Indian Journal of ChemistryVol. 35A,September 1996,IJP. 721-727

Advances in Contemporary Research

Stationary coupled cluster functionals in molecular property calculation

Sourav Pal" & Nayana VavalPhysical Chemistry Division, National Chemical Laboratory, Pune 411 OOg

Received 15 February 1996; revised 16 May 1996

Dr. Sourav Pal

Dr. Sourav Pal works in the area of electronic structure theo-ry. His area of expertise is in the study of electron correlationto spectroscopic energies, structure and molecular properties.He is a recepient of the INSA and CSIR Young ScientistAward and Fellow of the Indian Academy of Sciences, Banga-lore.

Miss Nayana Vaval

Miss Nayana Vaval is the Ph.D student of Dr. Sourva Pal.

IntroductionCoupled cluster (CC) methods'>' have been

widely developed for the accurate calculation ofenergy and energy derivatives'>, The analytic eval-uation of energy derivatives is one of the foremostachievements of quantum chemistry. The first andsecond derivatives of energy with respect to thenuclear coordinates are known as gradients andHessians respectively. Similarly, various derivativesof energy with respect to the electric/ magneticfield are known as molecular properties like di-pole moment. Energy derivatives are important inthe determination of molecular structure and pro-perties. In this article we present some contempor-ary aspects of the subject relating to calculation ofmolecular properties using the CC based methods.A considerable amount of work has been carriedout by our group in recent years on the direct sta-tionary methods for calculation of molecular pro-perties. We have limited our discussion in this arti-cle to closed shell like situations dominated by arestricted Hartree Fock configuration. It has beenrealized that efficient incorportion of electroncorrelation and proper choice of basis set are ofcrucial importance for the accurate calculation ofmolecular properties. These effects are more pro-nounced for the nonlinear properties. CC meth-ods, using exponential wave operator, incorporatedynamical aspects of electron correlation efficient-ly. A proper use of the CC method can providesize- extensive values of molecular properties.

From the computational point of view the sim-plest and the most straight forward way of calcu-lating various energy derivatives is the numericalfinite field or finite perturbation method. In thismethod the nth order property is obtained numer-ically by the method of finite difference approxi-mation. Though convenient, this method is not

722 INDIAN J CHEM, SEe. A, SEPTEMBER 1996

very accurate, particularly in the case of higher or-der properties. Therefore, analytic energy deriva-tives are desirable in view of this. Considerableprogress has been made in this front in the lastdecade.

Various versions of the CC based methods, i.e.non-variational C C, equation of motion (EOM)CC and different forms of stationary CC havebeen used to obtain the molecular properties. Thefirst CC based response approach for the deriva-tive calculations was formulated by Monkhorst" ina non-variational framework. In this method theexplicit expression for the first derivative of energydepends on the first derivatives of the cluster am-plitudes. However, using a Z-vector technique",Bartlett and coworkers eliminated the dependenceof the energy derivatives on the cluster amplitudederivatives in favour of an extra set of h-p de-exci-tation amplitudes. This is a simplification whichcan be incorporated for higher order properties,too. In general, (2n + 1) rule can be incorporatedin the evaluation of higher order properties. How-

• ever, any method based on stationary principle sa-tisfies this rule naturally.

Different structures of stationarity have been at-tempted in recent years by Bartlett and cowork-ers8,9, Jorgensen and coworkers'P-" and Pal andcoworkers+:". Their methods differ from eachother in their theme and details. Non-variationalCC equations can also be obtained from the sta-tionary method with the help of Lagrange multipli-er technique I5. In this method the NVCC equ-ations for the cluster amplitudes are kept as con-straints. While this technique can be shown to in-corporate a (2n+ 1) rule for the cluster ampli-tudes, it also involves additional set of Lagrangemultipliers, which satisfy (2n+ 2) rule. This tech-nique is a general one and can retain any particu-lar structure of the CC equations by Lagrangemultiplier constraints. The cluster amplitudes andthe derivatives may be obtained by making an ap-propriate energy functional stationary. This classof methods can be called as direct stationarymethods.

Pal has developed a direct stationary methodfor the evaluation of the amplitude derivatives'<.The stationary conditions of the derivative energyfunctional with respect to the unperturbed clusteramplitudes provide us with the amplitude deriva-tives. In this respect, choice of an energy function-al is of vital importance. Expectation value and bi-orthogonal functionals have been used by Pal andcoworkers'r-'? for the evaluation of amplitudederivatives with respect to the external fields andconsequently the molecular properties.

However, one can use other CC based function-als like unitary CCI8, which have been studied inthe quantum chemistry literature. We review herethe use of expectation value and bi-orthogonalfunctional. We discuss the convergence propertiesof the two functionals. The size-separation fea-tures of the CC based functionals is also an im-portant issue to be addressed. While the bi-ortho-gonal functional is fully extensive, the response ap-proach using the expectation value approach is notfully extensive. We discuss the origin of the size-separation errors in the molecular properties uptothe first hyperpolarizability.

Stationary response approachIn the direct stationary response approach'? we

first define the energy functional E(A) in the pres-ence of electric field whose strength of perturba-tion is denoted by A.The molecular properties areevaluated as various derivatives of it with respectto A at A= O. The basic idea of the stationary re-sponse approach is as follows

Consider a functional E which depends on thevariables X, which have parametric dependence onA.E= G{X(A)} ... (1)

We can use a Taylor's series expansion for E aswell as the variables {X} in terms of A aroundA= O. Thus we have

E (A) = E(O)+ AE(1) + A 2 E(2) + .

X(A) = XiO) + AXil) + A 2 Xi2) + .

... (2a)

... (2b)

A series of equations may be defined by takingthe derivatives of E(i) with respect to XV) equal tozero for i ~f. It has been shown by Pal12 that ifthe amplitudes and their derivatives are truncatedto a uniform degree, then

... (3)

provides an identical set of equations for a fixedvalue of (i-f). Hence, the following set of equ-ations is sufficient to obtain the variables and theirderivatives.oE(i)

ox'0)=O i=O, 1,2,3, ...

The above equations provide us with the variablesas well as their derivatives in a hierarchical fash-ion. The variables in our case are the coupled

... (4)

PAL et al.: STATIONARY COUPLED CLUSTER FUNCTIONALS IN MOLECULAR PROPERTY CALCULATION 723

Prop

Table I-LiH dimer results for size-separation tesr'LiHb [(LiHh/2jC

SXCCSD ECCSD SXCCSD ECCSD

Quad Cubic Quad Cubic Quad Cubic Quad CubicfIz 1.985 1.976 1.982 1.995 1.985 1.976 1.982 1.995«: 13.66 9.81 14.33 13.00 13.67 9.77 14.32 12.74

fizzz -771.0 - 277.7 - 852.9 -651.1 -777.1 - 275.9 - 852.9 - 651.6

"All results are in atomic units; molecule along the Z axisbFor this model test we have taken the Li-H distance in the monomer as 2.0 au'Two monomers of LiH are separated by 900 atomic unit; dimer results are divided by 2

cluster amplitudes. The use of Eq. (4) can provideus with the cluster amplitudes up to any desiredorder n and the properties up to (2n+ l)th order.We will review in the next section the expectationvalue functional and the extended coupled clusterfunctional.

FunctionalsIn the direct stationary method one starts from

an energy functional and calculates the variouscluster amplitudes and the derivative amplitudesthrough the use of Eq. (4). The form of this func-tional in the presence of the external field is givenbyEq. (5)

E(A)=<~oleT+(A.) H(A)eT(J.)I~o)L ... (5)

where L indicates the inclusion of linked or con-nected terms only. The functional is' a hermitianbut a nonterminating series in cluster amplitudesand thus has to be terminated for practical applic-ations. There are various ways of enforcing the,truncation on the series. Perturbation guided trun-cation can be enforced using the approach ofBartlett et all", In their approach, called XCC(n)approach, all the terms whose minimum contribu-tion is upto a certain order n are included in thefunctional. This approach is size-extensive at eachorder as there are no disconnected terms. Trunca-tion can be done on the power of TIT+ terms inthe energy expression. Such a fixed power basedtruncation scheme was used by Pal et al.2o for thecalculation of energy. A similar fixed power trun-cation scheme has been used for response pro-perties, too. In the initial application of this byGhose et aL 16a a quadratic truncation scheme wasused. However, it was realized that this truncationscheme is particularly unsuitable for higher orderproperties. Subsequently, a cubic truncationscheme was used by' Vaval et aLl6b. But, at this le-

vel, the equation for the amplitudes i.e. Eq. (4) willcontain disconnected tenns. The appearance ofthese terms in the equations introduces the size-in-extensivity errors. These inherent problems in theexpectation value functional. may be avoided bythe use of a bi-orthogonal functional of Arponenand Bishop?'. The form of this functional is,

where ljJ and ljJ' are parametrized differently andbear a bi-orthogonal relation. It can be seen thatthe stationarity of the above equation with respectto Q leads to the nonvariational coupled clusterequations of Cizek'. Arponen suggested a parame-trization of ljJ' in a fully exponential form using ahole-particle de-excitation operator T. This isknown as the extended coupled cluster functional.The ECC functional in the presence of electricfield may be written as

E(-A) = < ~o I eL(J.) (H( A) eL(J.) )L I ~o) DL ... (7)

where L indicates the linked or connected termsand DL indicates the double linking, which meansthat the ~ vertex to the right of the H must beconnected to H and the i: vertex must be eitherconnected to the H from the left or to two differ-ent ~ vertices. The double linking nature of theseries makes the series naturally terminating, butstill the natural termination is at quite a high orderand thus it has to be truncated for practical appli-cation. Double linking ensures that there are nodisconnected diagrams in the cluster amplitudes ortheir derivatives. Due to the natural termination ofthe series we expect the higher order properties tosaturate quickly. The double linked or ECC typeparametrisation has been used recently by Mu-kherjee and coworkers in a superoperator resol-vent framework+',

724 INDIAN J CHEM, SEe. A, SEPTEMBER 1996

Fig. I-The shape of disconnected block for 111)in E(l)

Separability properties of the functionalsAs discussed earlier, the equations in the SXCC

model contain disconnected terms in general. Thedisconnected pieces arise from the appearance ofthe "nondoubly linked" terms in the XCC expres-sion, leading to a loss in size-separation. In a re-

. cent study", we have identified the "nondoublylinked" terms in the XCC equations and using anexample of separated LiH dimers showed the nu-merical loss of size-separation. We reproduce inTable 1 from our work in ref. 23 the values ofproperties for LiH monomer and separate LiH di-mers using the SXCCSD and ECCSD method.The basis used for Li is 4S2P and for H it is 2S.In the separated dimer calculation, two LiH mon-omers are separated by a distance of 9000 au.The results were obtained using the cubic trunca-tion scheme. As expected, in the ECC methodcomplete separation is achieved. In the SXCCSDmethod there is a loss of separability. For dipolemoment values the loss of extensivity is nonexist-ent. The error of the size-separation increases fornonlinear properties.

The perturbation analysis of disconnected termsmay be done. This study has been reported in ourrecent work-". We reproduce the essential conclu-sions of the work. The equation for the nO) ampli-tudes is obtained by making the energy expressionstationary with respect to the ~O) amplitudes. Up-to the cubic truncation, the energy expressionswhich lead to the disconnected terms in this cate-gory are (a) Tt(O)VT~Orr~O)and (b) nO)T((O) v nO).When differentiated with respect to the Tt(O) ver-tex, both these terms contribute the disconnectedpieces to the nO) amplitude equation at the mini-mum of (a) fifth and (b) fourth order in perturba-tion. In both of these expressions for energy theTt vertex is exclusively connected to the T, ver-tex. Thus when the Tt vertex is differentiated (de-leted), these expressions lead to the disconnectedpieces. A typical disconnected block contributing

Fig. 2- The shape of disconnected block for 1V)

to the nO) amplitudes is shown in Fig. 1. Similarly,the equation for nO) amplitudes is obtained bymaking the energy expression stationary with re-spect to the nO)+. amplitudes, i.e., by solving[oE(O)];1o Tt(O)] = O.

In any linked expression for the energy, no rightvertex can be exclusively connected to the T( ver-tex. Hence, the differentiation of the Tt vertex inthese linked energy expressions can not lead tothe disconnected pieces explicitly. The only waythe disconnected pieces can explicitly appear inthe equations determining the nO) amplitudes isthrough the energy expressions which are them-selves disconnected (i.e., a product of two or moreclosed diagrams). However, such an expressionfor energy is not possible. On the other hand theequation for the nO) amplitudes contains the nO)amplitudes which have disconnected pieces. Sucha block has been schematically shown in Fig. 2.These disconnected pieces lead to the product ofan open one hole particle block and a closed dia-gram in the equation for nO}amplitudes. However,

I in the initial results (see Table 1) obtained by ourgroup using the XCC method, the energy expres-sion contained terms the minimum contributionsof which were only upto the fourth order. Thisconsideration eliminates the possibility of the ap-pearance of those terms (of the types a and b re-ferred to before) which lead to disconnectedpieces in the equation for the nO) amplitudes andthus the nO) amplitudes in our applications remainfully connected.

Since the expression of E(I) is explicitly con-nected, we notice in Table 1 the correct separabil-ity of the first order properties like dipole mo-ments. But the general problem of separability willremain in the stationary method using the XCCfunctional when we include additional terms in theexpression for E. For obtaining the derivative am-plitudes, similarly we consider in E(1) all quadratic

PAL et al: STATIONARY COUPLED CLUSTER FUNCTIONALS IN MOLECULAR PROPERTY CALCULATION 725

Table 2-Dipole moment values of HF at different Table 3-Polarizability values for HF molecule at differentgeometries-" geometries-"

Distance SCF SXCCSD ECCSD Distance SXCCSD ECCSD

Quad Cubic Quad Cubic quad cubic quad cubic1.7328 0.811 0.760 0.769 0.755 0.767 1.7328 4.595 3.796 5..25 4.232.00 0.922 0.836 0.855 0.821 0.846 2.00 6.552 4.945 7.85 6.072.22 1.002 0.859 0.885 0.852 0.897 2.22 8.730 6.014 10.88 7.482.24 1.018 0.869 0.896 0.854 0.901 2.24 8.947 6.108 11.21 7.622.26 1.026 0.871 0.899 . 0.855 0.905 2.26 9.168 6.202 11.54 8.412.28 1.034 0.873 0.901 0.856 0.909 2.28 9.294 6.295 11.89 7.83

2.5995 1.162 0.863 0.909 0.822 0.953 2.5995 13.600 7.631 18.99 11.53

"All results are in au; molecules along Z axis aAlI results are in au; molecules along Z axis"Basis set used in DZP; ad for F is 1.60; ap for H is 0.75 bBasis set used is DZP; a d for F is 1.60; a plor H is 0.75

terms and only those cubic terms the nummumcontributions of which are upto the fourth orderin the V perturbation. The differentiation of theT( vertex in the expression for the E(I) providesthe equation for the 1\1) amplitudes. The leadingterm in E(I) is T( 6 and this contributes to the1\1) amplitudes in the zeroth order in the V per-turbation. A similar consideration will show thatthe leading term in the equation for the Vi) ampli-tudes is of the first order. Since the perturbativeimportance of 1\1) amplitudes is higher than thatof the corresponding TI amplitudes, our prescrip-tion of choosing E(1) now involves terms whichhave the possibility of "nondoubly linked" termsand thus in our applicarions'P-" the disconnectedpieces remain in the Vi' amplitudes. The mostdominant such piece is disconnected 61\1) arisingfrom the Tf 61\1) term in the expression for E(1).

The dominant contribution of this term is at thesecond order. As in the case of the ground stateTI amplitudes, the equation for the derivative 1\1)amplitudes does not contain any explicit termwhich is disconnected. But the disconnected piecesin the equation for the 1\1) amplitudes appearthrough the disconnected Vi' amplitudes. Themost important term in this category is theT((OlVVP1\O) terms in the one body equation.The minimum perturbation order at which the dis-connected pieces appear in the equation for the'4.1)amplitudes is the second order and hence theconsequent contribution of the disconnectedpieces of the 1iJ) amplitudes in the equation forthe 111)amplitudes is at the fifth order.

Let us now discuss the effect of these discon-nected 1\1) and '4.1)amplitudes in E(2) and E('3).

The most important term in E(2) is the(;oI61V)I;o)v Disconnected 1\1) amplitudes

make E(2) unlinked. These 1\1) amplitudes contri-bute to the polarizability value at the fifth order.Similarly the most important terms in E(3) are (a)(~ol'4.I)+61V)I~oh (bX~ol1\I)+61\1)I~o)L and (c)(~ol'4.1)+61\1)1~O)L' The terms of the type (a) willbe fully unlinked only when the amplitudes of the'4.1)+as well as '4.1)operators are both disconnect-ed. The disconnected amplitudes of '4.1)are at thesecond order as previously discussed and hencethey contribute to the term (a) at the fourth orderonwards. However, since the disconnected 1\1)amplitudes are of the shape shown in Fig. 2 hav-ing closed diagrams as factor, any appearance ofthese amplitudes immediately make E(3) unlinked.The connected and disconnected pieces of 1\1)amplitudes contribute at the zeroth order andfourth order respectively. Thus the separability er-ror for the term (b) is the least at the fourth order.However, for the term (c) a disconnected block of'4.1)amplitudes can contract with the 6 to form aclosed diagram as one factor. From our perturba-tive considerations the term of the type (c) contri-butes to the separation error atleast at the secondorder in perturbation. This is the most dominantterm and this explains the large separation errorsfor hyperpolarizability values in Table 1.

Accuracy and quality of resultsIn some of our recent worksI6,17-23we made a

detailed numerical analysis of the performance ofthe two functionals by calculating properties uptothe first hyperpolarizability for some model sys-tems like HF, IAl state of CH2, CH;-, LiH etc. Weused two different truncations schemes of thefunctionals: (a) quadratic and (b) cubic. We ob-served in all cases that at the quadratic level theresults obtained by the use of the two functionals

726 INDIAN J CHEM, SEe. A, SEPTEMBER 1996

0-9!j

::>ac

~wEoEwO.8S'0Q.

s0-80

.~ .2-00 2'2~ 2·!50OistQncf! in au

Fig. 3-The change in .dipole moment with internuclear separ-ation of HF molecule m double zeta plus a set of polarizationbasis (DZP) using different levels of theories discussed in thetext. The curve with hollow circles corresponds to the ECCquadratic approximation whereas the one with filled circles isfor SXCC in quadratic approximation. The curve with traingleshows the ECC in cubic approximation and that with inversetraingles corresponds to the SXCC with cubic approximation

XCCSD and ECCSD are same. This is expectedsince at the quadratic truncation, the terms in thetwo function~s are almost identical and the equa-tion for the ~ amplitudes is nearly the adjoint ofthe one for the ~ amplitudes. However, the differ-ence between the doubly linked functional and ex-pectation value functional begins to appear at thecubic level onwards. Hence, the results obtained atthe cubic level using these two functionals arequite different. We find that the results at the cu-bic truncation using bi-orthogonal ECC functionalare very close to the full CI (or model exact) re-sults. For the first order properties. e.g. dipole mo-ments, the results of both the XCC and ECCfunctionals even at the quadratic level are close tothe model exact ones. Naturally, the results ofthese first order properties do not change on go-ing from quadratic truncation to the cubic trunca-tion. The variation of the results between theXCC and ECC functionals as well as between thequadratic and cubic truncation for each of thesetwo functionals is greater for higher order propert-ies. This has been documented in our earlier stud-ies. However, in most of the reported cases the re-sults obtained using the cubic truncation of theECC functional are saturated.

Most of the examples above are for equilibriumsituations. To study the performance of thesemethods at situations away from equilibrium, wepresent here numerical results of dipole moments,polarizability and first hyperpolarizability of HFmolecule using a double zeta basis augmented bya set of d functions with exponent 1.60. The re-

20~·--------------------------~

:::J 15\J

c>--1:= 10aoN

O~~ ~~ ~~ ~ -J1, 5 2·00 2-25 2-50

Dj_s~e In Q!JFig. 4-The polarizability values at different geometries

(labeling is same as in Fig. 3)

sults are presented at different internuclear dis-tances. Tables 2 and 3 present the dipole momentand polarizability values along the molecular axisat various internuclear distances. At the equilibri-um the trends are similar to the ones discussed.However, these trends change at large internuclearseparations. For dipole moments, we observe thatthe results obtained at the quadratic level usingthe XCC and ECC functionals are quite differentat large separations. The same is true for the cubicresults at large internuclear separations. Fig. 3gives the dipole moment values calculated usingdifferent functionals discussed at different inter-nuclear separations of the HF molecule. Similarly,Fig. 4 plots the change in polarizability (along themolecular axis) values at different internuclear dis-tances. We observe that the polarizabilities ob-tained using the XCC and ECC functionals differsignificantly at both the quadratic and cubic levelseven around the equilibrium. All the values in-crease with the internuclear separation. The largerdifference in the polarizability values calculatedusing the SXCC and ECC functionals is to be ex-pected. From our experience of the results aroundequilibrium, it may be safe to conclude that cubicECC results are the saturated ones.

AcknowledgementThe authors acknowledge the financial support

from the Department of Science and Technology(DST), New Delhi (DST grant No SP/Sl/H-15-94).References

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