Vibration-Internal Rotation-Overall Rotation Interactions in CH3OH

JOURNAL OF MOLECULAR SPECTROSCOPY ARTICLE NO.

188, 190–199 (1998)

MS987525

Vibration-Internal Rotation-Overall Rotation Interactions in CH3OH Development and Application of the Separation Transformations to the Zeroth Order Kinetic Energy C. Richard Quade Department of Physics, Texas Tech University, Lubbock, Texas 79409 Received August 5, 1997; in revised form December 30, 1997

The zeroth order kinetic energy is developed for the vibrating-internally rotating-rotating CH3OH molecule using the general theory of Guan and Quade for large amplitude internal motion–vibration–rotation interactions in molecules. The R and T transformations are applied, respectively, to separate internal rotation from the other vibrations and overall rotation from the other vibrations in zeroth order. All zeroth order kinetic energy coefficients are calculated from the geometry and atomic masses of the CH3OH molecule. The physical significance of the two transformations is discussed in detail. This paper reports the results of the first segment of the many segments necessary in the calculations for full solution of the problem. q 1998 Academic Press INTRODUCTION

Microwave spectroscopy has proven to be a powerful method for determining molecular properties that are related to the overall rotation of molecules. For example, from the empirical rotational constants the moments of inertia may be calculated, which in turn depend upon the molecular structure—bond lengths and bond angles. In some cases these have been determined up to one part in 10 4 . The rotational spectra are sensitive to centrifugal distortion effects. Further, satellite rotational lines are observed for the molecules in excited vibrational states. Of interest in the present work is the property that many molecules exhibit internal rotation about a chemical bond. This torsional motion has lower frequency than other vibrations, and the hindering potential energy has symmetry that leads to tunneling between equivalent molecular conformations. Coupling of the angular momentum of internal rotation with that of overall rotation leads to a splitting in the lines in the rotational spectra that is sensitive to the potential energy barrier hindering the internal rotation. Sometimes torsional–rotational transitions between the tunneling states fall within the microwave region and are detectable with an appropriate change in dipole moment. The general theory of internal rotation–rotation interactions for molecules with a symmetric internal rotor along with applications was reviewed many years ago by Lin and Swalen (1), and since that time it has been applied in its various forms in the determination of barriers to internal rotation for many molecules. Quade and Lin (2a) and Liu and Quade (2b), as well as others, have developed theories for internal rotation when the internal rotor is not a symmetric top. In general these

models of a rigid rotor with a rigid internal rotor have met with a high degree of success, for both symmetric and asymmetric internal rotors, in determining the intramolecular potential energy hindering the internal rotation with the analyses for symmetric internal rotors actually giving a much better spectroscopic fit than those for asymmetric internal rotors. However, in spite of its successes, this simple model has had its shortcomings. First, it does not account for the shift in the satellite lines with excited torsional state. Very simply, the model shows that the splittings become larger with excited state but the center of gravity should remain fixed. Actually the center of gravity shifts by an amount as large or even larger than the splittings. Second, even in the ground states for molecules with light internal rotors, such as –OH (3a, b) or –SH (3c, d) groups, some of the splittings cannot be accounted for by this simple model. Third, for the asymmetric internal rotors, the fact that the vibrational frequencies depend upon the angle of internal rotation contributes to the effective potential energy for the internal rotation. In other words, many effects observable in the spectra of molecules depend upon the interaction of both internal rotation and overall rotation with the other vibrations. These are often referred to as effects due to nonrigidity. The earliest models for –CH3 type molecules incorporating nonrigidity into the effective internal rotation–overall rotation Hamiltonian were developed by Kivelson ( 4). In these works a semi-quantitative theory was developed that contained coupling terms that could be determined empirically that depend upon the internal angular momentum and the threefold symmetry of the internal rotor. From the shifts and splittings of satellite lines for methyl silane, Kivelson was able to determine the barrier to internal rotation.

190 0022-2852/98 $25.00 Copyright q 1998 by Academic Press All rights of reproduction in any form reserved.

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Two alternative approaches were used to study the A–E splittings of the J Å 0–1 transitions for many isotopic species of symmetric methyl alcohol (3a). On the one hand, Hecht and Dennison (5) found that there was a large –OH rocking Coriolis interaction and considered the mixing of this vibrational mode with internal rotation. On the other hand, Swan and Strandberg (6) considered a vibration–internal rotation–rotation model for methyl alcohol consisting of inplane vibrations with two rigid groups vibrating with respect to each other. In an empirical fit of the data, Hecht and Dennision did well with a minimum of parameters except for one isotopic species. The fit by Swan and Strandberg was only qualitatively correct. In the early 1960’s, Kirtman (7) derived a formal theory for vibration–internal rotation–rotation interactions in molecules with symmetric internal rotors from what may be called ‘‘first principles.’’ Using perturbation theory he obtained spectroscopic coefficients very similar to those developed by Kivelson. The parametric results of the Kirtman theory have been used by many investigators in the analysis of microwave spectral data. Subsequent to the Kirtman work, Hougen (8) and then Bunker (9) developed some formal transformations for vibration–internal rotation–rotation interactions in molecules with symmetric internal rotors. For the asymmetric internal rotors, Quade (10) developed a theoretical model with very limited use in spectroscopic analyses. Although it is clear that interaction with other vibrations is larger for the asymmetric internal rotors than for the symmetric ones, there are more terms and these terms have lower symmetry. Recently, there has been some important and exciting analysis and prediction of the microwave to submillimeter wave spectra of CH3OH using the models that include the Kirtman terms for nonrigidity. This work has allowed for an extremely accurate determination of the spectroscopic coefficients of CH3OH and some other methyl-symmetric isotopic species. Herbst et al. performed the pioneering modeling of this problem (11a). Recently it has been improved by Takagi et al. (11b) with significant applications by Lees et al. (11c) and Hougen et al. (11d). This research is significant to the author’s work but at the latter stages where empirical spectroscopic parameters are utilized with the theoretically calculated parameters to extract information on the Coriolis coupling and the intramolecular forces. Recently Hougen (12) has renewed his study of vibration–internal rotation–rotation interactions in molecules with the intent of applying the theory to the analysis of the spectra of CH3COH. His modeling of the problem is different than that of Kirtman, but there is the expectation of getting spectroscopic constants from first principles. In the mid 1970’s Quade developed theories for, first, vibration–rotation interactions (13) and, second, large amplitude internal motion–vibration interactions (14) using curvilinear internal coordinates for the vibrational degrees

of freedom. In the first case, rather than use the Eckart conditions for the definition of the molecular axis system to reduce the zeroth order Coriolis coupling, a modified Nielsen transformation was developed. In the second case, rather than using the Sayvetz condition to separate internal rotation from vibration in zeroth order in the kinetic energy, a transformation was developed that used (G 01 ) 0 , including both the large amplitude coordinate and the other vibrational coordinates. In the mid 1980’s Guan and Quade (15) combined the two approaches to develop a general theory for large amplitude internal motion–vibration–rotation interactions, which was subsequently applied to the water molecule (16). In the present work this theory of Guan and Quade is applied to the internal rotation–vibration–rotation interaction problem for CH3OH. The idea is to develop the parametric coefficients, essentially those of Kivelson or Kirtman, from first principles in terms of molecular geometry and vibrational constants with the expectation of determining additional intramolecular forces from the empirical values. This application is a large task and will be done in stages over the next few years. This paper, the current work, derives the zeroth order kinetic energy and the necessary transformation coefficients, which are then applied to transform the kinetic energy in zeroth order to the vibration-uncoupled state. The second segment of this work will derive the internal coordinate dependence of the moments of inertia, of the vibrational coefficients, of the Coriolis coupling coefficients, and of the torsional coefficients and then transform them to the zeroth order separated basis. The third segment will be the solution of the vibrational problem to obtain the effective internal rotation–rotation Hamiltonian for reduction of the experimental data. Then, finally, relaxation of the tilt of the internal rotation axis as the internal rotation coordinate varies will be considered. In comparison to the Guan and Quade approach to that of Kirtman some things should be noted. First, Guan and Quade define internal rotation to be just another curvilinear internal motion in the molecule. This means, as is customarily assumed in the one-dimensional models, that the internal rotation is defined with respect to a chemical bond. In the case of methyl alcohol it is well established that the internal rotation axis is tilted with respect to the CO bond. What we do is fix this tilt angle so that it does not change during the internal rotation. (As mentioned above relaxation of this constraint will be considered in a subsequent work.) On the other hand, Kirtman defines the internal rotation axis so that it passes through the center of mass of the vibrationally distorted top. Further, his Sayvetz condition is a requirement that the vibrations of the top do not contribute to the internal angular momentum of the molecule. A further constraint condition must be introduced to fix the direction of the axis of internal rotation that is not necessarily related to a chemical bond. Second, Guan and Quade use the modified Nielsen transformation to separate vibration from rota-

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C. RICHARD QUADE

FIG. 1. Illustration of the internal coordinates for CH3OH.

tion in zeroth order while Kirtman uses the Eckart conditions. Quade (17) has shown that these two approaches to the separation are equivalent. However, the Nielsen transformation is more suited to the use of curvilinear internal coordinates for the vibrational degrees of freedom. In both cases a Van Vleck perturbation transformation is used to isolate the effective internal rotation–rotation Hamiltonian that does make use of the property that the torsional frequency is much lower than those of the other vibrations. In the present work, first the atomic coordinates are defined from which the zeroth order kinetic energy is derived for an arbitrary molecular coordinate system. Next the R transformation is developed and applied to separate internal rotation from the other vibrations in zeroth order. Then the other vibrations are separated from rotation in zeroth order by the T transformation. The result is the zeroth order kinetic energy where vibrations are separated from both internal rotation and overall rotation, while internal rotation and overall rotation remain strongly coupled to each other.

OH

bond stretch,

g

COH angle bend,

CO

bond stretch,

,1 or S1

CH1 bond stretch,

,2 or S2

CH2 bond stretch,

,3 or S3

CH3 bond stretch,

j1

zCH1 angle bend,

j2

zCH2 angle bend,

j3

zCH3 angle bend,

b1

H2CH3 angle bend projected on a plane ⊥ to z,

b2

H1CH3 angle bend projected on a plane ⊥ to z,

b3

H1CH2 angle bend projected on a plane ⊥ to z,

t

the dihedral angle of H1C along the z axis or internal rotation axis to the intersection of the z axis with the OH bond and then on to H. [1]

There are 13 coordinates defined above; however, the b’s satisfy the relationship b1 / b2 / b3 Å 2p.

Often other angles are used for the internal coordinates of the methyl group. For example, these might be

ATOMIC COORDINATES

The internal coordinates specifying the atomic positions for methyl alcohol, CH3OH, are illustrated in Fig. 1. These coordinates are curvilinear and internal. The tilt of the axis of internal rotation with respect to the C–O bond is denoted by z . In this work the tilt is assumed to be fixed throughout the internal rotation. The tilt cannot be considered an independent variable without increasing the number of independent coordinates from 3N 0 6 to 3N 0 5 for the internal degrees of freedom, vibration plus internal rotation. However, this tilt might be expected to show a small internal rotation, that is t, dependence, and it is hoped to investigate this point in a subsequent work. The internal coordinates used in the development of the kinetic energy for the vibrating–rotating–internal rotating molecule are as follows with the /z axis along the internal rotation axis and through the methyl group:

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u1

H2CH3 bending angle,

u2

H1CH3 bending angle,

u3

H1CH2 bending angle,

[3a]

and f1

H1C( 0z) bending angle,

f2

H2C( 0z) bending angle,

f3

H3C( 0z) bending angle.

[3b]

Constraint relations between the ui , ji , and bi may be expressed as cos j1 cos j3 Å cos u2 0 sin j1 sin j3 cos b2 cos j1 cos j2 Å cos u3 0 sin j1 sin j2 cos b3

[4]

cos j2 cos j3 Å cos u1 0 sin j2 sin j3 cos( b2 / b3 ), where the additional constraint relation, Eq. [2], has been

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CAL OF VIB-INTROT-ROT INTERACTIONS IN CH3OH

x1 Å ,1 sin j1 sin t,

used. Again it should be noted that the tilt z remains fixed throughout the internal motions of the molecule. The kinetic energy is given by

y1 Å 0,1 sin j1 cos t, z1 Å ,1 cos j1 ,

2T Å ∑ mi (dRi /dt) 2space ,

[5]

x2 Å ,2 (sin j2 sin b3 cos t / sin j2 cos b3 sin t ), y2 Å ,2 (sin j2 sin b3 sin t 0 sin j2 cos b3 cos t ),

where

z2 Å ,2 cos j2 , (dRi /dt)space Å v 1 Ri / Rg i

[6]

x3 Å ,3 ( 0sin j3 sin b2 cos t / sin j3 cos b2 sin t ), y3 Å 0,3 (sin j3 sin b2 sin t / sin j3 cos b2 cos t ),

with Rg i Å (dRi /dt)body .

Since the Ri are the positions of the atoms with respect to the center of mass of the molecule, there is no translation– rotation nor translation–internal motion coupling so the velocity of the center of mass, Rg CM , is not included in Eq. [6]. With Eq. [6], the kinetic energy becomes

[11]

The vector r is obtained from Mrx Å mH (x1 / x2 / x3 ), Mry Å mH (y1 / y2 / y3 ) / mO (CO)y / mH (OH)y , [12]

and

2T Å

z3 Å ,3 cos j3 .

[7]

∑ mi ( v 1 Ri ) 2 / ∑ mi Rg 2i / 2vr∑ mi Ri 1 Rg i . [8]

All vectors Ri are with respect to the center of mass of the molecule with the Z axis parallel to the axis of internal rotation, the z axis. For purposes of calculation, it is easiest to specify all coordinates with respect to the carbon atom, C, and then transform to the center of mass system. Then ri is the position of the ith atom with respect to the C atom with the z axis along the internal rotation axis and then r is a vector from the C atom to the center of mass of the molecule, CM, with the result Ri Å ri 0 r .

[9]

In terms of ri and r , the kinetic energy becomes 2T Å ∑ mi ( v 1 ri ) 2 0 M( v 1 r) 2 / ∑ mi rh 2i 0 Mrh 2 [10] / 2vr∑ mi ri 1 rh i 0 2Mvrr 1 rh , where M is the total mass of the molecule. Relative to the carbon atom, C, the atomic coordinates are

Mrz Å mH (z1 / z2 / z3 ) / mO (CO)z / mH (OH)z . To obtain the kinetic energy coefficients, Eqs. [11] and [12] are differentiated with respect to the time and substituted into Eq. [10]. See Appendix I. ZEROTH ORDER KINETIC ENERGY MATRIX

The kinetic energy coefficients in Eq. [10] are composed of three parts. The first part is the pure rotational kinetic energy given in terms of the vi and vi vj and the coefficients and products of inertia with respect to the center of mass for the X, Y, Z molecular coordinate system. The second part is the vibrational and torsional kinetic energy. The third part contains the Coriolis coupling between the overall rotation and internal motions. As calculated in Eq. [10], this interaction is not small—that is, it has contributions in zeroth order that will be removed in subsequent sections. For the calculation of the coefficients in the kinetic energy, it is necessary to specify a molecular structure for the methyl alcohol molecule. Over the years we have found the preferable structure to be that determined by Venkateswarlu and Gordy (18). In terms of the variables in Eq. [11], this structure gives CH Å 1.0961A,

CO Å 1.4271A,

(OH)y Å ,OH sin( g / z ) 0 ,CO sin z ,

OH Å 0.9558A,

g Å 108.8757,

(OH)z Å ,OH cos( g / z ) 0 ,CO cos z ,

j Å 70.0937,

( u Å 109.0327 ),

(CO)y Å 0,CO sin z ,

b Å 1207,

(CO)z Å 0,CO cos z ,

( g / z Å 112.2007 ), Copyright q 1998 by Academic Press

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z Å 3.3257,

[13]

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C. RICHARD QUADE

[ ∑ mi rh 2i 0 Mrh 2 ] 0

with atomic masses mH Å 1.007825 au,

Å 3.2114th 2 / 0.9658(Sg 2x / Sg 2y ) / 0.9968Sg 2z

mC Å 12.00rrrau,

and

[14]

/ 1.2042( jg 2x / jg 2y ) / 1.1098jg 2z / 0.3400( bg 2x / bg 2y ) / 2(1.5139) th ( 0s bg y 0 cbg x ) / 0.7136(s 2bg 2y / c 2bg 2x

mO Å 15.994915 au. It is helpful to simplify the kinetic energy for the vibrations by two transformations of the coordinates for the methyl group. The first transformation is to symmetry coordinates with the result q

/ 2sc bg y bg x ) / 7.9760Sg 2CO / 0.9761Sg 2OH / 0.8917gh 2 / 2(0.1529)Sg OH Sg CO / 2(0.4276) gh Sg CO 0 2(0.0167) 1 (Sg y jg y / Sg xjg x ) 0 2( 00.0334) 1 Sg zjg z 0 2(0.0266) 1 (Sy bg y / Sxbg x ) 0 2(0.0106)( jg y bg y / jg xbg x )

S\ Å 1/ 6(2S1 0 S2 0 S3 ),

0 2( 00.3150) Sg z Sg CO 0 2( 00.0071) Sg z Sg OH

S⊥ Å 1/ 2(S2 0 S3 ),

0 2( 00.0166)Sg zgh 0 2(0.9535) jg z SCO 0 2(0.0214)

q

q

Sz Å 1/ 3(S1 / S2 / S3 ),

[15a]

q

1 jg z SOH 0 2(0.0501) jg zgh 0 2( 00.0357)Sg y Sg CO

Sj\ Å 1/ 6(2Sj1 0 Sj2 0 Sj3 ),

0 2(0.0338)Sg y Sg OH 0 2( 00.0132) Sg y gh

Sj⊥ Å 1/ 2(Sj2 0 Sj3 ),

0 2( 00.0142) jg y Sg CO 0 2(0.0134) jg y Sg OH

q

q

Sjz Å 1/ 3(Sj1 / Sj2 / Sj3 ),

[15b]

q

0 2( 00.0052) jg y gh 0 2( 00.0277) bg y Sg CO 0 2(0.0214) bg y Sg OH 0 2( 00.0083) bg y gh .

Sb\ Å 3/2(Sb3 / Sb2 ), q

Sb⊥ Å 1/ 2(Sb3 0 Sb2 ).

[15c]

However, even with the basis of Eqs. [15], a sin t and cos t dependence remains in the vibrational kinetic energy terms for the methyl group. These may be removed by a t-dependent transformation that essentially leaves the vibrational displacements behind as the methyl group internally rotates. This second transformation is

Sx Å sin tS\ / cos tS⊥ ;

[16a]

Sjy Å 0cos tSj\ / sin tSj⊥ , Sjx Å sin tSj\ / cos tSj⊥ ;

[16b]

In Eq. [17], the seventh term, s stands for sin t and c cos t. Also, the units for the coefficients of th 2 , gh 2 , bg 2i , th bg i , jg i bg i , gh ji , and gh bg i are auA 2 ; for Sg 2i , Sg i Sg j are and for Sg i gh , Sg i jg i , Sg i bg i are auA. The terms from the Coriolis coupling in zeroth order

[16c]

In Eqs. [15] and [16], the S represents the small changes in the internal coordinates but can also be taken as a transformation of velocities when the th terms from Eqs. [16] are not included. The th terms introduce an additional Si dependence into the torsional coefficient that will be included with other higher terms further along in this work. From the molecular structure, after the transformations, we obtain the following for the zeroth order vibrational and torsional portions of the kinetic energy:

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/ ( 01.8204) jg y / ( 00.8121) bg y / (0.6713)Sg OH / ( 00.4623)Sg CO / ( 01.1132) gh

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[18]

Å 0.8501Sg x / 1.8204jg x / 0.8121bg x

[ ∑ mi ri xrh i 0 Mrxrh ] 0z

Å 1.5139( 0sbg y 0 c bg x ) / ( 00.0187)Sg x / 3.2114th / ( 00.0074) jg x / ( 00.0118) bg x .

In Eq. [18], the units for the coefficients of Sg i are auA and for jg i , bg i , and gh are auA 2 . The zeroth order coefficients and products of inertia are

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for jg 2i , au;

Å 0.0096Sg z / ( 00.0290) jg z / ( 00.8501)Sg y

[ ∑ mi ri xrh i 0 Mrxrh ] 0y

Sby Å cos tSb\ 0 sin t Sb⊥ , Sbx Å 0sin tSb\ 0 cos tSb⊥ .

/ 2(1.5139) th ( 0sbg y cbg x ).

[ ∑ mi ri xrh i 0 Mrxrh ] 0x

Sy Å 0 cos tS\ / sin tS⊥ ,

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I0xx Å 21.273 auA 2 I0yy Å 20.523 auA 2 I0zz Å 3.9612 auA

[19]

t* Å t /

2

I0yz Å 0.0648 auA 2 . The product of inertia I0yz is sufficiently small that the principal moments of inertia, Iyy and Izz , are the same as I0yy and I0zz to the accuracy of Eq. [19]. Care must be taken when internal and external motions are frozen to reduce the effective number of degrees of freedom in the kinetic energy. For example, the correct rotational Hamiltonian, H 0R , is obtained by setting the Sg i , jg i , bg i , gh , and th equal to zero. However, the vibrational motions do contain angular momentum in zeroth order and therefore the Coriolis coupling contributes in zeroth order to the vibrational G 01 matrix elements when the rotational motion is frozen. A similar argument applies to the internal rotation– vibration interactions. These interactions of vibration with rotation and internal rotation are removed by the T and R transformations of the next sections. THE R TRANSFORMATION SEPARATING INTERNAL ROTATION FROM THE OTHER VIBRATIONS

The remainder of this work is based upon the theoretical development by Guan and Quade (15) to separate both the internal rotation and overall rotation from the vibrations in zeroth order. The first separation, the separation of this section, is a transformation that removes the zeroth order G 01 matrix elements that connect internal rotation with the other vibrations. The second separation, that of the next section, separates overall rotation from the other vibrations by removing the zeroth order Coriolis coupling. These are call the R and T transformations, respectively (19). Even after these transformations, the internal rotation remains coupled to overall rotation in zeroth order. How this is handled depends upon how the internal rotation–rotation problem is approached. For methyl alcohol, usually the internal axis method (IAM) is used for the solution of this problem. Eqs. [17] and [18] of the previous section contain the zeroth order kinetic energy coefficients from vibration, including torsion, and the zeroth order Coriolis interactions. Within the notation of Ref. (15), these are the Ytt and Xit , respectively. The elements of the R transformation are given by Rtt ( t ) å (G 01 ) 0tt /(G 01 ) 0tt

[20]

with Rtt = Å dtt = ,

t, t * x t

Rtt Å 1 Rtt Å 0,

[21a] [21b]

t x t.

As indicated in Refs. (14) and (15), after the R transformation, a change in variables

[21c]

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[22]

is made that gives the St-dependent R transformation, itself defined in terms of R. The key point of Eq. [20] is that it is necessary to have the vibrational kinetic energy matrix, (G 01 ) 0 , in zeroth order for the zero overall angular momentum state of the molecule. There are at least two ways to get this matrix. The first is to invert the 3N 0 6 Wilson G 0 matrix, including torsion that neglects the interaction with rotation. The physical significance of this is that in momentum space Px , Py , Pz are the components of the total angular momentum, including that from vibration, and may be set equal to zero to freeze out the rotational motion. However, the vibrational motion cannot not be frozen out by setting the Pt equal to zero. The second method, the one that we use in the present work, is to remove the zeroth order Coriolis interaction by a T transformation ( Ref. ( 20 ) , using the results form Eqs. [17 ] – [19 ] ) . Removal of the zeroth order Coriolis coupling gives (G 01 ) 0tt = Å Ytt = 0 ∑ Xit Xit = /Iii ,

[23]

i

including t, t * Å t and t Å t with t * Å bx and by . The only matrix elements of (G 01 ) that are needed to develop the R transformation are (G 01 ) 0tt Å 0.6079 auA 2 , (G

01 0 t bx

)

[24a] 2

Å (0.0096 0 0.2866c) auA ,

[24b]

and (G 01 ) 0t by Å ( 00.2866s) auA 2 ,

[24c]

which give for the only nonvanishing Rtt Rtbx Å 0.0158 0 0.4714c

[25a]

Rtby Å 00.4714s.

[25b]

and

Two things should be noted from these results. First, (G 01 ) 0tt is the reduced moment of inertia for the torsional motion; that is, all interactions between internal rotation and overall rotation have been removed in zeroth order—the limit of a rigid molecule. Second, the internal rotation only interacts with two vibrational modes, bx and by . These modes are the only ones that contribute to the internal angular momentum of the molecule. In zeroth order, the R transformation defines the new ki-

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netic energy coefficients, Y *tt , Y *tt , Y *tt = , and Xit , for the internal motions. From Ref. (21) we see that

Under the transformation of Eq. [32], the zeroth order Y 9tt = and X 9it become (22), in addition to Eq. [31],

[26a]

X 9xt Å X 9yt Å 0

Y *tt Å Ytt 0 YttRtt

Y *tt Å Ytt

[26b]

X 9zt Å X *zt Å Xzt Å 3.2114 auA

[33b]

Y *tt = Å Ytt = / YttRtt Rtt = 0 Ytt Rtt = 0 Ytt = Rtt

[26c]

Y 9tt Å Y *tt Å Ytt Å 3.2114 auA 2

[33c]

X *it Å Xit 0 XitRtt

[27a]

Y 9tt Å Y *tt 0 ∑ r it X *it å 0

[33d]

X *it Å Xit .

[27b]

(G 01 ) 0tt = Å Y 9tt = 0 ∑ X *it X *it = /Iii .

[33e]

Application of Eqs. [25] – [27] with Eqs. [17] and [18] gives the transformed coefficients Y *tt Å Ytt Å 3.2114 auA , 2

and

Application of Eqs. [33c and e] gives the zeroth order kinetic energy of the 3N 0 6 vibrations, including torsion, 2T 0£,t Å 3.2114th 2 / 0.9305Sg 2x / 0.9318Sg 2y / 0.9968Sg 2z

Y *bxbx Å 0.3408 auA 2 ,

/ 1.0427jg 2x / 1.0484jg 2y / 1.1097jg 2z / 0.3077bg 2x

Y *byby Å 0.3400 auA 2 , Y *bxby Å 0;

/ 0.3090bg 2y / 7.9659Sg 2CO / 0.9549Sg 2OH

[28]

/ 0.8335gh 2 / 2(0.1675)Sg OH Sg CO / 2(0.4034)

X *zt Å Xzt Å 3.2114 auA , 2

1 Sg COgh / 2(0.0351)Sg OHgh / 2( 00.0921)Sg xjg x

X *zbx Å 00.0624 auA 2 , X *zby Å 0.

/ 2( 00.0894)Sg y jg y / 2(0.0334)Sg zjg z

[29]

/ 2( 00.0605)Sg xbg x / 2( 00.0591)Sg y bg y / 2(0.0004)Sg z Sg y / 2( 00.0827) jg xbg x

For all of the remaining vibrations, Y *tt = Å Ytt =

[30a]

X *it Å Xit

[30b]

/ 2( 00.0801) jg y bg y / 2( 00.0012) jg zSy / 2( 00.0025) jg zjg y / 2(0.0004)Sg zbg y / 2(0.3152)Sg z Sg CO / 2(0.0068)Sg z Sg OH / 2(0.0171)Sg zgh / 2( 00.9541) jg z Sg CO

THE T TRANSFORMATION SEPARATING THE 3N 0 7 VIBRATIONS FROM OVERALL ROTATION

/ 2( 00.0205) jg z Sg OH / 2( 00.0516) jg zgh / 2(0.0173)Sg y Sg CO / 2( 00.0070)Sg y Sg OH

In this section a second transformation, the T transformation (13, 15) is applied to the zeroth order kinetic energy of the previous section to separate the vibrations from overall rotation, exclusive of torsion. The transformation is determined by the condition X 9it å 0,

/ 2( 00.0313)Sg y gh / 2( 00.0254) jg y Sg CO / 2(0.0440) jg y Sg OH / 2( 00.0900) jg y gh / 2(0.0100) bg y Sg CO / 2(0.0042) bg y Sg OH / 2( 00.0341) bg y gh .

[31]

that is the zeroth order Coriolis coupling of the 3N 0 7 other vibrations with rotation is made equal to zero. The transformation coefficients are given by r it ( t ) Å X *it /Iii Å (Xit 0 XitRtt /Iii )Éu 0t ,

t Å t* .

[32]

It should be noted that Eq. [34] is approximate to the extent that the coefficients and products of inertia have not been taken to the principal axis system before the R transformation. That is, Eqs. [18] and [19] have not been transformed. We actually have done this transformation, which provides additional accuracy in some of the terms of the

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[34]

/ 2( 00.0011) jg zbg y / 2(0.0008)Sg zjg y

as given in Eqs. [17] and [18].

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order 0.00005. When rounding is taken into account, the terms in Eqs. [34], [24], [25], [28], and [29] are accurate to 1 unit at the fourth place past the decimal. This accuracy is good enough for all terms except those involving internal rotation and overall rotation, which will be treated in the next section in zeroth order. Of course, the reason that it is not necessary to go to the principal axis system before the R transformation for CH3OH is that Iyz is so small. ZEROTH ORDER VIBRATION–INTERNAL ROTATION– ROTATION KINETIC ENERGY IN MOMENTUM SPACE

The purpose of the R and T transformations is to remove the zeroth order coupling between the internal rotation and other vibrations and the Coriolis coupling between the other vibrations and overall rotation. These transformations do not affect the torsion–rotation interactions in zeroth order. The transformation from velocity to momentum space is easily facilitated. First, we have for the 3N 0 7 vibrations 2T 0£ Å P /G 0 P,

[35]

m0zz Å 0 m0zt Å 1.33404(auA 2 ) 01 Å 1348.36 GHz m0yz Å 0 m0yt Å 0.004214(auA 2 ) 01 Å 4.259 GHz m0tt Å 1.64543(auA 2 ) 01 Å 1663.10 GHz.

[38]

In Eq. [38], the coupling between internal rotation and overall rotation is very large for CH3OH. Traditionally this coupling has been removed by the transformation to the internal axis system (26). In the notation of Lin and Swalen, Eq. [37] would not have the sum multiplied by 12 so their coefficients would be one-half those in Eq. [38]. DISCUSSION

In the previous sections the vibration–internal rotation– rotation zeroth order kinetic energy has been derived for CH3OH with an arbitrary molecular axis system defined such that Z is parallel to the axis of internal rotation and X is perpendicular to the –COH plane. Two transformations have been made, the first separating vibrations from internal rotation in zeroth order. This R transformation amounts to the definition of a new internal rotation variable

where G 0 Å [(G 01 ) 0 ] 01

t* Å t /

[36]

from Eqs. [33e] and [34]. It turns out the G 0 may also be obtained by the method of Wilson (23) by deleting the row and column pertaining to the torsion. However, it should be kept in mind that the dihedral angle t is not defined as the C–O chemical bond because of the tilt of the internal rotation axis. The significance of the R transformation is that it separates or factors the G 0 and (G 01 ) 0 matrices into the torsional element that becomes noninteracting with the remaining 3N 0 7 by 3N 0 7 block. The G 0tt = elements are unaffected by the R transformation, while G 0tt Å [(G 01 ) 0 ] 01 . At the same time, the (G 01 ) 0tt matrix elements are given by the Y 9tt = in Eq. [33e]. The zeroth order torsional–rotational kinetic energy in momentum space can be obtained by the method of Ref. (24) or alternatively by the method of Lin and Swalen (25) for symmetric internal rotors. We use the latter approach for the desired accuracy of the torsion–rotation coefficients, higher than would be obtained by neglecting the principal axis transformation in the R transformation. Then we obtain T 0R ,t Å 12 ∑ m0ij pi pj

[37]

with i, j Å x, y, z, and m0yz , m0yt , and m0zt the only nonvanishing interaction terms. From Eqs. [19] and [33c] we obtain m0xx Å 0.047009(auA 2 ) 01 Å 47.514 GHz m0yy Å 0.048740(auA 2 ) 01 Å 49.263 GHz

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[39]

The dihedral angle t was fixed by molecular geometry, while t* now depends upon the vibrational displacements of the methyl group through Sbx and Sby . The Rtt were determined by the condition of separating internal rotation from the other vibrations in zeroth order in the kinetic energy. For internal rotation in molecules with symmetric internal rotors, the physical significance of the R transformation is that there be no internal angular momentum due to vibration in zeroth order. The second transformation, the T transformation, removes the zeroth order Coriolis coupling of the 3N 0 7 vibrations with overall rotation. Again, the initial molecular axis system was fixed with respect to the equilibrium molecular geometry of certain atoms but after the T transformation become ‘‘floating,’’ depending upon the vibrational displacements in such a manner that the zeroth order overall angular momentum from the vibrations vanish. However, as of yet an internal axis transformation has not been made to separate internal rotation from overall rotation in zeroth order. It should be noted that although there is a t dependence in the zeroth order vibrational kinetic energy even after the transformation to symmetry coordinates projected on the molecular plane, this t dependence disappears, as expected, after the R and T transformations. Therefore in zeroth order the vibrational energies will be independent of the angle of internal rotation for the case of a symmetric internal rotor. Finally, the results of Eqs. [33] – [38] along with the velocities and T-transformation coefficients found in the Appendices will be necessary for the next segment of this work. All things

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considered, it is expected that in the calculation of the St and St St dependence of the kinetic energy coefficients will be extremely tedious, as can be seen in Ref. (15). Not included in Ref. (15) is the additional St dependence from the time differentiation of Eqs. [16]. Work has been started on the internal coordinate dependence of the moments of inertia, Coriolis coupling coefficients, vibrational kinetic energy coefficients, and vibration– internal rotation interaction coefficients but it is not yet to the stage of application of the R and T transformations.

yh 3 Å 0,g 3 (sin j3 sin b2 sin t / sin j3 cos b2 cos t )

APPENDIX I

This Appendix contains the coefficients for the T transformation. They will be needed in the next segment of this work.

This Appendix gives the velocities of the atoms with respect to the carbon atom. These results will be needed in the next paper of this work to obtain the St dependence of the Coriolis interaction and the St , St St = dependence of the vibrational kinetic energy.

0 ,3[(cos j3 sin b2 sin t / cos j3 cos b2 cos t ) jg 3 / (sin j3 cos b2 sin t 0 sin j3 sin b2 cos t ) bg 2 / (sin j3 sin b2 cos t 0 sin j3 cos b2 sin t ) th ]

zh 3 Å ,g 3 cos j3 0 ,3 sin j3jg 3 . APPENDIX II

r (CO )z Å 0,g CO cos z

r (OH )z Å ,g OH cos( g / z ) 0 ,g CO cos q 0 ,OH sin( g / z ) gh

x r jz Å 00.0014

y r jx Å 0.0887

x r Sy Å 00.0400

y r bx Å 0.0396

r Sx OH Å 0.0316

r zjx Å 00.0019

REFERENCES

/ sin j1 cos tth )

yh 1 Å 0,g 1 sin j1 cos t 0 ,1 (cos j1 cos tjg 1 0 sin j1 sin tth )

zh 1 Å ,g 1 cos j1 0 ,1 sin j1jg 1 xh 2 Å ,g 2 (sin j2 sin b3 cos t / sin j2 cos b3 sin t ) / ,2[(cos j2 sin b3 cos t / cos j2 cos b3 sin t ) jg 2 / (sin j2 cos b3 cos t 0 sin j2 sin b3 sin t ) bg 3 / ( 0sin j2 sin b3 sin t / sin j2 cos b3 cos t ) th ]

yh 2 Å ,g 2 (sin j2 sin b3 sin t 0 sin j2 cos b3 cos t ) / ,2[(cos j2 sin b3 sin t 0 cos j2 cos b3 cos t ) jg 2 / (sin j2 cos b3 sin t / sin j2 sin b3 cos t ) bg 3 / (sin j2 sin b3 cos t / sin j2 cos b3 sin t ) th ]

zh 2 Å ,g 2 cos j2 0 ,2 sin j2jg 2 xh 3 Å ,g 3 ( 0sin j3 sin b2 cos t / sin j3 cos b2 sin t ) / ,3[( 0cos j3 sin b2 cos t / cos j3 cos b2 sin t ) jg 3 / ( 0sin j3 cos b2 cos t 0 sin j3 sin b2 sin t ) bg 2 / (sin j3 sin b2 sin t / sin j3 cos b2 cos t ) th ]

1. C. C. Lin and J. D. Swalen, Rev. Mod. Physics 31, 841–892 (1959). 2. (a) C. R. Quade and C. C. Lin, J. Chem. Phys. 38, 540–550 (1963); (b) M. Liu and C. R. Quade, J. Mol. Spectrosc. 146, 238–251 (1991). 3. (a) P. Venkateswarlu, H. D. Edwards, and W. Gordy, J. Chem. Phys. 23, 1195–1199 (1955); (b) R. K. Kakar and C. R. Quade, J. Chem. Phys. 72, 4300–4307 (1980); (c) T. Kojima and T. Nishikawa, J. Phys. Soc. Jpn. 10, 240 (1955); 12, 680 (1957); (d) R. E. Schmidt and C. R. Quade, J. Chem. Phys. 62, 3864–3874 (1975). 4. D. Kivelson, J. Chem. Phys. 22, 1733–1739 (1954); 23, 2230–2235 (1955); 23, 2236–2243 (1955). 5. K. T. Hetch and D. M. Dennison, J. Chem. Phys. 26, 48–69 (1957). 6. P. R. Swan and M. W. P. Strandberg, J. Mol. Spectrosc. 1, 333–378 (1957). 7. B. Kirtman, J. Chem. Phys. 37, 2516–2539 (1962). 8. J. T. Hougen, Can. J. Phys. 42, 1920–1937 (1964). 9. P. R. Bunker, J. Chem. Phys. 47, 718–739 (1967). 10. C. R. Quade, J. Chem. Phys. 44, 2412–2523 (1966). 11. (a) E. Herbst, J. K. Messer, F. C. De Lucia, and P. Helminger, J. Mol. Spectrosc. 108, 42–57 (1984); (b) J. Tang and K. Takagi, J. Mol. Spectrosc. 161, 487–498 (1993); (c) L. H. Xu, M. S. Walsh, and R. M. Lees, J. Mol. Spectrosc. 179, 269–281 (1996); (d) L. H. Xu and J. T. Hougen, J. Mol. Spectrosc. 173, 540–551 (1995); 169, 396–409 (1995). 12. J. T. Hougen, J. Mol. Spectrosc. 181, 287–296 (1997). 13. C. R. Quade, J. Chem. Phys. 64, 2783–2795 (1976). 14. C. R. Quade, J. Chem. Phys. 65, 700–706 (1976). 15. Y. Guan and C. R. Quade, J. Chem. Phys. 84, 5624–5638 (1986). 16. Y. Guan and C. R. Quade, J. Chem. Phys. 86, 4806–4823 (1987).

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r zSx Å 00.0047

r gx Å 00.0523

xh 1 Å ,g 1 sin j1 sin t / ,1 (cos j1 sin tjg 1

/

x r by Å 00.0382

r Sx CO Å 00.0217 r zbx Å 00.0158.

r (OH )y Å ,g OH sin( g / z ) 0 ,g CO sin z / ,OH cos( g / z ) gh

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x r jy Å 00.0856

r (CO )y Å 0,g CO sin z

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C. R. Quade, J. Chem. Phys. 82, 2509 (1985). P. Venkateswarlu and W. Gordy, J. Chem. Phys. 1200–1202 (1955). Reference 15, pages 5626 and 5628. Reference 13, Eq. [14]. Reference 15, Eqs. [16].

22. 23. 24. 25. 26.

Reference 15, Eqs. [27] and [31]. E. B. Wilson, Jr., J. Chem. Phys. 7, 1047 (1939). Reference 15, Eqs. [34] – [37]. Reference 1, Eq. (2–25). Reference 1, Section II.A.2.

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