Electron partitioning

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 ... 

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. 

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. 

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... 

The various contributions to the energy of a molecule were specified in Eq. (47). However, the fact that the electronic partition function was assumed to be equal to one should not be overlooked. In effect, the electronic energy was assumed to be equal to zero, that is, that the molecule remains in its ground electronic state. In the application of statistical mechanics to high-temperature systems this approximation is not appropriate. In particular, in the analysis of plasmas the electronic contribution to the energy, and thus to the partition function, must be included. 

Castillo et al. [27]. This behaviour may be an effect of the electronic partition in this type of molecules because of the fluorine substitution in the alkyl chain, which is responsible for the unique surface activity. 

Now, what if we abandon the orbital-by-orbital electron partitioning in favor of a description based on the stationary ground-state electron density p(r) Clearly, this will oblige us to redefine the coie-valence separation. In sharp contrast with what was done in orbital space, we need a partitioning in real space. Let us begin with isolated atoms. 

In practice, it proves more convenient to work within a convention where we define tire ground state for each energy component to have an energy of zero. Thus, we view 1/eiec as the internal energy that must be added to I/q, which already includes Eeiec (see Eq. (10.1)), as the result of additional available electronic levels. One obvious simplification deriving from this convention is that the electronic partition function for the case just described is simply eiec = 1, Inspection of Eq. (10.5) then reveals that the electronic component of the entropy will be zero (In of 1 is zero, and the constant 1 obviously has no temperature dependence, so both terms involving eiec are individually zero). 

We see that the partition function of a molecule is the product of the contributions of the translational, rotational, vibrational, and electronic partition functions, which we can calculate separately, as discussed next. We will see in Section 8.5 that any thermodynamic quantity of interest can be derived from the molecular partition function. Thus it is important to be able to evaluate q. 

In most cases, excited electronic energy levels lie high above the ground-state energy relative to ksT, and the population in the upper levels is negligibly small. In these cases the electronic partition function reduces to one term ... 

In some instances, multiple electronic states of a molecule are possible, and an electronic partition function must be evaluated. One would use an explicit summation over electronic energy levels, as in Eq. 8.50. No simple general theory is available to account for all the manifold of electronic energy states. 

As a numerical example, consider calculation of the electronic partition function for the H atom, using explicit evaluation of the summation in Eq. 8.50 (truncated after two terms)... 

Thus, even in this very high temperature example, excited electronic energy levels make a negligible contribution to the electronic partition function. 

The simplest QCE model incorporates environmental effects of cluster-cluster interactions by (1) approximate evaluation of the excluded-volume effect on the translational partition function >trans (neglected in Section 13.3.3) and (2) explicit inclusion of a correction A oenv) for environmental interactions in the electronic partition function qiQiec. Secondary environmental corrections on rotational and vibrational partition functions may also be considered, but are beyond the scope of the present treatment. 

As in (13.89), a nonzero amf brings complex nonlinearity into the electronic partition function,... 

The electronic partition function can be evaluated by summing over spectroscopically determined electronic states, but as the electronic energy-level separations are large, the number of molecules in excited electronic states is negligibly small at ordinary temperatures and the electronic partition function is unity and will be ignored henceforth. 

Note that the 1 /N term is assigned to the translational partition function, since all gases have translational motion, but only molecular gases have rotational and vibrational degrees of freedom. The electronic partition function is usually equal to one unless unpaired electrons are present in the atom or molecule. 

Finally, the electronic partition function is considered. The zero of energy is chosen as the electronic ground-state energy. The spacings between the electronic energy levels are, normally, large and only the first term in the partition function will make a significant contribution that is,... 

Chandrakumar, K.R.S. and Pal, S., Study of local hard-soft acid-base principle effects of basis set, electron correlation, and the electron partitioning method, J. Phys. Chem. A, 107, 5755-5762, 2003. 

PROBLEM 5.3.12. Show that in a CE, in the absence of degeneracy, the single-molecule electronic partition function is... 

---