Note the difference between energetic properties such as U, P and H, which all depend on derivatives of Q, and entropic properties such as A, S and G, which depend directly on Q. For simplicity, we will use U and A for illustrations in the following, but other quantities such as H and S can be treated completely analogously.

In order to calculate the partition function q (Q), one needs to know all possible quantum states for the system. In principle, these can be calculated by solving the nuclear SchrÃ¶dinger equation, once a suitable potential energy surface is available, for example from solving the electronic SchrÃ¶dinger equation. Such a rigorous approach is only possible for di- and triatomic systems. For an isolated polyatomic molecule, the energy levels for a single conformation can be calculated within the rigid-rotor harmonic-oscillator (RRHO) approximation, where the electronic, vibrational and rotational degrees of freedom are assumed to be separable. Additional conformations can be included straightforwardly by simply offsetting the energy scale relative to the most stable conformation. An isolated molecule corresponds to an ideal gas state, and the partition function can be calculated exactly for such a system within the RRHO approximation, as discussed in Section 13.5.

For a condensed phase (liquid, solution, solid) the intermolecular interaction is comparable to or larger than a typical kinetic energy, and no separation of degrees of freedom is possible. Calculating the partition function by summing over all energy levels, or integrating over all phase space, is therefore impossible. It is, however, possible by sampling to estimate differences in Q and derivatives such as d ln Q/dT from a representative sample of the phase space, as discussed in Section 13.6.

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