Partition function quantum

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 starting point is the quantum partition function Z in the coordinate representation ... 

The ratio of the quantum partition functions (Eq. (4-29)) for two different isotopes can be obtained directly through free energy perturbation (FEP) theory by perturbing the mass from the light isotope to the heavy isotope. Consequently, only one simulation of a given isotopic reaction is performed, while the ratio of the partition function, i.e., the KIE, to a different isotopic reaction, is obtained by FEP. This is conceptually and practically an entirely different approach than that used previously [23]. 

Wong K-Y, Gao J (2008) Systematic approach for computing zero-point energy, quantum partition function, and tunneling effect based on Kleinert s variational perturbation theory. J Chem Theory Comput 4(9) 1409-1422... 

With neglect of the quantum effects that arise from the exchange of identical particles [147], (8.66) gives the exact quantum partition function in the limit P — oo. For finite P, Qp((3) is the canonical partition function of a classical system composed of ring polymers. Each quantum particle corresponds to a ring polymer of P beads in which neighboring beads are connected by harmonic springs with force... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

Following Fey nman s original work, several authors pmsued extensions of the effective potential idea to construct variational approximations for the quantum partition function (see, e g., Refs. 7,8). The importance of the path centroid variable in quantum activated rate processes was also explored and revealed, which gave rise to path integral quantum transition state theory and even more general approaches. The Centroid Molecular Dynamics (CMD) method for quantum dynamics simulation was also formulated. In the CMD method, the position centroid evolves classically on the efiective centroid potential. Various analysis and numerical tests for realistic systems have shown that CMD captures the main quantum effects for several processes in condensed matter such as transport phenomena. 

For example, the classical-like phase space trace of this distribution function over the scalars x and p gives the quantum partition function in Eq. (5). However, in... 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

Quantum Corrections. The obvious way to introduce quantum corrections in eq. 10 would be to interpret Za and Z as quantum partition functions however, this neglects tunneling (Z+, being the partition function of a system constrained to the top of the activation bar-... 

The quantum partition function, eq.(35), of a system can be obtained through either Monte Carlo or molecular dynamics simulations. In PIMD simulation, the corresponding Lagrangian for such a classical isomorphic system can be given as ... 

There is an uncertainty in pressure of an order of 0.1 MPa depending on whether classical or quantum partition function is used for vibrational motions. Clearly, equation (14) gives too low dissociation pressure. Therefore, the influence of the guest molecule on the host lattice is fairly large and cannot be neglected[22, 24, 33]. Comparison with experiment will be made below. 

In Section IV.A, we have shown that the quantum partition function in D dimensions looks like a classical partition function of a system in (D+ 1) dimensions, with the extra dimension being the time. With this mapping and allowing the space and time variables to have discrete values, we turn the quantum problem into an effective classical lattice problem. 

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]... 

This expression is clearly isomorphic to <2 PI iiP/h2fi2 plays the same role (and has the same dimensions) as the force constant k, and F(xJ)/P plays the same role as U xm). In many dimensions, the quantum partition function becomes isomorphic to a solution of necklaces, the beads of which experience an external potential created by the positions of all the other beads... 

The quantum partition function in Eq. (1.1) is then obtained by the integration of the centroid density over all possible configurations of the... 

It should be noted that the centroid density is distinctly different from the coordinate (or particle) density p( ) = ((jlexp(-)3//) (5f). The particle density function is the diagonal element of the equilibrium density matrix in the coordinate representation, while the centroid density does not have a similar physical interpretation. However, the integration over either density yields the quantum partition function. 

In order to apply equation (1.94) to experimental data one must replace the classical partition functions by the corresponding quantum partition functions. Then, one obtains... 

From Eqs (3.37) and (3.38), it is obvious that for P = 1 one obtains the classical partition function and hence classical thermodynamics. The exact quantum partition function and quantum thermodynamics are obtained in the limit P —> oo however, in practice, it often suffices to take quite a small value of P to obtain accurate quantum results. 

In the sum over states formula, excited states for the vibrational modes need to be included up to convergence. A more convenient integral expression is provided by classical or semiclassical theories. At high temperatures and low frequencies, the vibrational motion behaves increasingly classically and the semiclassical Wigner-Kirkwood expression is an excellent approximation to the quantum partition function [78] for low-frequency vibrations at pyrolysis temperatures. The semiclassical... 

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