Indian Journa l of Chemistry Yoi.39A. Jan-March 2000, pp. 196-200

Calculation of vibrational spectra by the coupled cluster method -Applications to H2S

M Durga Prasad School of Chemistry, University of Hyderabad,

Hyderabad 500 046, India Emai l : mdpsc @uohyd .ernet.in

Received 5 Novemba / 999; accepted 17 December / 999

The coupl ed cluster method is used to calculate the spectra l intensities and mean displacement va lues of the normal coordinates for H2S. Good agreement is found between the CCM resu lt s truncated at S4 level and converged variational resu lts.

1. Introduction Several methods have been developed in recent years

for solving the Schrodinger equation associated with anharmonic molecul ar v ibrations 1 · 1 ~. Semic lass ica l me-chani cs has been used as the basis of several of such approaches 1 -~ . Quantum mechanica l self-consistent field approximation has been in voked by some authors4-7. The use of different coordi nate systems and the poss ibility of carrying out variationall y converged configurat ion interacti on (Cf) studies in an appropriate basis has been explored by several workers in the fie ld7-D While ac-curate, such procedures require the constructi on and sub-sequent cliagonali zation of large matrices whose size in-creases ex ponentiall y with the number of degrees of free-dom . This bas is set bottl eneck provides the motivation fo r the development of alternati ve approaches such as the canonica l van Vleck perturbation theory (CYPT) 1 ~ that do not require so much of the computer resources.

Some time ago, we suggested the use of the coupled cluster method (CCM) for the description of anharmonic molecular vibrat ions15.It had been used ex tensively in the electronic structure theory with considerab le suc-cess1 r,_20 . Its applicat io-ns to vibrati onal systems have been fewer, and were mostly limi ted to its convergence prop-etties in one-d imensional anharmonic osc ill a t ors 2 1 - 2 ~. The CCM posits an ex ponenti al ansa tz to the wave op-erator th at transforms the reference wave function to the exact ground state. Earlier studies have show n that thi s non- I in ear representation of the wave operator leads to a better convergence pattern fo r the ground state energy in terms or the number of cluster matrix elements than

the CI approach which uses a linear basis set expansion in the Hilbert space. The transiti on energies have been obtained via the coupled c luster linear response theory25 -27 (CCLRT) in all these works. The overall qual-ity of the results was found to be quite impressive par-ticul arl y for relatively low values of the truncation pa-rameters.

The goal of the present work is to calculate the spec-tral intensities by the CCM approach along with the mean displacements of the normal coordinates in different states to see if they too are reproduced to the same leve l of accuracy as the transiti on energ ies . This question assumes particular sign i fica nee si nee as Kal fus and Altenbokum pointed out , the CCM wave functi on for the vibrational systems is not normali zab l e2~ if the clu s-ter operator is truncated beyond three boson level. Con-sequently, it is not possible to evaluate the expectation va lues or the transition mat ri x e lements by the t rae! i-ti onal wave function route. However, within the frame-work of CCM such quantiti es can be evaluated as ex-pectati on values or transiti on matrix elements of effec-tive operators2x. The requisite theory is rev iewed in Sec-ti on 2. Computational results for H

2S are presented in

Secti on 3. It is found that not withstanding the loss of normali zability of the wave functions , the expectation va lues and spectral intensities are well reproduced by the CCM approach.

2. Theory Ignoring all rotation-vibration interactions, the vibra-

tional hami ltonian for a molecu le of N vibrational modes

PRASAD: YIBRATIO AL SPECTRA OF H1S BY COUPLED CLUSTER METHOD 197

can be written as H = L p,2/2 + V (q) . (2. 1)

Here, q; are the mass weighted normal coordinates, and P; are the associated momenta. We assume that the po-tential energy V can be approximated by a quartic poly-nomial in the normal coordinates.

... (2 2)

A few comments on the nature of the hamiltonian (2.1 ) are in order. The full vibrational hamiltonian in normal coordinates includes. in addition to the terms in (2. 1 ), Cori oli s coupling and the Watson term. Though small , the Corio! is coupling is known to mi x states of a given poly ad strongly at higher orders. However, our purpose is to assess the reliability of the CCM rather than repro-duce experimental data accurately. Consequent ly, this approx imation to the vibrational Hamilton ian should not have any effec t on the final conclusions. We note in passing, that the Coriolis cou pling for 1 = 0 states can 3lso be treated by the CCM approach at the same level as the quarti c part of the potential since v n,iolis contains operators of the type qipiqkp1• Extensions to 1

1 0 how-ever require inclusion of additional degrees of freedom corresponding to rotation . but the bas ic approach rema ins the same.

The coupled cluster approac h to molecul ar vibrat ional energy levels consists or three steps. The first step is the construction of a reference gaussi an fu ncti on for the ground state variati onally.'"

I 0 > = cxp 1-S w, (q, - q , ")~ 1.

The harmoni c osc ill ator ladder operators a; and a;+ are now defined with respect to thi s optimi zed vacuum stale.

a, = Ow,f2 (q, - q," + d/dw,q,) , (2.4a)

a,+ = Ow,f7. (q, - q," - d/dw,q,). (2.4h)

and the Hamiltonian is rewritten in terms of these op-eratoJ s. By definiti on, the vacuum state of Eq.(2.3) sat-isfi es

a, I 0 > = 0 . = cxp (S) I 0 > , '" (2.6) and the cluster operatorS is expanded as

(2 7)

Substituting Eq. (2.6) in the Schrodinger equation for the ground state

. .(2 .8)

premultiplying by ex p (-S) and projecting on various n-boson states one obtains in the usual fashi on

.. . (2 .9a)

. (290)

where

He~, = ex p (-S). H.exp (S). ...(29c )

Eq. (2 .9b) represents an infinite set of cou pled non-lin-ear equations which on so lu tion yield the cluster matri x elements { s,} or Eq.(2.7). The exact ground state en-ergy E11 is obtained directly from Eq. (2 .9a) once s; arc known.

In the final step, the excited state energies are obtained by using the coupl ed cluster linea r response theory 25·27

Formally, one writes the exc ited states as the resul t of the ground state wave operator acting on a linear combi-nati on of the n-hoson states.

le,> = exp (SJ n, 10> . (2. I Oa)

. .. (2.10b)

The coeffi cients in the linear ex pansion and the transi-tion energies are then obtained by in vok ing the eq uati on of moti on fo r the exc itat ion operator.

... (2.10c)

In pract ice, thi s in vo lves diagonali zing Hor1

in the n"(= L:n.) :;tO boson state manifold2x. Since H r· isj· ust a sim i-

' l' l

la rit y tran sfo rm of th e ori g inal Ha miltoni an (v ia Eq. (2.9c)), its e igenvalues are identical to th ose of the ori ginal Hamiltoni an.

We nex t turn to the calculation of the expectari on val-ues and spectral in tensities by the CCM approach. While the eigenvalues of H and H rr are identical the ir einen-

........ ~ ' 0

vectors are not. However, they are reLtted through the

198 INDIAN J CHEM, SEC. A , JAN - MARCH 2000

same similarity transformation used to construct the ef-

fective Hamiltonian Herr(ref.28). Since Herr is not mani -festly hermitian , its left and right e igenvectors

.. . (2. 11 a)

... (2. 11b)

are not identical. They are re lated to the e igenvectors of the original Hamiltonian vi a the re lations

I 'Pi> = exp (S) I Ri >N,. . .. (2. 12a)

< 'Pi I= N1< Li I exp (-S) , .. . (2. 12b)

upto some normalization constants N, and Nr Conse-quently, the expectation va lue of an operator 0 for a g iven state \}'; can be written as

if IL> and IR >are normalized such that I I

... (2 .14)

Here,

Ocrr = cxp (-S).O.exp (S). ... (2. 15)

is the effecti ve operator assoc iated wi th 0 . Similarl y, the square of the absolute transition matri x e lement (rep-

resenting, e.g. the spectra l intensity) is given by

... (2. 16)

Equations (2. 13)-(2. 16) a re the working equations for obtaining the expectation values and spectral intensi-

ties in the CCM approach .

3. Results and Discussion We have app lied the CCM to ca lculate the vibrational

levels, assoc iated spectra l intens iti es, and the mean di s-placement values of the normal coordinates of H,S mol-ecule. It is a typ ical local mode molecule due to the large di sparity in the masses of the central and te rminal atoms. As a consequence, the two stretch modes are nearl y degenerate and coupl e strong ly in hi ghe r orders. In addition, due to the re lative ly low barrie r to in ver-sion, the bending mode is quite anharmoni c . Thus this molecule provides a stringent test for the app licability of the CCM. The potential energy surface is take n from

the work of Kuchitsu and Morino. 29 It is an empirical surface and does not contain Coriolis terms. Based on our earlier studies 1\ we have restricted the cluste r op-erator S to conta in at most four c reation ope rators throughout these calculations. The non-linear equ ati ons (2.9b) were solved by quasi- linearli zation. The expan-sion for the excited states was also limited such that states containing more than four bosons were not included in it. In addition, a ll te rms in Horr that contain more than four ladder operators we re neglec ted. The coordinate

operators q; are sing le boson operators. Consequentl y, the corresponding effective operators contain no more than three ladder operators when S is restricted to S

4.

Since these are of lower rank than the effective Hamil -tonian , we have retained it in full. The dipole moment function of H

2S is not known . Since a ll the three modes

are IR-acti ve, we have taken the dipole operator as a linear combinati on of the normal coordinate operator . The spectral intens ity for the tran siti on between the ground state and nth exc ited state \}'" is defined as

.. . (3 . 1)

It was then calcul ated by Eq . (2. 16). As noted in the introduction, our goal is to tes t the

re liab ility of the CCM approach to the molecular vibra-tion e igenstates. A compari son w ith experimental data would not be appropri ate for this purpose since defic ien-c ies in the potentia l energy surface itse lf could be a ma-jor source of error. For thi s reason , we have compared the CCM results aga inst converged vari ational results rather than ex perimental results. The varia tional calcu-lations were carried out in harmonic oscillator bas is and converged when 6, 12, and 6 bas is functions were used for the three modes respective ly.

The results ofCCM calculation of the vibrati onal tran-s ition energ ies and the associated spec tral intensit ies are presented in Tabl e I and compared with nume rica ll y converged C l va lues . O nl y those states whose spec tral intens ity is greate r than 0.001 are inc luded in the list. As can be seen, the CCM performs quite we ll . The larg-est e rror in the transit ion energ ies is about 19 cm·1 for the state I 0 I . Sandwiched as it is between 02 1 and 031 states , this state is on the border of the basis se t we have used for the const ruc ti o n of the excited states, and as such can be ex pected to have the largest e rror. T he rror in the spectra l intensities is a lso quite small -the largest be ing about four percent for the state 0 I 0. Even the small intensiti es in the overtone and combinati on bands

PRASAD: YIBR ATIO AL SPECTRA OF H2S BY COUPLED CLUSTER METHOD 199

Table I - Vibrational transi tion energies and spectral intensities of H2S

- I -I State w CCM (em ) w CI (e m ) PCCM Per

010 11 72 11 72 0.500 0.5 18 020 2339 2336 0.008 0.009 100 2629 2630 0.49 1 0.500 001 264 1 2642 0.497 0.508 200 5244 5228 0.004 0.004 101 52 18 5237 0.006 0.004 002 5280 5275 0.002 0.00 1

Table 2 - Mean displacements of normal coordinates in different states or H2S

State ccM « -Jr >CI CCM CI

000 1.738 1.734 0 10 1.485 1.470 020 1.1 88 1. 16 1 100 3. 53 1 3.555 030 1.742 0.81 7 11 0 3.262 3434 040 1.829 0.465 120 2.196 3.230 200 4.683 5. 11 8 002 5.008 5.286 00 1 3.539 3.5n 011 3.3 15 3.386 02 1 1.882 3.217 10 1 5.066 5.050

are fairly well reproduced by the CCM . A curious trend is not iceable in these results. The intensit ies of the fun -damental bands are consistentl y underes ti mated , whil e the intens ities of the overtone and combination bands are generally overest imated. Since we have approx i-mated the dipol e operator to be linear in the normal co-ordin ates, mechani cal anh armonicit ics alone are respon-sible fo r these intensity borrowing effec ts. In terms of the wave function s the present results impl y that the CCM is underest imat ing the component of the one boson states in the fundamental states , wh il e overestimating it in the hi gher states. We have noticed a similar trend in other molecules al so implying that it is a general tendency in the coupl ed cluster approach.

The results of CCM ca lcul ation of the expectat ion va lues of the nom1al coordinates for some of the low lying states are presented in Table 2. As can be seen the CCM performs quite we ll except in the case of a few states. The state 040 is a typica l example_ Orthogonal ity to the lower states forces thi s state to have such mean displacements . Since, it is the last state in manifold of

0.052 0.053 0.089 0.086 0. 125 0.130 0.3 17 0.32 -0 034 0.197 0.3 14 0.349 -0.042 0.302 -0.4 16 0.355 0.5 12 0.539 -0.22 1 -0. 13 1 -0.106 -0. 108 -0 040 -0.08H 0.288 -0.06') 0. 143 0. 179

the bending mode exc itation, it lacks the fl ex ibility to pass on the burden of orthogonality to the coefficients of hi gher states. In our earlier work , we had speculated that the large errors in the energies of some states are due to such incorrect descr ipti on of the mean di spl ace-ments. The CCM overestimates the energy of the 040 state by 46 cm-1, thu s providing support to our specula-ti on.

In summary, we have ca lcul ated the spectral intensi-ties and mean displ acements of normal coordinates of H2S by the CCM approach. There is a good agreement between the CCM results and variati onally converged cr results giving ri se to the ex pectati on that the method can be used to simulate vibrational spectra. Th is is en-cmu·aging because, the computati onal resources required to carry out the CCM calculations is qu ite small. The method is parti cularl y attractive for app licat ions to large systems because the cluster operator is additively sepa-rabl e and hence the com putat ional requirements do not increase exponenti ally as in the case of C[ methods .

The other aspect of the present study is regardi ng the

200 I DIAN J CHEM , SEC. A, JAN - MARCH 2000

CCM approach itself. As mentioned in the introduc-tion , the wave function implicit in the CCM approach has infinite norm if the cluster operator is allowed to have terms beyond the three boson creation operators. Such a wave function cannot describe the properties of a bound state such as a vibrational states. The present study indicates that if one follows the effecti ve operator route to calculate physical quantities of interes t, these quest ions do not arise. The CCM is essentially a Fock space approach . The wave operator that maps the re fer-ence wave function to the exact eigenstate is in vertible in the Fock space. Given thi s situation , perhaps one shou ld not look to the underl ying wavefunctions at a ll but only consider the effective operators to ca lculate the physical quantities of interest.

Acknowledgement Financial support from DST is gratefu lly acknow l-

edged . I am grateful to the referee for his he lpful com-ments.

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