Theoretical methods Hamiltonian operator

The established methods of valence bonds and molecular orbitals (MO), including the method of linear combinations of atomic orbitals (LCAO), which have been so successful in the treatment of molecular systems, need further refinement when applied to crystals. The preference for the method of valence bonds in the case of solids is not accidental because it yields clearer results. In contrast, calculations dealing with the simplest molecules are currently tackled usually by the method of molecular orbitals, including the method of absolute, purely theoretical, quantiun-mechanical calculations, which is adopted in those cases when a sufficiently precise form of the Hamiltonian operator can be obtained for the system being considered. 

The idea in perturbation methods is that the problem at hand only differs slightly from a problem that has already been solved (exactly or approximately).The solution to the given problem should therefore in some sense be close to the solution to the already known system. This is described mathematically by defining a Hamiltonian operator that consists of two parts, a reference (Ho) and a perturbation (H ). The premise of perturbation methods is that the H operator in some sense is small compared with Ho. Perturbation methods can be used in quantum mechanics for adding corrections to solutions that employ an independent-particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT). 

The simple and extended Hiickel methods are not rigorous variational calculations. Although they both make use of the secular determinant technique from linear variation theory, no hamiltonian operators are ever written out explicitly and the integrations in Hij are not performed. These are semiempirical methods because they combine the theoretical form with parameters fitted from experimental data. 

There are several theoretical approaches that can be used to calculate the dynamics and correlation properties of two atoms interacting with the quantized electromagnetic held. One of the methods is the wavefunction approach in which the dynamics are given in terms of the probability amplitudes [9]. Another approach is the Heisenberg equation method, in which equations of motion for the atomic and held operators are found from the Hamiltonian of a given system [10], The most popular approach is the master equation method, in which the equation of motion is found for the density operator of an atomic system weakly coupled to a system regarded as a reservoir [7,8,41], There are many possible realizations of reservoirs. The typical reservoir to which atomic systems are coupled is the quantized three-dimensional multimode vacuum held. The major advantage of the master equation is that it allows us to consider the evolution of the atoms plus held system entirely in terms of atomic operators. 

Throughout this paper, we have seen that algebraic techniques often provide extremely simple numerical results with small computational effort. This is particularly true in the preliminary phases of one-dimensional calculations, where almost trivial relations can be found for the initial guesses for the algebraic parameters, as shown in Sections II.C.l and III.C.2. However, it is also true that as soon as real calculations of more complex vibrational spectra are requested, the problem of adapting the various algebraic Hamiltonian and transition operators to suitable computer routines must be resolved. The construction of a computer interface between theoretical models and numerical results is absolutely necessary. Nonetheless, it is rather atypical to discuss these problems explicitly in a theoretical paper such as this one. However, the novelty of these methods itself justifies further explanation and comment on the computational procedures required in practical applications. In this section we present only a brief outline of the development of algebraic software in the last few years, as well as the most peculiar situations one expects to encounter. 

Extensive introductions to the effective core potential method may be found in Ref. [8-19]. The theoretical foundation of ECP is the so-called Phillips-Kleinman transformation proposed in 1959 [20] and later generalized by Weeks and Rice [21]. In this method, for each valence orbital (pv there is a pseudo-valence orbital Xv that contains components from the core orbitals and the strong orthogonality constraint is realized by applying the projection operator on both the valence hamiltonian and pseudo-valence wave function (pseudo-valence orbitals). In the generalized Phillips-Kleinman formalism [21], the effect of the projection operator can be absorbed in the valence Pock operator and the core-valence interaction (Coulomb and exchange) plus the effect of the projection operator forms the core potential in ECP method. 

The exact quantum theoretical treatment of the dispersion effect involves quantizing matter and electromagnetic fields as well. The coupled electron-photon system is to be treated on the basis of quantum electrodynamics. Using the method of second quantization, it is possible to build up the total Hamiltonian from an electron Hamiltonian H, a photon Hamiltonian and an electron-photon interaction operator Hin,. The dispersion energy between two particles now results in fourth order perturbation. Each contribution is due to the interaction of two electrons with, fwo photons. 

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