Pseudopotential eigenstates

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... 

It will be useful later to notice that the core states arc eigenstates of the Hamiltonian, so that // 11> can be replaced by c>, but we can most clearly see the appropriateness of the empty-core model of the pseudopotential by leaving /f as a kinetic and potential energy. Wc have seen that the pscudopotential always enters our calculations in a matrix element between plane waves, so wc write such a matrix clement as... 

The non-adiabatic quantum simulation procedures we employ have been well described previously in the literature, so we describe them only briefly here. The model system consists of 200 classical SPC flexible water molecules," and one quantum mechanical electron interacting with the water molecules via a pseudopotential. 2 The equations of motion were integrated using the Verlet algorithm with a 1 fs time step in the microcanonical ensemble, and the adiabatic eigenstates at each time step were calculated with an iterative and block Lanczos scheme. Periodic boundary conditions were employed using a cubic simulation box of side 18.17A (water density 0.997 g/ml). 

It is important to recognize the power of the pseudopotential in this formalism. Without pseudopotentials, one has to solve the Kohn-Sham equations for all the electrons in the system. This is a very difficult exercise as the energy and length scales for the core and valence electrons eigenvalues and eigenstate differ by orders of magnitude. (Only years after the Yin and Cohen paper were all-electron structural energy calculations implemented.)... 

This example demonstrates practically independent development of ECPs for molecules and crystals when for the same ECP property different terms are used. The work by PhiUips and Kleinman (PK) [473] is an important step in the ECP apphcations for sohds. PK developed the pseudopotential formalism as a rigorous formulation of the earher empirical potential approach. They showed that ECP that has the plane-wave pseudo wavefunctions as its eigenstates could be derived from the all-electron potential and the core-state wavefunctions and energies. Thus a nonempirical approach to finding ECP was introduced. 

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