Partition function inversion

The ratio of symmetry numbers s s° in equation 11.40 merely represents the relative probabilities of forming symmetrical and unsymmetrical molecules, and ni and nf are the masses of exchanging molecules (the translational contribution to the partition function ratio is at all T equal to the power ratio of the inverse molecular weight). Denoting as AX, the vibrational frequency shift from isotopically heavy to light molecules (i.e., AX, = X° — X ) and assuming AX, to be intrinsically positive, equation 11.40 can be transated into... 

Table 11.9 Third-order polynomials relating rednced partition functions to inverse of squared absolute temperature x = 10 X (from Clayton and Kieffer, 1991).

Combining the results of Kieffer s model and of laboratory experiments, Clayton and Kieffer (1991) obtained a set of third-order equations relating the reduced partition functions of various minerals to the inverse of the squared absolute temperature (table 11.9) according to... 

For a unimolecular reaction like nitrogen inversion the translational, rotational and vibrational partition functions per degree of freedom may be assumed not to differ greatly in the initial and transition states. Then F /Fj may be reduced to l// where fv is the partition function for one vibrational degree of freedom. In (1 // ) is of the order of — 1 to 0 and should not change much with temperature 2.18). 

The partition function is the Laplace transform of the density of states, with the Laplace transform variable conjugate to energy being 1 /IcT. In principle therefore the density of states may be obtained by finding the inverse Laplace transform of the partition function. 

Methods for calculating these quantities are discussed in detail in the book by Holbrook, Pilling and Robertson [29]. These methods fall into several basic categories - classical approximations, inversion of the partition function, direct count methods and Monte Carlo methods, each of which is introduced briefly. 

The cosech function are expanded as a power series in kT. Inversion of the Laplace transform then gives a series solution for the density of states, whose first term is the same as the Marcus-Rice correction, Eq. (85), and whose subsequent terms are corrections containing smaller powers of -f- E. Ultimately the expansion of the partition function gives terms with negative powers of kT, which invert to delta functions and derivatives of delta functions, but truncating the series before this happens gives good smooth approximations to the density of states. 

In this section, we present briefly some very general features of the statistical mechanics of quantum systems. For a given system with a Hamiltonian H, the main quantity of interest is the partition function Z((3), where (3 is the inverse temperature... 

In the path integral approach, the analytical continuation of the probability amplitude to imaginary time t = —ix of closed trajectories, x(t) = x(f ), is formally equivalent to the quantum partition function Z((3), with the inverse temperature (3 = — i(t — t)/h. In path integral discrete time approach, the quantum partition function reads [175-177]... 

While this allows Monte Carlo simulations to be performed, the errors increase exponentially with the particle number N and the inverse temperature [3. To see this, consider the mean value of the sign (s) = Z/Z, which is just the ratio of the partition functions of the frustrated system Z = W(C) with weights W C) and the unfrustrated system used for sampling with... 

Z = Yhc partition functions are exponentials of the corresponding free energies, this ratio is an exponential of the differences Af in the free energy densities (s) = Z/Z = exp —(3NAf). As a consequence, the relative error As/ s) increases exponentially with increasing particle number and inverse temperature ... 

The partitioning functions have been chosen variously as Gaussians, atomic-like densities (proatoms, [30]), and inverse powers of r the latter is implemented in the DV program. One advantage of this approach, as emphasized by Becke [31], is that classical product rules can be applied to the overlapping multicenter density, to obtain integrals of high precision. 

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