How to make approximation of rotational partition function of diatomic linear molecules?
Using the rigid rotor approximation to the level energies, and such other appropriate assumptions, we can approximate rotational partition function, $Q_{\mathrm{rot}}$, of linear molecules as follows: \begin{eqnarray} Q_{\mathrm{rot}} & = & \sum_{J=0}^{\infty}{(2J+1)\exp(-\frac{E_J}{kT})}\ & \simeq & \sum_{J=0}^{\infty}{(2J+1)\exp(-\frac{hB_0J(J+1)}{kT})} \ \end{eqnarray} In the book, Microwave Molecular Spectra (Gordy & Cook, 1984), Chapter 3, Equation 3.64, the authors made approximation as: $$Q_{rot}\simeq\frac{kT}{hB_0}+\frac{1}{3}+\frac{1}{15}(\frac{hB_0}{kT})+\frac{4}{315}(\frac{hB_0}{kT})^2$$ Also, there is another approximation made by <PERSON>, R. S. 1988, Journal of Chemical Physics, 88, 356: $$Q_{rot}\simeq\frac{kT}{hB_0}\exp(\frac{hB_0}{3kT})$$
And we can see from the picture that $Q_{\mathrm{rot}}\simeq\frac{kT}{hB_0}+\frac{1}{3}$ is also a good approximation. Then I can guess how to derive <PERSON>'s approximation, but I cannot derive Gordy & Cook's approximation.
What I have got are as follows:
Since $$n\exp(-x(n+1)n)+\int_{(n+1)n}^{(m+1)m}{\exp(-xy)}dy+(m+1)\exp(-x(m+1)m) \ \leq\sum_{J=n}^{m}{(2J+1)\exp(-x(J+1)J)} \ \leq\sum_{i=n^2}^{m^2+2m}{\exp(-xi)}$$ hence, $$\int_{0}^{\infty}{\exp(-xy)}dy=\frac{1}{x} \ \leq\sum_{J=0}^{\infty}{(2J+1)\exp(-x(J+1)J)} \ \leq\sum_{i=0}^{\infty}{\exp(-xi)}=\frac{1}{1-\exp(-x)}$$
i.e., $$\frac{kT}{hB_0}<Q_{\mathrm{rot}}<\frac{1}{1-\exp(-\frac{hB_0}{kT})}$$
While the latter one approximates $\frac{1}{1-\exp(-\frac{hB_0}{kT})}\approx \frac{kT}{hB_0}+\frac{1}{2}+\frac{1}{12}\frac{hB_0}{kT}-\frac{1}{720}(\frac{hB_0}{kT})^3$ when $T$ is rather high.
Are there anyone knowing how to derive <PERSON> & <PERSON>'s approximation? Please tell me and I appreciate it very much.
Besides, another little question, are such approximations still useful as nowadays computers are greatly powerful?
Thanks! 565