Perturbation theory partition function

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Using the above partitioning into the Rayleigh-Schrddinger perturbation theory (RSPT) allows the perturbed reference function to be written as,... 

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. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

The ratio of the quantum partition functions (Eq. (4-29)) for two different isotopes can be obtained directly through free energy perturbation (FEP) theory by perturbing the mass from the light isotope to the heavy isotope. Consequently, only one simulation of a given isotopic reaction is performed, while the ratio of the partition function, i.e., the KIE, to a different isotopic reaction, is obtained by FEP. This is conceptually and practically an entirely different approach than that used previously [23]. 

Wong K-Y, Gao J (2008) Systematic approach for computing zero-point energy, quantum partition function, and tunneling effect based on Kleinert s variational perturbation theory. J Chem Theory Comput 4(9) 1409-1422... 

In the canonical partition function of (5.1), we have for simplicity ignored combinatorial prefactors. Free energy perturbation theory [12] relies on evaluating effectively the ratio of the partition functions to obtain the free energy difference between the initial and final states corresponding to coupling parameters A = 1 and 0 (see also Chap. 2),... 

In order to leam more about the nature of the intermolecular forces we will start with partitioning of the total molecular energy, AE, into individual contri butions, which are as close as possible to those we defined in intermolecular perturbation theory. Attempts to split AE into suitable parts were undertaken independently by several groups 83-85>. The most detailed scheme of energy partitioning within the framework of MO theory was proposed by Morokuma 85> and his definitions are discussed here ). This analysis starts from antisymmetrized wave functions of the isolated molecules, a and 3, as well as from the complete Hamiltonian of the interacting complex AB. Four different approximative wave functions are used to describe the whole system ... 

Fickett in "Detonation Properties of Condensed Explosives Calculated with an Equation of State Based on Intermolecular Potentials , Los Alamos Scientific Lab Rept LA-2712 (1962), pp 34-38, discusses perturbation theories as applied to a system of deton products consisting of two phases one, solid carbon in some form, and the other, a fluid mixt of the remaining product species. He divides these theories into two classes conformal solution theory, and what he chooses to call n-fluid theory. Both theories stem from a common approach, namely, perturbation from a pure fluid whose props are assumed known. They differ mainly in the choice of expansion variables. The conformal solution method begins with the assumption that all of the intermolecular interaction potentials have the same functional form. To obtain the equation of state of the mixt, some reference fluid obeying a common reduced equation of state is chosen, and the mixt partition function is expanded about that of the reference fluid... 

Contrary to the previously described supermolecular approach, perturbation theory treatment allows for the partition of the interaction energy into physically interpretable components. The most frequently used method for this purpose is symmetry-adapted perturbation theory (SAPT) [13]. More recently, great effort has also been invested in the development of DFT-SAPT [14-16], In the present contribution, we use the variational-perturbational scheme [17-20], In this approach, the intermolecular interaction energy components are determined based on the wave functions of the subsystems evaluated in the dimer-centered basis set. Thus, both interaction energy and its components are BSSE-free. More details about this scheme can be found elsewhere [21-23]. The total intermolecular interaction energy at the MP2 level of theory can be expressed as follows ... 

The basic theoretical framework for understanding the rates of these processes is Fermi s golden rule. The solute-solvent Hamiltonian is partitioned into three terms one for selected vibrational modes of the solute, including the vibrational mode that is initially excited, one for all other degrees of freedom (the bath), and one for the interaction between these two sets of variables. One then calculates rate constants for transitions between eigenstates of the first term, taking the interaction term to lowest order in perturbation theory. The rate constants are related to Fourier transforms of quantum time-correlation functions of bath variables. The most common... 

Flexible RRKM theory and the reaction path Hamiltonian approach take two quite different perspectives in their evaluation of the transition state partition functions. In flexible RRKM theory the reaction coordinate is implicitly assumed to be that which is appropriate at infinite separation and one effectively considers perturbations from the energies of the separated fragments. In contrast, the reaction path Hamiltonian approach considers a perspective that is appropriate for the molecular complex. Furthermore, the reaction path Hamiltonian approach with normal mode vibrations emphasizes the local area of the potential along the minimum energy path, whereas flexible RRKM theory requires a global potential for the transitional modes. One might well imagine that each of these perspectives is more or less appropriate under various conditions. 

For the multitude of cases in quantum chemistry in which the omnipresence of interelectronic interactions render the use of one-electron models inadequate, simple consideration of the formalism and the meaning of perturbation theory that is based on well-defined zero-order reference wave-functions indicates that not all terms play the same role as regards their contribution to the eigenfunction of each state. Therefore, depending on the problem under consideration, it may be possible, to a good and practical approximation, to partition the total wavefunction in such a way to... 

This paper describes a simple finite order PT for low spin OSS states using the spin-free ROHF wave function as a reference. Relying on the UGA formalism the method retains the features of a SR theory. Both MP and EN partitionings of the Hamiltonian are considered. Similarly as with the CS and HS OS perturbation theories, we find the MP partitioning to give better results than the EN one. The focus of the paper is on the OSS states and the calculations are carried out at the second and third order levels. The capabilities of this approach are illustrated on the low lying OSS excited states of CH2, NHj and H2O. An extension of the theory to the MP4 level and to the HS OS states will be given elsewhere. 

Another popular approach to the correlation problem is the use of perturbation theory. Fq can be taken as an unperturbed wave function associated with a particular partitioning of the Hamiltonian perturbed energies and wave functions can then be obtained formally by repeatedly applying the perturbation operator to Probably the commonest partitioning is the M ller-Plesset scheme, which is used where Fq is the closed-shell or (unrestricted) open-shell Hartree-Fock determinant. Clearly, the perturbation energies have no upper bound properties but, like the CC results, they are size-consistent. 

The central idea of thermodynamic perturbation theory is that the potential energy function can be partitioned in a convenient way i.e., one can write... 

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