Schrodinger equation theoretical background

Ab initio methods compute solutions to the Schrodinger equation using a series of rigorous mathematical approximations. These procedures are discussed in detail in Appendix A, The Theoretical Background. 

In this chapter, the most important quantum-mechanical methods that can be applied to geological materials are described briefly. The approach used follows that of modern quantum-chemistry textbooks rather than being a historical account of the development of quantum theory and the derivation of the Schrodinger equation from the classical wave equation. The latter approach may serve as a better introduction to the field for those readers with a more limited theoretical background and has recently been well presented in a chapter by McMillan and Hess (1988), which such readers are advised to study initially. Computational aspects of quantum chemistry are also well treated by Hinchliffe (1988). 

In the same formula, h is the step size of the integration and n is the number of steps, i.e. w is the approximation of the solution in the point x and Xn = XQ + n h and Xq is the initial value point. The authors, based on the theoretical background developed in [164], obtained four RKN methods (1) two methods based on the fifth order method [175] and (2) two methods based on the sixth order method [144] developed by Dor-mand, Mikkawy and Prince. For the methods based on [175] the authors constructed two methods with first and second exponential order. For the methods based on [144] the authors constructed two methods with first and second exponential order. They have applied the new developed methods to the radial time-independent Schrodinger equation for the calculation of the eigenvalues. The authors are used the following potentials the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential. From the results the authors proved the efficiency of the new developed methods. 

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