Partition function Classical

Using these assumptions and a classical partitioning function, integrated over the coordinates and momenta of aH molecules, a universal function was defined ... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Chymotrypsin, 170,171, 172, 173 Classical partition functions, 42,44,77 Classical trajectories, 78, 81 Cobalt, as cofactor for carboxypeptidase A, 204-205. See also Enzyme cofactors Condensed-phase reactions, 42-46, 215 Configuration interaction treatment, 14,30 Conformational analysis, 111-117,209 Conjugated gradient methods, 115-116. See also Energy minimization methods Consistent force field approach, 113 Coulomb integrals, 16, 27 Coulomb interactions, in macromolecules, 109, 123-126... 

Figure 10. Arrhenius plot of the thermal rate constants for the 2D model system. Circles-full quantum results. Thick solid (dashed) curve present nonadiabatic transition state theory by using the seam surface [the minimum energy crossing point (MECP)] approximation. Thin solid and dashed curves are the same as the thick ones except that the classical partition functions are used. Taken from Ref. [27].

Using the matrix method of calculation (Li), it is possible to show that the classical partition function of the system is ... 

The discussion will be limited to systems at temperatures which are high enough so that we need only consider the classical partition function Q in calculating the Helmholtz free energy A, where, of course... 

Early in the development of VTST calculations on simple three atom systems compared rates obtained by exact classical dynamics with conventional TST and VTST, the same potential energy surface and classical partition functions being used throughout. These calculations confirmed the importance of eliminating the recrossing phenomenon in VTST. While TST yielded very much larger rate constants than the exact classical calculations, the VTST calculations yielded smaller rate constants, but never smaller than the exact classical values. 

N-particle systems. The classical partition function Z (T) of a canonical ensemble of N molecules is given by [184] (Hirschfelder et al. 1956)... 

Assume that the activated complexes are non-linear. Determine the temperature dependence for hv C fc///, corresponding to classical partition functions for the harmonic vibrational degrees of freedom, as well as for hv 3> ksT. 

Finally, the classical partition function for N interacting molecules, that is, the classical limit of Eq. (A.2), takes the form... 

Since we replaced the classical partition functions in GR ° and G ° by quantum mechanical ones, we have included quantum effects on all degrees of freedom of the reactants and all but the missing degree of freedom at the transition state. One then includes quantum effects on the remaining degree of freedom by a transmission coefficient k, thereby replacing Equation (3.15) by... 

Armed with these basic equations, classical partition functions and state densities for various types of motion may be evaluated either directly or via the quantum results. 

For heavy particles or high temperatures, the quantum correction (square brackets) converge against 1, and the classical partition function is recovered. 

Instead of the quantum mechanical partition function for a harmonic oscillator, equation (28) may be replaced by the classical partition function... 

There are two corrections to equation (12) that one might want to make. The first has to do with dynamical factors [19,20] i.e., trajectories leave Ra, crossing the surface 5/3, but then immediately return to Ra. Such a trajectory contributes to the transition probability Wfia, but is not really a reaction. We can correct for this as in variational transition-state theory (VTST) by shifting Sajj along the surface normals. [8,9] The second correction is for some quantum effects. Equation (14) indicates one way to include them. We can simply replace the classical partition functions by their quantum mechanical counterparts. This does not correct for tunneling and interference effects, however. 

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. 

To further speed up this approach, one can replace the expensive explicit-solvent simulations with implicit ones. Statistical mechanical theory gives the Helmholtz free energy A, apart from the scaling constant of the classical partition function that cancels out in binding energy calculations, as... 

The second and by far more serious problem with an application of Eq. (5.2) involves the probability density p r X) which is a priori unknown. An inspection of Ek. (2.117) reveals that p (r ,x) depends on the (classic) partition function, which involves the configuration integral as Eq. (2.118) shows. However, the partition function itself is unknown such that the probability with which O (r ) needs to be sampled at points remains undetermined. 

Note that the expression (1.60) is not purely classical since it contains two corrections of quantum mechanical origin the Planck constant h and the M. Therefore, Q defined in (1.60) is actually the classical limit of the quantum mechanical partition function in (1.59). The purely classical partition function consists of the integral expression on the rhs of (1.60) without the factor (h3NM). This partition function fails to produce the correct form of the chemical potential or of the entropy of the system. 

The condition required for the applicability of the classical partition function, as given in (1.60), is... 

Using (1.76) in the classical partition function (1.67), we immediately obtain... 

The dependence of both p and S on the density p through In p is confirmed by experiment. We note here that had we used the purely classical partition function [i.e., the integral excluding the factors h3NM in (1.60)], we would not have obtained such a dependence on the density. This demonstrates the necessity of using the correction factors h3NM even in the classical limit of the quantum mechanical partition function. 

This is the condition for which the classical partition function is valid. 

That magnetism is a purely quantum mechanical phenomenon was discovered already in the 30s by Bohr and van Leeuwen, who showed that a classical partition function based on the Hamiltonian of electrons in a magnetic field does not depend on the magnetic field, and therefore ... 

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