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On the partition function of free internal rotation. A semiclassical approach
Volume 151, number
1,2
CHEMICAL
PHYSICS LETTERS
7 October
1988
ON THE PARTITION FUNCTION OF FREE INTERNAL ROTATION. A SEMICLASSICAL APPROACH Hui-Yun PAN The Department of Chemistry, Zhengzhou University, Zhengzhou, Henan, People? Republic of China 1988; in final form 2 1 March I988
Received 4 February
On the basis of the semiclassical approximation, a new formula is proposed to evaluate the partition function of free internal rotation. This formula has correct limiting behavior and for a given temperature it gives accurate numerical values for the partition function and related thermodynamic functions.
1. Introduction
moderate B (0% 1) it gives numerical values which differ considerably from the exact values [ 3 1. In this paper a new formula is proposed on the basis of the Korsch semiclassical approximation. Besides its correct limiting behavior, for a given (Tthis formula gives numerical values nearly identical to the exact values.
As is well known, for high temperature T and/or large reduced moment of inertia I,- the partition function of free internal rotation 4e.YzKt = F cxp(-crj2)=2 ,= --oD
f
exp(-ok*)-1
k=O
=22(a)-1,
(1)
2. Korsch semiclassical approximation [4] Using the Poisson sum formula, we get
where p= l/kT,
a=@z*/2I,,
reduces to its classical limit #l qo = (2nl,//W)“%
(Ir/o)“‘.
This formula is widely used in the literature, but as the temperature and/or moment of inertia diminish, the deviation of q. from qFXBEL increases, ultimately for o+cq q. breaks down altogether. In a recent series of papers [ 1,2 ] Slanina called attention to this situation and suggested a formula to evaluate the partition function qexac,_Slanina’s formula is q, = (x/a)‘/‘+
- (a/a)‘%q
iM=g exp( co
-2xiA40!)
m
=
f
exp(-2CVo!)QM,
@)
)
number
n may be easily taken
(4)
M=--m
where
1 t exp( -a)
(5) (3)
where 0 is the error function. 4, has correct limiting behavior for a-0 and WCC respectively, but for @’Symmetry
=
(2)
mto account
and I=h ( m+ a) is a continuous variable identified with the classical action
which is
if
necessary
0 009-2614/88/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
35
Volume 15 1,number
CHEMICAL
I,2
The lowest-order term M=O is recognized as the classical partition function, the terms with M# 0 give the quantum corrections. Evaluating the M# 0 integral in eq. (5 ) by the saddle point method, we get#2
Table 1 Partition function
y
>
rotation
with 9,), 9,, 9y and
90
91
9x
9CXK-1
0.1
5.60
5.51
1
1.77
1.65
0.56
I .OO
5.60 1.77 1.oo
5.60 1.77 1.00
0
4c*act=21(a)
,
for free internal
1988
4U.C,
10
-jH(Z,)t
7 October
PHYSICS LETTERS
where w(Z) = aH/X and IM is the saddle point value of I. Substituting eq. (6) into eq. (4), we get the semiclassical approximation to f: =
- 1
,,=c q0exp(-~M2qi) cc
3. New formula for the partition function of free internal rotation
=
-Yx
Applying the semiclassical approximation (eqs. (4)-(6)) to the partition function qexac, (eq. (2)), or more concretely, to A(a)=
E
exp(-uk*)
,
h=O
and noticing that in our case Z=P,=mi,
a=o,
m=O, Al, f2 )...,
H(Z)=Z2/21r=m2ri2/2Zr, Z,w= 2xilW,/pTi
(7)
we get #3
where q. is the classical partition function (2). Formula (9) is the semiclassical approximation to the partition function qcnac,.It is easily seen that for u+O qsc tends to its classical limit qo; on the other hand, as D increases the convergence of the series (9 ) slows down, more and more terms are needed to give reliable numerical results, in the limit UP+COthe sum (9) tends to qcxac, approximately. For usual applications, especially in the practically important region 0.1
(10)
Numerical results are given in tables 1 and 2. For comparison, WCalso list the results obtained with qcl,
(8) and
” A numerical error in Korsch’s original formula has been corrected. *’ The extra term “f” in the expansion A( a) is due to the choice (Y= 0. Formula (9) can also be obtained directly through a Theta transformation, but as a general method for evaluation ofpartition functions the Korsch semiclassical approximation seems preferable.
36
(9)
>
q1 and qenacl 13 lTable 2 Greatest differences between partition functions or thermodynamic functions with 9, and 9.%.?,. For comparison, the corrcspondmg values with 9>‘ are also listed Partition function or thermodynamic function
ISvalues corresponding to greatest differences
9
II,IRT SJR c,,. JR
91
9%
9ww
0.941
1.701
1.827
I.827
2.601 2.224 3.7
0.254 0.338 0.337 0.461 0.593 0.593 0.444
0.6 I3
0.6
IS
Volume I5 1, number I,2
CHEMICAL
PHYSICS LETTERS
Evidently, the agreement between qECand qexac,is excellent, whereas the differences between 4, and 4WBC,are considerable. All values obtained with qsc are calculated with its three-term expression eq. ( 10).
4. Remarks In conclusion two remarks should be made: (i) In the semiclassical formula ( 10) (or more gcncrally formula (9)), only the classical partition function q. appears. Once the temperature T and the reduced moment of inertia 1, are given, qo, hence qs,, are easily evaluated. So formula ( 10) is really very convenient to use.
7 October
1988
(ii) For rr< K, the semiclassical expansion (9) converges more quickly than the direct summation in qcxact (eq. ( I ) ), further numerical investigation indicates that in this critical region, only three terms in expansion (9) are necessary to yield an accuracy of five decimal places, hence eq. ( 10) merits special recommendation.
References [ 1 ] 2. Slanina, J. Phys. Chem. 86 ( 1982) 4782. [2] 2. Slanina, Intern. J. Quantum Chem. 27 ( 1985) 691. [3] Z. Slanina, .I. Phys. Chem. 90 (1986) 2957. [4] H.J.Korsch, J. Phys. A 12 (1979) 1521.
37
1,2
CHEMICAL
PHYSICS LETTERS
7 October
1988
ON THE PARTITION FUNCTION OF FREE INTERNAL ROTATION. A SEMICLASSICAL APPROACH Hui-Yun PAN The Department of Chemistry, Zhengzhou University, Zhengzhou, Henan, People? Republic of China 1988; in final form 2 1 March I988
Received 4 February
On the basis of the semiclassical approximation, a new formula is proposed to evaluate the partition function of free internal rotation. This formula has correct limiting behavior and for a given temperature it gives accurate numerical values for the partition function and related thermodynamic functions.
1. Introduction
moderate B (0% 1) it gives numerical values which differ considerably from the exact values [ 3 1. In this paper a new formula is proposed on the basis of the Korsch semiclassical approximation. Besides its correct limiting behavior, for a given (Tthis formula gives numerical values nearly identical to the exact values.
As is well known, for high temperature T and/or large reduced moment of inertia I,- the partition function of free internal rotation 4e.YzKt = F cxp(-crj2)=2 ,= --oD
f
exp(-ok*)-1
k=O
=22(a)-1,
(1)
2. Korsch semiclassical approximation [4] Using the Poisson sum formula, we get
where p= l/kT,
a=@z*/2I,,
reduces to its classical limit #l qo = (2nl,//W)“%
(Ir/o)“‘.
This formula is widely used in the literature, but as the temperature and/or moment of inertia diminish, the deviation of q. from qFXBEL increases, ultimately for o+cq q. breaks down altogether. In a recent series of papers [ 1,2 ] Slanina called attention to this situation and suggested a formula to evaluate the partition function qexac,_Slanina’s formula is q, = (x/a)‘/‘+
- (a/a)‘%q
iM=g exp( co
-2xiA40!)
m
=
f
exp(-2CVo!)QM,
@)
)
number
n may be easily taken
(4)
M=--m
where
1 t exp( -a)
(5) (3)
where 0 is the error function. 4, has correct limiting behavior for a-0 and WCC respectively, but for @’Symmetry
=
(2)
mto account
and I=h ( m+ a) is a continuous variable identified with the classical action
which is
if
necessary
0 009-2614/88/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
35
Volume 15 1,number
CHEMICAL
I,2
The lowest-order term M=O is recognized as the classical partition function, the terms with M# 0 give the quantum corrections. Evaluating the M# 0 integral in eq. (5 ) by the saddle point method, we get#2
Table 1 Partition function
y
>
rotation
with 9,), 9,, 9y and
90
91
9x
9CXK-1
0.1
5.60
5.51
1
1.77
1.65
0.56
I .OO
5.60 1.77 1.oo
5.60 1.77 1.00
0
4c*act=21(a)
,
for free internal
1988
4U.C,
10
-jH(Z,)t
7 October
PHYSICS LETTERS
where w(Z) = aH/X and IM is the saddle point value of I. Substituting eq. (6) into eq. (4), we get the semiclassical approximation to f: =
- 1
,,=c q0exp(-~M2qi) cc
3. New formula for the partition function of free internal rotation
=
-Yx
Applying the semiclassical approximation (eqs. (4)-(6)) to the partition function qexac, (eq. (2)), or more concretely, to A(a)=
E
exp(-uk*)
,
h=O
and noticing that in our case Z=P,=mi,
a=o,
m=O, Al, f2 )...,
H(Z)=Z2/21r=m2ri2/2Zr, Z,w= 2xilW,/pTi
(7)
we get #3
where q. is the classical partition function (2). Formula (9) is the semiclassical approximation to the partition function qcnac,.It is easily seen that for u+O qsc tends to its classical limit qo; on the other hand, as D increases the convergence of the series (9 ) slows down, more and more terms are needed to give reliable numerical results, in the limit UP+COthe sum (9) tends to qcxac, approximately. For usual applications, especially in the practically important region 0.1
(10)
Numerical results are given in tables 1 and 2. For comparison, WCalso list the results obtained with qcl,
(8) and
” A numerical error in Korsch’s original formula has been corrected. *’ The extra term “f” in the expansion A( a) is due to the choice (Y= 0. Formula (9) can also be obtained directly through a Theta transformation, but as a general method for evaluation ofpartition functions the Korsch semiclassical approximation seems preferable.
36
(9)
>
q1 and qenacl 13 lTable 2 Greatest differences between partition functions or thermodynamic functions with 9, and 9.%.?,. For comparison, the corrcspondmg values with 9>‘ are also listed Partition function or thermodynamic function
ISvalues corresponding to greatest differences
9
II,IRT SJR c,,. JR
91
9%
9ww
0.941
1.701
1.827
I.827
2.601 2.224 3.7
0.254 0.338 0.337 0.461 0.593 0.593 0.444
0.6 I3
0.6
IS
Volume I5 1, number I,2
CHEMICAL
PHYSICS LETTERS
Evidently, the agreement between qECand qexac,is excellent, whereas the differences between 4, and 4WBC,are considerable. All values obtained with qsc are calculated with its three-term expression eq. ( 10).
4. Remarks In conclusion two remarks should be made: (i) In the semiclassical formula ( 10) (or more gcncrally formula (9)), only the classical partition function q. appears. Once the temperature T and the reduced moment of inertia 1, are given, qo, hence qs,, are easily evaluated. So formula ( 10) is really very convenient to use.
7 October
1988
(ii) For rr< K, the semiclassical expansion (9) converges more quickly than the direct summation in qcxact (eq. ( I ) ), further numerical investigation indicates that in this critical region, only three terms in expansion (9) are necessary to yield an accuracy of five decimal places, hence eq. ( 10) merits special recommendation.
References [ 1 ] 2. Slanina, J. Phys. Chem. 86 ( 1982) 4782. [2] 2. Slanina, Intern. J. Quantum Chem. 27 ( 1985) 691. [3] Z. Slanina, .I. Phys. Chem. 90 (1986) 2957. [4] H.J.Korsch, J. Phys. A 12 (1979) 1521.
37