Core-valence separation

Relative Hartree-Fock (HF) energies (eV) of LS-states of Ce and Ce with respect to the 4f 5d 6s G ground state. Frozen-core errors (eV) in these relative energies are given for 4,12 and 30 valence electron systems. The core was taken from the neutral Ce atom in its ground state [92]. 

Besides the reduction of frozen-core errors when going from large-core to medium-core or small-core potentials also the valence correlation energies obtained in pseudopotential calculations become more accurate since the radial nodal structure is partially restored [97,98]. Clearly the accuracy of small-core potentials is traded against the low computational cost of the large-core po- 

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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 ... 

Core-Valence Separation in the Study of Atomic Clusters... 

CORE-VALENCE SEPARATION IN THE STUDY OF ATOMIC CLUSTERS... 

Inspection of these integrals indicates that they are amenable to a space partitioning— like that involved in the atomic real-space core-valence separation described in Chapter 3—simply by selecting the appropriate limits of integration. Briefly, we can approach the study of core and valence regions with the help of Cl wavefunctions. 

The average local electrostatic potential V(r)/p(r), introduced by Pohtzer [57], led Sen and coworkers [58] to conjecture that the global maximum in V(r)/p(r) defines the location of the core-valence separation in ground-state atoms. Using this criterion, one finds N values [Eq. (3.1)] of 2.065 and 2.112 e for carbon and neon, respectively, and 10.073 e for argon, which are reasonable estimates in light of what we know about the electronic shell structure. Politzer [57] also made the significant observation that V(r)/p(r) has a maximum any time the radial distribution function D(r) = Avr pir) is found to have a minimum. 

In the following we derive an expression for without bothering about a physically valid core-valence separation, treating E " as if it were a continuous function of rt-The acceptable discrete solutions of E " are selected afterward. 

This equation achieves a core-valence separation. The terms in brackets are certainly individually zero at the limits A = 0 and A = 0, but this does not warrant that these terms are individually zero for other values of A , that is, that there is an N satisfying a meaningful core-valence separation. We shall tentatively proceed with... 

The latter equation achieves a separation of the core and valence contributions. The terms in brackets are individually zero at the hmits = 0 and N = 0. Here we postulate that physically meaningful core populations exist that allow such a core-valence separation and proceed with... 

In short, the core-valence partitioning in real space offers the great advantage of being naturally best suited in problems concerned with real-space atom-by-atom decompositions of molecules. Yet, although serving different purposes, and however different they may seem, real-space and orbital-space core-valence separations appear for what they are two facets of the same reality. The route to this result was not overly exciting, I am afraid, but the final result certainly justifies our patience. 

Applying Gauss theorem, leading to the Politzer-Parr core-valence separation in atoms [61]... 

Recall that the core-valence separation in molecules is described in real space [83], as any atom-by atom or bond-by-bond partitioning of a molecule is inherently a real-space problem. Equation (10.6) does indeed refer to a partitioning in real space (as opposed to the usual Hartree-Fock orbital space), both for ground-state isolated atoms or ions and for atoms embedded in a molecule, with N = 2c for first-row elements. 

The core-valence separation is very large, 10 hartrees for C. 

The scheme for achieving the core-valence separation which we wish to discuss in this Report is embodied in the idea of a pseudopotential . The hamiltonian for the electronic part of the wavefunction can be symbolically expressed within the Bom-Oppenheimer ( clamped nucleus ) approximation as... 

Core-Valence Separability and the Formal Derivation of Pseudohamiltonians... 

Having reviewed the theoretical background to the core-valence separation, we now turn to the practical implementation of the theory. Starting from equations (31)— (34) we note that the valence pseudo-orbitals are eigenfunctions of an equation which can be written as... 

Huzinaga was the recipient of the 1994 John C. Polanyi Award of the Canadian Society for Chemistry. In his award lecture he described his model potential method, which deals only with the active electrons in molecular and solid state calculations. An invited review article,59 based on his 1994 Polanyi Award lecture, chronicles his efforts to develop a sound theoretical framework for the core-valence separation of electrons, a problem Van Vleck and Sherman60 once referred to as the nightmare of the inner core. ... 

Mukherjee/91/ initially proved LCT for incomplete model spaces having n-hole n—particle determinants, showing also at the same time the validity of the core—valence separation. The corresponding open-shell perturbation theory of Brandow/20/ for such cases leads to unlinked terms and a breakdown of the core-valence separation, which used IN for O. Mukherjee emphasized that it is essential to have a valence-universal wave operator O within a Fock space formulation/91/ such that it also correlates the subduced valence sectors. Later on,... 

The electron density (10) is the so-called diagonal element of a more general quantity, the (spinless) one-electron density matrix, P(r, r ), defined in exactly the same way except that the variables in it carry primes - which are removed before the integrations. The reduction to (11), in terms of a basis set, remains valid, with a prime added to the variable in the starred function. For a separable wavefunction, the density matrices for the whole system may be expressed in terms of those for the separate electron groups in particular, for a core-valence separation,... 

Special emphasis is placed on (i) the validity of a core/valence separation, with and without freezing of the core orbitals (ii) the quality of a perfect-pairing approximation, where appropriate, and the need for resonance mixing as the geometry changes (iii) the effect of various constraints during the optimization of the orbitals - and the way in which they affect the qualitative picture of the origin of the bonds. 

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