Schroedinger wave equation field

The foregoing discussion of valence is. of course, a simplified one. From ihe development of the quantum theory and its application to the structure of the atom, there has ensued a quantum theory of valence and of the structure of the molecule, discussed in this hook under Molecule. Topics thal are basically important to modem views of molecular structure include, in addition to those already indicated the Schroedinger wave equation the molecular orbital method (introduced in the article on Molecule) as well as directed valence bonds bond energies, hybrid orbitals, the effect of Van der Waals forces and electron-dcticiem molecules. Some of these subjects are clearly beyond the space available in this book and its scope of treatment. Even more so is their use in interpretation of molecular structure. [However, sec Crystal Field Theory and Ligand.)... 

Any description of the free atom begins with the Schroedinger wave equation for a single electron in the field of a positive point charge +Ze ... 

We cannot solve the Schroedinger equation in closed fomi for most systems. We have exact solutions for the energy E and the wave function (1/ for only a few of the simplest systems. In the general case, we must accept approximate solutions. The picture is not bleak, however, because approximate solutions are getting systematically better under the impact of contemporary advances in computer hardware and software. We may anticipate an exciting future in this fast-paced field. 

Electron-electron repulsion integrals, 28 Electrons bonding, 14, 18-19 electron-electron repulsion, 8 inner-shell core, 4 ionization energy of, 10 localization of, 16 polarization of, 75 Schroedinger equation for, 2 triplet spin states, 15-16 valence, core-valence separation, 4 wave functions of, 4,15-16 Electrostatic fields, of proteins, 122 Electrostatic interactions, 13, 87 in enzymatic reactions, 209-211,225-228 in lysozyme, 158-161,167-169 in metalloenzymes, 200-207 in proteins ... 

Auxiliary Formulae— The first of the equations (7) represents the wave motion in the atom not subjected to an external field. This problem has been exhaustively treated by Schroedinger and its solution is known to be... 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

Now consider that, at certain time to, the system of aU electrons is in the antisymmetric state. Group theory states that the system will be permanently in the antisymmetric state. Thus, instead of N classes of solutions, only one has to be considered. Then, one can make a proper formulation that considers only the space of antisymmetric wave functions. As shall be seen in a forthcoming section, this formulation is called second quantization representation and simplifies several solutions of the many-particle Schroedinger equation. Further, when no external magnetic field is present, there is spin symmetry and the eigenstates are classified according to the eigenstates of and S. ... 

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