Ab initio rovibrational states of the D3h and C2v isotopomers of Li3+

SPECTROCHIMICA ACTA

PART A

E L S E V I E R Spectrochimica Acta Part A 51 (1995) 1827-- 1835

A b init io rov ib ra t iona l states o f the D3h a n d C2v i so topomer s o f LiJ-

Feng Wang *, Ellak I. von Nagy-Felsobuki

Department of Chemistry, Tile University of Newcastle, Callaghan, NSW 2308, Australia

Received 22 January 1995; accepted in final form 14 March 1995

Abstract

Rovibrational states for all four isotopomers of the ground electronic state of LiJ- have been calculated by a variational solution of the rectilinear displacement coordinate Eckar t -Watson Hamiltonian. The rovibrational calculations were performed using a recently developed method based on a combination of a numerical finite-element method (FEM) with grid techniques, and took full account of anharmonicity and of Coriolis vibrat ion-rotat ion coupling. The discrete ab initio potential energy surfaces of Searles and von Nagy-Felsobuki (J. Chem. Phys., 95, (1991) 1107, denoted SDCI/Full) and of Dunne et al. (Spectrochim. Acta Part A, 43 (1987) 699, denoted SDCI/FC) were employed. Our rovibrational energies of 7Li~- and 7Li26Li ÷ based on the older SDCI/FC potential converged to results which Henderson et al. (Spectrochim. Acta Part A, 44 (1988) 1287) obtained from that surface using a body-fixed scattering coordinate nuclear Hamilto- nian. The differences between the results obtained from the SDCI/Full and SDCI/FC potential energy surfaces reflect the errors in the older (SDCI/FC) surface.

I. Introduction

Alkali metal vapors are basic materials in a number of technological processes. For example, Stwalley and Koch [1] have suggested that alkali metal vapors will be useful in the development of infrared (IR) lasers, thermoelectric solar energy converters, fusion reactors and UV lamps. Sodium and potassium trimers have been observed in supersonic beams [2]. Its electron spin resonance spectrum has shown that the sodium trimer is a chemically bound species rather than a van der Waals' complex [3]. Charged alkali trimers such as Naj- have been identified by pump-probe laser techniques [4]. Moreover, gaining a full understanding of alkali metal clusters should provide important insight into the nucleation process from a free atom to the metallic state [5].

Rovibrational spectra of polyatomic molecules exhibit frequency and intensity patterns which cannot be adequately interpreted using classical arguments or approximate quan- tum treatments. Although there are many theoretical studies of vibrational band origins, only a limited number of calculations of rovibrational states have been reported for rotationally excited states of polyatomic molecules, particularly for excited vibrational states. Studies o f the v i b r a t i o n - r o t a t i o n in terac t ion require realist ic me thods for describ- ing their dynamic behavior. In this regard, an Eckart-Watson Hamiltonian for non-linear

* Corresponding author. Current address: Guelph-Waterloo Center for Graduate Work in Chemistry, Univer- sity of Waterloo, Ontario N2L 3GI, Canada. Fax: 1-519-746-0435. e-mail: [email protected].

0584-8539/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0584-8539(95)01442-X

/ 1828 Feng Wang, E.I. yon Nagy-Felso~uki/Spectrochimica Acta Part A 51 (1995) 1927-1935

triatomic molecules was introduced by Carney, Porter and co-workers [6-8], and further extended by Searles and von Nagy-Felsobuki (SF) [9], who applied it to low-lying vibrational levels of the LiJ-, Li2Na ÷, LiNaJ- and KLiNa ÷ alkali metal cations.

In order to calculate rovibrational states of non-linear triatomic molecules, we have developed a four-step rovibrational approach [10,11] to solve the rovibrational eigenvalue problem. That treatment of the dynamics has been tested by the ab initio calculation of the rovibrational states of H20 ÷ [11], which were compared with experiments [12,13] and with ab initio calculations of Weis et al. [14]. Moreover, an application of this computational method to NaJ- [15] yielded excellent agreement with the ab initio calculations of Carter and Mayer [16]. Subsequently, the method was used to predict the IR rovibrational spectra of alkali methal triatomic cations, such as NaJ- [15], Li2K + [17], K2Li + [18], KNaJ- [19], Li2Na +, LiNa2 and KLiNa + [20].

Theoretical studies of rovibrational states of LiJ- began in 1978 when Gerber and Schiimacher investigated its vibrational frequencies using self-consistent field (SCF) and configurations interaction (CI) methods [21]. The fundamental vibrational frequencies of LiJ- were also studied by Martin and Davidson using the HONDO program suit [22] and by Martin et al. using pseudo-potential calculations [23]. In 1986, Vojtik et al. calculated the lowest six vibrational energy levels of 6Li~- and 7Li~- using a "diatomics-in-molecule" model potential energy surface [24]. Not long ago, a 104 point valence CI discrete ab initio potential energy surface, which was calculated using the frozen core (FC) apprrox- imation, was generated by Dunne et al. (denoted SDCI/FC) [25], and was employed in further calculations of the vibrational band origins and other properties of LiJ- [26,27]. The SDCI/FC potential energy surface was also employed by Henderson et al. (HMT) in their scattering coordinate nuclear Hamiltonian method calculations of the rovibrational spectra of 7LiJ- and 7Li26Li + [5]. Recently, a higher quality 50 points (SDCI) potential energy surface of Li ; which did not use the frozen core approximation (denoted SDCI/Full) was generated by SF [9], who employed the new surface to calculate the low-lying vibrational band origins of LiJ-.

The purpose of this paper is to present more accurate predictions of the rovibrational eigenvalues of all four isotopomers of Lif using the SDCI/Full surface. In previous work, HMT [5] used the less accurate ab initio potential energy surface SDCI/FC to perform calculations for the two most abundant isotopomers of Li ; , 7Li~- and 7Liz6Li +, which have natural abundances of 80% and 18% respectively. In addition, SF [9] reported rotationless band origins for homogeneous Li~- on the higher quality SDCI/Full potential energy surface. The present work therefore presents rovibrational states for all possible isotopomers of Li~- ~lTLi+3, 6Lif , 7Li26Li + and 6Li27Li +) implied by this SDCI/Full surface.

2. Computational procedure

The method for treating the dynamics used in this study is similar to the one used in a series of studies of alkali metal cations [15,17-20], and also on H20 + and its C2v symmetry isotopomers [11]. In this approach, a molecular symmetry dependent Eckart- Watson vibration-rotation Hamiltonian is used in the rectilinear displacement coordinates. The "pure" vibrational eigenvalue problem is solved variationally using finite-element method (FEM) and grid techniques, such as Gaussian quadrature and the Harris et al. (HEG) [28] scheme. The latter has been shown to be equivalent to the discrete variable representation (DVR) method [10,29,30]. Those vibrational eigenfunc- tions are then used as basis functions for representing the trial rovibrational wavefunctions. The rovibrational calculations take into account a full description both of the anharmonicity and of Coriolis vibration-rotation coupling.

The full rovibrational eigenvalue problem is given by

/S/rvl~'/rv = ( / / r ib "hi- /S/rot -~ /~/c)lJt/rv = gkI'/rv

Feng Wang, E.I. yon Nagy-Felsobuki/Spectrochimica Aeta Part A 51 (1995) 1827-1835 1829

where the rovibrational Hamiltonian/:/rv is a sum of the "pure" vibrational Hamiltonian /:/vib, the rotational Hamiltonian /:/rot and the Coriolis coupling operator/:/c- The trial rovibrational wavefunction ~rv can be constructed from a configuration product of vibrational basis functions ~P~ and the plus and minus combinations of the usual symmetric-top rotor function R~K.

It~rJvM E + = C, Ks °en R ZKM =.K

Vibrational band origins for the four isotopomers were calculated as follows. For the SDCI/Full potential surface, a P(6,4) order Pad~ approximant with a Dunham expansion variable gave a good fit with the (zZ)l/2 of 1.4 × 10 9 [9]. For the homogeneous isotopomers of LiJ-, D3h Eckart-Watson Hamiltonian was used in the S-coordinate system [6,7], whereas the Czv Eckart-Watson Hamiltonian was used in the t-coordinates for the heterogeneous isotopomers of Li~-. A three dimensional vibrational configuration basis was spliced from the one-dimensional (1D) FEM solutions and was restricted using a nodal cut-off criterion: that is, all products with no more than 13 nodes were included, which gave a total of 560 basis functions. At each coordinate, 1000 finite-elements were used in the construction of the 1D wavefunctions in following domains: [ - 1.75, 3.25], [-2.00, 4.50] and [-2.60, 2.60]. The integrals were evaluated using a 16 point Gaussian quadrature scheme, whereas the potential integrals were evaluated using the DVR method [28]. Finally, the secular determinants were constructed and diagonalized variationally to give vibrational wavefunctions and eigenenergies, both of which were required for the further rovibrational problem [10].

The rovibrational Hamiltonian [8] is simply the representation of the full Hamiltonian in vibrational basis set

f/rv = <q/n I/'~/rv[kIdm >

can be written as [10]

f/rv ~ 1 Ev+~(8~x+Bv, , ) ( f l2+[B=z , ^~ ~(Bx= + B~)ln.

1 _ .. ^.2 l~ivi~i,.) iFl~i . + 2 ( B ~ B,.v)(IZI]. - I-I ~) + B,.y (l~I,. IZIy + +

where Ev is the vibrational eigenvalue matrix which is diagonal in the vibrational representation, I~I is the total angular momentum operator and rI~ the ~-component of the total angular momentum operator, B,~ are rotational constants (~ = fl) and centrifugal distortion constant (~ # fl, where ~, fl = x, y, z) and F is the Coriolis coupling matrix. Here B=p and F can be calculated in the vibrational representation [8]

(B=a >.,. = ½(W, [/t=p [Wm>

where /Ga denote the reciprocal of the instantaneous effective inertial tensor operators I'=a, (~, fl = x, y and z) and q~(i = 1, 2 and 3) are the rectilinear displacement vibrational coordinates, which stand for the {Si} and {t~} coordinates for molecules with D3h and C2~ symmetry respectively, introduced in Refs. [6-8].

Matrix elements of the rovibrational Hamiltonian to which Coriolis coupling contributions might be complex if ordinary symmetric-top functions, OJXM, were used as rotational basis functions. Therefore, a transformation was employed to ensure that the rovibrational super-matrix contains only real matrix elements [8]:

1 R-~K'M = ~ (f~ JK'M "q- f~ JK'M)

1 RJK'M -- i t7-~/2 (~JK'M -- (~JK'M)

1830 Feng Wang, E.L yon Nagy-Felsobuki/Spectrochimica Acta Part .4 51 (1995) 1827-1835

here i = ~ and K'>~ 0. As a result, an nvi b × (2J+ 1)-order supermatrix (where nvib is the number of vibrational basis functions) can be constructed for any given combination of nvi b and J. Diagonalizations of the resulting rovibrational supermatrices give the rovibrational eigenstates.

3. Results and discussion

Differences between the SDCI/Full and SDCI/FC potential energy surfaces can be obtained by the calculated rovibrational states. The older 104 point SDCI/FC surface [25] for LiJ- gives a Li-Li bond length of 3.000/i. and an energy minimum of -22.20506 Eh; whereas the 50 point SDCI/Full potential energy surface [9] gives a Li-Li bond length of 2.982/~ and an energy minimum of -22.25613 Eh. Based on the rotationless calculations [25] of the SDCI/FC surface, we obtained the rotational energies at 0.796 and 1.064 cm -1 for the 10 and 11 levels of 7LiJ -, and 0.810, 0.869 and 1.123 cm -~ for the 1 lo, Ill and lol levels of 7Li26Li +. These calculations are equivalent to the energies obtained by HMT using the same SDCI/FC surface [5]. Because the vibration-rotation calculations are variational, difference between the calculated rovibrational energies using the two potential surfaces reflect errors of the SDCI/FC ab initio potential energy surface.

The lowest 20 vibrational energy levels for the four homogeneous and heterogeneous isotopomers have been obtained using the higher quality SDCI/Full potential surface [9]. The rotationless energy levels are assigned to (v~ v2 133) according to the "weight" of the dominant configuration of the vibrational wavefunction [9]. Table 1 gives the zeroth-order vibrational frequencies @ol, co2, co3) and anharmonic constants (X~, i, j = 1, 2 and 3; Y22) of the LiJ homogeneous isotopomers by fitting the lowest 20 vibrational energy levels to an energy expression as was applied previously to NaJ- [15], while that used for heterogeneous isotopomers was the same as that used for H20 + [11,14]. It can be seen that the 7Li~- isotopomer has consistently smaller values of vibrational frequencies and anharmonic constants than its 6Li~- counterpart. For the C2v isotopomers, 6Li27Li + has larger zeroth-order vibrational frequencies than those of 7LiE6Li ÷. However, the anharmonic constants of the CEv isotopomers do not have such a tendency.

Rotational constants (Bxx, Byy and B=), centrifugal distortion constant (Bxy), and Coriolis coupling (F) matrices spanned by the lowest five vibrational states are given in Table 2 for the D3h symmetry isotopomers (7Li~- and 6Li3+), and in Table 3 for the C2v symmetry isotopomers (TLi26Li + and 6Li27Li+). For the former, B~x( = Byy) gives the leading rotational contributions to the rovibrational Hamiltonian supermatrix, while B== is only about half the value of Bxx. The diagonal rotational constants are clearly the dominant elements. The centrifugal distortion constant Bxy, however, is expected to be

T a b l e 1

Z e r o - o r d e r v i b r a t i o n a l f r e q u e n c i e s a n d a n h a r r n o n i c c o n s t a n t s o f t h e 7Li~- a n d 6Li~- i s o t o p o m e r s

S D C I / F u l l s u r f a c e

( c m - 1) u s i n g

C o n s t a n t 7Li~- 6 .+ L13 7Li26Li + 6Li 27Li +

co I 309 .2862 333 .6724 320 .5484 328 .3353

co2 233.0301 251 .4566 245 .9932 251 .3163

co3 - - 244.6031 249 .9010

X~ - 1 .0826 - 1.2697 - 1.5068 - 1 .4826

;(22 - 1 .3486 - 1.5808 - 2 . 8 7 8 9 - 2 . 7 3 7 1

Z33 - - - 2 . 4 2 7 9 - 2 . 3 5 5 8

Z~2 - 4 . 3 5 5 8 - 5 .0647 - 5 .6970 - 5 .9987

;(t 3 - - 5 .3979 - 5 .3856

;(23 - - - 3 .7756 - 3.5081

~22 0 .6990 0 .8069 - -

Feng Wang, E.L yon Nagy-Felsobuki/Spectrochimica Acta Part A 51 (1995) 1827-1835 1831

Table 2 Rotational constants, centrifugal distortion constant and Coriolis coupling matrix elements of the 7Li+ and 6Li~ isotopomers (cm-i)

7Li ~- 1 1 0.5396 0.2679 0.0000 0.0000 2, 3 1 0.0368 0.0000 0.0000 0.0000 2, 3 2, 3 0.5441 0.2658 0.0000 0.0000 4 1 0.0000 0.0000 - 0.0368 0.0000 4 2, 3 0.0000 0.0000 0.0050 0.5250 4 4 0.5340 0.2658 0.0000 0.0000 5 1 0.0312 0.0157 0.0000 0.0000 5 2, 3 -0 .0045 0.0000 0.0000 0.0000 5 4 0.0000 0.0000 0.0045 0.0000 5 5 0.5364 0.2663 0.0000 0.0000 6, 7 1 0.0072 0.0018 0.0000 0.0000 6, 7 2, 3 0.0370 0.0000 0.0000 0.0000 6, 7 4 0.0000 0.0000 - 0.0370 0.0000 6, 7 5 - 0.0001 0.0005 0.0000 0.0000 6, 7 6, 7 0.5386 0.2637 0.0000 0.0000

6Li ~- 1 l 0.6278 0.3115 0.0000 0.0000 2, 3 1 -0 .0444 0.0000 0.0001 0.0000 2, 3 2, 3 0.6333 0.3088 0.0000 0.0000 4 1 O.O001 0.0000 0.0444 0.0000 4 2, 3 0.0000 0.0000 0.0063 0.6094 4 4 0.6207 0.3088 0.0000 0.0000 5 1 - 0.0377 - 0.0189 0.0000 0.0000 5 2, 3 -0 .0057 0.0000 0.0000 0.0000 5 4 0.0000 0.0000 0.0057 0.0000 5 5 0.6238 0.3095 0.0000 0.0000 6, 7 1 0.0090 0.0022 0.0000 0.0000 6, 7 2, 3 -0 .0447 0.0000 0.0001 0.0000 6, 7 4 0.0001 0.0000 0.0447 0.0000 6, 7 5 0.0001 -0 .0006 0.0000 0.0000 6, 7 6, 7 0.6265 0.3062 0.0000 0.0000

a Symmetric matrices, b Anti-symmetric matrix.

small for eigenstates near the potential energy minimum so that the diagonal terms are not necessarily larger in value than the off-diagonal ones. Tables 2 and 3 reflect this anticipation. Finally, diagonal elements of the Coriolis coupling matrix are not apprecia- ble because for an Eekart-Watson Hamiltonian the Coriolis coupling necessarily gives small diagonal matrix elements.

Table 4 reports rovibrational energies up to J ~< 5 in the lowest four vibrational states of 7Li~ and 6Li~-. As expected for an oblate top, the rotational energy increases with J and decreases with increasing values of K. The rotational spacings are similar for the different vibrational states. As a result, zeroth-order rovibrational calculations which assume that the rotational energies are the same on different vibrational states are approximately valid. Hence, the zeroth-order approximately may be a good one for low J on the low-lying vibrational states. As J increases or if the rotational energies lie on higher excited vibrational states such an approximation breaks down. Further, the rotational spacings decrease as vibrational excitation increases, and the higher the Jr value the more apparent the decrease. Similarly, Table 5 gives the rotational energies of the heterogeneous isotopomers in their ground vibrational states. Rovibrational energies are labeled using the Mulliken convention, Jxo. re, for the C2v symmetry isotopomers. The energies increase with Ka, but decrease as Kc increases.

The heterogeneous C2v symmetry isotopomers 7Li26Li + and 6Li27Li + are asymmetric- tops with Ray's asymmetric parameter values of x = 0.6241 and 0.6411, respectively. The

1832 Feng Wang, E.1. yon Nagy-Felsobuki/Spectrochimica Acta Part .4 51 (1995) 1827-1835

Table 3

Rotational constants, centrifugal distortion constant and Coriolis coupling matrix elements of the 7Li26Li + and 6Li27Li + isotopomers ( c m - t )

I) i Uj B x x a n y y a Bzz ~" B,:y a F b

7Li26Li + 1 1 0 .5384 0 .5979 0 .2813 0 .0000 0 .0000

2 1 - 0 .0394 0 .0384 - 0 .0023 0 .0000 0 .0000

2 2 0.5321 0 .6035 0 .2788 0 .0000 0 .0000

3 1 0 .0000 0 .0000 0 .0000 - 0.0391 - 0 .0015

3 2 0 .0000 0 .0000 0 .0000 0.0051 0 .5470

3 3 0 .5430 0 .5915 0.2791 0 .0000 0 .0000

4 1 - 0 .0277 - 0 .0392 - 0 .0166 0 .0000 0 .0000

4 2 - 0 .0042 0 .0042 0 .0000 0 .0000 0 .0000

4 3 0 .0000 0 .0000 0 .0000 - 0 . 0 0 5 5 0 .0649

4 4 0 .5359 0 .5936 0 .2800 0 .0000 0 .0000

5 1 - 0 .0078 - 0 .0072 - 0 .0019 0 .0000 0 .0000

5 2 0 .0475 - 0 .0464 0 .0029 0 .0000 0 .0000

5 3 0 .0000 0 .0000 0 .0000 0 .0293 - 0 .0246

5 4 0 .0002 - 0 .0004 0 .0005 0 .0000 0 .0000

5 5 0 .5318 0 .6029 0 .2766 0 .0000 0 .0000

6Li27Li + 1 1 0 .6277 0 .5682 0.2961 0 .0000 0 .0000

2 1 - 0 . 0 4 1 0 0 .0420 0 ,0023 0 .0000 0 .0000

2 2 0 .6215 0 .5725 0 .2937 0 .0000 0 .0000

3 1 0.0000 0.0000 0 .0000 0.04 17 - 0 .0015

3 2 0 .0000 0 .0000 0 .0000 - 0 .0062 - 0 .5759

3 3 0 .6332 0 .5620 0 .2934 0 .0000 0 .0000

4 1 0 .0414 0 .0298 0 .0177 0 .0000 0 .0000

4 2 0 .0060 - 0 .0058 0 .0000 0 .0000 0 .0000

4 3 0 .0000 0 .0000 0 .0000 - 0 .0046 - 0 .0648

4 4 0 .6229 0 .5656 0 .2942 0 .0000 0 .0000

5 1 - 0 .0088 - - 0 .0076 - 0.0021 0 .0000 0 .0000

5 2 0 .0480 - - 0.0491 - 0 .0027 0 .0000 0 .0000

5 3 0 .0000 0 .0000 0 .0000 - 0 .0338 0 .0230

5 4 - - 0 .0003 0 .0005 - 0 .0006 0 .0000 0 .0000

5 5 0.6231 0 .5702 0 .2912 0 .0000 0 .0000

a Symmetric matrices, b Anti-symmetric matrix.

former is in good agreement with x ~ 0.62 of HMT [5]. Table 6 gives the rotational constants (in the ground vibrational state) of the four isotopomers. The rotational constants of 7Li~- and 7Li26Li + are in good agreement with those obtained by HMT [5] but it should be noted that different potential energy surfaces were used. Table 7 compares the lol, 111 and 110 rovibrational energies of 7Li26Li + on the lowest four vibrational states with those obtained by HMT [5] using the SDCI/FC potential energy surface. Because the two dynamics calculation methods agree well if the same ab initio potential energy surface (SDCI/FC) is employed, the discrepancies given in Table 7 are due to the differences between the potential energy surfaces employed. The rovibrational discrepancies (A~v) are mainly due to the difference in the calculated vibrational energy differences (Avib) , but the rotational discrepancies (Arot) are very small. This implies that the SDCI/FC potential surface is close to the "true" potential only in the vicinity of the energy minimum, but that it diverges from the "true" potential energy surface elsewhere.

4. Conclusions

The rovibrational states of 7Li~-, 6Li~-, 7Li26Li + and 6Li27Li + have been calculated using the rectilinear displacement coordinate Eckart-Watson Hamiltonian method and the SDCI/Full ab initio potential energy surface. Our rotational energies of 7Li~- and

Feng Wang, E.I. yon Nagy-Felsobuki/Spectrochimica Acta Part A 51 (1995) 1827-1835

Table 4 Rotational energy levels of the low-lying vibrational states of the 7Li~ and 6 .+ LI 3 isotopomers (cm- I)

1833

JK/Evib 7Li~ 6Lif

(000) (011 ) (100) (020) (000) (O1 l) (lO0) (020)

0.000 a 227.451 302.773 450.837 0.000 b 244.917 326.025 485.098 11 0.807 0.805 0.803 0.802 0.939 0.936 0.933 0.933 1 o 1.079 1.078 1.073 1.077 1.256 1.254 1.248 1.253 22 2.151 2.141 2.138 2.132 2.501 2.490 2.486 2.478 2 t 2.966 2.961 2.948 2.956 3.450 3.444 3.428 3.438 20 3.237 3.234 3.218 3.231 3.766 3.762 3.742 2.758 33 4.030 4.009 4.006 3.989 4.687 4.661 4.657 4.635 32 5.388 5.376 5.356 5.363 6.267 6.252 6.227 6.235 31 6.203 6.194 6.166 6.186 7.215 7.205 7.170 7.194 30 6.475 6.466 6.437 6.460 7.532 7.522 7.484 7.514 44 6.445 6.409 6.407 6.373 7.495 7.450 7.447 7.405 43 8.346 8.321 8.297 8.297 9.708 9.676 9.646 9.647 42 9.703 9.689 9.646 9.315 11.237 11.269 11.215 10.823 41 10.518 10.505 10.456 9.317 12.235 12.219 12.158 10.825 40 10.791 10.775 10.727 9.668 12.553 12.533 12.473 11.243 55 9.395 9.340 9.340 9.284 10.926 10.856 10.856 10.787 54 11.840 11.797 11.770 11.758 13.771 13.718 13.683 13.669 53 13.741 13.709 13.660 12.226 15.983 15.944 15.882 14.203 52 15.096 15.080 15.008 12.228 15.560 17.542 17.449 14.206 51 15.911 15.892 15.817 13.684 18.508 18.485 18.391 15.913 50 16.185 16.158 15.090 14.700 18.828 18.793 18.709 17.081

a Zero-point energy of 7Li; is 384.544 cm-1. b Zero-point energy of 6Li~ is 414.614 cm ~.

7Li26Li + using the SDCIfffC potential energy surface have reproduced the rotational energies reported by Henderson et al. [5] using that potential energy surface. As the rovibrational calculations are variational and yield equivalent results, the dicrepancies between the two rovibrational calculations can be expected to reflect the real improve- ment of the present potential energy surface.

Table 5 Rotational energy levels of the ground vibrational state of the 7Li26Li + and 6LiJLi+ isotopomers (cm -~)

JKa, Kc/Evib 7Li26Li + ~Li27Li + JKo. l(c/Evib 7Li26Li + 6Li27Li + (000) a (000) b (000) a (000) b

lol 0.820 0.864 423 8.759 9.221 111 0.879 0.924 422 10.095 10.635 l lo 1.136 1.196 432 10.223 10.758 202 2.252 2.371 431 10.804 11.383 212 2.262 2.380 44i 11.392 11.972 2 N 3.033 3.198 440 11.487 12.077 22j 3.211 3.375 505 9.863 10.382 220 3.418 3.596 515 9.863 10.382 303 4.230 4.452 514 12.425 13.080 3t 3 4.231 4.453 524 12.426 13.080 3t2 5.625 5.925 523 14.391 15.153 322 5.670 5.968 533 14.416 15.176 321 6.358 6.700 532 15.654 16.495 331 6.714 7.060 542 15.924 16.575 33o 6.862 7.217 54t 16.384 17.257 4o4 6.766 7.121 55t 17.250 18.126 414 6.766 7.121 55o 17.307 18.191 413 8.753 9.215

a Zero-point energy of 7Li26Li + is 394.550 cm -1. b Zero-point energy of 6Li27Li + is 404.719 cm -I .

1834 Feng Wang, E.L yon Nagy-Felsobuki/Spectrochimica Acta Part .4 51 (1995) 1827-1835

Table 6 Comparisons of the rotational constants in the ground vibrational states of 7Li+ 6Li+, 7Li26Li + and 6Li27Li + (cm -I)

Constant This work ~ HMT b [5] This work a HMT b [5] 2 "+

L I 3 7 L i 2 6 L i + 6 ' + L13 7Lif 6Li27Li + 7Li 26Li +

A =Bxx 0.6277 0.5396 0.5319 0.6326 0.5995 0.5908 B=Byy - - - 0.5681 0.5492 0.5320 C=Bxy 0.3115 0.2679 0.2641 0.2994 0.2847 0.2779

SDCI/Full potential energy surface was employed [9]. b SDCI/FC potential energy surface was used [25].

Table 7 Comparisons of the lot, l tl and 11o energies in the lowest four vibrational states of the 7Li26Li + with the calculations of HMT [5] (cm-1)

(vl v2 v3) Jxa. rc This work HMT [5] (SDCI/FulI) (SDCI/FC)

E (SDCI/FulI) - E(SDCI/FC)

Arv Avib Arot

(000) lol 0.820 0.810 0.010 0.000 0.010 111 0.879 0.869 0.010 0.000 0.010 11o 1.136 1.123 0.010 0.000 0.010

(010) lol 233.747 231.888 1.859 1.245 0.614 lit 233.747 231.917 1.830 1.245 0.585 llo 233.759 232.527 1.232 1.245 -0.013

(001) lol 233.759 233.100 0.659 1.238 -0.579 Ill 233.781 233.105 0.676 1.238 --0.562 lto 234.302 233.047 1.255 1.238 0.017

(100) lol 311.824 308.106 3.718 3.709 0.009 Ill 311.882 308.163 3.719 3.709 0.010 llo 312.138 308.416 3.722 3.709 0.013

Acknowledgements

We wish to thank Professor R.J. Le Roy for reading and commenting on the manuscript. Useful suggestions from Dr. G.C. Corey are appreciated. The calculations were performed using a VAX 3100 work station and a VAX/VMS 6620s. Thanks should be given for the support of the Computing Center, the University of Newcastle. Finally, we wish to acknowledge an Overseas Postgraduate Research Award and a University of Newcastle Postgraduate Research Award (Australia) held by F. Wang.

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Ab initio rovibrational states of the D3h and C2v isotopomers of Li3+