A system of non-interacting molecules

at sufficiently high temperatures, for N non-interacting identical and indistinguishable particles (e.g., an ideal gas), the partition function can be written in the form 

The evaluation of the molecular partition function can be simplified by noting that the total energy of the molecule may be written as a sum of the center-of-mass translational energy and the internal energy, E = Etrans + Emt, which implies 

These partition functions can be evaluated quite readily. In the following it turns out to be convenient to choose the zero of energy such that it coincides with the zero-point level of the quantized energy levels. To that end, we note that, if we subtract 

Here the sum is the partition function with the energy measured relative to the zero-point level and Q is obtained after multiplication by exp(—Eo/kBT). The Boltzmann distribution Eq. (A.4) can be written in the form 

the partition functions differ by the factor exp(—Eo/kBT), whereas the Boltzmann distribution is invariant to such a shift of the energy scale in the standard 

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Partition function (q, z) - For a single molecule, q = Y,.g.exp EjkT), where e, is an energy level of degeneracy the Boltzmann constant, and T the absolute temperature the summation extends over all energy states. For a system of non-interacting molecules which are indistinguishable, as in an ideal gas, the canonical partition function Q = qN/Nl Pascal (Pa) - The SI unit of pressure, equal to N/m. [1]... 

The classical kinetic theoty of gases treats a system of non-interacting particles, but in real gases there is a short-range interaction which has an effect on the physical properties of gases. The most simple description of this interaction uses the Lennard-Jones potential which postulates a central force between molecules, giving an energy of interaction as a function of the inter-nuclear distance, r. 

Systems of chemical interest typically contain particles in molar quantity. Mathematical modelling of all interactions in such a system is clearly impossible. Even in a system of non-interacting particles, such as an ideal gas, it would be equally impossible to keep track of all individual trajectories. It is consequently not a trivial matter to extend the mechanical description (either classical or non-classical) of single molecules to macrosystems. It would be required in the first place to define the initial state of each particle in terms of an equation... 

It should be noted that if the system consists of a gas of non-interacting molecules then... 

The calculations were based on the Kohn-Sham density functional theory (60). This equation has become a popular method for calculating the molecular properties of organic molecules. The Kohn-Sham equation is the Schrodinger equation of a system of non-interacting particles, typically electrons (61). 

The third and final approach to the electron correlation problem included briefly here is density functional theory (DFT), a review of which has been given by Kohn in his Nobel lecture [38], The Hohcnberg Kolin theorem [39] states that there is a one-to-one mapping between the potential V(r) in which the electrons in a molecule move, the associated electron density p(r), and the ground state wave function lP0. A consequence of this is that given the density p(r), the potential and wave function lf 0 are functionals of that density. An additional theorem provided by Kohn and Sham [40] states that it is possible to construct an auxiliary reference system of non-interacting... 

Solvent potential. The averaged solvent electrostatic field, , is important for inhomogeneous media, such as enzymes, membranes, miscelles and crystalline environments systems. Due to the existence of strong correlations, such a field does not cancel out. This factor becomes an important contribution to solvent effects at a microscopic level. In a study of non-rigid molecules in solution, Sese et al. [25] constructed a by using the solute-solvent atom-atom radial distribution function. Electrostatic interactions in three-dimensional solids were treated by Angyan and Silvi [26] in their self-consistent Madelung potential approach such a procedure can be traced back to a calculation of . An earlier application of the ISCRF theory to the study of proton mechanisms in crystals of hydronium perchlorate both [Pg.441]

The molecular mechanics method, often likened to a ball and spring model of the molecule, represents the total energy of a system of molecules with a set of simple analytical functions representing different interactions between bonded and non-bonded atoms, as shown schematically in Figure 1. 

More recent work with cosolvency in dilute systems seems to indicate that the magnitude of the solubility enhancement is linear up to some 10-20% cosolvent fraction [55,172,184,250-262]. At very low concentrations of cosolvent, the assumption of non-interaction between the cosolvent and water cannot hold. In dilute solutions the individual cosolvent molecules will be fully hydrated and, as a result, will disrupt the water network structure. If the total volume disrupted is regarded as the extended hydration shell, and if Sc is the average solubility within this shell, then the overall solubility Sm in the water-cosolvent mixture will be approximated by... 

The procedure for obtaining SMOs starts from the occupied and virtual canonical molecular orbitals (CMOs) of the super-system it uses a bridge in order to transform the CMOs towards a prescribed set of orbitals. The criterion prescribes, that the overlap between the constmcted orbitals tj/ of a super molecule (SM) should be maximal with the initial, canonical orbitals (j) of the non-interacting molecules, the bridge thus implies an overlap criterion ... 

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