On the rotational partition function for tetrahedral molecules

J. Qume

Spectrosc.

Radiat.

Transfer.

Vol.

10, pp.

1335-1342.

Pergamon

Press 1970. Printed in Great

Britain

ON THE ROTATIONAL PARTITION FUNCTION FOR TETRAHEDRAL MOLECULES K. Fox Jet Propulsion Laboratory,* California Institute of Technology, Pasadena, California 91103 and Department of Physics and Astronomy,t

The University of Tennessee, Knoxville, Tennessee 37916

(Received 4 June 1970)

Abstract-A rigorous calculation of the quantum-mechanical

rotational partition function for tetrahedral XY, molecules yields Q, = (l/12)(21,+ l)4rr’/2a-“‘2 exp(u/4), where I, is the spin of the Y nucleus, and a = Bhc/kT. This result is accurate to 1 per cent or better for all values of B and Tsuch that c( < l/5.

I. INTRODUCTION THE PtutTITIoN

function, or “sum over states,” is important in a wide variety of physical and chemical applications. (l) Recently, for example, the quantum-mechanical rotational partition function Q, was required as a function of temperature T in a determination of the abundance of methane in the upper atmosphere of the planet Jupiter.‘2*3’ The purpose of the present paper is to elucidate the calculation of Q, as a function of T for tetrahedral molecules. Emphasis will be placed on the inclusion of nuclear-spin statistical weight factors in a rigorous way, and on the evaluation of infinite sums in closed form with rigorous error bounds. For a tetrahedral molecule, the spin I, of the Y nucleus may take the value l/2 as in CH, , 1 as in CD,, or 3/2 as in Ccl,. The total nuclear-spin degeneracy (21, f 1)4 must be distributed among the rotational states of the molecule according to the generalized Pauli exclusion principle for identical fermions or bosons. The group-theoretical derivation of the expression for Q, is given in Section II. Q, consists of infinite sums which depend on T. Except for extremely low temperatures, these sums converge slowly. The Euler-Maclaurin summation formula’4) was used previously to obtain asymptotic expansions for the sums in QI.(sV6)In Section III, an alternative method is presented in which each of the sums is expressed in terms of derivatives of theta functions,“) and transformations are applied to speed the convergence of the sums. A closed-form expression for Q, results.

* Work done while N.R.C.-NASA. 7 Present address.

Resident Research Associate (1967-69). 1335

1336

K. Fox

In the final Section, the form of Q, derived in Section III is compared with earlier results. Several recent applications are discussed. II. ROTATIONAL

PARTITION FUNCTION

The basic expression for the quantum-mechanical

rotational

partition function is(‘)

Q, = 2 gJ exk-E.&T) J=O

(1)

where J is the rotational quantum number corresponding to the total angular momentum, g, is the statistical weight (including nuclear spin) of the state of rotational energy E,, k is Boltzmann’s constant, and T is the absolute temperature. The energy of a rotational state is taken to be E, = BJ(J + l)hc

(2)

where B is the rotational constant for a spherical top, h is Planck’s constant, and c is the speed of light in vacuum. Vibration-rotation interactions which contribute small corrections to E, are neglected in the calculation of Q, . The statistical weight g, may be calculated from group-theoretical considerations. A tetrahedral XV, molecule belongs to the crystallographic point group q.(l) This group has one-, two-, and three-dimensional irreducible representations labelled A, E, and F, respectively.* The (21, + 1)4 nuclear-spin functions can be classified according to these reps. This is an essential step in the construction of a total quantum-mechanical wave function which has the correct transformation properties under an interchange of identical nuclei. Table 1 gives the number of occurrences (1,8,9)of each rep for I, = 0, l/2, 1, and 3/2. The nuclear-spin functions must be combined with the rotational functions in such a way that the product functions transform like A. The rotational functions for a spherical top are (23 + 1)-fold degenerate with respect to the projection of the total angular momentum J on a molecule-hxed z-axis. The number of occurrences of each rep of & is given for the rotational functions by the formulas(“) in Table 1. Product functions of type A can be produced only by combining spin and rotational functions belonging to the same rep. Reduction of group rep products results in the spin-rotation statistical weight. For example, for I, = l/2, g; = (5/12)[W+1+3(-1)J+(16/J3)sin(2J+l)~/3] +(2/12)[25+1+3(-1)J-~l6/~3)sin(2J+1)~/3]+(3/4)[25+1-(-1)J],

(2a)

TABLE ~.NUMBEROFOCCURRENCESOFREPSOFT~FORSPINANDROTATIONALF~JNC~ONSOF TETRAHEDRAL

1,

0

A E

1 0

F

0

l/2

1

312

5

15 6 18

36 20 60

1 3

XY,

MOLECULES(“*-lO)

J

(1/12)[2.J+1+3(-1)J+(16/,/3)sin(2J+1)x/3] (1/12)[25 + 1 + 3( - l)‘-#6/,/3) sin (2J+ l)rr/3] (1/4)[2J+1-(-1)J]

* For the purpose of the present work, it is not necessary to distinguish between A, and A,, or F, and F2. “Irreducible representation” is abbreviated to “rep”.

On the rotational

partition function for tetrahedral molecules

1337

or g; = (4/3)(2.Z + 1) + (- l)J + (16/343) sin(2.Z+ 1)x/3.

(2b)

The additional factor of 2 for E-type products arises from E x E containing two A reps instead of just one as in A x A and F x F. Expressions similar to equation (2b) are readily obtainable from Table 1 for the other values of I,. The spherical top wave functions are also (2.Z+ 1)-fold degenerate with respect to the projection of J on a space-fixed z-axis. Therefore, gJ = tw+ lki* (3) Then, for I, = l/2, Q, = J~o{(4/3)(25+1)2+(-l)J(2~+1)+(16/3~3)(2~+l)sin(U+l)n/3} x exp[ - BJ(.Z + l)hc/kT].

(4)

For the other nuclear-spin cases, Q, has the same form with only a change in purely numerical coefficients. There are then three essentially different infinite sums to be evaluated. This is done in the next Section. III. EVALUATION

OF SUMS

The three basic sums to be considered are Cl = f

(W+1)2exp[-a.Z(.Z+1)],

(54

(-l)J(2.Z+l)exp[-crJ(J+1)],

(5b)

J=O

u2 = f J=O

cr3 = 2 (W + 1) sin[(W + l)rc/3] exp[ - cr.&Z+ l)],

(5c)

J=O

where a = Bhc/kT.

(54

The infinite sums, considered as infinite series, converge, as can be proven by standard tests.‘“) The sums in equation (5) were evaluated previously(5*6) as functions of T by means of the Euler-Maclaurin summation formula. (4) In this approach, each sum is expressed as an integral plus an asymptotic series. However, there is, in principle, some difficulty in evaluating the remainder term when the original sum is infinite in range.(4’ An alternative approach, which is presented in this Section, is to employ the theory of theta functions.“’ Through the use of appropriate derivatives and transformations of the relevant theta functions, the original sums are converted to more rapidly converging ones.* As a result, closed-form expressions, with rigorous error bounds, are obtained for the sums.

* This technique was applied, in Ref. 5, to the sum 0,. However, Furthermore, the sums s2 and cJ were not considered.

the differentiation

was not done correctly.

K. Fox

1338

There are four theta functions which may be defined as follows:(‘) 8,(z, 7) =

exp(n27ri7+2niz),

f

(64

“=-al

8,(z, 7) = 8,(z-+7t, T),

VW

O,(z, 7) = -i exp(iz +&iz)0,(z ++rc?rz -$t,

T),

(k)

and

e2tz,

7)

=

exp(iz

+$ni7)O,(z

+&CT,

7).

(64

In equation (6), the imaginary part of the complex number 7 must be greater than zero. The formulas which will be used to speed the convergence of the infinite series expressions are based on the Jacobi transformation:‘7,‘2’*

e,tz, T) E (-

i7)-

l/2

exp(z2/7G7)0,(z/7, -

l/7).

(7)

Similar transformations can be written for the other theta functions. T~LKE(") has given many useful explicit relations for theta functions and their derivatives, including the transformed forms ; graphs of theta functions and their derivatives have also been presented. In what follows, several formulas in the notation of Ref. 15 will be used, and the connection with the notation of WHITTAKERand WATSONis 0&I, X) (Ref. 15) = &(z = ~5, 7 =

ix)

(Ref. 7),

1 = 1,2,3,4.

(8)

The first of the basic sums in equation (5) can be written cI = (- 1/2x2) exp(a/4)&(0, a/rc) where primes indicate differentiation

(9)

with respect to 5. Use of the relations

.* &(O, x) = - 27rx- ‘Q2(0,x) + 87t2x- 5’2.$I ( - l)“n2 exp( - an2/x)

(10)

and e,(o, X) = 2x- 1/2 l/2 + f n=l

(11)

1

(- 1)” exp( - rrn2/x)

leads to 1+ 2

0 1=7C 1/2a-312 exp(a/4) [

(- 1)“(2- 4Z2n2a- ‘) exp( - rt2n2/a) .

II=1

1

(12)

The factor a in the exponential of equation (5a) has, in effect, been replaced by x2/a in equation (12). As a result, for small a, the latter series converges much faster. * Prof. Bellman states in Ref. 12 that equation (7) “has amazing ramifications. . it is not easy to 6nd another identity of comparable significance.” He points out the intimate relation between the Jacobi transformation and the Poisson summation formula ; also, see Refs. 13 and 14.

On the rotational partition function for tetrahedral molecules

1339

For a < 1, $i (-lYexp(-

x2n2/a) < f exp( -x2n2/a) II=1

< f exp( -rc2n2) II=1

c f exp( -9.86n2) < 5 10-4”2 < 2 x 10-4. n=l n=l The other sum in equation (12) requires more detailed consideration the denominator of its coefficient. j,

(- l)?r2n2a- ’ exp( - n2n2/a) c

f

n=l

rc2n2a- ’ exp( -x2n2/a)

(13)

because of the a in

= 2 a,, n=l

(14)

where a, is the general term of the series of positive terms in equation (14). The ratio of successive terms is a,, Ja, = (1 + n- 1)2exp[ - a2(2n + 1)/a] I 4 exp( - 3x2/a) < 4exp(-3x2)

< 4e-29

< 10-12.

(15)

The leading term a, has its maximum at a = x1” so that for a < 1, a, < rt2 exp( -x2)

< 5.5 x 10W4.

(16)

of equations (12) through (16) then yields, for a < 1,

The combination

frl = d12a-3’2 exp(a/4)[1 +,c,],

(17a)

Cl < 3 x 10-3.

(17b)

where the correction term is The basic sum e3 in equation (5) will be evaluated for a < l/5. In anticipation of this, then, for a < l/5, Cl < 10-l*.

(17c)

The next basic sum in equation (5) can be written a2 = (1/2x) exp(a/4)8;(0, a/x).

(18)

Use of the relation e;(O, x) = 2nx-3’2 E~o(-l)Y2n+l)

exp[ - x(2n + 1)2/4x]

(19)

leads to a2 = rr3’2a-3’2 exp(a/4) f (- 1)“(2n+ 1) exp[ - x2(2n + 1)2/4a]. n=O

(20)

For a c l/5, “$o(-lR2n+l)

exp[ - x2(2n + 1)2/4a] < f (2n + 1) exp[ - n2(2n + 1)2/4a] n=O b,. n=O

c $ (2n + 1) exp[ - 5x2(2n + 1)2/4] = f n=O

(21)

1340

K. Fox

The ratio of successive terms in the last series is b,, Jb,

= [(2n + 3)/(2n + l)] exp[ - lOrr’(n + l)] I 3 exp( - 10rc2) < 10-42,

(22)

and the leading term is b, = exp(-5rc2/4)

< 5 x 10W6.

(23)

Then, for a < l/5, fs2 = n1/2a-3’2 exp(a/4)c,

(24a)

where c2 <2x10-5.

(24b)

The final basic sum in equation (5) can be-written rr3 = (- 1/27c)exp(a/4)B2(1/3, or/n).

(25)

Use of the relations f12(c,X) = -2~r~-le,g,

m x)+21cx-3/2 exp( - nc2/x) nzI ( - 1)“(2n)exp( - nn2/x) sinh(2nrrc/x) (26)

and*

e,g,

X) =

X-

1’2exp(-nr2/x)+x-1’2”~~(-1)7exp[-n(n-i)2/x]+exp[-~(~+~)2/~]} (27)

leads to cr3 = n3’2a- 3’2 exp(a/4)5{ exp( - rr2/9a) -

f ( - 1)“(3n- 1) n=l

x exp[ - n2(3n - l)2/9a] + f (- 1)“(3n+ 1) exp[ - lr2(3n + l)2/9a]}. II=1

(28)

Consider first the two infinite series in equation (28). For a < l/5, $r (-lY(3n-s)exp[-~2(3n-s)2/9al

< $I (3n-~)exp[-57t~(3n-s)~/9]

G f

n=l

d,,

(29)

where& = fl. d,+ r/d, = [(n+ 1 -$s)/(n-&I

exp[- 5rr2(2n + 1 -$)I

I (5/2) exp( - 35x2/3) < 3 x 10-so (30)

and d, I 2 exp(-20x2/9)

< 7 x lo-lo.

* Equation (11) is the special case of equation (27) corresponding

to c = 0

(31)

On the rotational partition function for tetrahedral molecules

1341

The net contribution of the two infinite series to the bracketed expression in equation (28) is then no more than 2 x lo-‘. The remaining term has the value, for a < l/5, exp( - n2/9a) < exp( - 5x2/9) < 0.0042.

(32)

Then, for a < l/5, cr3 = &2a-3’2 exp(a/4)c3

(33a)

where c3

<

0.0044.

WW

Inspection of equation (4) shows that a must be less than l/5 to ensure that o3 contributes less than 1 per cent to Q,. From equations (4), (17), (24), and (33) it follows that Q, = (4/3)7r”2a-3/2 exp(a/4),

I, = l/2

(34a)

with an accuracy of 1 per cent or better for a < l/5. Similarly Q, = (27/4)rt”2a-3/2 exp(a/4),

I, = 1

(34b)

Q, = (64/3)rr”2a-3/2 exp(a/4),

I, = 312;

(34c)

and

these results are accurate to somewhat better than 1 per cent for a c l/5. Note that the coefficients in equation (34) have the form (l/12)(21,+ 1)4. IV. DISCUSSION

In a recent determination of the abundance of CH, in the upper atmosphere of the planet Jupiter, (2*3)it was necessary to calculate Q, as a function of T. This was done by numerically evaluating equation (4), with(‘@ B = 5.240 cm-‘, for 40°K I T I 400°K. When the results were plotted on a log-log scale in 10” intervals, it was found that a straightline fit yielded Q, x 0.1 15T’j2. This is within 1 per cent of the result given by equation (34a), except for the low temperature range where the exponential factor correction is a few percent. It has been suggested ~9~) that the asymptotic expression for Q,, multiplied by the nuclear-spin degeneracy factor (21, + 1)4 and divided by the “symmetry number” 12 for tetrahedral XY, molecules, yields a correct expression for the quantum-mechanical rotationalpartition function. This is borne out by equation (34). Furthermore, a numerical evaluation of the analogue of equation (4) for CD, with I, = 1 and(“) B = 2.633 cm- ’ yields Q, x 1.63T312 in agreement with equation (34b). The rigorous forms for Qr have been used recently in the theory of collision-induced rotational spectra of tetrahedral XY, molecules. Cl*A’) This theory has been used to deduce the octopole moment R of CH, ,(20*21)CD4,(20) and CF,(“) from experimental data. The resultant values of Szdepend on the use of accurate values of Q,. It has been suggestedog’ that temperature studies of collision-induced rotational spectra would be of interest in elucidating the collision dynamics. Closed-form expressions for Q, as a function of T will be useful in the analysis of such spectra.

1342

K. Fox

Finally, it is interesting to note that sums of the type in equations (5b) and (14) have been evaluated exactly in closed form (23)for special values of a. However, the calculation of Q, as a function of T requires a more general result. tAcknowledgements-The

numerical calculations. publication results.

author is grateful to Dr. I. OZIER and to Dr. J. S. MARG~LISfor verifying several Dr. M. GELLERand Dr. M. M. SAFFRENkindly communicated some of their pre-

REFERENCES 1. G.

HERZBERG, Molecular

2. 3. 4. 5. 6. I. 8. 9. 10. 11. 12. 13. 14. IS. 16. 17. 18. 19. 20. 21. 22. 23.

Spectra

and Molecular

Structure

II. Infrared and Raman Spectra

of Polyatomic

Van Nostrand, Princeton (1945). J. S. MARG~LISand K. Fox, Astrophys. J. 158, 1183 (1969). J. S. MARGOLISand K. Fox, J. atmos. Sci. 26,862 ( 1969). F. B. HILDEBRAND, Introduction to Numerical Analysis, McGraw-Hill, New York (1956). I. E. VINEY, Proc. Camb.phiI. Sot. 29,142,407 (1933); the second reference contains a correction of a computation error in the first reference. L. S. KASSEL,Chem. Rev. 18,227 (1936). E. T. WHITTAKER and G. N. WATSON,A Course of Modern Analysis, Cambridge University Press, London (1927). E. B. WILSON. JR., J. them. Phys. 3,276 (1935). E. G. KAPLAN,Soviet. Phys. JETP 10,747 (1960). K. FOX and I. OZIER, J. them. Phys. 52,5044(1970). E. D. RAINVILLE,Infinite Series, Macmillan, New York (1967). R. BELLMAN.A Brief Introduction to Theta Functions, Holt, Reinhart and Winston, New York (1961). E. C. TITCHMARSH, Introduction to the Theory of Fourier Integrals, Oxford University Press, London (1937). K. FOX, Am. J. Phys., to be published. F. T~LKE, Praktische Punctionenlehre, Vol. 2. Springer-Verlag, Berlin (1966). K. T. HECHT,J. molec. Spectrosc. 5, 390 (1960). R. A. OLAFSON, M. A. THOMASand H. L. WELSH,Can. J. Phys. 39,419 (1961). I. OZIER and K. FOX, Phys. Lett. 27A, 274 (1968). I. OZIERand K. Fox, J. Chem. Phys. 52, 1416 (1970). G. BIRNBAUM and A. ROSENBERG, Phys. Lett. 27A, 272 (1968). S. WEISS,G. E. LEROIand R. H. COLE,J. them. Phys. 50,2267 (1969). A. R~XENBERG and G. BIRNBAUM, J. them. Phys. 48, 1396 (1968). M. GELLERand M. M. SAFFREN,to be published. Molecules,