but tedious in Ihe general case [ 123, so only the case of a single detenninar&d Qc is esplicit!y presented. As it is usually convenient to introduce basis set expansions, they are invoked now. Let {c) denote the set of N spinorbit& of ah, and {IJ) be the valence spin-orbitals in terms of which $,, is expanded. The {c> and {u] can be taken to be orthanormal without loss of generality. For instance, if 8 represents a ‘S atomic state with dje a ‘S core state, then {u) involves only a set of excited s spin-orbitais. ,Tfius, in order to form a complete set, {c: u} must be augmented with a set of “excited” spin-orbit& {e> which would include all excited p, d, f, .._ orbitals in the 2S example. The exact x of (3.3) is ‘&en a linear superposition of deter&Giants that contains any number of c + u’ and/or c + e excitations from the set of primary configurations {Co}, where C.denotes the N-spinorbitals of,@,, along with excitations of the form cu -+ ee’, etc. More explicitly, iEc_r denotes the removal of an electron from spin orbital c, the Nelectron ~on~gurations inx are c’WVJ’, c-Lc’~~CW~U’~, . . .. 0ulv2 . . . UN, c-‘&t?, c-lc’-i~e’~ c-LcF-i&dC?, ‘. * Actually in the w of m-u.ltideterminznta.I 4’c, the ~cffi&nts of the individual Slater determinants must also be

: varied [12]_ :.

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I November 1974

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u’ ivhereB is ~e’~~c~ ‘her& of ~e‘s~t~ and 3? is the exact effe&ve .on~-e~~ctron ha~ton~~ ‘w&h acts only in the space spanned by the {VI b~sis,.From ear_ lier work [S; lo] it follows that

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Foc+ke P~~~~~e~~ one-electron &uniltbnian (2.1). with& the ~~~‘b~s~(~~si~ augmented by’&e de~r~ti?~ ($3) for the. cc) set, by addle the term Z;,@A!?JcI $0 tk,e .ri+t in f3. I i);) The, rern~~ part involving X; i$.tFr&ed ,the .%orreh&n” part farwant 6f a %fier te;?rri,Fen (3.11) is substituted into (3.7)’ and ‘@5) is used, there results the exact v&nce-eIed., tron S~~r~d~~er.~quation : ~~~~~~~~~~

.”

.v :, ~‘__ .. ., where x has been decoapdsed into its parentage. form x P X&,~,. vacation of(3.4) with respect to the
‘.‘._‘:

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. wlie?e x, s&isfies the Schrtlding~er-like’equations. ’

:.

(3.12)

._I.

I

which shc@d form t% basis for any subsequent ap-

~r~x~a~i~n in wfiich ~~-ern~i~~~ models are introdu&d. .~e,constr~n~ that ‘&e {v) be ~~o~~n~ to the -:c)set is e&y removed, leaving the exact psetidoequation ;

‘, (W + ypRK}A& =E,&‘,

,,

'.

whiri Vig isstill giv& by (2.6) and AU by (2.4) with .’ c E. Qp,,Thus, given the exact effective valence electron equation (3.12) and a c~~ve~~ent choice of the core fu~pion ec, the P~ips-~~i~~ procedure can ir,nmediately be em@oyed. i;he .terms in xv correct for ariy inadequacies inherent in the approbate nature of ihe chosen @,, and they produce all correlation effects as nated above. Eq. (3.9) can be formally solved’to give

“ ijx$ = (~~~-~~A~}, :

(3.14).

’ ‘.

.‘. wh& the operator (E -J?)-l is defined as that with: ‘in the subspace spanned by the or~o~on~.co~~l~

ment Q = l:- P to the spa& dete~ed : jecto: : :_

by the pro:

(3.15).

_subje,ct,to the, constraint that xU be ~~~ogon~ to alI _the~A,,), I.: .' ‘.&&fJ

ii (3,. &.I, ."

Thus, we

may wiite

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., nf (324) and (3.25) in (3.8) Gads to the ~~a~‘. .resu& ',. '1, Use

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{u~~~~ur)i ~~,1H4~~(~Q-~~)-~~A”~}

,(3&)

._ which has ni$ heen’related to pseudopoterrtial

theoF despite its’obvious n&e:Eq. (3.1&j ~e~ateiy impli& thatthe energy de~~Rde~~ of the model

~c~~io~o~~~~~~ V

.~o~~~~ [email protected] energy.

:

: ‘u,u’: ‘v,d :. depeiidence ~~.~~~~~ also. arises froni%e energy * .aj ‘f&k ixac$ abstrdct effe~ti~~.o~~~le~tro~ h~to~~~. depeildence of @e second ter!r$&.(3;16);e.&, of. T$ pa+.of (Q 1) ~~~~~~ Vi, jHlA>? js t$e,,Hartr,ee-; ~ore’,~ol~~tio’~ effects, et& Eq:(3;16) can be ex;.~ . . . _. ,:* .. ,. .I’ I ., . : :. .- : _. .” ‘. :“-, _. .‘. _’ :,, : .. : ; _: ,.:. _ 1. ,. .’ : ,, .’ _: ;._:. : ‘I.. : ,’ :.’ :.: ,. ..-

Volunie 29, number 1

CHEMICAL PHYSICS LETTERS

1 Nuvcmber 1974

panded in a perturbation expansion, given some decomposition

WQ=Qf&Q+WQ,

(3.17)

in “Lhestandard fashion, and the resulting perturbative series may be expressed in terms of Feynmari diagrams. Alternatively, (3.9) may be approximately solved by-a s!ight modification of standard configuration interaction methods or their equivalents in terms of cluster functions [IO, 121. The case of manyelectron pseudopotentials follows analogously to the one-electron case. Again fcr simp.licity Qp,is taken as above. The many-electron valence wavefunction is written 2s (3.18) .where I’= u1u2... u, is an rr-electron valence configuration, the sum in (3.18) is only over distinct V, and 9Q, is the n-electron antisymmetrizer. Av is the Sister determinant corresponding to the configuration CV, so the exact counterparts of (3.6)-(3.10) are as follows: (3.6’)

(3.7’)

WU&y>

E)Ixv)

+HIAr;>=

= 0,

0,

all v’, ‘v’,

(3.9’) (3.10’)

respective!y, etc. for (3.11) and (3.12). The requirement that 9, be orthogonal to the core can be lifted as in the one-electron case. Let D denote the set of distinct~+electron configurations that involve UCIenst one core spin-orbital {c) with the remain&g electrons chosen from the {u} set. The nonorthogonal pseudowavefunction.

with ED en appropriately defmed energy. The exact many-electron operator JCCy may be resolved into its one-, two-, . .. . n-eIectron parts

iollowing methods presented earlier [9, LO, 121, so they need not be repeated here. Suffice it to say that, for instance, the Xrare dependent on the configuration of the remaining (n,-1) electrons, etc., as well as the energy E. If thz basis set {c, u} is not compIete, the remaining complementary functions {e) Icad to SC$ having screen@ between the u-type eleczons. As in actual applications it may be convenient, or be of theoretical interest, to limit the size of the {u) set, then this screening may be important. In the case of semi-empirical valence theories where the {u) set is a minimum basis set of valence shell orbit&, this “screening” represents an important qualitative effect [lOI. The recent ab initio calculations of Iwata and Freed on the correlation contributions to t!!e twoeLectron pi electron hamiltonian of ethylene [13] explicitly demonstrate that computations involving tke xU part of V’ are in fact possible, and it is expected that similar computations and theoretical analyses will be useful in further advanctirg our understanding of the approp riate choice of manyelectron model pseudopotentials. For instance, it follows from the fact that, say,
Acknowledgemeat

is the solution to the exact pseudoequation (3.13) with Xv ,and A, from (3.7’) and (2.4’) andpRK is ., .given by

This research is supported by NationaI Science Foundation Grant GP-28 135 81. I am grateful to the C&ille and Henry Dreyfus Foundation for a TeacherScholar Grant. 147 .-

,, ‘Volr;me29,hnb&1 ‘. : : ‘. “.. ;, &efetinief ; :‘.‘, ‘.

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CHEMICAL PtiYS!CS LI?IT!3RS

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.j8]

: ‘- ‘I f 2j. f.~.~&-iki;; A. H&i &a ,%A. Rice, Advan. Ch+% Fhys{ .I6 (1969) 283,~and‘ikferences therein. ‘. ,;, [j], hLE:Schwktz and J.D. ~tita.lb, 3..LThernb~y;~57 ‘. .’

Phy.r..3 (1974) 463; C&n: Ph$s.,ytt&s’24 275. :.y [ll] S. F+inaga and A.A. &&tu,S. Chern’Phys, ‘. $543.

,141 M.E. S,cfiwart?, ‘Bern. Phys. L&w

21 (1973) 314; I.D. S&&l&i, J.TJ. Huang and M.E.Schwaitz, J. ::: .‘: Zhem’. Phys..N! (1974) 2252. ‘. [51 ‘RI. Kleiner Vd’R. Mc%‘eeny, Cheq. P&s. Let&s 19 :’ (1973) r176. 161 L.R. Kahn and W.A.‘Goddard III; 1,‘Chem. ihis. 56 ,-.

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Department of ‘chemistry, ’ ‘, .i+dia@ Institute of Te&nol&y,‘. ,’ Bombay, I.ridia.

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B.P. Pad&a; S.N. Gtipta and V3.rish1~1, Chetitial effects oti the ~-absorption spectrhm of platinum; Chem. Whys. Letters 27 (1974) 224.

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