Partition function formula

The molecular Hamiltonian can be split into the field-independent part and the part linear in the magnetic field [4] 

The zero-field Hamiltonian H0 includes the spin-orbit and spin-spin interactions and eventually the crystal field operators the first-order Zeeman term n comprises the orbital and spin contributions. The matrix elements of the full Hamiltonian in the chosen basis set are 

Powder averaging is achieved using the magnetic fields [4] 

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Fig. 3. Functions in the integrand of the partition function formula Eq. (6). The lower solid curve labeled Pq AU/kT) is the probability distribution of solute-solvent interaction energies sampled from the uncoupled ensemble of solvent configurations. The dashed curve is the product of this distribution with the exponential Boltzmann factor, e AJJ/kT r the upper solid curve. See Eqs. (5) and (6).

Here SU = AU — AC/is the electrostatic contribution to the solute-solvent interactions. The final expression is another partition function formula,... 

The PDT partition function formula (9.5) does not require that AUa adopt a specifically simplified form such as additive contributions over pairs of molecules involved. AUa is simply... 

For small p the contribution of paths with large x (n 0) to the partition function Z is suppressed because they are associated with large kinetic-energy terms proportional to v . That is why the partition function actually becomes the integral over the zeroth Fourier component Xq. It is therefore plausible to conjecture that the quantum corrections to the classical TST formula (3.49a) may be incorporated by replacing Z by... 

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

The reader who is less familiar with the theory of grand partition functions may directly proceed to Eqs. 12a and 13. The physical basis of these formulas and the significance of the quantities CK% will then become apparent in the subsequent paragraph is the vapor pressure (or fugacity) of solute K and y i is the probability of finding a K molecule in a cavity of type i. 

One may also try to evaluate solvation-free energies by using the corresponding partition function through the formula (Ref. 4)... 

The nudear partition function does not usually contribute to the partition function and can therefore be taken as unity. We shall ignore this contribution in the following. Finally, an overview of the formulae for partition functions is given in Tab. 3.2. 

The first term on the right is the formula for the chemical potential of component a at density pa = na/V in an ideal gas, as would be the case if interactions between molecules were negligible, fee is Boltzmann s constant, and V is the volume of the solution. The other parameters in that ideal contribution are properties of the isolated molecule of type a, and depend on the thermodynamic state only through T. Specifically, V/A is the translational contribution to the partition function of single a molecule at temperature T in a volume V... 

Use the formula Eq. (2.44) for chemical potential to show that the equilibrium number abundance of a nucleus i with mass number A, = Z, + /V, partition function u, and binding energy B, with respect to free protons and neutrons is given by... 

The formulae for the partition function of a molecule in the rigid-rotor-harmonic-oscillator approximation are summarized in Table 4.1. 

The formulae given in Table 4.1 for the molecular partition functions enable us to write the partition function ratio qheavy/qiight or q2/qi where, by the usual convention, the subscript 2 refers to the heavy isotopomer and 1 refers to the light isotopomer if heavy and light are appropriate designations. Then, a ratio of such partition function ratios enables one to evaluate the isotope effect on a gas phase equilibrium constant, as pointed out above. Before continuing, it is appropriate to... 

Schrodinger equation. When the molecule is too large and difficult for quantum mechanical calculations, or the molecule interacts with many other molecules or an external field, we turn to the methods of molecular mechanics with empirical force fields. We compute and obtain numerical values of the partition functions, instead of precise formulas. The computation of thermodynamic properties proceeds by using a number of techniques, of which the most prominent are the molecular dynamics and the Monte Carlo methods. 

The JE (Eq. (40)) indicates a way to recover free energy differences by measuring the work along all possible paths that start from an equihbrium state. Its mathematical form reminds one of the partition function in the canonical ensemble used to compute free energies in statistical mechanics. The formulas for the two cases are... 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

For a linear molecule with a small moment of inertia (e.g., H2), Eq. 8.81 will not be valid. Starting with Eq. 8.78, derive an expression for (E — Eq)v01 f°r the case where the high-temperature limit is not valid, that is, when an explicit term-by-term summation is needed to evaluate the rotational partition function. Use the derived formula to evaluate (E — Eq )rot for IV = A = 1 mole of H2 at 298 K. Compare the result with the high-temperature limit prediction. Find the percent difference in the two results. 

In order to calculate thermodynamic activation parameters, we need to know how to evaluate the translational, rotational, and vibrational parts of the partition functions. This can be accomplished by means of the standard formulas of statistical mechanics (see, for example, Dole, 1954). 

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