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Torsional lattice vibrations in molecularcrystalsD.A. Oliver a & S.H. Walmsley aa William Ramsay and Ralph Forster Laboratories, University CollegeLondonVersion of record first published: 15 Dec 2010.

To cite this article: D.A. Oliver & S.H. Walmsley (1969): Torsional lattice vibrations in molecular crystals,Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 17:6, 617-626

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MOLECULAR PHYSICS, 1969, VOL. 17, No. 6, 617-626

Torsional lattice vibrations in molecular crystals II. Benzene

by D. A. OLIVER and S. H. WALMSLEY

William Ramsay and Ralph Forster Laboratories, University College London

(Received 27 May 1969)

Atom-atom interaction potential energies derived from various sources have been used to calculate the torsional lattice vibration frequencies of crystalline benzene. The results are compared with available experimental data. The usefulness of this type of model potential is discussed.

l . INTRODUCTION

In the first paper in this series [1] the theory of torsional lattice vibrations was discussed and a suitable form of displacement coordinate established. In making calculations of lattice frequencies of molecular crystals, two main types of potential function have been used. In the first the intermolecular interaction is emphasized and maximum use made of known molecular properties such as quadrupole moment and polarizability. This approach was used in earlier work on carbon dioxide [2]. In the second the overall intermolecular potential is further sub-divided into atom-atom interactions, which are usually only dependent on the interatomic separation. The parameters appearing in these atom-atom potentials have to be determined empirically and this has been most extensively investigated by Kitaigorodskii [3] and more recently by Williams [4]. In this paper the second approach is considered for the calculation of the torsional vibrations of a crystal of a typical aromatic hydrocarbon, benzene.

In w the crystal structure of benzene is described and vibrational symmetry coordinates defined. In w an outline of the calculation is given. The interatomic potentials are introduced in w and the results presented and discussed in w

2. CRYSTAL STRUCTURE AND SYMMETRY COORDINATES

There is one known crystal phase of benzene. Its structure has been determined by Cox [5] at 270 ~ using x-ray diffraction and by Bacon et al. [6] at 218 and 138~ using neutron diffraction. The crystal is orthorhombic, belonging to space group Pbca (D2~15). There are four molecules in each unit cell at the corner and centres of the faces. The dimensions of the unit cell are summarized in table 1. Details of the molecular orientation at two temperatures are given in table 2, in the form of direction cosines between the crystal fixed axes a, b, c parallel to the unit cell sides and a set of molecule fixed axes x, y, z. z is perpendicular to the molecular plane; x lies along a carbon hydrogen bond direction which is close to the b crystal axis and y is chosen to complete an orthogonal right-handed system.

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618 D . A . Oliver and S. H. Walmsley

Table 1.

Temperature

a

b C

270~

7-460 9.666 7.034

138~

7'39 9 "42 6 '81

Lattice constants in ~ngstr6ms of crystalline benzene at various temperatures.

270~ X y z

a - 0"2783 -0"6476 0"7094 b 0"9601 - 0.1674 0"2238 c -0-0262 0.7434 0"6684

138~ x y z

a --0"3233 --0.6551 0-6830 b 0"9457 --0.1978 0"2580 c - 0'0339 0.7293 0"6834

Table 2. Direction cosines of molecules in crystalline benzene at various temperatures.

T h e molecules are each located at inversion centres in the crystal and the different molecules in the unit cell are related by a series of 180 ~ screw rotations parallel to the crystal axes and glide reflections through perpendicular planes. The direction cosines are taken to refer to a molecule at the origin (000) and the set of molecules parallel to this is called set 1. Correspondingly, the molecules at (�89 �89 0), (0, �89 �89 and (�89 0, �89 have different orientations and the sets of molecules parallel to them are labelled 2, 3 and 4 respectively. The molecule fixed axes in the different sets are chosen to be interconverted by 180 ~ rotations, so that they are all right handed. In this way, set 1 is converted into set 2 by a screw rotation about an axis parallel to a, and into sets 3 and 4 by corresponding rotations about the b and c axes respectively. The general arrangement is illustrated in the figure.

The calculations reported later in the paper lead to those lattice vibrations which are allowed in Raman spectroscopy. These are all of the zero wave vector type in which each unit cell vibrates exactly in phase and they are all molecular torsions. For such vibrations, it is possible to make a further symmetry classification in terms of the factor (or unit cell) group, which in this case is isomorphous with the point group Dzh. The irreducible representations of this group are given in table 3. Vibrations belonging to representations with subscript g are Raman allowed.

Since there are three torsional degrees of freedom per molecule and four molecules per unit cell, there are twelve zero wave vector torsional lattice vibrations in all. When these are classified in terms of the factor group D2h, it is found that there are three vibrations belonging to each of the irreducible representations Ag, Big, B2g and B3g.

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Torsional lattice vibrations in molecular crystals 619

b.

I I

I I

I I

\ \

+ ---kx \ \

/ 3 /

/ /

/ __/_.

+

x

/ \ / " \

- - \ \ \ ,,4 \ \ \

- - \ ) , + \ / \ /-

_ \ . . . . / +

II a

Geometry of benzene crystals. The centres of molecules denoted by dashed lines lie half a unit cell dimension above the ab plane. Signs denote the sense of tilt out of the ab plane.

A g Big B2g Bag Au Blu B2u Bau

E C~ ~

1 1

- 1 - 1

1 1

- 1 - 1

C2 ~

1 - 1

1 - 1

1 - 1

1 - 1

C2 c

1 - 1 - 1

1 1

- 1 - 1

1

1 1 1 1

- 1 - 1 - 1 - 1

~bc

1 1

- 1 - 1 - 1 - 1

1 1

aC a

1 - 1

1 - 1 - 1

1 - 1

1

~ab

1 - 1 - 1

1 - 1

1 1

- 1

Table 3. Characters of irreducible representations of group D2h.

Corresponding s y m m e t r y coordinates may be cons t ruc ted as follows. Molecular displacement coordinates describing torsions may be chosen as infinitesimal rotations about principal inertial axes of the molecule [1], such

(1) as the set x, y , z . These are denoted by O a k where a = x, y , z and the coordinate

refers to the kth molecule of lth uni t cell. k = 1, 2, 3, 4 labels the translationally equivalent set. T h e zero wave vector coordinates are linear combina t ions of the fo rm:

where N is the n u m b e r of uni t cells in the crystal. linear combinat ions of these coordinates may be

(2 .1)

Using the screw rotations, cons t ruc ted which have

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620 D.A. Oliver and S. H. Walmsley

irreducible symmetry in the factor group. Different values of a are not mixed by this procedure and the results are given in (2.2):

r = �89 + 0a(2) + 0a(3) + Oa(4)] A g ,

r + - - B i g ,

r - + - B2g,

r - - + B3g.

(2.2)

The first coordinate is written in full and belongs to the representation Ag. The succeeding coordinates differ only in the signs of the coefficients and the representation. In each case a-- x, y, z.

3. TORSIONAL LATTICE VIBRATIONS

The theory used in the calculation of lattice vibration frequencies has been given at length in previous papers [1, 7]. Only a very brief summary is given. The torsional lattice frequencies are found by solving a 3 • 3 secular determinant for each of the symmetry types in (2.2):

I Fab( i ) -- $abla47r2u 2 ] ---- 0. (3.1)

Here F a b ( i ) = [ ~ 2 U / ~ r 1 6 2 is the force constant relating the potential energy U to the symmetry coordinates; Ia is the moment of inertia of axis a and v is the vibration frequency.

The force constants are most easily expressed in terms of the molecular

coordinates 0a(~). Expressions for these force constants in terms of a general

central atom-atom potential are given in equations (5.2) and (5.8) of [1] and will not be repeated here.

The explicit form of the interatomic potential used is:

U(S ) = A exp ( - BS ) - CS -6, (3.2)

where S is the interatomic distance and .4, B, C are constants. To complete equation (5.2) of [l], the explicit expressions for the first and second derivatives with respect to S are:

( 8 U/ S S ) = - , 4 B exp ( - B S ) + 6 C S -7, (3.3)

(82 U/ S S 2) = A B 2 exp ( - B S ) - 42CS -8. (3.4)

The coordinate transformations in (2.1) and (2.2) enable the molecular force constants to be converted to symmetry force constants.

4. THE INTERATOMIC POTENTIAL

The calculation outlined in the previous section depends on the values of the constants M, B, C appearing in the interatomic potential (3.2). In aromatic hydrocarbons there are three types of atom pair: carbon-carbon, carbon-hydrogen and hydrogen-hydrogen and one set of constants is required for each.

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Torsional lattice vibrations in molecular crystals 621

The most extensive work has been carried out by Kitaigorodskii [3], who has sought to relate the structures of a wide range of hydrocarbon crystals in terms of one set of ' universal ' constants for each of the three atom pairs. A similar type of investigation was made by Williams [4] who found the constants which gave the best fit to 77 properties of 11 aromatic hydrocarbon crystals. Th e properties used did not include vibration frequencies. Both of these workers give a number of sets of values reflecting variations in the method of processing or the range of properties included. In table 4 one set of constants of each is given. The Williams set show a particularly close fit to the observed crystal structure of benzene.

A B C

Williams

H-H C-H C-C

2"81 6"54

51"76

3 "74 3 "67 3 "60

2"50 9"66

37"19

H-H C-H C-C

Kitaigorodskii

28-19 28.19 28-19

4"86 4"12 3 "60

3 "64 9"83

24" 03

Bartell-Crowell

H-H C-H C-C

4"58 11 "17 27" 20

4"08 3 "81 3"55

3 "42 9"04

23" 90

Table 4. Parameters in atom-atom potentials. A, units of 10 -1~ ergs; B, units of A -1, C, units of 10 -12 ergs ~6.

Crowell [8] has used his measurements on the compressibility of graphite to obtain a carbon-carbon potential. There is a wide range of estimates in the literature of the hydrogen-hydrogen potential. Th ey may be combined with the Crowell potential to give new sets of constants. A corresponding carbon-hydrogen potential may be estimated by taking appropriate mean values:

Acri = ( A c c A aa) 1/2,

B c a = �89 + BAH),

C c a = ( C c c C a a ) 1/2.

The results using a potential of this kind fluctuate widely and in some cases imaginary frequencies are found. I f the hydrogen-hydrogen potential due to Bartell [9] is used, the results are physically realistic and the constants are also given in table 4.

5. RESULTS AND DISCUSSION

The zero wave vector torsional lattice frequencies calculated using the three potentials mentioned in the previous section are collected in table 5.

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622 D . A . Oliver and S. H. Walmsley

As

Big

B2g

B~o

Williams

94 76 43

134 93 48

100 87 81

131 85 61

Calculated

Kitaigorodskii

107 78 58

144 105 66

Bartell-Crowell

98 72 52

134 98 60

105 97 83

Experiment [11]

4~ 138~

(90) 79 57

136 128 (86) (79) 69 61

(90) 90 79

136 128 107 100 (86) (79)

(100) 86 64

113 110 88

140 102 71

132 92 62

(100) 100 86

Table 5. Frequencies in cm -1 of zero wave vector torsional lattice vibrations, for the 138~ structure. The experimental values in brackets are suggested frequencies for transitions which are forecast to be of low intensity in Frfihling's calculation [10].

T h e earliest work on the Raman spectrum of crystalline benzene is that of Frfihling [10] working at 273~ This work gave information about the polarization of the Raman spectrum which enabled the factor group symmetries to be determined. A summary is given in table 6. Frfihling also used a simplified

Frequency

105 Big B3g

69 Ao 63 As

B~g 42 A s

Symmetry Torsion axis

Table 6. Experimental Raman frequencies (cm -1) at 273 ~ and their assignment by Friihling [10].

intensity theory to estimate the axis of torsion in the molecule and considered three possibilities corresponding to the x, y and z axes in this paper. The theory assumes that the molecular polarizability is unchanged during torsions and that the intensity arises f rom changes in the projection of the molecular polarizability along the crystal axes. The intensity also depends on the difference between the polarizability along the two principal axes perpendicular to the axis of torsion. In this model, therefore, there is no intensity associated with z-axis torsions and other qualitative arguments are used to suggest that the 35 and 69 cm -1 bands are of this type.

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Torsional lattice vibrations in molecular crystals 623

The most recent work is by Ito and Shigeoka [11]. They have measured the Raman spectrum of benzene and fully deuterated benzene at various temperatures down to that of liquid helium. The frequency shift on deuteration confirms that the motions being observed are torsions. Th e frequencies are temperature dependent but there is very little change below 77 ~

Although twelve lattice vibrations are in principle allowed in the Raman effect, only six frequencies are observed at liquid helium temperatures. Ito makes his assignment on the basis of his own and previous work, and his frequency values at 4~ and interpolated values for 138~ are given in table 5. He makes use of calculations by Harada and Shimonouchi. Thei r first calculation [12] was made for the 270 ~ structure. I t takes into account only repulsive interactions between hydrogen atoms and a fit is obtained with experiment by adjusting the two potential parameters. In the second calculation [13] the 138~ structure is used. Following the same method it is necessary to add a carbon-hydrogen repulsion term to obtain agreement with experiment. The method of calculating the force constants differs from that used in the work reported here. (8U/88) is assumed to be zero and only linear variation of the infinitesimal rotations is assumed. This amounts to neglecting the first term on the right-hand side of equation (5.2) in [1]. Also, in the calculation given here, the potential parameters are empirically determined to give agreement with properties other than lattice vibration frequencies.

The three sets of results given here, although they use constants derived from different sources, show a very similar pattern to each other and all lead to frequencies in the same order of magnitude as the experiments.

The diagonalization of (3.1) also leads to the corresponding normal coordinates for the vibrations. I f the off-diagonal elements are small, the vibrations correspond closely to oscillation about one of the axes x, y and z. Th e calculated normal coordinates for the Williams potential are given in table 7.

Ag 94 76 43

Big 134 93 48

B2g 100 87 81

B3g 131 85 61

Calculated Experimental [11]

Cx Cu Cz Axis

-0"55 -0"77 +0"29

+0"32 +0"92 +0.18

+0"77 +0-23 +0"37

+0"31 - 0 "62 - 0 "71

-0"62 +0 "63 +0"50

+0"93 -0"27 -0"19

+0.42 -0-43 -0"84

+0"93 +0.21 +0"19

+0"56 -0"09 +0-82

+0"16 -0 -29 +0"97

+0"48 -0 .8 8 +0"39

+0"19 -0 .75 +0"68

(90) 79 57

128 (79) 61

(90) 9O 79

128 100 (79)

y X

Z

y X

Z

y Z

X

Y Z

X

Table 7. Normal coordinates of torsional lattice vibrations for the Williams )otential. Cx, Cu, Cz are defined in (5.1) and (5.2). The calculated and experimental frec uencies

(in cm -1) are shown in columns z and 6.

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624 D.A. Oliver and S. H. Walmsley

The results for the other two potentials are qualitatively similar. refer to inertia reduced coordinates defined in (5.1):

~a(i) = V/Ia (~a(i),

in which i=0 , 1, 2, 3 in the notation of (2.2). the form:

Z ca4~a(i). f~

The coefficients

(5.1)

The normal coordinates have

(5.2)

Ito follows Friihling in assigning the torsions as being associated with the x, y, or z axis only. His classification of the frequencies in this way is also included in table 7. The calculated values of the normal coordinates suggest that there is considerable mixing in many of the vibrations. Where this is not the case, as in the Big and the highest B~g frequencies, there is agreement with Ito's assignment. The biggest point of difference arises in the Bzg vibrations. The highest calculated frequency is close to being an x-axis torsion and Ito gives this assignment to the lowest frequency.

In the intensity calculations of Friihling already referred to, highly simplified values of the molecular orientation were used. In particular it was assumed that the x molecular and the b crystal axes are parallel. This leads to the result that for no torsion is there any change in the bb component of the crystal polarizability. Two of Friihling's frequencies, at 42 and 69 cm -1, are intense in this component and are assigned by him as z-axis torsions. The calculation has been repeated using the correct direction cosines and it is now found that there is a significant bb component associated with a y-axis torsion. On the other hand, there is no significant intensity associated with any B2g vibration. The strongest Raman line Friihling observes is of this type at 63 cm -1. The simple rigid polarizability model of Raman intensities does however lead to the other intense feature of the spectrum. It predicts that there will be two intense transitions of symmetries Big and B3g close to 130 cm -1, in agreement with experiment, and that these are y-axis torsions.

The three potentials used all lead to frequencies which are in reasonable agreement with experiment. They are, however, different in at least one important respect. The theory used in calculating the vibration frequencies assumes that the potential energy is expanded about a minimum in the function, so that linear terms vanish. This is necessarily true for non-totally symmetrical vibration coordinates, but in benzene there are three totally symmetric (Ag) torsions. The Williams potential has a minimum within 1 ~ of the 138 ~ observed crystal structure. The other two potentials are a 6 to 9 ~ rotation from the minimum. For this reason the Williams potential is preferable.

As a check the infra-red frequencies were calculated using the Williams potential and compared with available experimental data [13, 14]. The results are summarized in table 8. The infra-red measurements were not made using single crystals nor was polarized light used so that there is no experimental evidence on the symmetry species of the vibrations. However, the fit with the calculation using the assignment suggested by Harada and Shimonouchi [13] is very close indeed. The experimental value of 53 cm -1 is taken from the work of Chantry [14], who finds a very broad band at 173 ~ round this frequency and it is tentatively assumed here that two vibrations are involved.

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Torsional lattice vibrations in molecular crystals 625

The calculations of the torsional lattice vibrations have been repeated for the 270 ~ and the results are given in table 9. The values for all three potentials agree very closely with experiment: it is interesting that the match obtained is much better than at the lower temperature.

In conclusion the atom-atom potential gives a satisfactory account of the torsional lattice vibrations, in so far as a comparison can be made with experiment.

Table 8.

Symmetry Calculated Experimental Reference

Au

Blu

Bzu

Bau

91 61 53

82 69

96 56

93 53

85 70

94 53

94 53

[13] [13]

[13] [14]

[13] [14]

Calculated zero wave vector translatical lattice frequencies for the 138~ structure using the Williams' potential and comparison with experiment.

Symmetry

Ag

Big

B2a

B3a

Williams

74 63 28

107 73 33

77 66 63

105 66 49

Kitaigorodskii

79 63 37

108 78 45

82 80 64

105 74 54

BarteU- Crowell

75 60 35

106 76 43

79 74 43

104 69 50

Experimental [10]

69 63 42

105

63

105

Table 9. Calculated zero wave vector torsional lattice vibrations for the 270~ structure and comparison with experiment.

This is in marked contrast to the use of an electrostatic potential for carbon dioxide [2], in which the calculated values were only about half the experimental values. T h e fo rm of the potential is complete ly empirical. T h e r e is some theoretical justification [15] for sub-dividing the repulsive interact ion be tween benzene molecules into a sum of separate a t o m - a t o m interactions. T o adopt a similar p rocedure for the attractive potential is m u c h more dubious, especially

M . P . RR

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626 D.A. Oliver and S. H. Walmsley

as benzene in particular is characterized by its highly delocalized ~r electrons. Regarded as a purely empirical function the atom-atom interaction does seem very useful and the large amount of experimental data available for aromatic hydrocarbons makes the determination of realistic parameters possible.

Benzene itself is a very complicated molecule to use for a calculation of this kind. The approximations made in the theory are those introduced in an earlier paper [7]: the rigid molecule approximation, the harmonic approximation and the pair potential approximation. In respect of the first of these, it was assumed that the molecule was a regular hexagon. It would be interesting to investigate the breakdown of this approximation for benzene and see how far this affects the lattice vibration frequencies. The small deviations from a regular hexagon observed in the crystal structure determination, suggest the importance of a low-lying E2g molecular vibration and Bannerjee [16] has estimated the importance of this in calculating the lattice energy.

The theory is also only strictly applicable at 0 ~ The effect of raising the temperature is observed to be small at first and for the torsional frequencies in benzene is still negligible at 77~ They have decreased by about 10 per cent at 140~ and 30 per cent at 270~ The temperature is taken into account to an unknown degree by using observed structural data. A formal extension of the theory to take temperature into account is possible, but is probably much too complicated for benzene at this stage.

One of us (D. A. 0.) would like to acknowledge the receipt of an SRC studentship during the tenure of which this work was carried out.

REFERENCES

[1] OLIVER, D. A., and WALMSLEY, S. H., 1968, Molec. Phys., 15, 141. [2] WALMSLEY, S. H., and POPLE, J. A., 1964, Molec. Phys., 8, 345. [3] KtTAIGORODSKII, A. I., 1966, J. Chim. phys., 63, 9. [4] WILLIAMS, D. E., 1966, J. chem. Phys., 45, 3770. [5] Cox, E. G., 1958, Rev. mod. Phys., 30, 159. [6] BACON, G. E., CURRY, N. A., and WILSON, S. A., 1964, Proc. R. Soc. A, 279, 98. [7] WALMSLEY, S. H., 1968, J. chem. Phys., 48, 1438. [8] CROWELL, A. D., 1958, J. chem. Phys., 29, 446. [9] BARTELL, L. S., 1960, J. chem. Phys., 32, 827.

[10] FROHLINC, A., 1951, Annln Phys., 6, 401. [11] ITO, M., and SHICEOKA, T., 1966, Spectrochim. Acta, 22, 1029. [12] HARADA, I., and SHIMONOUCHI, T., 1966, J. chem. Phys., 44, 2016. [13] HARADA, I., and SHIMONOUCHI, T., 1967, J. chem. Phys., 46, 2708. []4] CHANTRY, G. W., GEBBIE, H. A., LASSIER, B., and WYLIE, G., 1967, Nature, Lond.,

214, 163. [15] CRAIG, D. P., MASON, R., PAULINC, P., and SANTRY, D. P., 1964, Proc. R. 8oc. A,

286, 98. [16] BANNERJEE, K., 1967, Molec. Phys., 12, 385.

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