J. Phys. Chem. 1995, 99, 6441-6451
6441
Coupled-Cluster Based Linear Response Approach to Property Calculations: Dynamic Polarizability and Its Static Limit B. Datta, P. Sen, and D. Mukherjee" Theory Group, Department of Physical Chemistry, Indian Association for the Cultivation of Science, Calcutta 700 032, India Received: September 8, 1994; In Final Form: January 26, 1995@
In this paper is described in detail a time-independent version of the coupled-cluster based linear response theory (CC-LRT) for computing second-order molecular properties. It utilizes a coupled-cluster representation of both the ket and bra functions for the ground state that are conjugates of each other, while for representing the excited functions-which enter the spectral representation of the response function-it employs coupledcluster based ansatz which generate ket and bra excited functions that are bi-orthogonal to each other as well as to the corresponding ground state functions. Emphasis has been given to the important practical problem of avoiding the tedious sum-over-state formula for second-order properties such as the dynamic polarizability by way of implicitly inverting a dressed Hamiltonian matrix in a set of elementary bi-orthogonal bases which are much simpler than those representing eigenvectors for the excited states. It is shown that the elementary bi-orthogonal bases used for the excited space in our formulation respect strict orthogonality with the ground state function even for the truncated, approximate version of CC-LRT. It is also proven that the theory generates size-extensive as well as size-consistent values of the dynamic polarizability for a closed-shell system that is composed of noninteracting closed-shell subsystems. As numerical applications, the first results using this formalism are reported for LiH, BeH', HF, H20, HC1, and H2S.
1. Introduction
Kobayashi et a1.8bin a comparative study vis-&vis the method of ref l l a . The coupled-cluster methodology has emerged in recent years A coupled-cluster based polarization propagator theory was as a powerful correlated method for electronic structure calculaalso suggested by Oddershede and co-workers,I2which also can tions. Originally developed for closed shells,'J the method has be utilized for computing the dynamic polarizability. In this also been extended to handle quasi-degenerate or open-shell approach, the inner projection manifold of operators in the state^^.^ in an effective Hamiltonian framework. Since it is now superoperator resolvent based formulation uses the Dalgaard widely documented that electron correlation plays a significant type of product basis, which leads to a coupling of the so-called role in shaping properties other than total e n e r g i e ~ ,it~ is advanced and retarded components of the propagatorresulting worthwhile to explore the efficacy of the coupled-cluster based in a doubling of the dimension of the inner projection manifold. methods in generating such properties as dynamic polarizability A consistent basis, based on the unitary coupled-cluster ansatz, and its static limit. was suggested some time ago by Prasad et al.,I3 which leads to For the static case, a finite field perturbation theory using the desired decoupling of the advanced and retarded compoperturbed orbitals in the closed-shell coupled-cluster approach nents. Related developments in the same direction using the has already been applied with considerable success,6 although unitary coupled-cluster representation for the ground state were its generalization to encompass the dynamic property remains also made by Mukherjee and Kut~e1nigg.l~Datta et aLl5 have to be explored. Coupled-cluster based linear response theories recently suggested another consistent scheme for propagators (CC-LRT), developed in various contexts by several author^,^-^ using the extended coupled-cluster parametrizationI6 for the seem to be a promising approach to study both static and ground state. dynamic polarizability. A time-dependent version of the Insofar as the MBPT expansion of the ground state may be formalism was first suggested by M ~ n k h o r s and t ~ ~ was later viewed as the perturbative solution of the CC equation for the extended by Dalgaard and M ~ n k h o r s tand ~ ~ by Koch and ground state, it is pertinent here to mention also the secondJorgensen.8 A related time-independent version was given by order MBPT approaches to dynamic polarizability by Rice and Mukherjee and Mukherjeega and Ghosh et aLgb A more Handy,I7 Wormer and Hettema,'* and Sasagane et al.I9 We convenient formulation, much along the line of refs 9a,b, has may also mention extension of these methods to dynamic recently been given by Kundu and Mukherjee,Io who also hyperpolarizabilities by Rice and HandyI7 and Sasagane et al.I9 presented the first dynamic polarizability values of He, Be, and We shall present in this paper a detailed theoretical treatment CH+ using their modified version of CC-LRT for polarizability. of the CC-LRT for dynamic polarizability, which was briefly Stanton and Bartlett1Iahave developed a coupled-cluster based reported in ref 10, and its molecular application to several equation of motion method' I b for computing excitation energies molecules of interest. The theoretical developments would and transition moments, which is conceptually very similar to the earlier CC-LRT method of Mukherjee and M ~ k h e r j e e . ~ ~ particularly highlight three important aspects: (a) the advantage of the use of a bi-orthogonal projection manifold to compute These quantities can be used directly in the sum-over-states the polarizability that bypasses the tedious sum-over-states formula for polarizability, as has been done very recently by formula; (b) the demonstration that this bi-orthogonal operator basis naturally makes use of the intrinsic nonhermitian nature * Author to whom correspondence should be addressed. Abstract published in Advance ACS Abstracfs, March 15, 1995. of the eigen problem for the excited states in CC-LRT; and (c) @
0022-365419512099-6441$09.0010
0 1995 American Chemical Society
Datta et al.
6442 J. Phys. Chem., Vol. 99, No. 17,1995
proof of the size extensivity of the dynamic polarizability using the modified CC-LRT, which is essential in the study of large systems. The numerical results presented in the paper illustrate the efficacies of all three aspects mentioned above.
2. Time-Independent Version of CC-LRT for Excitation Energies: A Brief R&umk We present first a concise description of the CC-LRT in the time-independent version?,'0 which in our opinion is both direct and provides an easier access to the second-order response properties of interest. In CC-LRT one starts with the ground state wave function Ilyo), written with the coupled-cluster ansatzI**
In CC-LRT, we intend to use a particular ansatz for ($kl which utilizes the underlying nonhermitian structure of the matrix of the eigen problem of CC-LRT to preserve the relation 2.7 above. This choice is dictated naturally from the structure of CC-LRT working equations generating Wl values. The Schrodinger equation for ( V k ) can be written in the equation-of-motion form for CC-LRT as
[H, wL1 I
= OkWL IVO)
where w k represents singlet excitation energies. Since T and Wl are both built from the same sets of operators { q j } , which commute among themselves, we may write eq 2.8 in a modified form
Lii, wil I@& = O k where T consists of various nh-np excitations out of the closedshell Hartree-Fock function, which is taken as the vacuum. The excited state IVk) are generated from IVo) by a statespecific excitation operator w;:
All of the Wl operators are constructed as linear combinations of the various nh-np excitation operators out of 40,since they themselves form a complete basis for constructing WL-provided the identity operator (corresponding to n = 0 excitation) is included in the set. In what follows, we shall adopt the more conventional nonorthogonally spin-adapted version of the CC-LRT, which uses the various spin-free excitation operators in normal order for constructing both T and Wl values. Thus, for illustrations, the single- and doubleexcitation operators in spin-free form are (2.3a)
where the braces denote normal ordering with respect to r#w, the labels p,q and a$ signify particle and hole orbitals, and u is the quantum number for the spin. We shall henceforth use the generic symbol qt for any member of the nh-np excitation operator of the type illustrated above. Using the set {qt}, the operators T and Wi can be expressed as
(2.8)
wL
140)
(2.9)
where we have premultiplied eq 2.8 by exp(-T)_and have introduced the nonhermitian "dressed" Hamiltonian H,defined as
ii = exp(-T) H exp(T) -
(2.10)
with EO as the exact ground state energy. The operators { q r } are neither norm-conserving nor orthogonal in the sense 'I]
= ( @ o ~ ~ I q ~ ~ @ o )' I ]
(2.11)
so that it is not advantageous to use the conjugate functions (&lqr for projections in eq 2.9. We shall rather use the biorthogonal counterparts {ql},defined as (2.12) such that
Now projecting eq 2.9 onto the functions (&I the eigen problem Rxk
= OGk
= (qh)Iiji, we have (2.14)
where R is the matrix with elements
and x k is the column of coefficients {Xik}. The eigenvalues Ok represent transition energies for k f 0. The root wo is just zero, corresponding to the excitation energy of the ground state itself. Since R is nonhermitian, the vectors x k are not orthogonal to each other, although the I?&) functions are so by definition for an exact theory. The left eigenvectors &, defined via With the ansatz for IVo) and IVk), eqs 2.1 and 2.2 above, the functions Iqo) and Ilyk) are not normalized to unity, though they are orthogozal for th_e exact theory. We define the conjugate functions (Val and ( V k l as
%@ = 2,
m k
generate a potentially complete basis for representing the bra functions, which are, how_ever, not related in a simple way to the conjugate functions (Vkl introduced in eqs 2.6a and 2.6b. Since the set of functions {&I}, V k including k = 0, is complete, the left eigenvectors of R are also complete and may be taken to be bi-orthogonal to the vectors { x k } :
which now include the entire normalization constants in them, so that we have
(gklV,)= 6 k [
b' k,lincluding k,Z = 0
(2.7)
(2.16)
(2.17) Let us now introduce the bra functions
J. Phys. Chem., Vol. 99, No. 17, 1995 6443
CC-LRT Approach to Property Calculations
where
w k
*
is given by (2.19)
which by construction satisfy the following bi-orthogonality relation: &klql)
tion, however, and worked instead with the choice eq 2.24a. For this the orthogonality relation between ($01 and / q k ) , k 0, is valid for the exact functions, but, when using a truncated scheme, we must use this constraint from the outset. This is due to the fact that (&I in the representation 2.24a does not come out naturally from the left eigenvector 2 0 , as it would have from the choice 2.24b. This is the price we pay for preserving the size extensivity of the second-order properties. We want to emphasize that the polarizability expression formulated by Kundu and Mukherjee’O is size extensive, as we shall demonstrate below, and any CC-LRT derived from an EOM-type of formulation does not necessarily have to use the choice 2.24b for ($01.
=x z k i
($01qiq;I$O)
x~l
IJ
= x E k ? i l = 6kl
(2.20)
I
We may thus identify ( & I with &I and find that ( & I be equivalently represented in two ways as ($kl
= ($01 exp(Tt) ($kl
w,/(qklqk)
= ( 4 0 l w k exp(-T)
can also
3. Transition Moments in CC-LRT In the computation of transition moments in CC-LRT we need to evaluate two quantities:
(2.21a) = ($01
lqk)
(3.1)
D m = w~l~O)l-lDklJ
wk(R+
(4*3)
kt0
(3.7) It should be noted that the expectation value expression (@ole" d q? eTl40) has been factored out in eq 3.7 into a linked term containing p,T, d, and q? and another term in which 4;is not joined to d. This factorization is essential for the simple expression ultimately obtained in eq 3.7. The symbol (...)L indicates a completely linked term. The quantity Dm is simply given by
For computing the dynamic polarizability, we shall sidestep the sum-over-states formula and use instead the elementary bases exp(T)I40)} and ((4014i exp(-T)l, from which {WlI and {Wk} are constructed. The Q-space projectors can be written either with the sets { l q k ) } and for k o or with the elementary bases above for i f 0:
{d'
{(qkl}
*
Q = CwL exP(T) I&) (401W k exP(-T)
A somewhat nontrivial point in eq 4.3 is the fact that it does not really matter whether we now expand Wl using (41') or {qy} as the bases in computing the expression (401 f i k ( f i w 1)Wl I&). In fact, we find that
+
(401 w k ( H + w
l)wLl
40>=
140)[c(&14i(H+w I JfO
e ( d O 1
4i('+
1'
140)
(301 q 1 1 q 0 ) l X k ? ~ k
1JtO
+
where we find that the second term is zero since (401&(fi w 1) I&) vanishes due to the fact that the T matrix elements have been found out in the coupled-cluster theory by equating them to zero. Thus we have the validity of eq 4.3. The difficulty with the expression of ai,(w) is the need to compute all of the eigenvectors of the matrix R for k f 0. This sum-over-states formula requires both extensive computer time and a storage of the entire matrix R, which can be very large. A way to circumvent this problem is to use the elementary functions (47' eTl&)} and {(@oo(qre-T) for i f 0 as biorthogonal pairs. Thus we can rewrite a+(w) using eqs 3.9b and 2.15 as
(3.9a)
k#O
Q=
&:'~ X P ( T1) 4 ~04~0l ) 4i ~xP(-T)
(3.9b)
it0
We should use the (4): basis for expanding in eq 3.9 above to maintain the explicit orthogonality of (lyol with the set { l q k ) } , k f 0.
4. Expression for Dynamic Polarizability in CC-LRT
Using manipulations similar to what is done in eq 3.7, we find that
a+(@> = C(40~e fldqiI eT 14~0)~ [ R + wlijl
x
if0
The traditional expression for the dynamic polarizability tensor a [ap,]is given by
(401 q j
e-Td eT140) (4.5)
This formula is much simpler and is more convenient to use. Defining the amplitudes and &, as
gi
%i
= (401 en&;
eTI40)L
(4.6a)
$0
= (401 C?i e-T'
eT140)
(4.6b)
we find that the components ai,(o) of This expression really corresponds to that of a polarization propagator, where we have explicitly written the contribution of its advanced and retarded parts. We shall illustrate our manipulation in CC-LRT leading to the working expressions by considering only the component a+(w). The treatment for a-(w) is entirely analogous. a+(w) is first transformed as follows:
a+ are given by (4.7)
The implied inversion of the large matrix ( R
+
w 1) in eq 4.7 can be bypassed by solving a simultaneousequation of the form
(R
+
0
l)Zy(o)=? I
(4.8)
and zV(o) is the desired where 3 is the column matrix linear response column ( R w)-'d". Using the preconditioned conjugate gradient method,19 the solution of eq 4.8 can be performed in an efficient manner. The iterative method updates a trial z", by the formulas involving ( R UJ a l v ( w )is thus evaluated from the expression
+
+ l)zr.
Using the expression for ~ q o ) ,I q k ) , (&I and 2.2, 2.21b, and 2.24a, we have
(Gkl
(4.9a)
for eqs 2.1, if0
CC-LRT Approach to Property Calculations
J. Phys. Chem., Vol. 99, No. 17, 1995 6445
ai,,(w)is similarly given by (4.9b) A similar method of using elementary orthogonal functions to simplify the expression for second-order properties in a CI context was first utilized by Iwata.20 It may be noted that our derivation for the expression of the dynamic polarizability has as the starting point the Fourier transform of the polarization propagator, where we have used the CC-based representation of the wave functions of both the ground and excited states. The advanced and retarded parts are treated separately from the outset. This should be contrasted with the CC-based polarization propagator approach of Geertsen and OddershedeI2-where only the ground state was written in the CC form, while the excited states were represented in the space of Dalgaard type of product basis operators containing both excitation and deexcitation operators qt and qi. This latter is needed to treat the advanced and retarded components in a compact manner. Recently there have also appeared both CC-baseds and M B P T - b a ~ e d ~formulations, ~x~~ where the dynamic polarizability is extracted in a time-dependent context as the first derivative of the perturbed expectation value of the dipole moment operator with respect to the electric field amplitude of the monochromatic field.”,’* Here, the perturbed ground state is expressed as a double perturbation series with respect to both electron correlation and the extemal field. The starting point is thus a timedependent expectation value and the effect of extemal perturbation is manifest both on the ket and the bra components of the ground state. In particular the CC-based approaches of ref 8 and ours9 are ultimately equivalent when the exact ground state function is used. They would remain nontrivially different when approximate expressions for the ground state are used to compute the polarizability. This stems essentially from our use of the adjoint of an approximate I ~ o )with the choice 2.24a, while eq 2.24b is the choice for ($01 explicitly used in ref 12.
5. Size Extensivity and Size Consistency of a(o)Using CC-LRT
&;
In our formation a(o)is computed as a sum over and and q0(used in Zk) are Since both connected entities, it might seem at fist sight that the extensivity of ap”(w)is rather trivially manifest. This, however, is not so: we have an implied inverse matrix [ R w 13-’ in the expression for To,whose separability we have to explicitly examine to prove connectedness of a(w)using our method. To prove the connectedness, we use formally the sum-overstates expression for a(o)in eq 4.3. For any truncated choice of the bases {q?} and {&}, we have a corresponding set of approximate {Wl} and {Wk}. We use as an illustration the CCSD scheme for I ~ O and ) the truncation of {q:’} and { i j i } up to 2h-2p operators. The nature of the proof is, however, quite general and can be easily extended to arbitrary truncation schemes. It has recently been shown by Mukhopadhyay et a1.22 that the energy differences wk, computed by diagonalizing the matrix R, display rather peculiar separability property as a system gets separated into two noninteracting subsystems (denoted A and B). Since the CC equations for Iy@ as well as the CC-LRT equations are invariant with respect to unitary transformation involving occupied and virtual orbital separately, we may assume that we have localized the occupied and the virtual orbitals onto the subsystems A and B. H,b, and LY can also be written as additively separable operators. The matrix
zo,as in eq 4.9.
+
Figure 1. Detailed structure of the sub-block RAB. The hatched circle isdicates an H vertex. The label on H indicates the orbital labels on H that can connect the components. The hatched diagonal entries are just the sum of H vertices labeled by either A or B that can cause transitions AiBj AiB,.
-
of R involves three sub-blocks, AA, BB, and AB. AA involves only the subsystem A. BB implies the same for B, and A 3 indicates where there are excitations on both sites. Since is a connected operator, each sub-block is noninteracting with the other, and as a result the values of ~ c ) kfor the noninteracting subsystems will be ut,wf, and wfB-the latter are the values when both subsystems are excited. In ref 22, the nature of LO? has been analyzed, which we recall here. We can have single excitations where a hole and a particle can be on two different sites. We can also have double excitations of the type AB, which are of two categories: in one we have single excitations on each site, in another we have either a hole or a particle on one site and the rest of the excitation on another site. Denoting these various types of mixed excitations A;Bj, where i a n d j refer to total hole or particle occupancies on A and B (example, AlBl one h or p on A, the other p or h on B; A1B3 implies one h or p on A, the rest h-2p or 2h-p on B; etc.), we have shown the detailed structure of the sub-block of AB involving various AiBj in Figure 1. It should be noted that only the entry A2B2 involves single excitations on both sites. Figure 1 shows that the component A2B2 is decoupled from the rest-implying that W? as a sum of single excitations on each site is a possible solution of CC-LRT. However, the other charge-transfer type of excitations are also solutions. ‘For the energies w t and wf, the corresponding Wi and W k are just WIAl@ and WIBIWk, The generate product excitation operators of the form WiABIWk, which may be written as
OF
(5.la)
(5.lb) ;€A jsB
For the system, viewed as the supersystem of the two noninteracting subsystems, the polarizability expression takes the form
+
Do,(ofB w)-’Dko (5.2) keAB
The first sum in eq 5.2 can be written simply as a sum of a$(@)and a:(w). In the second= sum, DO^ and Dm should involve connected terms involving d and 2connecting functions
6446 J. Phys. Chem., Vol. 99, No. 17, I995
Datta et al.
TABLE 1: Correlation Energy and Polarizability Values in au Using Full- and Truncated-CCSD Schemes (A) For LiH
truncated CCSD E,,, = -0.03637
a ,
w
30.60 30.97 32.14 34.32 38.03 44.44 56.76 87.91 309.73
0.0
(e)
(0
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
(g)
Figure 2. (a) to (c): Some typical diagrams of 2. The triangular vertex
corresponds to the dipole operator. The filled-in circle with lines going to the left corresponds to T. The vertices with unfilled squares with lines going to the right are the ijtoperators. The loops involving the square vertices would not have the usual loop-rule contribution. Thus parts a and c of Figure 2 have no factors for the loops, while Figure 2b would have a factor of 2 stemming from the loop containing the_dipole and the T-vertex. Parts d-g show some typical diagrams of d. The vertices with filled-in circles going to the right are the fl operators. The vertices with unfilled squares with lines going to the right are the q: operators. All of the diagrams will have a factor of 2 for each loop.
(401and W p 140). Since W p involves product operators of the form 41’and 4; with i a n d j on separate sites_(eq 5.lb), it is impossible to construct connected quantities do,U and &,o involving operators dp and LY which are labeled as either A or B but not both simultaneously. The second sum in eq 5.2 hence does not contribute to a(o), and we have
indicating both size extensivity and size consistency of our CCLRT formulation. It should be noted that Kobayashi et aLgb have also suggested a formulation for a(w) using the response approach which is also size extensive, but its formulation is different and the resulting expression rather lengthy. Our use of the function ($01 as (401exp(Tt)/(qolqo) has led to a simpler expression of a ( w ) while preserving size extensivity.
6. Applications
A. Truncation Scheme. For the molecular applications, we have used the standard CCSD truncation scheme (T = TI f T2) for representing the ground state IWo). With the H m e e Fock orbitals for the ground state, T I is rather small and we have checked the efficacy of a somewhat truncated version of CCSD where we have included terms involving contractions of H with T I , T2, TlT2, and i$ The differences in the computed polarizability tend to become smaller for larger molecules, justifying the use of the latter truncated scheme for bigger systems. We have included up to 2h-2p operators in the (41) basis, i.e. (41) = (qia €I3 qia,qg} for representing the operators WL and W k . The resulting matrix R is then of the same dimension as in the CI space of h-p and 2h-2p excitations out of $10, with the difference that H , rather than H , figures in the expression for the elements of R,,. The one- and two-body components of ?I are computed and stored, using the expressions previously given by Mukherjee and M~kherjee.~” The threebody terms of H are computed on the fly as required and immediately used in the computation in the step implied in eq
full CCSD E,,, = -0.03645
a,,
a,
a,,
27.19 27.60 28.95 3 1.67 37.08 50.13 118.51
30.87 31.26 32.51 34.87 38.94 46.14 60.68 102.37 1189.39
26.82 27.25 28.65 3 1.57 37.61 54.00 242.82
(B)For HF and HCI HF
HC1
truncated CCSD full CCSD truncated CCSD full CCSD E,,,= -0.21164 E,,, = -0.21065 E,,, = -0.17535 E,,,= -0.17476 w
a,
a,
a,
a,;
a,
a,,
a ,
a,
0.0
5.44 5.57 6.08 7.66
6.68 6.82 7.28 8.27 10.58
5.41 5.55 6.06
6.70 6.85 7.33 8.37 10.87
16.89 17.65 20.83
18.81 19.48 21.84 27.76
16.88 17.64 20.86
18.82 19.49 21.87 27.87
0.1 0.2 0.3 0.4
4.8. For fZ, all of the internal loops yield a factor of 2 (see, for example, section 6B). The expression for d;b involves a terminating series for the transformed operator LY = exp(-T) dp exp(T), and the entire expression has been used in our computations. Figure 2 displays !ome typical diagrams appearing in The expression for &i is, however, a nonterminating series, necessitating a truncation scheme for its evaluation. We have retained in our calculations all terms which are at most linear in Tt and T. For computing the transformed operator LY is computed first, and all of the internal loops contribute a factor of 2. = (401 &LY I+o) is computed next, by contracting the external lines of LY with 4i. There are no factors for the loops created by these final contractions, as explained in section 6B. B. Loop Rules in the Spin-Free Formulation. Some care should be exercised in computing the various expressions involved in &, etc. In the transformed operator LY is computed first, and this involves contractions of dp and exp(T), both of which have only q/qt operators. Since (4:) and {qi} are not bi-orthogonal bases, their contractions lead to the usual “loop rule” for all of the internal loops. When evaluating To,however, we need to contract the i j i operators on the bra side, Le. (&)I&, with all of the external lines of LY. Representing the action of LY on 140)as
zo.
zo,
z0,
z0
z0
e,
I
we find that
=q
(6.2)
and we have no extra loops for the final contractions involving the extemal lines of LY and the lines of iji. Thus, the expression of @ as such becomes the amplitude @o. The case of is, however, quite different. It involves all of the connected terms that can be generated as completely
&
CC-LRT Approach to Property Calculations
J. Phys. Chem., Vol. 99, No. 17, 1995 6447
A Y
-
CC-FULL -CC-TRUNCAED
d
i 25
I 25
0.00
0.05
0.10
0.15
0.00
0.05
FREQUENCY
FREQUENCY (0.u.)
0.10
0.15
(o.u.1
w for LiH
Figure 4. Plot of azL(m)for LiH using full- and truncated-CCSD schemes.
contracted quantities by joining powers of sitting on the left of dp, powers of T sitting on the right of it, and the operator q: coming from the ket q: 1400). Since all of the operators involved contain only q/qt operators, we have the usual loop rules: any term in will have a factor of 2 for each loop in the corresponding diagram. Thus, while one is interpreting the expressions for & and in Figure 2, this aspect should be borne in mind. C. Results and Discussion. We have applied the present formulation to the molecules LiH, BeH+, HF, H20, HC1, and H2S. The dynamic dipolar polarizabilities, a(w),and the static value at OJ = 0 have been computed. In all of our calculations we have used the polarized basis sets or MIDI basis sets proposed by Sadlej2' and by Sadlej and Urban2' which seem especially tailored for the calculation of properties such as polarizability. For the linear molecules, the Z-axis has been taken as the internuclear axis, while for the triatomic molecules the C2 symmetry axis is taken to coincide with the Z-axis of the coordinate system and the molecules lie in the XZ-plane. All of the geometries used here are the ones given by Sadlej.*' In Table 1 we have shown the values of the ground state correlation energy as well as a, and a,, values for the linear molecules LiH, HF, and HCl. Both the full-CCSD results and the truncated-CCSD results have been displayed. As the results indicate, the difference between the full- and truncated-CCSD schemes decreases with the size of the molecule, and it appears that the truncated-CCSD scheme would be a viable method for larger molecules. We display in Figure 3 the plot of variation of axx(w)and a,,(o)for LiH up to the first excitation threshold. Only the truncated-CCSD results are shown. We have chosen
LiH as the example molecule since for this we have found the largest difference in the values for the two schemes, so that we may observe the difference in the trends. Figure 4 shows the plot of a,,(o) for LiH using the full- and truncated-CCSD schemes. The trends are quite similar, although the actual values differ somewhat. For the discussion of our results we have grouped the molecules as 4-electron systems (LiH and BeH+), 10-electron systems (HF and H20), and 18-electron systems (HCl and H2S). We have quoted the dipolar polarizability values, both static and dynamic (wherever possible), calculated by various authors. Though many of these calculations have been done with rather large bases as compared to the ones used by us, we nevertheless give the values here to get a feel for the efficacy of our methodology. However, for all of the molecules, we have compared our results with those obtained by Sadlej et aL2' at various levels of approximation, since we have done all of our calculations with the same basis as theirs. (A) LiH and BeH+. The a,, and az, components of the polarizability for both molecules have been computed up to the first excitation threshold. For w = 0 we have also obtained the static value. The basis used for H is the [6s4p/3s2p] basis of Sadlej et a1.,21while for the Li and Be atom the basis is of type [lOs6p4d&3p2d]. It should be mentioned here that the first s exponent of H has been corrected to 33.685014 in place of 33.865014 (see ref 21d). In Table 2A we report the static polarizability values for both molecules, together with other pertinent results for comparison. For LiH our static a, value of 30.6 au and azZvalue of 27.19 au compare favorably with the CCSD values of 29.98 and 26.73
Figure 3. Plot of variation of a, and up to the first excitation threshold.
%[
3,
azzwith frequency
Datta et al.
6448 J. Phys. Chem., Vol. 99,No. 17, 1995
TABLE 2: Static Polarizability Values in au (A) For L M and BeHt
a, molecule
a,,
present
others
present
others
LiH
30.60
27.19
BeH+
11.96
(29.98, 30.01, 27.40, 28.32, 28.94)," 29.40,b 29.76,' 37.31,d 29.30' (1 1.923, 11.93, 11.21, 1 1.48, 11.65)"
(26.73, 26.81,24.10, 24.91, 25.49)," 22.96; 26.36,' 31.98: 26.30' (15.34, 15.38, 13.66, 14.21, 14.58)"
15.43
(B) For HF and HzO
a, molecule present HF
H20
5.44
10.35
others
aY, others
present
5.03;f5.37,*5.33,g5.18,h5.28,' 4.94,' (4.93,4.38)," (5.33,5.32),' (5.25, 5.34)m 10.31 f 0.01," 10.17," 10.09,g 9.87," 9.81,'9.48,'(10.05, 10.06),' 10.13: (10.03, 10.36)q (9.85, 9.71)'
9.59
9.91
a,,
0.010,0.020
9.55 f 0.09," 9.59,"" 9.57,g 9.30," 8.94,' (9.70,9.71),' 9.76,P (9.00, 9.74),9 (9.39,9.16),'8.98'
present
others
6.68
6.51,"f6.50,g6.45,h6.56,'6.10,J
10.02
(5.80, 5.70): 6.40,'(6.43, 6.44)m 9.91 f 0.02," 9.82," 9.84,g.O 9.64,' 9.22; 9.71,' 9.93: (9.94, 9.93),9 (9.50,9.35),' 9.22s
(C) For HCl and H2S
an
a,,
molecule
Present
others
present
HCl
16.89
H2S
24.43
(16.77, 16.60, 16.83)," (16.95, 16.74, 16.48, 16.65),'(17.37, 17.22, 17.14, 17.20)" (24.21, 24.04, 24.16)"
25.23
az: others
(25.14, 24.92, 25.13)"
present
others
18.81
(18.44, 18.34, 18.36): (18.51, 18.46, 18.28, 18.42),r (1 8.75, 18.63, 18.58, 18.64)" (24.20, 24.02, 24.16)"
24.49
a Reference 21. Reference 19. Reference 23. Reference 24. e Reference 25. f Reference 26. g Reference 27. Reference 28. ' Reference 29. Reference 30. Ir Reference 31. Reference 32. Reference 12. Reference 34. Reference 35. P Reference 36. 9 Reference 37. Reference 38. Reference 39. ' Reference 40. * Reference 41.
'
+
au, respectively, of Sadlej et a1.21 Since the CCSD T(CCSD) values (approximate inclusion of triples) are rather close to the CCSD results-being 30.01 and 26.81 au, we surmise that an approximate inclusion of T3 in a CCSD T(CCSD) scheme in our formulation would not significantly change our results. This would also probably be true for the dynamic values as well. The MBPT values of Sadlej et alV2I reported at various orders, are 27.40 and 24.10 au (MBPT-2), 28.31 and 24.91 au (MBPT3), and 28.94 and 25.49 au (MBPT-4). The difference between their MBPT-3 and MBPT-4 results also seems to indicate the relative smallness of the triples. Sasagane et al.I9 have calculated the dipolar polarizability using the MCTDHF approximation at various intemuclear distances. Their values at the experimental bond lengths, R = 3.015 au, the one used by us, give a, = 29.40 au and a, = 22.96 au with a [4~2pld/2slp]basis. An MCSCF calculationz3 with a [7s5p2d/4s4p] basis yields a, = 29.76 au and a,, = 26.36 au. Hyams et aLZ4have calculated the polarizability using perturbed spin-coupled wave functions. They have made a detailed basis set study. Their largest basis set (46 CGTFs), an [8s5p/8s5p] basis, gives a,, = 37.31 au and a, = 31.98 au. Roos and Sadlej,25who have used CASSCF wave functions, obtained a,, = 29.3 au and a,, = 26.3 au. For the dynamic polarizability, Sasagane et al.I9 have given a plot of a(@) versus the frequency w up to the third excitation threshold, at the experimental intemuclear distance. A comparison of our values at various frequencies with their plot shows a very close agreement. For the molecule BeHf, our static values are 11.96 and 15.43 au for a,, and a,,, respectively. Due to the very limited data for this molecule, we only give the values obtained by Sadlej et aL2' at various levels of approximation. Their a, and a, values at the CCSD and CCSD T(CCSD) approximation are 11.92 and 11.93 au for a, and 15.34 and 15.38 au for aZ2. The MBFT results are 11.21 (MBPT-2), 11.48 (MBPT-3), and 11.65
+
+
au (MBPT-4) for a, and 13.66 (MBPT-2), 14.21 (MBPT-3), and 14.58 au (MBPT-4) for a,,. Figure 5 gives a plot of the variation of ax,and a,, as a function of w for BeH+ upto the first excitation threshold. Only the truncated-CCSD results are plotted. (B)HF and HzO. For HF we have calculated the a, and ar,components of the polarizability, while for H20 the a,, a,,, and a,, components have been obtained. The basis for H is again [6s4p/3s2p], and those for fluorine and oxygen atoms are [ lOs6p4d/5s3p2d]. In Table 2B we give the static polarizability values for both the molecules together with other pertinent results. For the HF molecule our static a,, value is 5.44 au and a,, is 6.68 au, which compares well with the experimental values26 of a, = 5.03 au and a,, = 6.51 au Sadlej et aL2I who have carried out a fourth-order MBPT calculation, give a,, as 5.37 au and a,, as 6.51 au. A similar calculation by Diercksen et al.27yields a, = 5.33 au and a,, = 6.50 au. An SDQ-MBFT(4) calculation of Bartlett and Purvis2*gives a, = 5.18 au and a,, = 6.45 au. The older CEPA results of Wemer and M e ~ e r , ~ ~ who have used a rather large basis, give ax,= 5.28 au and a,, = 6.56 au. A time-dependent perturbation theory20 using MCSCF wave functions yields for a, 4.94 au and for a,, 6.10 au. Guan et aL3I have made a density functional calculation. Their values for a, and a,, are 4.93 and 5.80 au, respectively. Their CHF calculations yields a,, = 4.38 au and azI= 5.70 au . Recently Pluta et al.32 have used the finite field method, making a numerical differentiation of the total perturbed energy to yield the polarizability. Their CCSDT- 1 calculation correct through second-order gives a, = 5.33 au and a,, = 6.40 au. A full fourth-order CCSDT-1 calculation yields a, = 5.32 au and a,, = 6.40 au. Here a point to be noted is that they have used the MIDI basis of Sadlej et aL2' with slightly modified p and d shells, however, for F, where they have lifted the contractions,
J. Phys. Chem., Vol. 99, No. 17, 1995 6449
CC-LRT Approach to Property Calculations
TABLE 3: Dynamic Polarizability Values in au for €IF
2601
(A) For HF
a , present
ref 30
present
ref 30
0.0
5.44 5.57 6.08 7.66 22.07
4.94 5.04 5.40 6.36
6.68 6.82 7.28 8.27
6.10 6.21 6.58 7.36
21.00
10.58 22.69
9.04
0.1 0.2 0.3 0.37 0.4 0.5
213
a::
w
(B) For HzO
a, present
w
others
present
0.0 10.35 0.04
10.64 (10.11, 9.96); 9.71"
0.12 0.15 11.03 10.02" 0.2 11.63 10.50," (10.99, 10.81),b 10.71'
9
0.00
G.20 FREQUENCY(O.U.)
0.10
030
Figure 5. Plot of variation of an and a, with frequency w for BeH+ up to the first excitation threshold. and similarly on the p shell for H. Despite the small difference in the basis set quality, our results compare favorably with theirs. Sekino and BartlettI2 have earlier reported an a, values 5.25 and 5.34 au and a,, values of 6.43 and 6.44 au. The latter work employs the MIDI basis of Sadlej et aL2' Calculations for the dynamic polarizability are rather limited; however, a few calculations for the HF molecule are available. Reinsch30 has reported the a, and a, values at a number of frequencies. In Tables 3A we have given our results alongside other available data. Our values compare reasonably with those of R e i n ~ c h . ~ ~ For the H20 molecule, the static a,, a?,, and a,, values computed by us are, respectively, 10.35, 9.91, and 10.02 au. The experimental values34are, respectively, 10.31 f 0.01,9.55 f 0.09, and 9.91 f 0.02 au. The MBPT(4) result of Sadlej et aL2' for a,, is 10.17 au, that of a?, is 9.59 au, and a,, is 9.82 au. Those of Diercksen et aLZ7are a,, = 10.09 au, a,, = 9.57 au, and a,, = 9.84 au. The SDQ-MBFT(4) approximation of Bartlett and pur vi^^^ yields a,, = 9.87 au, ayJ= 9.30 au, and a,, = 9.84 au. The CEPA29results give a,, = 9.81 au, a?, = 9.59 au, and a,, = 9.64 au. Reinsch30 has reported a,, as 9.48 au, a,, as 8.94 au, and azzas 9.22 au. However the CCSDT-1 (up to second order) of Pluta et al.32are a,, = 10.05 au a,, = 9.70 au, and a,, = 9.71 au, while their complete CCSDT-1 calculation yields a,, = 10.06 au, a,, = 9.70 au, and azz= 9.71 au. Clearly the effect of inclusion of triples does not significantly change the values. M a r o u l i ~has ~ ~ carried out a complete fourth-order MBFT calculation which gives a,, = 10.13 au, a,, = 9.76 au, and a,, = 9.93 au using a rather large basis comprising of 84 CGTFs. A multireference singles and doubles CI calculation of Spelsberg
9.940 13.47 11.18,"(12.25, 11.87),* 11.36'
10.38 (9.82, 9.66),* 9.51" 10.75 9.93" 11.79 10.63," (1 1.02, 10.83),* 10.63'
a , present
0.0
16.89
a
9.29"
(C) For HCI
w
0.006 0.01 0.1 0.2 0.3
others
10.02
9.03" 10.25 10.46 (9.86, 9.61): 9.33"
0.08
10
present
9.91 9.99
0.05 10.42 9.53" 0.1
a,
a>> others
a,, ref 40
present 18.81
16.84 17.05 17.65 20.83
ref 40 18.54 18.73
19.48 21.84 27.76
Reference 30. Reference 38. Reference 39.
and Meyer3' gives a, = 10.03 au, a,, = 9.00 au, and a,,= 9.44 au. A CEPA calculation done by the same authors yields a,,= 10.36 au, a?, = 9.74 au, and a,, = 9.93 au. A complete active space (CAS) calculation of Sanchez et al.38 using two sets of active orbital (labeled CAS-1 and CAS-2) yields for CAS-1 a, = 9.85 au, a,, = 9.39 au, and a,, = 9.50 au, while the CAS-2 calculation gives a, as 9.71 au, a,,as 9.16 au, and a,, as 9.35 au. Using the same active space as in CAS-2, Jorgensen et al.39gives a, = 9.64 au, a,, = 8.98 au, and a,, = 9.22 au. Sanchez et al.38have also computed the dynamic polarizability at two frequencies, w = 0.1 and 0.2 au. We give in Table 3B our dynamic polarizability values, comparing them with other available data.38,39Our values at the frequency 0.1 au are a,, = 10.64 au, a,, = 10.46 au, and a,, = 10.38 au. Sanchez et al.,38using CAS-1, give a,, = 10.11 au, a,, = 9.86 au, and a,, = 9.82 au, while the CAS-2 values are 9.96, 9.61, and 9.66 au for a,,, a,, and azl,respectively. Jorgensen et al.39give at w = 0.2 au a, as 10.71 au, a,, as 11.36 au, and a,, = 10.63 au, whereas our values are 11.63, 13.47, and 11.79 au for axx, a,,, and azz,respectively. In Figures 6 and 7 we plot the variation of all the components of a with w for HF and H20,respectively. Only the truncated CCSD results are plotted. (C) HCI and H2S. Here again we compute ax,and a,, for HC1 and a,,, a,,, and a;, for H2S. We give the static polarizability values for both molecules in Table 2C. The basis sets used for C1 and S are [14slOp4d7~5p2d].It may be mentioned here that the exponent on the s function
6450 J. Phys. Chem., Vol. 99, No. 17, 1995
Datta et al.
17
15 ?
?
0
v
z=! m
4
13
3
e
B
5
11
d
0.00
0.20
0.40
0.00
FREQUENCY (a.u.) Figure 6. Plot of variation of an and a: with frequency w for HF up to the first excitation threshold.
for the C1 atom has been corrected to 44.795 instead of 47.795.2'd The basis on H is the same as used before. Our results for HC1 are a, = 16.89 au and a,, = 18.81 au. The many body perturbation calculations at various orders by Sadlej et al.*' give at MRPT-2 a, as 16.77 au and a, as 18.44 au; the MBPT-3 results are a, = 16.66 au and a, = 18.34 au, while the fourth-order results are a, = 16.83 au and a,, = 18.36 au. Hammond and Rice40 have made an extensive basis set study for HCl using the second-order Mbller-Plesset perturbation theory (MP2) for the electron correlation. They have also carried out a singles and doubles excitation coupled-cluster calculation (CCSD) and also a CCSD with a perturbative estimation of connected triple excitations (CCSD(T)). Their largest basis yields for the MP2 calculation a, as 16.95 au and a,, as 18.51 au. A medium-sized basis (designated A4) gives for MP2 a, as 16.74 au and a,, as 18.46 au. Their CCSD results are 16.48 and 18.28 au for a, and a,, respectively, while their CCSD(T) results are a, = 16.65 au and a, = 18.42 au. Maroulis4' has also carried out a Mbller-Plesset perturbation theory calculation at various R. At the internuclear distance, R = 2.4086 au, the same as used by us; their MP2 values for a,, and aZzare 17.37 and 18.75 au. Their MP3 results are a, = 17.22 au and az,= 18.63 au. A DQ-MP4 calculation yields a,, = 17.14 au and a,, = 18.58 au. The SDQ-MP4 calculation gives a, = 17.20 au and a,, = 18.64 au. Hammond and Rice40 have also calculated the dynamic polarizability at two frequencies, w = 0.0055 and 0.01 au. However, as our calculation does not include these two frequencies, we could not make a direct comparison. Nevertheless, we have given these values along with those obtained by us in Table 3C.
9
0.00
G.20 FREQUENCY (a.u.1
0.10
0.so
Figure 7. Plot of variation of a, and azzwith frequency w for H20 up to the first excitation threshold.
For the H2S molecule, polarizability calculations are rather scarce. Our calculation yields for the static case a, = 24.43 au, a,, = 25.23 au, and a,, = 24.49 au. The values of Sadlej et a1.*' at various levels of approximation are as follows: MBPT(2) a, = 24.21 au, a,, = 25.14 au, and a,, = 24.20 au; MBPT(3) a, = 24.04 au, a,, = 24.92 au, and a,, = 24.02 au; MBPT(4) a,, = 24.16 au, a,, = 25.13 au, and a,, = 24.16 au. The two sets of results are clearly comparative. Since to our knowledge there are no previously reported dynamic polarizability values for this molecule, we could not make any comparison with the results obtained by us. Figures 8 and 9 give the variation of a,, and a,, for HC1 and a,, a,,, and a,, for H2S with w,respectively. As is evident from the various results presented, the role of triples is not very significant for polarizability calculations for the molecules studied by us. However, we are at present carrying out calculations for the polarizability with the inclusion of triples to get a feel of the effect of this inclusion in our calculation for a wider set of systems. 7. Concluding Remarks
In this paper, we have described in detail a coupled-cluster based linear response theory for computing second-order properties-in particular the dynamic polarizability and its static limit. The method leads to a size-extensive and size-consistent expression for the polarizability and would thus remain viable for large systems. The method has been applied to compute the components of a ( w ) for several molecules. The results are very encouraging, which indicates that CC-LRT will turn out to be a powerful method for computing second-order properties. Generalizations to encompass nonlinear properties are underway and will be communicated in the near future.
CC-LRT Approach to Property Calculations
J. Phys. Chem., Vol. 99, No. 17, 1995 6451 dedicate this paper to Professor C. N. R. Rao on the happy occassion of his 60th birthday.
References and Notes
15
0.00
C.20
0.10
0.50
FREQUENCY (0.u.)
Figure 8. Plot of variation of an and up to the first excitation threshold.
0.00
0.10
a,,with frequency w for HC1
0.20
0.50
FREQUENCY (a.u.1
Figure 9. Plot of variation of an and a, with frequency w for H2S up to the first excitation threshold.
Acknowledgment. We thank the University Grants Commission (New Delhi) for financial support. It is a pleasure to
(1) Cizek, J. J. Chem. Phys. 1966, 45, 4256. Paldus, J.; Cizek, J.; Shavitt, I. Phys. Rev. A 1972, 5, 50. (2) Bartlett, R. J. Annu. Rev. Phys. Chem. 1981,32,359 and references cited therein. (3) See, for example: (a) Lindgren, I.; Mukherjee, D. Phys. Rep. 1987, 151, 93. (b) Mukherjee, D.; Pal, S. Adv. Quantum Chem. 1989, 20, 291. (c) Paldus, J. Methods in Computational Molecular Physics; Wilson, S., Diercksen, G. H. F., Eds.; Plenum: New York, 1992. (4) See the entire issue of Theor. Chim. Acta 1991, 80, 71-507, for an exhaustive survey of the various facets of the coupled-cluster approach. ( 5 ) See, for example: (a) Urban, M.; Cemusak, I.; Kello, V.; Noga, J. Methods of Computational Quantum Chemistry; Wilson, S., Ed.; Plenum: New York, 1987; p 1. (b) Jankowski, K. Methods in Computational Quantum Chemistry; Wilson, S., Ed.; Plenum: New York, 1992; p 5. (6) Sekino, H.; Bartlett, R. J. Int. J. Quantum Chem. 1984, S18, 255. Commeau, D. C.; Bartlett, R. J. Chem. Phys. Left. 1993, 207, 414. (b) Maroulis, G.; Thakkar, A. J. J . Chem. Phys. 1990, 93, 652. (c) Castro, M. A.; Canuto, S . Phys. Rev. A 1993, 48, 826. (7) (a) Monkhorst, H. J. lnt. J . Quantum Chem. 1977, S11, 403. (b) Dalgaard, E.; Monkhorst, H. J. Phys. Rev. A 1983, 28, 1217. (8) (a) Koch, H.; Jorgensen, P. J . Chem. Phys. 1990, 93, 3333. (b) Kobayashi, R.; Koch, H.; Jorgensen, P. Chem. Phys. Lett. 1994, 219, 30. (9) (a) Mukherjee, D.; Mukherjee, P. K. Chem. Phys. 1979, 37, 325. (b) Ghosh, S . ; Mukherjee, D.; Bhattacharyya, S . Chem. Phys. 1982, 72, 961. (10) Kundu, B.; Mukherjee, D. Chem. Phys. Lett. 1991, 179, 468. (11) (a) Stanton, J. F.; Bartlett, R. J. J . Chem. Phys. 1993, 98, 7029. (b) Geertsen, J.; Rittby, M.; Bartlett, R. J. Chem. Phys. Lett. 1989, 164,57. (12) (a) Geertsen, J.; Oddershede, J. J. Chem. Phys. 1986, 85, 2112. (b) Geertsen, J.; Eriksen, S.; Oddershede, J. Adv. Quantum Chem. 1991, 22, 167. (13) Prasad, M. D.; Pal, S.; Mukherjee, D. Phys. Rev.A 1985,31, 1287. (14) Mukherjee, D.; Kutzelnigg, W. In Lecture Notes in Chemistry; Kaldor, U., Ed.; Springer Verlag: Heidelberg, 1989; Vol. 52. (15) Datta, B. Mukhopadhayay, D.; Mukherjee, D. Phys. Rev. A. 1993, 47, 3632. (16) See, for example: (a) Arponen, J. Ann. Phys. 1983, 151, 311. (b) Arponen, J.; Bishop, R. F.; Pajanne, E. Phys. Rev. A. 1987,36,2519; 1987, 36, 2539. (17) (a) Rice, R.; Handy, N. J . Chem. Phys. 1991, 94,4951. (b) Rice, R.; Handy, N. Int. J. Quantum Chem. 1992, 43, 91. (18) Wormer, P. E. S.;Hettama, H. J . Chem. Phys. 1992, 97, 5592. (19) Sasagane, K.; Aiga, F.; Itoh, R. J. Chem. Phys. 1993, 99, 3738. (20) Iwata, S. Chem. Phys. Lett. 1983, 102, 544. (21) (a) Sadlej. A. Collect. Czech. Chem. Commun. 1988.53, 1995. (b) Sadlej, A.; Urban, M. J . Mol. Strucf. 1991, 234, 147. (c) Sadlej, A. Theor. Chim. Acfa 1991, 79, 123. (d) Sadlej, A. Private communication. (22) Mukhopadhyay, D.; Mukhopadhyay, S.; Chaudhuri, R.; Mukherjee, D. Theor. Chim. Acta 1991, 80, 441. (23) Bishop, D. M.; Lam, B. Chem. Phys. Lett. 1985, 120, 69. (24) Hyams, P. A,; Gerratt, J.; Cooper, D. L.; Raimondi, M. J . Chem. Phys. 1994, 100, 4417. (25) Roos, B. 0.; Sadlej, A. J. Chem. Phys. 1985, 94, 43. (26) Zeiss, G. D.; Meath, W. J. Mol. Phys. 1977, 33, 1155. (27) Diercksen, G. H. F.: Kello, V.: Sadlei. A. J. Chem. Phvs. Lett. 1983. 95, 26. Diercksen, G. H. F.; Roos, B. O.;"Sadlej, A. J. In;. J . Quantum Chem. 1983, S17, 265. (28) Bartlett, R. J.; Purvis, G. D., 111. Phys. Rev. A 1979, 20, 1313. (29) Wemer, H. J.; Meyer, W. Mol. Phys. 1976, 31, 855. (30) Reinsch, E. A. J . Chem. Phys. 1985, 83, 5784. (31) Guan, J.; Duffy, P.; Carter, J. T.; Chong, D. P.; Casida, K. C.; Casida, M. E.; Wrinn, M. J . Chem. Phys. 1993, 98, 475. (32) Pluta, T.; Noga, J.; Bartlett, R. J. Int. J . Quantum Chem. 1994, S28, 379. (33) Jorgensen, P.; Oddershedde, J.; Albertsen, P.; Beebee, N. H. F. J. Chem. Phys. 1978, 68, 2533. (34) Murphy, W. F. J. Chem. Phys. 1977, 67, 5877. (35) Bartlett, R. J.; Purvis, G. D., 111. Phys. Rev. A 1981, 23, 1594. (36) Maroulis, G. J . Chem. Phys. 1991, 94, 1182. (37) Spelsberg, D.; Meyer, W. J. Chem. Phys. 1994, 101, 1282. (38) Sanchez, A. M.; Jensen, H. J. Aa.; Jorgensen, P. Chem. Phys. Lett. 19991, 186, 379. (39) Jorgensen, P.; Jensen, J. J. Aa.; Olsen, J. J . Chem. Phys. 1988, 89, 3658. (40) Hammond, B.; Rice, J. E. J . Chem. Phys. 1992, 97, 1138. (41) Maroulis, G. Mol. Phys. 1991, 74, 131. JP9424 17V
