Ground state partition functions

Same as 1(b) when the /th-state vibrational partition function, Q [, and the /th-state rotational partition function, Q[y2Ltt equal to the respective ground-state partition functions. In this case the partition function... 

We further introduce the ground-state partition function... 

In addition, the ground-state partition function [Eq. (79d)] is given by... 

Although the relation between the vibrational g factor and the derivative of electric dipolar moment, equation (10), is formally equivalent to the relation between the rotational g factor and this dipolar moment, equation (9), there arises an important distinction. The derivative of the electrical dipolar moment involves the linear response of the ground-state wave function and thus a non-adiabatic expression for a sum over excited states similar to electronic contributions to the g factors. The vibrational g factor can hence not be partitioned in the same as was the rotational g factor into a contribution that depends only on the ground-state wave function and irreducible non-adiabatic contribution. Nevertheless g "(R) is treated as such. A detailed expression for ( ) in terms of quantum-mechanical operators and a sum over excited states, similar to equations (11) and (12), is not yet reported. 

In view of Eq. (4-2) we can take y = dyt without loss of generality. Since the functions are constructed as antisymmetrized products of states for the d-electron manifold and the ground state wave function, Plo, for the L subset of electrons, there are functions in the set A, describing all the excited states of the L electrons, i.e. we can write these functions as antisymmetrized products of the Pm , and Wtr discussed in Sect. 3 and 4 with / 0. The remaining functions in the set A, describe all partitions between the d-... 

The OOA was not designed for and does not apply to temperature dependencies of any kind in JT crystals. In particular, one cannot expect a reasonable estimate of the temperature of phase transitions in crystal lattice (structural), electron orbital, and/or spin system. This follows from the partitioning procedure that includes averaging over vibrational degrees of freedom. One can see the same reason from another perspective. The pseudo spin of a JT site, as the basic concept used in the OOA, operates in the basis of degenerate ground state wave functions. Excited vibronic states are beyond the pseudo spin setup. Therefore, in the OOA, by its very definition, temperature population of excited states does not make sense. 

Table III demonstrates the effect on the ionization potentials of including single excitations from HF> in the ground-state wave function for N. Simons basis set is employed in these calculations. Table III exhibits the small, but nonnegligible, effect of retaining third-order terms involving singly excited configurations in 0> in the partitioned EOM equation it...

The basic idea underlying Kohn-Sham theory [4] is the construction of a mode system of noninteracting quasi-particles for which the density is the same as that of the interacting system. As such the ground-state energy functional E[p] is partitioned as... 

Fig. 5.5 Ratio of the vibrational partition functions (Za and Zj) for the ground and excited states of a system with a single vibrational mode when the vibrational frequency in the excited state (Ofe) is 3/2 (curve 1) or Ifi (curve 2) times the fiequency in the ground state (v. The abscissa is the vibrational energy in the ground state (hVc relative to k. The dashed curve shows the ground-state partitirai function (Z )...

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

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 ... 

At low temperatures, the system will be entirely in the ground state, and the partition function approaches 1 in the limit of T —> 0 ... 

The above two examples illustrate that the value of the partition function is an indicator for how many of the energy levels are occupied at a particular temperature. At T = 0, where the system is in the ground state, the partition function has the value q = 1. In the limit of infinite temperature, entropy demands that all states are equally occupied and the partition function becomes equal to the total number of energy levels. 

Note that the zero of energy is now the bottom of the potential, and the ground state -the lowest occupied level - lies Vihv higher. As partition functions are usually given with respect to the lowest occupied state, we shift the zero of energy upward by Vihv to obtain... 

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

It is instructive to illustrate the relation between the partition function and the equilibrium constant with a simple, entirely hypothetical example. Consider the equilibrium between an ensemble of molecules A and B, each with energy levels as indicated in Fig. 3.5. The ground state of molecule A is the zero of energy, hence the partition function of A vnll be... 

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. 

Calculate the vibrational partition function with respect to the vibrational ground state (i.e. the lowest occupied state) and the fraction of molecules in the ground state at 300, 600 and 1500 K for the following molecules, using kTjh= 208.5 cm at 300 K ... 

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