near-dissociation expansion representation of large...

JOURNAL OF MOLECULAR SPECTROSCOPY 109, 352-367 (1985)

Near-Dissociation Expansion Representation of Large Spectroscopic Data Sets: The B(311iu) - X(‘Z,‘) System of I2

JOHN W. TROMP’ AND ROBERT J. LE ROY

Guelph- Waterloo Centre for Graduate Work in Chemistry, University of Waterloo, Waterloo, Ontario N2L 3GI, Canada

The utility of near-dissociation expansion (NDE) functions for efficiently representing large spectroscopic data sets is demonstrated by an application to 14712 lines of the visible B-X absorption spectrum of 12. In addition to providing a more compact representation of the input data than is obtainable using traditional methods, this approach has a unique ability to provide reliable predictions for vibrational levels lying above the highest one observed. Considerations governing choice of the number of parameters to include in an expansion and of the number of significant digits to quote for each of the resulting parameters are critically discussed. Q 1985 Academic Press, Inc.

I. INTRODUCTION

A continuing problem in molecular spectroscopy is the determination of compact and accurate expressions to summarize the information comprised by the myriad of line frequencies associated with a given electronic molecular transition. Such expressions should ideally satisfy the following four criteria: (i) Accuracy: they should be able to reproduce all of the available data to within the experimental uncertainties. (ii) Physics: at least some of their parameters should have physical significance. (iii) Compactness: within the limits allowed by criterion (i), they should require the smallest possible number of empirical parameters. (iv) Extrapolation ability: they should be able to make reliable predictions for cases well beyond the range of the data on which they are based. While traditional methods (I, 2) satisfy the first two of these criteria reasonably well, their performance is more dubious with regard to compactness and they are unquestionably deficient with regard to predictive ability. For diatomic molecules, the present paper describes and illustrates an alternative approach to this problem which is at least as good as the traditional method with regard to criteria (i) and (ii), and offers great advantages in the areas of compactness and extrapolation ability.

The usual energy level expression for a vibrating rotating diatomic molecule with no electronic angular momentum along the molecular axis is

E(u, J) = G(v) + B,[J(J + l)] - D,[J(J + l)]’

+ H”[J(J + I)]3 + L,[J(J + 1)]4 + M”[J(J + 1)]5 + * * - ) (1)

i Natural Sciences and Engineering Research Council of Canada Postgraduate Scholar. Present address: Department of Chemistry, University of California, Berkeley, Calif. 94720.

0022-2852185 $3.00 CopyrigJ~t 0 1985 by Academic Pi-es, Inc. All rights of reproduction in any form resewed.

352

NEAR-DISSOCIATION EXPANSIONS 353

where G(u) is the rotationless vibrational energy, B, is the inertial rotational constant. andD,,H,,,. . . etc. are centrifugal distortion constants (1, 2). The more convenient notation used herein defines K,,,(o) as the coefficient of [J(J + l)]” in this expansion:

Ko(v) = G(v) (2)

K,(v) = & (3)

K2(u) = -D, (4)

K,(v) = Ho (5)

etc.

This allows Eq. (1) to be written as

mmal EC% J) = c K?l(WV + 1)1”, (6)

m=O

and the line frequencies associated with a given electronic transition as

Y = 2 K’,(v’)[J’(J + I)]” - 2 K~(u”)[J”(J” + l)]“, (7) m=O m=O

where single primes refer to the upper electronic state and double primes to the lower. The advantages of the K,,,(u) notation are its compactness and the fact that it removes the somewhat artificial difference in sign between the definition of D, and those of the other centrifugal distortion constants.

The traditional approach to spectroscopic analysis either treats the various K,(V) values for the individual vibrational levels as the parameters to be determined, or represents them by the familiar Dunham (1) power series in (U + l/2):

K,(V) = 2 Y&21 + i,‘. (8) I=0

The first of these procedures has no real predictive ability for unobserved vibrational levels and is not very economical with regard to the number of parameters, since several K&V) values may have to be tabulated for each vibrational level of both the initial and final electronic states.

The custom of representing the vibrational dependence of the K,(V) constants by the simple power series of Eq. (8) arose from Dunham’s demonstration (I) that the coefficients of a Taylor series expansion about II = -4 could be related to the coefficients of a power series expansion of the potential about its minimum. However, insofar as virtually any other functional form could also be used to define Taylor series coefficients at IJ = -3, the single power series of Eq. (8) is no more

physically correct than any other. Moreover, polynomials are notoriously unreliable extrapolating functions, a point which was dramatically demonstrated for rotational and centrifugal distortion constants [K,(u) for m 2 I] in Ref. (3). That work also showed that Dunham expansions are often very zmeconomical with regard to the number of parameters required to represent a given set of K,,,(u) values.

354 TROMP AND LE ROY

It was clear from Dunham’s work (I) that centrifugal distortion constants [K,,,(D) for m 2 21 are not independent parameters, since they are determined by the potential energy function, which is in turn determined by a knowledge of only K,,(u) and K,(u). However, use of the Dunham approach to determine arbitrary centrifugal distortion constants is a sufficiently intimidating procedure that in practice it is only used to describe them at the potential minimum (2). In recent years, however, a number of methods have been developed which allow one to calculate distortion constants for any given potential energy curve.2 As a result, it is now a fairly straightforward matter to require that the distortion constants used in representing spectroscopic data be “mechanically consistent” with the associated vibrational energies J&(V) and rotational constants K,(v). This is done by performing the fits to the data to determine the Z&,(V) and K,(V) functions with the distortion constants held fixed at values computed from a potential curve generated from prior estimates of K,,(V) and K,(v), an iterative procedure which converges rapidly. The resulting availability of a consistent and accurate set of distortion constants should allow reliable extrapolation to high J for a given u, but of course does not itself yield a compact representation for the raw data which can provide realistic vibrational extrapolations. The latter problem is one which the present paper proposes to solve using “near-dissociation expansions” (NDE).

II. NEAR-DISSOCIATION EXPANSIONS

The approach recommended in the present work is that the vibrational dependence of all K,,,(u) constants be represented by “near-dissociation expansions,” functions which explicitly incorporate the correct limiting near-dissociation behavior. The basis for these expressions is the fact that for diatom vibrational levels lying near dissociation, the level density (5) rotational (6) and centrifugal distortion constants (3, 7, 8), and many other properties (9) depend mainly on the long-range part of the intermolecular potential. At sufficiently large distances, the long-range potential attains the limiting form

V(r) N D - CJr”. (9)

Substituting this approximation into the semiclassical expressions for various molecular properties (8) yields simple analytic expressions for their limiting near- dissociation behavior. For the K,,,(u) constants of Eqs. (2)-(6) these expressions are, for m = 0,

Kc(u) = D - &(n)(u, - u)~“‘+~), (10) and, for m > 1,

K,“(u) = X,(n)(u, - u)tZn’(“-2)-2ml, (11)

where D is the dissociation limit, u)D is the effective (in general, noninteger) vibrational index at dissociation, and X,(n) = ~~(n)/[~L”(C,)2]“(“-2), were ~1 is the reduced mass of the diatom and x,(n) are explicitly known numerical constants (5-8). The superscript “co” is used to denote the fact that these are limiting near-

* The method devised by Hutson (4) is the most accurate and generally applicable of the methods for

calculating centrifugal distortion constants reported to date, and it was used in the present work.


dissociation expressions which in general are only quantitatively reliable at energies very close to dissociation [except for the vibrational energy term m = 0, which due to a fortuitous cancellation of errors can sometimes be trusted over a more extended range (IO)].

For those few molecules for which reliable data are available for levels lying extremely close to dissociation, it may be possible to experimentally determine the values of the integer n and the potential constant C,, appearing in Eqs. (9)-( 11) (5, 8, 11-13). However, for virtually any diatomic species, the value of n is readily obtained from a knowledge of the electronic states of the molecule and the atomic dissociation products (5, 13, 14, 15),3 and reliable theoretical or semiempirical estimates of C,, may often be generated (14. 15). Thus, the limiting near-dissociation behavior of these spectroscopic constants, or at least their functional form, can be known even in the complete absence of experimental data.

Exploitation of this knowledge of the limiting functional behavior of these properties is the key feature of the present approach. For a neutral molecule dissociating into neutral atoms, n is usually 5 or 6. In this case, all of the centrifugal distortion (m 3 2) constants must approach infinity at dissociation (i.e., as &, - v), since the power of (uD - U) appearing in Eq. (11) is negative. As was shown in Ref. (3), an expression which does not build in this singular behavior will be unable to either provide the most compact representation possible for such constants or to extrapolate reliably. Similarly, the compressed spacings of vibrational levels near dissociation implied by Eq. (10) cannot readily be properly represented by the traditional polynomial in (v + 3). This is a very serious drawback of the traditional approach, since a vibrational energy extrapolation is required to determine both the molecular dissociation energy and the number and energies of missing levels.

The procedure recommended here is that the v dependence of all K,(z)) constants be represented by empirical expansions about the limiting behavior of Eqs. (10) and (11). For the vibrational energy and the various rotational constants considered separately, the advantages of this approach have been well demonstrated elsewhere (3, 16-18). The purpose of the present work is to illustrate the ease with which these expansions may be applied directly to large experimental data sets, and to demonstrate the economy of representation achieved in this way.

The near-dissociation expansions used in the present work have the form, for m = 0,

&(v) = D - &(n)(v,, - U)2n’(n-2kXp{ 2 a(o(vr, - U)‘}.

/= I (12)

and, for m >, 1,

Km(z)) = KE(u) exp{ 2 a/rn(vD - v)‘}.

/= I (13)

In previous NDE analyses of vibrational energies (16-18), polynomials or ratios of polynomials in (UD - u) were used in place of the exponential function appearing in Eq. (12). However, for the rotational and centrifugal distortion constants,

3 Brief summaries of the rules governing the determination of n and C, were presented in Refs. (5) and

(13).


exponent expansions have been found to be relatively more stable and to give somewhat more compact representations (3). While the exponential form does not appear to have any real advantages over even simple polynomials for the vibrational energies, for the sake of consistency it is used for all of the K,,,(V) constants in the fits reported here.

Equations ( 12) and (13) are not linear functions of the parameters D, no, and {a,,}, so realistic trial values of these variables are required to initiate the nonlinear least-squares fits to the experimental data. These are most readily obtained from preliminary fits to the various sets of K,,,(V) constants, considered separately. For m = 0 and 1 the required input values of K,(V) may be obtained from direct fits of Eq. (7) to experimental data, while for m Z= 2 they may be computed from the preliminary potential curve implied by the resulting J&(u) and K,(v) values.

Since the value of the vibrational intercept 2)u (and, if appropriate, also the value of C,) is most reliably obtained from the vibrational energies (6, ZO), it may be treated as a known constant in the expressions for the other (m 2 I) constants. The only unknowns in the expressions for the latter are therefore the exponent expansion parameters {a,,}. Realistic initial estimates of these m 2 1 expansion parameters are readily obtained from fits of (un - V) polynomials to log[K,(v)/K,“(u)J. Similarly, trial values of D, uu, and {a,,} may be obtained from fits to analogous approximate linearized versions of Eq. (12) (I 7).

III. APPLICATION TO THE B(311&,) - X(‘Zg+) SYSTEM OF Iz

A. Nature of the Experimental Data and Outline of Previous Analyses

Using Fourier transform spectroscopy, Gerstenkom and Luc have measured and assigned the frequencies of 1472 1 lines belonging to 149 bands of the B - X absorption spectrum of IZ (19-21). Those data involve X-state vibrational levels for V” = O-9 and B-state levels 2)’ = O-72, with the rotational quantum number ranging up to J = 200. For the B state, these levels fill virtually the whole potential well, 0’ = 72 lying less than 20 cm-’ from dissociation, so these measurements comprise a reasonably demanding prototype of a large data set for testing the present method.

In an initial analysis of the data for 139 of these bands, Luc (19) used direct fits to Eq. (7) to yield experimental values of K;(u”) and KS(V) for V” = O-9, of Kh(n’) for m = l-3 and o’ = l-62, and of the 139 band origins [Kb(zl’) - K;((u”)]. In a subsequent step he fitted those constants to Dunham expansions, Eq. (8), subject to the overall constraint that the resulting polynomial expansions would accurately reproduce the raw data [see footnote 4 of Ref. (3)]. However, Luc’s analysis treated the centrifugal distortion constants as independent parameters instead of requiring them to be mechanically consistent with the associated &(u) and K,(u) values, and his truncation of the rotational expansions at mkax = 2 and mAax = 3 means that extrapolations to very high J may not be trusted. Moreover, the resulting polynomial expansions for K*(v’) for m = l-3 have been shown (3) to be completely unreliable beyond the highest level included in the fit (in that case, 2)’ = 62).

In more recent work, Hutson et al. (21) analyzed the data for all 149 bands using a mechanically consistent set of calculated centrifugal distortion constants in the


iterative manner described at the end of Section I. However, their final result for the B state consists of a table of Z&,(V’) values for m = O-5 and 2)’ = O-72.4 These 438 parameters are not a very compact way of representing the raw data, and they cannot themselves provide predictions for transitions involving the 15 higher vibrational levels V’ = 73-87. The present work therefore reanalyzes the data using near-dissociation expansions to represent all of the K,(v’) constants.

In the Hutson et al. (21) analysis, the rotational expansions of Eq. (7) were truncated at mkax = 3 and mhax = 5. Since the transitions considered all involve X-state levels of low v”, for which all centrifugal distortion constants are very small, the former is certainly not a significant source of error. On the other hand, their initial fits did not adequately represent some of the lines with high J. They attributed this fact to the neglect of yet higher order (m 2 6) B-state distortion constants, and found that most of the discrepancies were removed upon fixing the K;(u’) constants used in the fits at values 2.6 times larger than those actually calculated. Although it now appears that this empirical fix-up procedure is probably not entirely sound (see Section WC), its validity has no bearing on the present illustration of the utility of NDE representations, so their procedure of scaling the highest-order K~(v’) parameter by an empirically determined constant factor is adopted here.

The experimental data, kindly provided to us on magnetic tape by S. Gerstenkorn, consisted of 14721 assigned line frequencies. Since the analysis of Hutson et al. (21) was performed on 14703 lines, we eliminated a group of nine lines for which the observed vs. calculated deviations were noticeably larger than for the others. However, we saw no obvious cut-off criterion for selecting only nine more lines for elimination, so the present fits were performed on 147 12 lines.

B. Near-Dissociation Expansion Analysis

For the B(311&J state of IZ, n = 5 (5, 13-15) and a range of theoretical and experimental values of C, have been reported (11, 13, 22-25). The present work used the best experimental estimates of these quantities (29, C, = 2.884 X lo5 cm-’ AS and Q, = 87.324. The former defines the X,,,(S) constants of Eqs. (lO)- (13); the resulting values, rounded off to a physically significant number of digits, are listed in Table I. The only limiting near-dissociation theory parameter actually varied in the fits was therefore the dissociation energy D of Eq. (12). In principle, it might appear preferable also to allow vu to vary in this global fit. However, the growth of interparameter correlation with the number of parameters being varied means that a more reliable value of this physically significant extrapolation intercept will probably be obtained from a local fit which considers only some of the highest observed levels.

In the present fits, the B-state energy levels were represented by Eq. (6) with ,

m max = 5 and all of the Kk(u’) functions represented by NDE functions, Eqs. (12)

4 Dunham polynomials for these &(u’) and K’,(v’) values were also reported in Ref. (ZJ), but since they

were not obtained from direct fits to the experimental data, they may not reproduce the measured

frequencies in an optimum manner. Note also that in the published table (2J), the last two Y,,, coefficients

shown actually belong at the bottom of the Y,! column.


TABLE I

Parameters of NDE Representation for the Vibration-Rotation Level Energy of B(311&)-State I2 (Additional necessary parameters are k’ 5 = 2.368, or, = 87.324, and D = 20043.2077 cm-‘, the latter expressed relative to the ground state zero point level. The X,,,(5) values are in cm-‘).

m= 0 m= 1 m= 7

X,(5) 2.08 XILF 2.67 X10-b -3.49 X1O-6

Y,lll 1.566 285 179x10-* -5.840 657 45~10-~ -1.949 143 57x10“

a2,rn -3.657 148 809x10-" 3.444 113 54x10-3 1.444 873 99x10-2

a3,m 3.598 901 190x10-' -1.314 789 49x10-4 -6.968 433 4 x10-"

a4,m -1.944 993 937x10-5 2.102 586 36~10-~ 1.454 449 6 x10-5

d5.m 6.457 493 671x10“ 2.086 861 6 x10-* 1.443 533 1 x10-7

'6,m -1.399 454 177x10-8 1.500 335 3 x10-9 -1.594 053 1 x10-E

a7,m 2.027 343 59 x10-" 2.525 735 x10-" 4.144 241 x10-'0

a8,m -1.954 112 0 x10-" -1.329 382 x10-13 -5.764 45 x10-12

a9,m 1.206 529 5 x10-" -1.415 26 x10-'5 4.652 92 x10-"

YO,lll -4.328 32 x10-" 2.627 8 x10-" -2.060 6 x10-16

Yl,m 6.874 XlO‘~~ -1.575 x10-'9 3.89 x10-'9

?2,m 3.49 x10-22

m=3 m=4 m=5

Xd5) -4.03 xlo-a -1.21 X1O-9 -4.92 x10-"

a1.m -2.593 108x10-' -2.116 224x10-' -2.700 11x10-'

a2,m 2.349 595x10-2 8.622 70 x~O-~ 1.225 18x10-*

a3.m -1.398 999x1O-3 -2.696 75 x10-' -4.395 7 x10-r

a4,rn 4.782 798xlO-5 4.872 5 x~O-~ 9.072 x10-6

a5,m -9.571 46 x10-' -4.533 x1o-8 -9.41 x10-8

a6,m 1.107 21 x10-8 1.654 x10-'0 3.75 x10-'Q

a7,m -6.860 x10-"

%,m 1.763 X10-IS

and (13), with X,(5) and nD held fixed at the values mentioned above. Expressions for the K,(u’) constants for m 3 2 were then obtained by fitting Eq. (13) to the computed centrifugal distortion constants of Hutson et al. (21) for u’ = l-72. Since their values of those constants were determined through a procedure which required them to be mechanically consistent, no further iterative correction of them should be required (this point was tested and verified). Direct fits to the 147 12 experimental transition frequencies were then used to determine D and the K&(v’) and K’,(v’)

expansion functions. As was mentioned at the end of the preceding section, this fit varied an additional empirical scaling parameter (k;) which multiplied the expression for K;(v’) in an attempt to compensate for the truncation of the rotational expansion at rnhax = 5.


Two questions which must always be addressed when performing least-squares fits are how to weight the data and how many parameters to vary in the fits. The most appropriate answers to these questions are sometimes somewhat less than obvious, particularly when the quantity of interest is obtained from calculations, and is not an experimental observable. This is the case for the centrifugal distortion constants appearing in the present analysis.

For the X- and B-state centrifugal distortion constants, the present approach to the first of these questions was to define the weights in terms of the sensitivity of the raw data to the values of the calculated K,(V) constants. In particular, when J,,&v) is the highest observed J state for vibrational level V, the statistical weight associated with the corresponding calculated K,(V) value was defined in terms of the squared inverse of the parameter “sensitivity”

Gn(~)l = A E,,,l[J,,,(u)(J,,,(u) + 1 )I”‘, (14)

where the constant AE,,,,, is the largest shift of predicted line frequencies due to errors in the analytic representations of the K,(o) values which may be tolerated. In the present work, where the raw data were reported with a precision of 0.0001 cm-’ and the global fit corresponded to a standard error of ca. 0.0012 cm-‘, we set

AC,,, = 0.0005 cm-‘. At the same time, for certain variables [Kb(v’) and K;(v’)] the weights had to be increased above the values l/s[K’,(~‘)]~ in order to assure that the fitted and calculated constants agreed to at least one significant digit. The original criterion did not ensure this at low u’, since the contributions of K)~(D’) and Kk(v’) to the rotational energy there are negligible even at J = Jmax(d).

With the weights defined as described above, the number of parameters used in the fit may be determined by the approach to unity of the root mean square value of the quantity AK,(u)/s[K,(v)], where AK,(v) are the residual discrepancies in the predictions of the fitted function. If this root mean square value has the value F, it means that on average the predicted values of K,(u) disagree with the values used as input to the fits by F times the associated s[&(v)] values.

In the present work, the above approach was used to define the weights and select the number of {a,,} parameters used to represent the centrifugal distortion constants for both the X and B states. Moreover, its application to preliminary values of the m = 0 and 1 constants [G(v) and B,] yielded initial trial values of the parameters in the expansions representing these variables. For the latter, however, the final choice of the number of parameters to be retained in the expansion was determined by examining the quality of the resulting global fit to the 147 12 observed line frequencies.

Although other data are available for higher levels, the Gerstenkorn-Luc work (19, 21) only observed X-state levels zl” = O-9. For the sake of simplicity, the present analysis therefore used the simple Dunham expansions of Eq. (8) to represent the X-state K”,(v”) constants in Eq. (7). Following Hutson et at. (21), five Y;:, parameters were used in fits to the calculated m = 2 and 3 distortion constants of Ref. (20), while four Y;I, parameters were used for each of Kt(v”) and K’;(v”) in the global fits. Note that the leading term in the Dunham expansion for the X-state vibrational


energy is Y10 X (u + l/2), since any constant term in the ground state energy expression is absorbed into the fitted value of the B-state parameter D.

C. Results and Comparisons

With the centrifugal distortion constants held fixed at values generated from expressions determined in the manner described above, the 147 12 reported transition frequencies were fitted to Eq. (7) to determine optimized expressions for the B- and X-state &(u) and K,(v) functions. As indicated, the spectroscopic constants for the X state were all represented by Dunham expansions, Eq. (8) while those for the B state were represented by the NDE functions of Eqs. (12) and (13). The resulting fit had only 33 free parameters: D, k;, 11 am and 12 alI coefficients for the B state, plus 4 Y, and 4 Y,, X state parameters. The resulting optimized parameter values are listed in Tables I and II, together with the constants defining the associated centrifugal distortion constant expressions.

The numbers of significant digits quoted for the constants listed in Tables I and II were determined subject to the following considerations. First and foremost, the (rounded off) numbers reported must be able to reproduce the predictions yielded by the raw “best-fit” parameter values to within a small fraction of the experimental uncertainty. At the same time, the reported parameters should have as few significant digits as possible, both for the sake of simplicity (e.g., to minimize the possibility of coding errors affecting their subsequent use by others), and because one would prefer not to have to quote parameters to a precision much higher than the accuracy suggested by the parameter uncertainties implied by the fit. The first of the above conditions is well satisfied here, as the rounding affected the root mean square error in the predicted line frequencies by less than 0.04%. As is often the case, the parameter uncertainties implied by the present fit are mostly due to the high degree of interparameter correlation, so the effect of rounding off a given parameter is largely compensated for by small correlated changes in others.

The standard error associated with the present global fit was u = 0.00124 cm-‘, and the value obtained for the parameter scaling the K;(v’) function was k; = 2.368. In Table III these values, together with the number of parameters varied in the fit

TABLE II

Dunham Expansion Parameters y/m for Levels u” = O-9 of X(‘Zd)-State I2 (all in units of cm-‘)

m=O 1 2 3

'0.m 3.736 835 25x10-z -4.535 83x1o-9 -5.133 5x10-16

'1,. 214.522 255 -1.139 395 x1o-k -2.303 71~10-~' -8.998 7dO“'

'2.m -0.608 901 4 -2.789 9 x1o-7 -6.453 35~10-'~ -2.571 9x10-"

'3,m -0.001 118 4 -5.12 x10-9 2.088 26~10-~' 9.153 6dO-*'

Y4.m -0.000 013 61 -8.470 a4x1O-'6 -1.223 2x10-"


TABLE 111

Comparison of Features of Present and Hutson et al. (21) Analyses, and the Number of Parameters Associated with a “Comparable” Dunham Expansion Treatment

present Hutson et Dunham work al. (21) expansions

standard error o/cm-' 0.00124 0.00131

scaling factor k; 2.38 2.6

No. free parameters in fit 33 164 34

No. parameters required to reproduce input data

81 472 105

and the number of parameters required to reproduce the input spectrum, are compared with analogous quantities for the analysis of Hutson et al. (21).5

As is shown by Table III, the present approach yielded a slightly smaller standard error and a slightly different value of the k; scaling parameter than did the analysis of Hutson et al. (21). The latter difference probably merely reflects both the slight difference (by 9 out of 14712) in the number of lines fitted in the two analyses and the fact that in Ref. (22) k; was determined by trial and error rather than by a free fit. The small difference in the two u values is also not very significant, and most likely arises from the slightly less systematic way in which the ground state rotational constants Ky(u”) were treated in Ref. (21). However, these CJ values do clearly illustrate one of the key premises of the present paper, that an analysis based on near-dissociation expansion expressions for the molecular constants can represent a large data set with an accuracy at least as good as that obtainable using the traditional polynomial expansion or term value representations.

The remaining entries in Table III demonstrate one of the advantages associated with the use of NDE functions, a considerable economy both in the number of parameters varied in the fit and in the number required to reproduce the raw input data. The contrast is of course most marked for the analysis of Hutson et al. (21) who did not represent their K,,,(u) values by functional forms. However, this advantage in compactness of the NDE approach is also retained if Dunham expansions are used to represent both the X- and B-state levels. To illustrate this point more explicitly, the number of Dunham parameters required to represent the B-state K,(r)‘) constants of Ref. (21) was determined using the same criteria applied in the NDE function analysis described above. Table IV compares the numbers of required parameters determined in this way with those associated with the present NDE analysis. As was seen in Ref. (3), the compactness advantage of the NDE functional form becomes most pronounced for the higher-order centrifugal distortion constants, since their increasingly drastic singular behavior at dissociation (7. 8) can never be fully represented by polynomials.

5 No overall standard error of fit was quoted by Hutson et al. (21). but a value for their analysis was computed here using predictions generated from their tabulated constants.


TABLE IV

Comparison of the Numbers of Dunham and NDE Parameters Required to Represent the Calculated &State Constants of Ref. (24 to Equivalent Precision

No. NDE No. Dunham Constant Parameters Parameters

Kofv) = G(v) 11 12

KI(v) = Bv 12 12

K2(v) = - Dy 11 13

K~(v) = HY 8 13

K,(v) = L 6 17 Y

&(*I = Mv 6 > 18

TOTAL 54 > 85

The other advantage of NDE functions is the fact that they should provide particularly realistic predictions for the properties of higher vibrational levels not included in the analysis. This point is examined in Table V. While the data sets

TABLE V

Comparison of Present (NDE-Function) and Polynomial Extrapolated Properties of E-State Levels U’ = 78-80 with Both Recent Experimental Results (24) and the Predictions of the “Constrained-Intercept” Polynomials of Hutson et al. (21) (energies in cm-‘)

" = 78 79 80

G(v): experiment (3) 20 039.609 20 040.729 20 041.569

NDE extrapolation 20 039.622 20 040.745 20 041.596

polynomial extrapolation 20 038.581 20 038.948 20 038.630

Hutson et al. (21) 20 039.604 20 040.713 20 041.548

B,:

_____

0,:

._.

experiment (24) 0.003 738 0.003 291 0.002 837

NOE extrapolation 0.003 747 0.003 294 0.002 846

polynomial extrapolation 0.003 642 0.003 124 0.002 580

Hutson et al. (21) 0.003 855 0.003 464 0.003 100

______---__--_________------------____---__---___----_-____~_____-

experiment (24) 2.85x10-' 3.25x10-' 3.72~10-~

NDE extrapolation 2.86x10-7 3.28~10-~ 3.83~10-~

polynomial extrapolation 2.74x10-' 3.07x10-7 3.46x10-'

-H” experiment (24) 3.27x10-"' 4,69x10-" --

NDE extrapolation 3.13x10-" 4.52x10-" 6.84x10-"

polynomial extrapolation 2.21x10-" 2.76~10‘" 3.45~10“'


used in the present and Hutson et al. (21) analysis only included observations of B- state levels up to u’ = 72, more recent measurements have yielded reliable results for levels up to u’ = 80 (24). Table V compares those new experimental results for 21’ = 78-80 with predictions yielded by the present approach and by the “comparable” Dunham expansion representations referred to above. Although this involves only a short extrapolation of six levels or 15 cm-‘, the substantial advantage of the NDE- functions is immediately obvious.

For the sake of completeness, Table V also shows the predictions implied by the Dunham expansions for G(v’) and B,, reported by Hutson et al. (21). They are not equivalent to the present “comparable” Dunham expansion, since the Hutson et al. (21) functions were constrained to approach the correct limiting values determined from an independent near-dissociation theory analysis. In spite of this, their B, predictions are significantly worse than those yielded by the present analysis, and although their G(v’) extrapolation appears comparable (or even slightly better) than the present one, their G(u’) function goes through a maximum between U’ = 86 and 87, even though they attempted to build a limit of (23) z)n = 87.18 into their analysis! Thus, even when constrained to have the correct intercept, Dunham expansions yield less reliable high-v extrapolations than do NDE functions. It seems clear, therefore, that NDE functions should be the “method of choice” for representing large data sets of this type.

As a final point of interest, Fig. 1 presents histograms of the residual deviations yielded by both the present and Hutson et al. (21) fits; the ordinate on this diagram is truncated at +50, and all transitions for which the deviations were larger than this are included in the last segment at the boundary. If the errors were normally distributed, the histograms would fall on the solid curves; clearly, both deviate substantially. In fact, this figure shows that the bulk of the data has residuals corresponding to a standard error of ca. 0.0007 cm-‘, while a small fraction of it has a much larger standard error associated with it. This fact may be accounted for in two ways. (i) The fits treated the experimental transition frequencies as all having the same experimental uncertainties. However, Hutson et al. (21) point out that weaker lines have smaller uncertainties, and hence that they should have been given less statistical weight in the least-squares fits. Thus, these histograms are superpositions of different normal distributions corresponding to different experimental uncertainties. (ii) The model used to fit the data has inadequacies, due both to truncation of the rotational power series and to discrepancies associated with the quantum mechanical calculation of centrifugal distortion constants on a semiclassical RKR potential (see below). This would lead to systematic deviations which would not be characterized by a normal distribution. The truth for this system is probably some linear combination of these two explanations. However, these results clearly demonstrate the importance of weighting experimental data by appropriate uncertainties, even in very large scale fits.

D. The k; Constant and Truncation of the Rotational Expansion

A salient physical question raised by both the present and Hutson et al. (21) analyses concerns the significance of the empirical k; scaling parameter introduced


600.

I calculated - observed I

FIG. 1. Histograms of residual deviations yielded by the present and Hutson et a/. (21) analyses of the B-X spectrum of 12.

in an attempt to compensate for truncation of the B-state rotational expansion. At first glance this seems to be a fairly reasonable procedure, since as is shown by Fig. 2 of Ref. (ti), the various centrifugal distortion constants for this state all have the same qualitative behavior as functions of u’. However, the value obtained for this scaling parameter [2.36 here or 2.6 in (241 seems too large for this explanation to be acceptable. To examine this question further, fits to the entire B-X data set were repeated with m’,, in turn set at 5 and 6, and with the scaling factor kh for the highest-order distortion constants alternately fixed equal to unity or determined by the fit. The results obtained are summarized in Table VI. The fact that the improvement in the u values for the kh = 1 fits with increasing mmax is quite small, together with the fact that the fitted kA value more than doubles and the associated c value increases slightly when mmax increases from 5 to 6, indicates that introduction of the k2 scaling factors accounts for more than just neglect of higher-order centrifugal distortion constants.

A possible explanation of the magnitude of the k& scaling factors is that they reflect the inconsistency of applying an exact quantum mechanical method of calculating centrifugal distortion constants (4)2 to a potential energy curve obtained by the RKR procedure, which is a first-order semiclassical method. We must recall that the whole purpose of introducing mechanically consistent calculated centrifugal distortion constants is to allow determination of the best possible K&v) and K,(u)


TABLE VI

Standard Errors (0) and Scaling Factors (kA) Associated with Various Fits to the 147 12 Lines of the B-X Spectrum

m' max kr;l

5 1 0.00177

6 1 0.00163

5 2.368 0.00124

6 5.267 0.00133

functions, which in turn are used to construct the RKR potential curve. Since the RKR procedure is exact within the first-order WKB approximation, eigenvalues and rotational constants calculated from an RKR curve using thefirst-order JWKB quantization condition will agree exactly with the experimental input data. However, since the first-order WKB approximation is not exact, quantum mechanical eigenvalues and B”‘s calculated from this same RKR curve will be slightly in error. These same errors may be expected to arise in centrifugal distortion constants calculated quantum mechanically from an RKR potential. Thus, it seems reasonable to attribute the magnitude of the kA scaling factors to the cumulative effect of errors in the quantum mechanically calculated centrifugal distortion constants.

There are two possible ways of avoiding this dependence on the nonphysical k, scaling constant. The first is to obtain a better-than-first-order RKR potential from the K,(u) and K,(v) data, either by using one of the higher-order RKR methods which have been proposed (8, 26, 27). or by empirically correcting an initial trial potential until it correctly mimics the quantum-calculated eigenvalues (28, 29). However, these techniques are fairly complex and computationally relatively expen- sive. A more appealing approach would be to calculate the centrifugal distortion constants from the ordinary RKR curve using a first-order semiclassical procedure. Since the K,(u) and K,(u) values calculated in this way must agree exactly with the input experimental values, it seems reasonable to expect the corresponding distortion constants to be quite good too. This approach is computationally very convenient, since the semiclassical distortion constants may be generated with the same ease normally associated with other first-order JWKB calculations (7).

IV. CONCLUSIONS

The present study of the B-X spectrum of 12 clearly demonstrates the advantages of using near-dissociation expansions for representing spectroscopic data. In the context of separate treatments of the various molecular properties, previous work had shown this superiority of near-dissociation expansions with regard to extrapolation ability and compactness (3, 16-18). The present work shows that these advantages are retained when these expansions are used in direct fits to large sets of experimental data, and that the quality of the resulting fit is at least as good as that obtainable using traditional term value expansions.


A minor drawback of near-dissociation expansions is that they are slightly more complex than the simple Dunham polynomials in (V + 4). In particular, they are not linear functions of the parameters un, X,,,(n), and {a,,>, and fractional powers of (vu - V) are required for species whose long-range potential dies off as r-5. The former means that fits to data must proceed iteratively, rather than by a single pass, and that realistic initial trial values of these parameters must be generated by some approximate scheme. However, these types of difficulties are readily accommodated within a properly written computer program, and since such fits would always be performed on a computer, these apparent complications should make little difference to the user. In view of the substantial advantages with regard to compactness and (vibrational) extrapolation ability yielded by near-dissociation expansions, it seems reasonable to suggest that they should eventually replace the traditional (U -t i) polynomials for use in the routine analysis of spectroscopic data.

ACKNOWLEDGMENTS

We are grateful to Dr. S. Gerstenkom for kindly providing us with a computer tape of his assigned data for the B-X spectrum, and to both he and Dr. J. M. Hutson for helpful discussions.

RECEIVED: June 19, 1984

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