Exact partition function

C vibrational partition function (exact for Morse oscillators) ... 

For a given Hamiltonian the calculation of the partition function can be done exactly in only few cases (some of them will be presented below). In general the calculation requires a scheme of approximations. Mean-field approximation (MFA) is a very popular approximation based on the steepest descent method [17,22]. In this case it is assumed that the main contribution to Z is due to fields which are localized in a small region of the functional space. More crudely, for each kind of particle only one field is... 

Notice that the associated spin model has the following three properties (1) it is, in general, anisotropic (i.e. a-2 / CI3), (2) its set of coupling constants hi, hij, /1123) are interdependent (this should be obvious from equation 7.63, which provides a parameterization of each of these seven constants in terms of our original four independent conditional probabilities, aj (equation 7.58)), and (3) its partition function, Z, can be calculated exactly. 

This is the translational partition function for any particle of mass m, moving over a line of length I, in one dimension. Please note that this result is exactly the same as that we calculated from quantum mechanics for a particle in a one-dimensional box. For a particle moving over an area A on a surface, the partition function of translation is... 

These are exact expressions for the configuration integrals. Alternatively, we can write the partition function as... 

With neglect of the quantum effects that arise from the exchange of identical particles [147], (8.66) gives the exact quantum partition function in the limit P — oo. For finite P, Qp((3) is the canonical partition function of a classical system composed of ring polymers. Each quantum particle corresponds to a ring polymer of P beads in which neighboring beads are connected by harmonic springs with force... 

The acceptance criteria for the Gibbs ensemble were originally derived from fluctuation theory [17]. An approximation was implicitly made in the derivation that resulted in a difference in the acceptance criterion for particle transfers proportional to 1/N relative to the exact expressions given subsequently [18]. A full development of the statistical mechanics of the ensemble was given by Smit et al. [19] and Smit and Frenkel [20], which we follow here. A one-component system at constant temperature T, total volume V, and total number of particles N is divided into two regions, with volumes Vj and Vu = V - V, and number of particles Aq and Nu = N - N. The partition function, Q NVt is... 

For an ideal gas, the functional F[p(r)] is known exactly. Because V = 0, the partition function and density profile are given, respectively, by... 

The partition function, Z(4>y), cannot be calculated exactly. It could be rewritten using the integral representation of the functional Dirac delta function and evaluated within the saddle place approximation. The calculations lead to the following expression [36,126,128] ... 

The incompressibihty constraint 5(A + < >B — 1) has been explicitly included in partition function (37), and the continuous chain model, Eq. (19), is being used. Q is the partition function of an independent single chain subjected to the fields Ua and Ub, and it can be evaluated exactly. One writes the partition function as Q = f drq(r, 1), where... 

We have so far four different versions of the APM the crude version and the refined versions I, II, and III. Their common features are obviously that their partition functions (i) reduce to the exact form (8) for a one-component system, and (ii) depend on the function q T,v) alone. 

Calculation of the partition function, even for the simple models described, cannot be carried out exactly. For details of the approximations used the reader is referred to the original papers fil-62>. From our point of view the most interesting results are ... 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

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,... 

Early in the development of VTST calculations on simple three atom systems compared rates obtained by exact classical dynamics with conventional TST and VTST, the same potential energy surface and classical partition functions being used throughout. These calculations confirmed the importance of eliminating the recrossing phenomenon in VTST. While TST yielded very much larger rate constants than the exact classical calculations, the VTST calculations yielded smaller rate constants, but never smaller than the exact classical values. 

The total electronic potential energy of a molecule depends on the averaged electronic charge density and the nonlocal charge-density susceptibility. The molecule is assumed to be in equilibrium with a radiation bath at temperature T, so that the probability distribution over electronic states is determined by the partition function at T. The electronic potential energy is given exactly by... 

So far, the effects of the chain ends were neglected in our stochastic model for the restricted chain. Therefore, n must be much larger than the number of steps needed to form the largest excluded polygon. The partition function, which incorporates the chain-end effects and which could be also employed for exact statistical description of short non-self-intersecting chains can be obtained as follows Assume, as before, that we eliminate only lowest-order polygons of t steps. Therefore, the first t — 1 steps in the chain are described as a sequence of independent events. Eq (9), then, will be replaced by... 

We also calculated the partition function, based on Eq. (26), with n = t.13 In this case, Eq. (28) expresses an exact partition function for a short chain with no restrictions being imposed on the maximal range of interactions between chain elements. Our calculations showed that, for low values of x and t, the partition function is given almost exactly by the term which is proportional to the largest eigenvalue. It is found that 1 - Z (x)/Zj0>(x), with Z (0)(x) = (s, p )(l rj, is a very small... 

The adsorption of diatomic or dimeric molecules on a suitable cold crystalline surface can be quite realistically considered in terms of the dimer model in which dimers are represented by rigid rods which occupy the bonds (and associated terminal sites) of a plane lattice to the exclusion of other dimers. The partition function of a planar lattice of AT sites filled with jV dimers can be calculated exactly.7 Now if a single dimer is removed from the lattice, one is left with two monomers or holes which may separate. The equilibrium correlation between the two monomers, however, is appreciable. As in the case of Ising models, the correlation functions for particular directions of monomer-monomer separation can be expressed exactly in terms of a Toeplitz determinant.8 Although the structure of the basic generating functions is more complex than Eq. (12), the corresponding determinant for one direction has been reduced to an equally simple form.9 One discovers that the correlations decay asymptotically only as 1 /r1/2. 

We noted in Section 8.2 that only half the values of j are allowed for homonuclear diatomics or symmetric linear polyatomic molecules (only the even-y states or only the odd- y states, depending on the nuclear symmetries of the atoms). The evaluation of qmt would be the same as above, except that only half of the j s contribute. The result of the integration is exactly half the value in Eq. 8.64. Thus a general formula for the rotational partition function for a linear molecule is... 

The present state in the theory of time-dependent processes in liquids is the following. We know which correlation functions determine the results of certain physical measurements. We also know certain general properties of these correlation functions. However, because of the mathematical complexities of the V-body problem, the direct calculation of the fulltime dependence of these functions is, in general, an extremely difficult affair. This is analogous to the theory of equilibrium properties of liquids. That is, in equilibrium statistical mechanics the equilibrium properties of a system can be found if certain multidimensional integrals involving the system s partition function are evaluated. However, the exact evaluation of these integrals is usually extremely difficult especially for liquids. 

Often at this point additional approximations are made, such as expanding In Q in cumulants and only keeping the first few terms. Such expressions are unnecessary because the exact expression for Q can be evaluated. The partition function is written as Q = J dnr/fr, ), where... 

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