Theory perturbation, formal

K s> mentioned earlier, these formulations are applicable to structureless particles (bare ions). The only one of them that may be easily extended to complex ions is the perturbative formulation, either in the form of Bethe s theory for atomic targets or Lindhard s theory (dielectric formalism, DF) for the electron gas model. In addition, a comprehensive semiclassical approach, which extends the Bohr model to complex ions, has been developed more recently by Sigmund et al. [26]. 

Consequently, within the spin free JV-electron Hilbert space, the MBPT for OSS states can be formulated as a SR theory with the spin free reference o), Eq-4, and the UGA representation of the perturbed and unperturbed Hamiltonians. The resulting theory has formally the same appearance as the spin orbital based MBPT theories for CS or HS OS cases. 

In the present paper we investigate the convergence properties of the RS, SRS, and HS perturbation series for He2 and HeH2 molecules, i.e. for the interaction of two ground-state two-electron systems. These perturbation formalisms correspond to none, weak, and strong symmetry forcing, respectively. In Sec. II the perturbation equations of the RS, SRS, and HS theories are briefly summarized. In Sec. Ill the computational details of the calculations are presented. The numerical results are presented and discussed in Sec. IV. 

It is worth noting that the convergence pattern of the polarization series for He2 is very similar to that found for Hj (9) and H2 (18). Thus, at the distances of the van der Waals minimum the Rayleigh-Schrodinger perturbation theory provides only a part of the interaction energy (15), and in practical applications symmetry-adapted perturbation formalisms must be used. 

Using perturbation theory [24] it is possible to derive expressions for the interaction energy of two molecules and write such energy as the sum of terms that have a useful physical meaning. One of these expressions has been provided by the IMPT method [25] (an acronym for Inter Molecular Perturbation Theory). Similar expressions can be obtained by employing other perturbation formalisms [26]. Within the IMPT theory [25], the interaction energy between two closed-shell molecules is the sum of the following five components ... 

The average treatment of electron correlation is inadequate in single determinant HF theory. Several schemes exist to remedy this, ranging from the explicit inclusion of Configuration Interaction to the use of perturbation theory. The formalism for improving on the HF approximation is well defined. However, the practical implementation of Cl is problematic. 

The first approximation to the many-electron theory of atoms and molecules was derived by solving Eq. (64) with the H.F. using operator techniques. The development of Brueckner s theory of nuclear matter and other many-body theories also made much use of perturbation formalism. 

There are two basic differences in [his] approach which permit all orders to be treated at once. First, the starting point is the Brillouin-Wigner BW) perturbation theory, whose formal structure is much simpler than that of the RS expansion. Secondly, we use a factorization theorem , which expresses the required energy-denominator identities in a simple and general form. ... 

In this chapter we have explored the fundamental theories and the cal-culational techniques for the properties of individual molecules. Generally, properties represent some behavior of the molecule or a response of a molecule to perturbations. Formally, these are all derivatives of the molecular energy. Solving for such derivatives is a natural generalization of the process of solving for state energies via the Schrodinger equation. Techniques exist today for the... 

This use of the unperturbed population to calculate the change in revenue is a crade example of a certain level of estimation (called first order ) in Rayleigh-Schrodinger perturbation theory. We will now proceed to develop the theory more formally in the context of wavefunctions and energies. The above example has been presented to encourage the reader to anticipate that there is a lot of simple good sense in the results of perturbation theory even though the mathematical development is rather cumbersome and unintuitive. 

In this context equations (50) and (53) can be considered forming a completely general perturbation theory for nondegenerate systems, although a recent development permits to extend the formalism to degenerate states [lej. 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

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