Normal coordinate treatments of internal-rotation vibrations

JOURNAL

OFMOLECULAR

Normal

SPECTROSCOPY

Coordinate

I&308-318

(1965)

Treatments Vibrations

of Internal-Rotation

T. MIYAZAWA AND K. FUKUSHIMA Institute

for Protein Research, Osaka University,

Kitaku,

Osaka, Japan

The general formulas of the C-matrix elements associated with internal-rotation coordinates are tabulated. The normal vibrations of the trams and gauche isomers of dichloroethane were treated by the use of the internal-rotation potential. I. INTRODUCTION

Internal-rotation coordinates as well as stretching and bending coordinates are necessary for complete treatments of the normal vibrations of molecules having internal-rotation axes. The internal-rotation coordinates for substituted ethane molecules have been derived previously (1). In the present study, a general method for normal coordinate treatments of internal-rotation vibrations was introduced and was applied to the trans and gauche isomers of dichloroethane. II. TORSIONAL

ANGLE

The torsional angle has been defined as the dihedral angle, 7, between the plane i, M, N and the plane M, N, i (Fig. l), with the restriction -7r < T s K (2). Actually in carrying out normal coordinate treatments, the torsional angle may be released from this restriction, so that - 00 < 7 < 00. We may now let AT be positive (a) if the projection of the bond [i, M in Fig. l(a)] closer to the observer (viewing from the left side of the molecule along the bond M, N) is rotated clockwise as well as (b) if the projection of the bond [i, M in Fig. l(b)] more distant from the observer (viewing from the right side of the molecule along the bond N, M) is rotated clockwise. Then the sign of AT (or 7) is invariant from whichever way the bond M, N is viewed. Some typical examples for 7 and AT are illustrated in Figs. l(c) and l(d). III.

INTERNAL-ROTATION

COORDINATE

The internal-rotation about the C-O bond may be discussed for the case of the methanol molecule (Fig. 2). There are three torsional coordinates, namely, A 71.4~4, Arzatx, and A7sag4 . From symmetry considerations, the torsional sym308

INTERNAL-ROTATION

M

CC1

VIBRATIONS

309

N

T= -n

CdJ

FIG. 1. Torsional angle (TLMN& and torsional coordiante (AT
coordinates

are derived

as

SI = (ATN +

AT24

S2 = (AT24 -

A~34)/2l’~,

Ss = (--A~14

+

ATU)/~~“,

+

AT24

+

(1) (2)

AT3r)/61’2,

(3)

where the subscripts A and /3 are omitted for simplicity. These symmetry coordinates &-S, for the methanol molecule are schematically shown in Fig. 2’ However, the latter two coordinates are expressible in terms of angle-bending coordinates. For example, if all the bond angles of the atom A are assumed tetrahedral, then 82

=

(4@2as

-

Sa = GWa41 -

On the other hand, of the angle-bending torsional symmetry C-O bond against angle is given by

2A#m

-

2Ah42 -

2Ah.42 &~AB

+

2A#ua + A42aa + A&b3~5)/30~‘~, (4) &‘2~d/1@‘~.

(5)

the coordinate S1 is not expressible as a linear combination coordinates. Accordingly S1 may be taken up as the genuine coordinate. When the CHSgroup is rotated about the the -OH group, the change (At,@) in the internal-rotation

AtAb = ( 2

i=l

A&4)/3

= SJ31’2.

(6)

The internal rotation mode about the C-C bond may be discussed for the case of the ethane molecule (Fig. 3). This molecule belongs to the point group L& and only the internal-rotation vibration belongs to the a, species. For this molecule, there are nine torsional coordinates, namely, ArIAB4 , AXUS , ATUB~ , , and A7snee (Fig. 3). From the symmetry AT2nrr4 , AT2ub, AT~ABG, fh~s4 , AQAB5 considerations, the torsional symmetry coordinates S1 for the ~2~ species are

MIYAZAWA

310

FIG. 2. Torsional

symmetry

FIG. 3. The torsional

coordinates

symmetry

(A~14

+

of the methanol

coordinates

found (by unitary transformation) SI =

AND FUKUSHIMA

A715 + A?,6 +

molecule.

of the ethane

molecule.

to be A.724 +

A725 +

A726

+

An4

+

An5

+

ke)/3,

(7)

where the subscripts A and B are omitted for simplicity. The symmetry coordinates for the e, species are X2 = (A715 8s

=

-

A742

(--A~15 +

-

‘h-42

Ar16 +

A~43 +

2Ar35 -

2Ar2~)/12~‘~,

- A716 + A743)/2,

(8) (9)

and those for the e, species are Sq = (--A725

+ 2Aq6 -

A~F, -

A724 +

Ana

+

A~4)/12~‘2,

(10)

INTERNAL-ROTATION

m M i

+

311

VIBRATIONS

4j

,c

n

N

FIG. 4. The internal rotation about the bond MN.

S’s = (4A714

-

2Ar25

-

2Ar~ + ATIK+ +

AT24

A734 -

+

ATIP,

2A735 -

2A~26)/36~‘~.

(11)

[Also there are four redundant coordinates (I).] The symmetry coordinates L&-S6 for the ethane molecule are schematically shown in Fig. 3. The torsional symmetry coordinates S2-S6 may be expressed in terms of bondangle bending coordinates (1). On the other hand, the coordinate S1 is not expressible as a linear combination of the angle bending coordinates. When one of the CH,groups is rotated about the C-C bond against the other -CH, group, the change in the internal-rotation angle (At,,) is given by

(12) The internal-rotation coordinate may now be formulated for a general case. In Fig. 4, the atom M is bonded with m atoms in addition to the atom N which, in turn, is bonded to 12 more atoms. For this molecule, there are mn torsional coordinates and hence are the same number of torsional symmetry coordinates. However, only the coordinate that is similar to the X1 coordinates in Eqs. (6) or (12), is the genuine torsional symmetry coordinat’e. The internal-rotation coordinate AtMwlv for the bond M, N is given by (13) For normal coordinate treatments, G-matrix elements are calculated from the s vectors of the coordinates concerned. With reference to Eq. (13), the s vectors for the internal-rotation coordinates are derived from the s vectors (2, S) for constituent torsional coordinates, AT.%xNj’s, which have been given by Eqs. (21-24) of Reference 2.1 1 Revision of Eq. (22) of Reference 2; the minus sign between the first term (of e12X e& and the second term (of era X e& should be replaced by the pZus sign.

MIYAZAWA

312 IV. G-MATRIX

ELEMENTS

AND

FUKUSHIMA

ASSOCIATED WITH COORDINATES

INTERNAL-ROTATION

The G-matrix elements associated with the internal-rotation coordinates, At’s, of Eqs. (6) and (12) were derived in the present study and are listed in Table I, where all the bond angles were assumed tetrahedral. Those formulas TABLE I

G Matrix Elements

INTERNAL-ROTATION TABLE I (Continued)

VIBRATIONS

313

I (Contlnuad)

INTERNAL-ROTATION

VIBRATIONS

315

FIG. 5. The notation of atoms for Table I. The tetravalent atoms, A, B, C, D, and E may be substituted with the divalent atoms, cy, P, 7, 6, and E,respectively.

are useful for normal coordinate treatments of saturated compounds consisting of Qetravalent atoms (carbon, silicon, germanium, etc.), divalent atoms (oxygen, sulfur, etc.), and any monovalent at.oms (hydrogen, chIorine, bromine, iodine, etc.). In Table I, Ar, A+, and At are the bond-stretching, angle bending, and internal-rotation coordinates, respectively, and P and p denote the reciprocal mass and reciprocal bond lengths, respectively. The Roman or numeric subscripts are illustrated in Fig. 5. The tetravalent aboms indicated by Roman letters in Fig. 5 may be replaced with divalent atoms denoted by Greek letters (A, B, C, D, and E to a, P, y, 6, and e, respectively) provided that the atoms indicated by numeraIs are removed correspondingly. The Roman subscripts for the At coordinates always denote tetravalent atoms whereas the Greek subscripts for the At coordinates always denote divalent atoms. The Roman subscripts which only appear for the stretching (AT) or bending coordinates (Ac#I) may well be replaced with Greek subscripts (divalent atoms) or with numeric subscripts (monovalent atoms). The numeric subscripts in Table I denote monovalent atoms. However, numeric subscripts may be replaced, in any case, with Greek subscripts (for divalent atoms) or with Roman subscripts (for tetravalent atoms). The general formulas in Table I are invariant on mut,ual exchange of numeric subscripts l-2-3, 4-5, and 6-7. The formulas for G(AtAB.Atec), G(At,,.At,c), on t’he mutual exchange of subscripts 6-7-D. and G(At,,. AtBC) are invariant The formula for G(AtAB.AtAB) is invariant on the exchange of subscripts 4-5-C. It may also be noted that, the general formulas in Table I are written so that they apply equally well even if the sign of 7 and At are reversed simultaneously from the definition as given in Section II. V. INTERNAL-ROTATION

VIBRATIONS

OF DICHLOROETHANE

The far infrared bands of the trans and gauche isomers of 1,2-dichloroethane have been observed at 134 cnl-1 and 124 cm-l, respectively, in the liquid state (4). Recently the normal vibrations of 1,2-dichloroethane were treated by Tasumi (5) wit.h the modified Urey-BradleyShimanouchi force field (6). In the

316

MIYAZAWA

AND FUKUSHIMA TABLE II

UREY-BRADLEY

POTENTIAL

K(C-C) K(C-H) K(C-Cl) H(C-C-H) H(C-C-Cl) H (H-C-H) H (H-C-Cl)

CONSTANTS~

(in md/A)

2.30 4.30 1.60 0.19 0.10 0.41 0.08

OF DICHLOROETHANE~

X0 T” F’ F(CCH) F(C-CCI) F(H-CH) F(H.CCl)

0.04 0.08 -F/10 0.54 0.60 0.07 0.72

8 K: stretching constant, H: bending constant, F: repulsion constant, H: intramolecular tension, and T: trans-coupling constant. b Reference b. 0 In md.A. TABLE III THE

FREQUENCIES (cm-i) AND POTENTIAL OF THE A, VIBRATIONS OF THE TRANS

ENERGY ISOMER

DISTRIBUTIONS (PED in %) OF 1,2-DICHLOROETHANE

Frequency Calc.

Obs.a 3005 1124 768 134

3010 1162 737 0

PED(II)b

VII

VII- YI

3010 1177 770 134

0 15 33 134

C& va (100)

At(O) At(3) At@)

CH2 twist (94) CH2 rock (93)

At@)

s References 4 and 7. b Y.: antisymmetric stretching.

present study, however, the internal-rotation account:

potential2 was also taken into

V (internal rotation) = Y(At)z/Z.

(14)

The potential constants derived by Tasumi are cited in Table II. The G-matr@ elements were calculated by the use of the bond lengths of r(C-C) = 1.54A, = 1.088 and tetrahedral bond angles. The r(C-Cl) = 1.76ij and r(C-H) a, frequencies of the trans isomer and the a frequencies of the gauche isomer are shown in Tables III and IV, respectively. In these tables, the frequencies calculated without the internal rotation potential, Eq. (14), are shown in the column headed with v1 while the frequencies calculated with the internal rotation potential are shown in the column headed with VII . 2 A useful method for approximately correcting for the anharmonicity of the internalrotation potential has been described, in part, by Miyazawa and Pitzer [J. Chem. Phus. 30, 1076 (1959)l. More detailed discussions on the method of correction based on the GF matrix method will be reported separately.

INTERNAL-ROTATION

317

VIBRATIONS

TABLE

IV

THE FREQUENCIES (cm-l) AND POTENTIAL ENERGY DISTRIBUTIONS (PED in %) OF THE A VIBRATIONS OF THE GAUCHE ISOMER OF 1,2-DICHLOROETHANE Frequency PED(II)b

Calc.

Ohs.”

VII

VII -

VI

3005 2957 1429 1264 1207 1031 943 651 265

3000 2957 1446 1311 1212 1014 928 645 223

3000 2957 1446 1311 1219 1014 965 647 260

0 0 0 0 7 0 37 2 37

At(O) At(O) At(O) At(O) At(l) At(O) At(8) At(O) At (31)

124

0

124

124

At@))

8 References 4 and 7. b vh: symmetric stretching;

va: antisymmetric

CHz vs(lC@) CH2 v,(lOO) CH2 bend (102)

CHZ wag@@, CH2 twist(17) CHf twist(78), CH2 wag(21) C-C streteh(ll0) CH, rock(87) G-Cl stretch(ll4) C-C-Cl bend(65) C-C-Cl bend(30)

stretching.

The potential energy distributions (in %) for the internal symmetry coordinates were also calculated as shown in Tables III and IV, where only the fractions greater than 10 % are listed. The fractions of the potential energy associated with the internal-rotation coordinates are shown in the parentheses which follow the symbol At. Trans Isomet The internal-rotation constant for the trans isomer was adjusted to be Ycc = 0.130 mdyne,A with reference to the observed frequency of 134 cm-‘. As indicated by the potential energy distributions, this vibration is almost exclusively associated with the internal rotation mode. Comparing the calculated frequencies, ~1 and yIr , it is remarkable to note that for the so-called CH2 rocking vibration ~11 - VI is as great as 33 cm-l. In accordance with this difference, the fraction of the energy associated with the internal rotation mode amounts to 8%. Therefore for refined treatments of the CH, rocking vibrations, t’he internal-rotation potential about the CH2-CH, bond may not be neglected. The same conclusion has been derived from the normal coordinate analysis of polyet,hylene (1). Gauche Ismne? The internal-rotation constant of the gauche isomer was adjusted to be Yoo = 0.160 mdyne. A with reference to the observed frequency of 124 cm-‘. The potential energy distributions as listed in Table IV indicate that the C-C-Cl bending mode (A) and the internal-rotation mode of the gauche isomer strongly

318

MIYAZAWA

AND

FUKUSHIMA

couple with each other, giving rise to the bands at 265 and 124 cm-l. Accordingly the contributions of the bending potential terms may never be overlooked in the calculations of int#ernal rotation barriers from the observed frequencies of the gauche isomer. The authors wish to express their gratitude to Prof. Takehiko Shimanouchi and Dr. Mit,suo Tasumi of the University of Tokyo for informative discussions. They also thank Miss Yoshiko Ideguchi for her technical assistance. RECEIVED March

9, 1964 REFERENCES

1. M. TASUMI, T. SHIMANOUCHI, AND T. MIYAZAWA, J. Mol. Spectry.

11,422 (1953). C. DECIUS, AND P. C. CROSS, “Molecular Vibrations,” p. 61. McGraw-Hill, New York, 1955. E. B. WILSON,JR., J. Chem. Phys. ‘7, 1047 (1939); 9, 76 (1941). I. ICHISHIMA,H. KAMIYAMA,T. SHIMANOUCHI, ANDS. MIZUSHIMA,J. Chem. Whys. 29, 1190 (1958). M. TASUMI,paper presented at Annual Meeting of the Chemical Society of Japan, Kyoto, 1962. T. SHIMANOUCHI, Pure Appl. Chem. 7, 131 (1963). S. MIZUSHIMA,“Structure of Molecules and Internal Rotation.” Academic Press, New York, 1954.

2. E. B. 3.

4. 6. 6. 7.

WILSON,

JR., J.