Electronic and Nuclear Partition Functions

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

The nudear partition function does not usually contribute to the partition function and can therefore be taken as unity. We shall ignore this contribution in the following. Finally, an overview of the formulae for partition functions is given in Tab. 3.2. 

Partition functions of a diatomic molecule (per degree of freedom) 

---

The integration on the rhs of (1.67) extends over all possible locations and orientations of the N particles. We shall refer to the vector XN=Xt,..., XN as the configuration of the system of the N particles. The factor q, referred to as the internal partition function, includes the rotational, vibrational, electronic, and nuclear partition functions of a single molecule. We shall always assume in this book that the internal partition functions are separable from the configurational partition function. Such an assumption cannot always be granted, especially when strong interactions between the particles can perturb the internal degrees of freedom of the particles involved. 

Consider a solute s with internal rotational degrees of freedom. We assume that the vibrational, electronic, and nuclear partition functions are separable and independent of the configuration of the molecules in the system. We define the pseudo-chemical potential of a molecule having a fixed conformation Ps as the change in the Helmholtz energy for the process of introducing s into the... 

If the nucleus has an odd mass number, the overall wave function is anti-symmetrical with regard to the nuclei it is the product of all the translational, rotational, vibrational, electronic and nuclear wave functions. With all the translational, vibrational and electronic wave functions being symmetrical, we only have to consider the rotational and nuclear functions, one of which must be symmetrical and the other anti-symmetrical or vice versa. The g(g-l)/2 wave functions with anti-symmetrical nuclear spin must have corresponding symmetrical rotational wave functions, i.e. with even values of j the g(g+l)/2 wave functions with symmetrical nuclear spin must have corresponding rotational wave functions, i.e. with odd values of j. The combined nuclear-rotational partition function will therefore be ... 

In order to illustrate the consequences of equation (70), it will be assumed that the partition functions for the reactants and the complex can be expressed as products of the appropriate numbers of translational, rotational and vibrational partition functions. For simplicity we shall also neglect factors associated with nuclear spin and electronic excitation. If = total number of atoms in a molecule of species i and = 0 for nonlinear molecules, 1 for linear molecules, and 3 for monatomic molecules, then the correct numbers of the various kinds of degrees of freedom are obtained in equation (70) by letting... 

The partition function for a molecule is formed of the partition functions for individual types of energy increments (motions), i.e. from the translational, rotational, internal rotational (free rotation, hindered rotation), vibrational, electronic and nuclear spin partition functions... 

It is, nevertheless under certain conditions, possible to measure them, as was done for instance for H on tungsten [194], The nuclear partition function is unity and the problem left is to estimate the electronic partition function of the adsorbed atom since that now includes in this scheme the adsorption energy of the atom. Inserting the expression for the partition functions from Equations (4.41) and (4.10) in Equation (4.43) we find straightforwardly, utilizing the ideal gas law and Stirlings approximation ... 

Assuming the adsorbed molecule behaves as a three-dimensional harmonic oscillator and that the internal partition function (for electronic and nuclear energies) remains unchanged upon adsorption, A° may be expressed as ... 

Internal partition functions are relative to the vibrational, rotational, electronic and nuclear movements. We usually find that these movements are independent, although we know that this is not always so between the vibrations and rotations. We also find that the forces exerted on the molecnle by the exterior have no inflnence on these internal degrees of freedom. 

Recognizing that molecules are an important part of chemistry, we will define a molecular partition function, Q, that is the product of partition functions from various energies of a molecule translational, vibrational, rotational, electronic, and nuclear. 

There is also a simple relationship between the pressure of a monatomic gas and its kinetic energy, which can be considered solely as energy of translation. (We are ignoring electronic and nuclear energies, as we did in our original discussion of partition functions of monatomic gases.) Because the classical expression for kinetic energy is... 

In an elegant paper, by Moleslq and Moran, a fourth-order perturbative model is suggested and developed for the study of photoinduced IC. The authors stress that in case of a similar timescale for the electronic and nuclear motions, a second-order perturbation scheme, a la Fermi, will fail. Additionally, the model, as suggested here, in the case of a dominant promoting mode, can exclusively be parameterised from experimental data. The method is based on a three-way partition of a model Hamiltonian—system, bath and system-bath interaction. Subsequent use of a time correlation function approach facilitates the evaluation of rate formulas. This analysis is applied to a three-level model system containing a ground state, an optical active excited state and an optical dark state, the latter two share a CDC. In their paper the model is used to analyse the initial photoinduced process of alpha-terpinene. The primary conclusion of the study is that the most important influence on the population decay (Gaussian versus exponential) is the rate at which the wavepacket approaches the CIX of the two exeited states. 

Here, represents the contribution of all other internal motions of the molecule to the molecular partition function (rotations, vibrations, electronic and nuclear spin motions). For atomic liquids, this term can be taken as being equal to 1. 

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... 

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 internal motion partition function of the guest molecule is the same as that of an ideal gas. That is, the rotational, vibrational, nuclear, and electronic energies are not significantly affected by enclathration, as supported by spectroscopic results summarized by Davidson (1971) and Davidson and Ripmeester (1984). 

Nuclear Spin Effects on Rotation. There is an interesting effect on the rotational partition function, even for the hydrogen molecule, due to nuclear spin statistics. The Fermi postulate mandates that the overall wavefunction (including all sources of spin) be antisymmetric to all two-particle interchanges. A simple molecule like (1H1)2, made of two electrons (S = 1/2) and two protons (spin 7=1/2), will have two kinds of molecule ... 

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

---