The calculation of molecular parameters for a molecule with an internalrotorYun-Bo Duan, Li Wang, Xudong T. Wu, Indranath Mukhopadhyay, and Kojiro Takagi Citation: J. Chem. Phys. 111, 2385 (1999); doi: 10.1063/1.479616 View online: http://dx.doi.org/10.1063/1.479616 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v111/i6 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 6 8 AUGUST 1999

The calculation of molecular parameters for a moleculewith an internal rotor

Yun-Bo Duana)

Department of Chemistry, The Ohio State University, Columbus, Ohio 43210and Institute for Computational Science and Engineering, Ocean University of Qingdao,Shandong 266003, China

Li WangDepartment of Physics, Toyama University, Toyama 930-8555, Japan

Xudong T. WuDepartment of Chemistry, The Ohio State University, Columbus, Ohio 43210

Indranath MukhopadhyayLaser Programme, Center for Advanced Technology, Indore 452 013, India

Kojiro TakagiDepartment of Physics, Toyama University, Toyama 930-8555, Japan

~Received 1 March 1999; accepted 17 May 1999!

A derivation for the formulas to calculate centrifugal distortion constants, based on a recentformulation@Duan and Takagi, Phys. Lett. A~1995!# of centrifugal distortion effects for a moleculecontaining a threefold symmetric internal rotor, is presented. Some constants which are independentof the barrier derivatives, especially the constants representing interactions between torsion androtation, are given in terms of molecular structural parameters and force constants. These calculatedconstants are helpful in the reduction of the Hamiltonian and in the analysis of observed transitions.The derived formulas are applied to numerical calculations of the centrifugal distortion constants ofmethanol. It is shown that most of the calculated constants are in good agreement with thoseobtained from the fitting to experimental data. ©1999 American Institute of Physics.@S0021-9606~99!01530-5#

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I. INTRODUCTION

Recently, a formal theory of the centrifugal distortion fa molecule containing a threefold symmetric internal rowas formulated by Duan and Takagi.1,2 This theory presentsa method for introducing the centrifugal distortion terms ina model Hamiltonian for a molecule with the internal rotoThis theory also relates the parameters used in the fiobserved data with fundamental molecular parameterstained in the original Hamiltonian. From the distortion costants determined experimentally, some information ccerning the vibrational potential constants may be obtainThis information can be used to supplement or check vibtional data. Conversely, when the force constants and sttural parameters are given for the molecule, some ofdistortion constants may be calculated by using these forlas. This is helpful in the reduction of Hamiltonian and in tanalysis of the observed torsion-rotational spectrum. Toknowledge, such a calculation has not been published prously.

In the present paper, a detailed description of themerical calculation of the centrifugal distortion parameterspresented for a molecule with an internal rotor. The quaties that are needed for the calculations are structural paeters, the first and second inertia derivatives and the baderivatives with respect to vibrational coordinates.1 At thecurrent stage, only those distortion constants that are in

a!Tel: 614-688-8180; electronic mail: yduan@chemistry.ohio-state.edu

2380021-9606/99/111(6)/2385/7/$15.00

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pendent of the barrier derivatives are calculable becausedata are available that relate the changes in barrier tochanges in the molecular geometry. In our application ofobtained formulas for the distortion constants, all of the cculable constants up through the fourth order are determnumerically using known force constants for CH3OH and arecompared with experimental results. It is shown that mosthe calculated constants are in good agreement with thobtained from the fitting to experimental data.

II. THEORETICAL MODEL

A. The Hamiltonian

For a molecule containing aC3v symmetric internal ro-tor attached to an asymmetric rigid frame with a planesymmetry, which is referred as CH3OH-type molecule in thispaper, a frame-fixed axis system with its origin at the cenof mass of the whole molecule is chosen in whicha axis isparallel to the symmetric axis of the internal rotor,b axis isparallel to the plane of symmetry, andc axis is perpendicularto the plane of symmetry. The internal rotation coordinategis chosen as the torsional angle of the internal rotor wrespect to the frame. Following the earlier study by Duan aTakagi,1 the torsion-rotational Hamiltonian is written schmatically as

H5H21H41¯ , ~2.1!

whereH2n represents a sum of terms involving angular mmentum operators~Pa , Pb , Pc , andPg! and torsional po-

5 © 1999 American Institute of Physics

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2386 J. Chem. Phys., Vol. 111, No. 6, 8 August 1999 Duan et al.

tential operators~sin 3fg and 12cos 3fg! to the 2nth power.For the calculation presented in this paper, where only teup to H4 will be considered, the Hamiltonian is given by,

H251

2 (ab

m00abPaPb1

V3

2~12cos 3g! ~2.2!

with

V35V3e2

1

2 (i j

Vi3~ f 21! i j Vj

3 ~2.3!

and

H451

4 (abgd

tabgdPaPbPgPd

1(ab

PaPb@Qab2;2~12cos 3g!

1Vab2;2 sin 3g#1

V6

2~12cos 6g!, ~2.4!

with

tabgd521

2 (i j

m i0ab~ f 21! i j m j 0

gd , ~2.5!

2Qab2;252

1

2 (i j

m i0ab~ f 21! i j Vj

32(i j

m i1ab~ f 21! i j Vj

3 ,

~2.6!

2Vab2;252

1

2 (i j

n i1ab~ f 21! i j Vj

3 , ~2.7!

V65V6e1

1

8 (i j

Vi3~ f 21! i j Vj

3 . ~2.8!

In Eqs. ~2.3! and ~2.5!–~2.8!, ( f 21) i j is the i j th element ofthe inverse force constant matrix so that the vibrationaltential energy of the molecule may be written as

V51

2 (i j

3N27

f i j QiQj . ~2.9!

The internal displacement coordinatesQi andQj do not nec-essarily have to be normal coordinates. The internal boangle coordinates will be employed in our calculation. TcoefficientsV3

e andV6e are the torsional potential constants

the equilibrium,Vi3 are the first-order derivatives of barrie

with respect to internal displacement coordinateQi ,

Vi35S ]V3

]QiD

e

. ~2.10!

Finally m i fab andn i f

ab are the coefficients in the expansionthe inverse inertia tensormab , specifically speaking,mab isexpanded for a CH3OH-type molecule as

mab5m00ab1(

iQi H m i0

ab1(f 51

`

@m i fab~12cos 3f g!

1n i fab sin 3f g#J 1

1

2 (i j

QiQj H m i j 0ab

1(f 51

`

@m i j fab~12cos 3f g!1n i j f

ab sin 3f g#J 1¯ ,

~2.11!

where subscriptsi and f are used in the expansion coefcients asm i f

ab and n i fab , respectively. The expansion coeffi

cients can be determined from

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s

-

d-et

m00ab5mab

e , ~2.12!

m i fab52

3

p E2p/3

p/3imab

1 cos 3f gdg, ~2.13!

n i fab5

3

p E2p/3

p/3imab

1 sin 3f gdg, ~2.14!

m i0ab5

3

2p E2p/3

p/3imab

1 dg2(f 51

`

m i fab , ~2.15!

wheremabe is the equilibrium value ofmab , and imab

1 is thefirst-order partial derivative ofmab with respect to thei thinternal displacement coordinate,

imab1 5S ]mab

]QiD

e

. ~2.16!

B. The derivatives of mab

Using the method proposed by by Kivelson and Wilso3

the partial derivatives appearing in Eq.~2.16! are expressedas

]m

]Qi52m

]I

]Qim, ~2.17!

wherem5I 21. An important observation is that the partiderivatives of components of the inertia matrix needs to sisfy the Eckart and Sayvetz conditions.3–5

In the present work, the four-dimensional inertial tensI is written as1

I5S I aa 2I ab 2I ac 2I ag

2I ab I bb 2I bc 2I bg

2I ac 2I bc I cc 2I cg

2I ag 2I bg 2I cg I gg

D , ~2.18!

where the inertia components of the molecule (I ab) (a,b5a,b,c) are determined as in Ref. 6,

I ab5(i 8

mi 8~ea3r i 8!•~eb3r i 8!

1(i 9

mi 9~ea3r i 9!•~eb3r i 9!, ~2.19!

I ag5I aa8 1~ea3ea!•(i 8

mi 8~ l i 8e

3d i 8!, ~2.20!

I gg5I ag . ~2.21!

In Eqs. ~2.19!–~2.21!, all single-primed quantities are assciated with the internal rotor and all double-primed quanties with the framework. The vectorsr i 8 and r i 9 are drawnfrom the center of mass of the entire molecule to massesmi 8andmi 9 , respectively, andl i 8 is the vector from the center omass of the internal rotor to a massmi 8 and is given byl i 85r i 82%8, where%8 is the vector from the center of massthe entire molecule to that of the internal rotor. The secoterm in Eq. ~2.20! results from the Eckart and Sayveconditions.6,7 The inertia components of the internal rot(I ab8 ) (a,b5a,b,c) are relative to the center of mass of thinternal rotor and are defined as

I ab8 5(i 8

mi 8~ea3 l i 8!•~eb3 l i 8!. ~2.22!

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2387J. Chem. Phys., Vol. 111, No. 6, 8 August 1999 Calculation of molecular parameters

If one usesd i 85r i 82r i 8e and d i 95r i 92r i 9

e to represent thedistortion of the molecule from its equilibrium configuratiod i 8 and d i 9 cannot be arbitrary valued since the Eckart aSayvetz conditions must be satisfied. It is shown by Kirtm6

that an unallowed set of displacements,d i 8* andd i 9

* , can beconverted to an allowed set of displacementsd i 8 and d i 9which satisfy these conditions by adding appropriate ritranslation and rotation of the entire molecule. The first-orincrements in the components of the inertia tensorI in Eq.~2.17! can be expressed in terms ofd i 8

* andd i 9* as

dI aa52dBaa1dhb@ I age 1I ga

e #2dhg@ I abe 1I ba

e #

12dt@ I bc8e~12dac!2I cb8e~12dab!#, ~2.23!

dI ab5dBab1dBba1dhbI bge 2dhaI ag

e 1dhg@ I aae

2I bbe #1dt@ I ca8

edba2I ba8edaa1~ I cc8e2I bb8e!dga#,

~2.24!

dI gg52dBaa8 12dhbI bc8e22dhcI ab8e , ~2.25!

dI ag5dBaa8 1dBaa8 1dhaI ab8e2dhaI ab8e 1dhb~ I aa8e

2I aa8e!1dtI ca8e1~ea3ea!

•(b

S dAb81(g

I bg8e dhg1dtI ba8e Deb , ~2.26!

where

dBab5(i 8

mi 8~ea3d i 8* !•~eb3r i 8

e!1(

i 9mi 9~ea3d i 9

* !

•~eb3r i 9e

!, ~2.27!

dBab8 5(i 8

mi 8~ea3d i 8* !•~eb3 l i 8

e!, ~2.28!

de521

M S (i 8

mi 8d i 8* 1(

i 9mi 9d i 9

* D , ~2.29!

dha52(b

~ I 21!abe ~dAb1dtI ba8e !, ~2.30!

dt52@rI aa8e#21FdAa82(ab

dAa~ I 21!abe I ba8e G , ~2.31!

with

r 512~ I aa8e!21F(ab

I aa8e~ I 21!abe I ba8e G , ~2.32!

dA5(i 8

mi 8~r i 8e

3d i 8* !1(

i 9mi 9~r i 9

e3d i 9

* !, ~2.33!

dA85(i 8

mi 8~ l i 8e

3d i 8* !. ~2.34!

In Eqs. ~2.23!–~2.34!, the superscript (e) indicates equilib-rium values, (I 21)e is the inverse ofI e, andM is the mass ofthe entire molecule. These relations allow the displacemed i 8* and d i 9

* , to be entirely arbitrary while the system stsatisfies the Eckart and Sayvetz conditions. To find a dertive ]I ab /]Qi , it is only necessary to find a set of incrementsd i 8

* andd i 9* which produce an incrementdQi to thei th

internal coordinate and leave other internal coordinates wtheir equilibrium values. To do that, we must know the dplacement transformation relation,X5AR, whereX is a col-

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d

dr

ts,

a-

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umn vector with a set of (3N) displacement coordinates iCartesian axis system as elements andR is a column vectorwith a set of (3N27) independent internal coordinateselements. Here the internal rotation coordinateg is separatedfrom the (3N27) modes of vibration based on one-largamplitude internal-rotor model.1,2 The matrixA can be con-structed byAi j 5(kBikGk j

21/Mii ,8–11 whereM is a diagonalmatrix of mass andB is the transpose of the transformatiomatrix B with (3N27)33N elements. The matrixG is theWilson G-matrix which is independent ong.

III. APPLICATION AND DISCUSSION

For methanol, the HamiltonianH in Eq. ~2.1! is invariantunder the groupC3v(M ) operations. By the use of the commutation relations for angular momentum the HamiltonianHis first reduced to an effective form with new coefficients thare functions of the distortion constants and the rotatioconstants. The functional form and the corresponding coficients of the terms of the effective Hamiltonian up to fourorder are presented in Tables I and II, respectively. Tablshows that all the second order constants exceptB8 can becalculated, and the 22 constants, out of the 33 fourth orconstants, are independent on the barrier derivativetherefore are calculable. In the calculation of the constalisted in Table II for methanol, we used the molecular modand the geometric parameters given by Lees and Bak12

The vibrational potential function and force constantstaken from those given by Margottin-Maclou.13 The calcu-lated constants are given in Table II.

In order to compare the calculated constants with thobtained from a fitting to observed data, the Hamiltonian wreduced, by a torsion-rotational contact transformatH5exp(iS)H exp(2iS) with a function S, into a reduced

TABLE I. The effective torsion-rotation terms up to the fourth order.

Order Coefficient Operatora Order Coefficient Operator

2 B1 Pa2 4 T14 $Pc

2 ,Pa%Pg

B2 Pb2 T15 $Pc

2 ,Pb%Pg

B3 Pc2 T16

$Pa ,Pb%(12cos 3g)

B4 $Pa ,Pb% T17 $Pa ,Pc%sin 3gB5 PaPg T18 $Pb ,Pc%sin 3gB6 PbPg T19 Pa

2(12cos 3g)B7 Pg

2 T20 Pb2(12cos 3g)

B8 (12cos 3g) T21 Pc2(12cos 3g)

4 T1 Pa4 T22 Pa

2Pg2

T2 Pb4 T23 Pb

2Pg2

T3 Pc4 T24 Pc

2Pg2

T4 $Pa3 ,Pb% T25 $Pa ,Pb%Pg

2

T5 $Pb3 ,Pa% T26 Pa$Pg,12cos 3g%

T6 $Pa2 ,Pb

2% T27 Pb$Pg,12cos 3g%T7 $Pa

2 ,Pc2% T28 Pc$Pg ,sin 3g%

T8 $Pb2 ,Pc

2% T29 PaPg3

T9PaPc

2Pb

1PbPc2Pa

T30 PbPg3

T10 Pa3Pg T31 $Pg

2,12cos 3g%T11 Pb

3Pg T32 Pg4

T12 $Pa2 ,Pb%Pg T33 (12cos 6g)

T13 $Pb2 ,Pa%Pg

a$A,B%5AB1BA.

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i

2388 J. Chem. Phys., Vol. 111, No. 6, 8 August 1999 Duan et al.

form of H. The general expression forS is given by Duanet al.1,2 and terms through the third order are presentedTable III.

The coefficientsmabe are written in matrix notation as

TABLE II. Coefficients for the effective Hamiltonian~MHz!.

B15Ae134tbcbc2

12tabab2

12tacac

671095.929

B25Be134tacac2

12tbcbc2

12tabab

24662.605

B35Ce134tabab2

12tbcbc2

12tacac

23770.313

B45Dabe 1

14(2tabcc2taaab2tabbb2taccb) 23029.43706

B5522Ae212(taccg1tabbg) 21342189.99

B6522Dabe 2

12(tabag1tbccg) 6058.783

B75Fe 828632.091

B8512V3

a8

T1514taaaa

2181.4367

T2514tbbbb

20.049189

T3514tcccc

20.045307

T4512taaab

215.5283

T5512tabbb

20.058006

T6514(2tabab1taabb) 22.087622

T7514(tacac1taacc) 20.37244

T8514(2tbcbc1tbbcc) 20.04690

T9512(2taccb1tabcc) 20.084366

T105taaag 725.516T115tbbbg 0.147152

T12512(2tabag1taabg) 47.4969

T13512(2tabbg1tbbag) 4.27777

T14512(2taccg1tccag) 0.74533

T15512(2tbccg1tccbg) 0.08065

T1652Qab2;2

T1752Vac2;2

T1852Vbc2;2

T1952Qaa2;2

T2052Qbb2;2

T2152Qcc2;2

T22512(2tagag1taagg) 21088.108

T23512(2tbgbg1tbbgg) 25.21496

T24512(2tcgcg1tccgg) 21.42653

T25512(2tagbg1tabgg) 247.8833

T265Qag2;2

T275Qbg2;2

T285Vcg2;2

T295taggg 725.406T305tbggg 31.9070

T31512Qgg

2;2

T32514tgggg

2184.3499

T33512V6

a8

TABLE III. Terms for contact transformation.

Order Coefficient Operator Order Coefficient Operator

1 Fc1 Pc 3 Fbcg

3 $Pb ,Pc%Pg

3 Fccc3 Pc

3 Fcgg3 PcPg

2

Faac3 $Pa

2 ,Pc% Qc1;2 Pc(12cos 3g)

Fbbc3 $Pb

2 ,Pc% Va1;2 Pa sin 3g

Facb3 PaPcPb1PbPcPa Vb

1;2 Pb sin 3g

Facg3 $Pa ,Pc%Pg Vg

1;2 $Pg ,sin 3g)

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n

me52S Ae Dabe 0 2Ae

Dabe Be 0 2Dab

e

0 0 Ce 0

2Ae 2Dabe 0 Fe

D , ~3.1!

TABLE IV. Theoretical formulas for the constants in transformedH.

B15B1813B58Va1;224B48(Fccc

3 2Faac3 2Fbbc

3 )24(B282B38)Fabc3

1~3T42T522T9!Fc1

B25B2814B48(Fccc3 2Faac

3 2Fbbc3 )14(B182B38)Fabc

3

2~3T52T422T9!Fc1

B35B3822B48(Fbbc3 1Faac

3 )24(B182B28)Fabc3 23(T42T5)Fc

1

B45B48132B58Vb

1;22(B182B28)(Fccc3 2Faac

3 2Fbbc3 )

1(T12T222T622T722T8)Fc1

B55B5816(B58Vg1;21B78Va

1;2)1(B382B28)Fbcg3 1B48Facg

3

1~T122T11/2!Fc1

B75B78(1112Vg1;2)

B85B88127B78Vg1;2

T15T114B48Faac3 12T4Fc

1

T25T224B48Fbbc3 22T5Fc

1

T35T3

T45T412(B282B18)Faac3 12B48Fabc

3 12(T62T1)Fc1

T55T512(B282B18)Fbbc3 22B48Fabc

3 12(T22T6)Fc1

T65T612B48(Fbbc3 2Faac

3 )12(B282B18)Fabc3 13(T52T4)Fc

3

T75T71B48(3Fccc3 24Faac

3 )12(B182B38)Fabc3 1T9Fc

1

T85T82B48(3Fccc3 24Fbbc

3 )12(B382B28)Fabc3 2T9Fc

1

T95T913(B282B18)Fccc3 14(B382B28)Faac

3 14(B182B38)Fbbc3

12~T82T7!Fc1

T105T1014B48Facg3 12T12Fc

1

T115T1124B48Fbcg3 22B58Fbbc

3 22T13Fc1

T125T1212B48Fbcg3 12(B282B18)Facg

3 2B58Faac3 1(2T1323T10/2)Fc

1

T135T1322B48Facg3 12(B282B18)Fbcg

3 2B58Fabc3 2(2T1223T11/2)Fc

1

T145T1422B48Facg3 12(B182B38)Fbcg

3 1B58Fabc3 1T15Fc

1

T155T1512B48Fbcg3 12(B382B28)Facg

3 1B58(2Fbbc3 23Fccc

3 /2)2T14Fc1

T165T161(B282B18)Qc1;22

32B58Vb

1;21(T202T19)Fc1

T175T171(B182B38)Vb1;213B58Qc

1;2/21B48Va1;223B88Facg

3 1T18Fc1

T185T181(B382B28)Va1;21B48Vb

1;223B88Fbcg3 2T17Fc

1

T195T1912B48Qc1;223B58Va

1;212T16Fc1

T205T2022B48Qc1;222T16Fc

1

T215T21

T225T2212B48Fcgg3 12T25Fc

1

T235T2322B48Fcgg3 22B58Fbcg

3 22T25Fc1

T245T2412B58Fbcg3

T255T251(B282B18)Fcgg3 2B58Facg

3 1(T232T22)Fc1

T265T2623B58Vg1;223B78Va

1;21T27Fc1

T275T2723B78Vb1;22B58Qc

1;2/22T26Fc1

T285T2813B78Qc1;223B88Fcgg

3 1B58Vb1;2/2

T295T291T30Fc1

T305T302B58Fcgg3 2T29Fc

1

T315T3116B78Vg1;2

T325T32

T335T3323B88Vg1;2

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n

l m

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, toian,

theco-

2389J. Chem. Phys., Vol. 111, No. 6, 8 August 1999 Calculation of molecular parameters

where the matrix elements ofme are defined as those giveby Lin and Swalen.10

The first contact transformation withS15Fc1Pc is used

to eliminate the termPbPg so that the resultant torsionaHamiltonian has only diagonal (DK50) matrix elements.After carrying out the transformation withS1 , the secondorder Hamiltonian becomes

H285B18Pa21B28Pb

21B38Pc21B48$Pa ,Pb%1B58PaPg

1B78Pg21B88~12cos 3g! ~3.2!

with

B185B112B4Fc1 , ~3.3!

B285B222B4Fc1 , ~3.4!

B385B3 , ~3.5!

B485B41~B22B1!Fc1 , ~3.6!

B585B51B6Fc1 , ~3.7!

B785B7 , ~3.8!

Downloaded 17 Apr 2013 to 198.91.37.2. This article is copyrighted as indicated in the abstract. R

B885B8 , ~3.9!

whereFc15(B6 /B5)'24.531023 for methanol.

The reduction of the fourth order terms originates frothe transformation withS3 . The formulas for the coefficientsidentified by a tilde in the transformed Hamiltonian up to tfourth order are given in Table IV, where the quartic contbutions coming from the higher-order term reduction areglected. Eleven constraints used to evaluate the elevenstants of S3 in Table III can be used to eliminate thcorrelation caused by the transformationS3 . As discussed inRef. 15, the reduction scheme is not unique. Howeverguarantee the convergence of the transformed Hamiltonthe absolute values of the coefficients ofS3 should be se-lected to be as small as possible. In the present work,following eleven equations are used to define the elevenefficients ofS3 ,

T21T322T850, ~3.10!

T41r2T2550, ~3.11!

TABLE V. Comparison between calculated results and fitting ones.

Operatora Constant Calculated Herbstet al.b fit Present fit ~O–C!/Oc

Pa2 A 127 617.363 127 630.7538~1583! 127 588.009~7262! 20.02

Pb2 B 24 648.498 24 691.1785~2751! 24 687.383 90~1588! 0.16

Pc2 C 23 770.104 23 765.3680~2738! 23 762.294 34~6866! 20.03

$Pa ,Pb% Dab 2111.152 277.6655~4671! 2118.161~1449! 5.92Pg

2 F(cm21) 27.640 27.6335~3533! 27.632 718~1076! 20.0312(12cos 3g) V3(cm21) 373.083 950 7~9174! 373.134 06~3488!12(12cos 6g) V6(cm21) 20.80~fixed! 21.046 19~3836!

r~unitless! 0.8099 0.809 745 111 7~1452! 0.812 893 8~1237! 0.37P4 DJ3102 4.7570 4.9768~284! 5.023 865~5749! 5.31P2Pa

2 DJK3101 2.939 2.860 82~3103! 2.763 296~7675! 26.36Pa

4 DK 1.181 1.058 349~231 60! 1.282 1591~58625! 7.882P2(Pb

22Pc2) dJ3103 1.127 1.668 94~6986! 32.4

$Pa2 ,Pb

22Pc2% dK 0.017 20.750 71~119 32!

P2Pg2 GV 23.551 23.535 184~248 2! 23.555 173~2227! 0.11

P2PaPg LV3102 3.697 4.010 84~307 62! 9.334 5~1297! 60.4P2(12cos 3g) FV 271.431 05~252 1! 271.576 494~8926!Pa

3Pg k1 1.255 23.88~fixed! 24.862 0~1473! 126Pa

2Pg2 k2 247.483 280.0~fixed! 274.280~1174! 35.6

PaPg3 k3 2128.041 2132.0~fixed! 293.406~2083! 237.0

Pg4 k4 2184.350 2249.0~fixed! 2184.35~fixed!

Pa2(12cos 3g) k5 161.0~fixed! 224.84~1145!

Pa$Pg,12cos 3g% k6 884.0~fixed! 22127.2~1014!$Pa ,Pc%sin 3g k78 2238.72~1793!

$Pg2 ,Pb

22Pc2% c1 21.649 21.227 22~9944! 21.580 310~8434! 24.37

(Pb22Pc

2)(12cos 3g) c2 1.139 2~2378!$Pg

2 ,$Pa ,Pb%% Dab 0.101 71 26.860 2~8270! 101$PaPg ,$Pa ,Pb%% dab 20.163 06 20.8276~4469! 80.3$Pa ,Pb%(12cos 3g) dab 293.796~2947!P4Pa

2 HJK3104 22.816 4~4998!P2Pa

4 HKJ3104 9.425 00~1.66993!Pa

6 HK3102 21.071 236~92898!$Pa

4 ,Pb22Pc

2% hK 21.071$Pg

4 ,Pb22Pc

2% c33102 21.469 5~6582!P2Pg

4 M v3103 2.1925~1171!

aWherePg is defined asPg1rPa .bReference 14.cO represent the results in column 5, while C represent the results in column 3. The ratios are multiplied by 100.

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2390 J. Chem. Phys., Vol. 111, No. 6, 8 August 1999 Duan et al.

T52rT11/250, ~3.12!

T92rT15/250, ~3.13!

T1250, ~3.14!

T1450, ~3.15!

T1850, ~3.16!

T2750, ~3.17!

T2850, ~3.18!

T3050, ~3.19!

T3150. ~3.20!

The resulting Hamiltonian is given in Table V. The relatiobetween the empirical constants in the reduced Hamiltonand the theoretical constants are collected in Table VI. Tcomparison between calculated constants and empirical fiones is shown in Table V. The third column of Table V givthe calculated results from the present work. The fourth cumn gives the experimental results fitted by Herbstet al.,14

where the root-mean-square~RMS! derivation is 1.2 MHzusing 28 constants. The fifth column is our fitting resuwhere the data set is the same as that used by Herbstet al.,14

the values of fixed constants are taken from the presentculated ones, and the RMS derivation is 0.36 MHz usingconstants. The sixth column shows the difference in percage between the fifth column and the third column. Fr

TABLE VI. The relations between empirical and theoretical constants.

A5B12r2B7

B5B2

C5B3

Dab5B4112r2T25

F5B712V35B812V65T33

r5B5 /(2B7)

DJ5238(T21T3)2

14T8

DJK534(T21T3)2(T71T6)1

12T81rT132(r2/2 )(T231T24)

DK52T11T71T62T81r3T292r4T32

2r(T132T10)1 (r2/2) (T231T2422T22)

dJ514(T32T2)

dK514(T22T3)1

12(T72T6)1 (r/2)T131 (r2/4) (T242T23)

GV512(T231T24)

LV5T132r(T231T24)

FV512(T201T21)

k15T102T131r(T231T2422T22)13r2T29214r3T32

k25T22212(T231T24)23rT2916r2T32

k35T2924rT32

k45T32

k55T19212(T201T21)

k65T26

k785T31

c1514(T232T24)

c2512(T202T21)

Dab512T25

dab5T16

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Table V it is found that the second-order constants,A, B, C,Dab , F, andr, are in excellent agreement with the expemental constants and the fourth-order constants,DJ , DJK ,DK , dJ , Gv , c1 , k2 , k3 , and k4 , are generally in goodagreement with experimental results. However, there ilarge difference between the calculated results and our lafitting results for the constantsLV , k1 , Dab , anddab . Thediscrepancy might be due to the high degree of correlaamong the large number of parameters needed in the ansis, to the reduction scheme used, or to the approximathat the quartic contribution from the reduction of higheorder terms has been neglected. In addition, Eqs.~3.16!–~3.19! are not used in the present calculation and thencontribution from the reduction involving distortion of barier is dropped due to the lack of barrier derivative informtion. It can be shown that the calculated constants are setive to the reduction scheme. To reduce the effect causereduction, the absolute values of coefficients of the transmation operatorS must be taken as small as possible amust satisfy thesmallnessrequirement onS.15 This conclu-sion agrees with the discussion about correlation problemdescribed in Ref. 15. In the present reduction scheme, uEqs.~3.10!–~3.20!, the seven coefficients in the transformtion operator are determined as shown in Table VII.checking the order of magnitude of these calculated coecients we can judge whether the selected reduction schemappropriate or not.15 The desirable reduced Hamiltoniashould be able to yield a good fit of the experimental dwithin the measurement accuracy using a minimum numof terms.

It is interesting to compare the fit obtained here wHerbst et al.’s fit to the same data set of methanol. In thpresent fit, the constant,k4 , is fixed to the calculated valuesUsing 27 terms of Hamiltonian, a total of 470 transitionsthe A and E symmetry levels involving torsional quantumnumberv t<2 and rotational quantum numberJ<9 are glo-bally fitted to a RMS derivation of 0.36 MHz. In the fittinby Herbstet al.14 the RMS derivation is 1.2 MHz with 28terms, where some constants are fixed to values taken fprevious fitting. The comparison between the two fittinshows that the calculated constants are helpful for analysobserved transitions. Experience with fitting indicates tsome of lower-order torsion-rotational parameters arestable and must be fixed when theK-changing information inexcited torsional states is not available. Ideally, the consshould be fixed to its exact value as far as possible if it habe fixed, where the calculated value may be a better cadate. Fixing the constant to the calculated value can impr

TABLE VII. Values of coefficients in contact transformation.

Constant Value

Faac3 3105 21.3

Fbbc3 3108 24.9

Facb3 3107 1.4

Fccc3 3108 28.5

Facg3 3105 2.7

Fbcg3 3107 24.4

Fcgg3 3105 22.6

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2391J. Chem. Phys., Vol. 111, No. 6, 8 August 1999 Calculation of molecular parameters

the physical significance of other constants obtained byfit. Finally using the procedure presented in this work, infmation on the force field and related parameters can betained from centrifugal distortion constants determined bfit to observed transitions.

ACKNOWLEDGMENTS

One of the authors~Y.B.D.! wishes to acknowledge thpostdoctoral fellowship by the Japan Society for the Promtion of Science~JSPS! and the support from Office of Research of the Ohio State University. The authors are gratto Dr. A. B. McCoy for her helpful discussion. This researwas supported in part by a Grants-in-Aid for scientific rsearch from the Ministry of Education, Science and Cultof Japan~Nos. 09490014!. The computation was performeon the IBM RS 6000 Scalable Powerparallel systems offormation Center at Toyama University.

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