Schrodinger equation perturbative development

In his classical paper, Renner [7] first explained the physical background of the vibronic coupling in triatomic molecules. He concluded that the splitting of the bending potential curves at small distortions of linearity has to depend on p2A, being thus mostly pronounced in n electronic state. Renner developed the system of two coupled Schrodinger equations and solved it for n states in the harmonic approximation by means of the perturbation theory. 

There is another manner in which perturbation theory is used in quantum chemistry that does not involve an externally applied perturbation. Quite often one is faced with solving a Schrodinger equation to which no exact solution has been (yet) or can be found. In such cases, one often develops a model Schrodinger equation which in some sense is designed to represent the system whose full Schrodinger equation can not be solved. The... 

In the perturbative "transfer Hamiltonian approach developed by Bardeen 58), the tip and sample are treated as two non-interacting subsystems. Instead of trying to solve the problem of the combined system, each separate component is described by its wave function, i tip and i/zj, respectively. The tunneling current is then calculated by considering the overlap of these in the tunnel junction. This approach has the advantage that the solutions can be found, for many practical systems, at least approximately, by solution of the stationary Schrodinger equation. 

The Statistical Rate Theory (SRT) is based on considering the quantum-mechanical transition probability in an isolated many particle system. Assuming that the transport of molecules between the phases at the thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between the two phases 1 and 2 , R 2, was developed by using the first-order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. 

Our task is to find approximate solutions to the time-independent Schrodinger equation (Eq. (2)) subject to the Pauli antisymmetry constraints of many-electron wave functions. Once such an approximate solution has been obtained, we may extract from it information about the electronic system and go on to compute different molecular properties related to experimental observations. Usually, we must explore a range of nuclear configurations in our calculations to determine critical points of the potential energy surface, or to include the effects of vibrational and rotational motions on the calculated properties. For properties related to time-dependent perturbations (e.g., all interactions with radiation), we must determine the time development of the... 

With the development of material science, fine chemistry, molecular biology and many branches of condensed-matter physics the question of how to deal with the quantum mechanics of many-particle systems formed by thousands of electrons and hundreds of nuclei has attained unusual relevance. The basic difficulty is that an exact solution to this problem by means of a straight-forward application of the Schrodinger equation, either in its numerical, variational or perturbation-theory versions is nowadays out of the reach of even the most advanced supercomputers. It is for this reason that alternative ways for handling the quantum-mechanical many-body problem have been vigorously pursued during the last few years by both quantum chemists and condensed matter physicists. As a consequence of... 

The first target of quantum chemistry was how to solve the SchrOdinger equation for electronic motions in molecules. To address this challenge, the Hartree-Fock method (Hartree 1928) and its variational method (Slater 1928), molecular orbital theory (Hund 1926 Mulliken 1927), and the Slater determinant (Slater 1929) were developed, resulting in the Hartree-Fock method (Fock 1930 Slater 1930), which is accepted as the precursor of quantum chemistry. Soon afterward, the configuration interaction (Cl) method (Condon 1930), M0ller-Plesset perturbation method (Mpller and Plesset 1934), and multiconflgurational SCF method (Frenkel... 

The Schrodinger equation for the one-electron atom (Chapter 6) is exactly solvable. However, because of the interelectronic-repulsion terms in the Hamiltonian, the Schrbdinger equation for many-electron atoms and molecules cannot be solved exactly. Hence we must seek approximate methods of solution. The two main approximation methods, the variation method and perturbation theory, will be presented in Chapters 8 and 9. To derive these methods, we must develop further the theory of quantum mechanics, which is what is done in this chapter. 

The special properties of nonlinear Schrodinger equations, like Eq. (5.13) has been discussed by Sanhueza et al. [165], and a special perturbation theory has been developed for their solution [166]. 

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