Schrodinger equation accurate solution

It is now a generally accepted view that electrons and atomic nuclei are the fundamental particles of chemistry and that the time-dependent Schdrdinger equation is the central equation for the study of molecular structure and dynamics, and thus also for chemical reactions in general. In spite of the tremendous advances in the power and speed of electronic computers and in generally available sophisticated software for finding adequate approximate solutions to the Schrodinger equation, accurate treatments are still limited to rather simple systems, for which predictive results can be obtained from theory alone. Nevertheless much can be achieved by approximations to the time-dependent Schrodinger equation and even with approximate solutions to approximate equations. 

For small molecules, the accuracy of solutions to the Schrodinger equation competes with the accuracy of experimental results. However, these accurate ab initio calculations require enormous computation and are only suitable for the molecular systems with small or medium size. Ab initio calculations for very large molecules are beyond the realm of current computers, so HyperChem also supports semi-empirical quantum mechanics methods. Semi-empirical approximate solutions are appropriate and allow extensive chemical exploration. The inaccuracy of the approximations made in semi-empirical methods is offset to a degree by recourse to experimental data in defining the parameters of the method. Indeed, semi-empirical methods can sometimes be more accurate than some poorer ab initio methods, which require much longer computation times. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

In his first communication23 on the new wave mechanics, Schrodinger presented and solved his famous Eq. (1.1) for the one-electron hydrogen atom. To this day the H atom is the only atomic or molecular species for which exact solutions of Schrodinger s equation are known. Hence, these hydrogenic solutions strongly guide the search for accurate solutions of many-electron systems. 

The study of behavior of many-electron systems such as atoms, molecules, and solids under the action of time-dependent (TD) external fields, which includes interaction with radiation, has been an important area of research. In the linear response regime, where one considers the external held to cause a small perturbation to the initial ground state of the system, one can obtain many important physical quantities such as polarizabilities, dielectric functions, excitation energies, photoabsorption spectra, van der Waals coefficients, etc. In many situations, for example, in the case of interaction of many-electron systems with strong laser held, however, it is necessary to go beyond linear response for investigation of the properties. Since a full theoretical description based on accurate solution of TD Schrodinger equation is not yet within the reach of computational capabilities, new methods which can efficiently handle the TD many-electron correlations need to be explored, and time-dependent density functional theory (TDDFT) is one such valuable approach. 

The central problem in any quantum mechanical model is finding accurate solutions to the time independent form of the Schrodinger equation,... 

Moreover, as the Thomas-Fermi model for atomic systems becomes more accurate when increasing Z (asymptotically exact in the large Z-limit) with respect to the non-relativistic solution of Schrodinger equation, but relativistic effects increases with Z does, the inclusion of these is demanded for its application. 

The concept of numerical convergence is quite separate from the question of whether DFT accurately describes physical reality. The mathematical problem defined by DFT is not identical to the full Schrodinger equation (because we do not know the precise form of the exchange-correlation functional). This means that the exact solution of a DFT problem is not identical to the exact solution of the Schrodinger equation, and it is the latter that we are presumably most interested in. This issue, the physical accuracy of DFT, is of the utmost important, but it is more complicated to fully address than the topic of numerical convergence. The issue of the physical accuracy of DFT calculations is addressed in Chapter 10. 

The exchange-correlation functional for the uniform electron gas is known to high precision for all values of the electron density, n. For some regimes, these results were determined from careful quantum Monte Carlo calculations, a computationally intensive technique that can converge to the exact solution of the Schrodinger equation. Practical LDA functionals use a continuous function that accurately fits the known values of gas(/i). Several different... 

Momentum-space methods, pioneered by McWeeny, Fock, Shibuya, Wulfman, Judd, Koga, Aquilanti and others [4,17-26] provide us with an easy and accurate method for constructing solutions to the Schrodinger equation of a single electron moving in a many-center Coulomb potential... 

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