The analysis of the rotational spectrum of methanol to microwave accuracy

JOURNAL OF MOLECULAR SPECTROSCOPY 13+395-411

( 1989)

The Analysis of the Rotational Spectrum of Methanol to Microwave Accuracy FRANK

C. DE LUCIA, ERIC HERBST, AND TODD ANDERSON

Departmentof Physics,Duke University,Durham, North Carolina 27706 AND PAUL HELMINGER Departmentof Physics, Universityof South Alabama, Mobile, Alabama 36688 A method for the analysis and characterization of a large number of rotational-torsional ttansition frequencies of methanol (CH,OH) to microwave accuracy is presented. It is based on our earlier work which used a direct diagonalization of an internal axis method Hamiltonian. In the work reported here the analyses of the A and E symmetry species were separated and careful attention was paid to the selection of cross terms between rotational and torsional operators to be included in the Hamiltonian. In order to facilitate comparison, the previous data set, which contained 470 rotational transitions, was also used for this new work. In our original analysis the root-mean-square (mw) deviation was I .2 MHz. This same data set was subsequently reanalyzed by Nakagawa, Tsunekawa, and Kojima with an effective Hamiltonian based on a Watson-like transformation with a larger number of adjustable parameters. Their approach reduced the rms deviation to -0.6 MHz, but substantial model error remained. In the work reported here rms deviations of 0.065 MHz and 0.062 MHz for the A and E symmetry states, respectively, have been achieved. These deviations approximate experimental error, indicating that the model error has now been reduced from _ 1 MHz to essentially zero. Of perhaps more importance,our new approach has significantly enhanced the predictive powers of the model and provides a means for an accurate characterization of the rotational-torsional spectrum of methanol. o 1989 Academic Press. Inc.

I. INTRODUCTION

This work was motivated by the desire to have a theoretical model for the analysis of the rotational-torsional spectra of methanol and methanol-like internal rotors, the model error of which would be small compared to the accuracy of microwave measurements (GO. 100 MHz) and which could be used to calculate “complete” spectral maps over a significant range of quantum states. Models for the rotational spectra of other classes of molecules, such as asymmetric rotors, which have these desirable properties have proved to be very important both for applications such as radio astronomy and atmospheric science and also as bases for fundamental considerations of their spectroscopic properties. Experimental and theoretical work on the rotational-torsional spectrum of methanol has a rich history which has been summarized up through 1984 in our earlier paper 395

0022-2852189 $3.00 Copyright 0 1989 by Academic PRS.

inc.

All rights of reproduction in any form reserved.

396

DE LUCIA ET AL.

on methanol ( I ). Most notable in the early work are the pioneering theoretical studies of Dennison and co-workers (2-5) in which the internal axis method (IAM) was developed and utilized, and the millimeter-wave spectral work of Lees and Baker (6) in which large numbers of rotational-torsional transitions were analyzed. In our earlier paper, we used an extended IAM treatment based closely on the work of Lees and Baker (6) to fit 470 lines of the rotational-torsional spectrum of methanol involving the quantum numbers J d 8 and u1 G 2. All transitions in the analysis involved changes in rotational quantum numbers only. The overall root-mean-square (rms) deviation of the fit was 1.18 MHz and was obtained by varying 2 1 out of a total of 34 parameters. Although this fit did not achieve an rms deviation comparable to normal microwave accuracy (GO. 100 MHz), it was the first attempt at a global least-squares fit to a significant portion of the methanol data and provided a limited capability for predicting the frequencies of transitions not in the analysis. Our work on methanol was followed by similar studies on the isotopic variants “CH30H and CHjOD ( 7-9). Since the time of our initial work on methanol, significant new developments have been reported. In 1987, Nakagawa et al. (10) published a new analysis of the set of rotational-torsional transition frequencies collected by us ( I ) . These authors used symmetry considerations and a Watson-like contact transformation to obtain a fourth-order effective Hamiltonian without indeterminate terms. In addition, they included more matrix elements off-diagonal in torsional quantum number ( ut). Their analysis of the data resulted in an improvement in the rms deviation to 0.57 MHz, still far above standard microwave accuracy. The authors were able to make an even more substantial improvement in the rms deviation to the rotational-torsional spectrum of methyl mercaptan compared with our work ( 1I ). Another important development has been the extension of the analysis of the methanol spectrum to significantly higher rotational quantum numbers. Using the same IAM approach as Herbst et al. (I), Sutton and Herbst (12) managed to fit 783 rotational-torsional lines out to J = 22 by varying 32 out of a total of 34 parameters. Their analysis required supercomputer implementation and has proved useful to astronomers interested in predictions of rotational-torsional spectral lines with large values of the rotational quantum number J. Sutton and Herbst (12) were also able to fit series of spectral lines that could not be fit in the analysis of Herbst et al. (1) because their larger data set allowed them to vary more parameters. Still, the rms deviation of their fit was 4.4 MI-Ix, clearly not comparable to microwave accuracy. The difficulties encountered in fitting the spectrum of methanol to microwave accuracy via an analysis based on a quantum mechanical Hamiltonian have given rise to an alternative procedure, hrst utilized by Pickett et al. ( 23). In this procedure, spectral lines are fit by suitable subtraction of empirical power series expansions for the energy levels. Pickett et al. (23) managed to fit 270 lines to microwave accuracy by varying 104 coefficients. More recently, Moruzzi et al. ( 14) have used the procedure to analyze a vast number of methanol lines which they measured with a high-resolution Fourier transform spectrometer in the range 8 to 100 cm -’ to an accuracy of = l-2 MHz for unblended lines. These lines were analyzed along with higher resolution microwave data, primarily at lower frequencies. A glance at a small fraction of the 6725 assigned and analyzed lines in their table shows the quality of the fit to be variable but certainly not to be of microwave accuracy even for those lines measured to such accuracy. A

ROTATIONAL

SPECTRUM

OF METHANOL

397

power series expansion has also been used to analyze the Fourier transform spectrum of methanol between 950 and 1100 cm-’ (IS). In this paper we present the results of a new quantum mechanical analysis of the low J (J Q 8) rotational-torsional spectrum of ‘2CH30H in its first three torsional states. Basedon the sign&ant successof Nakagawaet al. ( 10) in improving our earlierquantum mechanical analysis of the methanol spectrum, it was the goal of this work to be able to fit the data to microwave accuracy (x0.100 MHz) and to have the results of the analysis characterize the spectrum in a fashion similar to modem analyses of light asymmetric rotors. In other words, we would like to predict accurately transitions that are within the general range of the quantum numbers of the observed transitions and to be able to extrapolate with reasonable, but decreasing accuracy, as the quantum numbers of the calculated transitions move away from those of the experimental data set. These goals appear to have been met for the data set contained in Ref. ( 1) by use of a significantly expanded number of carefully chosen parameters and the separation, at least in our current procedure, of the previous global analysis into two distinct treatments-one for the A symmetry sublevels and one for the E symmetry sublevels. In the next section, we review briefly the IAM method used and the particular parameters added. II. THEORY

The IAM approach used in Lees and Baker (6) and incorporated into Herbst et al. (I) has once again been utilized. Shown in Table I, the IAM Hamiltonian can be subdivided into rotational, torsional, and distortion terms. The rotational Hamiltonian is the standard rigid-rotor expression used for near-prolate symmetric tops (symmetry axis labeled by a) with the addition of a cross term between the angular momentum operators P. and Pa made necessary by the use of a nonprincipal axis system. It contains the four rotational constants A, B, C, and Dab. The torsional Hamiltonian is the standard IAM expression in which both the torsional angular momentum P, and P. appear because of the unusual boundary conditions chosen (16). It contains the torsional-rotational constant F and potential constants V3and V, which, when multiplied by suitable trigonometric functions, describe the height of the potential barrier to internal rotation and its angular dependence. The simplifying approximation is made that the torsional motion is one-dimensional; that is, it is described by one angle y. An expression for the parameter p in the torsional Hamiltonian is given in Ref. (6). The distortion Hamiltonian, which is composed of a wide assortment of coupling terms among angular momenta and potential energy expressions, contains all of the remaining parameters. These parameters can be subdivided into groups. The “standard” distortion constants DJ, DJK, DK, by, 6,, HJ, HJK, HKJ, HK, h.,, h.JK,hK, LK, LJK, and PK multiply terms in which end-over-end angular momentum operators are multiplied by one another. The three last constants multiply bigb powers of angular momentum operators and were not present in our previous Hamiltonian ( 1). In addition to the standard distortion constants, there are distortion-like expressions often referred to as “interaction” terms needed to describe the coupling between torsion and end-over-end angular momentum not reduced to zero by the transformation generating the nondistortion parts of the Hamiltonian, as shown in detail by Kirtman ( 17). In Table I, the great majority of these interaction terms are grouped according

398

DE LUCIA ET AL. TABLE I Rotation-Torsion

Hamiltonian”

H = ‘%ot + Htors + “dist

El

rot

= lR(B+CJ

(P,2+Pc2)

+A_P*2

+ & H tom I E (pY+Opa)*

Hdist - -RJ P4 _ OJK P2Pa2

+ 1, ti + tr,, +ilx

* - P,2]) +&

Se2P,2[P

+ ( 1 - cody)

(1 - cos3y) + l/z&

Pa4 _ & S(Pa’[‘b*

+ QJ

P2Pp4 + 4

s(Pn4[Pb2

s ,‘$ P2 +& Pa2 +q

Bb2.PC2)

S(P,Pb)

+ lI2q

_ 4

p4P,,*

+ l/2&-w

_ PC?)

(1. cos6y)

_24 (pz[pb2-Pc21)

Pa6 + h, P‘t (pb2 - PC’,

. PC*]) + LK Pa* + Lx

(Pb2 - PC2, + dab S(paPb, ‘f,,

p4Pa4+

PK PalO

ti + 0, P2Pa2

- 2c6 p2(Pb2 - P 2) - cg S(P 2[Pb2 -P~])+h,,~P~+hKtP21’~} l SlP,+&P*)2

{~c~+k2P.;+4(Pb*-P,2~+~~bP,Pb+CP4+y,PP~2

_ 2c5 P2(Pb2 - PC*, - c8 Pa2 (pb2 - Pc2, + h,* tiPa + S(P,t?,,+

pp,))

{

-20, P2(Pb2 - PC’, - c,~ P,*(Pb2. + S(PY+aPa,4

+ ha

P2Pa4

&, P2 +kl Pa2 + c4 (Pb2 - Pc2, + s.&, PpPb + 1” +

{

4

+4

Pc2, + h,3 tiPa2

+ hK3 P2P,4

} + 2, P2Pp2 1

(Pb2-P,2)+Y,P2+(M),bP,Pb}

a The script symbol S refers to the symmetrization of operator expressions. For some factored expressions, the symbol is included before one factor and need not apply to all terms multiplying the factor.

to the torsional expression. Four different torsional expressions-( 1 - cos 3y), (P-, + PP~)*, PdP, + PPA, and (P,+ pPd4- are multiplied by an assortment of endover-end angular momentum operators such as P2,Pf,etc., each term possessing a multiplicative parameter or constant and each term properly symmetrized. Together there are now 37 different interaction terms as compared with 14 in the previous work and they contain end-over-end angular momentum operators through sixth order in the overall exponent. The current grand total of 60 available parameters is almost double the previous total of 34. However, many of these terms were not required in our final analysis and a number that were included make only small contributions. These small contributions are now observable because of the high accuracy of the fit. The letter designation of some interaction parameters is based on historical context whereas others have been newly named by us. To help keep track of what is new and what is old, we have underlined all of the parameters in Table I included in Ref. ( I ) . Parameters in addition also varied in Ref. ( 1) are written in italics. In greatly expanding the number of available interaction terms, we have been guided by both empirical considerations (the need for specific higher order terms) and analogy with normal centrifugal distortion. We have not used a contact transformation as was done in Ref. (10) and we make no pretensions that our Hamiltonian is indeed a complete one. As in Ref. (I), the overall Hamiltonian has been evaluated seriatim. First, the

ROTATIONAL

399

SPECTRUM OF METHANOL

torsional Hamiltonian was diagonal&d using a product basis set consisting of 2 1 free rotor functions in the torsional degree of freedom of the type exp[ i( 3k + a)~] where -lO
TABLE II Transitions That Are Blends or Violate VTOR

SPECIES

I’

I I 2 2 0 0 I I I 1 2 2 0 0 0 0 0 0 I I 0 0 2 0 0 0 0

E A A A E E E A A A A E A A A A E E A A E E E A

1 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 5 6 6 6 6 6 7 8 8 8

A

E E

-

K’ 0 I 2 2 -2 2 -2 2 3 3 0 -2 4 4

P’

+

+ + + +

3 3 -2 2

+ -

3 3 -2 2 0 I 4 4 -2

+ -

+

t

I”

t t t t t t

-i 2 2 2 2 2 2 3 3 3 3 3 5 5 4 4 4 4 5 5 5 5 5 7 8 8 7

c t c t t t t t t t t + t t t t c t t t t

-

K” 1 2 2 2 -2

p”

+ +

^ L

-2 2 3 3 0 -2 3 3 3 3 -2 2 3 3 -2 2 0

+ + + + +

+

I

+

3 -3 -I

-

LoopCriteria FREQUENCY a 169689.570 554oS2.OS6 144728.700 144728.700 145126.370 145126.370 144728.700 192963.470 192963.470 192963.470 192757.650 192756.310 636393.568 636393.568 241832.950 241832.950 241904.119 241904.407 289429.150 282429.750 290307.563 290307.563 289429.120 23347.530 636304.355 524740.167 811445.210

OBSXALC. 0.610 -0.626 0.343 -0.375 -0.011 0.152 -0.011 -0.288 0.130 0.128 0.096 0.188 -0.240 0.274 0.220 -0.166 -0.048 -0.220 0.446 0.750 0.243 -0.183 -0.579 0.551 I.115 -0.3 14 0.558

a

DE LUCIA

400

ET AL.

TABLE III CHIOH A Transitions (MHz) VTOR

I’

K’

0

I

1

0

I

1

0

I

0

I

I

0

2

I

0

0

1

1

0

2

I

0

2

I

0

2

2

0

2

2

1

2

1

0

2

0

0

2

1

0

2

1

1

2

0

I

2

1

1

2

I

2

2

0

2

2

I

2

2

1

0

2

1

0

2

0 I

2

0 2

P

t

+ + + + + + + + + + + + + + + +

J”

K’

p”

1

0

+ + + + + + + + +

e

1

1

t

0

0

t

0

0

t

0 0

0

t t

2

0

t

2

1

t

2

1

t

2

1

t

2

2

t

I

0

t

I

1

t

I

1

t

I

0

t

I

1

t

0

1

1

t

1

c

1

0 I

t

1

I

t

1

t

1

0 1

t t

1 1

I 1

2

2

1

2

I

1

3

2 1

t

0

t

3

0

3

1

t

3

0 1

0

3 3 3

2

t t t

3

3 3

3 1

3 3

1

0

0

0

3

1

0

3

1

0

3

2

0

3

2

1

3

0

1

3

1

1

1

2

3 3

0 0 0 1 1

2 3

0

2

3

2

3

1 1

0 0

3 3

0

0

3 3

2 2

0

3

1

0

3

3

0

3

3

0

1

+ + + + + + + + + + + + + + +

t +

3 3 3 3

I 1 2 2 2

+

3 2

t

2

e

2

I

t

2

2

t

2

2

t

2

e

2

0 1

t

2

L

t e

2

0 I

+

t

2 2

2 0 I

I

t +

2

0

2

I

t t

2 2

I I

+

2

2

+

2

2

t t

2 2

2

1

3

2

1

3 4

2

t

2

I

0

1

e

4

0

0

4

1

+

4

I

0

4

2

+

4

I

0 0 0 0

4 4 4 4 4 4 4 4

2

+ t t + +

4 4 4 4 4

I

t t +

4 4 3

t t e e t e +

3 3 3 3 3 3 3 3 3 3 3 3 3

0 I I 0 0 0 0 0 0 1 1 1 I 2 2 2 2

4 4 4 4 4 4 4 4 4 4 4 4 4

3 3 4 4 I I 0 1 1 2 3 3 0 I 1 2 I I 2 2

+ + + + + + + + + + +

t + + + + t

I

2 2 3 3 2 2 0 I I 2 3 3 0 1 I 2 1 I 2 2

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

FREOUENCY M3366.890

RESIDUAL

-0.064

834.261

-0.018

48372.456

-0.014

48257.490

0.184

48192.120

0.047

350905.119

-0.020

304208.350

-0.018

2502.778

-0.CO5

484cc4.740

0.044

481504.232

0.048

553763.582

-0.067

96741.420

0.030

95914.290

-0.016

97582.830

0.026

%513.660

-0.017

%3%.010

-0.048

%588ml

-0.015

%383.130

0.058

%l89.670

-0.092

96553.780

0.139

398446.920 -205791.270

-0.055

579il84.700

OS69

-457367.889

-0.018 -0.119

-457463.899

-0.011

305473.520 5005.321

0.026 -0.W

485263.263 480269.321 251905.812

0.007 0.041 0.078

251917.050

-0.019 0.024

553624.453 554202.988 145103.230

0.065 0.018

143865.790

-O.MI

146368.300

-0.038

145133.460

0.026

145124.410

0.059

144768.200

0.022

144589.820

-0.036

144878.580

-0.017

144571.970

0.046

144281.780 144827.610

-0.074

445571.414

O.CC6 0.037

-156602.420

-0.047

629140.493 626626.302

0.092 0.038

-335133.513

0.062

397039.O!xl

-0.059

397041.410

-0.029

409035.534

-0.063

-409323.292

0.135

307165.940

0.030

8341.640

0.006

486940.837

0.012

478633.272 251866.579 251900.495 636420.231 636420.231

0.025 0.057 o.w5 0.138

553437476 554402.514 193454.390 191810.4!XI 195146.760 193488.030 193471.520 193471.520 193019.870 192778.520 193163.330 192965.530 192370.480 193098.290 L92613.090 192613.090

0.266 0.022 -0.041 O.lXCl -0.011 -0.045 -0.040 0.070 -0.041 -0.005 -0.059 -0.550 -0.024 -0.119 0.152 o.cQ2 -0.012

ROTATIONAL SPECTRUM OF METHANOL TABLE III-Continued VTOR 2 2 0 0 0 0 0 0 0 0 I 1 0 0 0 0 0 0 I I 0 0 0 0 0 0 0 1 I I 1 I I 1 2 2 2 2 2 2 2 0 0 0 0 0 0 I I 0 0 0 0 0 0 0 I 1 0 0 0 0 0 0 0 0 0 0 0 I I I 1 L 2 2 2 2

I'

K'

F'

4 4 4 4 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 s 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

3 3 I 0 2 2 I 1 2 2 2 2 1 I 2 2 3 3 1 I 0 I I 2 2 4 4 0 I 1 2 2 3 3 0 I I 2 2 3 3 I 0 2 2 I I 2 2 1 I 2 2 3 3 4 L I 0 I I 2 2 3 3 4 4 5 5 0 I L 2 2 0 1 1

+

6

z

+

+ + + + + +

c c e t + t e c t t c c

c t e

+ + + + + + + + + + + + 4

c c c + t

c c + e t t e t t t t t e + t t e t

+

t t

+

c +

+

c

+

e

+

e

+

+

+

+ + +

+ +

+

e t t t + t t e t c e c t + t t + e t

+

+

+

+

e + + + t + t

+

e

+

t e t

+ +

c + c t

1"

K"

3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

3 3 0 I I 1 2 2 3 3 I 1 0 1 I 1 2 2 2 2 0 I I 2 1 4 4 0 I 1 2 2 3 3 0 I I 2 2 3 3 0 I 1 I 2 2 1 I 0 I I I 2 2 3 2 2 0 I I 2 2 3 3 4 4 5 5 0 I I 2 2 0 1 I 2

F--

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

FREQUENCY 192739.440 192139.440 492278.713 .I07013.850 678785.455 673145.931 -293463.990 -285111.150 -58394.940 -58429.010 -360661.433 -361236.476 M9290.400 12511.ooO 489036.955 476605.142 251811.882 25189OXll 553201.597 554651.031 241791.430 239746.250 243915.830 241887.700 241842.320 241806.510 241806.510 241267.880 24096c.56!l 241441.240 241192.810 2411%.350 241198.2% 241198.290 240938.940 2Ao454850 241364.120 2Ao757.910 x0757.910 240916.160 240916.160 538570.582 -57032.920 728862.523 720441.236 -247228.693 -234683.390 -312247.354 -313203.428 311852.640 17513.341 491550.827 4741%.319 251738.520 251895.728 636363.843 552915.356 554947.481 290110.666 287670.835 292672.8% 290264.150 290184.690 290189.510 290190.540 290161.140 290161.140 2wl45.crx 290145.090 289511.110 289134.050 2a9710.460 289414.030 2a9420.240 289114.820 288533.640 20%24.280 288897.110

RESIDUAL -0.023 -0m3 0.047 -0.075 0.073 -0.065 0.119 -0029 0.003 0.008 0.136 -0.LW6 0.066 -0.221 0.045 0.008 -0.060 0.028 0.028 0.044 0.045 0.029 0.021 0.W 0.013 0.065 0.056 0.050 0.023 -0.025 -0.M3 .om3 0.003 .0.003 0.052 -0.031 0.021 0.010 -0.019 0.012 0.013 0.085 -0.u29 -0.054 -0.262 -0.034 -0.0X3 -0.038 0.010 0.056 -0.021 -0.001 0.011 0.118 0.030 -0203 0.023 0.001 o.w2 0.060 -0.025 0.061 -0.003 -0.039 -0.009 -0.160 -0.157 0.017 0.017 0.004 0.008 -0.033 O.OM -0.030 -0.032 0.054 -O.ClXl 0.036

401

DE LUCIA ET AL.

402

TABLE III-Continued VTOR

J’

K’

P

2 2 2 0 0 0 0 0 0 0 I 1 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 I 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 1 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 a 8 8 8 8 8 8 8 8

2 3 3 I 2 2 I 1 2 2 2 2 1 2 2 3 3 4 4 1. 0 I 1. 2 2 3 3 4 4 5 5 1 0 2 1 1 3 2 2. 2 2. I 2 2 3 3 4

+ + +

E, 1 1 2 2 3 3 4 4 5 5 I 0 1 1 3 3 2 2. 3 3 2 2

+ + + + + + + + + + + + + + + + + +

t

t t t t t t + t t t t t t t t t t t t + t t t t t t t t t t t t c t t t t t t

+

t t

-

t t

; + -

t t t

; +

c

; + + + + + + + +

t

t t + + t t t t

c t t t t t t t t t c

+ +

t t t t

J”

K”

5 5 5 5 5 5 5 5 5 5 5 5 7 7 7 7 7 7 7 7 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 8 8 8 8 8 8 8 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7

2. 3 3 0 1 1 2 2 3 3. 1 1 0 I 1 2 2 3 3 2 0 I 1 2 2 3 3. 4 4 5 z 1 1 2 2 2 3 3 1 1. 0 1 1 2 2 3 2 0 1 1 2 2 3 3 4 4 5 5 0 1 2 2 2 2 3 3 1 4 1 1

I’-

+ + + + + + + + + + + + + + + + + ; + + + + + + + + + + + + + + + + + + + + + + + + -

FREQUENCY 288897.110 289087.370 289087.370 584449.713 779380.507 766710.345 .201445.590 -183853.ooO 38452.690 38293.500 -263793.856 -265224.400 314859.550 494481.683 471420.477 251641.667 251923.631 636337.419 636333.470 555291.142 [email protected] 335582.005 3d1415.500 338639.939 338512.762 338540.795 338543.206 338512.762 338512.762 336486.337 338486.337 629921.337 44069.490 830349.412 -156l27.700 -132621.940 5X7277.688 86903.060 86615.760 -215302.201 -217299.202 318318.793 497828.231 468293.771 251517.262 251984.702 636311.690 555680.931 386681.920 383477.880 39u141.730 387014.800 386824.430 386885.540 386890.120 386859.820 386859.820 386820.010 386820.010 674990.423 95169.440 -111289.620 -80993.160 638523.486 638817.830 135376.760 134896.960 -24945l.911 -249443.402 -166773.278 -169427.231

RESIDUAL -0.014 -O.osl -0.M8 -0.174 0.063 -0.037 -0.WJ9 -0.227 0.028 0.194 -0.533 0.025 0.031 0.020 -0.005 -0.050 -0.023 0.020 -0.085 -0.cn3 -0.031 -0.025 -0.147 0.118 -0.103 -0.025 0.068 0.116 0.129 0.007 om7 0.084 0.001 -0.079 -0050 -0.120 0.010 0.096 0.138 0.011 -0.w2 -0.166 0.017 0.016 0.074 -0.075 0.012 -0.012 -0.110 -0m9 0.261 0.058 -0.030 -0.043 -0.053 -0.077 -0.043 -0.073 -0.073 -0.027 -0.049 -0.069 0.056 0.046 0.103 -0.125 0.014 -0.079 -0.077 0.038 -0.1328

quantum number J < 8 in individual torsional states through u, = 2. The data set of Herbst et al. (I) was chosen to facilitate comparison with earlier approaches. However, the higher accuracy of the analysis reported in this paper has allowed us to detect 27

ROTATIONAL

SPECTRUM

OF METHANOL

403

lines that should be either removed from the analysis or given lesser weight. These are listed in Table II. All of these either do not satisfy loop conditions (6) or can clearly be identified as blends. No lines were removed solely because they did not fit. The errors associated with all of these lines are small enough that they had essentially no effect on previous analyses and could be left in this analysis with no great effect. In previous treatments of the methanol spectrum, the spectra involving the A and E symmetry species have been included in one fit and a common set of molecular constants obtained. However, the calculation of the spectra of the A and E species involves the diagonalization of separate matrices. Because of this, it was decided to divide the analysis into two pieces at the onset. The separation allows for more rapid calculation of the spectrum of each symmetry subspecies, an important advantage in the “bootstrap” process associated with the assignment and analysis of complex spectra. Along with the additional parameters, it has allowed us to remove the model error of previous approaches. It may well be that if the torsional problem were handled in greater detail by using a more realistic potential function and incorporating other degrees of freedom, there would be less of a need for the large set of interaction parameters and the decoupling of the A and E symmetry subspecies. 111. RESULTS

Tables III and IV show the data sets for the A and E symmetry species as well as the residuals (measured - calculated frequencies) obtained from the spectral constants of Table V. In Table V, all constants fixed to zero are excluded and, for comparison, our previously determined spectral constants (I) are also included. Inspection of the residuals in Tables III and IV shows that the rms deviations of 0.065 and 0.062 MHz, respectively, are representative of the entire fits, and that no obvious systematic residuals, characteristic of model errors, exist. The fit to the 226 A symmetry species transitions required the variation of 31 out of the 60 parameters in our current model, whereas the fit to the 217 E symmetry species transitions required the variation of 32 parameters. There is clearly still substantial redundancy in the fits. Comparison of the newly determined parameters with previously determined spectral constants is hazardous since the newly included interaction terms can be sufficiently important to alfect the value of previous constants significantly. An extreme example of the difficulty is the case of the rotation constant D,t, which was determined to be -77.7 (0.5) MHz in our earlier work (I). In this work, the D,bp,pb term in the Hamiltonian has been supplemented with a variety of interaction terms (those with the constants dab,Aa,,, bob, and ( AA)ab) containing this angular momentum operator multiplied by different torsional expressions. One can indeed think of these interaction terms as representing the torsional dependence of the Dab rotational constant. In any event, their inclusion changes the meaning and value of Dab. For the A species, the new value of D,,h is + 110.99 (2.1) MHz, while the interaction constants Aob = -40.1 (0.4) MHz and b,h = - 16.3 (0.4) MHz are well determined. For the E species, the new value of Dab is -111.5 (I .7) MHz, while the interaction constants dab = 288.4 (2.1) MHz, Aah = -20.7 (0.6) MHz, Sab= -8.1 (0.3) MHz, and (AA),, = 0.51 (5) are all well determined. The other three rotation constants (A, B, C) , and the torsional constants (F, p, V,) are not changed dramatically from our earlier work for either the

404

DE LUCIA ET AL. TABLE IV CH,OH E Transitions (MHz) VTOR

I’

K’

t

0

1

0

1

0

1 1 I 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3

0 I 0 0 0 -1 1 0 I 2 -I 0 1 -I 0 1 -1 0 I -2 0 -1 1 0 2 1 -2 0 1 2 3 0 -1 0 1 -1 0 I 2 -L 0 I 2 -2 0 -1 1 0 2 1 3 1 0 4 -2 0 1 2 3 0 -3 -2 -* 0 I 2 3 -3 -2 -1 0 1 2 3 -I 0 1 2

+ t t + t + t t + t + t t + t t t t t t + t t t t t t t t t t t t t t t t t

I 2 0 L 0 0 0 0 0 0 1 1 1 2 2 2 0 cl 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 1 I 1 2 2 2 2 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 I 1 1 1 1 I 2 2 2 2

: 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

c c c t c c c c c c c + + c + e +

c c e c c c e t (_

e e t c e t t t e t t t + t

J’

K”

FREQLENCY

RESIDUAL

1

-1

1

0

0

0

137270.700 16.WsO.190 48377.wo 48247.890 48178.cm [email protected] .141441.2BO 157276MO 165061.140 24934.380 %739.390 %744.580 96755.510 96501.660 96493.590 96492.200 %391.w+l 96355.550 %334.990 52al79.066 254015.340 -60531.420 261805.710 -63305.630 121689.850 -93196.710 423468.663 157272.470 165099.310 24928.700 530547.277 189692.150 145097.470 145093.750 145131.880 144750.2O!l 144736.300 144734.520 144733.320 144583.910 144530.560 l44499.760 144579.860 568566.054 302369.9wJ -12178.600 310193.m -19967.300 17CC60.630 120197.520 675773.439 4l9S5.810 334426.5% 524947.234 423538.258 157246.1cQ 165190.530 24933.470 53fx10.288 189694.39u 193488.990 193511.210 19344 1.620 193415.370 193506.6al 1935tl.210 193474.330 1929527W 192%2.1(x) 1!?2w6.030 192974.360 192972.080 192973.650 192957.680 192773.580 1927U2.240 192661.130 192768.140

0.123 0.010 0.1% 0.329 -0.282 0.011 O.IzO 0.023 0.014 0.020 0025 0021 -0Ml3 osm 0.058 0.037 -0.019 0.054 -0.019 -0024 -0.042 0.038 -0.003 0.012 -O.cEz 0.058 O.CKKl 0.127 O.od8 O.m, 0.050 -0.001 01124 -0.022 O.&Xl 0.084 -O.Ml 0.0% -0.114 -0.062 0.106 0.037 ~0.079 -0.055 0.111 -0.029 -0.035 0.062 0.050 O.ooO -0.009 0.026 -0.031 0.021 -0.021 0.027 0.032 -0.014 0.010 0.041 -0.w7 -O.Ml 0.005 0.025 0.020 -0.154 -0.084 0.015 -0.010 -0.027 0.022 -0.066 -0.253 0.010 -0.027 0.094 0.043 -0.039

0

0

:

0

0 0 -*

0

2 2 2 I 1 I 1 I I 1 L 1 1 1 I I L I I 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

0

I -1 0 L -1 0 I -1 0 I -I -I 0 0 L I 0 -1 -1 0 1 2 1 -1 0 1 -I 0 1 2 -1 0 I 2 -1 -1 0 0 1 I 2 2 0 .: -I -1 0 1 2 1 -3 -2 -I 0 1 2 3 -3 -2 ,I 0 : 3 -I 0 1 2

ROTATIONAL

SPECTRUM

405

OF METHANOL

TABLE IV-Continued RESIDUAL

I'

K'

+

1"

K'

FREQUENCY

0

4

-I

0

4

0

4

3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5

-2

-2 0 -1 l 0 2 1 3 2 0 -I -2 0 l 2

+

-230027.060 616979.984 350687.730 36169.240 358605.8cil 28316.030 21a440.050 168577.860 724121.576 -337135.873 382666.474 524908.133 423675.155 151178.970 165369.440 24959.080

-0.012 O.O!Xl 0.042 -0.031 -0.043 -0.052 -0.014 -0.020 -0.065 -0.009 -0.021 -0.029 -0.035 -0.031 0.074 -0.016

53Q549.27l 18%%.750 241813.257 241852.352 U1767.224 241700.219 241879.073 241843.646 241829.646 241179.900 241187.400 241238.lrYl 241205.990 241203.690 241210.680 241166.530 240936.730 240958.800 240869.490 240817.940 240952.070 -2aw94.9al -181771.050 665442.450 398946.230 S4521.210 407069.553 266838.130 216945.Mx) 772453.803 -288705.567 51569.280 4309m.319 524860.820 423912.6% 157048.620 165678.770 25018.140 530454.695 290117.815 290162.430 29Q209.700 290069.824 289939.477 290248.762 290213.238 290183.210 290138.8% 289399.390 289402.490 289475.610 289427.600 289443.930 289355.020 289111.350 289138.820 28w31.2w) 288%9.290 289130.620 -234698.450 -133605.500 713982.470

0.05I 0.117 0.037 o.cQ7 -0.031 0.037 0.023 0.041 0.037 0.084 0.051 0.035 a04 -0.020 -0.254 O.lXO 0.101 -0.136 0.045 -0.045 -0.003 -0.032 -0.026 0.001 -0.026 0.028 om5 -0.016 0.033 -0.080 0.048 -0.081 -0.024 -0.032 -0.036 -0.008 0.090 0.019 0.033 0.107 -0.016 -0.056 0.049 0.075 0.045 0.054 -0.056 0.014 0.003 0.089 0.051 -0.028 0.266 0.015 -0.091 -0.063 0.038 -0.010 0.089 0.044 -0.085 -0.037

YTOR

0

4

0 0 0

4

0

4

0

4

0

4

4 4

I

4

0

5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6

0

0 0 0 0 1 0 0 0 0

0 0 0

1 I I I L I I

2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 I 1 2 2 2 2 2 0 0 0

3 0 4 -3 -I 0 I 3 4 -3 -2 -I 0 l 2 3 -2 -I 0 1 2 -3 -I -2 0 -1 I 2 1 3 2 1 0 4 -2 0 1 2 3 -5 -4 -3 -I 0 I 3 4 5 -3 -2 -I 1 2 3 -2 -1 0 1 2 -3 -1 -2

e t t + t t t t t + + t c t t +

c + e t t t c t t + t t + e e c t t t t e c e + c c t t t + t t c t t + t t t t e t c + t e t t t t + c e t e t c t t e t

-1 -I 0 0 I I

2 2 3 I -3 -I -I 0 I 2 l -4 -3 -I 0 1 3 4 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 4 -2 -I -1 0 0 1 2 2 3 0 l -3 -I -1 0 l 2 -5 -4 -3 -I 0 l 3 4 5 -3 -2 -1 l 2 3 -2 -1 0 l 2 -4 -2 -1

406

DE LUCIA

ET AL.

TABLE IV-Continued VTOR

0 0 0

I'

K'

6

-L

6

I

6

0

0

6

0

6

1

0

6

3

0

6

2

I I

6

I

2

6

0

0

7

-l

0 0

1

0

7

I

0

7

2

0

7

3

0

7

-5

0

1

-4

0

7

-3

0

7

-2

0

7

-1

0

7

0

0 0

7

1

7

2

0

7

3 4

t

J"

t t t + t t t t t + t t + t + t t t + t t t t

5

K"

0

132890.790

5

0

455618.116

0.033

5

I

124569.970

-ox!67

FREQUENCY

RESIDUAL

5

1

315266.830

-0.w7

5

2

265289.650

0.029

5

2

820762.501

5

3

-2AQ241.502

-0.023

5

0

99730.900

-0.095

5

0.097

1

479126.480

0.147

7

-3

524804.969

-0.071

7

-1

156828.520

O.MM

7

0

166169.210

0.070

7

1

7

2

6

338456.499

0.007

6

-5 4

338504.099

-0.132

6

-3

338559.928

-0.115

6

-2

338722.940

-0.010

6

-1

338344.628

0.001

6

0

338124.502

-0.017

6

1

338614.999

0.020

6

2

338721.630

-0.048

3 4

338583.195 338530.249

-0.025

5 4

338475.290

0.018 -0.006

-186300.950

-0.141

25124.880 5M316.196

0.060 -0.009

e

6 6

5

t

6

t

6

7

-3 -1

6 6

-I

-85567.970 162635.948

0.135

-2

t t

-2

7 7

-I 1

t

6

0

181296.030

t t

6

0 1

504293.685

0.031 0.026

0

7

0

0

7 7

0 0

0 0

7

0

7

6

172445.950

0.266

0.111

0

7

0 2

+

6

I

363739.820

0.022

0

7

1

t

6

2

313596.840

-0.018

0

1

3

t

6

2

869037.809

-0.073

0

7

2

t

6

-0x.55

7

I

t t

6

3 0

-191733.050

1

147943.630

-0.038

6

1

527342.624

0.050

-1 -1

424857.263 156488.950

-0.042 0.055

I

7

:

ii

0 0 0 I 0

8

-2 0 1

8

2

t

8

1

25294410

0.129

8

3

t

8

2

530123.294

-0.041

8

0

t

-0.208

+

-5

386788.660

-0.074

8

-5 4

8 7

169mJ.490

8

0

+

7

4

386837.700

0.279

0 0 0 0 0 0

8

+

7

-3

386901.950

-0.067

8

-3 -2

t

7

387153.550

-0.026

8

-1

t

7

-2 -1

386587.270

-0.056

8

0 1

t

7

0

386247.660

-0.041

t

7

1

386977.140

-0.048

2

t

7

2

387146.560

-0.089

3 4

+ + t

7 7 7

3 4

-3 -1

t

1

5 -4

t

7

t + t

7 7 7

8

-1 1 2 1

+

8

0

t

7 7

0

8 8 8

0

8

0

8 8

0

0 0

0 0

1 1

8 8 8 8

0

5

t t t

8" 8

0

1

166898.650

0.024

386953.820

0.041

386869.410 386802.300 -137%3.O@l

0.023 -0.157

-2

-37703.720

0 0 3 0

229758.160 553146.2% -143169.500 1%146.210

0.009 -0.046 -0.031 0.056 -0.007

I

575547.119

0.033

-0.037

A or E symmetry species although the newly determined constants for the E species do differ more from our previous results, as can be seen in Table V. This pattern does not necessarily hold for the distortion and interaction constants. In order to facilitate comparisons among the fits, the values of k,-I+, were fixed at the same values as in our earlier analysis. An important measure of a spectral analysis is its ability to predict lines that are not included in the fit. As a test we have removed 20 lines from the analyses and predicted them on the basis of an analysis of the remaining lines. These lines were

ROTATIONAL

SPECTRUM OF METHANOL

407

chosen to include the ones that should be the hardest to predict, that is, high J, high K lines, K changing transitions, and lines in the excited torsional states. Table VI shows the results. It can be seen that the differences are comparable with expected experimental error. We conclude that it is now possible to calculate “complete” spectral maps for methanol within the range of the experimental data and that, with care, modest extrapolations should be possible. Is it really necessary to decouple the analyses of the A and E symmetry states as we have done here? Preliminary tests with our current program have yielded an rms deviation to the global data set of Ref. (I) of -0.160 MHz with the variation of 49 parameters. This represents a reduction of almost an order of magnitude in the model error of our previous approach and a factor of 3-4 improvement on the result of Nakagawa et al. ( 10). Evidence exists that a modest expansion of our current program would also provide analyses with no evidence of model error. Although this is of significant theoretical interest, it may well be that separate A and E analyses will remain the method of choice for the analysis and characterization of methanol and methanol-like spectra. It is interesting to consider taking this process one step farther and to perform analyses on individual torsional states. Because this requires fewer constants and at the same time is numerically stable, we have found in several preliminary studies that this is a good way to start the bootstrap procedure that is inherent in the measurement, assignment, and analysis of complex spectra. IV. DISCUSSION

The most important result of this work is that it provides a straightforward means of analyzing the spectrum of methanol (and presumably similar species like methyl mercaptan) to microwave accuracy in a way analogous to the methods that have been developed over the years for asymmetric rotors. As a practical matter this means that complete spectral maps with statistically meaningful calculated uncertainties can be obtained from an experimental subset of the spectrum. Furthermore, theory provides a redundancy and cross check against measurement or assignment error that is unavailable in a purely experimental spectrum. In exchange for the characterization accuracy obtained for methanol two prices are paid. Although substantial redundancy is retained, the total number of constants used to characterize the A and E spectra is larger than with a global analysis. This first price is more apparent than real, because many of the additional higher order constants are a result of the added precision of the fit which allows (and requires) a more care&l centriI$al distortion and interaction treatment because the small contributions of higher order parameters are now determinable. The second price is that the separation of the problem into A and E symmetry parts is an acknowledgment that the model has moved away from its geometric roots and that the parameters are becoming to a greater or lesser extent fitting parameters. It is interesting to consider the historical context of our approach to the spectrum of a complex and floppy species. Many years ago Wilson and Howard (28) and Nielsen (19) formulated general theories of centrifugal distortion for asymmetric rotors. The theories were closely coupled to the geometry and force constants of the molecules and provided relationships among these and spectroscopic observables such as rotational constants, centrifugal distortion parameters, vibrational frequencies, and the

408

DE LUCIA ET AL. TABLE V Spectral Constants of CH30H” A

Constant

Species

E Species

Previous

Value

b

(MHz unless noted) A

127631.7339(242)

127631.5055(157)

E

24685.96292(10083)

24694.4626(5836)

24684.1785(275’)

C

23763.62765(10365)

23755.2881(5816)

23765.3680(2738)

Dab

110.9888(2.0851)

F (cm-‘)

v3 (cm”)

FY

-0.8O(fued)

-71.55603(441)

16.7177(2009)

D, x lo*

1

DK kl k2 L3 k4 % ‘6

-0.8qfixed) -71.43105(2521)

-3.559603(1301)

-3.535184(2482)

8X671(13467)

4.01084(30762)

5.08701(571)

5.08124(624)

4.9768(284)

2.94771(2307)

2.88313(469)

2.86082(3103)

1.262907(1992)

1.146639(3601)

1.058349(23160)

-3.8(flxed)

-3.Effixed)

-3.8(fixed)

-8O.O(fixed)

-(lO.O(fixed)

-8O.O(fixed)

-13*.O(fixed)

-132.O(fixe.d)

-132.O(fixed)

-249.0(fixed)

-249.qfixed)

-249.o(fixed)

lbl.qfixed)

16Lqfixed)

16l.O(lixcd)

884.0(fixcd)

884.O(lixcd)

884.O(iixed)

s, x 104

11.2904(3031)

6KX 10’

-8.31145(1.04956)

cl c2 H,K x IO4 HK, x lo4

-1.745366(45686)

-1.22722(9944)

0.89245(13976)

-2.00348q362507)

1.1392(2378)

f” x 104

8.433(813)

c3 x 102

O.O(tixed)

9.42500~166993)

-I .96915(28994) -0.682732(18592)

-I .07 t236(92898)

O.O(lixed)

-592119(87581)

O.O(tixed)

O.O(tixed) -1.469X6582)

-4.25410(75599)

-0.928105(116277)

represent

-2.8164(4998)

O.o(fixed)

O.O(tixed)

-2.52008(44548)

in parentheses

O.O(Bxed) -7.5071(1.1932)

-1.209498(26794)

o.qfim$

hK x lo*

20.2564(1.7393) 1.94605(30363)

-18.87YX2.3321)

HK x IO’

a Uncertainties

0.8097451117(1452) 373.0839507(9174)

-0.77856q3431)

-3.562297(904)

L” x 102

27.63354173(3533)

OXC9746365q438) 373.06&%626(26016)

-71.300360036)

G”

-77.6655(467 1)

27.63316276(6910)

0.8097457727(316) 373.0881039(2203)

V6 (cm-‘)

D,K x 10

-111.5105(1.7013)

27.633738302(8856)

p (unitless)

for

127630.7538(1583)

lo

deviations.

-1.91685(23437)

The number

of listed significant

each constant is necessary to reproduce calculated frequencies to l-10 kHz.

b Reference (1).

figures

ROTATIONAL SPECTRUM OF METHANOL

409

TABLE V-Continued Constant (MHz unless noted)

A Species

E Species

dab

o.qexcd)

288.36Q2(2.0663)

A ab

40.11431(40862)

-20.68134(64366)

6

-16.3033(4038)

-8.1222(2928)

gy x 104

O.O(lixed)

0.4343(1039)

1” x 104

O.O(fixed)

0.5552(2172)

0, x 102

1.6819(4369)

O.O(tixed)

y, x 103

o.qfixcd)

2.1418(1500)

ab

I, x 103 c5x104 C8 x 10’ CgXIOl hK, x 10 hK2 x 10

3 4

hK3 x 104

LIK x 106 M,r

(M)ab

lo4

-8.52825(17441)

O.O(fincd)

1.@681(551)

0.0(lixcd)

l.l43c9(13366)

O.O(lixcd)

-0.9251q69967)

O.O(lincd)

2.86563(34948)

O.O(lixed)

6.85703(31220)

0.8424(1589)

o.qtixed)

0.6530(1075)

4.7835(9214) O.O(lixed) O.O(lixed)

Previous Value b

o.qfmd) 21292X5228) 0.50854q50791)

like, Indeed, much of the thrust of molecular spectroscopy has been devoted toward the inversion of spectroscopic parameters to obtain molecular properties. However, modern spectroscopic techniques produce bodies of data with an accuracy that far exceeds that of the theory (except perhaps for simple molecules like diatomics) which binds the fundamental properties of a molecule to its spectroscopic features. Spectroscopists must then utilize analysis techniques that are not so tightly coupled to basic molecular properties. For example, current practice in asymmetric rotors is to fit the rotational spectrum of each vibrational state to its own set of rotational and distortion constants. In this case, for the lowest order constants, relations to the geometric basis exist (but not uniquely ) , but the operators that are the coefficients of the higher order constants were obtained by logical extension of the lower order terms in the power series (20), An alternative procedure would be to build into the Hamiltonian a suitable dependence on vibrational state so that effectively the rotational and distortion constants change with vibrational state. This second procedure is what is accomplished in our extended IAM Hamiltonian. The interaction terms contain the torsional dependence of the more “fundamental” rotational and distortion terms. Although they can be justified and given explicitly by basic theory ( I7), they are used here to reduce data sets of high accuracy so that additional spectml lines can be calculated accurately. It is unclear at the present time how and if the number of interaction terms needed

410

DE LUCIA ET AL. TABLE VI Observed Transitions Predicted by Fit VTOR

SPECIES

1'

K’P

t

I"

K"

0

A

4

t

E A A A

5 6 6 6

t

3 4

I

I I I 2 2 0 I 0 0

A E E E E

6 6 6 7 7

2 3 1 2 3 3 5 3 -3 -4

0 0 0 0 0 1 0 I 0 0

A A E A A A E E A E

8 8 8 8 8 8 8 8 8 8

t t t c t t t t t t t t t t t t t t

5 s 5 5 5 5 6 7 7 7 7 7 7 7 7 7 8 8

+ +

5 5 4

+ -

3 3 2 -3 0 3 3

-

+

3 I 2 3 3 5 3 -3 -3 5 5 4 2 4 I 4 I 2 2

P"

+ +

+

FREQUENCYa

0BS.-CALC.3

673745.931 241166.530 289710.460 289414.030 289087.370 289087.370 290138.890

-0.073 -0.013 -0.035 0.028 -0.033 -0.042 0.017

289335.020 338559.928 524804.969 386820.010 386820.010 386869.410 638817.830 -249443.402 -169427.231 -137903.060 575547.119 251984.702 530123.294

-0030 -0.119 -0.104 -0.197 -0.197 0.029 O.lc4 -0.096 -0.054 -0.009 -0012 -0.081 -0.091

in our current treatment can be significantly reduced. Perhaps a careful treatment of higher order terms in the Hamiltonian akin to the complete fourth-order Hamiltonian previously derived (10) can achieve such a reduction. The inclusion of more offdiagonal terms in u,, the torsional quantum number, might be useful. Another possibility previously alluded to is a more accurate treatment of the torsional problem. Careful ab initio treatments of the internal rotor HSSH (22, 22) show that at least one other geometrical parameter (the SSH bond angle) changes substantially as the internal rotor undergoes torsional motion. It is probable that similar effects occur in other internal rotors, in which case the torsional Hamiltonian should include them. An inaccurate torsional Hamiltonian clearly affects the final rotational-torsional eigenstates obtained. Convergence in any spectroscopic model is always an important issue. We see no evidence at this time that our approach cannot be extended to transitions of higher J and K, subject to limitations similar to those on other species with similar rotational constants. It will be interesting to see how rapidly higher order centrifugal distortion and interaction parameters will need to be added to the analysis with increasing Jand K in the data set. In any event, the spectrum of methanol is so rich that substantial redundancy in the fit should not be a problem even if the high order centrifugal distortion parameters that are now common in the analysis of light asymmetric rotors are required. In addition to extending the model to include transitions of higher J and K, it is of interest to determine whether or not a global model of the methanol spectrum that achieves microwave accuracy can be obtained. As mentioned above, currently we have not yet succeeded in obtaining such a model although there is every indication that with a reexamination of the interaction parameters included in the Hamiltonian we will be able to achieve this goal. In summary, a means of characterizing the spectrum of methanol to microwave

ROTATIONAL

SPECTRUM OF METHANOL

411

accuracy has been presented. This method should prove to be especially valuable for applications, such as radio astronomy, where it is important to have complete spectral maps of assigned lines. In addition, our analysis, by accurately characterizing a large body of data with substantial redundancy, should provide usefkl information for attempts at reconciling all methanol data in terms of a single global theoretical model. ACKNOWLEDGMENT We acknowledge the support of the National Science Foundation for our program in laboratory spectroscopy through Grant CHE-85 1533 I. RECEIVED: November

14,

1988 REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

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