2.8. Molecular Statistical Mechanics

Overview and Learning Objectives¶

Molecular statistical mechanics often builds the total molecular partition function as a product of translational, rotational, vibrational, and electronic factors. This section collects high-temperature rigid-rotor formulas for different molecular shapes, reviews vibrational and electronic partition functions, and summarizes standard textbook approximations used to connect molecular constants to thermodynamic properties. We use rotational temperatures $\Theta_{\text{rot}}$ and vibrational temperatures $\Theta_{\text{vib}}$ , which package molecular constants into energy scales expressed in Kelvin (i.e., divided by Boltzmann’s constant $k_{\text{B}}$ ).

Learning objectives:

Identify how molecular shape (linear, spherical top, symmetric top, asymmetric top) affects the rotational partition function.
Use the rotational symmetry number $\sigma$ and rotational temperatures $\Theta_{\text{rot}}$ to write high- $T$ expressions for $q_{\text{rot}}$ .
Write the harmonic-oscillator vibrational partition function for diatomic and polyatomic molecules in terms of $\Theta_{\text{vib}}$ .
Interpret the electronic partition function and justify the ground-state approximation $q_{\text{elec}}\approx g_1$ when excited states are high in energy.

Core Ideas and Derivations¶

Rotational Symmetry¶

The rotational partition function $q_{\text{rot}}$ depends on:

The rotational constants (encoded as rotational temperatures $\Theta_{\text{rot},A}$ , $\Theta_{\text{rot},B}$ , $\Theta_{\text{rot},C}$ , etc.).
The rotational symmetry number $\sigma$ .
The temperature $T$ .

Below is a summary table of approximate expressions for rigid rotors in the high-temperature limit $\bigl(T \gg \Theta_{\text{rot}}\bigr)$ . Here, $\Theta_{\text{rot}}$ and $T$ are both expressed in Kelvin, so ratios like $T/\Theta_{\text{rot}}$ are dimensionless. In the symmetric-top expressions below, we take $\Theta_{\text{rot},A}=\Theta_{\text{rot},B}$ .

Table 1:Rotational Partition Functions of Rigid Rotors

Molecular Class	Symmetry Number ( $\sigma$ )	Rotational Partition Function ( $q_{\text{rot}}$ )	Examples ( $\sigma$ )
Heteronuclear Diatomic (Linear)	1	$\frac{T}{\Theta_{\text{rot}}}$	CO, NO, HF, HCl, etc.
Asymmetric Polyatomic (Linear)	1	$\frac{T}{\Theta_{\text{rot}}}$	HCN, N₂O, FCN, etc.
Homonuclear Diatomic (Linear)	2	$\frac{T}{2\,\Theta_{\text{rot}}}$	H₂, O₂, N₂, etc.
Symmetric Polyatomic (Linear)	2	$\frac{T}{2\,\Theta_{\text{rot}}}$	CO₂, CS₂, XeF₂
Spherical Top	12 ( $T_d$ ) or 24 ( $O_h$ )	$\frac{\sqrt{\pi}}{\sigma}\,\biggl(\frac{T}{\Theta_{\text{rot}}}\biggr)^{3/2}$	CH₄, SiH₄, SF₆, etc.
Symmetric Top	Varies with symmetry	$\frac{\sqrt{\pi}}{\sigma}\,\sqrt{\frac{T^3}{\Theta_{\text{rot},A}^2\,\Theta_{\text{rot},C}}}$	C₆H₆ ( $\sigma=12$ ), NH₃ ( $\sigma=3$ ), XeF₄ ( $\sigma=8$ ), etc.
Asymmetric Top	Varies with symmetry	$\frac{\sqrt{\pi}}{\sigma}\,\sqrt{\frac{T^3}{\Theta_{\text{rot},A}\,\Theta_{\text{rot},B}\,\Theta_{\text{rot},C}}}$	H₂O ( $\sigma=2$ ), NO₂ ( $\sigma=2$ ), SO₂ ( $\sigma=2$ ), most large molecules

Nonlinear Rigid Rotors¶

Below is a visualization comparing spherical top, symmetric top, and asymmetric top molecules. The ellipsoid semi-axes are scaled by the three principal rotational temperatures $\Theta_{\text{rot},i}$ , highlighting how anisotropy in the rotational constants maps onto the 3D “rotational profile.”

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from myst_nb import glue

u = np.linspace(0, 2*np.pi, 100)
v = np.linspace(0, np.pi, 100)
u, v = np.meshgrid(u, v)

def surface_points(a, b, c):
    """Return x, y, z for an ellipsoid with semi-axes a, b, c."""
    x = a * np.cos(u) * np.sin(v)
    y = b * np.sin(u) * np.sin(v)
    z = c * np.cos(v)
    return x, y, z

# Rotational temperatures (sample values)
theta_ch4 = (7.54, 7.54, 7.54)     # Spherical top
theta_nh3 = (19.9, 19.9, 9.6)      # Symmetric top
theta_h2o = (40.1, 20.9, 13.4)     # Asymmetric top

fig = plt.figure(figsize=(18, 6))

# 1) CH4: Spherical top
ax1 = fig.add_subplot(1, 3, 1, projection='3d')
x1, y1, z1 = surface_points(*theta_ch4)
surf1 = ax1.plot_surface(x1, y1, z1, color='skyblue', alpha=0.4, edgecolor='none')
ax1.set_title("Spherical Top")
ax1.set_box_aspect((1,1,1))
ax1.set_xlim(-7.54, 7.54)
ax1.set_ylim(-7.54, 7.54)
ax1.set_zlim(-7.54, 7.54)
ax1.view_init(elev=20, azim=35)
ax1.set_xlabel('$x$')
ax1.set_ylabel('$y$')
ax1.set_zlabel('$z$')
ax1.set_xticks([])
ax1.set_yticks([])
ax1.set_zticks([])

# Add a CH4 molecule to the plot (schematic coordinates)
positions = np.array([
    [ 0.0000,  0.0000,  0.0000],
    [ 0.6276,  0.6276,  0.6276],
    [ 0.6276, -0.6276, -0.6276],
    [-0.6276,  0.6276, -0.6276],
    [-0.6276, -0.6276,  0.6276]
]) * 7.54/2

for i, pos in enumerate(positions):
    if i == 0:
        ax1.plot(*pos, 'o', markersize=10, markeredgecolor='black', markerfacecolor='black')
    else:
        ax1.plot(*pos, 'o', markersize=5, markeredgecolor='black', markerfacecolor='white')
    if i > 0:
        ax1.plot([positions[0][0], pos[0]],
                 [positions[0][1], pos[1]],
                 [positions[0][2], pos[2]], color='black')

ax1.text(7.54/4, -7.54/4, 7.54/16, 'CH$_4$', color='black', fontsize=12)

# 2) NH3: Symmetric top
ax2 = fig.add_subplot(1, 3, 2, projection='3d')
x2, y2, z2 = surface_points(*theta_nh3)
surf2 = ax2.plot_surface(x2, y2, z2, color='salmon', alpha=0.4, edgecolor='none')
ax2.set_title("Symmetric Top")
ax2.set_box_aspect((1,1,1))
ax2.set_xlim(-19.9, 19.9)
ax2.set_ylim(-19.9, 19.9)
ax2.set_zlim(-19.9, 19.9)
ax2.view_init(elev=20, azim=35)
ax2.set_xlabel('$x$')
ax2.set_ylabel('$y$')
ax2.set_zlabel('$z$')
ax2.set_xticks([])
ax2.set_yticks([])
ax2.set_zticks([])

positions = np.array([
    [ 0.0000,  0.0000,  0.0000],
    [ 0.0000, -0.9377, -0.3816],
    [ 0.8121,  0.4689, -0.3816],
    [-0.8121,  0.4689, -0.3816]
]) * 19.9/2

for i, pos in enumerate(positions):
    if i == 0:
        ax2.plot(*pos, 'o', markersize=10, markeredgecolor='black', markerfacecolor='skyblue')
    else:
        ax2.plot(*pos, 'o', markersize=5, markeredgecolor='black', markerfacecolor='white')
    if i > 0:
        ax2.plot([positions[0][0], pos[0]],
                 [positions[0][1], pos[1]],
                 [positions[0][2], pos[2]], color='black')

ax2.text(19.9/4, -19.9/4, 19.9/16, 'NH$_3$', color='black', fontsize=12)

# 3) H2O: Asymmetric top
ax3 = fig.add_subplot(1, 3, 3, projection='3d')
x3, y3, z3 = surface_points(*theta_h2o)
surf3 = ax3.plot_surface(x3, y3, z3, color='plum', alpha=0.4, edgecolor='none')
ax3.set_title("Asymmetric Top")
ax3.set_box_aspect((1,1,1))
ax3.set_xlim(-40.1, 40.1)
ax3.set_ylim(-40.1, 40.1)
ax3.set_zlim(-40.1, 40.1)
ax3.view_init(elev=20, azim=35)
ax3.set_xlabel('$x$')
ax3.set_ylabel('$y$')
ax3.set_zlabel('$z$')
ax3.set_xticks([])
ax3.set_yticks([])
ax3.set_zticks([])

positions = np.array([
    [ 0.0000,  0.0000,  0.1173],
    [ 0.7572,  0.0000, -0.4692],
    [-0.7572,  0.0000, -0.4692]
]) * 40.1/2

for i, pos in enumerate(positions):
    if i == 0:
        ax3.plot(*pos, 'o', markersize=10, markeredgecolor='black', markerfacecolor='red')
    else:
        ax3.plot(*pos, 'o', markersize=5, markeredgecolor='black', markerfacecolor='white')
    if i > 0:
        ax3.plot([positions[0][0], pos[0]],
                 [positions[0][1], pos[1]],
                 [positions[0][2], pos[2]], color='black')

ax3.text(40.1/4, -40.1/4, 40.1/16, 'H$_2$O', color='black', fontsize=12)

plt.tight_layout()
plt.show()
plt.close(fig)

Approximate visualization of rotational shapes: (left) a spherical top (CH₄), (middle) a symmetric top (NH₃), and (right) an asymmetric top (H₂O). The ellipsoid semi-axes are scaled by the chosen $\Theta_{\text{rot},i}$ values along each principal axis.

Vibrations of Polyatomic Molecules¶

For a diatomic molecule approximated as a harmonic oscillator with characteristic vibrational temperature $\Theta_{\text{vib}}$ , the vibrational partition function is

q_{\text{vib}} \;=\; \frac{e^{-\Theta_{\text{vib}}/(2T)}}{\,1 - e^{-\Theta_{\text{vib}}/T}\,}.

(1)

For a polyatomic molecule with $\alpha$ normal modes (where $\alpha = 3N - 6$ for a nonlinear molecule and $\alpha = 3N - 5$ for a linear molecule), the vibrational partition function factors into a product of each mode’s contribution:

q_{\text{vib}} \;=\; \prod_{j=1}^\alpha \; \frac{\,e^{-\Theta_{\text{vib},j}/(2T)}\,}{\,1 - e^{-\Theta_{\text{vib},j}/T}\,}.

(2)

Here, $\Theta_{\text{vib},j}$ is the vibrational temperature of the $j$ -th normal mode.

Electronic Partition Function¶

The electronic partition function is

q_{\text{elec}} \;=\; \sum_{i=1}^{\infty} g_i \, e^{-\beta\,\varepsilon_i} \;\;\underset{\varepsilon_1 = 0}{\longrightarrow}\;\; g_1 \;+\; g_2\,e^{-\beta\,(\varepsilon_2 - \varepsilon_1)} \;+\;\cdots,

(3)

where $g_i$ is the degeneracy of the $i$ -th electronic state, $\varepsilon_i$ is that state’s electronic energy, and $\beta = 1/(k_{\text{B}} T)$ . Taking $\varepsilon_1=0$ as the reference, most ground-state-dominated situations satisfy $\varepsilon_2 - \varepsilon_1 \gg k_{\text{B}} T$ , so the higher-lying states contribute negligibly and $q_{\text{elec}}\approx g_1$ .

Summary¶

The table below collects the main partition function formulas for atoms and molecules under typical textbook approximations (high- $T$ rotation, harmonic vibration, and negligible population of excited electronic states):

Table 2:Partition functions for various systems

System	$q_{\text{trans}}$	$q_{\text{rot}}$	$q_{\text{vib}}$	$q_{\text{elec}}$
Atom	$\frac{V}{\Lambda^3}$	—	—	$g_1$
Diatomic	↓	$\frac{T}{\sigma\,\Theta_{\text{rot}}}$	$\frac{e^{-\Theta_{\text{vib}}/(2T)}}{\,1 - e^{-\Theta_{\text{vib}}/T}\,}$	↓
Polyatomic	—	—	—	—
Linear	↓	same as diatomic (with adjusted $\sigma$ )	$\prod_{j=1}^{\alpha}\,\frac{e^{-\Theta_{\text{vib},j}/(2T)}}{1-e^{-\Theta_{\text{vib},j}/T}}$	↓
Nonlinear	—	—	—	—
Spherical	↓	$\frac{\sqrt{\pi}}{\sigma}\,\bigl(\tfrac{T}{\Theta_{\text{rot}}}\bigr)^{3/2}$	↓	↓
Symmetric	↓	$\frac{\sqrt{\pi}}{\sigma}\,\sqrt{\frac{T^3}{\Theta_{\text{rot},A}^2\,\Theta_{\text{rot},C}}}$	↓	↓
Asymmetric	↓	$\frac{\sqrt{\pi}}{\sigma}\,\sqrt{\frac{T^3}{\Theta_{\text{rot},A}\,\Theta_{\text{rot},B}\,\Theta_{\text{rot},C}}}$	↓	↓

The translational partition function $q_{\text{trans}} = \frac{V}{\Lambda^3}$ (with $\Lambda$ the thermal de Broglie wavelength) applies to all gas-phase systems. For many molecules, the total partition function is well approximated as the product

q \;\approx\; q_{\text{trans}}\;q_{\text{rot}}\;q_{\text{vib}}\;q_{\text{elec}},

(4)

with each factor evaluated using the formulas above, assuming the relevant excited states are not significantly populated beyond the ground state.

Worked Example¶

High- $T$ rotational partition function for a spherical top (CH $_4$ )¶

For a spherical top,

q_{\text{rot}} \approx \frac{\sqrt{\pi}}{\sigma}\left(\frac{T}{\Theta_{\text{rot}}}\right)^{3/2}.

(5)

Using the sample rotational temperature in the section visualization for CH $*4$ , $\Theta*{\text{rot}}=7.54\ \mathrm{K}$ , and the tetrahedral symmetry number $\sigma=12$ :

Compute $T/\Theta_{\text{rot}}$ at $T=300\ \mathrm{K}$
$\frac{T}{\Theta_{\text{rot}}}=\frac{300}{7.54}=39.8.$
(6)
Evaluate $q_{\text{rot}}$
$q_{\text{rot}} \approx \frac{\sqrt{\pi}}{12}(39.8)^{3/2} =\frac{1.772}{12}\,(39.8)\sqrt{39.8} \approx 37.$
(7)

Result. At room temperature, the rotational partition function of CH $_4$ is large, indicating many thermally accessible rotational states.

Concept Checks¶

Why do nonlinear molecules typically have $q_{\text{rot}}\propto T^{3/2}$ in the high- $T$ limit rather than $q_{\text{rot}}\propto T$ ?
What is the physical meaning of the symmetry number $\sigma$ , and why does it reduce $q_{\text{rot}}$ ?
When is it reasonable to approximate $q_{\text{elec}}\approx g_1$ ?
For a polyatomic molecule, why is $q_{\text{vib}}$ a product over normal modes?

Key Takeaways¶

Total molecular partition functions are built as products of translational, rotational, vibrational, and electronic factors.
Rigid-rotor high- $T$ formulas depend on molecular symmetry and rotational temperatures.
Harmonic-oscillator vibration introduces $\Theta_{\text{vib}}$ and freezes out at $T\ll\Theta_{\text{vib}}$ .
Electronic contributions are often dominated by the ground state unless excited states lie within $k_{\text{B}}T$ .