Partitioned total energy values

Figure I. Partitioned total energy values (in hartree) calculated for three different water hexamer using basis set 6-31G/d.

The results presented in this work show that in the linear structured water dimer the partitioned energy terms calculated for the proton donor and acceptor molecules are significantly different (except the kinetic energy). The electron structure of the proton donor molecule was found more compact than that of the acceptor subsystem, when compared their (partitioned) total energy EM values. This result is in an excellent agreement with our pre-vious results obtained on the separated molecular orbital energies [17]. 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

The equilibrium constants Kf are not measurable and we must resort to statistical thermodynamics to estimate these values theoretically. The partition function (Q) is a quantity with no simple physical significance but it may be substituted for concentrations in the calculation of equilibrium constants (Eqns. 4 and 5) [5], (It is assumed that there is no isotopic substitution in B.) Partition functions may be expressed as the product of contributions to the total energy from translational, rotational and vibrational motion (Eqn. 6). 

As with a molecular dynamics simulation, a Monte Carlo simulation comprises an equilibration phase followed by a production phase During equilibration, appropriate thermodynamic and structural quantities such as the total energy (and the partitioning of the energy among the various components), mean square displacement and order parameters (as appropriate) are monitored until they achieve stable values, whereupon the production phase can commence. In a Monte Carlo simulation from the canonical ensemble, the temperature and volume are, of course, fixed. In a constant pressure simulation the volume will change and should therefore also be monitored to ensure that a stable system density is achieved. 

In this scheme, the INT is partitioned into electrostatic, exchange, repulsion, polarization, and dispersion. Let us now look at the mathematical derivation of the various components. For a molecular system (AB) with wavefunction O and total energy Hamiltonian operator H, the expectation value is given as... 

While molecular dynamics simulations are usually performed at a constant total energy, energy partitioning arguments allow one to associate an effective temperature with the average value of the total system kinetic energy ... 

The above two examples illustrate that the value of the partition function is an indicator for how many of the energy levels are occupied at a particular temperature. At T = 0, where the system is in the ground state, the partition function has the value q = 1. In the limit of infinite temperature, entropy demands that all states are equally occupied and the partition function becomes equal to the total number of energy levels. 

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