Hamiltonian matrix transition metals

Extension of pseudopotential theory to the transition metals preceded the use of the Orbital Correction Method discussed in Appendix E, but transition-metal pseudopotentials are a special case of it. In this method, the stales are expanded as a linear combination of plane waves (or OPW s) plus a linear combination of atomic d states. If the potential in the metal were the same as in the atom, the atomic d states would be eigenstates in the metal and there would be no matrix elements of the Hamiltonian with other slates. However, the potential ix different by an amount we might write F(r), and there arc, correspondingly, matrix elements (k 1 // 1 r/> = hybridizing the d states with the frce-eleclron states. The full analysis (Harrison, 1969) shows that the correct perturbation differs from (5K by a constant. The hybridization potential is... 

Several quantum mechanical calculations have been made for electron transfer processes between metals and atoms or ions " in gaseous medium. In all the cases, the considerations concern the transition of electrons from a metal state to a bound atomic state or to a free continuum state or vice versa. The calculations of transition probabilities in the cited works have been based on Fermi s golden rule of time-dependent perturbation theory. However, it was pointed out by Gadzuk that the use of the golden rule usually presents a difficult problem if an estimate of the transition probability is desired, because it requires evaluation of a matrix element one must specify initial and final state (wave functions) and an interaction. This is not as straightforward as it seems. In a transition, e.g., between an atomic and a conduction band metal state, the initial and final states are eigenfunctions of different Hamiltonians. It seems meaningless to evaluate matrix elements, if the initial and final states are solutions of different Hamiltonians. 

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