Born-Oppenheimer breakdown effects in the determination of diatomic internuclear potentials: Application of a least-squares fitting procedure to the HCl molecule

JOURNAL

OF MOLECULAR

SPECTROSCOPY

117,36 1-387 (1986)

Born-Oppenheimer Breakdown Effects in the Determination of Diatomic internuclear Potentials: Application of a Least-Squares Fitting Procedure to the HCI Molecule J. A. COXON Deparlment of Chemistry,Dalhousie University,Halifax, Nova Scotia B3H 4J3. Canada This paper describes a least-squares fitting procedure for the reduction of measured line positions in ‘Z diatomic spectra to effective internuclear potentials. The procedure, which is based on firstorder perturbation theory/HeIlman-Feynman theorem, can be applied to data within a single electronic state, or to such data in combination with those of an electronic transition. Following recent theoretical work on Born-Oppenheimer breakdown, a modified, or effective, vibrationrotation Hamiltonian is employed to take account of non-adiabatic effects. An unlimited amount of data can be fitted simultaneously, with appropriate weighting. Results are described from an application of the procedure to the extensive data available for H”C1 (X’Z’), including those from a recent study of the B - X system. Combined with similar results from vibration-rotation data for H”C1 and D35C1,the effective potentials for the ground states of the three isotopes lead to a determination of the Born-Oppenheimer potential for HCI (X’Z’). The synthetic vibrationrotation spectrum of D”C1, obtained from the eigenvalues of the calculated effective Hamiltonian for this isotope, is found to be in excellent agreement with the precise experimental data from Fourier transform SpeCtrOsCopy. Q 1986 Academic Pres, Inc. INTRODUCTION

The application of numerical techniques to problems in diatomic spectroscopy has been an important factor in the development of this field in the last decade or so. Well-known examples of formerly tedious or impossible calculations that are now performed on a routine basis would include the determination of spectroscopic “constants” by least-squares fitting of a large set of measured line positions, the calculation of Rydberg-Klein-Rees (RKR) (1-3) potential functions, and numerical solution of the radial Schroedinger equation. Despite such important developments, however, there remains a problem that is fundamentally disturbing: the “end product” of a comprehensive spectroscopic investigation of a diatomic species can be regarded as the internuclear potential energy functions of the various electronic states; and yet there are few examples where the eigenvalues of such potentials are found to reproduce the experimental spectra to within the accuracy of the measurements. Such discrepancies are particularly severe for light molecules, especially hydrides. Although this problem has been realized for many years, and is hence the subject of an extensive literature, there has been relatively little success in designing improved procedures for real data. For ‘2 states, to which the present paper is restricted, there are two distinct effects that can contribute to the discrepancies between the calculated and experimental eigenvalues; these are (i) neglect of higher order terms in the WKB semiclassical quantization condition, and (ii) breakdown of the Born-Oppenheimer approximation. 361

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362

J. A. COXON

Difficulties associated with the solution of the radial wave equation by the WKB method have been understood in principle since Dunham’s pioneering treatment (4) of the rotating diatomic oscillator, which was developed within the Born-Oppenheimer approximation. As in more recent formulations (5), the quantization condition can be written as u + 4 = (l/x@) r”

[E - U,&)]“*dr + (/3/96a) 9;, U’$(r)[E - U,&)]-3’2dr +

-

-

-,

bin (1)

where v is the integer vibrational quantum number, fl* = (h */2~), rmi, and r,, are the inner and outer classical turning points for the level o of total energy E, U,&) is the potential energy function of the rotating molecule in the electronic state n, and r is the contour of integration enclosing the real line with E > U,,.,(r);this latter term is commonly labeled as either a second-order or a third-order phase integral. As shown by Dunham for the general rotationless potential, U,(r) = (B&8/4$)(1

+ six + a*x* + * - *),

(2)

where x = (r - rC)/r,, and Y,, = B,,,/w,,, the associated eigenvalues can be written concisely as En(v, J) = 2 Y&v + $)k[J(J + l)]‘. (3) If the second- and higher order terms on the right-hand side of Eq. (1) are negligible, as is usually assumed, the resulting approximate first-order quantization condition leads to a direct equivalence between the Dunham coefficients (Y& and the molecular constants (Y,iO = o,,, Y&i = B,,, Ynl, = a,,, etc.). Equation (1) is central to a calculation of U,,,,(r)by the RKR inversion procedure. Although some authors have attempted to take account of the higher order terms of Eq. (I), most RKR calculations are performed with the first-order quantization, 0 + f = (l/7$)

rm.%X [E - Udr)]‘/*dr,

s hin

which leads to Eqs. (5) and (6) for the turning points rmin(U,J) and r,,,&u, .I), r,,,=(2),J) - rmin(t),J) = 28 J

l/red%

ug

[E,,(u, J) - E,,(D’,J)]-“*do’

4 - l/r,,&% 4 = (VP) s’ BdUEn(~, w

4 - En(U’,J)l-“2dU’,

(6)

where B,,.,(u) = aEn( J)/tl[J(J + l)]lJ, and u. is the vibrational quantum number at the potential minimum. In the first-order approximation, u. = -1; however, many authors have employed the Kaiser (6) approach in which u. is calculated from the second-order Dunham Y, constant. Although this modification is introduced with the intent of partial compensation for the absence of higher order terms in Eq. (4), it has been shown recently (7) that results can actually be inferior to those of an unmodified first-order calculation. In any event, the use of Eq. (4) rather than Eq. (1) as the basis of an RKR calculation can never be wholly satisfactory, and leads to significant

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

363

errors when precise data are available for light molecules. It is these same cases that also require consideration of Born-Oppenheimer breakdown effects. In the adiabatic approximation, this is quite straightforward. Diagonal corrections for nuclear motion lead simply to different effective potentials, u$r)

= u%(r) + AU,,(r, M,, Mb),

(7)

for different isotopes of the same molecule with atomic masses M, and Mb. A not so readily accommodated problem is the influence of non-adiabatic terms from mixing with other electronic states. This is a topic receiving particular emphasis in the present work, and, in general, will be a matter of increasing importance, especially for high vibrational levels, as the quality of experimental data continues to improve. As discussed later in more detail, a consequence of significant non-adiabatic mixing is that the traditional type of effective vibration-rotation Hamiltonian, H,(r) = (1/2p)Pf + p’J(J + 1)/r* + U$!!(r),

(8)

cannot reproduce the terms of a ‘Z state even if the effective rotationless potential of the electronic state n, UC&m(r),includes J-independent non-adiabatic contributions. Instead, the effective Hamiltonian must be modified to include an additional J-dependent term, H,(r) = (1/2~)Ps + P’J(J + l)[l + q(r)]/r’

+ U!$!(r).

(9)

As a consequence, the experimental rotational constants (BJ are no longer defined with their usual mechanical significance as the expectation vaiues ( i/rz)V, but contain in addition a non-mechanical component from non-adiabatic mixing. This occurs in much the same way as in states with A > 0, where in addition to similarly modified definitions of the rotational parameters, such mixing with (or distant perturbation by) other electronic states leads to the observable effect known as A doubling. The present article, which describes a new procedure for the inversion of spectroscopic line positions to internuclear potentials, arises as an extension of recent work (8, 9) from this laboratory on the spectroscopy of hydrogen chloride. The work of Coxon and Ogilvie (8) was concerned with existing literature data for the lower levels of H3’Cl, H37Cl, D3’Cl, and D37Cl in the X’Z+ ground state. Some of these data are in the form of very precise line positions from Fourier transform vibration-rotation spectroscopy. The objective was to take account of both the first-order WKB approximation and Born-Oppenheimer breakdown according to the analytical formalism of Watson (10); in this approach, the data for several isotopes can be reduced simultaneously to a set of first-order isotopically invariant energy parameters (U,) and associated mass-scaling parameters (A,). Using known relationships between the Uk, and the parameters {Uif of the Born-Oppenheimer potential in the form of Eq. (2), the overdetermined set of {ai, i = l-S} could be estimated from a weighted leastsquares procedure, and shown to yield a potential essentially identical, as expected, to that from a first-order RKR calculation employing the U,,, and with v. = - 1. In the more recent work of Coxon and Roychowdhury (9), precise data for the higher vibrational levels (u” < 17) of the ground state of H3’Cl have become available from an analysis of the B - X system photographed at high resolution. Since the B-state molecules were formed with a rotationally hot distribution, the spectrum was complex

364

J. A. COXON

in appearance due to the overlapping structure of several bands. The assigned lines include transitions to numerous quasibound levels of the ground state lying above the dissociation limit. As discussed in the analysis (9) of these data, a conventional leastsquares estimation of the molecular parameters, which required the use of a truncated data set ignoring data on high-J levels, and subsequent generation of the first-order RKR potentials of the B and X states, could only be regarded as an approximate initial treatment. An extension of the approach of Ref. (8) was ruled out by the practical difficulty of reducing the data to sets of VWand Aklfor the two states; the large number of highly correlated VWthat are required for levels approaching a dissociation limit leads to serious instability in the least-squares matrix inversion. Direct reduction of the line positions to the Dunham potentials without explicit consideration of the Ukj (or Y,,) might be a somewhat more promising approach. This procedure has been employed successfully by Maki and Lovas (II) in the reduction of precise infrared diode laser data on AlF and KF. However, it is not obvious how such a direct inversion could be adapted to take account of the effects of Born-Oppenheimer breakdown. NUMERICAL

PROCEDURE

Before setting out the computational method of the present work, it is helpful to outline the various approaches that have been employed previously for the estimation of improvements to first-order RKR potentials. Direct inclusion of the second (third)order semiclassical phase integral of Eq. (1) was first discussed by Vanderslice et al. (12). An equivalent numerical approach was developed later by Kirschner and Watson (13), and applied to the ground state of carbon monoxide. However, it would appear that the complexity of the method, as well as the possibility that even higher order corrections are required for the higher vibrational levels, have discouraged its general usage. The same authors (13) also attempted to improve the second-order RKR curve for CO (X’Z’) with an iterative procedure in which discrepancies between the experimental E, and B, values, and those obtained by numerical solution of the radial equation with gradually improving potentials, were introduced as corrections in a subsequent second-order RKR calculation. However, since convergence could not be achieved, and results varied with the particular choice of polynomial representation for E,,this method was eventually abandoned. Several authors have considered means of elimination of the higher order effects as an alternative to their inclusion. Huffaker (14), and more recently Watson (IO), pointed out that if the second (and higher) order contributions to the experimental Yk,of Eq. (3) are first subtracted, use of the corrected Yklin a first-order RKR calculation is equivalent to the inclusion of the higher phase integrals. This approach was central to the work of Coxon and Ogilvie (a), although in this case, the use of mass invariant U, from several isotopes led to 17&~~)(r). In the recent work of Schwartz and Le Roy (7), it was shown that the contour integral of Eq. ( 1) could be eliminated by considering data for two isotopes simultaneously. Expressions for the turning points then have the same structure (7) as the familiar first-order results of Eqs. (5) and (6). Unfortunately, however, this simplest of methods requires that the two isotopes have identical potentials; since Born-Oppenheimer breakdown will most often be a concern of parallel importance in cases where the first-order WKB approximation is inadequate, it appears that this method will be of limited practical utility. Finally, there are two approaches

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

365

based on first-order perturbation theory. In the numerical Inverse Perturbation Analysis (IPA) method developed by Kosman and Hinze (15) and later by Vidal and Scheingraber (16) hereinafter referred to as KH and VS, respectively, errors in the (u, J) term values of an approximate trial potential are minimized by estimation of an appropriate correction potential, AU&). The more recent analytical approach of Gouedard and Vigue (17) is based on the semiclassical inversion procedure of Watson (18). The elegantly simple principle here is to regard the errors AE, and AB, of the trial potential as the expectation values (X,) and distortion constants (Xn,) of a correction potential X(r). A fair degree of success was demonstrated for relatively heavy molecules; in the case of CO (X’Z’), however, improvements in the AE, and AB, residuals were not so dramatic. A general criticism that can be raised for methods (13, 14, 17) aimed merely at obtaining a potential reproducing the input parameters is that these parameters themselves are always subject to varying error, depending on the method of their estimation. Although a particular set of estimated YUmay represent the experimental line positions very successfully, the necessarily truncated expansion of Eq. (3) implies that the fitted Y,, are always to some extent effective parameters, which vary according to the J range of the spectroscopic data. This can be a severe problem for high-lying vibrational levels, such as in the present case (9) of HCl (X’Z’). It was principally for this reason that the essential idea of the IPA method (15,16) held the most promise for developing a new approach employing the weighted raw frequency data, rather than any derived quantities. In addition, however, it has been found possible to devise an extension of the IPA method in order to take account of the non-adiabatic effects in Eq. (9) from Born-Oppenheimer breakdown. As indicated earlier, the use of Eq. (9) as an effective vibration-rotation operator is based on a publication of Watson (10) concerned with isotopic relationships between experimental Y,,. It is well known that other prominent groups (19-22) have also previously made important contributions to the understanding of Born-Oppenheimer breakdown effects; however, the comprehensive treatment of Ref. (10) is in a form that can be readily adapted in numerical procedures. Eq. (32) of Watson’s paper (10) is an expression for the energy corrections, AE,,(v, J) from both adiabatic and non-adiabatic effects, A&(u, J) = C (m,lMi)(&I{P~J(J

+ l)B?(r)lr2

+ Sl”‘(r))I&),

(10)

which must be added to the eigenvalues of the zeroth order Hamiltonian, H;(r) = (l/2/.@:

+ /3,2J(J + l)/r2 + U$?(r),

(11)

defined with the atomic reduced mass, we. The summation of Eq. (10) is taken over both nuclei. The functions @“j(r) and s?)(r), are defined in terms of three other functions, Q”)(r), Ry)( r), and S?(r), where Qy’( r) and B?)(r) represent non-adiabatic effects,

Q?‘(r) = (2/mJ c’ I(n’IPzAn)121[~n(r) - unlr)l+ zi

(12)

P(r) = (llmr*) C’ {I(~‘lLil~)l’ + I(~‘l~yil~)12}l[~~(~) - u,O-)l+ zi7

(13)

n’

n’

366

J. A. COXON

and S?)(r) is the diagonal matrix element, S?‘(r) = (1/2me)(njPZj + (L$ + L:i)/r21n)*

(14)

Since @(r) and @j(r) are not completely distinguishable, Watson suggested that they be defined such that dP)(r,) = 0. An expression for the total effective Hamiltonian can now be obtained by combining Eq. (11) with the first-order corrections of Bq. ( 101, fly-) = PZ/2pL, + &/?)J(J + l)[l + q”(r)] + Ufyr), (15) where (16) 4n(r) = C (me/Mi)df”)(f? and U$(r) = Up(r) + 2 (m,/Mi)Sy)(r).

(17)

of a ‘Z-‘Z band system (and/or vibration-rotation/ Individual line positions (Vn,v,Sn-v-Jn) pure rotational transitions within a ‘z1 state) can be represented as differences in the upper and lower state terms, Vn’“*J,“.“.J. = E n’tu -

&VT,

(18)

where En, are the eigenvalues of Htin(r). If L$‘r(r) in Eq. ( 15) is replaced by an initial approximate function, say U,RKRfrom an RKR calculation, and the usual BomOppenheimer expression for H,t is employed, but with the atomic, rather than nuclear, reduced mass, the new eigenvalues are given by En”jyl =

E~UJ -

AEn,

(~!%RbU~)

(19)

A&,,,

where =

+ gnOMJ+

1>1@iff),

AU,(r) = U;ff(r) - U;KR(r),

(20)

(21)

and ‘Cl(r) = ~,2~~(~)lr2.

(22)

It follows, therefore, that individual line positions calculated from the eigenvalues of Eq. (19) with a trial potential(s) are given by =E$s%'u'J'n"v'J"(caJc)

Efv@$ = Eny,sy- E,puuJ*- (AE,,,“,J,- AE,,.,.J*).

(23)

The residuals, AV = v - vtcalC), are correspondingly obtained as Av,,,,~,,“~“J= (J/!%%AG(r)

+ g&)J(J

+ l)M~!%) - (~~~~~lAU,~r) + g,
(24)

In a manner similar to the formulations (IS, 16) of the IPA method, AU&), AU,,{r), g”(r), and g,.(r) can be estimated in a least-squares procedure if they are each represented as a linear combination of some functions&(r), Au(r) = 2 CL(r)

(25)

BORN-OPPENHEIMER

BREAKDOWN

367

EFFECTS

In that case, the coefficients c{, c;, d:, and dr, given by the set of linear equations, Au,,‘~~~~,,.,,“.,~ = C c:(j-:)

+ 2 d:(j-;)

i

i

- 2 c:(f;)

- C d:(f;),

i

(27)

i

where (1;) = (1c/:YiA(r)i+ZY j,

(28)

are grossly overdetermined. As discussed previously (15, 16), the functions AU(r) and g(r) can be fitted over some data-dependent range, ri < r =S r2. In addition, an appropriate choice of the functions A(r) must be made. KH employed Legendre polynomials, Pi(x); VS used Gaussian-weighted Legendre polynomials, P;(x)exp(-x2”), with 1 < rr < 5. In both cases, x = - 1 at r = rI and x = + 1 at r = rz. However, the later workers (VS) argued that a linear interpolation of x as adopted by RI-I tends to optimize the fitted corrections only at the outer turning points of the rotationless potential; instead, VS defined a non-linear interpolation, (r - r&r2 - rJ x = (r2 + rI)(re + r) - 2rlr2 - 2r,r ’

(29)

such that x = 0 at r = r,. The same choice of reduced coordinate x has been adopted in the present work. However, the use of weighted Legendre polynomials in preliminary work was later abandoned in favor of the simpler power series representation, fi(r) = xi. In addition, it was decided that separate functional forms for AU(r) would be employed for r < r, and r > r,. Finally, to avoid discontinuities and erratic behavior of AU(r) and g(r) in the regions beyond rl and r2, the corrections employed damping functions in these regions. Summarizing, AU(r) = C c,;x’;

r,
r,

(30)

AU(r) = C czixi;

r, =S r < r2

(31)

r,
(32)

g(r) = 2 dix’;

9(r) = W,)exp(xl d(r) =

44r2)ewtx

- x); -

x2);

r2

r < r,

(33)

r>

(34)

r2,

where 4(r) is AU(r) or g(r). Some comment on the choice of rl and r2 is appropriate. KH (15) noted that AU(r) can be best determined in the region where the wavefunctions titi are “large.” However, VS (16) found that better results were obtained for a smaller r range, with rl and r2 defined as the turning points of the highest vibrational level for which data are considered. In the present work it has been found essential to use a larger r range, as in Ref. (15); somewhat arbitrarily, however, rl and r2 were defined as the points where &r) had decreased to a value of 20% of $&(rJ and &(rb), where $&rJ and Mrb) are the amplitude maxima near the inner and outer classical turning points of the highest v level considered for J = 0 and f = J,,,, respectively.

J. A. COXON

368

To summarize, then, the fitting procedure developed in this work is a weighted least-squares reduction of the measured line positions of an electronic band system and/or vibration-rotation spectrum according to Eq. (27), which is based on a recent theoretical treatment (IO) of Born-Oppenheimer breakdown. The trial line positions are calculated with Eq. (23) from the eigenvalues of first-order RKR curves with the approximate vibration-rotation Hamiltonian having q(r) = 0. The entire procedure has been developed both for a small laboratory computer (PDP 1l/23) and for the popular IBM/PC microcomputer. RESULTS

AND DISCUSSION

Results obtained from the procedure given in the previous section are described first for H3%J1, for which the most extensive data are available. All the vibrationrotation and pure rotation data considered by Coxon and Ogilvie (8), as well as the more recent B - X data (9), have been fitted simultaneously. Each segment (j) of the data was assigned a weight of (1/2GJ2, where Gj are mostly the estimated standard deviations from unweighted least-squares fits of Y,, to the data for individual bands. For the B - X bands, these fits employed constrained values of some of the higher order YiI and Y’h, so that the sj values are generally somewhat larger than the unrealistically low values from “free” fits, as in Table 1 of Ref. (9). For the remaining data, Gj given in Ref. (8) were employed. The use of 2Zj, rather than Gj, in the assignment of weights was adopted in an attempt to take account of relative systematic error in the data. Based on experience with the method of direct fitting (merging) (22, 23) of Dunham parameters to line positions, which involves absolute wavenumbers across an entire band system, a typical result for high quality data is that the fitted parameters reproduce the data, on average, to within about two standard deviations. Since any approach concerned with deriving internuclear potentials, as in the present work, also depends on the relative accuracy of the line positions, some account of systematic error is clearly desirable. A further point requiring proper consideration in the present fitting procedure was revealed in preliminary work. After AU(r) have been estimated in the fit, the rotationless eigenvalues of the corrected potentials, URKR(r) + AU(r), can be calculated. It was found that although the spacings of the calculated E, matched closely the experimental AG(v + $) values, as expected, the absolute values of & were not in agreement with the experimental values (9). For the ground state, for example, there was an almost constant discrepancy of -0.1 cm-‘. The origin of this problem lies in the fact that there is no information content within the data on the zero-point energies, so that AU(r) around r, are free to vary without regard for the absolute vibrational energies. It was concluded therefore that optimum results will be obtained if the zeropoint energies are introduced as additional “data” to the line positions. The values of Ref. (9) were employed in the present work, and are closely reproduced by the final effective potentials for the X and B states. The first-order RKR turning points for the B and X states obtained in Ref. (9), with extrapolations beyond the inner and outer turning points of the highest levels, served to define the trial potentials. Since the eigenvalues of the u” = 17 level, and to some extent the u’ = 6 level, are extremely sensitive, particularly with increasing J, to the

BORN-OPPENHEIMER

BREAKDOWN

369

EFFECTS

extrapolations of the outer limbs beyond the rk, (u = 17) and rkax (V = 6) turning points, these extrapolations required special consideration. For the X’Z+ state, for example, additional RKR turning points were generated for the rotating molecule with J = 15 and J = 19; this led to a significant and quite reliable extension to r = 3.137 A for J = 19 over the r = 2.878 A, V” = 17 outer turning point for J = 0 (9). Extrapolation beyond r = 3.137 A was performed from these last two data points according to Eq. (35) U(r) = D, - C,/r6 - Cs/r8 - C,,fr”,

(35)

where D, = 37 243 cm-’ from thermochemical data, Do = 35 759 cm-’ (24) and C’S and Cs were fixed at calculated values (25) of Cs = 1.59 X lo5 cm-’ A6 and C8 = 1.89 X lo5 cm-’ A8. A similar approach was taken for the B state, with RKR calculations for J = 16 and 23, D, estimated as - 11 390 cm-‘, and extrapolation performed with the simpler form, U(r) = D, - Cn/rn. (36) In order to reduce errors in the eight-point Lagrangian interpolations of the RKR turning points to the required point-wise potentials on grids of equally spaced r values, the RKR calculations were performed additionally for hypothetical levels with noninteger quantum numbers, u + $, o + 4, and u + a. Table I summarizes the ranges of r (r, and r2) over which the correction potentials AU(r) (and g”(r)) were fitted for the two states, the radial mesh stepsizes, and the r ranges (rmin and rmax) used for numerical solution of the Schroedinger equation. Plots of&(r) for J = 0 and J = 19, and the form of Ux(r) near the dissociation limit, are shown in Fig. 1. A total of 1998 line positions were employed in the final fit according to Eq. (27). Results obtained in preliminary work with all d: and d; fixed at zero were very unTABLE I Fit Parameters” for the B’Z+ and X’Z+ States of H3’Cl

rl

(A)

0.9

1.75

1-2

(A)

3.3

3.55

rmin

rnax AI

a

(A)

(A)

0.7

1.4

(A)

4.2

3.9

0.0025

0.0025

Cln

n

=

l-6

n

=

l-4

c2n

”

=

1-Y

”

=

l-7

dn

”

=

l-5

rl and r2 define the ranges over whrch tho potentials are corrected. rmln and rmax define the ranges employed in solution of the radial wave squation. & i6 the mesh size. cln and czn define inner and outer limb correction potontrals. d, define the function y(r), 6-e text.

370

J. A. COXON Levels of

H35Cl (Xi Near Dfsvxlotlon

r tbnsstrm)

FIG. I. Internuclear potentials of H”CI (X’S?‘) for J = 0 and J = 19. $2 are shown for u = 16 and 17 of the J = 0 potential and for 11= 17 of the J = 19 potential. These illustrate the choice of r, = 0.9 and r2 = 3.3 8, defining the range over which the ttial potential is adjusted in the least-squares procedure.

satisfactory. Relatively good fits without the inclusion of rotationally dependent nonadiabatic terms were only achieved when data for higher J levels were excluded, especially for the highest D” levels. The fit quality would improve as the data set was increasingly truncated. It was found, however, that virtually the entire data set could be fitted very satisfactorily when the function g”(r) was introduced to the model. Introduction of g’(r) alone was not helpful; similarly there was no significant further improvement when g’(r) was also included. This is in accord with the expectation that Born-Oppenheimer breakdown in a hydride will be particularly significant either for very precise data on low vibrational levels, or for less precise data at high J for levels approaching a dissociation limit, where the energy separation from other electronic states becomes small. In the case of H35C1,both types of data are present for the X1 Z+ ground state, but neither exist for the B’Z+ state. There was some uncertainty in the work concerning the required numbers of terms in the expansions of Eqs. (30)-(34). Slightly smaller standard deviations were in fact achieved for a larger number of parameters than was retained in the final fit. However, in such cases, the fitted corrections AU’(r) and AU”(r) were closely similar over most of the r ranges, but became physically unreasonable at small or large r. It was clear that such unwarranted increased flexibility in the model served only to reduce the residuals for a few of the lines having the highest (u’, J’) or (u”, J”); it was concluded that for such data, the model was not quite adequate, either through marginally unsatisfactory extrapolations of the potentials beyond r2, the need for values of r2 somewhat larger than those of Table I, or the failure of first-order perturbation theory for levels close to dissociation. The standard deviation of the final fit, which had 32 adjustable parameters (AT, and the 3 1 parameters indicated in Table I), was c = 0.74. This very satisfactory result indicates that the entire data set was fitted, on average, with somewhat smaller residuals than was anticipated from the assigned weights. Results for two of the bands posing particularly stringent demands on the method are given in Tables II and III. The measurements for the 2-O vibration-rotation band (Table II) from Fourier transform spectroscopy (26) are exceptionally precise: a value of i = 0.00009 cm-’ was estimated in Ref. (9). Errors in the line positions calculated from the ground state first-order

BORN-OPPENHEIMER

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EFFECTS

371

TABLE 11

Least-Squares Fit for the 2-O

Band’ of H3’C1 (X1X+)

R(J) (en-l)

P(J) (cm-l)

J

Vobs

AV

Res.

%bs

AV

Res.

5687.6504

0.1080

0.0000

1

5647.1051

0.1084

0.0002

5706.0940

0.1073

0.0000

2

5625.0282

0.1080

0.0000

5723.3021

0.1064

0.0001

3

5601.7659

0.1074

-0.0000

5739.2627

0.1050

-0.0001

4

5577.3312

0.1064

-0.0001

5753.9647

0.1034

-0.0001

5

5551.7380

0.1053

-0.0000

5767.3970

0.1015

-0.0001

0

6

5524.9997

0.1038

0.0001

5779.5488

0.0993

-0.0001

7

5497.1303

0.1020

0.0001

5790.4098

0.0968

-0.0001

a

5468.1440

0.0998

-0.0000

5799.9700

0.0939

-0.0000

9

5438.0555

0.0975

0.0001

5808.2198

0.0907

0.0000

10

5406.8787

0.0944

-0.0002

5815.1497

0.0871

-0.0001

11

5374.6294

0.0916

0.0001

5820.7515

0.0835

0.0002

12

5341.3215

0.0881

-0.0001

5825.0158

0.0793

0.0001

0 bbs are the experinontal line porrtzons from Ref. (26). with an estimated uncertainty t&j of 20 = 0.00018 cm-f. AV are defined by AV = V,,bs- kale. where k=,,lcare obtained from the eigenvolue6 E"J of the first-ordsr RKR potential for H%l (Xl&:+). The columns labelled "Res." contain the residuals between the observed and fitted values of AV.

RKR curve are -0.1 cm-‘, and are reduced to 0.00007 cm-’ on average with the fitted AU”(r) and g”(r) functions. The 0- 12 B - X band (9) represents a situation (Table III) of extensive rotational development involving a fairly high-lying vibrational level of the X state (the vibrational energy for V” = I2 is about 80% of the well depth). In such a case, the residuals from the eigenvalues of the trial potentials show a larger dependence on J, especially at the highest J. It should be emphasized that the absolute values of the residuals for the B - X bands depend on the particular choice of the trial value of T’,; the results in Table III are for a TL (trial) which differed by 0.16 cm-’ from the final fitted value (T, = 77 307.126 cm-‘). The magnitudes of the residuals calculated from the fit parameters are again much reduced; the mean of the magnitudes of the residuals (0.052 cm-‘) is less than half the estimated accuracy of the measurements (0.137 cm-‘). It is noted, however, that although the residuals of the fit are satisfactorily small, they are not completely free of systematic variation with J. In view of the large spectral range covered by an individual B - X band, it is likely that this arises in part from systematic error within the data. However, it is also probable that a contribution to the systematic trends arises from some slight inadequacy of the model (first-order perturbation theory; arbitrary functional form for g”(r)). A particularly interesting observation in Table III is that three lines with the highest values of

372

J. A. COXON TABLE III Least-Squares Fit for the 0- 12 Band’ of H3’C1(B - X) P(J) (cm-l)

J

vobs

13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

47617.474 47499.131 47374.012 47242.534 47104.947 46961.278 46811.868 46656.892 46496.698 46331.553 46161.665 45987.428 45809.168 45627.211 45442.086 45254.080 45063.865 44871.448 44677.839 44483.496 44289.071 44095.338 43903.050 43713.219 43526.931 43345.746 43171.199 43005.614

AY

-1.394 -1.329 -1.443 -1.506 -1.467 -1.512 -1.530 -1.590 -1.606 -1.590 -1.631 -1.653 -1.671 -1.723 -1.670 -1.646 -1.433 -1.519 -1.432 -1.307 -1.152 -0.932 -0.729 -0.496 -0.276 0.135 0.575 1.116

R(J) (cm-l)

Res.

vobs

0.033 0.132* 0.050 0.019 0.088 0.070 0.076 0.036 0.035 0.061 0.024 -0.002 -0.032 -0.105 -0.083 -0.102 0.054 -0.104 -0.106 -0.091 -0.070 -0.010 -0.002 -0.004 -0.069 -0.009 -0.007 -0.035r

47874.886 47773.654 47665.545 47550.824 47429.623 47302.054 47168.464 47029.012 46883.920 46733.469 46577.932 46417.602 46252.846 46083.968 45911.280 45735.234 45556.220 45374.735 45191.262 45006.335 44820.526 44634.536 44449.058 44264.979 44083.238 43904.942 43731.498 43564.587 43406.606

AY

-1.441 -1.479 -1.554 -1.567 -1.568 -1.643 -1.653 -1.667 -1.703 -1.738 -1.777 -1.821 -1.819 -1.805 -1.828 -1.825 -1.825 -1.783 -1.709 -1.606 -1.494 -1.329 -1.156 -0.924 -0.655 -0.370 -0.020 0.384 0.985

Res.

0.047 0.048 0.014 0.040 0.076 0.037 0.059 0.074 0.064 0.048 0.023 -0.013 -0.011 -0.005 -0.046 -0.072 -0.112 -0.125 -0.121 -0.107 -0.104 -0.072 -0.060 -0.022 0.011 0.009 0.008 -0.028' 0.0038

* VObs (lrothe experirental line positions from Ref. (3). with an estimated uncertainty (this work) of 2% = 0.137 cm-l. AV clre defined by AV = Vobs - V,,lc. rhero vcalc are cbtalnod fron the eigenvalues Envy of the first-order RKR potontlals for H35C1 XIE' and BIC*. The colunns labolled "Res." contain the residuals between the observed and fitted values of AV. LlIlW flagged with an asterisk were excluded from the fit.

J, P(41), R(40), and R(4 1), were unnecessarily excluded from the fit, but yet are very reliably reproduced by the estimated parameters. (It has to be emphasized that no other band with I)” = 12 extends to such large J; in fact, the highest J level included

r,

FIG. 2. The fitted correction, AU”(r) = V”(r) - URKR(r)for the X’Z+ state of HS5CI.The correction at is arbitrarily zero.

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

313

1 5.00AU

cm-l o.oo-

-5.00I

-10~oo-21

3.10 r A 3.55

FIG. 3. The fitted correction, AU’(r) = V&(r) - U”(r) for the B’Z+ state of H”Cl. At r,, the correction is arbitrarily zero, and shows a physically unreasonable discontinuity in slope.

in the fit from other u” = 12 bands is from the R(33) line of the 1-12 band.) This predictive ability of the present approach was not limited to the 0” = 12 level, and is of course far superior to any extrapolation that can be performed in the usual way from the molecular constants. As indicated in Fig. 1 of Ref. (9), J = 40 and 41 of 2)” = 12 are actually quasibound levels for which extrapolated energies from Eq. (3) are particularly unreliable. Plots of AU”(r), AU’(r), and q”(r) given by the estimated parameters are shown in Figs. 2-4. The AU”(r) correction of the ground state potential lies in the range 54 cm-’ over most of the range of r where it is fitted (0.9-3.3 A). Only at very small r, r < 0.93 A, is the magnitude of the correction much larger. This part ofthe potential, corresponding to the inner turning points of the highest levels, u” = 14-17, is very steep, and thus particularly sensitive to errors in the B, used as input to the RKR calculation, as well as to error in the first-order approximation. For the excited state (Fig. 3) a similar situation holds, except that in this case the correction becomes large for large r values also, near and beyond the outer turning point of the highest level (v’ = 6) of the data set. The estimates (9) of B; and Bb are subject to much larger error than those for 2)’G 4. After the trial potentials had been corrected, rotational term values for the two states were calculated using the effective Hamiltonian of Eq. ( 15) and used to calculate

0

1039 1 -2 1

-+

FIG. 4. The fitted function q(r) for the X’Z+ state of H35Cl. This function describes the fitted J-dependent energy corrections from non-adiabatic interactions (see. the text).

J. A. COXON

374

TABLE IV Term Values (cm-‘) for HCl (X’Z’)

H35Cl J

0 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 lb 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

v=o

1483.8798 1504.7581 1546.5020 1609.0862 1692.4727 1796.6110 1921.4379 2066.8778 2232.8427 2419.2323 2625.9339 2852.8227 3099.762 3366.603 3653.186 3959.338 4284.875 4629.604 4993.317 5375.799 5776.820 6196.144 6633.521 7088.692 7561.389 8051.333 8558.235 9081.798 9621.717 10177.674 10749.345 11336.400 11938.495 12555.283 13186.41 13831.50 14490.20 15162.11 15846.86 16544.06 17253.29 17974.17 18706.27 19449.18 20202.47 20965.72

"=I

4369.857 4390.128 4430.656 4491.417 4572.373 4673.474 4794.658 4935.851 5096.965 5277.902 5478.549 5698.784 5938.472 6197.466 6475.607 6772.726 7088.640 7423.158 7776.075 8147.178 8536.240 8943.028 9367.293 9808.781 10267.225 10742.349 11233.869 11741.488 12264.905 12803.805 13357.867 13926.762 14510.15 15107.69 15719.02 16343.78 16981.60 17632.11 18294.91 18969.63 19655.85 20353.18 21061.20 21779.50 22507.64 23245.20

v=2

7151.8631 7171.5302 7210.8521 7269.8041 7348.3490 7446.4375 7564.0081 7700.9868 7857.2877 8032.8128 8227.4521 8441.0838 8673.5741 8924.778 9194.538 9482.687 9789.044 10113.420 10455.612 10815.409 11192.587 11586.912 11998.142 12426.022 12870.288 13330.666 13806.875 14298.620 14805.601 15327.506 15864.02 16414.80 16979.53 17557.85 18149.42 18753.86 19370.82 19999.91 20640.75 21292.94 21956.10 22629.81 23313.65 24007.21 24710.05 25421.74

V=3

v=4

v=5

9830.657 9849.724 9887.847 9945.001 10021.149 10116.242 10230.219 10363.008 10514.522 10684.666 10873.329 11080.392 11305.722 11549.174 11810.594 12089.815 12386.659 12700.936 13032.447 13380.981 13746.317 14128.223 14526.457 14940.766 15370.889 15816.553 16277.476 16753.368 17243.928 17748.846 18267.80 18800.47 19346.52 19905.59 20477.34 2lObl.41 21657.42 22265.00 22883.75 23513.29 24153.21 24803.10 25462.53 26131.09 26808.32 27493.80

12406.707 12425.177 12462.105 12517.467 12591.226 12683.333 12793.729 12922.340 13069.083 13233.860 13416.564 13617.076 13835.262 14070.981 14324.079 14594.388 14881.733 15185.926 155Ob.768 15844.049 16197.55 16567.038 16952.274 17353.005 17768.97 18199.90 18645.51 19105.51 19579.60 20067.47 20568.81 21083.27 21610.53 22150.24 22702.04 23265.57 23840.45 24426.30 25022.73 25629.34 26245.71 26871.43 27506.07 28149.18 28800.32 29459.02

14880.149 14898.021 14933.755 14987.325 15058.695 15147.818 15254.632 15379.066 15521.036 15680.446 15857.189 16051.144 16262.182 16490.159 16734.923 16996.308 17274.138 17568.225 17878.37 18204.37 18545.99 18903.02 19275.21 19662.30 20064.04 20480.15 20910.35 21354.35 21811.85 22282.54 22766.09 23262.17 23770.44 24290.56 24822.16 25364.86 25918.30 26482.07 27055.79 27639.02 28231.37 28832.38 29441.61 30058.60 30682.89 31313.98

H35Cl J

0 1 2 3 4 5 6 7 8 9 10 11

V=b

17250.73 17268.00 17302.54 17354.31 17423.26 17509.41 17612.63 17732.87 17870.05 18024.07 18194.83 18382.20

v=7

19517.75 19534.42 19567.75 19617.71 19684.27 19767.38 19866.98 19982.99 20115.34 20263.93 20428.64 20609.37

"=8

21680.00 21696.05 21728.16 21776.28 21840.40 21920.44 22016.37 22128.10 22255.55 22398.62 22557.21 22731.19

YE9

23735.54 23750.97 23781.83 23828.08 23889.70 23966.62 24058.80 24166.16 24288.60 24426.05 24578.37 24745.46

”

=

10

25681.61 25696.41 25725.98 25770.31 25829.36 25903.07 25991.40 26094.25 26211.55 26343.19 26489.06 26649.04

" = 11

27514.56 27528.68 27556.93 27599.26 27655.65 27726.04 27810.36 27908.54 28020.49 28146.11 28285.28 28437.86

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

375

TABLE IV-Continued

H35Cl J

v=6

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

18586.05 18806.25 19042.64 19295.05 19563.31 19847.22 20146.60 20461.23 20790.90 21135.36 21494.38 21867.70 22255.07 22656.21 23070.83 23498.63 23939.32 24392.58 24858.08 25335.49 25824.45 26324.62 26835.61 27357.05 27888.55 28429.71 28980.10 29539.31 30106.88 30682.36 31265.28 31855.15 32451.47 33053.70

v=7

20805.97 21018.32 21246.24 21489.58 21748.16 22021.79 22310.28 22613.40 22930.94 23262.66 23608.33 23967.67 24340.44 24726.34 25125.10 25536.40 25959.94 26395.40 26842.44 27300.72 27769.87 28249.54 28739.33 29238.84 29747.68 30265.42 30791.61 31325.81 31867.54 32416.30 32971.60 33532.88 34099.61 34671.18

v=.s

v=9

22920.43 23124.80 23344.12 23578.24 23826.98 24090.14 24367.52 24658.91 24964.08 25282.80 25614.81 25959.86 26317.67 26687.95 27070.42 27464.76 27870.65 28287.76 28715.74 29154.23 29602.86 30061.24 30528.96 31005.61 31490.76 31983.94 32484.68 32992.50 33506.87 34027.25 34553.06 35083.69 35618.49 36156.77

24927.17 25123.37 25333.88 25558.55 25797.19 26049.61 26315.59 26594.92 26887.37 27192.69 27510.62 27840.91 28183.26 28537.38 28902.97 29279.70 29667.23 30065.22 30473.31 30891.10 31318.22 31754.22 32198.70 32651.18 33111.20 33578.24 34051.78 34531.26 35016.08 35505.59 35999.12 36495.91 36995.17 37495.98

" = 10

" = 11

27010.75 27212.17 27427.08 27655.27 27896.56 28150.73 28417.55 28696.79 28988.18 29291.46 29606.36 29932.57 30269.79 30617.69 30975.93 31344.15 31721.98 32109.02 32504.86 32909.06 33321.15 33740.67 34167.08 34599.85 35038.38 35482.06 35930.22 36382.11 36836.95 37293.84 37751.79 38209.66 38666.10

28603.73 28782.72 28974.66 29179.37 29396.66 29626.32 29868.12 30121.82 30387.17 30663.91 30951.75 31250.38 31559.50 31878.77 32207.83 32546.32 32893.84 33249.97 33614.28 33986.30 34365.53 34751.44 35143.48 35541.02 35943.41 36349.94 36759.79 37172.09 37585.83 37999.83 38412.71 38822.78 39227.86 39624.91

v = 16

" = 17

34747.88 34757.89 34777.89 34807.84 34847.69 34897.34 34956.70 35025.66 35104.06 35191.74 35288.52 35394.16 35508.44 35631.08 35761.78 35900.18 36045.93 36198.58 36357.67 36522.64 36692.88 36867.66 37046.14 37227.30

35725.02 35733.86 35751.51 35777.92 35813.03 35856.72 35908.88 35969.35 36037.95 36114.46 36198.62 36290.16 36388.73 36493.96 36605.41 36722.58 36844.86 36971.57 37101.86 37234.69 37368.70 37501.95 37630.88 37751.56

26822.98

H35Cl J

" = 12

0

29229.64

1 2 3 4 5 6 7

29243.08 29269.93 29310.17 29363.77 29430.66 29510.78 29604.06 29710.39 29829.67 29961.78 30106.58 30263.92 30433.64 30615.56 30809.48 31015.20 31232.50 31461.13 31700.84 31951.36 32212.39 32483.64 32764.76

a

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

" = 13

30820.32 30833.01 30858.38 30896.40 30947.02 31010.19 31085.84 31173.88 31274.21 31386.72 31511.28 31647.74 31795.95 31955.72 32126.86 32309.17 32502.42 32706.37 32920.75 33145.29 33379.69 33623.62 33876.73 34138.66

" =14

32278.11 32290.01 32313.77 32349.39 32396.80 32455.95 32526.77 32609.15 32702.99 32808.17 32924.54 33051.94 33190.21 33339.14 33498.52 33668.11 33847.68 34036.93 34235.58 34443.29 34639.72 34884.50 35117.19 35357.37

Y =15

33592.39

33603.40 33625.40 33658.36 33702.23 33756.94 33822.40 33898.x 33985.15 34082.17 34189.41 34306.70 34433.83 34570.57 34716.69 34871.90 35035.91 35208.40 35389.00 35377.33 35772.95 35975.39 36184.12 36398.56

J. A. COXON

376

TABLE IV-Continued

H35Cl J

" = 12

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

33055.42 33355.24 33663.84 33980.79 34305.65 34637.94 34977.15 35322.72 35674.07 36030.53 36391.38 36755.82 37122.95 37491.73 37860.96 38229.22 38594.71 38955.08 39306.80 39643.67

" = 13

34409.00 34687.34 34973.20 35266.09 35565.46 35870.72 36181.22 36496.22 36814.92 37136.41 37459.62 37783.31 38105.96 38425.60 38739.43 39042.61 39328.78

Y =14

35604.54 35858.17 36117.66 36382.38 36651.59 36924.48 37200.08 37477.29 37754.74 38030.67 38302.64 38566.50 38815.14

" =15

v = 16

36618.03 36841.79 37068.95 37298.46 37529.03 37759.00 37986.03 38206.17 38413.00

37409.88 37592.17 37771.72

" = 17

H37Cl J

0

1 2 3 4 5 6 7 a 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

v=o

1482.7643 1503.6112 1545.2924 1607.7827 1691.0441 1795.0264 1919.6664 2064.8890 2230.6062 2416.7181 2623.1124 2849.665 3096.238 3362.686 3648.846 3954.549 4279.610 4623.836 4987.021 5368.949 5769.393 6188.12 6624.87 7079.39 7551.42 8040.68 8546.87 9069.71 9608.88 10164.07 10734.97 11321.23 11922.51 12538.48 13168.77 13813.02

v=l

4366.6357 4386.8759 4427.3440 4488.0150 4568.8515 4669.8037 4790.8095 4931.7944 5092.6717 5273.3428 5473.6965 5693.6102 5932.9490 6191.567 6469.305 6765.994

7081.454 7415.492 7767.905 8138.480 8526.992 8933.206 9356.88 9797.75 10255.56 10730.02 11220.a7 11727.80 12250.50 12788.68 13342.00 13910.14 14492.76 15089.52 15700.06 16324.02

v=2

7146.6919 7166.3304 7205.5951 7264.4613 7342.8921 7440.8382 7558.2383 7695.0187 7851.0937 8026.3657 8220.7248 8434.0497 8666.2071 8917.052 9186.428 9474.168 9780.092 10104.011 10445.724 10805.oia 11181.671 11575.451 11986.11 12413.41 12857.07 13316.83 13792.39 14283.48 14789.79 15311.01 15846.82 16396.89 16960.89 17538.40 18129.30 18732.99

v=3

9823.691 9842.731 9880.800 9937.872 10013.911 10108.868 10222.683 10355.283 io506.583 10676.485 10864.882 11071.65 11296.67 11539.78 llaoo.83 12079.66 12376.10 12689.94 13020.99 13369.05 13733.89 14115.28 14512.97 14926.73 15356.28 15801.35 16261.67 16736.94 17226.07 17731.13 18249.43 18781.43 19326.79 19885.17 20456.22 21039.58

v=4

12398.10 12416.55 12453.42 12508.71 12582.36 12674.34 12784.58 12913.01 13059.55 13224.10 13406.55 13606.79 13824.67 14060.07 14312.82 14582.76 14869.71 15173.50 15493.91 15830.73 16183.76 16552.76 16937.49 17337.70 17753.12 1.9ia3.50 18628.53 19087.95 19561.44 20048.71 20549.42 21063.25 21589.87 22128.93 22680.07 23242.94

v=5

14870.06 14887.91 14923.59 14977.09 15048.36 15137.36 15244.03 15368.29 15510.07 15669.26 15845.77 16039.46 16250.21 16477.88 16722.32 16983.36 17260.82 17554.52 17864.26 18189.83 la531.01 18887.58 19259.28 19645.88 20047.11 20462.70 20892.37 21335.82 21792.76 22262.87 22745.84 23241.34 23749.02 24268.53 24799.52 25341.61

BORN-OPPENHEIMER BREAKDOWN EFFECTS

377

TABLE IV-Continued

H37Cl v=o

J

36

37 38 39 40 41 42 43 44 45

14470.86 15141.91 15625.78 16522.10 17230.45 17950.43 18681.64 19423.65 20176.04 20938.39

"=I

16961.03 17610.72 18272.70 18946.59 19631.97 20328.46 21035.64 21753.10 22480.40 23217.12

v=2

v=3

19349.18 19977.50 20617.56 21268.98 21931.35 22604.27 23287.33 23980.10 24682.15 25393.06

21634.87 22241.73 22859.76 23488.57 24127.75 24776.91 25435.61 26103.44 26779.95 27464.70

v=4

v=5

23817.15 24402.33 24998.09 25604.02 26219.72 26844.77 27478.73 28121.18 28771.65 29429.70

25894.43 26457.59 27030.68 27613.30 28205.03 28805.43 29414.05 30030.44 30654.14 31284.64

" = 10

" = 11

25666.58 25681.36 25710.90 25755.17 25814.15 25887.79 25976.01 26078.74 26195.90 26327.39 26473.10 26632.90 26806.65 26994.21 27195.41 27410.08 27638.03 27879.06 28132.95 28399.49 28678.44 28969.53 29272.51 29587.09 29912.98 30249.88 30597.45 30955.37 31323.26 31700.76 32087.48 32482.99 32886.88 33298.67 33717.89 34144.02 34576.52 35014.81 35458.26 35906.22 36357.95 36812.65 37269.46 37727.39 38185.29 38641.85

27499.10 27513.21 27541.42 27583.71 27640.04 27710.35 27794.58 27892.65 28004.48 28129.97 28268.99 28421.41 28587.10 28765.91 28957.65 29162.16 29379.23 29608.66 29850.22 30103.68 30368.78 30645.26 30932.83 31231.20 31540.06 31859.05 32187.85 32526.07 32873.33 33229.20 33593.26 33965.04 34344.04 34729.74 35121.58 35518.95 35921.20 36327.61 36737.39 37149.66 37563.41 37977.51 38390.57 38800.93 39206.45 39604.18

H37Cl J

0 1 2 3 4 5 6 7 a 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 3.5 39 40 41 42 43 44 45

v=6

17239.31 17256.56 17291.05 17342.76 17411.64 17497.65 17600.73 17720.81 17L)57.81 18011.63 18182.16 18369.28 18572.87 18792.78 19028.85 19280.94 19548.85 19832.40 20131.40 20445.63 20774.07 21118.90 21477.48 21850.34 22237.23 22637.88 23052.00 23479.30 23919.47 24372.20 24837.16 25314.03 25802.44 26302.05 26812.48 27333.37 27864.31 28404.90 2a954.73 29513.38 30080.40 30655.34 31237.72 31827.06 32422.86 33024.59

v=7

19505.17 19521.82 19555.11 19605.00 19671.47 19754.47 19653.94 19969.81 20101.% 20250.38 20414.88 20595.38 20791.73 21003.80 21231.44 21474.47 21732.73 22006.02 22294.15 22596.90 22914.05 23245.37 23590.62 23949.54 24321.86 24707.32 25105.61 25516.44 25939.51 26374.48 26821.03 27278.81 27747.47 28226.63 28715.91 29214.93 29723.26 30240.50 30766.20 31299.91 31841.15 32389.45 32944.28 33505.13 34071.42 34642.59

v=a

21666.42 21682.46 21714.52 21762.58 21826.62 21906.56 22002.37 22113.96 22241.25 22384.14 22542.53 22716.30 22905.31 23109.42 23328.48 23562.31 23810.75 24073.59 24350.64 24641.69 24946.50 25264.85 25596.48 25941.14 26298.54 26668.42 27050.47 27444.38 27849.84 28266.51 28694.05 29132.09 29580.27 30038.21 30505.49 30981.70 31466.40 31959.15 32459.48 32966.89 33480.86 34ocQ.85 34526.30 35056.58 35591.07 36129.06

v=9

23721.14 23736.55 23767.37 23813.57 23875.11 23951.94 24044.01 24151.24 24273.54 24410.81 24562.95 24729.84 24911.34 25107.31 25317.58 25541.99 25780.35 26032.48 26298.16 26577.17 26869.30 27174.28 27491.87 27821.80 28163.79 28517.54 28882.75 29259.10 29646.25 30043.86 30451.56 30868.97 31295.69 31731.32 32175.42 32627.53 33087.18 33553.88 34027.09 34506.25 34990.77 3Mao.01 35973.29 36469.88 36968.95 37469.63

J. A. COXON

378

TABLE IV-Continued H37Cl J

0 1

2 3 4 5 6 7 a 9 10 11 12 13 14 1s lb 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 II 42 43

" = 12

29213.98 29227.40

29254.22 29294.43 29347.97 29414.80 29494.84 29588.02 29694.25 29813.42 29945.40 3cmO.06 30247.25 30416.81 30598.56 30792.31 30997.84 31214.9s 31443.38 31682.89 31933.20 32194.03 32465.06 32745.98 33036.43 33336.05 33644.44 33961.20 34285.88 34617.99 34957.05 35302.49 35653.71 %010.08 36370.88 36735.30 37102.45 37471.32 37840.71 38209.21 38575.07 30935.99 39288.54 39626.73

" = 13

" = 14

30804.72 30817.40 30842.74 30880.72 30931.30 30994.42 31070.00 31157.96 31258.21 31370.63 31495.00 31631.43 31779.51 31939.1s 32110.16 32292.33 32405.44 32689.24 32903.47 33127.86 33362.10 33605.88 33850.04 34120.63 34390.84 34669.05 34954.80 35247.60 35546.89 35852.09 36162.56 36477.56 36796.31 37117.89 37441.26 37765.19 30088.17 30408.29 38722.82 39027.13 39314.60

32262.86 32274.75 32298.50 32334.08 32381.46 32440.57 32511.33 32593.66 32607.43 32792.54 32908.83 33036.15 33174.33 33323.17 33482.45 33651.95 33831.42 34020.50 34219.13 34426.76 34643.11 34867.01 35100.44 35340.57 35587.70 35841.31 36100.82 36365.57 36634.86 36907.86 37183.64 37461.09 37738.87 38015.25 38287.87 38552.77 38802.94

" = 15

33577.85 33580.05 33610.84 33643.79 33687.64 33742.32 33807.7s 33883.82 33970.42 34067.40 34174.60 34291.84 34418.92 34555.62 34701.69 34856.87 35020.85 3X93.31 35373.90 3Ss62.22 35757.85, 35960.31 36169.10 36383.61 36603.19 36827.10 37054.46 37284.24 37515.18 37745.63 37973.33 38194.52 38402.81

D3k J

0 1 2 3 4 5 6 7 e 9 10 11 12 13

v-0

1066.6041 1077.3880 1098.9526

1131.2912 1174.393s 1228.2463 1292.8327 3368.1327 1454.1229 1550.7764 1658.0634 1775.9504 1904.4009 2043.3749

v=l

3157.664 3168.223 3189.338 3221.002 3263.206 3315.93s 3379.174 3452.902 3537.096 3631.730 3736.773 3852.194 3977.95s 4114.018

v=2

5195.0351 5205.3708

v=3

7179.056 7189.169

5226.0389

7209.393

5257.0328 5298.3426 5349.9551 5411.8540 5484.0193 5566.4202 5659.0544 5761.8683 5874.8372 5997.9253 6131.0933

7239.720 7280.141 7330.643 7391.209 7461.820 7542.453 7633.082 7733.678 7844.207 7964.636 8094.924

" = 16

Y = 17

34734.50 34744.51 34764.50 34794.4s 34634.28 34883.93 34943.29 35012.24 35090.63 35178.31 35275.08 35380.74 35495.02 35617.68 35740.40 35886.85 36032.65 36185.38 36344.56 36509.65 36680.05 36855.03 37033.77 37215.25 37398.24 37581.09 37761.41

35713.3s 35722.19 35739.0s 35766.28 35801.40 35845.11 3sa97.30 3s957.01 36026.45 36103.01 36187.24 36278.86 36377.53 36402.09 36594.48 36711.82 36834.32 36961.30 37091.91 37225.14 37359.66 37493.61 37623.68 37745.48

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

379

TABLE IV-Continued

D35CI v=o

J

14

15 16 17 18 19 20 21 22 23 24 2!5 26 27 28 29 30 31 32

2192.82% 2352.7185 2522.9923 2703.5% 2694.400 309s.%C 3306.835 3528.181 3759.549 4ooo.869 4252.065 4513.062 4763.779 5064.134 5354.041 5653.411 5962.154 6280.176 6607.380

v=l

v=2

4260.339 4416.873 4583.570 4760.378 4947.243 5144.106 5350.905 5567.576 5794.051 6030.260 6276.129 6531.H12 6796.541 7070.922 7354.641 7647.611 7949.741 8260.937 8581.105

6274.2989 6427.4967 6590.6dO 6763.6711 6946.541 7139.190 7341.557 7553.579 7775.180 8006.314 8246.884 8496.823 0756.052 9024.490 9302.053 9588.653 9884.201 10188.604 10501.767

v=3

8235.030 8384.908 e544.511 6713.786 8892.68 9081.14 9279.09

D3'Cl J

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

v=o

1065.042 1075.794 1097.2% 1129.540 1172.516 1226.211 1290.609 1365.689 1451.428 1547.800 1654.775 1772,319 1900.396 2038.967 2187.989 2347.416 2517.1% 2697.283 2887.62 3008.14 3298.79 3519.90 3750.21 3990.84 4241.33 4501.58 4771.54 5051.10 5340.19 5638.73 5946.61 6263.75 6590.05

v=l

3153.102 3163.630 3184.684 3216.257 3258.338 3310.915 3373.971 3447.486 3531.437 3625.798 3730.539 3845.628 3971.028 4106.700 4252.6O2 4400.688 4574.909 4751.214 4937.55 5133.85 5340.06 5556.12 5701.96 6017.50 6262.68 6517.42 678i.64 7055.26 7338.19 7630.35 7931.65 8241.99 8561.28

v=2

5187.631 5197.937 5218.546 5249.451 5290.643 5342.108 5403.829 5475.789 5557.963 5650.325 5752.646 5865.494 5988.232 6121.022 6263.822 6416.587 6579.268 6751.814 6934.17 7126.28 7328.08 7539.51 7760.50 7990.98 8230.89 8480.14 8738.65 9o06.35 9283.15 9568.97 9863.72 10167.29 10479.61

v=3

7168.965 7179.050 7199.217 7229.458 7269.765 7320.124 7380.919 7450.931 7531.336 7621.710 7722.023 7032.242 7952.332 6082.255 0221.969 8371.429 8530.587 8699.391 8877.79 3065.72 9263.13

the wavenumbers of all the lines used in the fit. The fact that the residuals between the measured and calculated line positions were now small, and generally indistinguishable from the discrepancies of the fit (columns 4 and 7 of Tables II and III)

380

J. A. COXON TABLE IV-Continued

D37C1

D=Cl

J

33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 50 59 60 61 62 63 64 65

v=o

v=l

v=2

v=o

v=l

6943.667 7288.936 7643.081 8005.997 8377.573 8757.699 9146.26 9543.14 9940.22 10361.38 10782.49 11211.43 11648.08 12092.30 12543.96 13002.93 13469.07 13942.24 14422.31 14909.13 15402.57 15902.46 16408.68 16921.07 17439.48 17963.75 18493.74 19029.29 19570.25 20116.45 20667.74 21223.95 21784.93

8910.145 9247.957 9594.437 9949.479 10312.975 10684.814 11064.aa 11453.06 11849.24 12253.29 12665.10 13084.54 13511.47 13945.78 14387.33 14835.99 15291.62 15754.09 16223.26 16698.98 17ial.12 17669.52 18164.05 18664.56 19170.89 19682.69 20200.41 20723.30 21251.40 21784.55 22322.58 22865.35 23412.68

10823.593 11153.981 11492.829 11a40.030 12195.478 12559.062 12930.67 13310.18 13697.49 14092.47 14495.00 14904.95 15322.20 15746.62 16178.08 16616.45 17061.59 17513.37 17971.65 18436.20 10907.13 19384.05 19866.89 20355.50 20849.74 21349.46 21854.49 22364.69 22879.90 23399.96 23924.70 24453.97 24987.59

6925.40 7269.72 7622.90 7984.83 8355.39 8734.49 9122.00 9517.82 9921.82 10333.88 10753.88 lllal.70 11617.20 12060.27 12510.76 12968.55 13433.50 13905.47 14304.33 14869.93 15362.14 15860.80 16365.78 16876.91 17394.07 17917.09 18445.82 la980.11 19519.80 20064.73 20614.75 21169.70 21729.41

8889.43 9226.32 9571.86 9925.95 10288.47 10659.32 11038.38 11425.53 11820.67 12223.67 12634.40 13052.75 13478.58 13911.78 14352.21 14799.73 15254.22 15715.53 16183.53 lbb58.08 17139.04 17626.26 18119.60 labla.91 19124.03 19634.83 20151.15 20672.83 21199.72 21731.66 22268.48 22810.04 23356.17

v=2

10800.57 1113o.oa 11468.02 11814.30 121ba.81 12531.44 12902.07 132aO.M) 13bbb.90 14060.66 14462.35 14871.26 15207.45 15710.81 16141.19 16578.47 17022.51 17473.18 17930.34 18393.85 1.9ab3.56 19339.34 19821.04 20308.51 20801.61 21300.17 21804.05 22313.10 22827.15 23346.05 23869.65 24397.76 24930.25

demonstrates that the entire procedure is numerically self-consistent. The term values are listed in Table IV, along with those obtained later for H37C1,D3%1, and D37C1. The fitted q”(r) in Fig. 4 is given according to Eq. (16) as 4Mr)

= ~,[&-t(r)/~n

+

&W/~35CJ

(37)

If the unknown forms & and I&, are similar in magnitude, the second term on the right side of Eq. (37) becomes negligible; in other words, non-adiabatic corrections are due principally to the much lighter H atom. With this approximation, the corresponding function for D35C1(and D37C1)is &XJ(r) =

(~H/~Dh?kl(r).

(38)

Such a calculated q’&-, was employed as a constraint in a fit of the correction AU”(r) from the reliable data that are available (8) for D3%1. Seven parameters (cln, n = l-3; c2,,, n = l-4) were fitted to the residuals of 102 line positions calculated from the RKR potential for this isotope. The standard deviation was G = 0.58, showing that for these data (pure rotation and vibration-rotation transitions for u G 3), the measurements appear to be essentially free of systematic error. It is unfortunate that

ISORN-OPPENHEIMER

BREAKDOWN

381

EFFECTS

FIG. 5. Experimentally determined difference potentials, Udf(H%3) - vc8(d5Cl) (solid line) and L@(H35Cl) - CJti(H3’CI)(broken line) for the X’Z+ state. Arbitrarily, AU = 0.0 at r, for both curves.

the data are so limited that a determination of q;;cl(r) is not possible. An experimental verification of the isotopic relationship of Eq. (38) would be very desirable. A similar determination of the effective potential for H3’Cl (X’Z’) was also made from the most reliable data (8) available for this isotope. In this case, the trial potential was the effective potential found for H3’Cl, and q”(r) was assumed to be the same for both isotopes. Only four parameters (cl”, n = 1, 2; c22n,n = 1, 2) were required for a satisfactory fit (c = 0.62, 111 line positions). Plots of the difference potentials, Ueff(H3%1) - Ueff(D3%l) and Ueff(H3%l) - Ueff(H3’C1)are shown in Fig. 5 over the limited range of r for which U”ff(D35Cl)and Ueff(H37Cl)can be considered to be determined. It should be emphasized that the results in Fig. 5 do not include any constant electronic isotope shifts; the zero differences at r, are a consequence of the absence of any data that would determine such a shift. Despite this, however, it is clear that a very significant difference in the forms of the effective potentials exists for H35C1and D3’Cl even for relatively small shifis from the equilibrium internuclear separation. On the other hand, the difference between the potentials for H35Cl and H37Cl is extremely small, as expected; the pronounced discontinuity in the latter curve at r, is obviously physically unrealistic, and it would probably have been better to determine the AU”(r) correction for H3’Cl as a single form over the whole r range. However, given the very low magnitude of the Ueff(H3’Cl) - Ueff(H3’Cl) difference potential, this is not a serious problem. Having determined effective potentials for the three isotopes, H35Cl, H”C1, and D35Cl, a rather appealing test of the present methodology is afforded by the data that are available for D37C1.According to Eq. (17), we can write Ueff(H3’C1) = Urn(r) + UH(r)/MH + Ucl( r)/M35a

(39)

U”ff(D35C1)= Urn(r) + UH(r)/MD + Ua(r)/M35~~.

(40)

and It follows simply that U,(r) = m,&(r)

=

U”ff(H35Cl) - Ueff(D3’Cl) (l/MH

-

l/MD)

(41) ’

382

J. A. COXON TABLE V Observed and Calculated Line Position9 for D3’Cl P(J)

2 - o=

3 -

v =

o=

od

0.00051

0.00134

1.1

0 1 2 3 4 5 6 7 6 9 10 11 12 13 14 15

1 2 3 4 5 6 7 a

x 10-6 o

3.8 x 10-6 I

2.83 0.0014 0.0007 1.46 0.0008 1.61 0.0004 0.83 0.0004 0.87 0.0003 0.63 0.0010 2.05 0.0001 0.12 0.0003 0.49 0.0004 0.73 3900.5257 -0.0012 -2.28 3965.0971 -0.0001 -0.20 3949.2642 -0.0005 -0.99 3933.0333 0.0000 0.03

4111.8380 4100.6419 4009.0070 4076.9355 4064.4318 4051.4989 4038.1414 4024.3604 4010.1625 3995.5502

6081.7556 0.0010 0.72 6069.6763 -0.0008 -0.60 6056.9445 0.0023 1.70 6043.5546 0.0010 0.74 6029.5143 -0.0006 -0.45 6014.8321 0.0022 1.68 5999.5010 -0.0013 -0.95

R(J)

4132.8956 4142.7517 4152.1558 4161.1035 4169.5920 4177.6191 4185.1802 4192.2738 4198.8967 4205.0458 4210.7189 4215.9143 4220.6270 4224.0559 4228.6011 4231.8526

0.0005 0.0001 0.0005 0.0005 0.0003 0.0008 0.0004 0.0004 o.ooo4 0.0002 0.0002 0.0013 0.0010 0.0008 0.0031 0.0002

1.00 0.19 0.97 0.90 0.56 1.57 0.71 0.70 0.82 0.39 0.35 2.49 2.00@ 1.55 6.03* 0.42

6123.4328 6132.1611 6140.2248 6147.6077 6154.3085 6160.3212

0.0103 -0.0014 -0.0005 -0.0003 0.0007 -0.0006

7.68~ -1.01 -0.37 -0.25 0.52 -0.43

0 .oooo 0.0000

1.37 2.71

10.7524 21.5015

Walculatsd line poeitions were obtained from the eigmvsluer of Eq.(lS>: the effective potential MS given by Ueff(D37C1) = ugD(r) + UR(r)/iID4 uCl(r)lR37Cl. Lines flagged by an astorrsk YS~B excluded in the leastsquares fit of Ref. (8). bar the vibratmn-rotation data. $ are the standard deviations of leastsquares fits given in Ref.(&). =Data frow Ref.(z). dData from Ref.(a).

With a similar expression providing U,(r) from Ueff(H3%1) and Ueff(H3’C1),it is then straightforward to obtain Urn(r), and hence U”(r) for any isotope. This approach has been employed for D3’C1. Having calculated U’“(r) for this isotope, the eigenvalues given by Eq. (15) permit a comparison of wholly synthetic line positions with those obtained experimentally. Such comparisons are listed in Table V for two vibrationrotation bands where precise Fourier transform data are available (27), and for the microwave transitions in n = 0 (28). It is striking that there is remarkable agreement in all cases between the calculated and experimental transition energies; the magnitudes of the residuals are on average about half the estimated uncertainties (2~) for the entire D3’C1 data with u < 3 (8). The author is not aware of any alternative method with such apparently excellent predictive ability for the energy levels of a hydride. A more interesting test would of course be provided by a comparison for T3’C1 or T37C1,and

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

383

FIG. 6. Contributions to E,,Jfrom J-dependent non-adiabatic mixing in the X’Z+ ground state of H3’CI. The curves are calculated from the fitted function in Fig. 4.

it is hoped that the present results might stimulate high-resolution work on these isotopes. The plot in Fig. 4 of q(r) for the ground state of HCl represents the first experimental determination of the radial form of an g(r) function, Eq. ( lo), defined by Watson (10) in terms of off-diagonal interactions. As noted earlier, this function is not determined uniquely, being defined with an arbitrary zero value at r,. It would be of much interest to compare results of the type shown in Fig. 4 with an ab initio calculation. Since the contributions to the eigenvalues from the function in Fig. 4 are not immediately obvious, these are shown explicitly in Fig. 6. In accord with the radial form of Fig. 4, the shifts, AE,, = (uJlg(r)luJ)J(J + I), increase with increasing u, and are appreciable even for u = 0, especially at high-J values. The essential results of the present work are summarized in numerical form in Table VI. The effective potential for H3’Cl, and the Iunctions UH(T), U&r), and q(r) are listed over r ranges where their determinations are considered reliable. These data could be employed directly in other work, and permit, for example, the generation of U”(r). As expected, this latter potential is in excellent agreement with that determined by Coxon and Ogilvie (8) with Watson’s (10) analytical formulation. Values of the equilibrium separation r: for the effective potentials of HCl and DCl, and for Ueo(r) are compared in Table VII with results from previous work. The values from Ref. (8) are obtained by extrapolating the fitted Q,, for individual isotopes from u = -$ to the potential minima. The present results are determined from the numerical potentials. Clearly, there is excellent agreement. Equally satisfying is the agreement between the estimates of r, for VW(r), including that of Bunker (29). In summary, it is believed that the present work represents a promising new approach for the reduction of spectroscopic data on diatomic molecules. Although similar to the IPA method (15, 16), an unlimited set of line positions can be fitted directly with appropriate weights; the earlier approach employed an intermediate set of equally

384

J. A. COXON TABLE VI Numerical Data (cm-‘) for the Effective Vibration-Rotation r(A)

0.900 0.905

0.910 0.915 0.920 0.925 0.930 0.935 0.940 0.945 0.950 0.955 0.960 0.965 0.970 0.975 0.980 0.985 0.990 0.995 1.000 1.010 1.020 1.030 1.040 1.050 1.060 1.070 1.080 1.090 1.100 1.110 1.120 1.130 1.140 1.150 1.160 1.170 1.180 1.190 1.200 1.210 1.220 1.230 1.240 1.250 1.260 1.270 1.280 1.290 1.300 1.310 1.320 1.330 1.340 1.350 1.360 1.370 1.380 1.390 1.400 1.410 1.420 1.430 1.440 1.450 1.460

U=ff(H35Cl)

39944.785

38426.752 36951.619 35518.580 34124.546 32779.289 31477.233 30222.248 28996.877 27828.032 26672.842 25564.549 24491.428 23451.572 22445.859 21472.192 20530.060 19618.848 18737.410 17885.217 17061.597 15496.500 14036.883 12677.294 11412.897 10238.7872 9150.3509 8143.3076 7213.5163 6357.0558 5570.1993 4849.4078 4191.3197 3592.7407 3050.6349 2562.1158 2124.4396 1734.9973 1391.3070 1091.0078 831.8537 611.7076 428.5358 280.4027 165.4658 81.9710 28.2479 2.7061 3.8203 30.1537 80.3373 153.0584 247.0673 361.1740 494.2450 645.2002 813.0101 996.6936 1195.3153 1407.9834 1633.8475 1872.0968 2121.9578 2382.6930 2653.5988 2934.0041 3223.2688

103q

1.174 1.073 0.980 0.894 0.817 0.745 0.680 0.620 0.566 0.516 0.471 0.429 0.391 0.356 0.325 0.296 0.269 0.245 0.223 0.203 0.185 0.153 0.127 0.105 0.08741 0.07266 0.06053 0.05057 0.04240 0.03570 0.03021 0.02570 0.02199 0.01892 0.01637 0.01423 0.01242 0.01085 0.00948 0.00825 0.00713 0.00609 0.00510 0.00414 0.00320 0.00228 0.00135 0.00043 -0.00051 -0.00146 -0.00242 -0.00339 -0.00437 -0.00537 -0.00637 -0.00739 -0.00842 -0.00945 -0.01050 -0.01155 -0.01262 -0.01369 -0.01477 -0.01587 -0.01697 -0.01810 -0.01924

UH

-4.8428 -4.4612 -4.0924

-3.7406 -3.4112 -3.1034 -2.8117 -2.5432 -2.2904 -2.0526 -1.8282 -1.6154 -1.4125 -1.2180 -1.0293 -0.8431 -0.6578 -0.4714 -0.2822 -0.0889 0.0813 0.2310 0.3769 0.5146 0.6419 0.7585 0.8649 0.9619 1.0498 1.1296 1.2027 1.2704 1.3335 1.3926 1.4484 1.5012 1.5519 1.6008 1.6471

Hamiltonian of H’5Cl UC1

3.046 3.250 3.403 3.509 3.572 3.595

3.582 3.534 3.455 3.348 3.213 3.054 2.873 2.670 2.449 2.209 1.954 1.683 1.399 1.102 0.793 0.474 0.146 -0.014 -0.023 -0.027 -0.025 -0.019 -0.009 0.005 0.023 0.045 0.070 0.098 0.129 0.163 0.200 0.239 0.280 0.323 0.369 0.416

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

385

TABLE VI-Continued

r(A)

1.470 1.480 1.490 1.500 1.510 1.520 1.530 1.540 1.550 1.560 1.570 1.580 1.590 1.600 1.610 1.620 1.630 1.640 1.650 1.660 1.670 1.680 1.690 1.700 1.710 1.720 1.730 1.740 1.750 1.760 1.770 1.780 1.790 1.800 1.810 1.820 1.830 1.840 1.850 1.860 1.870 1.880 1.890 1.900 1.910 1.920 1.930 1.940 1.950 1.960 1.970 1.980 1.990 2.000 2.010 2.020 2.030 2.040 2.050 2.060 2.070 2.080 2.090 2.100 2.110 2.120 2.130

Wff

(H35Cl)

3520.7824 3823.9623 4138.2530 4457.1240 4782.0693 5112.6062 5448.2742 5788.6335 6133.2650 6481.7681 6833.7608 7188.8786 7546.7733 7907.1131 8269.5809 8633.8746 8999.7054 9366.7976 9734.8885 10103.727 10473.074 10842.701 11212.392 11581.938 11951.139 12319.804 12687.751 13054.809 13420.815 13785.616 14149.062 14511.011 14871.328 15229.884 15586.554 15941.221 16293,771 16644.101 16992.115 17337.719 17680.826 18021.341 18359.179 18694.259 19026.512 19355.876 19682.287 20005.686 20326.010 20643.219 20957.256 21268.034 21575.493 21879.620 22180.390 22477.765 22771.718 23062.202 23349.115 23632.432 23912.231 24188.498 24461.152 24730.176 24995.574 25257.308 25515.370

-0.02040 -0.02158 -0.02279 -0.02403 -0.02530 -0.02660 -0.02795 -0.02935 -0.03080 -0.03231 -0.03387 -0.03551 -0.03722 -0.03900 -0.04088 -0.04284 -0.04491 -0.04708 -0.04936 -0.0518 -0.0543 -0.0569 -0.0597 -0.0627 -0.0657 -0.0690 -0.0724 -0.0760 -0.0798 -0.0837 -0.0879 -0.0922 -0.0968 -0.102 -0.107 -0.112 -0.117 -0.123 -0.129 -0.135 -0.142 -0.148 -0.155 -0.163 -0.170 -0.178 -0.186 -0.195 -0.204 -0.213 -0.223 -0.232 -0.243 -0.253 -0.264 -0.275 -0.287 -0.299 -0.311 -0.324 -0.337 -0.350 -0.364 -0.379 -0.393 -0.409 -0.424

1.6912 1.7357 1.7804 1.8236 1.8669 1.9111 1.9537 1.9966 2.0418 2.0874

0.465 0.515 0.567 0.621 0.676 0.732 0.789 0.847 0.906 0.967 1.028 1.089 1.152 1.215 1.279 1.343 1.408 1.473 1.539

386

J. A. COXON TABLE VI-Continued

r(A)

2.140 2.150 2.160 2.170 2.180 2.190 2.200 2.210 2.220 2.230 2.240 2.250 2.260 2.270 2.280 2.290 2.300 2.310 2.320 2.330 2.340 2.350 2.360 2.370 2.380 2.390 2.400 2.410 2.420 2.430 2.440 2.450 2.460 2.470 2.480 2.490 2.500 2.510 2.520 2.530 2.540 2.550 2.560 2.570 2.580 2.590 2.600 2.610 2.620 2.630 2.640 2.650 2.660 2.670 2.680 2.690 2.700 2.710 2.720

UeffcH35Cl)

25769.825 26020.695

26267.914 26511.466 26751.390 26987.696 27220.363 27449.340 27674.572 27896.111 28114.060 28328.459 28539.298 28746.458 28949.792 29149.403 29345.543 29538.279 29727.497 29913.149 30095.285 30273.956 30449.193 30621.037 30789.581 30954.913 31117.022 31275.860 31431.445 31583.873 31733.230 31879.486 32022.607 32162.562 32299.342 32432.965 32563.575 32691.320 32816.282 32938.456 33057.832 33174.345 33287.949 33398.594 33506.536 33612.010 33715.261 33816.171 33914.647 34010.554 34104.016 34195.115 34283.990 34370.649 34455.126 34537.446 34617.622 34695.671 34771.603

103q -0.440 -0.456 -0.473 -0.491 -0.508 -0.527 -0.545 -0.564 -0.584 -0.604 -0.624 -0.645 -0.667 -0.689 -0.711 -0.734 -0.758 -0.782 -0.806 -0.831 -0.857 -0.883 -0.909 -0.936 -0.964 -0.992 -1.021 -1.050 -1.079 -1.110 -1.141 -1.172 -1.204 -1.236 -1.269 -1.303 -1.337 -1.372 -1.407 -1.443 -1.480 -1.517 -1.555 -1.593 -1.632 -1.671 -1.711 -1.752 -1.793 -1.835 -1.878 -1.921 -1.965 -2.009 -2.054 -2.100 -2.146 -2.193 -2.240

r(A) 2.730 2.740 2.750 2.760 2.770 2.780 2.790 2.800 2.810 2.820 2.830 2.840 2.850 2.860 2.870 2.880 2.890 2.900 2.910 2.920 2.930 2.940 2.950 2.960 2.970 2.980 2.990 3.000 3.010 3.020 3.030 3.040 3.050 3.060 3.070 3.080 3.090 3.100 3.110 3.120 3.130 3.140 3.150 3.160 3.170 3.180 3.190 3.200 3.210 3.220 3.230 3.240 3.250 3.260 3.270 3.280 3.290 3.300

lPff(H35Cl) 34845.440 34917.270 34987.165 35055.221 35121.491 35186.008 35248.801 35309.882 35369.267 35426.964 35482.977 35537.264 35589.858 35640.822 35690.199 35738.105 35784.804 35830.138 35874.134 35916.821 35958.228 35998.383 36037.316 36075.054 36111.626 36147.061 36181.388 36214.634 36246.829 36278.108 36308.895 36338.561 36367.149 36394.704 36421.267 36446.877 36471.572 36495.388 36518.361 36540.522 36561.904 36582.538 36602.451 36621.673 36640.229 36658.146 36675.448 36692.157 36708.298 36723.890 36738.956 36753.514 36767.584 36781.183 36794.330 36807.041 36819.332 36831.219

103q -2.288 -2.337 -2.386 -2.436 -2.487 -2.538 -2.590 -2.643 -2.696 -2.750 -2.804 -2.860 -2.915 -2.972 -3.029 -3.087 -3.145 -3.205 -3.264 -3.325 -3.386 -3.448 -3.510 -3.573 -3.637 -3.702 -3.767 -3.833 -3.900 -3.967 -4.035 -4.103 -4.173 -4.243 -4.313 -4.385 -4.457 -4.530 -4.603 -4.677 -4.752 -4.828 -4.904 -4.981 -5.059 -5.137 -5.217 -5.296 -5.377 -5.458 -5.540 -5.623 -5.706 -5.790 -5.875 -5.961 -6.047 -6.134

weighted term values, incorrectly assumed to be uncorrelated, and based on an arbitrary partial selection of the available data. In addition, the present method enables an allowance for non-adiabatic effects, which are estimated simultaneously. Further work will be undertaken to verify whether such estimated non-adiabatic contributions exhibit theoretical isotope behavior. To this end, data on the analogous B - X systems of HF and DF are currently under investigation.

BORN-OPPENHEIMER

BREAKDOWN

EFFECTS

387

TABLE VII Equilibrium Internuclear Separation (A) in the Ground State of HCl lJeff(HC1)

IJeff

ua0

1.274552

1.274585

1.274617

1.27455o

I.274583

1.27460a

Ref. this work (8)

1.27460(5)

(29)

1.274615(l)

(30)

ACKNOWLEDGMENTS Support for this work from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Dr. R. J. LeRoy is thanked for helpful criticism and encouragement. RECEIVED:

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