Partition function evaluation

Kaufman B 1949 Orystal statistics II. Partition function evaluated by Spinor analysis Phys. Rev. 65 1232... 

Table 4.1 Partition functions evaluated in the rigid rotor harmonic oscillator approximation... 

It can also be shown that Kct the equilibrium constant in terms of concentrations, is related to the partition functions evaluated per unit volume... 

Here the quantity is a configurational partition function evaluated... 

With the development of computational techniques, more accurate approaches have been used. Quantum-mechanical partition function evaluation of the second virial coefficient Rmm (T) has been performed by Mies Julienne (1982) for lithium and sodium, using the singlet and triplet potentials of Konowalow and Olsen. Holland et al. (1986, 1987) calculated mm(7 )> nd Z>m for lithium and sodium by accurately rep-... 

Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be... 

This is connnonly known as the transition state theory approximation to the rate constant. Note that all one needs to do to evaluate (A3.11.187) is to detennine the partition function of the reagents and transition state, which is a problem in statistical mechanics rather than dynamics. This makes transition state theory a very usefiil approach for many applications. However, what is left out are two potentially important effects, tiiimelling and barrier recrossing, bodi of which lead to CRTs that differ from the sum of step frmctions assumed in (A3.11.1831. 

This definition is based on and proportional to the g-expectation value. However, it is more useful since it is not necessary to evaluate the partition function to compute an average. 

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. 

In the semiclassical evaluation of the barrier partition function the integration goes along the whole imaginary axis in the c, plane (see fig. 21). 

Except for the nonlocal last term in the exponent, this expression is recognized as the average of the one-dimensional quantum partition function over the static configurations of the bath. This formula without the last term has been used by Dakhnovskii and Nefedova [1991] to handle a bath of classical anharmonic oscillators. The integral over q was evaluated with the method of steepest descents leading to the most favorable bath configuration. 

The partition functions vib. and for use in Eq. (5-23) can be evaluated by quantum mechanical arguments. We will subsequently require vib. which is given by Eq. (5-25), where v is the vibrational frequency. 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

The last equation follows from the definition of the partition function, eq. (16.2). Analogously to eq. (16.10) the free energy difference can be evaluated as an ensemble average. 

E. Evaluation of the Cell Partition Function by the Method of Lennard-Jones and Devonshire... 

The theory introduced by Lennard-Jones and Devonshire13 17 for the study of liquids provides a powerful method for the quantitative evaluation of the partition function of a solute molecule within its cavity.51 Because the application of this method to the present problem has been described in detail,62 we shall restrict ourselves to its most essential features. 

Because the quantity (/ 2/87r/r/c) is small at most T, the summation can be replaced by an integral over K in a procedure similar to that used to evaluate the rotational and translational partition functions earlier. The result is... 

One may also try to evaluate solvation-free energies by using the corresponding partition function through the formula (Ref. 4)... 

In the following, the MO applications will be demonstrated with two selected equilibrium reactions, most important in radical chemistry disproportionation and dimerization. The examples presented will concern MO approaches of different levels of sophistication ab initio calculations with the evaluation of partition functions, semiempirical treatments, and simple procedures employing the HMO method or perturbation theory. 

Quite similar equations can be formulated for AG and AH by use of the partition function f of the activated complex. It follows from equations (6) and (7) that AEp can only be evaluated if the partition functions and AEz are available from spectroscopic data or heat capacity measurements. However, if AG = AH, the entropy change AS equals zero, and if AEz also equal to zero, either AG or AH can then be identified with the potential energy change. If... 

The molecule reaches the transition state and from there it desorbs into the gas phase. To evaluate the rate constant we use the same procedure as above Write down the partition functions for the participating species, equalize the chemical potentials, and find an expression for the number of molecules in the transition state. Since it is much more practical to do this in terms of coverages we immediately obtain ... 

For molecular hydrogen the evaluation of.the vaq n tb pdyi nic tides from the partition functions is... 

From Eqs. (45) and (46) it is apparent that the calculation of the energy and heat capacity of a system depends on the evaluation of the partition function a a function of temperature. In the more general case of molecules with an internal structure, the energy distributions of the various degrees of freedom must bo determined. This problem is outlined briefly in the following section. 

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