Pram~na, Vol. 13. No. l, July 1979, pp. 15-24, ~) printed in India
Scattering contributions to the internal partition function of a diatomic molecular system*
B T A L U K D A R , M C H A T T E R J I and P B A N E R J E E Department of Physics, Visva-Bharati University, Santiniketan 731 235 MS received 14 November 1977; revised 30 March 1979
Abstract. An analytical expression for the phase shift contribution to the internal partition function for the Morse potential is derived by using an approximate Jost function. This function is shown to be a convergent sum. The numerical results obtained for Hz and HC1 show the partition function to be a monotonically increas- ing function of temperature. This observation agrees with the results of Rogers and co-workers.
Keywords. Internal partition function; diatomic molecules; phase-shift contribution.
T h e total energy o f a molecular system can be expressed in terms o f its c o n t r i b u t i n g energies:
(i) the energy associated with the translation of its m a s s t h r o u g h space, a n d (ii) energy arising due to the internal m o t i o n of the molecule.
Typically, f o r visualising the internal motion, the g e o m e t r y o f a diatomic m o l e c u l e m a y be a p p r o x i m a t e d b y t w o point masses attached by a massless spring. T h e m o l e - cule is free to r o t a t e a b o u t its centre of gravity a n d m a y also vibrate along the line o f the centres. It is c u s t o m a r y to associate a partition function with the internal contribution to the energy. This is c o m m o n l y k n o w n as the internal partition func- tion. T h e c o m p l e t e internal partition function requires scattering (phase shift) contributions in addition to the usual b o u n d state sum. T h e internal partition func- tion is given b y
Zint --- ~ 1 (2l -+- 1)
[zfl + zS].(1)
T h e b o u n d state a n d scattering state contributions zt n a n d zt s can be written as ( L a n d a u a n d Lifshitz 1959)
z/' = exp ( - (2a)
a n d
zs = wlexp ( - -
[3h2k~/2m) -d-k n, (k) dk.(2b)
*Work supported in part by the Department of Atomic Energy, Government of India.
16 B Talukdar, M Chatterfi and P Banerjee
Here E,~ represents the binding energy of the nth state with orbital angular momentum L Also fl=(KnT) -1 and n~(k), the phase shift of the lth partial wave. The object z f will be discontinuous when the density or temperature of the gas molecules changes so as to admit a new bound state. These discontinuities affect all the thermal pro- perties such as pressure and specific heat. The awkward discontinuities in Zint can be removed (Rogers et al 1971; Petschek 1971; Petschek and Cohen 1972) by taking proper account of the scattering contribution. It is therefore of some interest to evaluate zt s in a relatively non-complicated manner by using simple and physically founded assumptions for the molecular potential.
In this paper we present a method for computing zt s for diatomic molecular systems.
The vibration as well as the rotation of a diatomic molecule are excellently repre- sented by the Morse function
V(r) = D [exp (-- 2ax) -- 2 exp (-- ax)]. (3) We shall therefore base our treatment on the use of an approximate solution of the Schrrdinger equation for the potential given in (3). The symbol D stands for the depth of the Morse potential, x=(r--ro)/r o and a, a constant.
All these parameters depend on the typical molecule under consideration. The distance r o corresponds to the separation of the atomic centres when the potential energy is minimum. The potential V(r) is attractive for r > r 0 and produces a strong repulsion if the two nuclei approach each other closer than r 0. In § 2 we derive an analytic expression for z s. Based on this, numerical results are presented in § 3 for H~
and HCI. Usefulness of the present approach is also discussed.
2.I. Continuum partition function and Jost function
The proof regarding continuity (Petschek and Cohen 1972) of the internal partition function has been given by writing z s in terms of the Jost function. Using the relation between the S-matrix and Jost function
St = exp (-- 2in,) = fz (k)/fz (-- k). (4)
Equation (2b) can be written in the form
~ f+_~d lnf~(k) dk.
ztS --- ~t" exp (--
Here we have followed the convention of Newton (1966). The upper half of the k-plane refers to the physical sheet and the phase shift n~ is the negative of the phase of the Jost function. Equation (5) could very well form the basis for computing the scattering contribution to the internal partition function for a system consisting of interacting particles, provided the Jost function for all partial waves would be known for the interaction. Unfortunately, the Schr6dinger equation cannot be solved in dosed form for the so-called realistic potentials. For this reason the Jost function
lnternal partition function in molecular system 17 approach to the continuum partition function has been confined to the s-wave case only (Petschek 1971 ; Petschek and Cohen 1972). Reasonably, therefore, one may be interested to introduce approximation techniques for deriving the/-wave Jost func- tion in analytical form. Assuming that the atoms of a diatomic molecule are bound together by Morse type interactions, we derive an analytic expression for f~(k) by replacing the centrifugal term by an approximate one. This approach, in a natural way, incorporates the rotational correction to the Morse formula (Flfigge 1971).
One can deal with the problem without meeting any fresh mathematical difficulties as compared to the s-wave problem. Before such a study is made it will be useful to know if the approximate treatment of the centrifugal term affects the continuity of the partition function. To that end we partially integrate (2b), make use of the Levinson's (1949) theorem and use the resulting equation in (1) to write Zint in the form
Z IZ 1
Zint = I (21 -~- 1) n [exp (-- E, JKBT) -- 1] + mTrKT X
f o k dk n~(k) exp (-- k2/2mKT)l .
The third term in the right hand side of the above equation stands for the scattering contribution made by the zero energy part of the phase shift, which exactly cancels the zero energy part of the bound state sum and thereby removes the discontinuity.
Here we approximate the centrifugal term by a superposition of Morse-type function and thus convert the/-wave problem formally to an s-wave one to write simple analy- tical solutions. The Levinson's theorem is valid under extremely general conditions like the existence of first and second absolute moments of the interaction (Newton 1966). Fortunately, the effective potential obtained in our approximation satisfies these conditions. Thus our approximation of the centrifugal term will not affect the continuity of the partition function.
2.2. L-wave Jost function for the Morse potential
The/-wave Schr6dinger equation for the potential in (3) is given by
[d~Z -q- k Z - 2m----D-D;~2 [exp (-- 2 a x ) - 2exp ( - - a x ) ] _ l(l-br 1 . _ _ . )] z f~(kr) = O.
It is well known that for l>0, equation (1) does not admit of an analytical solution due to the presence of the singular term l(l+l)/r ~. To incorporate the effect of the centrifugal barrier we introduce the identity
1 _ 1 [1 q--x]-3 = 1 [C o q_ C1 exp (-- ~x) q- C 2 exp (-- 2ax)].
18 B Talukdar, M Chatterji and P Banerjee
This identity has been devised by Fltigge
et al(1967). Following their viewpoint we assume that in the region o f physical interest the two expressions in (7) deviate only in the order x a. We thus obtain
C o = 1 - - 3 l a + 3 1 a 2.(8a)
Ca = 4/a -- 61a a, (8b)
C• = - - 1/a -~- 3/a a.
The series in (7) attempts to include in some way the effect o f centrifugal term as a part o f the potential. This approach thus bears an analogy with the method o f Martin (1960) and o f Alfaro and Rossetti (1968) for solving the radial Schr6dinger equations with Yukawian potentials. Inserting (7) in (6) and changing the variable r to x we obtain
-q- k~r~ + 2r~exp (-- ax) -- y~ exp (-- 2ax) ~
(kxx) = O,(9)
l ( l +1)
where k~ = k 2 & - - -~o r" 2 - - - ' ( 1 0 a )
- 2mDr] C l ( l +
r~ - - h" - - l - - - T - - , ( 1 0 b )
2m Dr2o q_
y~ = h a C 2 l ( l +1). (lOc)
We transform the variable in (9) by substituting y ---- 8 exp (--
[y ddy d + k_~ [_ 2y 2 1 ]
a n d get
y ~y+ % ~" Y 8--a 2 - - -4 y~ f~ ( k t Y ) = O.
Here 3 -- 2y2 2 12mDr~
To reduce (12) to a known form we proceed as follows.
(i) We note that for large r (i.e. for small y) it has a solution
and (ii) for small r the equation has a solution of the form exp (-- ½y).
Internal partition function in molecular system 19
From (i) and (ii) we see t h a t the exact solution o f (12) can be put in the form
f z ( k l y ) = exp ( - - ½y) y-,k,,,/a GO'), (14)
where GO') is asymptotically normalised such that
f~ (kit) ~ exp
r - - - l , o o
Substitution o f (14) in (12) yields
yG" (y) + ( c - - y ) G' (y) -- aG(y)-- 0, (16)
where C = 1 --
and a = ½ -- (2~,~/8a ~) --
The general solution o f (16) is given by
G(y)= A 1 i F i ( a , c;
y ) + A 2 y 1-ciFi
( a - - c +1; 2 - - c ; x), (18) where iF1 (...) is the so-called confluent hypergeometric function. We note that only the first term in (18) together with (14) satisfies the asymptotic boundary condi- tion prescribed for the Jost function provided
A i = 8~x'0/a exp
Therefore the normalised Jost solution is
f t ( k l y )= 81kl'o/* exp
(iktro)exp ( - - ½y)y-ik~'0/a t F i ( a , c; y). (20) F o r 1~ 0, the point r - - 0 is a regular singular point o f the Schr6dinger equation a n d
l(l+1) multiplies the term o f the highest singularity in it. In other words, for small r, the centrifugal term dominates over the potential (Newton 1966) so that
f~(kr)behaves as r -t as r approaches zero. For higher angular momenta, therefore, the Jost function is obtained from the Jost solution by taking the limit
r ~ (kr)
r ---)- 0
and then multiplying the latter by the appropriate normalisation term. In our case we have incorporated the effect o f the centrifugal barrier by writing it as a part of the potential. This necessarily implies that the centrifugal term is no longer more singular t h a n the Morse potential. We therefore compute the Jost function from the behaviour o f (20) near the origin. The Morse Jost function is f o u n d to be
f l ( k l ) = exp ( - - ½Be a) tFi (a, c; Be*). (21) The s-wave limit (Talukdar
et al1975) and the high energy behaviour (Newton 1966)
20 B Talukdar, M Chatterji and P Banerjee
o f f , ( k 1 ) i m m e d i a t e l y follows f r o m (21). E q u a t i o n (21) can be well a p p r o x i m a t e d by replacing the hypergeometric function by its a s y m p t o t i c expansions
1F1 (a, c; z) ~ I" (c) e, z,_C. (22)
We h a v e verified this f o r the parameters o f the M o r s e potential given in table 1.
In this table the values o f the parameters Co, C 1 a n d 6"2 are also included. T o be m o r e specific the values o f 6e ~ for some of the partial waves are given in table 2. We h a v e numerically checked that the error is less t h a n 0 . 1 % for each o f the partial waves. In view o f (22), (21) takes the form
f~(kj) : H ( a , 3; ~'a) 3'k~'°/~ exp (iklro) x r [1 - (2il,1,.0/0`)]
F [½ - - (2~'2/80` 2) - - (iklrol0`)] (23)
where H ( a , a ; ~"1) = e x p [(½a8 tt) - - (lo.) - - (2~,~/3a2)] 3 - ½ - 2 ~ ' ~ [ 3 a z . (24) As noted b y Fltigge the a p p r o x i m a t i o n in (7) is quite reasonable for the c o m p u t a - tion o f b o u n d states of the Morse potential. O n e m a y ask: H o w g o o d is this a p p r o x i m a t i o n for estimating scattering contributions to the partition function 9 . U n d e r s t a n d a b l y , the validity o f the a p p r o x i m a t i o n depends on the region o f i m p o r t - ant k values which contribute significantly to the integrand o f (5). As a useful check one m a y c o m p a r e the phase of the a p p r o x i m a t e Jost function, which by definition is the p h a s e shift [n~(k)] with the a c c u r a t e value o f n~(k) o b t a i n e d b y numerically integrating (6). T o facilitate this we h a v e calculated the phase o f the Jost function in (23) with the help o f the standard d e c o m p o s i t i o n
F (x + iy) ---- ~: exp (iv),
Table 1. Values of Morse parameters and the constants C's E(eV) = E (cm -~) × 1.2398 × 10 -4
Molecule h ~/2 mro ~ cm-' cm -1 D a Co 6'1 6"2
H~ 60'8296 38292 1"440 0'363426 --0"1157407 0"7523148
HCI 10.5930 37244 2.380 0.26912 0-621425 0-1094555
Table 2. Values of 8e a Molecule • 1
0 I 2 3 4 5
Hz 147"08 147"25 1 4 7 ' 6 0 1 4 8 " 1 3 1 4 8 " 8 3 149"69 HCI 538'38 5 3 8 ' 4 0 5 3 8 " 4 3 5 3 8 " 4 9 5 3 8 " 5 5 538"64
Internal partition function in molecular system
for the gamma function with complex argument.
Here s e ---- r ( x ) II °° n = 0 I q- ( X
and rl = y I__~,-b ~ "Q° ( l _ l t a n - X Y )}
L.cn= l \n y x - I - n - - 1 '
with y=0-577215, the Euler constant. We have calculated the accurate value o f the phase shifts by employing the variable phase approach to the potential scatter- ing (Calogero 1967). This approach consists in transforming (6) to a first order non-linear differential equation, the solution of which yields in the asymptotic limit the value of the phase shift. The mathematical foundation of the variable phase approach is based on the well-known connection between second order differential equations and first order equations of the Ricatti type (Davis 1957). The numeri- cal integration of the non-linear equation under consideration was done by using the algorithms of the Runge-Kutta method with an appropriate stability check.
It is seen that for T ~ 0.2 Ry, the exact and approximate phases agree to within 1%. However the error increases significantly at temperatures higher than this.
Thus the approximation in (7) seems to be valid only at low temperatures.
The Jost function f,(kl) in (23) together with (5) can now be used to derive an analytical expression for the continuum partition function. To facilitate this we require (23) by introducing the well-known infinite product representation for gamma function
r ( l +z)
- - e x p (yz)II°°n=l (1
fl (kx) = H(a,8;
exp ~, [(--½) -- (2yl~/Se ~) -1-
(iklro/e) -~ (iktro/y)] ×[n - - ½ -- (2y21/Sa ~) --
IIn = 1
exp [(½) +
22 B Talukdar, M Chatterji a n d P Banerjee Inserting (26) in (5) we get
exp ( _ f l a ' r , 1 ( 1 + 1)~ fl~'k~l
iy 1 - F I N S + -F
cL n = 1
k,q- (ia/ro) In - - ½ - - (2y~/aa2)]
a o = 1
a t = b2~r-V2 = 0"936438 a~ : barr-]l~ = 0"329897 bl = (4--~)rY ~ = 2"064817 b 2 = btrrl/~--2 = 1"659793
b3 = - - 2 ~ r l n + 2 b t = 0"5847278 -- F o r large z, therefore,
W(z) ~ ~-112z-1
*Equations (30)-(32) are from Appendix A of Petschek and Cohen (1972)
n-~,,,-~/ J~' (28)
w h e r e W(z) = e x p ( z 9 (1 - - erfz), X = a~12mr~, ( 2 9 )
The convergence o f the sum in (28) m a y be explicitly demonstrated by using the P a n d e ' approximants* for W i.e.
W ( z ) = a°-Falz-Faaz~ (30)
1 + b t z + b 2 z ~ + b ~ z 8'
k 1 q- (ina/2ro) n a j
The integrals in (27) can be related to tabulated results to get z~ = ~ exp [-- t ( t + 1) C0~Xl ~-~ ( ~ + l u ~ + ~ ) ×
Internal partition function in molecular system 23 W i t h the h e l p o f (32) it is easy to see t h a t the nth t e r m Tn o f the s u m in (28) goes to zero as n ~ oo. It is interesting to n o t e t h a t W(z) evaluated b y m e a n s o f (30) does n o t introduce e r r o r m o r e t h a n 0-2 % for all physical values o f z. T h u s (28) together with (30) f o r m s a basis o f calculating scattering contributions to the internal p a r t i t i o n function o f a diatomic molecular system.
3. Results and discussion
Table 3 illustrates the results o f o u r partition function (zt s) calculation or H 2 a n d HCI. In this table we h a v e included the results for the lowest ten partial waves f o r T----0.002 R y to 0.2 R y varying in multiples o f ten. F r o m o u r results we could see t h a t the contributions o f z s to the total partition function b e c o m e p r o n o u n c e d at high temperatures. A similar observation has been m a d e b y Rogers et aI (1971). B o t h f o r H 2 a n d HC1 the values o f z s exhibit convergence with respect to the partial wave L T h e rate o f convergence in b o t h cases appears to be slow f o r higher t e m - peratures. This represents the general trend o f the n u m b e r s presented. H o w e v e r , T : 0 - 2 e n t r y f o r HC1 is a n exception which shows rapid convergence with 1. T h i s point is n o t clear to us a n d deserves to be closely examined. L o o k i n g into the table we f u r t h e r see t h a t the values o f zt s for H C I at T = 0 . 0 2 R y a n d 0.2 R y exhibit a relatively r a p i d convergence with l t h a n the corresponding n u m b e r s f o r H~. This b e h a v i o u r m a y be attributed to the existence o f a longer tail o f the H~ p o t e n t i a l as c o m p a r e d to t h a t associated with the H C I molecule. U n d e r s t a n d a b l y , a rela- tively l o n g tail m a k e s significant contributions to higher partial waves since the region o f the potential sampled by a scattering lies at larger distance f o r greater l.
Since the continuity o f the total partition function is associated with a c c u r a t e n u - merical values o f z s, we would venture to suggest t h a t special c a u t i o n m u s t be t a k e n t o c o m p u t e this frequently elided contribution to the former. O n e m a y t h e n rea- s o n a b l y a s k : H o w realistic is the model presented in this p a p e r ? O u r m o d e l h a s the obvious virtue o f simplicity a l t h o u g h it involves a n u m b e r o f a p p r o x i m a t i o n s . F o r example, we have used an a p p r o x i m a t e f o r m o f t h e / - w a v e M o r s e Jost function
Table 3. Values of ztS as a function of I and T for H2 and HC1
T in Rydberg H~ HC1
1 0.002 0"02 0.2 0-002 0-02 0-2
0 0"967 --0"830 --1"890 2"848 12"360 14"470 1 0"866 -- 0"459 -- 1-002 2"913 12-579 13-703 2 0"676 0'184 0"806 2"741 13-042 12.370 3 0"448 0'966 3"703 2-635 13"805 10.754 4 0"248 1-788 8"254 2"498 14-982 9"093 5 0"115 2"630 16"463 2"335 16-815 7'531 6 0"055 3.050 20-623 2-115 17"513 6"005 7 0"027 3"251 22"251 1 "890 17"050 4-651 8 0"012 3"373 23"470 1 "652 15"825 3"372 9 0"006 2"511 24"082 1-373 14-234 2-201 10 0.002 1 "955 23"423 1 "012 12"501 1 "152
24 B Talukdar, M Chatterji and P Banerjee
f~(ki) to derive an analytical expression for z/s. The estimation o f the partition f u n c t i o n based onf~(ki) is a convergent sum a n d has c o r l e c t limits. O u r expression is thus expected to yield reasonably accurate numbers. H o w e v e r we feel that our n u m b e r s should be compared with the results o f an exact numerical calculation.
Admittedly, the object in this paper is partly pedagogical to see h o w far one could go by using a simple analytical procedure; the inevitable numerical routine being i n v o k e d only at a later stage o f the game. Hopefully, by m a k i n g a few judicious a p p r o x i m a t i o n s we could focus attention on the basic conceptual aspect o f the p r o b l e m a n d obtain some physical weight without straying w a y off the m a r k with regard to the numerical accuracy.
A c k n o w l e d g e m e n t
One o f the authors (MC) is grateful to the University G r a n t s Commission for a research grant.
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