280
6 Molecular Systems
This improved expression for z rot (T ), first obtained by Kassel [4] in 1936, can be
employed for the calculation of more accurate values of the rotational partition
function at temperatures for which the classical expression does not yet suffice.
Indeed, this formula yields z rot (T ) = 10.3410 for T = 10 rot and z rot (T ) = 5.3475
for T = 5 rot , which gives essentially perfect agreement with the 11-term sum
for T = 10 rot and overestimates the 8-term sum for T = 5 rot by less than
0.01%. The Euler–Maclaurin summation formula (6.2.64) is what is referred to as
an asymptotic series [5], and is generally not a convergent series: this means that
some care must be exercised in its use.
A variant of the Euler–Maclaurin formula for z rot (T ) may be obtained if the
original summation expression (6.2.59) is reformulated by first completing the
square in the exponent to give
z rot (T ) = 2
∞
j =0
j +
1
2
e
−α(j +
1
2 ) 2 e
α/4 ,
for which an Euler–Maclaurin expansion gives z rot (T ) as
z rot (T ) = e
α/4 α
−1
1 +
α
12
+
7α 2
480
+
31α 3
8064
+ · · ·
,
(6.2.65)
a result now known as the Mulholland formula [6].
Finally, an asymptotic series that requires fewer terms to yield more accurate
values for z rot (T ) up to quite high temperatures (of the order of several thousand
Kelvin for many diatomic molecules) has been obtained by McDowell [7], who
noted that by treating the first two terms within the square brackets of Eq. (6.2.64)
as the start of the series expansion for e α/3 , writing z rot (T ) = e α/3 α −1 e −α/3 [1 +
1
3 α +
1
15 α 2 +
4
315 α 3 + · · · ], expanding e −α/3 , multiplying the two series and, finally,
collecting like powers of α, z rot (T ) is given as
z rot (T ) = e
α/3 α
−1
1 +
α 2
90
+
8α 3
2835
+ · · ·
.
(6.2.66)
Expression (6.2.66) has the double advantage that not only is there no linear
correction term, but also the coefficients of all higher-order powers are smaller than
those of the corresponding powers in both the Euler–Maclaurin and Mulholland
expressions: this means that in practice the series can be terminated sooner to attain
a specified accuracy. Indeed, McDowell found that values for the prefactor α −1 e α/3
alone underestimate the exact values for z rot (T ) by less than 0.05% for temperatures
of the order of 5 rot and by less than 0.005% for temperatures of the order of 20 rot .
An elegant operator treatment of these expansions has been given by Fernández and
Tipping [8].
6 Molecular Systems
This improved expression for z rot (T ), first obtained by Kassel [4] in 1936, can be
employed for the calculation of more accurate values of the rotational partition
function at temperatures for which the classical expression does not yet suffice.
Indeed, this formula yields z rot (T ) = 10.3410 for T = 10 rot and z rot (T ) = 5.3475
for T = 5 rot , which gives essentially perfect agreement with the 11-term sum
for T = 10 rot and overestimates the 8-term sum for T = 5 rot by less than
0.01%. The Euler–Maclaurin summation formula (6.2.64) is what is referred to as
an asymptotic series [5], and is generally not a convergent series: this means that
some care must be exercised in its use.
A variant of the Euler–Maclaurin formula for z rot (T ) may be obtained if the
original summation expression (6.2.59) is reformulated by first completing the
square in the exponent to give
z rot (T ) = 2
∞
j =0
j +
1
2
e
−α(j +
1
2 ) 2 e
α/4 ,
for which an Euler–Maclaurin expansion gives z rot (T ) as
z rot (T ) = e
α/4 α
−1
1 +
α
12
+
7α 2
480
+
31α 3
8064
+ · · ·
,
(6.2.65)
a result now known as the Mulholland formula [6].
Finally, an asymptotic series that requires fewer terms to yield more accurate
values for z rot (T ) up to quite high temperatures (of the order of several thousand
Kelvin for many diatomic molecules) has been obtained by McDowell [7], who
noted that by treating the first two terms within the square brackets of Eq. (6.2.64)
as the start of the series expansion for e α/3 , writing z rot (T ) = e α/3 α −1 e −α/3 [1 +
1
3 α +
1
15 α 2 +
4
315 α 3 + · · · ], expanding e −α/3 , multiplying the two series and, finally,
collecting like powers of α, z rot (T ) is given as
z rot (T ) = e
α/3 α
−1
1 +
α 2
90
+
8α 3
2835
+ · · ·
.
(6.2.66)
Expression (6.2.66) has the double advantage that not only is there no linear
correction term, but also the coefficients of all higher-order powers are smaller than
those of the corresponding powers in both the Euler–Maclaurin and Mulholland
expressions: this means that in practice the series can be terminated sooner to attain
a specified accuracy. Indeed, McDowell found that values for the prefactor α −1 e α/3
alone underestimate the exact values for z rot (T ) by less than 0.05% for temperatures
of the order of 5 rot and by less than 0.005% for temperatures of the order of 20 rot .
An elegant operator treatment of these expansions has been given by Fernández and
Tipping [8].