配分函数Partition function (statistical mechanics)

分子的总能量近似可分为平动、转动、振动和电子能量之和，因此分子配分函数可写成乘积 q = q_平·q_转·q_振·q_电。平动配分函数极大（常温下约 10³⁰ 量级）；双原子分子转动配分函数约为 kT/(hcB)；振动配分函数 q_振 = 1/(1 − e^(−hν/kT))，常温下接近 1；电子配分函数通常等于基态简并度。N 个不可区分分子的正则配分函数 Q = qᴺ/N!。

A molecule's energy is approximately the sum of translational, rotational, vibrational and electronic parts, so the molecular partition function factorises, q = q_trans·q_rot·q_vib·q_elec. The translational part is enormous (around 10³⁰ at room temperature); the rotational part of a diatomic is about kT/(hcB); the vibrational part, 1/(1 − e^(−hν/kT)), is close to 1 at room temperature; and the electronic part usually equals the ground-state degeneracy. For N indistinguishable molecules the canonical partition function is Q = qᴺ/N!.

内能 U = kT²(∂lnQ/∂T)_V；亥姆霍兹自由能 A = −kT·lnQ；熵 S = U/T + k·lnQ。平衡常数也可由各物种的配分函数计算：K = Π(qᵢ/N_A)^νᵢ·e^(−ΔE₀/RT)。由光谱数据（键长、振动频率）计算配分函数，即可从第一性原理得到熵、热容和平衡常数，这是量子化学程序计算热化学数据的方法。

Internal energy U = kT²(∂lnQ/∂T)_V; Helmholtz free energy A = −kT·lnQ; entropy S = U/T + k·lnQ. Equilibrium constants follow from the species' partition functions: K = Π(qᵢ/N_A)^νᵢ·e^(−ΔE₀/RT). Computing partition functions from spectroscopic data such as bond lengths and vibrational frequencies gives entropies, heat capacities and equilibrium constants from first principles, which is how quantum-chemistry programs produce thermochemistry.

玻尔兹曼和吉布斯奠定了配分函数的基础。德语名称 Zustandssumme（状态和）由普朗克在 1921 年左右使用，英文 partition function 由达尔文和福勒在 1922 年提出，意为能量在各状态之间“分配”的方式。过渡态理论（艾林，1935）用反应物和活化络合物的配分函数计算速率常数。

Boltzmann and Gibbs laid the foundations. The German name Zustandssumme, 'sum over states', was used by Planck around 1921, and the English partition function was introduced by Darwin and Fowler in 1922, referring to how energy is partitioned among states. Eyring's transition-state theory (1935) computes rate constants from the partition functions of reactants and activated complex.