Nonadiabatic Eigenvalues and Adiabatic Matrix Elements for all …scienide2.uwaterloo.ca/~rleroy/Pubn/87JMS-SchwartzH2.pdf · 2005. 8. 5. · JOURNAL OF MOLECULAR SPECI-ROSCOPY 121,420-439

JOURNAL OF MOLECULAR SPECI-ROSCOPY 121,420-439 (1987)

Nonadiabatic Eigenvalues and Adiabatic Matrix Elements for all Isotopes of Diatomic Hydrogen

CAREY SCHWARTZ* AND ROBERT J. LE ROY

Guelph- Waterloo Centre for Graduate Work in Chemistry, University of Waterloo, Waterloo, Ontario, Canada NZL 3Gl

For all bound and quasibound levels of the ground electronic state of ah six isotopes of diatomic hydrogen, wavefunctions obtained from the most recent ab initio potentials are used to calculate expectation values of the nuclear kinetic energy, of various powers of R, and of the average polarixability and polarizability anisotropy, together with the off-diagonal matrix elements of the polarizability required for predicting the intensities of Raman transitions for AJ = 0, +2 and Au = 0, -1, and -2. A scaling procedure for treating the nonadiabatic eigenvalue corrections is developed, which allows an extrapolation beyond results reported for HZ, HD, and Dr to yield nonadiabatic level shift predictions for the three tritium isotopes. Features of this procedure which take account of implicit centrifugal distortion effects lead to significant improvements in the agreement between theory and experiment. 0 1987 Academic FVCW, IX

I. INTRODUCTION

As well as being of considerable current interest (l-3), the molecular hydrogen system has long been a key meeting ground for theory and experiment. It is therefore important that the inventory of calculated properties be kept as complete and up to date as possible. The present paper addresses this need by reporting a new calculation of the properties of the ground electronic state of this system.

Some 15 years ago, one of us used the best (then) available relativistically corrected adiabatic potential energy curves for this system (4-6) to generate a compendium of calculated eigenvalues and expectation values for alI vibration-rotation levels of the ground electronic states of Hz, HD, and Dz ( 7).2V3 Uses of those results have included a calculation of the current best values of the thermodynamic properties of these species (8) from eigenvalues which incorporated a semiempirical nonadiabatic eigenvalue correction term (9, 10). However, no similarly complete tabulations have been reported for the tritium isotopes. Moreover, our knowledge of the clamped nuclei potential (11-13) and its radiative (14) and adiabatic (15) corrections, and of the nature of the nonadiabatic corrections to the level energies (15, 16) has distinctly improved since that time. Hunt et al. (I 7) used versions of these improved potentials

’ Present address: Michelson Laboratory, Physics Division, Naval Weapons Center, China Lake, Calif. 93555.

2 A complete tabulation of the results described in Ref. (7) is available as University of Wisconsin Theoretical Chemistry Institute Report WIS-TCI-387 (197 1) by R. J. Le Roy (see footnote 3).

3 If this report is not obtainable from its author or hi institution, a copy may be borrowed on request to the Editor of this journal,

0022-2852187 $3.00 420 Copyrieht 0 1987 by A- PBS, Inc. AU &d.s of repmdution in any form -d.

Hz ENERGIES AND MATRIX ELEMENTS 421

to calculate adiabatic eigenvalues and matrix elements of the quadrupole moment and polarizability for all six hydrogen isotopes; however, their work preceded the most recent improvements to the clamped nuclei potential (13), they did not take account of nonadiabatic corrections, and their results were limited to the quantum number range u = O-5 and J = O-5.

The present paper reports new calculations of nonadiabatic eigenvalues, of certain expectation values, and of off-diagonal matrix elements of the average polar&ability and polarizability anistropy, for all vibration-rotation levels of all six isotopes of diatomic hydrogen. These results are based on the most recent version of the ab initio potential for this system, and they utilize an isotopic scaling procedure for extrapolating beyond the nonadiabatic corrections calculated for HZ, HD, and Dz to obtain values for the three tritium isotopes. This extrapolation procedure also predicts a damping of the nonadiabatic eigenvalue corrections with increasing rotational quantum number J, which removes the sharp increase in the apparent discrepancy between theory and experiment for the highest rotational levels of Hz, recently pointed out by Dabrow- ski (28).

II. POTENTIALS AND PROCEDURES

As in previous studies, the present calculations proceed by first combining the current best clamped nuclei potential (13) with the relativistic (4) and radiative (14) corrections to obtain a mass-independent effective Born-Oppenheimer potential V,(R). The appropriately scaled adiabatic correction function (4, 15) A&(R) is then added to this function to yield the effective adiabatic potential for each isotope. The radial S&r&linger equation is then solved to obtain adiabatic eigenvalues and eigenfunctions for all bound and quaisbound vibration-rotation levels of the potential for each isotope, and the resulting eigenfunctions used to generate the desired expectation values and matrix elements. Nonadiabatic eigenvahte corrections determined using a procedure outlined below are then used to define the optimum theoretical energy for each level.

A. The Efective Adiabatic Potentials

The effective adiabatic potentials used in this work were constructed using a pre- scription analogous to that of Bishop and Shih (19); the recipe used to combine the various types of contributions is summarized in Table I. First of all, a smooth clamped nuclei potential VW(R) was generated on a selected 169~point grid by combining piecewise polynomial interpolation over the ab initio points with an inverse-power extrapolation at long range (20,21). Since use of an inappropriate interpolation scheme can introduce significant errors in the final calculated eigenvalues [e.g., see Fig. 1 of Ref. (IO)], considerable care was taken in choosing this interpolation procedure. The scheme chosen utilized both the potential values and their [directly calculated (I3)] hrst derivatives, and consisted of 4-point (7th order) interpolation over the quantity R4 X V,(r). Tests showed that use of other “good” procedures affected calculated eigenvalues by =~0.0003 cm-‘; note, however, that use of certain other superficially reasonable schemes (such as cubic spline interpolation on V,(R) itself) can produce interpolation-noise errors as large as 0.1 cm-’ in the calculated eigenvalues.

422 SCHWARTZ AND LE ROY

TABLE I

Recipe for Preparing Effective Potentials from Ab Initio Data

term range/bohr source & conents

Clamped 0.2 < R < 12. Ref.13 nuclei 12. T R - Refs.20 & 21

Relativistic 0.4 < R < 3.6 Ref.4 correction 3.67 R- See radiative correction

Radiative 0.4 < R < 3.6 correction 3.67 R -

Refs.4,14 & 22 with Bethe logarithim of 2.3 Decreasing exponential fitted to last two relativistic plus radiative corrections points

Adiabatic 0.4


TABLE II

Mass-Independent Clamped Nuclei VdR) and Adiabatic Con&ion AV&R) Contributions to the Effective Adiabatic Potentials for the Various Hydrogen Isotopes [See Eq. (1); Energies in Hartrees and Lengths in Bohr]

Av.&)

0.0003354522 0.0002349477 O.WOl775486 O.OWl458952 0.0001280241 O.OW1163783 O.OWlO681.51 O.OOW976211 O.OWO885113 0.0000710739 0.0000549880 0.0000511963 O.WW474915 O.OOW438770 0.WO0403570 O.OWO369461 O.WOO336340 O.OOW304182 O.WOO272955 0.0000242580 O.OOW213095 O.OWOl84495 O.WWl56774 0.0000129942 O.OWO103964 0.0000078823 o.WQOo54499 O.OOWO30967 O.Wf3XB8218

-0.oWWl3764 -0.m34994 -O&WOO55478 -0.OOWO75.266 -0.Wo0094374 -0.OOlMl12716 -0.OOWl30426 -0.0000147481 -0.OOOtll63880 -0.OWOl79629 -0.OWJ194762 -0. oLHm209283 -0.0ooO223206 -0.0000236538 -ON00249283 -0.OWO261459 -O&O00273078 -0.OOW284153 -0.OOUO2947W -0.oOW304724 -0.0000314232 -0.OWO323232 -0.0000331728 -0.0m339735 -0.WOO347264

) -0.ooW354325 -0.Wflo360935 -0 .WW367090

R VFJ#u &d(R)

1.975 -0.1403403475 -0.oow372794 2.OW -0.1381287982 -0.OWO378052 2.025 -0.1359018266 -0.WW382856 2.050 -0.1336626711 -0.OWO387230 2.075 -0.1314143556 -0.OWO391186 2.100 -0.1291597052 -0.oWO394735 2.125 -0.1269Oi3600 -0.OWU397911 2.150 -0.1246417879 -0.0ooO400684 2.175 -0.1223832954 -0.OOW403047 2.200 -0.1201280387 -0.OOW404996 2.225 -0.1178780341 -0.OOW406496 2.250 -0.1156351661 -0.0000407597 2.275 -0.1134011949 -0.0000408321 2.300 -0.1111777653 -0.OOW408691 2.325 -0.1089664136 -0.OOW408754 2.350 -0.1067685749 -0.OWO408487 2.375 -0.1045855887 -0.OOW407892 2.400 -0.1024187056 -0.0OWIO6970 2.425 -0.lW2690912 -0.OOW405709 2.450 -0.0981378320 -0.0OW404131 2.475 -0.0960259387 -0.0003402245 2.E.W -0.0939343514 -0.0000400063 2.525 -0.0918639432 -0.OOW397598 2.550 -0.0898155240 -0.0000394858 2.575 -0.0877898440 -0.OOW391851 2.600 -0.0857875971 -0.OOW388589 2.625 -0.0838094237 -0.OOW385076 2.650 -0.0818559131 -0.OOW381330 2.675 -0.0799276056 -0.WGU377365 2.7W -0.0780249953 -0.0000373191 2.725 -0.0761485321 -0.WOO368833 2.750 -0.0742986240 -0.OOW364281 2.775 -0.07247563% -0.OOW359537 2.EXlO -0.0706799066 -0.OWO354604 2.825 -0.0689117204 -0.OWO349473 2.850 -0.0671713385 -0.OGOO344167 2.875 -0.0654589848 -0.OooO338698 2.900 -0.0637748511 -0.K00333079 2.925 -0.0621190980 -0.OWO327334 2.950 -0.0604918559 -0.WW321469 2.975 -0.0588932270 -0.oWo315493 3.WO -0.0573232854 -0.0000309420 3.025 -0.0557820790 -0.OWO303257 3.050 -0.0542696299 -0.WW297016 3.075 -0.0527859352 -0.0WO290710 3.100 -0.0513309677 -0.00,0284348 3.125 -0.0499046772 -0.OOW277942 3.150 -0.0485069913 -0.OWO271502 3.175 -0.0471378163 -0.OWO265039 3.2W -0.0457970376 -0.OWO258563 3.225 -0.0444845212 -O.OWOZ520@4 3.250 -0.0432Wll41 -0.OOW245612 3.275 -0.0419436440 -0.OWO239156 3.300 -0.0407149215 -0.OOWZ32726 3.325 -0.0395137395 -0.0ooO226331 3.350 -0.0383398753 -0.oooO219980 3.375 -0.0371930901 -0.0ooO213681

R VBO(W &v,(R)

3.400 -0.0360731307 3.425 -0.0349797300 3.450 -0.0339126072 3.475 -0.0328714680 3.500 -0.0318560059 3.525 -0.0308659030 3.550 -0.0299Wmx 3.575 -0.0289604502 3.&M -0.0280444142 3.625 -0.0271523535 3.650 -0.0262839172 3.675 -0.0254387330 3.700 -0.0246164222 3.725 -0.0238165997 3.750 -0.0230388754 3.775 -0.0222828546 3.800 -0.02154813% 3.9W -0.0188142717 4.000 -0.0163887819 4.100 -0.0142455076 4.200 -0.0123586373 4.300 -0.0107031836 4.400 -0.0092553011 4.5w -0.0079925541 4.600 -O.W68940%8 4.700 -0.0059407010 4.800 -0.w51149174 4.900 -0.0044W9871 5.ooO -0.0037847427 5.100 -0.0032534874 5.200 -0.0027960570 5.300 -0.W24025893 5.4W -0.00206442% 5.500 -0.w1774W81 5.600 -0.0015247319 5.700 -0.w13108695 5.800 -0.0011274518 5.9M) -0.WO9701826 6.OW -0.0008353530 6.2W -0.ooO6206739 6.4W -0.OW4628405 6.600 -O.W03466@38 6.8W -0.OW2610535 7.OW -0.ow1977802 cow -0.oWO555537 9.OW -0.OWOl97618

10.000 -0.OWOO87476 11.m -0.oWW45033 12.OW -0.OWOO25453 13.WO -0.WOW15258 lO.OW -0.OWWO96W 15.WO -0.OOWOU6253 16.000 -0.OWWO4195 17.000 -0.OWOOO2888 18.OW -0.0ooO002033

-0.OWO207441 -0.OWO201270 -0.OOOO195173 -0.OOWl89158 -0.WWl83232 -0.ocm177403 -0AWOl71674 -0.OOWl66048 -0.WWl60532 -0.0000155129 -0aJWl49842 -0.OWOl44676 -O.OWOl3%32 -0.OOWl34716 -0.WWl29925 -0.0000125260 -0.KCQ120723 -0.0000103951 -0.WLXMl8987 -0.OOWO76036 -0.WWO64862 -0.OWW55294 -0.WWa47155 -0.OWW40270 -0.OQWO34457 -0.OWOO29565 -0.OOWO25451 -0.OOWO21978 -0.WOw19020 -o.o@mm43o -0.OWW14155 -0.WOW12147 -0.0oooO10378 -0.woooo9832 -0.oWWO7609 -0.oooooo6591 -0.0WuW5751 -0.WWW5060 -0.ooWW4495 -0.oowoo3641 -0.0oWW3017 -0.oooooO2484 -0.OcKm001944 -0.oooooo1351 -0.OCUMW456

:;$f&;;

-0hGVXHMOl8 -o.oWWcmM -0.WWWWO2 -0.0WrmWo1

“o:E o.OOWWWOo o.oowWOOw

for other isotopes would be reduced by the ratio of the reduced masses. The resulting mass-independent (i.e., without the HZ reduced mass factor implicitly included in the calculated values) correction function Al&(R) is also listed in Table II. The results in Table II then define the present recommended effective adiabatic potential Vi(R) for diatomic hydrogen isotopes “i” with nuclear reduced mass pi as

V,(R) = &KM) + ( 1 hiM f’-a&O. (1) The actual calculations of course require Vi(R) to be defined on a much finer mesh

than that associated with the grid used in Table II. However, the smoothing inherent


in the above procedure makes results relatively less sensitive to the details of this second stage of interpolation; for example, eigenvalues obtained using 8-, lo-, or 12- point piecewise polynomial interpolation over the 169 values of R2 X Vi(R) for Hz agreed to within 0.000 1 cm-l. Overall, then, we conclude that errors due to potential energy interpolation noise and extrapolation uncertainty should affect the eigenvalues reported below by no more than ~0.01 cm-r.

B. Adiabatic Eigenvalue and Matrix Element Calculations

Using standard numerical techniques (23, 24),4 the radial Schrijdinger equation was solved to obtain the adiabatic eigenvalues and eigenfunctions for all vibration- rotation levels of each isotope from the effective adiabatic potentials of Eq. (1). The radial integration stepsize was set at 0.0035 bohr for HZ, with the corresponding values for the other isotopes being scaled down by the ratios of the square roots of the reduced masses; these values assure eigenvalue convergence to better than 0.0005 cm-‘. Quasi- bound levels, those lying above the dissociation limit but behind a centrifugal potential energy barrier, were located using the Airy function boundary condition technique of Ref. (25), while their tunneling predissociation widths were calculated using the uniform semiclassical method (26) described in Ref. (27).

The reduced masses used in the radial Schriidinger equation and in Eq. (1) were based on the masses of the bare nuclei. They were obtained by taking the atomic masses from Ref. (28), subtracting the electron mass, and correcting relativistically for the binding energy of the electron. For the H, D, and T isotopes, respectively, the resulting nuclear masses are 1.007276470, 2.0135532 14, and 3.0 1550070 u, with uncertainties of 0.0 12,0.024, and 0.04 ppm, respectively.

In solving the radial Schrodinger equation, the value of h was introduced in atomic units by combining the electron mass with Rydberg’s constant (29), 219 474.63058 cm-’ to yield the factor AZ/2 = 60.1996933 (kO.045 ppm) cm-’ bohr’ u. When this term is combined with the reduced mass and used in the radial S&r&linger equation, a relative error of q in the physical constant factor h2/21 would introduce an error of q X ((KE) + J(J + l)(h2/2~)(R-2)} in the resulting level energy, where (ICE) is the expectation value of the radial kinetic energy of the nuclei and (Re2) the expectation value of the squared inverse of R for that level. The uncertainties in the physical constants (29) therefore introduce uncertainties of up to +O.OO 18 cm-’ in the adiabatic eigenvalues calculated for this system. This is much smaller than the uncertainties associated with construction of the effective adiabatic potentials from the ab initio points, so the latter is the source of the overall uncertainties of ca. +O.OlO cm-’ in the adiabatic level energies for these species.

Diagonal and off-diagonal matrix elements were calculated from the usual expression

(u’,J’lf(R)lu”,J”) = $+ Xo~yf(R)X”._,.dR.

4 An annotated listing of the program used in the present work is described in University of Waterloo Chemical Physics Research Report CP-230R3 ( 1985) by R. J. L.e Roy (see footnote 3). A copy of this program may be obtained from this author on request. Together with most other radial eigenvalue and Franck- Condon factor pmgrams in general use, it was originally based on the seminal program described in University of California Lawrence Radiation Laboratory Report UCRLlO925 (1963), authored by R. N. Zare.


If both levels are truly bound, the upper bound on this integral, R+, is effectively infinity, while if one or more of the levels are quasibound, this bound is set at the outermost (3rd) classical turning point of the (higher) quasibound level.

The isotropic polarizability G(R) and polarizability anistropy ha(R) functions used as f(R) in the matrix element calculations were based on the ab initio results of Ry- chlewski (30-32). Frequency-dependent polarizabilities corresponding to the wave- length 4880 a were obtained from the numerical results of Refs. (31, 32) by cubic spline interpolation. Calculated function values on the range 0.6 < R < 10 bohr were supplemented at the united atom limit (R = 0) by the He-atom polarizabilities of Glover and Weinhold (33), and in the long-range region by the functions of Buck- ingham d al. (34) with the frequency-dependent H-atom polarizability in the separated atom limit taken from Karplus and Kolker (35). Smooth hmctions of R were generated by using cubic splines to interpolate over the points on the range 0.0-10 bohr and extrapolating with the appropriate inverse-power function (34) at long range.

Published convergence tests (30, 36) and comparisons with other calculations (37, 38) suggest that the ab initio polarizability values should be converged to ca. 0.001 au. Numerical noise associated with the interpolation will amplify this uncertainty, particularly for both larger distances and for nonzero frequencies, for which the calculated function values are only available on a relatively coarse mesh (3Z, 32). However, such noise would be of much higher frequency than the oscillations of the radial wavefunction x”J(R), and hence it should be damped by the quadrature of Eq. (2) so our calculated expectation values and matrix elements should have an accuracy close to that of the ab initio results.

C. Nonadiabatic Corrections to the Level Energies

Most previous calculations of the nonadiabatic level shifts of molecular hydrogen (9, 10, 16, 19) have used an approximation due to Van Vleck (39), which suggests that this term is proportional to the expectation value of the (radial) nuclear kinetic energy times the inverse of the reduced mass5 In some cases (16, I9), a fit to experimental data for low-J levels was used to determine an empirical scaling factor which optimized the agreement with experiment. However, while this approach yields fairly good results for low-J levels, it overlooks the existence of both an additional contri- bution to the rotationless energies of the heteronuclear isotopes, and terms proportional to J(J + l), which become quite important at high J.

More recently, Wolniewicz (15) showed that for homonuclear hydrogen isotopes, the nonadiabatic corrections to the level energies may be expressed as

AE,,(u, J)= AE”(Z&+J(J+ l)A,,&)

and those for heteronuclear isotopes as

hE,(v, J) = AE”(Z& + AE”(I;u) + J(J+ l)[&(I&) +-M&)1 (4)

’ The nonadiabatic shift calculations presented in Refs. (16,19) were not formulated in this manner, but within first-order perturbation theory, the approaches used effectively reduce to this form


where the symbols Z,, Z,, II,, II,, identify the types of electronic states responsible for the coupling. He also reported numerical values of the AE” and A, constants for ail vibrational levels of (J = 0) Hz, HD, and D2 (15). In the present work, an isotope scaling procedure was developed which smoothes these results, extends them to the tritium isotopes, and introduces an implicit J dependence into his “rotationless” AE” values.

It is evident from the theory (15, 16) that the various contributions to the nonadiabatic eigenvalue corrections depend on the isotopic nuclear masses ml and m2 in the following way: AZ?‘(&) is proportional to l/p, &(I&) to 1/p2, AE”(Z,) to l/p&, and A,(&) to l/pi, where p= mlm2/(ml + m2) is the usual diatom reduced mass, and pal = mlm2/(ml - m2). The essence of the present isotopic scaling procedure is the ansatz that once this explicit mass dependence is taken into account, the various AE” and A, terms may be represented as smooth functions of the mass-reduced vibrational quantum number (40) 1 = (u + l/2)&. The validity of this assumption is illustrated by Fig. 1 and the lower segment of Fig. 2, where the quantities p2 X A&I,) and P X AE”(Z,) for H2 (round points), HD (triangular points), and D2 (square points) are plotted vs 9. The fact that the resulting points lie on a single smooth curve in each case provides convincing evidence of the validity of the proposed mass scaling procedure.6 Although values of the 2, and II, corrections were calculated for only one isotope (HD) (IS), for the sake of completeness, the quantities pctol X AE’(&) and pi X A”(&) are also plotted in Figs. 1 and 2.

The similarity between the q dependence of the AE” values seen in the lower segment of Fig. 2 and that of the expectation values of the nuclear kinetic energy (7) is of course the qualitative basis of the success of previous approximate respresentations of these corrections (9, 10, 16, 29). This suggests that rather than devise isotope- independent functions to fit the curves shown in the lower half of Fig. 2, one should incorporate the expectation value of the nuclear kinetic energy KE = (0, JIKE(u, J) into such functions. The advantage of the latter approach is demonstrated by the near constancy of the quantities ~1 X AE”(Z&(KE) and CL, X E”(Z,)/(KE) plotted in the upper half of Fig. 2. It is clear that substantially fewer empirical parameters are required to reproduce the functional behavior seen there, than that evidenced in the lower half of Fig. 2. In addition, because of the J dependence of the kinetic energy expectation values, this parameterization implicitly introduces a J dependence into our representation of the AE” values; it will be seen below that this results in significant improvements in the agreement between the theoretical and experimental level energies for high J.

In view of the above, the reported (15) AE” and A, values for Hz, HD, and DZ were fitted to the following polynomial expansions in V:

AWE& = [(UT JImIvy J)/PI IX h(Zd 71i (3

6 The fact that Table III of Ref. (15) shows identical values of A, (IQ for levels Y = 10 and 11 of Hz is clearly a typographical error, and an extrapolation to high u based on the results for HD and Dz shows thattbe values of A, (II,) tabulated (IS) for D = 12-14 Hz almost certainly belong to u = 11-13. However, the reported A, (IQ values for Hz levels II & 11 were omitted from both Fig. I and the fits described herein. Note added in proof: L. Wolniewicz ( 1986) conlirmed this conjecture and reported that his directly-calculated value ofA,&.) is -0.001061 cm-‘.


o-

. H2 A HD

-2 - . 02

-4

FIG. 1. For Hz, HD, and Dz, plot of the mass-scaled nonadiabatic eigenvalue correction constants A, vs mass reduced quantum numbers $ = (u + l/2)&; energies are in cm-’ and masses in amu.

i

0

-1

-2

0

-1 t

-2

i

4 . . . :

FIG. 2. For H2, HD, and 4, plot of the nonadiabatic eigenvalue correction constants AE” vs q = (u + 1/2)/G (with energies in cm-’ and masses in amu). Lower segment: using only reduced mass scaling of the AE” values; upper segment: using both reduced mass and kinetic energy scaling.


It was pointed out in Ref. (15) that the reported aE”(Z,) values for Hz levels D 3 7 and HD and Dz levels u > 8 included an empirical correction term. In the fit to Eq. (5), the hE”( &) values for the first half of these levels (V = 7- 10 for HZ, v = 8- 12 for HD, and u = 8-14 for Dz) were given one-fourth the statistical weight of the values for the lowest levels, while the corresponding relative weight used for the remaining levels was l/25. The coefficients of the resulting cubic polynomial for AE”(&) are listed in Table III, together with the coefficients {bi} of the polynomials representing the other nonadiabatic correction terms. (These polynomials define the smooth curves seen in Figs. 1 and 2.) A further weak 17 dependence was discerned in the deviations of the results for the different isotopes from this smooth curve; however, incorporating an extra pc-’ scaling term into Eq. (5) reduced the standard error of the fit by only 15%, so this additional systematic 7 dependence was not considered further. Most of the scatter of the directly calculated AU’(&) values about the curve defined by Eq. (5) appears to reflect real uncertainties in the former; the magnitude of this scatter appears to decrease significantly with increasing isotopic reduced mass. The resulting polynomial representation of L\E”(Z& represents the directly calculated values with weighted rms uncertainties of 0.014, 0.007, and 0.008 cm-’ for Hz, HD, and Dz, respectively.

The calculated A,!?‘(&) values for HD did not depend on an empirical correction of the type alluded to above, and the values themselves had much less scatter. As a result, the cubic polynomial of Eq. (6) and Table III fitted the reported values with a standard error of only 0.0007 cm-‘.

It is clear from Eqs. (3) and (4) that the importance of the A, terms grows with J(J + 1). As a result, our fits of the A, values to Eqs. (7) and (8) used a weighting scheme designed to give equal weights to “equivalent” high-J nonadiabatic shifts for the various isotopes. These (unnormalized) weights were [JdJo + 1)12, where J&u) is the value of the rotational quantum number required to shift level u to the dissociation limit. The maximum deviations from the resulting polynomial representations of A,@& and A,(&) correspond to energy shift discrepancies at J = JD(u) of 0.002 and 0.001 cm-‘, respectively.

In summary, therefore, the uncertainties in the adiabatic level energies, together with the apparent scatter in the AP(&) values lead to an overall estimated average uncertainty of ca. kO.0 15 cm-’ for the low-J nonadiabatic eigenvalues reported here. At high J these uncertainty estimates increase slightly because of uncertainties in both the calculated values of the A, constants and in our analytic representation of them. Other possible sources of error at high J include our neglect of centrifugal distortion

TABLE III

Polynomial Expansion Parameters (bi) of Eqs. (S)-(S), Defining Our Representations of the Nonadiabatic Level Shifts

i biok) bi( 2”) hi(nq) kin,)

0 -2.263x10-’ -9.592x1O-5 -9.151 x10-’ -8.381~10-~

: 3.744x10-t 1.101.10-; 5.671 x10-; 2.946x10-’

-~.;~;xUI-~ -9.697x1O-9 -1.1546x10- 2.654x10-’ 3 .x 3.200x10- 1.9628x10-7 -2.218x10-’ 4 ___ _-- -1.5326x10-8 1.840x10-* 5 _-- ___ 5.5306x10-lo ---

H2 ENERGIES AND MATRIX ELEMENTS 429

effects on the A, values, and possible inadequacies in our assumption that centrifugal distortion of the AE” values scale as the J dependence of (2), J(KF$, J). However, evidence of the validity of these assumptions is presented below.

III. RESULTS AND DISCUSSION

Using the procedure described above, nonadiabatic eigenvalues were calculated for all bound and quasibound vibration-rotation levels of HZ, HD, HT, Dz, DT, and T2, and the associated adiabatic wavefunctions were used to generate expectation values of the nuclear kinetic energy, of R, R2, and Re2, and of the average cllx = (a! + a&)/ 3 and anisotropy Aax = ((Y! - (Y:) of the frequency-dependent polarizability for both the static field (X = co) and X = 4880 A cases. Illustrative results for the three tritium isotopes are seen in Tables IV-VI; complete tabulations of these diagonal properties for all levels of all six isotopes, together with the tunneling predissociation widths of the quasibound levels, may be obtained from the authors on request.’

One of the reasons for initiating this work was an interest in making reliable predictions for the intensities of the Raman transitions of the various isotopes of diatomic hydrogen. Those intensities depend on off-diagonal matrix elements of the frequency- dependent polarizability anisotropy (for AJ = 0, +2) and average polarizability (for N = 0). For alI bound and quasibound vibration-rotation levels of all six isotopes of ground state hydrogen, we have therefore calculated these matrix elements for Au = 0, - 1, and -2 and AJ = 0, f2 using the frequency-dependent polarizability corresponding to X = 4880 A. Sample results of this type for the three common isotopes are seen in Tables VII-IX; complete tabulations of these 4880-A matrix elements for all six isotopes are given elsewhere (see footnote 7), while results for other frequencies may be obtained from the authors on request.

A. Polarizability Matrix Elements and Expectation Values

The expectation values of the static polarizability obtained here agree with those reported by Rychlewski (30) (for a more limited range of u and J) to within +0.0002 au for low u and +O.OOl au at higher u. The small discrepancies at low u lie within the range of uncertainties associated with the ab initio polarizability calculations, while their growth as u increases reflects the slightly improved potential (13) used here and our different treatment of the long-range part of the polarizability function. Ac- curate diagonal and off-diagonal Hz matrix elements for a wide range of frequencies but a very limited range of u and J were also reported by Rychlewski (31). Where direct comparisons can be made, they are essentially equivalent to the present results. Off-diagonal polarizability matrix elements for the static field case at a limited range of u and J have also been published by Hunt et al. (I 7). However, they were based on a less accurate polarizability function which was known over a narrower range of internuclear distance (38), and hence are superseded by the present work.

It is important to realize that the polarizability expectation values and matrix elements of interest here have a very significant frequency dependence. This is clearly illustrated by the large differences (of 520%) between the expectation values of the

’ A complete tabulation of the results described herein may be found in University of Waterloo Chemical Physics Research Report CP-30 1 (I 986) by R. J. Le Roy and C. Schwartz, copies of which may be obtained from the authors on request (see footnote 3).


TABLE IV

Nonadiabatic Eigenvalues and Certain Expectation Values for Levels of Ground State HT (Energies in cm-‘, Distances in Bohr, and Polarizabilities in Atomic Units)

STATIC A = 48806,

v J E < KE > CR> < R2,f < R-‘>-+ Aa, -L Aal h

1.9881 5.3740 2.0938 5.5499 1.9933 5.3803 2.0994 5.5567 2.0036 5.3931 2.1106 5.5702 2.0191 5.4122 2.1273 5.5905 2.0398 5.4375 2.1496 5.6176 2.0656 5.4692 2.1776 5.6513

2.3595 5.7559 2.3653 5.7626 2.3769 5.7760 2.3943 5.7961 2.4175 5.8228 2.4465 5.8560

2.5033 5.9624 2.5097 5.9695 2.5223 5.9839 2.5413 6.0054 2.5667 6.0340 2.5984 6.0696

0 0 -36512.166 0 1 -36432.713 0 2 -36274.295 0 3 -36037.874 0 4 -35724.869 0 5 -35337.128

884.34 1.439836 1.447756 883.51 1.441285 1.449205 881.86 1.444178 1.452097 879.39 1.448506 1.456424 876.12 1.454252 1.462169 872.05 1.461399 1.469313

2521.91 2519.50

1.518191 1.540892 1.519688 1.542389 1.522679 1.545378 1.527151 1.549849 1.533091 1.555788 1.540480 1.563176

1.415848 1.417300 1.420197 1.424531 1.430285 1.437440

1 0 -33077.320 1 I -33001.105 1 2 -32849.149 1 3 -32622.387 1 4 -32322.202 1 5 -31950.386

2514.68 2507.48 2497.93 2486.06

1.445603 1.447106 1.450106 1.454592 1.460549 1.467955

2 0 -29800.557 4003.82 1.599146 1.635756 1.476683 2.7604 6.1502 2.9496 6.3909 2 1 -29727.517 3999.87 1.600699 1.637309 1.478242 2.7668 6.1573 2.9566 6.3985 2 2 -29581.898 3991.99 1.603800 1.640410 1.481354 2.7796 6.1713 2.9708 6.4136 2 3 -29364.612 3980.22 1.608438 1.645050 1.486007 2.7989 6.1923 2.9921 6.4362 2 4 -29077.000 3964.59 1.614599 1.651215 1.492185 2.8245 6.2202 3.0205 6.4664 2 5 -28720.809 3945.17 1.622265 1.658886 1.499867 2.8566 6.2550 3.0560 6.5039

3 0 -26678.870 5333.81 3 1 -26608.957 5328.36 3 2 -26469.581 5317.48

1.683117 1.732900 1.509339 3.1875 1.684734 1.734518 1.510960 3.1945 1.687962 1.737751 1.514195 3.2085 1.692792 1.742589 1.519034 3.2295 1.699210 1.749018 1.525458 3.2575 1.707198 1.757022 1.533447 3.2925

6.5557 3.4311 6.8344 6.5630 3.4389 6.8424 6.5776 3.4546 6.8582

3 3 -26261.627 5301.21 3 4 -25986.404 5279.62 3 5 -25645.609 5252.78

4 0 -23710.043 4 1 -23643.226 4 2 -23510.027 4 3 -23311.313 4 4 -23048.356 4 5 -22722.810

6513.98 1.770624 1.832955 1.543889 3.6377 6.9703 3.9449 7.2911 6507.03 1.772315 1.834651 1.545579 3.6453 6.9779 3.9535 7.2994 6493.15 1.775693 1.838039 1.548953 3.6604 6.9930 3.9706 7.3159 6472.40 1.780748 1.843110 1.554000 3.6829 7.0156 3.9962 7.3406 6444.85 1.787467 1.649852 1.560701 3.7130 7.0457 4.0304 7.3735 6410.58 1.795834 1.858249 1.569036 3.7506 7.0831 4.0730 7.4144

6.5995 3.4782 6.8819 6.6286 3.5096 6.9135 6.6648 3.5488 6.9528

7.3914 4.4852 7.7581 7.3991 4.4943 7.7666 7.4147 4.5127 7.7837 7.4379 4.5401 7.8093 7.4687 4.5767 7.8433

5 0 -20892.705 7544.62 1.862327 1.936673 1.580738 4.1058 5 1 -20828.967 7536.14 1.864107 1.938461 1.582508 4.1138 5 2 -20701.918 7519.22 1.867663 1.942034 1.586041 4.1298 5 3 -20512.401 7493.90 1.872986 1.947384 1.591326 4.1537 5 4 -20261.657 7460.26 1.880065 1.954499 1.598345 4.1855 5 5 -19951.300 7418.43 1.888885 1.963367 1.607077 4.2251 7.5071 4.6224 7.8856

6 0 -18226.406 8423.97 1.959077 2.044978 1.620417 4.5841 7.8153 5.0450 8.2320 7 0 -15711.716 9147.90 2.061986 2.159038 1.663624 5.0619 8.2374 5.6144 8.7079 8 0 -13350.363 9709.44 2.172538 2.280378 1.711308 5.5250 8.6515 6.1756 9.1783 9 0 -11145.411 10098.17 2.292751 2.411038 1.764780 5.9557 9.0504 6.7003 9.6314

10 0 -9101.503 10299.30 2.425452 2.553840 1.825919 6.3308 9.4244 7.1545 10.0534 11 0 -7225.175 10292.55 2.574718 2.712811 1.897504 6.6224 9.7617 7.5078 10.4328 12 0 -5525.272 10050.48 2.746664 2.893957 1.983846 6.7987 10.0474 7.7294 10.7582 13 0 -4013.505 9536.16 2.950954 3.106706 2.092037 6.7885 10.2632 7.7731 11.0126 14 0 -2705.212 8699.77 3.203976 3.366984 2.234733 6.5270 10.3854 7.5711 11.1713 15 0 -1620.404 7473.85 3.536505 3.704636 2.437264 5.9589 10.3831 7.0104 11.1925 16 0 -785.207 5766.33 4.016767 4.185975 2.760486 4.9319 10.2141 5.8999 11.0047 17 0 -233.725 3455.29 4.853560 5.016475 3.409282 3.2822 9.8172 3.9951 10.5373 18 0 -6.531 530.66 8.1298 8.4319 6.33555 0.7027 9.1382 0.8392 9.6858

static and X = 4800 A polarizabilities seen in Tables IV-VI. Similar differences arise for the analogous off-diagonal matrix elements. Thus, results obtained for one frequency should not be lightly used for another.

Centrifugal distortion can also cause massive changes in these polarizability expectation values and matrix elements. While modest examples of this behavior appear in Tables IV-XI, much more dramatic effects are seen in the complete tabulations referred to in footnote 7. They show, for example, that the expectation values of the polarizability anisotropy and average polarizability for the highest rotational sublevel (J = 38) of


TABLE V

Nonadiabatic Eigenvalues and Certain Expectation Values for Levels of Ground State DT (Energies in cm-‘, Distances in Bohr, and Polarizabilities in Atomic Units)

STATIC A = 4880A

VJ E < KE >


TABLE VI

Nonadiabatic Eigenvalues and Certain Expectation Values for Levels of Ground State T2 (Energies in cm-‘, Distances in Bohr, and Polarizabilities in Atomic Units)

STAT I C x = 4880A

VJ E ’ -* ba .m L hl 8,

0 0 -37028.481 629.02 1.428352 1.433953 1.411433 1.9354 5.3183 2.0359 0 1 -36988.418 628.72 1.429074 1.434676 1.412157 1.9379 5.3215 2.0386 0 2 -36908.415 628.14 1.430518 1.436119 1.413602 1.9430 5.3278 2.0441 0 3 -36788.716 627.26 1.432681 1.438282 1.415767 1.9506 5.3373 2.0523 0 4 -36629.686 626.09 1.435559 1.441159 1.418648 1.9608 5.3500 2.0633 0 5 -36431.803 624.64 1.439147 1.444746 1.422239 1.9735 5.3658 2.0770

1 0 -34563.983 1820.96 1.483301 1.499561 1.432241 2.1917 5.5859 2.3179 1 1 -34525.075 1820.10 1.484040 1.500300 1.432982 2.1945 5.5892 2.3209 1 2 -34447.381 1818.37 1.485517 1.501776 1.434463 2.2000 5.5958 2.3269 1 3 -34331.141 1815.79 1.487730 1.503987 1.436682 2.2083 5.6056 2.3360 1 4 -34176.710 1812.36 1.490673 1.506930 1.439633 2.2194 5.6187 2.3480 1 5 -33984.558 1808.09 1.494344 1.510599 1.443312 2.2332 5.6351 2.3630

5.4900 5.4933 5.5001 5.5102 5.5236 5.5404

5.7784 5.7819 5.7889 5.7995 5.8135 5.8310

6.0752 6.0788 6.0861 6.0971 6.1116 6.1298

2 0 -32179.461 2 1 -32141.692 2 2 -32066.273 2 3 -31953.438 2 4 -31803.538 2 5 -31617.032

2933.88 1.539473 1.565920 2932.47 1.540230 1.566677 2929.64 1.541743 1.568189 2925.41 1.544010 1.570455 2919.77 1.547026 1.573471 2912.75 1.550787 1.577231

1.453665 1.454425 1.455944 1.458218 1.461244 1.465017

3 0 -29873.687 3969.45 1.596997 1.633214 3 1 -29837.042 3967.49 1.597774 1.633991 3 2 -29763.868 3963.58 1.599327 1.635544 3 3 -29654.396 3957.72 1.601653 1.637871 3 4 -29508.968 3949.93 1.604749 1.640968 3 5 -29328.036 3940.22 1.608609 1.644830

1.475783 2.7499 6.1401 2.9376 1.476563 2.7531 6.1436 2.9412 1.478122 2.7595 6.1506 2.9482 1.480456 2.7691 6.1612 2.9589 1.483562 2.7819 6.1752 2.9730 1.487434 2.7979 6.1927 2.9908

4 0 -27645.628 4928.91 1.656023 1.701641 1.498686 4 1 -27610.095 4926.42 1.656822 1.702441 1.499487 4 2 -27539.142 4921.44 1.658419 1.704039 1.501089 4 3 -27432.996 4913.98 1.660811 1.706434 1.503488 4 4 -27291.993 4904.06 1.663996 1.709621 1.506680 4 5 -27116.576 4891.70 1.667966 1.713597 1.510659

2.4638 5.8599 2.6187 5.8633 2.6220 5.8701 2.6286 5.8803 2.6384 5.8939 2.6515 5.9109 2.6678

2.4668 2.4728 2.4817 2.4937 2.5086

6.3798 6.3836 6.3912 6.4025 6.4176 6.4365

6.6921 6.6960 6.7038 6.7156 6.7312 6.7507

3.0493 6.4259 3.0527 6.4295

3.2744 3.2783 3.2859 3.2973 3.3126 3.3316

3.0595 6.4367 3.0698 6.4476 3.0834 6.4620 3.1005 6.4800

5 0 -25494.449 5813.12 1.716729 1.771418 1.522481 3.3609 6.7166 3.6282 7.0113 5 1 -25460.018 5810.09 1.717552 1.772243 1.523306 3.3645 6.7203 3.6323 7.0153 5 2 -25391.267 5804.05 1.719198 1.773892 1.524954 3.3717 6.7278 3.6404 7.0234 5 3 -25288.419 5795.01 1.721664 1.776363 1.527423 3.3826 6.7389 3.6526 7.0355 5 4 -25151.804 5782.97 1.724947 1.779653 1.530708 3.3970 6.7537 3.6689 7.0516 5 5 -24981.857 5767.97 1.729041 1.783756 1.534803 3.4151 6.7722 3.6893 7.0717

6 0 -23419.519 6622.48 1.779322 1.842789 1.547297 3.6832 7.0115 3.9970 7.3365 7 0 -21420.429 7356.94 1.844050 1.916031 1.573286 4.0142 7.3094 4.3788 7.6669 8 0 -19497.002 8015.92 1.911209 1.991467 1.600632 4.3514 7.6093 4.7712 8.0011 9 0 -17649.309 8598.25 1.981156 2.069476 1.629558 4.6912 7.9095 5.1715 8.3378

10 0 -15877.696 9102.14 2.054327 2.150510 1.660341 5.0294 8.2081 5.5754 8.6749 11 0 -14182.808 9525.04 2.131253 2.235116 1.693321 5.3608 8.5031 5.9764 9.0097 12 0 -12565.617 9863.56 2.212598 2.323966 1.728927 5.6797 8.7919 6.3647 9.3381 13 0 -11027.472 10113.31 2.299197 2.417896 1.767708 5.9789 9.0716 6.7285 9.6554 14 0 -9570.136 10268.73 2.392111 2.517966 1.810371 6.2493 9.3385 7.0556 9.9566 15 0 -8195.854 10322.86 2.492717 2.625535 1.857854 6.4807 9.5887 7.3359 10.2377 16 0 -6907.417 10267.07 2.602834 2.742393 1.911420 6.6655 9.8172 7.5597 10.4954 17 0 -5708.255 10090.71 2.724914 2.870938 1.972819 6.7916 10.0185 7.7140 10.7247 18 0 -4602.541 9780.64 2.862361 3.014489 2.044548 6.8230 10.1860 7.7830 10.9198 19 0 20 0 21 0 22 0 23 0 24 0

3.177789 2.130324 6.7348 10.3119 7.7452 11.0729 3.367931 2.235943 6.5306 10.3868 7.5717 11.1735 3.596142 2.371050 6.1623 10.3995 7.2197 11.2057 3.881724 2.553197 5.5869 10.3355 6.6238 11.1445 4.262160 2.818717 4.7582 10.1772 5.7010 10.9600 4.826591 3.260204 3.6030 9.9000 4.3715 10.6338

$5 8 -f&j 1894.03 27.95 22.18 5.753156 2:.;q2095 . l4”.$$;$23 0.0578 2.0054 8.53 9.4745 0.0671 2.4620 10.1360 9.00

-3595.327 -2692.717 -1902.071 -1232.281

-694.103 -300.518

9320.63 8690.59 7865.62 6814.62 5498.77 3871.63

3.020061 3.205335

:%:z’: 4:094824 4.665359

B. Comparison with Experimental Energies

The molecular hydrogen system has long been a key meeting ground for theory and experiment, so a careful examination of the nature of the remaining discrepancies


TABLE VII

For Ground State Hz, Matrix Elements of the Frequency-Dependent Polarizability (in Atomic Units) Required for Predicting the Intensities of Raman Transitions Excited by X = 4880 A Radiation

< v,JjAa~jv'.J'> < v,JJ&Jv',J >

dv=O dv-0 dv=O dv=-1 dv=-1 dv=-1 dv=-2 dv=-2 dv=-2 dv=-1 dv=-2 VJ E dJ=-2 dJ=O dJ=2 dJ=-2 dJ=O dJ=2 dJ=-2 dJ=O dJ=2 dJ=O dJ=O

0 0 -36118.074 0.0 2.1392 2.1514 0.0 0.0 0.0 0.0 0.0 0.0 00 0 1 -35999.587 0.0 2.1476 2.1677 0.0 0.0 0.0 0.0 0.0 0.0 ::i 0'0 0 2 -35763.700 2.1514 2.1646 2.1920 0.0 0.0 0.0 0.0 0.0 0'0 0 3 -35412.555 2.1677 2.1901 2.2246 0.0 0.0 0.0 0.0 0.0 i:: ::i 0:o

1 0 -31956.927 0.0 2.6507 2.6641 0.0 0.6622 0.7094 0.0 0.0 i::

0.7883 0.0 1 1 -31844.352 0.0 2.6606 2.6815 0.0 0.6642 0.7431 0.0 0.0 0.7895 0.0 1 2 -31620.254 2.6641 2.6803 2.7075 0.6209 0.6682 0.7790 0.0 0.7919 0.0 1 3 -31286.700 2.6815 2.7100 2.7423 0.5952 0.6742 0.8173 0.0 ii:: i:: 0.7955 0.0

2 0 -28031.088 0.0 3.2166 3.2309 0.0 1.0250 1.1029 0.0 -0.0072 0.0033 1.1510 -0.0711 2 1 -27924.296 0.0 3.2278 3.2489 0.0 1.0278 1.1580 0.0 -0.0074 0.0107 1.1527 -0.0714 2 2 -27711.728 3.2309 3.2504 3.2759 0.9556 1.0336 1.2163 -0.0169 -0.0078 0.0185 1.1559 -0.0718 2 3 -27395.385 3.2489 3.2842 3.3120 0.9120 1.0422 1.2779 -0.0229 -0.0084 0.0266 1.1607 -0.0726

3 0 -24335.689 0.0 3.8331 3.8479 0.0 1.3588 1.4693 0.0 -0.0285 -0.0100 1.4473 -0.1349 3 1 -24234.591 0.0 3.8457 3.8659 0.0 1.3623 1.5467 0.0 -0.0290 0.0033 1.4491 -0.1354 3 2 -24033.379 3.8479 3.8709 3.8930 1.2587 1.3693 1.6280 -0.0459 -0.0300 0.0171 1.4527 -0.1364 3 3 -23733.991 3.8659 3.9086 3.9291 1.1952 1.3797 1.7132 -0.0566 -0.0314 0.0316 1.4582 -0.1378

4 0 -20867.698 0.0 4.4912 4.5060 0.0 1.6768 1.8233 0.0 -0.0692 -0.0427 1.7036 -0.2100 4 1 -20772.252 0.0 4.5050 4.5230 0.0 1.6806 1.9250 0.0 -0.0700 -0.0238 1.7054 -0.2107 4 2 -20582.315 4.5060 4.5324 4.5486 1.5416 1.6881 2.0308 -0.0941 -0.0718 -0.0040 1.7091 -0.2123 4 3 -20299.765 4.5230 4.5736 4.5828 1.4550 1.6993 2.1409 -0.1093 -0.0745 0.0166 1.7145 -0.2146

5 0 -17626.119 0.0 5.1780 5.1918 0.0 1.9713 2.1581 0.0 -0.1370 -0.1035 1.9236 -0.3000 5 1 -17536.336 0.0 5.1927 5.2067 0.0 1.9750 2.2866 0.0 -0.1385 -0.0794 1.9253 -0.3011 5 2 -17357.698 5.1918 5.2219 5.2289 1.7959 1.9824 2.4191 -0.1688 -0.1414 -0.0543 1.9285 -0.3034 5 3 -17092.030 5.2067 5.2655 5.2586 1.6825 1.9933 2.5557 -0.1880 -0.1458 -0.0284 1.9332 -0.3068

6 0 -14612.257 0.0 5.8732 5.8848 0.0 2.2273 2.4587 0.0 -0.2393 -0.2006 2.1025 -0.4082 7 0 -11830.105 0.0 6.5383 6.5455 0.0 2.4156 2.6962 0.0 -0.3844 -0.3439 2.2274 -0.5378 8 0 -9286.901 0.0 7.1148 7.1149 0.0 2.4892 2.8221 0.0 -0.5848 -0.5485 2.2769 -0.6924 9 0 -6993.907 0.0 7.5434 7.5336 0.0 2.3957 2.7833 0.0 -0.8453 -0.8227 2.2263 -0.8710 10 0 -4967.499 0.0 7.7658 7.7423 0.0 2.0996 2.5415 0.0 -1.1361 -1.1406 2.0559 -1.0564 11 0 -3230.727 0.0 7.6885 7.6456 0.0 1.5693 2.0594 0.0 -1.3877 -1.4375 1.7458 -1.2105 12 0 -1815.600 0.0 7.1523 7.0796 0.0 0.7778 1.2979 0.0 -1.4947 -1.6111 1.2744 -1.2738 13 0 -766.456 0.0 5.8568 5.7354 0.0 -0.2466 0.2579 0.0 -1.2839 -1.4817 0.6380 -1.1435 14 0 -144.606 0.0 3.3379 3.1238 0.0 -1.1403 -0.7605 0.0 -0.5363 -0.7782 -0.0351 -0.6634

between the two is desirable. For the dissociation energies and the pure vibrational spacings of the three common isotopes, this question has been carefully addressed in Refs. (13, 15), while for the tritium-containing isotopes, comparisons are presented in Refs. (I, 2). Within the uncertainties discussed in Section II.C (largely associated with the AE”(&) corrections), our nonadiabatic eigenvalues for the pure vibrational (J = 0) levels of Hz, HD, and Dr agree with those reported by Kilos et al. (13). Their discrepancies with experimental are fairly small, and are not discussed in detail here. However, the situation is quite different at high J.

For all of the observed (u, J) levels of ground state HZ, HD, and Dr , the differences (calculated minus observed) between the present nonadiabatic vibration-rotation energies [J&,(t), J) - l&,(0, 0)] and the analogous experimental level spacings (18, 41, 42) are listed in Tables X and XI. For HD and Ds, the only analogous prior comparisons between theory and experiment had failed to incorporate nonadiabatic effects in the calculated energies, so the poor agreement then obtained (41, 42) was not surprising. However, in her recent reanalysis of the experimental data for Hz, Dabrowski (18) compared her updated experimental energies with theoretical values which did include nonadiabatic corrections. The discrepancies she obtained, shown in parentheses in Table X, were small for the pure vibrational energies but increased systematically with


TABLE VIII

For Ground State HD, Matrix Elements of the Frequency-Dependent Polarizability (in Atomic Units) Required for Predicting the Intensities of Raman Transitions Excited by X = 4880 A Radiation

tv,J(Aa~lv',J' >


TABLE IX

For Ground State D2, Matrix Elements of the Frequency-Dependent Polarizability (in Atomic Units) Required for Predicting the Intensities of Raman Transitions Excited by A = 4880 A Radiation

< v,Jl$,(v'.J'>

dv=O dv-0 dv=O dv=-I dv=-1 dv=-1 dv=-2 dv=-2 dv=-2 dv=-1 dv=-2 VJ E dJ=-2 dJ=O dJ=2 dJ=-2 dJ=O dJ=2 dJ=-2 dJ=0 dJ=2 dJ=0 dJ=O

0 0 -36748.349 0.0 2.0671 2.0732 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0 1 -36688.569 0.0 2.0713 2.0812 0.0 0.0 0 2 -36569.282 2.0732 2.0795 2.0933 0.0

i.i E E K?i 0.0

E 0"::

0 3 -36391.035 2.0812 2.0920 2.1094 0.0 010 0:O 010 0:O 0.0 010 i::

1 0 -33754.742 0.0 2.4174 2.4239 0.0 0.5413 0.5676 0.0 0.0 0.6563 0.0 1 1 -33697.072 0.0 2.4221 2.4324 0.0 0.5422 0.5860 0.0

00.:

Ki

010 0.6568 0.0

1 2 -33581.999 2.4239 2.4313 2.4451 0.5176 0.5439 0.6053 0.0 0.6578 0.0 1 3 -33410.055 2.4324 2.4452 2.4620 0.5025 0.5464 0.6255 0.0 0:O 0.0 0.6593 0.0

2 0 -30880.242 0.0 2.7955 2.8024 0.0 0.8188 0.8604 0.0 -0.0027 0.0024 0.9509 -0.0484 2 1 -30824.645 0.0 2.8006 2.8112 0.0 0.8200 0.8894 0.0 -0.0027 0.0060 0.9516 -0.0485 2 2 -30713.712 2.8024 2.8109 2.8244 0.7808 0.8224 0.9197 -0.0075 -0.0028 0.0097 0.9530 -0.0487 2 3 -30547.963 2.8112 2.8263 2.8420 0.7566 0.8261 0.9514 -0.0106 -0.0030 0.0135 0.9551 -0.0489

3 0 -28122.759 0.0 3.2004 3.2076 0.0 1.0668 1.1236 0.0 -0.0115 -0.0025 3 1 -28069.205 0.0 3.2061 3.2166 0.0 1.0683 1.1631 0.0 -0.0116 0.0038 3 2 -27962.353 3.2076 3.2173 3.2302 1.0144 1.0712 1.2041 -0.0201 -0.0119 0.0104 3 3 -27802.710 3.2166 3.2342 3.2482 0.9808 1.0757 1.2468 -0.0256 -0.0123 0.0172

4 0 -25480.643 0.0 3.6311 3.6385 0.0 1.3035 1.3764 0.0 -0.0280 -0.0150 4 1 -25429.110 0.0 3.6372 3.6476 0.0 1.3052 1.4268 0.0 -0.0282 -0.0058 4 2 -25326.294 3.6385 3.6494 3.6612 1.2356 1.3086 1.4790 -0.0404 -0.0287 0.0036 4 3 -25172.691 3.6476 3.6677 3.6794 1.1920 1.3137 1.5332 -0.0483 -0.0294 0.0134

5 0 -22952.701 0.0 4.0844 4.0918 0.0 1.5325 1.6227 0.0 -0.0539 -0.0370 5 1 -22903.174 0.0 4.0909 4.1008 0.0 1.5343 1.6848 0.0 -0.0543 -0.0250 5 2 -22804.366 4.0918 4.1040 4.1141 1.4478 1.5380 1.7489 -0.0702 -0.0550 -0.0127 5 3 -22656.760 4.1008 4.1236 4.1320 1.3930 1.5435 1.8150 -0.0804 -0.0561 0.0

6 0 -20538.231 0.0 4.5562 4.5636 7 0 -18237.066 0.0 5.0422 5.0492 8 0 -16049.615 0.0 5.5356 5.5421 9 0 -13976.943 0.0 6.0254 6.0309 10 0 -12020.847 0.0 6.4944 6.4982 11 0 -10183.972 0.0 6.9204 6.9220 12 0 -8469.946 0.0 7.2825 7.2810 13 0 -6883.561 0.0 7.5626 7.5574 14 0 -5430.997 0.0 7.7383 7.7285 15 0 -4120.111 0.0 7.7788 7.7631 16 0 -2960.825 0.0 7.6402 7.6169 17 0 -1965.632 0.0 7.2559 7.2221 18 0 -1150.258 0.0 6.5159 6.4671 19 0 -534.551 0.0 5.2702 5.2007 20 0 -143.336 0.0 3.3427 3.2395 21 0 -1.648 0.0 0.4162 0.0

1.1900 -0.0893 1.1908 -0.0894 1.1925 -0.0898 1.1949 -0.0902

1.4002 -0.1347 1.4011 -0.1350 1.4029 -0.1354 1.4056 -0.1361

1.5898 -0.1860 1.5908 -0.1863 1.5926 -0.1870 1.5954 -0.1880

0.0 1.7522 1.8612 0.0 -0.0924 -0.0716 1.7614 -0.2444 0.0 1.9591 2.0885 0.0 -0.1459 -0.1218 1.9148 -0.3110 0.0 2.1478 2.2992 0.0 -0.2169 -0.1901 2.0481 -0.3868

i:! 2.4296 2.3093 2.4844 2.6300 0.0 0.0 -0.3082 -0.4241 -0.2797 -0.3956 2.1573 2.2363 -0.4731 -0.5712

i:: 2.4896 2.4704 2.7168 2.7253 0.0 0.0 -0.5702 -0.7483 -0.5439 -0.7275 2.2764 2.2686 -0.6827 -0.8073

Ki 2.1428 2.3583 2.6416 2.4543 0.0 0.0 -0.9507 -1.1585 -0.9397 -1.1628 2.0822 2.2059 -0.9405 -1.0721

00.: 0:o

1.8111 1.3520 2.1490 1.7124 0.0 0.0 -1.3472 -1.4827 -1.3737 -1.5394 1.6201 1.8895 -1.1888 -1.2715 0.7593 1.1342 0.0 -1.5160 -1.6110 1.2674 -1.2925

0.0 0.0420 0.4160 0.0 -1.3788 -1.5186 0.8299 -1.2122

E -0.7314 -1.2629 -0.9992 -0.3869 0.0 0.0 -0.9866 -0.2794 -1.1694 -0.4740 -0.1152 0.3289 -0.9779 -0.5338 0.0 -0.5865 -0.5147 0.0 0.2160 0.1444 -0.1398 -0.0200

terms at highly J by factors as small as one-fifth, and the rate at which this damping factor changes grows rapidly with J. The improved agreement with experiment which the present treatment yields (see Table X) therefore provides convincing evidence of the validity of this scaling procedure.

In view of the success of the kinetic energy scaling introduced into the AE” corrections, it behooves us to examine the question of whether an analogous scaling should be introduced into the A, constants. Consideration of the nature of the interactions giving rise to these terms [Eq. (28) of Ref. (Z5)] suggests that the A,(&,) constants might vary with the exception value (Rm4) and the A,(&) constants with (R-*). However, dividing the directly calculated (15) A, values by these expectation values did not yield the same qualitative simplification illustrated by the difference between the lower and upper segments of Fig. 2. In particular, while the quantity PZ X A,(&)/ (R-*) becomes approximately constant (except for 7 B 20), CL* X LI,(II,)/(R-~) does not. Moreover, the lack of any pronounced Jdependence in the discrepancies seen in


TABLE X

For Ground State Hz, Comparison with Experiment (18) (Calculated Minus Observed, in cm-‘) of the Present Nonadiabatic Vibration-Rotation Energies [E,(u, J) - Em (0, O)] (Results in Parentheses are the Analogous Differences Reported by Dabrowski (18))

J v=o v=l V=2 v=3 v-4 v=5 V=6 v=7

0 o.oo( 0.00) O.Ol( 0.04) 0.06( 0.11) 0.02( 0.10) 0.07( 0.17) 0.03( 0.15) 0.04( 0.17) 0.06( 0.19) 1 -O.Ol(-0.01) -0.03( 0.00) -0.03( 0.02) -0.03( 0.04) O.Ol( 0.11) 0.03( 0.15) 0.04( 0.17) -O.Ol( 0.11)

: 0.02( 0.04) O.OO( 0.03) 0.06( 0.11) 0.03( 0.11) 0.06( 0.16) O.lO( 0.19) 0.07( 0.19) 0.04( 0.15) -0.02( 0.00 -0.04(-0.01) -O.Ol( 0.04) -0.06( 0.02) 0.04( 0.13) 0.03( 0.11) 0.03( 0.12) O.Ol( 0.10)

4 0.02( 0.06 O.OO( 0.03) 0.03( 0.07) 0.04( 0.11) 0.07( 0.15) 0.09( 0.14) 0.07( 0.13) 0.14( 0.19)

z -0.02( 0.03) -0.04(-0.03) 0.04( 0.07) O.Ol( 0.07) 0.03( 0.09) 0.03( 0.04) 0.06( 0.08) 0.07( 0.08) 0.14( 0.22) O.lO( 0.11) -O.OS(-0.06) -0.05( 0.01) 0.08( 0.13) 0.13( 0.12) 0.09( 0.05) O.lO( 0.05)

i -o.lo( 0.01) 0.07( 0.08) -0.06(-0.06) O.Ol( 0.06) O.Ol( 0.05) 0.04(-0.01) 0.02(-0.07) 0.02(-0.10) 0.21( 0.35) -0.20(-0.19) 0.00(-0.02) 0.04( 0.08) 0.23( 0.26) 0.09( 0.01) 0.13(-0.02) 0.06(-0.15)

9 0.15( 0.16) -0.07(-0.11) -0.04(-0.03) 0.04( 0.03) -0.10) 0.07(-0.14) 0.13(-0.17)

K -0.06(-0.05) -0.12(-0.18) O.lO( 0.08) 0.07( 0.03) -0.01) 0.21(-0.08) 0.17(-0.23)

0.03( 0.28) -0.15(-0.13) -O.OZ(-0.10) O.OO(-0.06) 0.06(-0.02) 0.17(-0.05) 0.15(-0.21) 0.22(-0.28)

:: -o.ol( 0.27) 0.17( 0.19) 0.05(-0.06) 0.04(-0.06) O.OS(-0.05) 0.03(-0.26) 0.14(-0.20) 0.09(-0.54) -0.03( 0.29) 0.09( 0.13) -0.15(-0.28) -0.09(-0.25) 0.02(-0.18) 0.12(-0.24) O.ll(-0.42) 0.30(-0.46)

:: 0.14( 0.21) -O.ll(-0.25) 0.06(-0.14) 0.28(-0.01) 0.24(-0.20) 0.20(-0.43) 0.22(-0.70)

0.15( 0.48) 0.02( 0.03) -0.09(-0.30) -0.09(-0.41) 0.06(-0.38) 0.16(-0.45) 0.22(-0.60) 0.41(-0.75)

:7" 0.02( 0.06) 0.01(-0.22) -0.03(-0.41) O.lO(-0.42) 0.13(-0.59) 0.37(-0.60) 0.33(-1.05) -0.05( 0.00) -0.18(-0.41) -0.13(-0.56) -0.09(-0.69) O.ll(-0.73) 0.18(-0.95) 0.38(-1.23)

:: -0.03( 0.39) -0.11(-0.04) -0.02(-0.26) -0.07(-0.54) 0.13(-0.56) 0.39(-0.58) 0.58(-0.76) 0.44(-1.41) 0.06( 0.51) -0.38(-0.30) -0.25(-0.51) -0.25(-0.77) 0.07(-0.69) 0.33(-0.76) 0.43(-1.12) 0.35(-1.77)

20 -0.20( 0.28) -0.04( 0.05) 0.02(-0.25) 0.07(-0.49) 0.27(-0.58) 0.50(-0.73) 0.68(-1.09) 0.39(-2.03) 21 -0.16( 0.35) -0.38 -0.28 0.04(-0.26 -0.07 -0.69 0 25 0 68 0 46 0 92 0.20 1.81 0.08(-2.94)

:: -0 281-O 181 0.17(-O 161 0 101-O 571 : 1: ’ 1 ’ 1: 1 1: 1 0 63 0.40 0.71 0.85 0.74 1.53 0.44( 3.03) -

-0.04( 0.52) -0:22(-0:lZ) 0.16(-0:20) 0:32(-0:42) 0.57(-0.58) 0.60(-1.17) 0.45(-2.37)

2'5" -0.19( 0.38) -0.52(-0.44) 0.07(-0.35) 0.40(-0.89) 1.23(-1.08) 0.46(-2.74) -0.39( 0.20) -0.26(-0.18) 0.36(-0.10) 0.32(-0.58) O.Ol(-1.43) 0.61(-1.97)

:7" -0.24(-0.19) 0.55( 0.03) 0.91(-1.10) 0.44(-2.45)

-0.65(-0.04) O.OO( 0.01) 0.58(-0.55) 0.60(-1.63) .,o

;; -0.67(-0.04) 0.22(-1.63) -1.37(-4.31) J v=8 v=9 " = 10 " = 11 " = 12 " = 13 " = 14

0 o.Ol( 0.12) 0.08( 0.15) O.ll( 0.19) 0.22( 0.35) 0.27( 0.26) 0.26( 0.32) 0.09( 0.02) 1 0.02( 0.11) 0.06( 0.11) 0.13( 0.19) 0.20( 0.31) 0.22( 0.17) 0.25( 0.26) O.ll(-0.02) 2 0.05( 0.12) O.ll( 0.14) 0.18( 0.20) 0.23( 0.28) 0.28( 0.16) 0.28( 0.20) 0.19(-0.07) 3 0.04( 0.11) 0.08( 0.06) 0.16( 0.11) 0.21( 0.18) 0.24( 0.02) 0.24( 0.03) 0.08(-0.39) 4 0.14( 0.13) 0.08) 0.26( 0.14 0.30( 0.16) 0.29(-0.07) 0.31(-0.09) 0.05(-0.81) 5 O.lO( 0.04) 0.25(-0.03) 0.28(-0.27) 0.24(-0.41) 6 0.17( 0.04) 0.23(-0.05) 0.28(-0.06) 0.30(-0.15) 0.21(-0.56) 0.23(-0.73) 7 0.16(-0.06) 0.19(-0.21) 0.27(-0.21) 0.31(-0.35) 0.32(-0.71) 0.23(-1.13) 8 0.13(-0.20) 0.26(-0.28) 0.13(-0.51) 0.35(-0.54) 0.31(-1.04) 9 0.23(-0.22) 0.29(-0.40) 0.35(-0.49) 0.39(-0.78) 0.33(-1.33) 10 0.22(-0.39) 0.36(-0.51) 0.24(-0.82) 0.34(-1.15) 0.41(-1.79) 11 0.41(-0.36) 0.28(-0.78) 0.37(-0.96) 0.33(-1.52) 12 0.46(-0.50) 0.47(-0.81) 0.42(-1.24) 0.31(-2.00) 13 0.35(-0.80) 0.42(-1.10) 0.37(-1.66) 0.28(-2.74) 14 0.53(-0.83) 0.50(-1.31) 0.27(-2.16) 0.05(-3.74) 15 0.43(-1.21) 0.44(-1.75) 0.33(-2.79) 16 0.70(-1.19) 0.49(-2.09) 17 0.47(-1.71) 0.271~2.93) 18 0.21(-2.32) 0.04(-3.76) 19 -0.26(-3.42) 20 -O.Ol(-3.69)

Tables X and XI suggest that the A, constants are not significantly a&&d by centrifugal distortion effects, at least not for the homonuclear isotope case (Hz) for which experimental results are available over a wide range of J. Thus, introduction of some sort of expectation value scaling into the A, constants does not seem to be justifiable at this time.

IV. CONCLUDING REMARKS

Through a carell treatment of the available ab initio results, the present study has yielded accurate effective adiabatic potentials for all isotopes of diatomic hydrogen [Eq. (1) and Table II] which incorporate relativistic and radiative corrections. On


TABLE XI

For Ground State HD and D2, Comparison with Experiment (41,42) (Calculated Minus Observed, in cm-‘) of the Present Nonadiabatic Vibration-Rotation Energies [E&J J) - E, (0, 0)]

v=o v=l v=z v=3 v=4 v-5 "16

J HO O2 HO D, HO D, HD O2 HO D, HD D, HD Dz

0 0.00 0.00 -0.02 0.01 -0.01 -0.35 -0.11 -0.12 0.00 -0.09 -0.14 -0.07 0.00 -0.15 : -0'05 0 00

-0:Ol

-0 0'06 01

0:06

-0 -0:04 02 0 0'07 00

0:08

0'00 0 18

0:06

-0 -0'23 31

0:02

-0'01 0 06

0:02

-0 -0'11 10

-0:OS

-0'02 0 09

0:06

-0 0'16 07

3 IO:09

-0'09 0 08

-0:Ol

-0'10 -0 07

-0:07

0'04 0 08

-0:06

-0 -0'16 10

-0:lO 4 -0.14 0.08 -0.03 0.10 -0.01 -0.09 -0.04 -0.10 -0.06 -0.16 -0.09 -0.25 -0.03 -0.09 5 -0.14 0.13 -0.17 0.18 0.07 -0.08 0.09 -0.23 0.05 -0.30 0.01 0.05 0.06 0.01 6 -0.59 0.14 0.25 -0.03 -0.07 0.00 -0.47 -0.08 0.09 -0.01 0.22 -0.08 -0.18 7 -0.25 0.04 -0.02 0.07 0.09 -0.50 0.06 -0.64 0.00 -0.24 -0.06 8 -0.01 0.17 0.61 0.54 -2.54 -2.97 0.61 -5.07 0.25 9 -" 18

v-7 v=8

J HD D, HO D2

0 -0.06 -0.10 -0.06 -0.14 1 0.05 -0.09 0.03 -0.12 2 -0.05 -0.12 -0.05 -0.12 3 0.03 -0.09 0.07 -0.08 4 -0.04 -0.14 -0.07 -0.13 5 0.04 -0.16 0.04 -0.11 6 0.04 -0.08 -0.03 0.05 7 0.06 0.28 -0.02 0.02 8 -0.14 0.12 0.23 9 -0.30

v-9

HD D2

-0.04 -0.07 0.03 -0.15 -0.05 -0.12 0.06 -0.07 -0.02 -0.15 0.10 -0.13 -0.04 -0.06 0.11 -0.18 0.04

” = 10

HO D2

-0.02 -0.13 0.11 -0.12 0.10 -0.12 0.07 -0.12 0.02 -0.13 0.11 -0.21 -0.10 -0.01 0.06 -0.27

" = 11 " = 12 " = 13

HD 0, HD D, HO O2

0.02 -0.16 0.06 -0.14 0.04 -0.15 0.06 -0.15 0.14 -0.13 -0.17 -0.12 0.03 -0.17 0.11 -0.16 0.12 -0.12 0.17 -0.18 0.07 -0.13 0.08 -0.13 0.06 -0.05 0.18 -0.18 -0.04 -0.16 0.00 -0.08 -0.01 -0.08 0.25 -0.15

0.05

0.06 -0.13 0.20 -0.13 0.05 -0.14 0.15 -0.15 0.17 -0.01 0.06 -0.09

-0.04 0.27

" = 14 " = 15 " = 16 Y = 17 " = 18 19 20 21 --~--- J HD D, HD D2 HD D, HD O2 02 02 0,x

0 0.05 -0.12 0.11 -0.09 0.00 -0.09 0.78 -0.08 -0.02 -0.03 -0.17 -0.12 1 0.22 -0.13 0.17 -0.12 0.08 -0.06 -0.72 -0.05 0.00 -0.02 -0.16 -0.48 : 0.32 0.08 -0.11 -0.13 0.16 0.05 -0.08 -0.09 0.00 0.09 -0.07 -0.02 -0.01 -0.02 -0.10

-0.08 -0.05 -0.03 -0.05 -0.09 4 0.24 -0.12 0.10 -0.02 -0.01 -0.01 -0.02 -0.01 -0.02 -0.14 ; _;*g :g;

0:36 0:02

0.13 0.11 -0.04 -0.03 -2.96 0.02 -0.08 -0.08 -0.03 -0.07 -0.14 0.00 -0.02 -0.05 -0.08

7 -0.08 -0.03 0.12 0.19 -0.05 8 -0.06 0.11 0.23 -0.10

applying standard numerical techniques to these potentials (2%25), one may readily generate reliable adiabatic wavehmctions which may be used for calculating expectation values and/or matrix elements of interest. In the present work, results of this type were obtained for the kinetic energy, for certain powers of R, and for the average polarizability and polarizability anisotropy. However, analogous results may readily be obtaining for any property known as a smooth function of the internuclear distance. A program for doing this may be obtained from the authors on request.

The procedure developed here for representing and extrapolating the nonadiabatic corrections to the eigenvalues allows reliable nonadiabatic level energies to be generated for all isotopes of diatomic hydrogen (including the exotic muonium species). This has allowed the present work to provide the first nonadiabatic level energy predictions for the tritium isotopes of this species. The kinetic energy scaling introduced for the AE” corrections has led to a remarkable improvement in the degree of agreement between theory and experiment in the case for which the most extensive data are available (see Table X). This in turn has improved our basic understanding of the


nature of the two types of adiabatic corrections, and of the way in which they are affected by centrifugal distortion.

As illustrated in Tables X and XI, the overall agreement between the theoretical and experimental vibration-rotation level energies of ground state hydrogen is now remarkably good, and little systematic behavior is discernable in the remaining discrepancies. However, the mean deviations of 0.14 (*0.22), 0.04 (*O. 15), and -0.08 (*O. 14) cm-’ for Hz, HD, and DZ , respectively,s are significantly larger.than the maximum errors of ca. kO.02 cm-’ believed to be associated with the present calculations. As was pointed out in Ref. (13) for case of the rotationless energies, it seems unlikely that this situation will be significantly changed by further improvements to the effective adiabatic potentials. The most likely source of further improvements to the theory would therefore seem to be the nonadiabatic corrections. In any case, it appears clear that diatomic hydrogen wilI continue to provide a stimulating challenge to theory and experiment for some time to come.

ACKNOWLEDGMENTS

We are pleased to acknowledge very helpful discussions with Dr. A. J. Tbakker, Dr. W. J. Meatb, Dr. J. F. Ogilvie, and Dr. D. M. Bishop, and thank Dr. L. M. Cbeung and Dr. J. Rychlewski for providing us with unpublished results. We are also very grateful to Dr. M.-C. Chuang for bringing to our attention an error in our treatment of the mass scaling of tbe nonadiabatic corrections. Tbis work was supported by the Natural Sciences and Engineering Research Council of Canada.

RECEIVED: June 12, 1986

REFERENCES

1. M.-C. CHUANG AND R. N. ZARE, J. Mol. Spectrosc. 121,38CUOO (1986). 2. D. K. VEIRS ANDG. M. ROSENBLATT, J. Mol. Spectrosc. 121,401-419 (1986). 3. C. SCHWARTZANDR. J. LEROY, J. Chem. Phys. 81,4149-4159 (1984). 4. W. Ko&s AND L. WOLNIEWICZ, J. Chem. Phys. 41,3663-3673 (1964). 5. W. KoZix AND L. WOLNIEWICZ, J. Chem. Phys. 43,2429-244 1 ( 1965). 6. W. Kct&s AND L. WOLNIEWICZ, J. Chem. Phys. 49,404-410 (1968). 7. R. J. LEROY, J. Chem. Phys. 54,5433-5434 (1971). 8. R. KOSLOFF, R. D. LEVINE, AND R. B. BERNSTEIN, Mol. Phys. 27,981-992 (1974). 9. J. D. POLLANDG. KARL, Cur&. J. Phys. 44,1467-1477 (1966).

IO. R. J. LEROY ANDR. B. BERNSTEIN, .I. Chem. Phys. 49,4312-4321 (1968). Il. W. K&os AND L. WOLNIEWICZ, Chem. Phys. Lett. 24,457-460 (1974). 12. W. K&OS AND L. WOLNIEWICZ, J. Mol. Spectrosc. 54,303-311 (1975). 13. W. KozGs, K. SZALEWICZ, ANDH. MONKHORST, J. Chem. Phys. 84,3278-3283 (1986). 14. D. M. BISHOPAMDL. M. CHEUNG, J. Chem. Phys. 69, 1881-1883 (1978). 15. L. WOLNIEWICZ, J. Chem. Phys. 78,6173-6481 (1983). 16. (a) P. R. BUNKERANDR. E. Moss, Mol. Phys. 33,417-424 (1977); (b) P. R. BUNKER, C. J. MCLARNON,

AND R. E. Moss, Mol. Phys. 33,425-429 (1977). 17. J. L. HUNT, J. D. POLL, ANDL. WOLNIEWICZ, Canad. J. Phys. 62, 1719-1723 (1984). 18. I. DABROWSKI, Cunud. J. Phys. 62,1639-1664 (1984). 19. D. M. BISHOP AND S.-K. SHIH, .I. Chem. Phys. 64, 162-169 (1976). 20. W. J. DEAL, Zntl. J. Quant. Chem. 6,593-596 (1972).

8 The few anomalously large deviations seen in Tables X and XI (i.e., the discrepancies for the 8 = 4, J =29andu=5,J=24levelsofH~,fortheu=l6,J=6levelofHD,andfortheJ=8,o=3-5levelsof Dz) are almost certainly due to experimental error, and hence are omitted from the averages.


21. J. F. BUKTA AND W. J. MEATH, Mol. Phys. 27, 1235-1239 (1974). 22. J. D. GARCIA, Phys. Rev. 147,66-68 (1966). 23. J. W. COOLEY, Math. Cornput. 15,363-374 (1963). 24. J. K. CASHION, J. Chem. Phys. 39,1872-1877 (1963). 25. R. J. LE ROY AND R. B. BERNSTEIN, J. Chem. Phy. 54,5 114-5 126 (1971). 26. J. N. L. CONNOR AND A. D. SMTH, Mol. Phys. 43, 397-414 (1981). 27. R. J. LE ROY AND W.-K. LIU, J. Chem. Phys. 69,3622-3631 (1978). 28. A. H. WAPSTRA AND G. AUDI, Nucl. Phys. A 432, l-54 (1985). 29. B. N. TAYLOR, J. Res. Nut. Bur. Stand. 90,91-94 (1985). 30. J. RYCHLEWSKI, Mol. Phys. 41,833-842 (1980). 31. J. RYCHLEWSKI, J. Chem. Phys. 78,7252-7259 (1983). 32. J. RYCHLEWSKI, private communication, 1986. 33. R. M. CLOVER AND F. WEINHOLD, J. Chem. Phys. 65,49 13-4926 (1976). 34. A. D. BUCKINGHAM, P. H. MARTIN, AND R. S. WATTS, Chem. Phys. Lett. 21, 186- 190 (1973). 35. M. KARPLUS AND H. J. KOLKER, J. Chem. Phys. 39, 1493-1506 (1963). 36. D. M. BISHOP ANLI L. M. CHEUNG, J. Chem. Phys. 72,5 125-5 132 (1980). 37. L. M. CHEIJNG, D. M. BISHOP, D. L. DRAPCHO, AND G. M. ROSENFUTT, Chem. Phys. Lett. 80,445-

450 (1981). 38. W. K&OS AND L. WOLNIEWICZ, J. Chem. Phys. 46, 1426-1432 (1967). 39. J. H. VAN VLECK, J. Chem. Phys. 4,327-338 (1936). 40. W. C. STWALLEY, .J. Chem. Phys. 63,3062-3080 (1975). 41. I. DABROWSKI AND G. HERZBERG, Can&. J. Phys. 54,525-567 (1976). 42. H. BRED~HL AND G. HERZBERG, Cunad. J. Phys. 51,867-887 (1973).

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Nonadiabatic Eigenvalues and Adiabatic Matrix Elements for all …scienide2.uwaterloo.ca/~rleroy/Pubn/87JMS-SchwartzH2.pdf · 2005. 8. 5. · JOURNAL OF MOLECULAR SPECI-ROSCOPY 121,420-439