Core-valence separation energy

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 ... 

Figure 7. Nonrelativistic orbital energies (Hartree-Fock, HF) and relativistise spinor energies (DLrac-Hartree-Fock, DHF) for valence shells of jgCe in the 4f 5d 6s ground state configuration from average level calculations [78], TTie horizontal dotted line corresponds to a core-valence separation based on one-particle energies.

The very simplest one is rooted in Politzer s formula = Vi/7 connecting Ek, the energy of atom k, to VklZk , the total potential at its nucleus with charge Zk- The 7 parameters are treated as constants for each type k =H, C. etc and their proper selection is known to give reasonably accurate results. The valence energy = — E " calculated on this basis (approximation A) for atom k is not a direct product of, but is consistent with, our core-valence separation in real space. 

In our real-space core-valence separation, defines the boundary of the region around nucleus k that contains the core electrons associated with it. The pertinent energy components are ... 

We leave the construction of the H matrix to consider core-valence separation. The treatment of the separation of the inner shells or cores from the valence orbitals is important in understanding what quantities we are trying to approximate with energy integrals deduced from atomic data. 

At last we can proceed with the derivation of the appropriate formulas for the core and valence region energies for both atoms and molecules. The atoms and their ions come first. This is another way of solving the question about the boundary separating the core and valence regions. 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

It is a simple matter to solve the Dirac equation in all four potentials, and the results for the 2 3/2 — 2si/2 transition in lithiumlike bismuth are collected in the first row of Table 1. We also give the 2si/2 and 2p i2 energies separately in Tables 2 and 3 respectively. We note that we will always drop any term contributing to the energy that affects only the core states. These cancel out of the transition we are studying, and also do not affect the valence removal energies. 

There is much precedent but no particular justification for omitting the core orbitals from the molecular calculation. To determine the consequences of (6) separating the core orbitals and electrons, we divide the set of atomic orbitals into two classes, Xp, core orbitals with energies Ep, and valence orbitals for which we retain the symbol . Unlike the Crrti, the coefficients of the core orbitals are not free for variation to minimize the energy but are determined by requiring that arbitrary admixture of the core orbitals in the valence molecular orbitals do not change the energy of the latter. The final matrix equation (6) is of the order of the number of valence orbitals, but the definitions of the S and H matrix elements are modified ... 

Incidentally, the same kind of technique is applicable to separate core electronic energies from valence electronic energies, and this can be used as numerical test of the relevance of our o—jt separation. Consider a ring of six lithium atoms and a distortion that keeps nn constant, as in 6a 6b. The valence electronic system is a set of six electrons moving over the field created by six positive charges of just one unit. 

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