Energy eigenvalues, orbital Schrodinger equation

These equations, derived from the Schrodinger equation of Quantum Mechanics, can be solved iteratively for matrices and jL, containing as elements the appropriately normalized molecular orbital (MO) coefficients and orbital energy eigenvalues of eq. 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

Thus, it is the ansatz that is separable that leads to the concept of orbitals, which are the one-electron functions ( )y found by solving the one-electron Schrodinger equations (fj + eigenvalues s. are called orbital energies. 

Here, the e, value is the orbital s energy eigenvalue. Equation (6.9) is remarkably similar to the original Schrodinger equation. Equation (6.2), but the wave functions have been replaced with the KS orbitals and the exchange and correlation terms have been isolated. Thus, we have replaced the iV-body coupled electronic wave function with a collection of uncorrelated wave fimctions while at the same time defining precisely what the uncertain many-body terms in need of approximation are. 

These are two hydrogen-like Schrodinger equations. The eigenvalues E and 2 are hydrogen-like orbital energies. The total electronic energy in the zero-order approximation is... 

In this chapter, we have discussed the quantum mechanics of electrons in diatomic molecules using the Born-Oppenheimer approximation, which is the assumption that the nuclei are stationary as the electrons move. With this approximation the time-independent Schrodinger equation for the hydrogen molecule ion, hJ, can be solved without further approximation to give energy eigenvalues and orbitals that depend on the intemuclear distance. 

Because of Schur s lemma, we should not- be surprised by this phenomenon. Any rotated version of a solution to Schrodinger s equation will yield another solution with the same eigenvalue (or energy level, or principal quantum number). So of course there is a third 2p orbital as indicated in Fig. 17.6 ... 

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