Molecular partitioning

The denominator in this expression is the molecular partition function ... 

The numbers iVj and N- are only equal if there are no degeneracies. The sum in the denominator runs over all available molecular energy levels and it is called the molecular partition function. It is a quantity of immense importance in statistical thermodynamics, and it is given the special symbol q (sometimes z). We have... 

Instead of formulating the reaction rate expression in terms of molecular partition functions, it is often convenient to employ an approach utilizing pseudo thermodynamic functions. From equation 4.3.29, the second-order rate constant is given by... 

The q terms are the molecular partition functions of the superscript species. For the transition state,, the vibration along which the reaction takes place is omitted in the partition function, q. ... 

Equilibrium constants in the model were evaluated from the partition functions of the intermediates, assuming a uniformity of sites. The molecular partition function... 

The authors recognize that the symbol q has previously been used for thermodynamic heat. In using the letter q to symbolize the molecular partition function, usual practice is being followed. This usage should not give rise to confusion. 

Here /i j3 is the chemical potential of the ideal gas at the standard pressure. It will be seen subsequently that qi for an ideal gas depends linearly on the volume V, so fif is a function only of the temperature. It does of course depend on the distribution of energy levels of the ideal gas molecules. The form of Equation 4.59 for the chemical potential of an ideal gas component is the same as that previously derived from thermodynamics (Equation 4.47). The present approach shows how to calculate m through the evaluation of the molecular partition function. Furthermore, the... 

The formulae given in Table 4.1 for the molecular partition functions enable us to write the partition function ratio qheavy/qiight or q2/qi where, by the usual convention, the subscript 2 refers to the heavy isotopomer and 1 refers to the light isotopomer if heavy and light are appropriate designations. Then, a ratio of such partition function ratios enables one to evaluate the isotope effect on a gas phase equilibrium constant, as pointed out above. Before continuing, it is appropriate to... 

Note the subscript C to indicate classical (or high temperature). In Equation 4.81 the p s are momenta and the q s the associated coordinates (not to be confused with q s previously used to symbolize molecular partition functions). In Cartesian coordinates dpjdqj = dpxidpyidpzidxidy1dzi with xi, yi, zi, the coordinates of atom... 

The second product is over the 3N—6(3N—5) normal mode frequencies of the ideal gas harmonic molecule to which Equation 4.78 applies. Thus the product over vibrations Equation 4.90 is indeed the quantum mechanical contribution to the molecular partition function for the ideal gas. 

This result indicates that (s2/si)f (compare Equations4.78 and 4.93) is just the quantum effect on the molecular partition functions of the normal mode vibrations. This result has now been derived without the explicit use of the Teller-Redlich product rule. 

For practical purposes the rules for diatomic molecules concerning even and odd J reduce to the statement that for homonuclear diatomic molecules the molecular partition function must be divided by two (s = 2), while for heteronuclear diatomic molecules no division is necessary (s = 1). The idea of the symmetry number, s,... 

The partition function ratios needed for the calculation of the isotope effect on the equilibrium constant K will be calculated, as before, in the harmonic-oscillator-rigid-rotor approximation for both reactants and transition states. One obtains in terms of molecular partition functions q... 

When treating polyatomics it is convenient to define an average molecular partition function, In = (lnQ)/N, for an assembly of N molecules. In the dilute vapor (ideal gas) this introduces no difficulty. There is no intermolecular interaction and In = (In Q)/N = ln(q) exactly (q is the microcanonical partition function). In the condensed phase, however, the Q s are no longer strictly factorable. Be that as it may, continuing, and assuming In = (In Q)/N, we are led to an approximate result which is superficially the same as Equation 5.10,... 

Figure 3 displays the molecular partition of the fragments for the three states previously discussed in the quantum mechanical section, at d= 6 k. Figure 3 A and 3 B respectively display the and 2 covalent states, and Figure 3 C shows the ionic 82 Charge Transfer state. It is worth examining the striking features of the molecular partitions in each case. In the A1 molecular partition, the disynaptic basin V(Cli, CI2), indicated by an arrow, corresponds to the Cl—Cl bond [22]. Two basins are found around Li, one corresponding to its core C(Li), and the second one, V(Li), to its valence odd electron (L shell). The 82 covalent state is characterized by two monosynaptic basins, Vi(Li) and V2(Li), located on both sides of the C(Li) basin in the molecular plane. They correspond to the half-filled 2p AO of Li. As when dealing with the previous state, the Cl atoms are bonded through a disynaptic basin, still noted V(Cli, CI2). In the ionic state, the Cl atoms are linked by a (3, -I) saddle point, or.

Fig. S. Part A display of the ELF molecular partition ofthe Ai state in 2 ) geometry, d = 6.00.. A(a) is the projection of the ELF in the molecular plane. A(b) (upper part) is a cut of the ELF isosurface (y = 0.8/ A(b) (lowerpart) shows the entire envelope of the same ELF isosurface. Part B corresponds to the 82 state, part C charge transfer state.

---