74
Vol. 62
SIDNEYGOLDEN
constitutes a unique advantage of the Rayleigh obtaining DA and QR are free from such effects. method; although the accuracy of the extrapolaAcknowledgments.-The author wishes to thank tion is not very high, the method is likely to be use- Dr. J. W. Williams for placing the facilities of his ful where material is in short supply. Calculations laboratory at the author’s disposal, and for his of this kind are susceptible to errors arising from helpful criticisms during the course of the work and the “Wiener” skewness a n ~ m a l y ,and ~ . ~accord~ ~ ~ ~ the preparation of the MS. He is also greatly iningly must be made only from exposures obtained debted to Dr. L. J. Gosting for many fruitful disa t relatively long times; the methods given for cussions, and to Dr. R. L. Baldwin, who gave valuable advice and assistance in the ultracentri(41) H.Svensson, Opt. Acta, 1, 25 (1954); 1, 90 (1954). JOURNAL, 61, 244 (1957). (42) L. G. Longsworth, THIS fuge experiments.
EVALUATION OF THE PARTITION FUNCTION FOR RESTRICTED INTERNAL ROTATION BY SIDNEY GOLDEN Chemistry Department, Brandeis University, Waltham, Massachusetts Received July 1, 1067 I
,
l h e partition function for restricted internal rotation is evaluated exactly for the model of two coaxial tops treated by Koehler and Dennison. The influence of nuclear-spin statistics is not taken into account and restricts the present results to temperatures which are not too low. The partition function which is obtained is shown explicitly to be symmetric in the moments of inertia of either top.
Introduction f n a series of papers, Pitzer and co-workers’ have presented a theory and extensive numerical tables which enable good estimates to be made of the thermodynamic functions for molecules with restricted internal rotation. In another series of papers, Halford2 has given an alternative theory which also enables good estimates to be made of the thennodyvamic functions of such molecules. In fact, whlle they differ in detail, both theories employ the same basic theory of the restricted internal rotator due to Dennison and ca-workers3and the results of both theories are generally in agreement. However, two important questions have been raised by Halford and these furnish the motivation for the present paper. First, the adequacy of the integration procedure employed by Li and Pitzer,’ which procedure approximates the average of the pdrtition function for restricted internal rotation over external rotational states, needs examination to assess the reliability of their theory. Second, since the model of two coaxial tops was treated originally by Koehler and Dennison3 in a manner which apparently introduced an asymmetric dependence upon one of the tops (or the other), the resulting statistical mechanical theory reflects an ambiguity related t o which top is employed. However, examination of the Koehler-Dennison3 theory discloses that the set of energy levels for restricted internal rotation do not have this ambiguity. The ambiguity which persists in the statistical mechanical theory, therefore, requires further examination. ( I ) K. 8. Piteer, J. Chehem. Phya., 5, 469 (1937); 14, 239 (1946);
K.9. Piteer and W. D. Gwinn, ibid., 10,428 (1942); J. E. Kilpatriok and K. S. Pitzer, ibid., 17, 1064 (1949); J. C. M. Li and K. S. Pitzer, THISJOURNAL, 60, 466 (1956); Bee, also, E. V. Ivaeh, J. C. M. Li and K. S. Pitzer, J. Chem. P h y s . , 28, 1814 (1955). (2) J. 0.Halford, ibid., 16, 645 (1947); 16, 410 (1948); 16, 444 (1950);18, 1051 (1950); 26, 851 (1957). (3) J. 8. Koehler and D. M. Dennison, Phye. Rev., 67, 1006 (1940); D.G.Burkhard and D, M, peqnjson, ibjd., 84,408 (1951).
In the present paper the essential result is an exact evaluation of the average of the partition function for restricted internal rotation over external rotational states for themodel of two coaxial tops with a symmetry number of three for internal rotation. It is expressed as a rapidly convergent series of terms depending upon the quantum number K which is associated with angular momentum about. the symmetry axis of the molecule. Certain of the terms are identical with those obtained by Li and Pitzer.l Retention of these alone, however, introduces the asymmetry indicated previously, for the exact expression can be shown to be symmetric with reference to either top. Mathematical Treatment.-The problem to be considered here is that of evaluating the quantity
where 8 = ha/8irzCkT
and t,he energy value for restricted internal rotation E K is~measured ~ from the lowest value; A is the moment of inertia perpendicular to the symmetric axis of the molecule; C = C1 Cz is the sum of the moments of inertia of the coaxial tops about the symmetry axis; u is the total symmetry number for over-all rotation; X is the symmetry number for internal rotation; the remaining symbols have their usual meanings. Now, Koehler and Dennison3 have shown that E K is~a 2~~ p e r i o d i cfunction of the quantity
+
Pr
= ( 2 H c l / s c ) K f 2ir&/s
(2)
where P, depends only upon r . For S = 3, Or assumes the values 0, 1, 2. Presumably for other values of X, pr may take on integral values 0 6 P r 6 s - 1. Because of the periodic character of
the E Kit ~is justifiable ~ to express m
Q(pr)
= Re
m
a,,, COS mpr
exp ( - - E K ~ ~ / / c = T)
=
75
PARTITION FUNCTION FOR RESTRICTED INTERNAL ROTATION
Jan., 1958
m=O
n=O
where the Fourier coefficients, am,are readily evaluated from Q(pr). Insertion of this expression into eq. 1 yields
1 1
- exp(2~im)
1
- exp(2~im/S)
8,m = pS, p an integer
= =
(3)
1
0, otherwise
Thus
Now This expression has been approximated by Ivash, Li and Pitzer,' and Li and Pitzerl with an integration over K. For the present purpose, however, eq. 4 will be converted into a rapidly convergent series with the aid of an identity due to P o i ~ s o n . ~This can be expressed as
c2/c = 1 - CI/C
(9)
and one can obtain
." "
p=O
exp {
L=-m
- ( P C Z / C+ L)2r2/6j (10)
where - L = K - p . Since the summatiom include all possible values of K and L, eq. 8 and 10 are identical in form and are thus symmetric in C1 exp (-n2y + inz) = (T/y)*/z and C2. Because 6 generally is small the series in n=-m +m K or L may be expected to converge rapidly. exp { - (3 2 ~ n ) ~ / 4(5)~ ) Remarks.-One may note that the terms in n=--m eq. 7 with m = 0, 1 and with K = 0, but without To employ this expression it proves convenient to the summation over T being carried out explicitly, represent QT in terms of imaginary exponentials. correspond to those obtained by Li and Pitzer.' If only these are,retained, the resulting expression Since for QT cannot yield a symmetric dependence upon 1 CI and CZ. On the other hand, Ivash, Li and Pitcos mx = - (exp (imz) + exp( - i m z ) ] 2 zerl have effected the summation over r in a manner similar to the present one. However, they have employed an integration procedure for effecting a summation over K. The net effect is to retain only those terms of eq. 8 with K = 0. Again, the symmetry in Cl and CZis lost as a result. m#O The present treatment tacitly neglects the influwhere a _ , = am. No difficulty is involved in inter- ence of nuclear-spin statistics and permits, thereby, changing the order of summation. Carrying this the summation over r and K to be carried out simply. As a result, the treatment may be presumed out and employing eq. 2 and 5, one obtains adequate for temperatures which are not too low. However, when spin-statistics are to be involved explicitly the summations mentioned may be expected to be more complicated since the over-all +m symmetry of the various states will depend upon eup { - (mCI/SC + K ) w / / s ] ( 7 ) K=-m both T and K . This feature has been stressed by Halford.% Now since the sum over K is independent of r, the From the point of view of practical calculations sum over the latter is carried out simply. One has one requires only the Fourier coefficients aPs, to evaluate where p is a non-negative integer. These, howS=1 8-1 ever, can only give an appreciable contribution cos ( 2 ? r m n / ~ ) Re {exp ( 2 ? r i m / ~ ) j * K ) z or (pCz/C L ) 2 is when either (pC1 C n =O n=O very much smaller than 6/n2. Since the latter generally is small these terms will make the great(4) See, for example, H. Bateman, "Partial Differential Equation8 est contribution when Cl/C is a rational fraction, an of Mathematical Physics," Dover Publications, New York, N. Y.. 1944, pp. 216-217. occurrence which may be termed accidental. +m
+
=i
+
+
