Configurational integral energy

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... 

Here, 6 is the Dirac delta function, U is the potential energy function, and q represents the 3N coordinates. In this expression, the integral is performed over the entire configuration space - each coordinate runs over the volume of the simulation box, and the delta function selects only those configurations of energy S. The N term factors out the identical configurations which differ only by particle permutation. It is worth noting that the density of states is an implicit function of N and V,... 

Pu depends on the quotient flj, / TT, the calculation of the configurational integral Z(N,V,T) is avoided. The change in potential energy of the system due to the trial move determines if the attempted new configuration is accepted. 

The partition function can be written as the product of K, the kinetic energy integral, and Z, the configurational integral,... 

The use of statistical calculations of configuration integrals to determine thermodynamic adsorption characteristics of zeolites dates back to the late 1970s (49). Kiselev and Du (22) reported calculations based on atom-atom potentials for Ar, Kr, and Xe sorbed in NaX, NaY, and KX zeolites. Then-calculations, which included an electrostatic contribution, predicted changes in internal energy in excellent agreement with those determined experimentally. The largest deviation between calculated and experimental values, for any of the sorbates in any of the hosts, was a little over 1 kJ/mol. 

Configurational-integral ratios of the form of Eq. (50) appear widely in the free-energy literature, but the underlying physical motivation is not always the same. The spectrum of possible usages is covered by writing Eq. (50) in the more general form... 

However, these results do not immediately carry over to the problems of interest here where (while PBCs are the norm) the ensembles are frequently open or constant pressure, and the systems do not fit in to the lattice model framework. Even in the apparently simple case of crystalline solids in NVT, the free translation of the center of mass introduces /-dependent phase space factors in the configurational integral which manifest themselves as additional finite-size corrections to the free energy these may not yet be fully understood [58, 97]. If one adopts the traditional stance, then, one is typically faced with having to make extrapolations of the free-energy densities in each of the two phases, without a secure understanding of the underlying form (jf . ..) of the corrections involved. 

In the above expression Z stands for the configuration integral which differs from the partition function Q by being calculated only over the coordinates of the molecules and not over their momenta. This is possible because the coordinate and momentum parts of the whole phase space are run over independently throughout the integration and to a simple quadratic form of the kinetic energy, which thus can be integrated immediately. 

The applicability of the dimensional model to Ionic and nonlonlc molecules containing as many as seven atoms Is gratifying. This demonstrates the Insensitivity of the entropy (as determined by the configurational Integral) to the exact form of the pair potential. This encouraged us to attempt similar correlations with homonuclear clusters. We will test whether the Xq species follow the prescriptions of the dimensional model for the corresponding MXjj species. For convenience, we will restrict ourselves to the entropy at 1000 K. The correlations at other temperatures and for the free energy functions are similar. 

Statistical thermodynamics of the electric double layer starts with modelling the electrolyte and the Interface. This can be done by specifying all inter-molecular and external Interactions in the phase space as a Hamiltonian. The notion of phase space was defined in sec. 1.3.9a and the Hamiltonian H was introduced in [1.3.9.11. As the kinetic part of the Hamiltonian does not contribute to the configuration Integrals, we sum only over the potential energies of the ions. In the Inhomogeneous system it Is customary to separate the interactions with the charged wall (the external" field) from the interlonlc ones. 

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