Eigenfunctions separated wells

The search for eigenfunctions and eigenvalues in the example of the simplest difference problem. The method of separation of variables being involved in the apparatus of mathematical physics applies equelly well to difference problems. Employing this method enables one to split up an original problem with several independent variables into a series of more simpler problems with a smaller number of variables. As a rule, in this situation eigenvalue problems with respect to separate coordinates do arise. Difference problems can be solved in a quite similar manner. 

The conclusion above that optimisation of the non-linear parameters in the AO basis leads to a basis with correct spatial symmetry properties cannot be true for all intemuclear separations. At R = 0 the orbital basis must pass over into the double-zeta basis for helium i.e. two different 1 s orbital exponents. It would be astonishing if this transition were discontinuous at R = 0. While considering the variation of basis with intemuclear distance it is worth remembering that the closed-shell spin-eigenfunction MO method does not describe the molecule at all well for large values of R the spin-eigenfunction constraint of two electrons per spatial orbital is completely unrealistic at large intemuclear separation. With these facts in mind we have therefore computed the optimum orbital exponents as a function of R for three wave functions ... 

The analysis of complicated NMR multiplet spectra (AB, A2B, where A and B have relatively close chemical shifts and AX, A2X, etc., where the A and B nuclei are well separated by chemical shift) is done by perturbation theory, involving nuclear spin eigenfunctions. 

The time-independent Schrodinger equation (SE) for a molecular system derives from Hamiltonian classical dynamics and includes atomic nuclei as well as electrons. Eigenfunctions are therefore functions of both electronic and nuclear coordinates. Very often, however, the nuclear and electronic variables can be separated. The motion of the heavy particles may be treated using classical mechanics. Particularly at high temperatures, the Heisenberg uncertainty relation Ap Ax > /i/2 is easy to satisfy for atomic nuclei, which have a particle mass at least 1836 times the electron mass. The immediate problem for us is to obtain a time-independent SE including not only the electrons but also the nuclei and subsequently solve the separation problem. 

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