Contact Transformations and Determinable Parameters in Spectroscopic Fitting Hamiltonians

Journal of Molecular Spectroscopy 199, 284 –301 (2000) doi:10.1006/jmsp.1999.8019, available online at http://www.idealibrary.com on

Contact Transformations and Determinable Parameters in Spectroscopic Fitting Hamiltonians Mirza A. Mekhtiev and Jon T. Hougen Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8441 Received September 16, 1999

In recent least-squares fits of torsion–rotation spectra of acetaldehyde and methanol it was found possible to adjust more fourth-order parameters than would be expected from traditional contact-transformation considerations. To investigate this discrepancy between theory and practice we have carried out numerical fitting experiments on the simpler three-dimensional (three-Eulerian-angle) asymmetric rotor problem, using J ⱕ 20 unitless energy levels generated artificially from a full orthorhombic Hamiltonian with quadratic through octic operators in the angular momentum components. Results are analyzed using the condition number ␬ of the least-squares matrix, which is a measure of its invertibility in the presence of round-off and other errors. When ␬ is very large, parameters must be removed from the fit until ␬ becomes acceptably small, corresponding to procedures which lead to reduced Hamiltonians in molecular spectroscopy. We find that under certain circumstances ␬ can be decreased to an acceptable level for Hamiltonians which are only partially reduced when compared to Watson A and S reductions. Some insight into this behavior is obtained from classical mechanics and from the concept of delayed contact transformations. Transferring this numerical and algebraic understanding to the more complicated fourdimensional methyl-top internal rotor problem supports the empirical observation that presently existing data sets for methanol and acetaldehyde are most efficiently fit using partially reduced Hamiltonians and further suggests that expanding the methanol data set to transitions involving levels of higher J, K, and v t would favor even more strongly the use of partially reduced fourth-order Hamiltonians. 1. INTRODUCTION

In a series of elegant papers (1–3) culminating in his classic review article (4), Watson showed that the number of parameters of each order determinable in least-squares fits using an asymmetric rotor Hamiltonian is given by the difference between two numbers, the larger of which corresponds to the number of possible terms of that order in a general Hamiltonian operator written in standard form, the smaller of which corresponds to the number of possible terms of order one less in a general contact transformation operator written in standard form. Some years later these same ideas were used by Nakagawa et al. (5) and others (6, 7) to find the number of determinable parameters of various orders in the torsion–rotation Hamiltonian for methyl top molecules. During essentially the same period in which Refs. (5–7) appeared, a series of large global least-squares fits of torsion– rotation levels in acetaldehyde (8 –11) and methanol (12, 13) were carried out in which convergent fits were reported using Hamiltonians containing two (13) or three (10, 11) more than the allowed number of 22 determinable fourth-order parameters. This apparent contradiction between the contact transformation results (5–7) and the least-squares fit results (8 –13) is the subject of the present paper. One of the questions which arises in any application of Watson’s ideas for Hamiltonian reduction using contact transformations is the correct assignment of orders of magnitude to

all terms in the Hamiltonian. Watson investigated this problem in detail in connection with his treatment of asymmetric rotors (2), but rather less attention has been devoted to an analogous investigation for molecules exhibiting internal rotation. Duan et al. have recently presented (14) a new analysis of higher order vibration–torsion–rotation interactions for a molecule with an internal rotor, in which the ratio of orders of magnitude for pure torsional terms and pure rotational terms is characterized by a variable parameter. A description of these general ideas can also be found in a thesis by Mikhailov (15), carried out under the direction of M. R. Aliev before his untimely death, and in a more recent publication by Mikhailov and Smirnov (16). In both of these works three vibration–rotation ordering schemes are compared (referred to (15) as the limiting schemes of Oka, Nielsen-Amat, and Watson-Aliev, respectively), in which the order assigned to components of the rotational angular momentum is varied relative to that assigned to vibrational coordinates and momenta. In the present paper we look at ordering from a slightly different point of view. We first examine both numerically and algebraically, by means of a number of artificial examples, how the well-studied asymmetric rotor problem is affected by (partial) breakdown of some of the mathematical conditions shown by Watson (2) to be necessary for application of contact transformation ideas to the question of determinable parameters. We then use this experience to try to understand what may be happening in our earlier global fit studies (8 –13) of the

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internal rotation– overall rotation problem in acetaldehyde and methanol. 2. ASYMMETRIC ROTOR PROBLEM NUMERICAL EXPERIMENTS

In this section we present results from a rather large number of numerical experiments involving least-squares fits to artificially generated data for a nominal asymmetric rotor problem. Our purpose here is not to reinvestigate the asymmetric rotor fitting procedures now in common use for real molecules. Our purpose instead is to use this well-understood problem as a vehicle for studying mathematical distortions which might shed light on what is happening in the more complicated internal rotation– overall rotation problem. These mathematical distortions fall into two categories. First, we consider Hamiltonians whose power series expansions exhibit less common convergence behavior, in the sense that e x converges for all x, even though the first few terms in the infinite series appear to be diverging when x ⫽ 10, or in the sense that the polynomial 1 ⫹ x ⫹ x 2 converges for all x when considered as an infinite series with all terms zero after the third. Second, we consider artificial data sets generated from these mathematically distorted Hamiltonians which differ qualitatively from those normally available from experiment in the following three points. (i) The asymmetric rotor energy levels we use are given far more absolute accuracy than usual, corresponding to relative errors of the order of one part in 10 10 or 10 11. (ii) The energy level set we use is far more complete than usual, since all K levels belonging to a given J manifold are included. (iii) The quartic centrifugal distortion effects in some of our artificial data sets are much larger than usual, particularly the effects of terms with ⌬K ⫽ 0. We emphasize again that we have not chosen these Hamiltonians and data sets to seek insight into conventional asymmetric rotors, rather we are using them as tools to look for insight into the internal rotor problem, and in particular to look for clues to any mathematical inconsistencies which could arise when rotational spacings and torsional vibrational frequencies are grouped together in the same order (5–7). A. Hamiltonian Notation To avoid confusion, we use the word “order” in this paper to indicate the position of a given term in some perturbationscheme classification, and the word “power” to indicate the sum of exponents of all angular momentum operators in that term. Thus, in the general expansion H ⫽ H0 ⫹ H1 ⫹ H2 ⫹ H3 ⫹ · · · ,

[1]

the subscript n indicates the order of the term H n in a perturbation classification sense. We also limit consideration to orthorhombic asymmetric rotor Hamiltonians, because each of

the three molecule-fixed components of the angular momentum operator belongs to a different symmetry species in the orthorhombic point groups. This simplifies our initial untransformed Hamiltonian, since all terms with odd powers of any given angular momentum component are zero by symmetry. Consider now an orthorhombic asymmetric rotor Hamiltonian with operators of the same power grouped together such that the term of order n contains rotational operators of power 2n ⫹ 2. H 0 then becomes the quadratic rigid rotor operator (which we write here in a I r representation, using Cartesian components of J): H 0 ⫽ AJ 2z ⫹ BJ x2 ⫹ CJ y2 .

[2]

H 1 is the full orthorhombic quartic centrifugal distortion operator, which we write here using cylindrical components J 2 , J z , J ⫾ (4) to simplify the programming in our various numerical experiments: Subscripts on the six coefficients T 2k,2l,2m thus indicate rotational operators containing factors of the form J 2k , J z2l (with selection rules ⌬K ⫽ 0) and J ⫾2m (with selection rules ⌬K ⫽ ⫿2m), 2 2 H 1 ⫽ T 400 J 4 ⫹ T 220 J 2 J 2z ⫹ T 040 J 4z ⫹ T 202 J 2 共 J ⫹ ⫹ J⫺ 兲 2 2 4 4 ⫹ J⫺ 兲其 ⫹ T 004 共 J ⫹ ⫹ J⫺ 兲, ⫹ 12 T 022 兵 J 2z , 共 J ⫹

[3]

where {A, B} ⬅ AB ⫹ BA. When the T 004 term in Eq. [3] (corresponding to R 6 or d 2 in other notations) is eliminated by a contact transformation, the A reduction (4) form of H 1 is obtained; when the T 022 term (corresponding to ␦ K in other notations) is eliminated, the S reduction (4) is obtained. H 2 and H 3 are the full orthorhombic sextic and octic Hamiltonians, with 10 and 15 coefficients, respectively, which can be written in the same general form H n ⫽ ⌺ kl C 2k,2l,0 J 2k J 2l z

[4]

2m 2m ⫹ 12 ⌺ k,l,m⬎0 C 2k,2l,2m J 2k 兵 J 2l z , 共 J ⫹ ⫹ J ⫺ 兲其,

where C ⫽ ⌽ and 2(k ⫹ l ⫹ m) ⫽ 6 when n ⫽ 2, and C ⫽ ⌳ and 2(k ⫹ l ⫹ m) ⫽ 8 when n ⫽ 3. When the three H 2 terms with 2m ⱖ 4 and the six H 3 terms with 2m ⱖ 4 in Eq. [4] are eliminated by contact transformation, the A reduction (4) forms of H 2 and H 3 are obtained; when the three H 2 and six H 3 terms with 2m ⱖ 2 and 2l ⫽ 0 are eliminated, the S reduction forms are obtained. Table 1 gives the set of unitless parameters used to generate the artificial energy levels for most of our numerical experiments. Values chosen for A, B, C in H 0 are similar in magnitude to A, B, C expressed in megahertz for typical small molecules. Relative magnitudes of the T 2k,2l,2m quartic terms in H 1 are consistent with T’s from Eq. (121) of Ref. (4) satisfying the approximate proportionality relations T xx ⬀ ⫺B 2 , T xy ⬀ ⫹BC, etc. The sextic and octic terms are just random sets of

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TABLE 1 Unitless Coefficients a for Generating Artificial Data Sets from the Full Orthorhombic Hamiltonian b of Eqs. [1]–[4]

refer to this convergence factor (or sometimes to the integer g itself) as the gap size between successive orders in our numerical experiments. Parameters from Table 1 were used in Eqs. [1]–[4] to calculate unitless rotational energy levels up to some J max. When the energies from a given calculation are rounded to three digits after the decimal, they generate an artificial data set of ( J max ⫹ 1) 2 levels with a root-mean-square (rms) round-off error of 0.0003 (analogous to measurement error in a very precise experimental data set), which can then be subjected to various trial least-squares fits. When these calculated energies are truncated (instead of rounded) to three digits after the decimal, they generate a similar data set with an rms truncation error of 0.0005, which can be used in conjunction with the rounded set to look for parameter determination instabilities in the least-squares procedure. B. Reduced Hamiltonian Fits

numbers. Exponents for the terms in H 1 , H 2 , and H 3 contain a variable scaling parameter g, which is used to adjust the convergence properties of this power series representation of the Hamiltonian. Terms of successive orders in Table 1 decrease approximately by a factor of 10 ⫺g . For convenience we

Table 2 shows standard deviations obtained from fits to J ⱕ 20 unitless data sets generated from Eqs. [1]–[4], using Watson A and S reduction quartic, sextic, and octic Hamiltonians. (In this table and elsewhere, the concise (but very nonstandard) shorthand notion Bq, Bqs, or Bqso, is used to indicate fits in which three quadratic, q quartic, s sextic, and o octic constants are adjusted.) The data sets for Table 2 were generated using the full set of 34 orthorhombic constants given in Table 1, together with a variable scaling factor 10 ⫺g which describes the approximate ratio between constants of different order.

TABLE 2 Standard Deviations from Least-Squares Fits a to Watson A and S Reduction Hamiltonians of Five J < 20 Unitless Energy Level Sets b

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The purpose of this short section and Table 2 is to remind the reader (in agreement with extensive spectroscopic experience) that Watson-reduced Hamiltonians give excellent fits to asymmetric rotor energy levels, even when the power series convergence scaling changes by four orders of magnitude. As expected, energy levels calculated from the more rapidly convergent Hamiltonians (larger g values in Table 2) require fewer parameters to achieve a fit to 0.0003 (round-off error) than do energy levels calculated from the more slowly convergent Hamiltonians. (The fit with g ⫽ 5 in Table 2 does not reach round-off accuracy because terms in J 10 are required in the reduced Hamiltonian for this gap size, and we did not include any J 10 terms in our computer program.) C. The Condition Number in Least-Squares Fits It is convenient when examining, in the spirit of earlier workers (e.g., 17–20), the stability of spectroscopic leastsquares fits, to make use of the condition number (21) ␬(A) of a real n ⫻ m rectangular matrix A, where n ⱖ m, which is defined in terms of the condition number ␬(A trA) of the real symmetric m ⫻ m square matrix formed from it. If ␭ i are the eigenvalues of A trA, then

␬ 共A trA兲 ⫽ ␭ max/ ␭ min ⱖ 0 [5]

␬ 共A兲 ⫽ 共 ␭ max/ ␭ min兲 1/ 2 . For the symmetric square matrix A trA, ␬ is a measure of the error matrix which must be added to A trA to make its determinant exactly zero and its inverse indeterminate. For the rectangular matrix A, ␬ is a measure of the linear dependence of its m columns. It is recommended (22) that rows and columns of the matrix A be scaled so that all its elements are of similar magnitude before determining ␬. Consider a linear least-squares problem and corresponding solution represented by the two equations Ax ⫽ b x ⫽ 关A trA兴 ⫺1 关A tr b兴,

[6]

where (using spectroscopic terminology) b i is the observations vector (i.e., i ⫽ 1, . . . , n runs over the set of observed minus calculated values in each iteration of the fit), x j is the constants vector (i.e., j ⫽ 1, . . . , m runs over the set of molecular parameters), A ij is the derivative matrix ⭸b i /⭸ x j , and A trA is the least-squares matrix. Scaling is possible here only for the m columns of A, since scaling the n rows corresponds to changing the weights of the data; we thus generated a scaled A matrix, A s , prior to determining ␬(A trA) and ␬(A) by normalizing each column to unit length, using the expression:

共 A s 兲 ij ⫽ A ij /关⌺ k A 2kj 兴 1/ 2 .

[7]

The condition number ␬(A trA) can be used to characterize the errors in numerical solutions of this problem with the following two inequalities (21), 储⌬x储/储x储 ⱕ ␬ 共A trA兲储⌬共A tr b兲储/储A tr b储

[8a]

储⌬x储/储x储 ⱕ ␬ 共A trA兲储⌬共A trA兲储/储A trA储,

[8b]

where Eq. [8a] applies when errors in A trA are negligible, and Eq. [8b] applies when errors in A trb are negligible. In these expressions, 储x储 and 储A trb储 represent the lengths (Euclidian norm (21)) of these two m-dimensional vectors, and 储⌬x储 and 储⌬(A trb)储 represent the lengths of the corresponding error vectors. The relative error in the m ⫻ m A trA matrix occurring in the second equation is defined more precisely in terms of matrix norms in Ref. (21). The first of these expressions indicates that the relative error in the molecular constants vector x returned by the least-squares routine is less than or equal to ␬(A trA) times the relative error in the modified data vector A trb; the second indicates that the relative error in the constants vector x is less than or equal to ␬(A trA) times the relative error in the A trA matrix. More sophisticated modern computational techniques (22) for solving Ax ⫽ b lead (23) to an analog of Eq. [8a], 储⌬x储/储x储 ⱕ 兵2关 ␬ 共A trA兲兴 1/ 2 ⫹ sin ␪␬ 共A trA兲其

[9]

⫻ 关1/cos ␪ 兴储⌬b储/储b储, where sin␪ is essentially [⌺ i (obs ⫺ calc) i2 ] 1/ 2 /[⌺ i (obs) i2 ] 1/ 2 . Note that relative errors in x determined from Eq. [9] depend on 公␬ rather than on ␬. Equations [8] above lead to the approximate rule (22) that the maximum number of significant digits returned in a leastsquares calculation equals the number of decimal digits in the floating-point word of the computer being used minus log 10(␬). Thus, when log 10(␬) is greater than the word size used, the least-squares calculation frequently diverges. Because, however, Eqs. [8a] and [8b] are both inequalities, they can provide only upper bounds for error estimates, so that the preceding rules are not strict. It is perhaps for this reason that limiting the comparison of ␬ values to closely related least-squares calculations has proved more useful to the authors than comparisons between least-squares calculations with very different data sets (size, quantum number distribution, measurement weights, etc.) and very different parameter sets (size, type of quantum mechanical operators, etc.). D. Partially Reduced Hamiltonian Fits In this section we explore least-squares fitting behavior for data sets calculated from the coefficients in Table 1 when the

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TABLE 3 Standard Deviations from Least-Squares Fits to S-Reduced a and Partially Reduced Hamiltonians b of Various J < 20 Energy Level Sets c

gap size is uneven, i.e., when the gap-scaling parameters g(T), g(⌽), and g(⌳) are not all the same. Even though an intuitive extension of the ideas in Eq. (2) of Kivelson and Wilson (24) suggests that uneven gap spacing is expected to be rare in the asymmetric rotor problem, we consider its mathematical consequences here because some analog of irregular gaps may arise when overall-rotation and internal-rotation degrees of freedom are combined to make one Hamiltonian. In particular, we focus on three closely related questions. (i) Under what circumstances can a number of higher order terms in a reduced Hamiltonian be replaced by fewer lower order terms in an unreduced Hamiltonian, without degrading the standard deviation of the fit? (ii) Will such a fit be stable enough to converge to a least-squares minimum? (iii) Will the molecular constants obtained from such a fit be stable with respect to small changes in the data set? Question (i) can be discussed with the help of Table 3, which shows standard deviations from a number of fits to rounded and truncated data (see Section 2A) calculated with gap-scaling

parameters satisfying g(T) ⱕ g(⌽) ⫽ g(⌳) ⫽ 6 or 7. (This corresponds to a 10 ⫺g gap size which is smaller for the quadratic to quartic step than for the quartic to sextic or sextic to octic steps.) When g(T) decreases by unity for constant g(⌽) and g(⌳), energy contributions from the B (quadratic), ⌽ (sextic), and ⌳ (octic) terms remain fixed, while energy contributions from the T (quartic) terms increase by one order of magnitude. Fits on the right of Table 3 are carried out with partially reduced Hamiltonians containing all six T coefficients; fits on the left are carried out with various reduced Hamiltonians for comparison. We conclude from the clear numerical evidence in Table 3 that it is mathematically possible to generate asymmetric rotor energy levels for which a fit to a partially reduced Hamiltonian is preferable to a fit to a fully reduced Hamiltonian, in the sense that fitting to a standard-reduced Hamiltonian requires relatively many higher order terms to achieve the desired overall standard deviation, while fitting to a partially reduced Hamiltonian requires fewer and lower order terms to achieve the same result.

DETERMINABLE PARAMETERS

Question (ii) can be discussed in terms of the condition number ␬. Since the least-squares fits in this paper were performed by inverting A trA, Eqs. [8] are appropriate. Relative errors in the energy level data set represented by the vector b rounded to three places after the decimal range from 10 ⫺8 to 10 ⫺11 for 1 ⱕ J ⱕ 20. Relative errors in the A trA matrix elements, which are computed in double precision arithmetic from a Hamiltonian which can be considered exact at each iteration step, should not be significantly worse than 10 ⫺12 to 10 ⫺14. Equation [8a] thus suggests that values of ␬(A trA) smaller than 10 8 should lead to no trouble, but that parameters returned by the least-squares routine when ␬(A trA) is larger than 10 10 should be treated with suspicion. (It sometimes happened in our many calculations, however, that ␬(A trA) values as large as 10 13 did not cause serious trouble, possibly because (i) Eqs. [8] give only upper bounds for relative errors in the parameters returned, or (ii) row scaling of A cannot be carried out in least-squares problems, so that ␬(A trA) values can be misleading.) In any case, values of ␬(A trA) for the very stable S-reduced (or A-reduced) Hamiltonian fits shown in Table 2 are near 6 ⫻ 10 2 (or 1 ⫻ 10 3) for B5 fits, 6 ⫻ 10 4 (or 1 ⫻ 10 6) for B57 fits, and 6 ⫻ 10 6 (or 1 ⫻ 10 9) for B579 fits (in the Bq(uartic)s(extic)o(ctic) notation introduced in Section 2B). Values of ␬(A trA) for the partially reduced fits in Table 3 also indicate relatively good stability. Furthermore, they behave as expected, increasing when more parameters are added to the fit and decreasing when g(T) decreases. Fit stability can also be determined from the correlation coefficient matrix, from the least-squares uncertainties assigned to the fitted parameters, or from the number of correct digits returned by the fit for each parameter, as we now describe below. When the scaling parameter g(⌳) ⫽ 7, Table 2 indicates that the maximum energy contribution from the ⌳ terms, E max(⌳), is negligible. The partially reduced Hamiltonian fits in the top half of Table 3 were therefore carried out without ⌳ terms (and sometimes with only six ⌽ terms). As a representative example, the B66 fit in the 677 row has a ␬ of 5 ⫻ 10 7; only seven correlation coefficients are greater than 0.999, and these seven are not greater than 0.9998; the worst relative uncertainty in the T parameters from this fit is 1 ⫻ 10 ⫺4; the fit returned eight correct digits for A, B, C and at least three correct digits for the six T’s. (It makes no sense to compare ⌽ constants in these rows with their original values, since one, two, or four of the 10 ⌽’s used to calculate the energy levels were set to zero in the fits. This procedure presumably also changes the T values by amounts comparable to the ⌽ values.) When the scaling parameter g(⌳) ⫽ 6, Table 2 indicates that E max(⌳) ⱕ 0.004 for J ⱕ 20. Nevertheless, the partially reduced Hamiltonian fits in Table 3 contain no ⌳ terms; instead they sometimes contain more than the seven determinable ⌽ terms allowed in a fully reduced Hamiltonian. As extreme examples, the B69 fits in the 466 and 366 rows, which use more than the fully reduced number of both T and ⌽ coefficients, return nine or more good digits for A, B, C and five or

289

more for the six T constants. Correlation coefficients are all less than 0.985. These last two fits are discussed in Section 3B using the idea of a “delayed” contact transformation. We conclude that for the data sets considered in Table 3, many partially reduced Hamiltonian fits are stable enough to converge quickly to well-defined least-squares minima. Question (iii) can be discussed by comparing parameters obtained from fits of corresponding rounded and truncated data sets, which should agree to a few standard deviations if the least-squares solution is stable. Thus, when the rounded and truncated B66 fits in row 677 or the rounded and truncated B68 fits in row 566 are compared (each of these rows has the highest ␬ in its group), we find (a) slightly larger overall standard deviations for the truncated data, (b) A, B, C, and T parameters with the same number of correct digits in corresponding rounded and truncated fits, and (c) agreement between parameters in corresponding rounded and truncated fits to within one to six standard deviations. From a spectroscopic point of view, pairs of corresponding fits are thus essentially identical, and we conclude that fits to partially reduced Hamiltonians can, under favorable circumstances, return accurate values for the molecular parameters. E. The Correlation Problem in Classical vs Quantum Mechanics It is interesting to note that from a classical point of view, the very large ␬(A trA) values and bad correlation matrices that frequently arise in fits to unreduced rotational Hamiltonians are caused by the fact that in quantum systems it is not possible to obtain more than 2J ⫹ 1 energy samples for given J. In the classical analog of our quantum system, we can imagine sampling energies at any number of points for fixed J. Consider the simple example of calculating classical energies from Eqs. [1]–[3] and the constants in Table 1 with g ⫽ 5 at six points for J ⫽ 1, namely J x ⫽ 1, J y ⫽ 1, J z ⫽ 1, J y ⫽ J z ⫽ 1/公2, J z ⫽ J x ⫽ 1/公2, and J x ⫽ J y ⫽ 1/公2, as well as at the corresponding six points for J ⫽ 2, namely J x ⫽ 2, J y ⫽ 2, J z ⫽ 2, J y ⫽ J z ⫽ 公2, J z ⫽ J x ⫽ 公2, and J x ⫽ J y ⫽ 公2. These 12 J ⫽ 1 and J ⫽ 2 energies (in contrast to the eight available in quantum mechanics), when rounded to three digits after the decimal, permit a least-squares fit to A, B, C in Eq. [2] and all six quartic centrifugal distortion constants in Eq. [3]. This fit is characterized by a 12 ⫻ 9 derivative matrix A, a ␬(A trA) of 5 ⫻ 10 2, and a correlation coefficient matrix with a maximum off-diagonal element of 0.75; it returns eight correct digits for A, B, C and four or more correct digits for all six T’s. It is also interesting to note, though the authors are unsure of its deeper significance, that the 2n ⫹ 1 terms of power J 2n in a fully reduced orthorhombic asymmetric rotor Hamiltonian is just equal to the number of energy levels permitted by quantum mechanics for J ⫽ n. It is thus always possible to introduce exactly one new parameter for each new energy level by adding one complete order of the fully reduced Hamiltonian for

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each unit increase in J. Such a Hamiltonian would give a stable and exact fit to the data, but it would have no predictive power whatever, and in such a situation we would say that the power series representation of the Hamiltonian does not converge. In the usual limit, where the power series representation of the Hamiltonian converges well ( g ⱖ 5 in Table 1), the highest power operators required for a good fit using a fully reduced Hamiltonian are very much less than 2J max, as illustrated by the J ⱕ 20 fits in Table 2. From the point of view of the preceding two paragraphs, we are trying in this work to generate and study situations where the power series representation of the full orthorhombic asymmetric rotor Hamiltonian converges in such a way that (at least for some powers of J) independent new quantum level information becomes available fast enough with increasing J to permit determination of many or all of the terms in an unreduced Hamiltonian before terms of the next higher order become important. 3. ASYMMETRIC ROTOR PROBLEM ALGEBRAIC CONSIDERATIONS

In this section we try to understand in some algebraic sense the numerical results obtained in the calculational experiments of Section 2, in the hopes that algebraic understanding of any unusual behavior in this simple model system will provide clues for useful directions of investigation in the torsion– rotation problem discussed in Section 4. Section 3A below deals with the creation of tails, a word used here to describe contributions to operators of higher (and lower) order generated from the contact transformation of a given operator. Section 3B deals with a nonstandard contact transformation, namely one whose application is delayed by one order in the Hamiltonian. Section 3C deals with the linear relation which can occur under certain circumstances between the expectation values of three quadratic and three quartic operators. A. Contact Transformation Tails It is well known (1– 4) that elimination of a given term from the Hamiltonian by reducing its coefficient to zero via a contact transformation results in the generation of many new terms multiplied by functions of the remaining coefficients, e.g., symbolically, elimination of the first term in an untransformed two-term Hamiltonian of the form c 1 A 1 ⫹ c 2 A 2 leads in general to an infinite transformed Hamiltonian of the form ⌺ iⱖ2 c i A i , where all terms with i ⱖ 3 can be thought of as the tail generated from the original term c 2 A 2 . Because the number of operators generated by contact transformations becomes large very rapidly, we consider here only tails generated from the three quadratic and six quartic terms in Eqs. [2] and [3] by commutators involving the S 1 function given by: 2 2 S 1 ⫽ s 111 共 J z J y J x ⫹ J x J y J z 兲 ⫽ 共s 111 /4i兲兵共 J ⫹ ⫺ J⫺ 兲, J z 其. [10]

This is equivalent to considering an unreduced Hamiltonian in Eq. [1] containing only H 0 and H 1 terms. (We use cylindrical notation here because such coefficients are often reported in spectroscopic fits.) The two most commonly used values for s 111 are (4) s 111 ⫽ ⫺T 004 /B 002

[11a]

s 111 ⫽ ⫹T 022 / 2B 020 ,

[11b]

corresponding to Watson’s A and S reduction, respectively. Table 4 shows the contributions to H 2 (taken to be zero in the untransformed Hamiltonian for this table) through terms in s 3 B and s 2 T, where s ⬅ s 111 , as generated by the contact transformation formalism (25). As expected, the coefficients B 200 and T 400 do not contribute to any ⌽’s, since the isotropic operators which they multiply in the untransformed Hamiltonian are unaffected by the contact transformation. Somewhat less intuitively obvious, however, coefficients of operators obeying ⌬K ⫽ 0 selection rules in the untransformed Hamiltonian, i.e., B 020 , T 220 , and T 040 , contribute to ⌽’s multiplying operators in the transformed Hamiltonian with ⌬K ⫽ 0, ⫾2, and ⫾4 selection rules. In a roughly equal-gap case, magnitudes of sB terms following a transformation would be of the same order as T terms in the original untransformed Hamiltonian, as implied by Eqs. [11] used to choose the value for s; similarly, magnitudes of all s 2 B and sT terms in Table 4 would be of the same order as the untransformed ⌽ terms. (The s 2 T and s 3 B terms in Table 4 then represent very small or negligible corrections.) We conclude from this analysis that trouble in the reduced Hamiltonian can arise if one of the B or T terms is anomalously large, since then the magnitude of its contribution to some of the ⌽ terms as a result of the contact transformation may be significantly larger than the magnitude of the ⌽ terms in the untransformed Hamiltonian, spoiling the convergence properties of the unreduced Hamiltonian power series. Table 4 indicates that even an apparently harmless term like D K K 4 ⬅ T 040 K 4 can, if it is large enough, give an anomalously large contribution to ⌽ 042, making elimination of the ⌽ 042 term by subsequent S reduction transformations more difficult. Similarly, a large AK 2 ⬅ B 020 K 2 term gives a large contribution to ⌽ 024, making removal of the ⌽ 024 term by subsequent A reduction transformations more difficult. We have found by numerical experiments fitting levels calculated from three B’s and six T’s that ␬(A trA) values for a B6 fit (with one more T parameter than is allowed in an A or S reduction Hamiltonian) decrease dramatically when even one T 2k,2l,2m (other than T 400 ) is made large. We further conclude that the magnitudes of terms in each order of the B and T tails are determined by the magnitude of the original T/B ⬃ s ratio. Thus, if the power series representation of an unreduced model Hamiltonian converges more rapidly for higher orders than for low, then the contact trans-

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291

TABLE 4 Contributions a to H 2 (i.e., Tails) Generated by Contact Transformation of H 0 and H 1 b

formation to eliminate one T term will destroy this property and reduce convergence at all orders to the slower low-order rate. When this latter convergence rate falls below some critical limit (which must depend on J max), it is better to carry out fits to data from the model Hamiltonian using the rapidly converging unreduced orthorhombic Hamiltonian containing all six of the original quartic coefficients as adjustable parameters. We believe this is the algebraic reason for some of the numerical results presented in Table 3. The above considerations make it seem likely that some type of unexpected tail behavior will arise in reduced Hamiltonians for internal rotor molecules, since, for example, V 3 ⫽ 400 cm ⫺1 and D ab ⫽ ⫺0.1 cm ⫺1 in acetaldehyde (11) have traditionally (5–7) been grouped together in the same perturbation order, and since the gap size between the potential energy terms V 3 and V 6 seems to be two orders of magnitude or so, while that between purely rotational terms is four or five orders of magnitude. B. Reduction by Delayed Contact Transformation Consider now the contact transformation procedure using a nontraditional ordering scheme for an orthorhombic Hamilto⬁ nian H ⫽ ¥ n⫽0 H n , where H 0 contains all angular momentum operators of powers J 2 and J 4, H 1 contains those of power J 6 ,

⬁ H n contains J 2n⫹4 for n ⱖ 2, and where S n in S ⫽ ¥ n⫽0 Sn contains as usual angular momentum operators of powers J 2n⫹1 . We also assume, as usual, that the series H n and S n both converge rapidly, insuring in particular that the final eigenfunctions are determined to very good precision in zeroth order only by the J 2 and J 4 terms in H 0 . Note that the model Hamiltonians with g(T) ⫽ 3 in Table 3 fit the description of this paragraph for J values around 30. For the H 0 chosen above, the commutator [S 1 , H 0 ] generates angular momentum operators of power 2, 4, and 6. Following traditional logic, operators of power 2 and 4 are returned to H 0 , and operators of power 6 are sent to H 1 . A suitable small value for s 111 could now be used to eliminate one term of power 6 in H 1 (so that in a sense the usual elimination of terms of power 4 by contact transformation has been delayed here until power 6). Continuing this procedure (2), we use the three parameters in S 2 to eliminate three terms of power 8 in H 2 , etc., leading to the final result that the number of determinable parameters of order n in an orthorhombic Hamiltonian with this nontraditional ordering scheme is 3n ⫹ 9 for n ⫽ 0 (i.e., 3 B’s and 6 T’s) and 3n ⫹ 6 for n ⱖ 1 (i.e., 9 ⌽’s, 12 ⌳’s, etc.). The result is therefore a reduced Hamiltonian containing one more T, two more ⌽’s, three more ⌳’s, etc. than the traditional A or S reduced Hamiltonian.

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This delayed contact transformation procedure differs in some details from the usual contact transformation (1– 4). For example, the removal of no T terms and only one ⌽ term from the untransformed Hamiltonian corresponds in Table 4 to canceling the ⌽ term on the left in one particular row by a suitable choice for s in the collection of T and B terms on the right in that row. It can be seen for the last two rows in Table 3, where a delayed contact transformation makes the most sense, that a value chosen for s based on one of its linear terms in a given row of Table 4 has both the necessary sign flexibility and also leads to s 2 and s 3 terms which represent only small corrections in that row. (In a traditional transformation, represented by Eqs. [11] and the 777 or 666 rows in Table 3, the s 2 B terms resulting from commutators of the form [S 1 , [S 1 , H(J 2)]] are not significantly smaller than the sT terms resulting from [S 1 , H(J 4)].) The use of reduced Hamiltonians obtained by delayed contact transformation is illustrated numerically by the B69 fits to 18 parameters shown in the last two rows of Table 3, which (i) float A, B, C, six T’s, and nine ⌽’s, (ii) are characterized by completely acceptable ␬ values around 10 6, and (iii) give standard deviations several orders of magnitude lower than those obtained from fits to a traditionally reduced Hamiltonian with 24 adjustable parameters. Furthermore, addition of the tenth ⌽ immediately produces an increase of ␬ to 10 14 and a divergent fit. C. Linear Relation between Matrix Elements Watson, in his first paper on the topic of reduced Hamiltonians (1), considered expectation values of the commutator i关AJ 2z ⫹ BJ x2 ⫹ CJ y2 , J x J y J z ⫹ J z J y J x 兴 ⫽ 4共C ⫺ B兲J 2z ⫹ 4共 A ⫺ C兲J x2 ⫹ 4共B ⫺ A兲J y2 ⫹ 2共C ⫺ B兲共 J x2 J y2 ⫹ J y2 J x2 兲

[12]

⫹ 2共 A ⫺ C兲共 J y2 J 2z ⫹ J 2z J y2 兲 ⫹ 2共B ⫺ A兲共 J 2z J x2 ⫹ J x2 J 2z 兲, over eigenfunctions of the Hamiltonian operator H 0 ⫽ AJ z2 ⫹ BJ x2 ⫹ CJ y2 appearing on the left side of the commutator bracket. Since this Hamiltonian is Hermitian, the expectation value of the left-hand side vanishes, and Watson obtained a linear relation among the six expectation values on the right, a relation of particular importance for the following reason. Consider a least-squares fit in which the Hamiltonian is written (in the spirit of Eq. [4]) as a sum of products of constant coefficients c i times operators h i , i.e., H ⫽ ⌺ ic ih i.

[13]

Then one can equate, at each iteration, the expectation value of a given operator over each eigenfunction from the previous

iteration with the derivative of the corresponding eigenvalue with respect to the coefficient of that operator, i.e., A ij ⬅ ⭸E i /⭸c j ⫽ 具⌿ Ei 兩h j 兩⌿ Ei 典,

[14]

where the derivatives A ij are elements of the matrix A in Eqs. [6] and the coefficients c i are elements of the column vector x, if a fit to energy levels is being carried out. A linear relation with constant coefficients among a set of expectation values on the right of Eq. [14] thus gives rise (1) to a linear relation with constant coefficients a j among the corresponding set of energy derivatives A ij on the left, i.e.,

冘 aA ⫽0 j

ij

[15]

j傺set

for all i. This is sufficient, as is well known, to cause the determinant of the matrix A trA to vanish and the least-squares problem to become singular. One of the steps in the argument above involves taking expectation values over eigenfunctions of precisely the Hamiltonian operator H 0 occurring in the commutator bracket in Eq. [12], since otherwise expectation values of the commutator will not in general vanish. Another step involves equating expectation values on the right of Eq. [12] with the derivatives needed for each iteration of the nonlinear least-squares fitting procedure. Trouble can arise because the correctness of this second step relies on taking expectation values over reasonably good approximations to the eigenfunctions of the instantaneous total Hamiltonian for that iteration cycle, rather than over eigenfunctions of H 0 . Table 5 presents results of some numerical experiments using wavefunctions from a Hamiltonian containing only A, B, C, and the six T’s in Table 1. The integer g in the first column determines the gap size and thus the relative importance of the quartic energy contributions as indicated in Table 1. The next two columns give a measure ⑀ of how far the expectation value of the right side of Eq. [12] departs from zero for eigenfunctions with J ⫽ 10 and J ⫽ 20. To obtain this measure ⑀ for given J, we have (i) adopted a simple intuitive scaling procedure in which the actual expectation value for a given eigenfunction is divided by the largest (in absolute magnitude) of the six individual expectation values on the right of Eq. [12], and (ii) taken the root-mean-square of all 2J ⫹ 1 such ratios (which vary by an order of magnitude within the J ⫽ 20 manifold). The J ⫽ 10 and J ⫽ 20 values for this measure suggest that it increases approximately as J 2 , as might be expected from the fact that the relative importance of quartic terms over quadratic terms increases approximately as J 2 . The next two columns of Table 5 give ␬(A trA) values for the least-squares problem of fitting all J ⱕ 10 or J ⱕ 20 energy levels to A, B, C, and the six T’s (nine parameters). (These ␬ values are determined from the A trA matrix in the first itera-

DETERMINABLE PARAMETERS

TABLE 5 Deviation of the Expectation Value of Eq. [12] from Zero a and Values of ␬(A trA) b as Quartic Contributions Approach Quadratic Energy Contributions in Table 1 c

Presented as ⑀, the root-mean-square of expectation values of the right side of Eq. [12] divided (for scaling) by the largest of the six terms in the sum (see text). b Calculated from derivatives with respect to A, B, C and the six T’s in Table 1 of all energy levels having J ⱕ 10 or J ⱕ 20, and evaluated at the parameter values used to generate the original data. c Energy levels are calculated from the A, B, C, and T values in Table 1, using the gap-size parameters g indicated in column 1 above. d The number of digits in the worst T returned by the fit which agree with digits in the original value of T used to generate the J ⱕ 20 data set.

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proportional to ␭ J 2 , in agreement with values in the second and third columns in Table 5. Further, if the first two columns of the A matrix are related by A k1 ⫽ A k2 ⫹ a k ␭ J 2 , where the second term is small and the a k are independent of ␭ and J, then the ␬ value of the 2 ⫻ 2 matrix (A trA) ij with i, j ⫽ 1, 2 is proportional to ( ␭ J 2 ) ⫺2 , in agreement with values in the fourth and fifth columns of Table 5. Unfortunately, the above arguments are not as useful as they appear to be, since they do not indicate what the proportionality constants in these expressions should be, making it impossible to relate expectation values of Eq. [12] a priori to ␬(A trA) values. Nevertheless, they suggest that some insight into the torsion–rotation problem might be gained by looking into the various analogs of the linear relation in Eq. [12] which arise there (5).

a

tion, i.e., from an A trA matrix calculated with input parameters equal to those used to generate the initial energy levels, so they can be evaluated even if the fit ultimately diverges.) The last column indicates the number of correct digits returned for the most poorly determined of the six T’s after the fit has converged. When g ⫽ 9, the initial J ⱕ 20 ␬ value is 2.8 ⫻ 10 13 and the fit diverges. When g ⫽ 8, the ␬ value is 2.8 ⫻ 10 11 and the fit converges, but the ␬ value increases, and only some of the adjusted T’s agree with their original values. By g ⫽ 6, the fit is rather well behaved, returning the initial ␬ value and at least five good digits for every T. Thus, as might be expected, when the scaled average expectation value of Eq. [12] becomes greater than 0.001 or so (i.e., when the number of significant figures in the “zero” of this expectation value becomes less than two or three), then in effect we no longer have a linear relation between expectation values of the six operators on the right of Eq. [12], and a least-squares fit which simultaneously adjusts A, B, C, and all six quartic distortion constants is no longer ill-conditioned. We find empirically from Table 5 that ⑀ ⬀ 10 ⫺g J 2 over a four-decade change in gap size, and that dln␬ /dln⑀ ⬇ ⫺2 for given J. These results can be rationalized as follows. Consider a Hamiltonian of the form h 0 ⫹ ␭ J 2 h 1 , where ␭ represents the 10 ⫺g factor in Table 1, and the J 2 factor represents the gain in importance of quartic terms over quadratic terms with increasing J. First-order perturbation theory then indicates that the deviation ⑀ of the expectation value of Eq. [12] from zero is

4. TORSION–ROTATION PROBLEM

In this section we attempt to transfer the experience gained from the artificial asymmetric rotor examples of the two previous sections to the more complicated reduction of the torsion–rotation Hamiltonian, the significant additional complication arising when the torsional degree of freedom is added to the three rotational degrees of freedom. The fundamental work on reduction of the torsion–rotation Hamiltonian is that of Nakagawa et al. (5), who defined an ordering scheme for the operators in this problem, and then considered the Hamiltonian reduction through fourth order; later workers (6, 7) extended this reduction to sixth and eighth orders. (To maintain consistency with this torsion–rotation ordering terminology from the earlier literature, based essentially on equating the order of terms with the total power of angular momentum operators appearing in terms of that order, we use it throughout Section 4. The reader should note, however, that second, fourth, sixth, and eighth order in this terminology are in many ways analogous to zeroth, first, second, and third order in the traditional perturbation terminology used in Sections 2 and 3 above.) In a more recent work, Duan et al. (14) considered reductions of the torsion–rotation Hamiltonian in which torsional operators are moved to lower orders than in Ref. (5); in particular, they developed a complete vibration–torsion–rotation formalism based on the idea that the nth power of a rotational angular momentum operator is of order 2n, while the nth power of the torsional angular momentum is of order 3n/ 2 or (3n ⫺ 1)/ 2 for even or odd n, respectively. Inspection of their Table 6 indicates (i) that terms of order 4, 5, and 6 in their scheme correspond exactly to those of order 4 in Ref. (5), and (ii) that the 22 terms in these orders in their reduced Hamiltonian exactly equals the 22 fourth-order terms in the reduced Hamiltonian of Ref. (5). Since the present investigation is concerned with controversy surrounding the use of 24 (13) and 25 (11) terms of this type, the work of Duan et al. does not seem directly applicable.

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TABLE 6 Methanol Parameters a (in cm ⴚ1) Obtained from Fits to Two Different Torsion–Rotation Hamiltonians

Parameters have been converted to the notation of Ref. (13), except for A zz ⫽ A ⫹ ␳ 2 F and B ␥z ⫽ ␳ F, since the large data set of 6656 transitions being fitted here was also obtained from Ref. (13). b A partially reduced Hamiltonian fit with a total of 54 floated parameters (including 23 floated fourth-order parameters), which gives ␴ ⫽ 1.074 and ␬ ⫽ 1.19 ⫻ 10 13. c A reduced Hamiltonian fit with a total of 58 floated parameters (including 22 floated fourth-order parameters), which gives ␴ ⫽ 1.074 and ␬ ⫽ 1.22 ⫻ 10 13. d A reduced set of fourth-order parameters, for comparison with those in column 2, obtained algebraically by applying the first contact transformation to the parameters in column 1 (see text). Only terms affected by the contact transformation are shown. a

A. Least-Squares Fits of Methanol Data from the Literature A fully reduced Hamiltonian for methanol (5) contains a maximum of 22 fourth-order terms. As a convenient shorthand characterization we thus refer to the first 22 fourth-order terms in any fit as legal terms, and to any C 3v -allowed terms above the limit of 22 as extra or illegal terms. Hamiltonians contain-

ing one or more illegal fourth-order terms will be called partially reduced. To compare partially reduced with reduced Hamiltonians for the torsion–rotation problem, we have undertaken a series of fitting experiments with the v t ⫽ 0 and 1 methanol data set used in the fit of Ref. (13), which had 56 floated and 8 nonzero fixed parameters, a weighted standard deviation ␴ ⫽ 1.03, and ␬ ⫽ 6 ⫻ 10 12. (It was originally

DETERMINABLE PARAMETERS

295

TABLE 7 Nonvanishing Contributions to the Fourth-Order Terms from the First Contact Transformation of the Torsion–Rotation Hamiltonian

a This set of parameters, used with the ( J b2 ⫺ J c2 ) operator representation and employed in Ref. (13), is given here to indicate the correspondence between those parameters and the parameters in the cylindrical operator representation. b Our notation, slightly modified from Ref. (5), for the cylindrical operator form of the parameters given in column 3. c Explicit algebraic expressions for the contributions to the corresponding original constants produced by the 11 s coefficients of the first contact transformation. d Numerical expressions based on second-order parameters from the partially reduced fit in Table 6.

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planned to remove the eight fixed parameters by floating them or setting them to zero in the next stage of global fitting, when v t ⫽ 2 data were added, but their presence was somewhat bothersome for the present investigation, so we initially devoted some time to refitting the v t ⫽ 0 and 1 data set without fixing any of the parameters to values other than zero, and only such fits are discussed here.) Parameter sets from fits to Hamiltonians with two different levels of reduction are given in reciprocal centimeters in columns 1 and 2 of Table 6 (column 3 will be discussed in the next section). The first column of numbers represents a set of constants obtained from a partially reduced Hamiltonian fit which has a total of 54 floated parameters (slightly better than in Ref. (13)), ␴ ⫽ 1.074 (slightly worse), and ␬ ⫽ 1.2 ⫻ 10 13 (slightly worse). (Note that this apparent degradation of the ␬ value is misleading, because the ␬’s here describe fits with no fixed parameters, whereas the smaller ␬ of the published fit (13) was achieved by fixing a number of important, but highly correlated, parameters to values returned in earlier trial fits with smaller numbers of floated parameters.) In the fit of the first column of Table 6, 23 fourth-order parameters were floated, which is one more than allowed in a fully reduced Hamiltonian. For comparison, the second column of numbers represents a set of constants obtained from a fully reduced Hamiltonian fit (with only 22 fourth-order parameters). This reduced Hamiltonian fit was able to achieve the same standard deviation of 1.074 with essentially the same ␬ value of 1.2 ⫻ 10 13, but it required four more parameters (58 altogether) than the partially reduced Hamiltonian fit in the first column. The fits in columns 1 and 2 represent the best possible combinations of parameters we were able to find for either partially reduced or reduced Hamiltonians, i.e., some parameters which noticeably improved the partially reduced Hamiltonian fit in the first column produced little or no improvement in the reduced Hamiltonian fit and vice versa. An interesting point is that a fit with the same set of parameters as in the first column but without the illegal D abK term gives the same ␬ but a worse ␴, indicating that this extra parameter plays little role in degrading the stability of the fit but helps to achieve the required ␴. Columns 1 and 2 in Table 6 show that five parameters in sixth order for this particular molecule and data set can be efficiently replaced by one illegal parameter in fourth order. In addition to these fits we attempted to push the partially reduced Hamiltonian model to the limit and include as many fourth-order terms as possible before the least-squares program starts to diverge. In this paragraph we discuss some general features of two of these fits. The first had a total of 57 floated parameters, including 30 fourth-order terms. This fit did not increase ␬ (with respect to Table 6) and further reduced ␴ to 1.060. The second fit had a total of 55 floated parameters, including 31 fourth-order parameters, i.e., nine more than allowed in a reduced Hamiltonian, corresponding to no reduction at all in the terms not involving sine and cosine operators. This fit gave (compared to Table 6) ␬ ⫽ 5.5 ⫻ 10 13 (4.6 times

worse) and ␴ ⫽ 1.082 (1% higher). In both of these extreme cases, however, some parameters were returned with one or more fewer significant digits than in the two fits of Table 6 and, as discussed in the next section, there are reasons to believe that some of the parameter values returned may be too large. Interestingly enough, adding the extra fourth-order terms above to the first column did not produce much improvement in the standard deviation. Once one illegal fourth-order term had been included, better results could be achieved by adding legal sixth-order parameters, i.e., instead of adding another fourth-order parameter (e.g., T 0004 or T c1001 in the notation of Table 7), two or more sixth-order parameters produces a better standard deviation without any deterioration in ␬ and determinacy. This suggests that even though up to a certain point adding a smaller number of illegal fourth-order parameters can be equivalent to adding a larger number of sixth-order parameters, adding more fourth-order parameters cannot account for missing higher order parameters if contributions of the latter are strongly present in the spectra. As might be expected, adding too many sixth-order parameters (i.e., adding four or more to the various fits discussed here) also deteriorates ␬, indicating that interorder in addition to intraorder linear dependencies among parameters are significant in methanol. B. Contact Transformation Questions As summarized in Table V of Nakagawa et al. (5), there are 11 independent s coefficients in the transforming S 3 function of the first contact transformation, which must be applied to 33 fourthorder terms of the C 3v torsion–rotation Hamiltonian in the RAM system. This is what led the authors of Ref. (5) to conclude that a maximum of 33 ⫺ 11 ⫽ 22 fourth-order terms are determinable in any global fit. To avoid ambiguity in the discussion below, we now specify the second-order (quadratic) torsion–rotation Hamiltonian and the third-order (cubic) S 3 function used in this section (a contact transformation using S 1 (5) has already been implicitly carried out here to arrive at the RAM system): H 2 ⫽ 12 共B xx ⫹ B yy 兲J 2 ⫹ 12 共2B zz ⫺ B xx ⫺ B yy 兲J 2z 2 2 ⫹ 14 共B xx ⫺ B yy 兲共 J ⫹ ⫹ J .⫺ 兲 ⫹ 12 B xz 兵 J z , 共 J ⫹ ⫹ J ⫺ 兲其

⫹ 2B ␥ z P ␥ J z ⫹ B ␥␥ P ␥2 ⫹ 12 V 3 共1 ⫺ cos 3 ␥ 兲 S 3 ⫽ s 0201 iJ 2 共 J ⫹ ⫺ J ⫺ 兲 ⫹ s 0021 i兵 J 2z , 共 J ⫹ ⫺ J ⫺ 兲其 2 2 3 3 ⫹ s 0012 i兵 J z , 共 J ⫹ ⫺ J⫺ 兲其 ⫹ s 0003 i共 J ⫹ ⫺ J⫺ 兲

⫹ s 2001 iP ␥2 共 J ⫹ ⫺ J ⫺ 兲 ⫹ s 1011 iP ␥ 兵 J z , 共 J ⫹ ⫺ J ⫺ 兲其 2 2 ⫹ s 1002 iP ␥ 共 J ⫹ ⫺ J⫺ 兲 ⫹ s sz sin 3 ␥ J z

⫹ 12 s sx sin 3 ␥ 共 J ⫹ ⫹ J ⫺ 兲 ⫹ 12 s cy i共1 ⫺ cos 3 ␥ 兲共 J ⫹ ⫺ J ⫺ 兲 ⫹ s ␥ s 兵P ␥ , sin 3 ␥ 其.

[16]

DETERMINABLE PARAMETERS

Two of the 33 fourth-order terms, i.e., those involving P ␥4 and P ␥3 J z , are unaffected by the contact transformation with S 3 ; explicit expressions for contributions to the other 31 terms are given in Table 7 below, which we now discuss in some detail. The second and third columns define the coefficients and operators used in this paper. (Since the cylindrical form of operators is extremely convenient for contact transformation manipulations in orders higher than second, as well as for operators with 兩⌬K兩 ⬎ 2, we use it here, in spite of the fact that results of most methanol fits have historically been presented using operators in Cartesian form.) The last three subscripts on the s nmpq and T nmpq coefficients are defined following Eq. [2]; the first numerical subscript n indicates the power of P ␥ ; the initial subscripts c or s, when they occur, indicate the presence of cos 3␥ or sin 3␥ factors, respectively. Column 2 of Table 7 gives the T coefficients used here; column 1 gives the traditional parameter used in the global fits (8 –13) which corresponds to each T. Column 4 gives algebraic contributions to the original (untransformed) T constants produced by contact transformation with the 11 s coefficients. Column 5 gives the same contributions evaluated numerically for methanol, using the secondorder constants from column 1 of Table 6. Note that the values in Table 6 correspond to Hamiltonians already transformed to varying degrees, where the transformations are implicitly defined by the parameters which have been set equal to zero in each fit vs the original (true) values for all parameters. Since the original values are unknown for this real example (in contrast to the artificial asymmetric rotor examples considered earlier), we are forced to use various sets of fitted values as starting points in the discussion below. The entries in Table 7 are arranged in five groups; the first four groups (n ⫽ 0–3) involve Hamiltonian operators with increasing powers of P ␥ ; the last group (with various n values) contains all cos 3␥ and sin 3␥ operators. Contributions from the s coefficients are distributed among the first four groups in such a way that the contact transformation parameter s nmpq , where n is the power of P ␥ , contributes only to P ␥n and P ␥n⫹1 terms in the Hamiltonian. The four parameters s ␥s , s sx , s sz , and s cy contribute only to the cos 3␥ and sin 3␥ group. This distribution of s parameters, in addition to the properties discussed below, imposes certain restrictions on which combinations of terms can be eliminated. Based on common spectroscopic experience, we know that eliminating some terms is a better choice than eliminating others. The theoretical criterion is how large a contact transformation tail is created by eliminating any particular term, since a large tail indicates that many higher order terms must be included to compensate for the removed term. From the asymmetric rotor calculations it is clear that the convergence gap between successive orders in the created tail is related to the size of the corresponding s nmpq coefficients in the S function, which in turn is governed essentially by the size of the fourth-order parameters to be removed divided by some com-

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bination of second-order constants. In the torsion–rotation Hamiltonian one must consider linear combinations of the various s coefficients contributing to a particular term, but visual inspection of column 5 of Table 7 suggests that among the terms not containing cos 3␥ or sin 3␥, T 0211 , T 0031 , T 0022 , T 0013 , T 1201 , T 1021 , T 1012 , T 1003 , T 2011 , T 2002 , and T 3001 might be the most suitable candidates for elimination since contact transformation contributions to these terms have large numerical factors in front of at least one s coefficient. In contrast, elimination of the T 0202 term, which is not removable at all in the asymmetric rotor case and which is removable here only because of a contact transformation contribution proportional to the very small B xz constant, would almost certainly create a bigger tail. We unfortunately do not know the values of the original untransformed T’s, but from the inequalities A Ⰷ B ⫹ C Ⰷ B ⫺ C, one might estimate that the magnitude of the T’s ranges from the largest for J z4 to the smallest for J ⫹4 ⫹ J ⫺4 . From this point of view, the best parameters to remove in the first group seem to be T 0211 and T 0013 . In addition, the value for T 0004 is expected to be quite small; therefore, even though the contact transformation contributions to the T 0004 constant involve very small numerical coefficients for s parameters, the latter might be of the same order or even smaller than the s parameters required to eliminate T 0022 . (Contrary to our initial expectations for this near prolate rotor, a fit to the S-reduced torsion–rotation Hamiltonian is no better than a fit to the A-reduced Hamiltonian.) Even though the thinking in this paragraph is unlikely to produce a “correlation-free Hamiltonian” (26), it should minimize the adverse effect of large tails. The fifth group of terms in Table 7, containing the cos 3␥ and sin 3␥ factors, may constitute a special case for methanol at its present stage of global fitting. Even though some of the coefficients on the right of Table 7 are rather large, the corresponding T’s are also expected to be quite large, since they represent centrifugal or structural variations of one kind or another in the several hundred reciprocal centimeter barrier to internal rotation. The resulting large s coefficients should make a number of illegal fourth-order terms determinable. The fact that they were not may well arise because only two torsional levels of A species and two of E species are included in our most complete global fit to date (13). Since the statements above represent only conjecture at this time, however, it will be interesting to see if new terms from this fifth group become determinable when v t ⱖ 2 torsional levels are added to the global methanol fit. In this paper we are concerned not only with the traditional question of which terms should be removed from the Hamiltonian, but also with the alternative question of which terms could be kept and determined in a least-squares fit. One way of answering this question theoretically in fourth order would be to start with a set of untransformed T’s (perhaps obtained from ab initio calculations) and then begin eliminating them one at a time, verifying after each elimination that the s coefficients

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chosen lead to acceptably small tails in the sense of Table 4. Elimination of additional terms would cease when the tails generated became larger than a certain threshold, implying that the remaining terms were all determinable. Unfortunately, since we have at the present time neither a set of untransformed ab initio T’s, nor complete algebraic expressions for all tails generated, this conceptually clean approach cannot be applied. Instead what is readily available is a collection of various empirical fits, as illustrated by the two in Table 6. For the present data set, we believe the first column represents the best fit, in the sense that it requires the least number of parameters, gives an excellent weighted standard deviation, and is characterized by a reasonable ␬ value. A step to the right in Table 6 brings one to a fully reduced Hamiltonian fit giving the same ␴ and ␬, but requiring four more adjustable parameters. This step corresponds to carrying out a complete contact-transformation reduction on the true torsion–rotation Hamiltonian, but it can be approximated algebraically by carrying out the first contact transformation H 4 ⫹ i[S 3 , H 2 ] on the torsion–rotation Hamiltonian defined by the parameters from the first fit in Table 6, i.e., by making use of column 5 of Table 7 to solve a set of 11 linear equations for the 11 unknown s coefficients which eliminate the nonzero T 0031 term in column 1 of Table 6 while keeping the other terms T 0211 , T 0013 , T 0004 , T 1201 , T 1003 , T 3001 , T c2000 , T c1010 , T c1001 , and T s1001 at their previously fixed value of zero. Values calculated from the expressions in Table 7 (but keeping more significant figures) for all fourth-order Hamiltonian coefficients in the second (i.e., the reduced) fit of Table 6 which are affected by this transformation are given in column 3 of Table 6. The complete contact transformation taking us from column 1 to column 2 is somewhat more complicated and involves a series of repeated transformations. Briefly, the term B ␥x becomes nonzero (1.4 ⫻ 10 ⫺4 in our case) after the S 3 transformation and it must therefore be reeliminated using the s y coefficient in S 1 , which rotates the coordinate system back to RAM. Then the S 3 transformation must be carried out again to clean up the mess created by s y , and so on, until fully converged. In addition, every contact transformation affects all orders (including the s y transformation, which, however, does not lead to a redistribution of Hamiltonian parameters among orders, i.e., does not generate tails, but only leads to a redistribution of Hamiltonian parameters within each order). Thus, the transformation responsible for the difference in the number of sixth-order terms in the fits of the two columns has also in principle had an impact on the fourth-order parameters. Exploratory iterative calculations with s y indicate that the first effect above is nearly negligible for the model Hamiltonians we have considered. Therefore, we believe that the first S 3 transformation alone should yield coefficients of sufficient accuracy to allow us to judge whether the partially reduced Hamiltonian fit produces meaningful results. As can be seen from Table 6, the values obtained algebraically (column 3)

from the partially reduced Hamiltonian fit (column 1) are rather close to the values returned in the reduced Hamiltonian fit (column 2), suggesting that the fits in columns 1 and 2 are in some sense consistent with each other. The two fits with 30 and 31 fourth-order parameters discussed in the previous section were also subjected to the S 3 -transformation test, but for these cases it failed. Values obtained for the transforming s coefficients were quite large, and some of the resulting transformed parameters were thus very different from the fitted values in column 2 of Table 6. This result seems to be due to an overly correlated (incipient indeterminate) least-squares procedure which caused the fitted parameters to become large, rather than to the fact that a single S 3 transformation for these larger parameters without subsequent iterations was not sufficient to produce meaningful numbers. (The latter factor might still be a part of the problem, however, since the B ␥x parameter regenerated after the S 3 transformation became quite large (i.e., 0.005 cm ⫺1, which is larger than D ab ), and the resulting s y ⬇ 3 ⫻ 10 ⫺4 needed to reeliminate it is bound to alter significantly some of the fourthorder parameter values.) It would be interesting to further investigate this problem by performing a more complete contact transformation on various partially reduced Hamiltonian fits to see exactly which tails are generated and how large they are, and also to see which unreduced sets of fitted parameters transform in a convergent iterative manner into the reduced set from column 2 of Table 6. But from an empirical point of view it will be even more interesting to see whether the reduced Hamiltonian fit of the second column (as suggested by orthodox contact transformation considerations) or the unreduced fit of the first column (as suggested by the present results) turns out to be a better guide to global fitting strategy when the current experimental data set for methanol is extended by five or six J values and one or two K and v t values. C. Linear Relations For the torsion–rotation problem there are 11 commutation relations analogous to the single commutation relation in Eq. [12] for the asymmetric rotor problem, whose expectation values vanish to good approximation when the power series representation of the Hamiltonian operator converges sufficiently rapidly. Equations [16] show that the following 11 commutation relations occur in [S 3 , H 2 ]:

2 2 关J 2 共 J ⫹ ⫺ J ⫺ 兲, H 2 兴 ⫽ B xz 关J 2 共 J ⫹ ⫹ J⫺ 兲 ⫺ 6J 2 J 2z ⫹ 2J 4 兴

⫺ 共B xx ⫺ B zz 兲J 2 兵 J z , J ⫹ ⫹ J ⫺ 其 ⫹ 2B ␥ z P ␥ J 2 共 J ⫹ ⫹ J ⫺ 兲

[17a]

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DETERMINABLE PARAMETERS

关兵 J 2z , J ⫹ ⫺ J ⫺ 其, H 2 兴

2 2 ⫺ J⫺ 兲, H 2 兴 关P ␥ 共 J ⫹ 3 3 ⫽ 2B xz 关P ␥ 共 J ⫹ ⫹ J⫺ 兲

2 2 ⫽ B xz 关3共 J ⫹ ⫹ J⫺ 兲 ⫺ 20J 4z ⫹ 12J 2 J 2z

⫹ P ␥ J 2 共 J ⫹ ⫹ J ⫺ 兲 ⫺ 32 P ␥ 兵 J 2z , J ⫹ ⫹ J ⫺ 其兴

2 2 ⫺ 兵 J 2z , J ⫹ ⫹ J⫺ 其 ⫹ 2J 2 ⫺ 10J 2z 兴 3 3 ⫹ J⫺ 其 ⫹ J 2 兵 J z , J ⫹ ⫹ J ⫺ 其兴 ⫹ 共B xx ⫺ B yy 兲关⫺兵 J z , J ⫹

⫹ 共B yy ⫺ B zz 兲兵 J z , J ⫹ ⫹ J ⫺ 其 ⫺ 共3B xx ⫺ B yy ⫺ 2B zz 兲 ⫻ 兵 J 3z , J ⫹ ⫹ J ⫺ 其 ⫹ 2B ␥ z P ␥ 兵 J 2z , J ⫹ ⫹ J ⫺ 其

[17b]

2 2 ⫺ 共B xx ⫹ B yy ⫺ 2B zz 兲 P ␥ 兵 J z , J ⫹ ⫹ J⫺ 其 2 2 2 2 ⫹ J⫺ 兲 ⫺ 32 V 3 i sin 3 ␥ 共 J ⫹ ⫺ J⫺ 兲 ⫹ 4B ␥ z P ␥2 共 J ⫹

[17g]

关i sin 3 ␥ J z , H 2 兴

2 2 关兵 J z , J ⫹ ⫺ J⫺ 其, H 2 兴

⫽ ⫺ 12 B xz i sin 3 ␥ 兵 J z , J ⫹ ⫺ J ⫺ 其

3 3 ⫽ B xz 关兵 J z , J ⫹ ⫹ J⫺ 其 ⫺ 7兵 J 3z , J ⫹ ⫹ J ⫺ 其

2 2 ⫺ 12 共B xx ⫺ B yy 兲i sin 3 ␥ 共 J ⫹ ⫺ J⫺ 兲

⫹ 3J 2 兵 J z , J ⫹ ⫹ J ⫺ 其 ⫹ 兵 J z , J ⫹ ⫹ J ⫺ 其兴

⫺ 6B ␥ z cos 3 ␥ J 2z ⫺ 3B ␥␥ 兵P ␥ , cos 3 ␥ 其J z

4 4 ⫹ 共B xx ⫺ B yy 兲关⫺共 J ⫹ ⫹ J⫺ 兲 ⫹ 2J 4 ⫺ 12J 2 J 2z

⫹ 10J 4z ⫺ 4J 2 ⫹ 14J 2z 兴 ⫹ 2共B xx ⫹ B yy ⫺ 2B zz 兲

[17h]

关i sin 3 ␥ 共 J ⫹ ⫹ J ⫺ 兲, H 2 兴 2 2 ⫽ B xz i sin 3 ␥ 共 J ⫹ ⫺ J⫺ 兲

2 2 2 2 ⫻ 关⫺兵 J 2z , J ⫹ ⫹ J⫺ 其 ⫹ 2共 J ⫹ ⫹ J⫺ 兲兴 2 2 ⫹ 4B ␥ z P ␥ 兵 J z , J ⫹ ⫹ J⫺ 其

[17c]

⫺ 共B yy ⫺ B zz 兲i sin 3 ␥ 兵 J z , J ⫹ ⫺ J ⫺ 其 ⫹ B ␥ z 关兵P ␥ , i sin 3 ␥ 其共 J ⫹ ⫺ J ⫺ 兲

关 J ⫺ J , H 2兴 3 ⫹

⫹ 2共B xx ⫺ B yy 兲共2P ␥ J 3z ⫺ 2P ␥ J 2 J z ⫹ P ␥ J z 兲

⫺ 3 cos 3 ␥ 兵 J z , J ⫹ ⫹ J ⫺ 其兴

3 ⫺

⫺ 3B ␥␥ 兵P ␥ , cos 3 ␥ 其共 J ⫹ ⫹ J ⫺ 兲

⫽ 3B xz 关共 J ⫹ J 兲 ⫹ J 共 J ⫹ J 兲 4 ⫹

4 ⫺

2

2 ⫹

2 ⫺

2 2 2 2 ⫹ J⫺ 兲 ⫺ 32 兵 J 2z , J ⫹ ⫹ J⫺ 其兴 ⫹ 共J⫹

关cos 3 ␥ 共 J ⫹ ⫺ J ⫺ 兲, H 2 兴

⫹ 32 共B xx ⫺ B yy 兲关兵 J 3z , J ⫹ ⫹ J ⫺ 其

2 2 ⫹ J⫺ 兲 ⫺ 6 cos 3 ␥ J 2z ⫹ 2 cos 3 ␥ J 2 兴 ⫽ B xz 关cos 3 ␥ 共 J ⫹

⫺ J 2 兵 J z , J ⫹ ⫹ J ⫺ 其 ⫹ 兵 J z , J ⫹ ⫹ J ⫺ 其兴

⫺ 共B xx ⫺ B zz 兲cos 3 ␥ 兵 J z , J ⫹ ⫹ J ⫺ 其

⫺ 共B xx ⫹ B yy ⫺ 2B zz 兲兵 J z , J ⫹ J 其

⫹ B ␥ z 关兵P ␥ , cos 3 ␥ 其共 J ⫹ ⫹ J ⫺ 兲

3 ⫹

3 2

3 ⫺

⫹ 6B ␥ z P ␥ 共 J ⫹ J 兲 3 ⫹

3 ⫺

[17d]

[17j]

关兵P ␥ , i sin 3 ␥ 其, H 2 兴 ⫽ ⫺6B ␥ z 兵P ␥ , cos 3 ␥ 其J z

2 2 ⫽ B xz 关P ␥2 共 J ⫹ ⫹ J⫺ 兲 ⫺ 6P ␥2 J 2z ⫹ 2P ␥2 J 2 兴

⫺ 共B xx ⫺ B zz 兲 P ␥2 兵 J z , J ⫹ ⫹ J ⫺ 其 ⫹ 2B ␥ z P ␥3 共 J ⫹ ⫹ J ⫺ 兲 ⫺ 32 V 3 兵P ␥ , i sin 3 ␥ 其共 J ⫹ ⫺ J ⫺ 兲

⫺ 3i sin 3 ␥ 兵 J z , J ⫹ ⫺ J ⫺ 其兴 ⫺ 3B ␥␥ 兵P ␥ , i sin 3 ␥ 其共 J ⫹ ⫺ J ⫺ 兲

关P ␥2 共 J ⫹ ⫺ J ⫺ 兲, H 2 兴

[17e]

关P ␥ 兵 J z , J ⫹ ⫺ J ⫺ 其, H 2 兴 ⫽ 2B xz 关4P ␥ J 2 J z ⫺ 8P ␥ J 3z ⫺ P ␥ J z 兴 3 3 ⫹ 共B xx ⫺ B yy 兲关⫺P ␥ 共 J ⫹ ⫹ J⫺ 兲 ⫹ P ␥ J 2 共 J ⫹ ⫹ J ⫺ 兲兴

⫺ 12 共5B xx ⫺ B yy ⫺ 4B zz 兲 P ␥ 兵 J 2z , J ⫹ ⫹ J ⫺ 其 ⫹ 12 共B xx ⫹ B yy ⫺ 2B zz 兲 P ␥ 共 J ⫹ ⫹ J ⫺ 兲 ⫹ 2B ␥ z P ␥2 兵 J z , J ⫹ ⫹ J ⫺ 其 ⫺ 32 V 3 i sin 3 ␥ 兵 J z , J ⫹ ⫺ J ⫺ 其

[17i]

[17f]

⫺ 3B ␥␥ 关2兵P ␥2 , cos 3 ␥ 其 ⫺ 9 cos 3 ␥ 兴 ⫹ 32 V 3 共1 ⫺ cos 6 ␥ 兲.

[17k]

Setting expectation values on the left of these equations to zero gives 11 linear relations among derivatives of energies with respect to various fourth-order parameters, permitting reduction of the total number in the Hamiltonian from 33 to 22, as shown in Ref. (5). As suggested by the asymmetric rotor calculations in Section 2, however, expectation values of these commutators do not in general vanish identically. Table 8 (which is analogous to Table 5) gives the root-mean-square average (over 兩K兩 ⱕ 14, corresponding to the actual molecular data sets) of the normalized J ⫽ 20 expectation values for these 11 equations obtained from E-species wavefunctions calculated using molecular pa-

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TABLE 8 Root-Mean-Square Averages for K < 14 of Normalized and Unitless J ⴝ 20 Expectation Values for the 11 Linear Relations in Eqs. [17a]–[17k] obtained from E-species Wavefunctions Calculated Using Molecular Parameters Adjusted in a Large Global Fit of v t ⴝ 0 and 1 levels in Methanol (column 1 of Table 6) and from Wavefunctions Calculated Using Molecular Parameters Adjusted in a Large Global Fit of v t ⴝ 0, 1, 2 Levels in Acetaldehyde (10)

was reduced to 0.0003. This value approaches a correspondence in Table 5 to g ⫽ 7 and to only two correct digits in the floated parameters, suggesting that at least one of the 11 illegal parameters should be removed. It seems, however, that it will be difficult to significantly reduce the majority of the methanol expectation values in Table 8 by taking appropriate linear combinations, so that we are inclined to conclude from the considerations of this section also, that many of the 11 illegal parameters will be quite determinable once the methanol data set is expanded to higher J, K, and v t values. We have attempted to discover which fourth-order terms in methanol are most responsible for the nonzero expectation values in Table 8 by recalculating the expectation values using wavefunctions obtained when various groups of fourth-order terms are artificially set to zero. It appears that almost all the fourth-order terms are large enough to cause significant deviations from zero in Table 8, although a first-order perturbation theory result (somewhat disguised notationally by the traditional ordering scheme for this problem (5)), 具⌿ 2 ⫹ ␭ ⌿ 4 兩关H 2 , A兴兩⌿ 2 ⫹ ␭ ⌿ 4 典 ⫽ ⫺具⌿ 2 兩关 ␭ H 4 , A兴兩⌿ 2 典, [18] shows that a fourth-order term ␭ H 4 in the Hamiltonian which commutes with an operator A generating one of the commutators in Eqs. [17a]–[17k] does not affect to order ␭ the expectation value of that particular commutator.

rameters obtained from column 1 of Table 6, as well as analogous information for expectation values over wavefunctions generated from a large global fit of v t ⫽ 0, 1, 2 levels in acetaldehyde (10). Many of the expectation values for methanol are on the order of 0.1– 0.8, which for J ⫽ 20 in Table 5 for the asymmetric rotor problem would correspond to a gap size characterized by g ⫽ 3 or 4. If one estimates a gap size in methanol from the ratio V 6 /V 3 ⫽ 0.004 or k 4 /F ⫽ 0.0004, one obtains g ⬇ 3, in reasonable agreement with the first estimate. On the other hand, some of the expectation values are as small as 0.002, which corresponds to g ⫽ 6 in Table 5. Naive comparison of all expectation values for methanol in Table 8 with Table 5 suggests that all 11 illegal parameters could be determined to six digits. This is in fact not true, however, since empirical fits floating more than nine illegal parameters, while not actually divergent, return very large and very poorly determined values for many parameters, suggesting that the results are essentially meaningless. Thus, particular linear combinations of the linear relations in Eqs. [17a]–[17k] may well exist, which would lead to smaller expectation values than those found in Table 8. Indeed, visual inspection of the J ⫽ 20 expectation values for E-species levels led to an empirical linear combination of Eqs. [17h], [17i], and [17k] for which the v t ⫽ 0 expectation value

5. DISCUSSION

In the first part of this work we addressed the question of whether conditions can be found and imposed on model asymmetric rotor Hamiltonians which limit the usefulness of reduced Hamiltonians for spectral fitting purposes. We feel that results from our numerical and algebraic calculations indicate that such conditions do exist, for the following reasons. First, in numerical least-squares fits using an asymmetric rotor Hamiltonian with centrifugal distortion terms and an artificial data set containing all K levels for each J up to a given J max generated from that Hamiltonian, it was possible to show empirically that when the quartic contributions are large enough (compared to the quadratic) all six input quartic centrifugal parameters could be recovered in the least-squares fits to five or six significant digits. Second, algebraic investigation of the linear relation permitting contact transformation reduction of the number of parameters in the quartic asymmetric rotor Hamiltonian shows that this linear relation between derivatives of energy levels with respect to fitting parameters in the Hamiltonian breaks down when quartic contributions are large enough. A similar algebraic investigation of the higher order terms generated by contact transforming a given quartic term to zero shows that removing a large quartic term may

301

DETERMINABLE PARAMETERS

require the introduction of a half dozen or more large sextic terms to achieve the same quality of fit. Given these observations on the simple asymmetric rotor problem, we examined the more complicated torsion–rotation problem for analogous behavior. Here because of the much larger number of parameters and much larger data sets, we did not generate artificial data sets, but restricted our consideration instead to actual molecular examples, focusing primarily on methanol. It again proved possible to recover with good numerical precision in least-squares fits a somewhat larger number of fourth-order terms than would be permitted in a fully reduced Hamiltonian, though for this experimentally measured data set we cannot prove that we recover the true values for these parameters. It also turns out that ratios of fourth-order to second-order parameters which would be uncommonly large in asymmetric rotor examples are commonplace in the torsion– rotation problem, suggesting that contact-transforming these fourth-order terms out of the Hamiltonian will generate large and extensive tails of higher order terms. Two avenues of useful future work suggest themselves at this time. First, numerical studies of internal rotor Hamiltonians using artificially generated data sets containing all quantum mechanically allowed transitions up to some given J specified with state-of-the-art experimentally attainable accuracies would settle the question of how many large fourth-order torsion–rotation input parameters can in principle be recovered in a least-squares fit and to what precision. Second, leastsquares fits of expanded data sets of actual molecules, particularly to higher torsional levels and higher K quantum numbers, would soon reveal whether fully reduced or partially reduced Hamiltonians provide the better fitting strategy for realistic modern torsion–rotation problems. Until this additional work has been carried out, the present results must be taken as suggestive rather than definitive.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

21. 22. 23.

ACKNOWLEDGMENTS The authors are very grateful to Dr. R. Boisvert for several pedagogical discussions concerning the condition number of a matrix. They are also indebted to Professor A. Bauder, Dr. W. J. Lafferty, Professor H. Ma¨der, and Dr. J. Ortigoso for helpful criticism of the manuscript.

24. 25. 26.

J. K. G. Watson, J. Chem. Phys. 45, 1360 –1361 (1966). J. K. G. Watson, J. Chem. Phys. 46, 1935–1949 (1967). J. K. G. Watson, J. Chem. Phys. 48, 181–185 (1968). J. K. G. Watson, in “Vibrational Spectra and Structure” (J. R. Durig Ed.), Vol. 6, Elsevier, Amsterdam, 1977. K. Nakagawa, S. Tsunekawa, and T. Kojima, J. Mol. Spectrosc. 126, 329 –340 (1987). O. I. Baskakov and M. A. O. Pashaev, J. Mol. Spectrosc. 151, 282–291 (1992); Opt. Spectrosk. 65, 1070 –1072 (1988). [in Russian] J. Tang and K. Takagi, J. Mol. Spectrosc. 161, 487– 498 (1993). I. Kleiner, J. T. Hougen, R. D. Suenram, F. J. Lovas, and M. Godefroid, J. Mol. Spectrosc. 148, 38 – 49 (1991). I. Kleiner, J. T. Hougen, R. D. Suenram, F. J. Lovas, and M. Godefroid, J. Mol. Spectrosc. 153, 578 –586 (1992). S. P. Belov, M. Yu. Tretyakov, I. Kleiner, and J. T. Hougen, J. Mol. Spectrosc. 160, 61–72 (1993). I. Kleiner, J. T. Hougen, J.-U. Grabow, S. P. Belov, M. Yu. Tretyakov, and J. Cosle´ou, J. Mol. Spectrosc. 179, 41– 60 (1996). L.-H. Xu and J. T. Hougen, J. Mol. Spectrosc. 169, 396 – 409 (1995). L.-H. Xu and J. T. Hougen, J. Mol. Spectrosc. 173, 540 –551 (1995). Y.-B. Duan, L. Wang, and K. Takagi, J. Mol. Spectrosc. 193, 418 – 433 (1999). V. M. Mikhailov, Thesis 1980, Institute of Spectroscopy of the Academy of Sciences of the USSR, Troitsk. [in Russian] V. M. Mikhailov and M. A. Smirnov, SPIE Proc. 3090, 0277-786X, 135–142 (1997). R. M. Lees, J. Mol. Spectrosc. 33, 124 –136 (1970). P. No¨sberger, A. Bauder, and Hs. H. Gu¨nthard, Chem. Phys. 1, 418 – 425 (1973). J. Demaison, R. Bocquet, W. D. Chen, D. Papousˇek, D. Boucher, and H. Bu¨rger, J. Mol. Spectrosc. 166, 147–157 (1994). K. Sarka, D. Papousˇek, J. Demaison, H. Ma¨der, and H. Harder, in “Vibration–Rotational Spectroscopy and Molecular Dynamics” (D. Papousˇek, Ed.), pp. 116 –238, World Scientific, Singapore, 1997. G. E. Forsythe and C. B. Moler, in “Computer Solution of Linear Algebraic Systems,” Chap. 8, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1967. J. J. Dongarra, C. B. Moler, J. R. Bunch, and G. W. Stewart, “LINPACK Users’ Guide,” Siam, Philadelphia, PA, 1990. G. H. Golub and C. F. Van Loan, “Matrix Computations,” Chap. 5, The Johns Hopkins University Press, Baltimore, MD, 2nd ed., 1989. D. Kivelson and E. B. Wilson, Jr., J. Chem. Phys. 21, 1229 –1236 (1953). D. Papousˇek and M. R. Aliev, “Molecular Vibrational/Rotational Spectra,” Academia, Prague, 1982. Y.-B. Duan, L. Wang, I. Mukhopadhyay, and K. Takagi, J. Chem. Phys. 110, 927–935 (1999).