Perturbative equations function expansion

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

With the perturbed wave functions and energies in hand, it is a straightforward matter to calculate oscillator strengths via Equation (61). The coefficients / , (i = 0, 1, 2,..., 15), in the expansion of /is-2p are listed in Table 10. In this case, however, we are limited to the order of the perturbation, viz. 15 in the present calculation (this limit can easily be changed by increasing the dimensions of the arrays in the computer programs used). 

This calculation is typically performed in some form of a restricted configuration interaction (Cl) expansion (CASSCF [complete active space self consistent field], MRCI [multireference configuration interaction]). The perturbation V is represented by the operators and The perturbed wave function F and energy E satisfy the equation... 

Calculation of a perturbed distribution function can be approached in various ways (1) direct solution of the Boltzmann equation for the distribution function in the perturbed system, (2) distribution-difference methods, (3) local calculations, and (4) normal-mode expansion methods. 

This result can be deduced from a number of perspectives such as renormalized perturbation theory [93], polymer density fimctional theory [95,96], and Per-cus functional expansion methods [97]. The medium-induced pair potential is determined by the direct correlation function and collective density fluctuations which are both functionally related to the intramolecular pair correlations via the PRISM equation. Hence, a coupled intramolecular/intermolecular theory is obtained. 

Many versions of perturbation theory have been proposed to overcome these problems. A large number of them rely on an expansion of the perturbation equations in powers of the overlap between the functions on A and those on B. This approach appears to work when small basis sets are used, but as the basis is improved, the overlap between the functions on the two molecules becomes larger, and the expansion ceases to converge. This failure of the overlap expansion occurs with quite modest basis sets. Accordingly it is necessary to use a method that deals explicitly with the natural non-orthogonal basis functions for the problem. 

Approximate many-electron wave functions are then constructed from the Hartree-Fock reference and the excited-state configurations via some sort of expansion (e.g., a linear expansion in Cl theory, an exponential expansion in CC theory, or a perturbative power series expansion in MBPT). When all possible excitations have been incorporated (S, D, T,. .., for an -electron system), one obtains the exact solution to the nonrelativistic electronic Schrodinger equation for a given AO basis set. This -particle limit is typically referred to as the full Cl (FCI) limit (which is equivalent to the full CC limit). As Figure 5 illustrates, several WFT methods can, at least in principle, converge to the FCI limit by systematically increasing the excitation level (or perturbation order) included in the expansion technique. 

A matrix formulation of these equations was developed in Section 14.1.2, using an orthonormal basis for the expansion of the perturbed wave functions (14.1.28). In this exercise, this matrix formulation is generalized to the expansion of the perturbed states ... 

This makes it desirable to define other representations in addition to the electronically adiabatic one [Eqs. (9)-(12)], in which the adiabatic electronic wave function basis set used in the Bom-Huang expansion (12) is replaced by another basis set of functions of the electronic coordinates. Such a different electronic basis set can be chosen so as to minimize the above mentioned gradient term. This term can initially be neglected in the solution of the / -electionic-state nuclear motion Schrodinger equation and reintroduced later using perturbative or other methods, if desired. This new basis set of electronic wave functions can also be made to depend parametrically, like their adiabatic counterparts, on the internal nuclear coordinates q that were defined after Eq. (8). This new electronic basis set is henceforth refened to as diabatic and, as is obvious, leads to an electronically diabatic representation that is not unique unlike the adiabatic one, which is unique by definition. 

Equation 24.14 provides an alternative definition of the electronic responses they are derivatives of the energy s relative to the field E. Note that the response of order n, the nth derivative of the response to the perturbation, is the n + 1th derivative of the energy relative to the same perturbation. Hence, the linear response a t is a second derivative of the energy. Because the potential (E) and the density (p) are uniquely related to each other, the field can be formulated as a function of the dipole moment p. The expansion of the field in function of p can be obtained from Equation 24.12 which can be easily inverted to give... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

If one could solve Eq. (203) exactly for exact energy— provided that the reference function is n-representable (e.g., is a normalized Slater determinant). The unitary transformation preserves the n-representability. Equation (203) is an infinite-order nonlinear set of equations and not easy to solve. However, the perturbation expansion terminates at any finite order. We have [6,12]... 

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... 

The equations of state and expansion functions for the perturbation theories are found in paper of Fickett... 

The parameter X has been embedded in the definition of Hp. The wave function from perturbation theory [equation (A.109)] is not normalized and must be renormalized. The energy of a truncated perturbation expansion [equation (A.110)] is not variational, and it may be possible to calculate energies lower than experimental. ... 

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