Electronic partition functions

Mciny of the theories used in molecular modelling involve multiple integrals. Examples include tire two-electron integrals formd in Hartree-Fock theory, and the integral over the piriitii >ns and momenta used to define the partition function, Q. In fact, most of the multiple integrals that have to be evaluated are double integrals. 

The electronic partition function of the transition state is expressed in terms of the activation energy (the energy of the transition state relative to the electronic energy of the reactants) E as ... 

MaxweU-Boltzmaim particles are distinguishable, and a partition function, or distribution, of these particles can be derived from classical considerations. Real systems exist in which individual particles ate indistinguishable. Eor example, individual electrons in a soHd metal do not maintain positional proximity to specific atoms. These electrons obey Eermi-Ditac statistics (133). In contrast, the quantum effects observed for most normal gases can be correlated with Bose-Einstein statistics (117). The approach to statistical thermodynamics described thus far is referred to as wave mechanics. An equivalent quantum theory is referred to as matrix mechanics (134—136). 

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

Following on the work of Kohn and Sham, the approximate functionals employed by current DFT methods partition the electronic energy into several terms ... 

With this set of energy levels, the electronic partition function is given by... 

We have seen that for the electronic partition function there is no closed form expression (as there is for translation, rotation, and vibration) and one must know the energy and degeneracy of each state. That is. 

Using equation (10.77), the electronic contribution to the heat capacity can be obtained by appropriate differentiation of the partition function ... 

In Eq. (44), gei(T ) is the ratio of transition state and reactant electronic partition functions [31] and the rotational degeneracy factor = (2ji + l)(2/2 + 1) for heteronuclear diatomics, and will also include nuclear spin considerations in the case of homonuclear diatomics. 

The first illustration of the concept of a partition function is that of a two-level system, e.g. an electron in a magnetic field, with its spin either up or down (parallel or anti parallel to the magnetic field) (Fig. 3.2). The ground state has energy Eq = 0 and the excited state has energy Ae. By substituting these values in Eq. (3) we find the following partition function for this two-level system ... 

Usually, we would choose the separate atoms in their ground state as the zero energy. The electronic partition function is then... 

Choosing the separate atoms as the zero energy, the electronic partition function of the hydrogen molecule is... 

Here we have utilized Eq. (147) and assumed that the electronic ground state of the transition state has been raised by AE (to refer partition functions to the transition state s own ground state) and qto-vih is referred with respect to the bottom of the potential, as in Fig. 3.10. Expression (156) shows that the adsorption rate per area is the collision number for that area times a factor So(T), the so-called sticking coefficient, which must always be smaller than one. The sticking coefficient describes how many of the incident atoms were successful in reaching the adsorbed state... 

Table 10.4 lists the rate parameters for the elementary steps of the CO + NO reaction in the limit of zero coverage. Parameters such as those listed in Tab. 10.4 form the highly desirable input for modeling overall reaction mechanisms. In addition, elementary rate parameters can be compared to calculations on the basis of the theories outlined in Chapters 3 and 6. In this way the kinetic parameters of elementary reaction steps provide, through spectroscopy and computational chemistry, a link between the intramolecular properties of adsorbed reactants and their reactivity Statistical thermodynamics furnishes the theoretical framework to describe how equilibrium constants and reaction rate constants depend on the partition functions of vibration and rotation. Thus, spectroscopy studies of adsorbed reactants and intermediates provide the input for computing equilibrium constants, while calculations on the transition states of reaction pathways, starting from structurally, electronically and vibrationally well-characterized ground states, enable the prediction of kinetic parameters. 

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