Schrodinger equation, obstacle

The development of an ab initio quantum molecular dynamics method is guided by the need to overcome two main obstacles. First, one needs to develop an efficient, yet accurate, method for solving the electronic Schrodinger equation for both ground and excited electronic states. Second, the quantum mechanical character of the nuclear dynamics must be addressed. (This is necessary for the description of photochemical and tunneling processes.) This section provides a detailed discussion of the approaches we have taken to solve these two problems. 

Conventional methods based on quantum mechanical models use matrix diagonalization to find a self-consistent solution of the time-independent Schrodinger equation. Unfortunately, the cost of matrix diagonalization grows extremely rapidly with the number of atoms in the system. Consequently, methods based on quantum mechanical models tend to be computationally expensive. As a result, the zeolite framework is often treated as a cluster instead of as a periodical system. To overcome this obstacle, hybrid models have been put forward in which the problem is circumvented the reaction center is described in a quantum mechanical way, whereas the surroundings are described in a classical way. ... 

Until this point, the consideration of electron-electron repulsion terms has been neglected in the molecular Hamiltonian. Of course, an accurate molecular Hamiltonian must account for these forces, even though an explicit term of this type renders exact solution of the Schrodinger equation impossible. The way around this obstacle is the same Hartree-Fock technique that is used for the solution of the Schrodinger equation in many-electron atoms. A Hamiltonian is constructed in which an effective potential of the other electrons substitutes for a true electron-electron repulsion term. The new operator is called the Fock operator, F. The orbital approximation is still used so that F can be separated into i (the total number of electrons) one-electron operators, Fi (19). 

In 1999, one of the authors got an inspiration that the Schrodinger equation might be able to be solved. He clarified the structure of the exact wave function and showed a method of obtaining the exact wave function by introducing the ICI method and its variants [3, 4]. However, there still existed a big obstacle, called... 

An ideal quantum device works in a reversible way due to the unitary evolution according to the Schrodinger equation. This does not constitute a problem. It has been shown that the necessary reversibility is not an obstacle to constructing any desired computer in an efficient way Universal computation can be done by reversible gates [13-15]. No essential additional expenditure in space and time is necessary [16-18]. 

Figure 2.9 illustrates the approximate dependence of the energy on the wave vector. The picture is very similar to the parabolic form of a free electron (see Eq. (2.23)) however, there are deviations (see the thick lines) as a result of the obstacles we have inserted (a,2a,3a etc.) . We remember that the Schrodinger equation is a wave equation. We expect diffraction effects at the relevant positions in the reciprocal space (k space) marked in Fig. 2.9 . In the case of a small box, it is true that e quadratically depends on a, but there are only a few discrete points. For a large box the function becomes continuous. Since we imagine our periodic soUd as composed (cf. Fig. 2.2) of small boxes forming a large box , we expect a behaviour according to Fig. 2.9.

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