Normal Vibrations and Intermolecular Forces of Crystalline Benzene andNaphthaleneIssei Harada and Takehiko Shimanouchi Citation: J. Chem. Phys. 44, 2016 (1966); doi: 10.1063/1.1726976 View online: http://dx.doi.org/10.1063/1.1726976 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v44/i5 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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THE JOURNAL OF CHEMICAL PHYSICS VOLUME 44, NuMBER 5 1 MARCH 1966

Normal Vibrations and Intermolecular Forces of Crystalline Benzene and Naphthalene

ISSEI HARADA AND T AKEHIKO SHIMANOUCIII

Department of Chemistry, Faculty of Science, University of Tokyo, Bongo, Tokyo, Japan

(Received 8 September 1965)

The GF matrix method was applied to the analysis of band splittings and lattice vibrations of crystalline benzene. The far-infrared spectrum of crystalline naphthalene was measured in the region from 300 to 50 cm-I and three infrared-active lattice vibrations were observed. A normal-coordinate treatment of the lattice vibrations of naphthalene was performed. Intermolecular force constants of non bonded hydrogen­hydrogen pairs and hydrogen-carbon pairs were obtained.

INTRODUCTION

THE analysis of the vibrational frequencies of molecular crystals provides important information

about the intermolecular interactions. The relationship between the splittings of the vibrational bands and the intermolecular forces has been studied by Snyder,! and Tasumi and Shimanouchi2 for polyethylene, and by Dows3.4 for methyl halides and ethylene.

The recent developments in the far-infrared meas­urement enable one to measure the lattice vibrations of molecular crystals. The obtained frequencies and the low-frequency Raman lines yield additional and more direct information concerning intermolecular forces.

The normal-coordinate treatments of molecular crys­tals have also been further developed. The G F matrix method has been applied to this problem and has been shown to be useful,2,5,6

In the present paper this method has been applied to crystalline benzene and naphthalene and the results are compared with the observed band splittings and lattice frequencies. For naphthalene, the far-infrared spectrum has also been measured, the observed three bands were assigned to the lattice vibrations and com­pared with the calculated values. These results give important information regarding the intermolecular interactions of these molecules.

PART I. BENZENE

Crystal Structure and Positions of Atoms

The symmetry of the crystalline benzene is ortho­rhombic and the space group is Pbca(D2h15) with the unit cell dimension of 00= 7.460, bo= 9.666, and Co

lR. G. Snyder, J. Mol. Spectry. 7,116 (1961). 2 M. Tasumi and T. Shirnanouchi, J. Chern. Phys. 43, 1245

(1965). 3 D. A. Dows, J. Chern. Phys. 32, 1342 (1960). • D. A. Dows, J. Chern. Phys. 36, 2836 (1962). 5 T. Shirnanouchi, M. Tsuboi, and T. Miyazawa, J. Chern.

Phys.35, 1597 (1961).

7.034 A at -3°C.7 There are four differently oriented molecules in the unit cell, viz., at the cell corners and at the centers of the three cell faces. The arrangement of the molecules in the crystal is shown in Fig. 1. In

\ \ i \. , (I.m,n,) \.! " (J + l,m.n)

"'/ '-<.,' "'/ l-< \ LI_\-\-_--\c--L-~-_'t\~. \

\ 0 }... \ a o }..

}. ", )... I', I .......... I I .......... ""

I '\ / \

\ \ \ \

FIG. 1. Arrangement of benzene molecules in crystal.

the figure, the molecules drawn in the dotted lines lie higher or lower than those of full lines, by a distance of co/2.

This crystal has the symmetry elements of E, C2IJ

,

C2b, C2c, i, ITa, ITb, and ITc. In Table I the results of the factor group analysis of the crystal are shown. The correlations of the symmetry species of the point group, the site group, and the factor group are shown in Table II. Each normal mode of the nondegenerate species of a single molecule splits into four vibrational modes belonging to the different symmetry species of the crystal and that of the degenerate species splits into eight normal modes of the crystal.

The Bravais unit cell of the crystal consists of four benzene molecules as is shown in Fig. 2. Each Bravais

6 T. Shirnanouchi and I. Harada, J. Chern. Phys. 41, 2651 (1964). 7 E. G. Cox, Rev. Mod. Phys. 30, 159 (1958).

2016

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V I BRA T ION S 0 F CRY S TAL LIN E BEN ZEN E AND NAP H T HAL ENE 2017

TABLE I. Symmetry species and the character table of crystalline benzene.'

D2hl6 E C2a C2b C2c i Ua Ub Ue

A. 1 1 1 1 Au 1 1 1 -1 -1 -1 -1 B I • 1 -1 -1 1 1 -1 -1 B I • 1 -1 -1 -1 -1 1 1 B2• 1 -1 1 -1 1 -1 1 -1 B2u -1 -1 -1 -1 B3• -1 -1 -1 -1 B3u -1 -1 -1 1 -1

a ir: infrared, R: Raman, f: forbidden, Ni: number of degrees of freedom belonging to i species, T.: number of over-all translations of Bravais unit cell,

unit cell is expressed by (1, m, n), where the indices 1, m, and n specify the position along the a, b, and c axes, respectively. Among the four molecules, I, II, III, and IV, in the Bravais unit cell shown in Fig. 2, III and IV lie higher than I and II by co/2. The jth hydrogen atom of the ith molecule in the Bravais unit cell (1, m, n) is denoted by H j i(1, m, n), where i= I, II, III, and IV andj= 1,2,3,4,5, and 6. Likewise the jth carbon atom of the ith molecule in the Bravais unit cell (1, m, n) is denoted by C/(l, m, n). The Cartesian coordinates of the atoms belonging to the Molecule I are tabulated in Table III. The origin of the coordinate system is located at the center of gravity of the molecule. The Cartesian coordinates of the atoms belonging to the Molecules II, III, and IV are given by [(aoI2) +a, (bo/2) -b, -c], [ -a, (bo/2) +b, (co/2) -c], and [(ao/2)-a, -b, (co/2)+c], respectively, where a, b, and c are the coordinates of the corresponding atom of the Molecule I.

TABLE II. Correlation among the point group, the site group, and the factor group of crystalline benzene.

D6h C. D2h 15

1

AIg

~Ag A2g BI

E2g g g

B2g

B2g

B3g EIg

Blu

u~::u B2u

Elu

A2u

B2u

B3u E2u

N. T. T;' R;' n. ir R

18 0 0 3 15 f aaa, abb, ace

18 0 3 0 15 f f 18 0 0 3 15 f abc

18 1 2 0 15 Ma f 18 0 0 3 15 f aea

18 1 2 0 15 Mb f 18 0 0 3 15 f aab

18 2 0 15 Me f

T;': number of translational lattice modes, Ri': number of rotational lattice modes, n.: number of intramolecular vibrational normal modes.

Displacement Coordinates

The following three coordinate systems are used in the calculation.

(A) Internal Coordinates (R and R')

The internal coordinate vector R of a single molecule consists of the stretchings of the CH and the CC bonds, the deformations of the angles HCC and CCC, the torsional displacements around the CC bonds, and the CH out-of-plane deformations (Fig. 3). Internal sym­metry coordinates Rs, used for the determination of the intramolecular potential, are listed in Table IV.

(I,m,,,)

FIG. 2. Bravais unit cell of crystalline benzene.

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2018 1. HARADA AND T. SHIMANOUCHI

TABLE III. Crystal-fixed Cartesian coordinates of the atoms of and the molecule I in angstroms. a

a b C

H, -0.690 2.382 -0.065

H2 -1. 737 0.831 1.565

Ha -1.046 -1.551 1.630

H. 0.690 -2.382 0.065

H5 1. 737 -0.831 -1.565

He 1.046 1.551 -1.630

C, -0.389 1.341 -0.037 (3)

C2 -0.978 0.468 0.881

C3 -0.589 -0.873 0.918 wherej= 1,2, and 3 and i=I, II, III, and IV.

C. 0.389 -1.341 0.037 C5 0.978 -0.468 -0.881

(C) Crystal-Fixed Cartesian Coordinates (Y)

Ce 0.589 0.873 -0.918 As the crystal axes are perpendicular to each other in this case, we use them for the crystal-fixed Cartesian

a The coordinates for the hydrogen atoms are calculated from those of the coordinate axes. The vector, yi(l, m, n), is composed carbon atoms given hy Cox.

The intermolecular internal coordinate vector R' consists of the changes of the intermolecular hydrogen­hydrogen distances ql, q2, qa, q4, and q5 shown in Fig. 2.

(B) Molecule-Fixed Cartesian Coordinates (X)

We fix the right-handed Cartesian coordinate system to each molecule. The origin of the system is located at the center of gravity of the molecule. The z axis is taken normal to the plane of the molecule, the y axis lies in this plane and passes through the point midway between the atoms H5 and H6, while the x axis is perpendicular to the y and the z axes and passes through the atom HI. (See Fig. 3.) The displacements of the atom H/(l, m, n) are denoted by LlXH/(l, m, n), LlYH/(l, m, n), and ilzH/(I, m, n) in this coordinate system. Then the Cartesian coordinate column vector Xi(l, m, n) is given by the Cartesian displacement coordinates of the atoms of the ith molecule, the row vector Xi being defined by

The site symmetry Cartesian coordinates X_i, where s is g or u, are defined as follows:

ilxH;Ui= 1/v2 (LlXH/- LlXHJ+3 i ) ,

LlYH;Ui= 1/v2 (LlYH/- LlYHJ+3i) ,

ilzHli= 1/v2 (LlZH/- LlZHi+3i ) ,

LlXcli = 1/v2 (LlXc/- LlXCi+3i) ,

LlYc/i= 1/11 (Llyc/- LlYCJ+3i) ,

y

1 H,

r,

tx2

H6

R, C2 C6

X Z

C, c,

c. '%"4, 'Z"~, T~, -r: H, 8.

H.

FIG. 3. Molecular parameters of benzene molecule, 'i, Ri, a;, Pi, 0;, T;, T;', T;", and T;"'. 0 is the angle of CH out-of-plane deformation and T, T', Til, and T'll are the angles of internal rota­tion, HCCH, HCCC, CCCH, and CCCC, respectively. The internal coordinates are defined as follows:

t;=ARi,

S;=A1;,

r/ti=R(Aa,- Aai+1)/Y1,

<1>;= R (2A{3;-Aa,-Aai+1) / (6)t,

'Y;= AO; sin{3;,

6;= (AT;+AT;'+AT;"+AT;'")/4,

Llzc/i= 1/v2 (Llzc/- LlZCJ+3i) , (2) where R is the equilibrium CC distance.

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V I BRA T ION S 0 FeR Y S TAL LIN E BEN ZEN E AND NAP H T HAL ENE 2019

of the displacement coordinates of the atoms in the ith molecule,

yi(l, m, n)

= (.<:lam', .<:lbm" .<:lcm i , ••• , .<:lacs', .<:lbC6i , .<:lccai ) (1, m, n).

The following relations among R, X, Yexist:

R=B",Xi,

Yi=TiX',

Xi=TiY·,

(4)

(Sa)

(Sb)

(Sc)

where B", and Ti are the transformation matrices. The intermolecular internal coordinate vector R' is given by

R"· B , .. (Yi) B '''(T' T·)(Xi) 13= y t) = y'1) i, J ,

yi Xi (Sd)

where By'ij is the transformation matrix determined by the relative arrangements of molecules in the Bravais cell. The matrix elements of T' are composed of the sets of the direction cosines between (ai, bi, ci ) and Xi, yo, Zi).

Ti= (:: ::: ::).

la' mat' na'

(6)

The symmetry of the Bravais cell yields the following relations among Tt, TII, TIlI, and Tlv.

( " mIl ") -n:I

, TIl = -121 -m21

-tal -mal -nal

Cli -mll -.") TIII= li m21 n21 ,

-Ii -m31 -n31

Cl" -mll -") TIV= -Ii -m2I (7) -n2 .

laI ml nal

The elements of TI are calculated from the positions of the carbon atoms given by Cox.7

(

0.64758 -0.27828 0.70937)

TI= 0.16742 0.96014 0.22382 . (8)

-0.74338 -0.02618 0.66836

TABLE IV. Internal symmetry coordinates of benzene.

SI = (1/6)~(tl+t2+t,+t4+t.+t6)

S2= (l/6)~ (SI +S2+S3+S,+S.+S6)

S3= (1/6)t('h+,p.+,p3+,p.+,p6+,p6)

S,= (1/6)'(01-0.+1),-0,+06-06)

S5= (1/6)~(1'1-1'2+1'3-1"+1'5-1'6)

S6= (1/12)t(2q,1-<I>2-<I>2+2q,,-<I>6-<f>s)

S7= (1/12)'(2S1-S2-S3+2S,-S6-56)

Ss= (1/12)~(tl-2t2+t3+t,-2t5+t.)

Sg= (1/2) (,p2-,p3+"'.-,p.)

SlO= (3/52)t(2')'1+1'2-1',-21"-1'6+1'6) + (1/39)!(lh+202+ 03 -0, - 286-06)

Sn = (1/6)-(1'1+1"+1'3+1',+1'6+1'6)

S'2= (1/6)1(</>1-</>2+</>3-<1>4+</>5-</>6)

S13= (1/6)1 (SI-S2+S3-S,+S6-S6)

S14= (1/6)!(tl-tz+t.-t4+to-t.)

S15= (1/6)1 (,pl-if;2+,p,-t/l4+if;6-'t/la)

S16= (1/2) (Ih -0,+04-06)

S17= (1/12)1(21'1-1'2-1'3+21"-1'5-1'6)

SIS= (1/12)1(2sl+S2-S3-2s4-S5+S.)

S19= (1/10)I(tl-13-t4+t6) + (l/20)1( -2</>1-</>2+<1>3+2</>.+ </>5-<1>6)

S20= (1/2) ("'2+if;,-t/l5-if;6)

The Cartesian symmetry coordinates X( S) are ob­tained from the site symmetry coordinates Xu. and XU'. For example, the .<:lXHj( cr) 's are expressed as follows:

.<:lXHj(Ao) = Ij2(.<:lxHl I+.<:lXHl II+.<:lXHl III+.<:lXHl IV),

.<:lXUj(Blg) = Ij2(.<:lxH/ I+.<:lxnl II_.<:lXHl III_.<:lXHl IV),

LlXHj(B2y) = Ij2(.<:lxH/ I_.<:lXHl n+.<:lXH/ III_.<:lXHl IV),

LlXHj(Bsg)= Ij2(.<:lxHl I_.<:lXHl II-.<:lxn/ III+.<:lXHl IV),

.<:lXHJ(A u ) = 1/2 (.<:lXH u I+.<:lxHju II+.<:lXH/ III+.<:lXH/' IV),

.<:lXHj(BJu) = Ij2(.<:lxHjU I+.<:lxHju II_ .<:lxHju III_ .<:lXn/ IV),

.<:lXHj(B2U) = Ij2(.<:lxHjU 1_ .<:lXH/' II+.<:lxHju III_ .<:lXHju IV),

.<:lxHj(Bsu) = Ij2(.<:lxH/' 1_ .<:lXH/L I1-.<:lXH/' III+.<:lxu/ IV).

(9)

Similarly, .<:lYUi(cr) , .<:lZUj(cr) , .<:lXCj(cr), .<:lYC;(cr) , and .<:lzCj(cr) can be obtained from the corresponding site symmetry coordinates.

Potential Functions

(A) Intramolecular Potential

For the normal-coordinate treatment of the benzene crystal the intramolecular force constants of the indi­vidual molecule are needed. The authors have studied the vibrational frequencies of CoH6 and CoDe in detail and obtained a set of force constants on the basis of a general and well-conditioned force field. These are also

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2020 I. HARADA AND T. SHIMANOUCHI

TABLE V. Potential energy matrix of benzene and benzene-d, in millidynes per angstrom.

SI 7.506726 S2 0.543810 5.214165

(5.381754) S3 S, Si S6 0.433149 S7 -0.089478 5.140911

(5.238200) Ss 0.024374 0.271905 S9 0.0 0.0 SIO Sa 0.294179 SI2 0.443387 SI3 -0.014912

Sa Sa SI6 0.333063 S17 -0.050910 0.286834 SIS

S19 S20

consistent with those obtained for a series of basic molecules, which are based on the local symmetry force field.s In this treatment we take all the diagonal elements of the F matrix functionally dependent on the internal symmetry coordinates, given in Table IV, to be independent. As for the off-diagonal elements for the in-plane vibrations, we assume that they are determined by the repulsive forces between the non­bonded atoms.

TABLE VI. Observed and calculated frequencies of benzene and benzene-ds in em-I.

1'1

1'2

"3

'" "i "6 "7 "8 "9 VIO

VII

VI2

VI3

"I' VIi

"16

VI7

"18

VI9

V20

Poba

993 3074 1350

703 995 607

3055 1600 1177 849 675

1010 3057 1309 1146 405 975

3068 1482 1037

Peale Ill'

990 3 3074 0 1350 0

703 0 996 -1 604 3

3055 0 1603 -3 1189 -12 850 -1 676 -1

1016 -6 3057 0 1299 10 1149 -3 402 3 977 -2

3068 0 1486 -4 1024 13

"obe

946 2303 1059 601 827 579

2275 1558 868 662 497 970

2285 1282 824 352 793

2288 1333 814

8 T. Shimanouchi et al. (to be published).

Vealo

949 -3 2303 0 1050 -9 600 826 1 585 -6

2275 0 1558 0 857 11 661 1 496 1 966 4

2285 0 1287 -5 821 3 352 0 789 4

2288 0 1347 -14 797 17

0.257482 0.395704

-0.073110 0.261909

6.043580 0.157692 0.276925

0.367931

5.077206 (5.124673)

3.000745 -0.182087 0.243228

5.199276 (5.337342) 0.367166 2.940000 0.0 0.057581 0.266400

The potential energy matrix elements thus deter­mined are shown in Table V and the observed and the calculated frequencies are listed in Table VI. The details of the procedure of calculations and the discus­sion of the meaning of the force constants will be re­ported elsewhere.

The transformation of the matrix, F., based on the internal symmetry coordinates, R., into Fx. based on the site symmetry Cartesian coordinates, X', is per­formed as follows:

F,,=B.F.B.,

R.=B.X=UBX,

Fx.= UxgF",Uxg+ UxuFx1Jxu.

If we solve the secular equation,

I M-IF""M-l-Ell =0

(lOa)

(lOb)

( 11)

at this stage, we obtain the eigenvalues for a single molecule and the six zero roots corresponding to the over-all translations and rotations.

(B) Intermolecular Potential

As for the intermolecular potential of nonpolar molecular crystals like the crystalline n-paraffin, only the short-range hydrogen-hydrogen interactions seem significant in the problem of vibrational frequencies. 1•2•4

Therefore, in the present treatment, only the H·· . H interactions are taken into account, in order to see how well this simple assumption can explain the vibra­tional frequencies of the crystalline benzene. As is mentioned later, the short-range hydrogen-carbon in­teraction must be taken into consideration in the case of naphthalene, but it needs not to be taken into account for benzene because it has no short-range hydrogen-carbon distance smaller than 3.2 A. We

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V I BRA T ION S 0 FeR Y S TAL LIN E BEN ZEN E AND NAP H T HAL ENE 2021

assume the following quadratic potential for the inter­molecular H···H distances:

V'=! L (12) whole crystal

where lk's are the force constants and qk'S are the distances between the two hydrogen atoms belonging to the different pairs of molecules. There are five kinds of H··· H distances, ql, q2, qa, q4, and q., which are smaller than 3.0 A, as shown in Fig. 2. The distances larger than 3.0 A are not taken into account, because the corresponding force constants are too small to significantly affect the vibrational frequencies.

The change of the distance, ilqAB, between the atoms A and B is expressed in terms of the Y coordinates as

ilqAB= (1/qAB) [(aA-aB) (ilaA-ilaB)

+(h -bB) (ilbA - ilbB) +(CA -CB) (ilCA-ilcB)]' (13)

With the use of Eqs. (7) and (8), ilqAB is expressed in terms of the X coordinates. Upon substituting it into (12), we obtain the intermolecular potential matrix Fx'.

Spectroscopically Active Vibrations

The spectroscopically active normal vibrations in a crystal have the phase factors of unity. Accordingly, the spectroscopically active Cartesian coordinate vector xt and the spectroscopically active Cartesian sym­metry coordinate vector XC S) t can be given by

xt = (liN) LLL:X(l, m, n), (14) I m n

and

X(S)t= (liN) LLLX(S) (l, m, n). (15) I m n

We construct the intra- and intermolecular potential energy matrices based on these spectroscopically active Cartesian symmetry coordinates. The F matrices for the spectroscopically active vibrations, F,,/ and F,,( S) t, are obtained by adding the corresponding elements of the intra- and intermolecular F matrices5 :

where

and

and F,,(S)t= L[Fx(u)t+F.,'(u)t],

"

where F x ( 17 ) t and F.,' ( (7) t are the minor rna trices factored by the use of the symmetry coordinates given in Eq. (9). The summation is over eight species, A,,, Blr" B2g, Bag, Au, BIu , B2u, and B3u•

If we solve the secular equation

all the spectroscopically active normal frequencies in the crystal are obtained.

Summary of Experimental Results

Hollenberg and Dows9 have observed the splittings of the absorption bands at 85° and l55°K. Since the crystal-structure analysis referred to above has been carried out at -3°C, the authors tried to measure the spectra at higher temperature with the use of the high-resolution infrared spectrometer. The spectra at 2500 K show that most of the bands are broadened at this temperature and the multiplet structures observed at lower temperature become less definite. The Pl&

band belonging to the b2u species is observed to split into a doublet, the band centers of which are 1147.2 and 1143.2 cm-l at 250°K. Hollenberg and Dows give the magnitude of this splitting to be 5.22 cm-1 at 85°K and 5.50 cm-1 at 155°K, as compared with the 4.0 cm-1 at 2500 K observed by the authors. The 1117 band belonging to the e2u species is observed as a triplet at 981.8, 975.3, and 973.5 cm-I • The magnitudes of the splittings observed by Hollenberg and Dows at 155° and 85°K for this band are different from these and, moreover, three weak bands appear in addition to the triplet.

On the whole, the splittings of bands are less definite at 2500 K and the magnitudes of these splittings do not show any regular change. Therefore, only the maximum splittings of the bands observed at 85°K are given in Table X. Some of the shapes of the bands observed by Hollenberg and Dows are reproduced in Fig. 5.

The low-frequency region of the Raman effect of the benzene crystal has been studied by Ichishima10 and Fruhling.ll Fruhling has observed four Raman lines at O°C in the low-frequency region, which are reasonably assigned to the symmetry species of the crystal as shown in Table IX. Ichishima showed that these fre­quencies change linearly with the change in the obser­vation temperature.

Results of Calculation and Force Constants

The procedures adopted in the present calculation are as follows. At first, the force constants, 11> 12, fa, h,

OJ. L. Hollenberg and D. A. Dows, J. Chern. Phys. 37, 1300 (1962).

10 I. Ichishirna, Nippon Kagaku Zasshi 70. 391 (1949). 11 A. Fruhling, Ann. Phys. (N. Y.) 6,401 (1951).

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2022 1. HARADA AND T. SHIMANOUCHI

TABLE VII. Short-range intermolecular H···H distances and the force constants.

-----~-- _._---

Interatomic Force constant distance (A) (mdyn/A)

Benzene

q1 2.628 0.016 q2 2.698 0.012 q3 2.764 0.010 q. 2.767 0.0096 q. 2.826 0.0080

Naphthalene

q, 2.405 0.031 q2 2.833 0.008 q3 3.076 0.003

q. 2.683 0.012 q. 2.776 0.009 qa 2.927 0.005 q7 2.955 0.005

and /6, were calculated from the hydrogen-hydrogen interactions given by Buckingham,12 MUller,13 and Bartelll4 and the frequencies of the lattice vibrations and the band splittings were calculated from these three sets of force constants. The frequencies of the lattice vibrations calculated were somewhat different from those observed. The frequencies obtained from Buckingham's equation were too high. Those obtained from MUller's and Bartell's equations were too low. Ac­cordingly, the five force constants were refined so as to yield the best fits with the frequencies, 35 and 63 cm-l

of Ag, 105 cm-l of Big, 63 cm-l of B2g, and 105 cm-l of B3g , the assignments for which are definitely given by Fruhling. The values of these force constants are taken so that they are consistent with each other too. The force constants thus obtained are listed in Table VII. The results of the calculation are shown in Tables VIII and IX. The calculated maximum splittings obtained from Table VIII are compared with the observed ones in Table X. On the whole, the force constants, giving the best fits for the lattice vibrations, also explain the magnitudes of the splittings of the absorption bands.

The correlation between the R···R distances and the values of the force constants obtained here is shown in Fig. 4. The curve in the figure is the second derivative of the repulsive part of de Boer's potential. The five force constants of benzene agree well with the curvature. This fact gives a further support to the conclusion ob­tained for the vibrational analysis of polymethylene crystal.2

12 R. A. Buckingham, Trans. Faraday Soc. 54, 453 (1958). 13 A. MUller, Proc. Roy. Soc. (London) AI54, 624 (1936);

ibid. AI7S, 227 (1941). 14 L. S. Bartell, J. Chern. Phys. 32, 831 (1960).

Lattice Vibrations

The agreement of the data shown in Table IX is excellent. All the observed lattice frequencies are straightforwardly explained by the reasonable set of the hydrogen-hydrogen force constants. The Raman line at 35 cm-l observed at ooe corresponds to the two calculated frequencies, 28 and 33 em-I. Recently, Ito15

studied the Raman spectra of crystalline benzene at 4.2°K and observed a doublet at 64 (m.) and 69 cm-1

(v.w.). Taking into account the frequency change due to the temperature change as given by Ichishima, one can reasonably assign the stronger line at 64 cm-1 to the Ag vibration calculated to be 28 cm-1 and the weaker line at 69 cm-I to the Big vibration calculated as 33 em-I.

Ito also observed a doublet at 100 cm-1 (w.) and 103 cm-I (v.w.) at 4.2°K which correspond to the line at 69 cm-l at ooe. The calculation gives two fre­quencies at 69 and 72 em-I and explains the splitting very well. The displacements of hydrogen atoms given by the calculated Ly matrix for the Raman active lattice vibrations reveal the nature of the vibrations as shown in Fig. 5.

Table IX includes also the calculated frequencies for the translational lattice vibrations. These vibrations are expected to appear in the far-infrared region, although they have not yet been measured. All the experimental results support the results of the calcula­tions given in Table IX.

mdyn/A 0.05

0.03

0.02

0.01

• Benzene

o Naphthalene

-deBoer

OLI_--L1_-L1 _--1-1 ~~.~_--L~ 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 A

FIG. 4. Values of H···H force constants obtained from crystal spectra.

15 M. Ito (private communication).

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V I BRA T ION S 0 F CRY S TAL LIN E BEN ZEN E AND NAP H T HAL ENE 2023

TABLE VIII. Calculated normal frequencies of crystalline benzene (-3°C) in cm-I •

B3a

(1) gerade species VI 990.6 990.5 990.2 989.8

1'2 3077.2 3076.2 3074.9 3073.5

V3 l352.6 1351.6 l356.6 l355.5

V4 709.1 704.4 708.8 704.8

V6 1014.9 1000.2 1013.6 1001.1

V6 { 604.1 604.3 604.7 604.6 603.2 603.4 603.3 603.9

V7 { 3057.1 3057.5 3059.1 3058.8 3054.8 3055.3 3055.0 3056.7

Vs { 1603.2 1603.2 1603.7 1603.9 1602.7 1602.9 1602.7 1602.6

V9 { 1198.2 1198.8 1194.4 1195.7 1191.6 1192.5 1189.6 1189.6

VIO { 861.4 874.0 862.1 873.1 856.1 860.2 856.6 860.0

Lattice modes 78.7 102.3 75.5 100.2 56.5 70.3 68.9 28.2 32.7 56.0

Splittings of Absorption Bands

(A) eH Out-oj-Plane Vibrations, Vu (au) and V17( e2u)

72.1 59.9

The calculated values of Vu are 706.1 (b2u), 685.5 (bau ) , and 683.2 em-I (bl .. ) , the magnitudes of the

TABLE IX. Observed and calculated lattice vibrational frequencies of crystalline benzene (-3°e).a

Symmetry species Vob, (cm-I)b Veale (cm-I)

Au 35 28 63 57

79

Bla (35) 33 70

105 102

B2a 63 56 (69) 69

76 Baa 60

(69) 72 105 100

Au 26 49 58

Blu 34 58

B2u 31 57

Bau 30 44

a The notations in the table are different from those used by Fruhling. The assignments of the frequencies given in parentheses are tentative.

b Reference 11.

(2) ungerade species Vu 706.6 683.2 706.1 685.5

VI2 1015.9 1015.8 1015.7 1015.3

VI3 3060.2 3058.9 3058.3 3056.9

Va 1302.1 l301.9 1302.9 1302.7

VI6 1156.2 1154.9 1162.2 1160.7

VI6 { 410.2 418.6 410.4 417.4 406.5 409.1 406.8 409.3

VI7 { 987.7 997.7 987.1 995.9 987.3 985.5 983.5 986.2

VIS f 3070.3 3070.7 3071.8 3071. 7 l 3067.4 3068.2 3067.8 3069.3

VI9 { 1489.8 1490.3 1487.3 1487.7 1487.0 1487.1 1485.9 1486.1

V20 { 1029.9 1029.3 1029.5 1031.3 1026.2 1027.9 1025.1 1024.3

Lattice modes 57.5 58.3 57.3 44.0 49.4 33.7 30.5 30.0 26.3 0 0 0

splittings being 20.6 and 2.3 em-I. The corresponding observed values are 25.3 and 1.7 cm-I at 8S oK and 24.1 and 2.5 cm-I at lSS°K. The agreement between the calculated and the observed values is satisfactory as shown in Fig. 6. In the case of the degenerate CH out-of-plane bending vibration, V17, the calculated values are 997.7 and 985.5 cm-1 (bIu ), 987.1 and 983.5 (b2u ), and 995.9 and 986.2 cm-I (bau ). The splittings are too complicated for the calculated frequencies to have one-to-one correspondence with the observed ones. Accordingly, we take the difference between the highest calculated frequency and the lowest one and compare it with the maximum splitting observed. They are 14 and 16 em-I, respectively, as given in Table X, and are in good agreement with each other.

TABLE X. Maximum frequency splittings of crystalline benzene in cm-I •

,dVobe AVcalc

VII 27 23 VI2 0.5(4) 0.5 Via 0 2 VI4 1.5(4) 1 VI6 5 7 VI6 15 12 VI7 16 14 VIS 4 VI9 6 4 V20 8 7

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2024

AI, \-3

.... /). ....... 'i'

f "i '-'1 \ \ -7

\ \. +7 )... ,10. ...

z:::~~ -,

60c .. 1

I. HARADA AND T. SHIMANOUCHI

:;;'-"1 1+3 I +16

\ ...... ""\ ~' , I Y I I

.( I ...... , I

I ,,"\ -16 r -3\ 1- 18 ;

,-+0 1

+"X-I:%, 7,'-Y . ./ , .. / "\

\ -14 1 1 1 )-,

~~ 1

--' -0

"13

,."

(\-15 .. ,,A +4 +<X,,\.,.,

I I ..( I

~' '+I}V.

-4 \ ........ ,

~ +18 /H5

,!~; ~""15 \ +18 }, ...... / -4

...... I \ +18

/-.'\ \. H/ -18 \ , .... ".../ >-, I

/ 'i +4

L: ' .J -15 -18 -15

~ ~' -5'.. ...... 1, +5

I ')-:> I I I I

,..( I

~ \ ....... \ +12 r ~

----- ~ '-::1' '. ,~;}!7

~"""·l "<\ \ \ \ \ \ f.. .. +5 ).. I-t-?- I

"'12\~ +17

100cm-1

FIG. 5. Displacements of hydrogen atoms for the Raman active lattice vibrations.

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V I BRA T ION S 0 FeR Y S TAL LIN E BEN ZEN E AND NAP H T HAL ENE 2025

1'ob,.

i

680 690 700

FIG. 6. Observed and calculated splittings. Vob •.

A"

710 em-I

B"

i

680

Vuk

/I i i

690 700

Veale.

.

A II i i

1300 1310

(B) eH In-Plane Deformations, P1s(b2u ) and P20(eIu)

AA 1140 1150

The observed and the calculated frequencies for PIS are shown in Fig. 6. The ratio of the intensities of the observed doublet is 2: 1, which coincides with the dis­distribution of the calculated frequencies. The mag­nitudes of the splittings are 5 cm-I for the observed frequencies and 7 cm-I for the calculated ones. For P20 also the calculated maximum splitting is in good agree­ment with the observed one as shown in Table X.

(e) Skeletal Vibrations

Of all the ring vibrations, P16(e2u), the ring out-of­plane deformation vibration, shows the largest splitting. The observed frequencies of PI6 are 419.2 and 404.5 em-I. The calculated values also split into two groups. For PI2, PI4, and PIg, the magnitudes of the splittings are small. The agreements are satisfactory for all the split­tings. According to Hollenberg and Dows, PI2 has a weak satellite band on the lower-frequency side and PI4 has one on the higher-frequency side, but the results of our calculation cannot explain these as the funda­mental bands.

(D) eH Stretching Vibrations

No splittings are observed for PIs(b1u). The calculated splittings are also very small. Since P1S(elu) is in Fermi

i i i

1320 em-I 1300 1310 em I

i I i II em I 1150 1160 1170 em -I

resonance with PI9+P8, the observed splittings can not be compared with the calculated ones.

PART II. NAPHTHALENE

Introduction

While studying the infrared spectra of crystalline naphthalene, we observed three weak absorptions in the low-frequency region which can be assigned to the three infrared-active translational lattice vibrations. With these and the Raman active lattice vibrations studied by Kastler and RoussetI6 and Ichishima,17 we performed a normal-coordinate treatment of lattice vibrations and studied the intermolecular forces.

For the complete normal-coordinate treatment of crystalline naphthalene, the intramolecular force con­stants are needed. However, even the vibrational assign­ments for this molecule have not yet been established. The vibrational spectra of the naphthalene molecule have been studied by Pimentel and his co-workers,18-21

16 A. Kastler and A. Rousset, J. Phys. Radium 2, 49 (1941). 17 I. Ichishima, Nippon Kagaku Zasshi 71, 607 (1950). 18 G. C. Pimentel and A. L. McClellan, J. Chern. Phys. 20,

270 (1952). IV W. B. Person, G. C. Pimentel, and O. Schnepp, J. Chern.

Phys. 32,230 (1955). 20 G. C. Pimentel, A. L. McClellan, W. B. Person, and O.

Schnepp, J. Chern. Phys. 23,234 (1955). 21 A. L. McClellan and G. C. Pimentel, J. Chern. Phys. 23,

245 (1955).

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2026 I. HARADA AND T. SHJMANOUCHI

c' = c sin f3

8

7

B),m,n+l

A),m,n+l

A),m,n

Mitra and Bernstein,22 Freeman and ROSS,23 and Scully and Whiffen.24 There are some ambiguities in the assignments.

In the present study the approximation method given in a previous paper is used.6 The treatment includes only the over-all translations and rotations and the information on the intramolecular potential energy is not needed.

Crystal Structure

The symmetry of crystalline naphthalene is mono­clinic and the space group is P21/a(C2h

6) with unit cell dimensions of ao= 8.235, bo= 6.003, Co= 8.658 A, and (3= 122°55' at room temperature.25 There are two mole­cules in the unit cell, viz., at the cell corners and at the center of the ab face. The arrangement of the molecules in the crystal is shown in Fig. 7. This crystal has the

2" S. S. Mitra and H. J. Bernstein, Can. J. Chern. 37, 553 (1959).

23 D. E. Freeman and I. G. Ross, Spectrochim. Acta 16, 1393 (1960) .

• 4 D. B. Scully and D. H. Whiffen, Spectrochim. Acta 16, 1409 (1960).

26 S. C. Abrahams, J. M. Robertson, and J. G. White, Acta Cryst. 2, 233, 238 (1949).

8

B),m,n

FIG. 7. Arrangement of naph­thalene molecules in crystal. qa is the distance between HaA(l, m, n) and H7A (l, m+ 1, n). q7'S are the distances between HaA (l, m, n) and HaA(l, m+l, n) and H.A(l, m, n) and H7A(l, m+l, n).

symmetry elements of E, C2b, (Ta, and i. In Table XI, the results of the factor group analysis are shown. The correlations among the symmetry species of the point group, the site group, and the factor group are shown in Table XII. Each gerade normal mode of a molecule is split into Ag and Bg and each ungerade mode into Au and Bu- Beside these, there are six rotational (3Ag and 3Bg ) and three translational (2Au and lBu) lattice vibrations.

Calculation of Lattice Vibrations

The spectroscopically active intermolecular potential energy matrix F.,'t of naphthalene may be calculated in the same way as the one for crystalline benzene. F/t is transformed into FTR, the matrix expressed by the coordinates corresponding to the over-all transla­tions and rotations of molecules, by the use of the L",TR

matrix as follows:

FTR= L",TRF",'tL",TR.

The eigenvalues of FTR give the lattice frequencies of naphthalene crystal. The calculation can be simplified according to the symmetry consideration. A similar method to that given in a previous paper6 can be applied.

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V I BRA T ION S 0 F CRY S TAL LIN E BEN ZEN E AND NAP H T HAL ENE 2027

TABLE XI. Symmetry species and the character table of crystalline naphthalene.·

C2h6 E C2b (Ta N.

A. 1 27

Au 1 -1 -1 27

B. -1 -1 27

B" -1 -1 27

• As for the notations, see Table I.

Intermolecular Potential

The results of the vibrational analysis of benzene suggest that the H···H repulsive forces play the main role in the vibrational problem of crystalline naphtha­lene. The calculation was first carried out by taking into account the H···H interaction only. However, there are several short-range intermolecular hydrogen­carbon distances in this crystal. Therefore, in the second calculation the forces between these short-range hydro­gen-carbon distances are taken into account.

Experimental Results

All the Raman active lattice frequencies have been observed by Kastler and Rousset.16 The authors studied the far-infrared spectra of the crystal using a vacuum and double-beam Hitachi FIS-1 far-infrared spectrom­eter. The sample used for the measurement was mixed with paraffin at about 55°C and was homogeneously spread over a small piece of polyethylene film. The spectrum obtained is shown in Fig. 8. The bands ob­served at the higher-frequency region are assigned to the skeletal deformation frequencies which are split owing to the crystal field. The bands at 98, 66, and 53 cm-l are assigned to the translational lattice vibra­tions.

TABLE XII. Correlations among the point group, the site group and the factor group of crystalline naphthalene.

T. T;' R;' no ir R

0 0 3 24 f £l:aa, abb, ace, aca

1 2 0 24 Mb f 0 0 3 24 f aab, aoe

2 1 0 24 Ma,M. f

Results of Calculation and Discussion

The H···H distances, ql, q2, and q3, the force con­stants of which contribute to the infrared-active vibra­tions, are shown in Fig. 7. We assumed a proper set of force constants for these distances, considering the results of the analysis of benzene and de Boer's poten­tial. The force constants are given in Table VII. The results of calculation are listed in Table XIII as Calc. I. Each calculated value is equal to about one-half of the corresponding observed frequency. In order to give a good agreement by using only these three H···H forces the values of the force constants have to be far larger than those expected from de Boer's potential. This may not be reasonable. There are two hydrogen­carbon pairs, which are closer than 2.9 A to each other, and additional three pairs which are closer than 3.1 A. We assumed forces for these pairs and adjusted the force constants so that the calculated frequencies of the ungerade species fit to the observed ones. The H···R repulsive force constants were fixed throughout the calculation. The force constants are given in Table XIV and the results are listed in Table XIII as Calc. II. There are four other R···R forces which contribute to the Raman active modes. The corresponding distances q4, q6, q6, and q7 are listed in Table X. For the frequencies of gerade species in Calc. II, the force constants for q4' •• q7 calculated from de Boer's potential are used in

% 100 ,-------------------,

~ / 98

50

175

~OLO----------~2~00~----------~10~0---750~cm~-·1

FIG. 8. Far-infrared spectra of crystalline naphthalene.

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2028 I. HARADA AND T. SHIMANOUCHI

TABLE XIII. Observed and calculated lattice frequencies of crystalline naphthalene (24°C) in em-I.

Paalo

Symmetry -------------species Pobs Calc. I Calc. II Calc. III

Au 98 52 97 104 53 21 54 58

Bu 66 37 74 76

AD 127 115 119 76 85 86 54 58 62

B. 109 82 93 74 70 77 46 42 47

addition to the R···R and C···R force constants referred to above. In Calc. II, the value of 82 cm-I of Bg is in poor agreement with the observed frequency, 109 em-I. Calc. III gives the frequencies calculated from the force constants of the above seven R··· R pairs and those of the 10 C···R pairs given in Table XIV, which are closer than 3.3 A.

The present method of calculation includes some approximations. There are uncertainties of a few wave­numbers in some of the observed frequencies. Further­more, the temperatures at which the infrared measur~­ment, the Raman measurement, and the x-ray analysIs were made are not exactly the same. The agreement between the calculated and the observed frequencies is satisfactory when the above factors are taken into account. It may be interesting to compare these C· .. R force constants with those obtained from the normal­coordinate treatment of the intramolecular vibrations. The C···R distances, RcR'S, and the Urey-Bradley force constants, FCR's, of ethane,8 ethylene,8 and benzene8 are as follows:

Ethane Ethylene Benzene Naphthalene In termolecular

RCH(A) 2.1 2.1 2.2

12.8 3.0 3.2

FCH(mdynj A) 0.375 0.389 0.353 0.045 0.012 0.004.

TABLE XIV. Short-range C···H distances and force constants of crystalline naphthalene.

Interatomic distance (A) Force constant (mdyn/ A)

2.817,2.820 0.045 (Calc. II and III)

3.026,3.034,3.049 0.012 (Calc. II and III)

3.141,3.207,3.237 0.004 (Calc. III)

3.242,3.294

The magnitudes of these values are consistent with each other.

CONCLUSION

The lattice vibrations and the band splittings of crystalline benzene were both successfully analyzed by the normal-coordinate treatment of the whole crystal, on the nonbonded hydrogen-hydrogen interactions being assumed as the intermolecular forces. For crystal­line naphthalene, on the other hand, the contributions of the nonbonded hydrogen-carbon interactions are comparable to those of the hydrogen-hydrogen interac­tion. All the lattice frequencies are explained reasonably from the force constants for these two kinds of inter­molecular forces.

The results of the analyses of these crystals support the view that the de Boer-type hydrogen-hydrogen interaction plays the dominant role in the vibrational problem of the crystalline hydrocarbons and, moreover, the hydrogen-carbon interaction is also important when the crystal has the short intermolecular C···R distances.

Recently, Craig, Mason, Pauling, and Santry calcu­lated the lattice energies of aromatic hydrocarbons26 and concluded that the repulsive interactions are of prime importance. This conclusion is in agreement with ours.

26 D. P. Craig, R. Mason, P. Pauling, and D. P. Santry, Proc. Roy. Soc. (London) A286, 98 (1965).

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