Total molecular partition function

But molecular gases also have rotation and vibration. We only make the correction for indistinguishability once. Thus, we do not divide by IV l to write the relationship between Zro[, the rotational partition function of N molecules, and rrol, the rotational partition function for an individual molecule, if we have already assigned the /N term to the translation. The same is true for the relationship between Zv,h and In general, we write for the total partition function Z for N units... 

In order to leam more about the nature of the intermolecular forces we will start with partitioning of the total molecular energy, AE, into individual contri butions, which are as close as possible to those we defined in intermolecular perturbation theory. Attempts to split AE into suitable parts were undertaken independently by several groups 83-85>. The most detailed scheme of energy partitioning within the framework of MO theory was proposed by Morokuma 85> and his definitions are discussed here ). This analysis starts from antisymmetrized wave functions of the isolated molecules, a and 3, as well as from the complete Hamiltonian of the interacting complex AB. Four different approximative wave functions are used to describe the whole system ... 

Some important systems, which certainly do not fulfill the assumptions of harmonic transition state theory are gas phase reactions. In the gas phase, there are zero-modes such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal mode analysis. For these species one can in a simple manner modify the terms going into the HTST rate by incorporating the molecular partition functions [3,119]. 

In a polyatomic molecule witli many vibrations, we simplify the vibrational partition function much as the original molecular partition function was simplified we assume that the total vibrational energy can be expressed as a sum of individual energies associated with each mode, in which case, for a non-linear molecule, we have... 

Let qs be the molecular partition function for an adsorbed species. Consider the adsorption of N molecules on some portion of the surface containing a total of M possible adsorption sites. The system partition function Qs of the collection of N adsorbed species is... 

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 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... 

Here E ( y1 ) stands for the single-particle contribution to the total energy, allowing for molecule interaction with the surface <2 is the heat released in adsorption of molecules z on the /Lh site Fj the internal partition function for the z th molecules adsorbed on the /Lh site F j the internal partition function for the zth molecule in the gas phase the dissociation degree of the z th molecule, and zz the Henry local constant for adsorption of the zth molecule on the /Lh site. Lateral interaction is modeled by E2k([ylj ), and gj (r) allows for interaction between the z th and /Lh particles adsorbed on the /th and gth sites spaced r apart. In the lattice gas model, separations are conveniently measured in coordination-sphere numbers, 1 < r < R. For a homogeneous surface, molecular parameters zz and ej(r) are independent of the site nature, while for heterogeneous, they may depend on it. 

For the surface, we calculate the Helmholtz free energy from Eq. (45) of Chapter 5 A = —RT In Q. We assume that surface molecules are distinguishable (by their position) and noninteracting, so that the system partition function is a product of N molecular partition functions. However, because we are not interested in which of the N out of a total of M surface sites are occupied, we must include a degeneracy factor of M /N M — A) . The energy of a molecule on the surface is taken as zero. 

This section contains the background for the combination of density functional theory and molecular mechanics. Following the basic philosophy of quantum mechan-ics/molecular mechanics approaches we partition the total system into at least two parts which can be treated simultaneously. The quantum mechanical subsystem is described using DFT and the classical subsystem is given by molecular mechanics. Based on the QM/MM approach we have that the total energy of the system is... 

The TST was developed originally by Eyring and others on the basis of statistical mechanics [see, e.g., Lasaga (1983) or Moore and Pearson (1981)]. The fundamental result is a bimolecular rate constant for an elementary process expressed in terms of (1) the total molecular partition functions per unit volume iqi) for reactant species and for the activated complex species (q ), and (2) tlie difference in zero-point potential energies between the activated complex and reactants (Eq) ... 

The quantization of transition state energy levels is not simply a mathematical device to add quantum effects to the partition functions. The quantized levels actually show up as structure in the exact quantum mechanical rate constants as functions of total energy [51]. The interpretation of this structure provides clear evidence for quantized dynamical bottlenecks, both near to and distant from the saddle points, as reviewed elsewhere [52]. Quantized variational transition states have also been observed in molecular beam scattering experiments [53]. Analysis of the reactive flux in state-to-state terms from reactant states to transition state levels to product states provides the ultimate limit of resolution allowed by quantum mechanics [53,54]. Quantized energy levels of the variational transition state have been used to rederive TST using the language of quantum mechanical resonance scattering theory [55]. 

Such representation is useful when it is impossible or impractical to make detailed calculations of partition functions, but when order-of-magnitude estimates would be helpful. Table 2.3 is a tabulation of representative values for these factors computed from equations (2-62), (2-64), or (2-65) and (2-66) using typical values for molecular constants in the temperature range from 300 500°K. The translational contribution per degree of freedom is much larger than rotation or vibration, however, in complicated polyatomic molecules it is possible for the total vibrational contribution to become large as (3iV — 6) becomes a large number. 

The concept of local properties, that is, properties of atoms or groups of atoms within one molecule, is a useful tool for the interpretation of abstract wave functions in terms of intuitive building blocks. In principle, any molecular property can be partitioned and distributed over a set of subsystems of a molecule, which then provides information for a quaHtative analysis and understanding of bonding. However, such a partitioning is not unique and requires additional ad hoc) assumptions. Still, within the realm of these assumptions, useful qualitative reasoning is possible. One important local concept for open-shell molecules is the decomposition of the total molecular electron spin S, ... 

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