Core-valence integrals

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 ... 

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 SDCI calculations are somewhat more involved in calculations of atomic real-space core-valence partitioning models because of the two-center integrals (2.10) and (2.11) that require definite integration limits to cover the appropriate core and valence subspaces. Foitunately, these calculations are greatly aided by most efficient standard techniques. 

The one-center approximation allows for an extremely rapid evaluation of spin-orbit mean-field integrals if the atomic symmetry is fully exploited.64 Even more efficiency may be gained, if also the spin-independent core-valence interactions are replaced by atom-centered effective core potentials (ECPs). In this case, the inner shells do not even emerge in the molecular orbital optimization step, and the size of the atomic orbital basis set can be kept small. A prerequisite for the use of the all-electron atomic mean-field Hamiltonian in ECP calculations is to find a prescription for setting up a correspondence between the valence orbitals of the all-electron and ECP treatments.65-67... 

Orbitals Core, Valence, Polarization, and Diffuse Basis Sets Molecular Integral Evaluation. 

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,... 

The expression for the anisotropic part of hyperfine coupling involves an integral over the spatial distribution of the unpaired electron, which is relatively easy to compute accurately even at a relatively low level of theory. The contact term, however, includes a delta-function that chips out the wave function amplitude at the nucleus point. The latter is quite difficult to compute both because standard Gaussian basis sets do not reproduce the wavefunction cusp at the nucleus point and because additional flexibility has to be introduced into the core part of the basis to account for the now essential core valence interaction. " ... 

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. 

Raffenetti, R.C. General contraction of Gaussian atomic orbitals core, valence, polarization, and diffuse basis sets Molecular integral evaluation, J. Chem. Phys. 1973, 58,4452. 

During the past three deeades, three main versions of the MCP method have been developed [1,53]. Version I is based on the local approximation. The core-valence Coulomb repulsion is a local interaction and can be satisfactorily approximated by a local potential function. For convenience of the integral evaluation, such a local potential function is chosen to be a linear combination of Gaussian type functions. The core-valence exchange operator is not a local operator. However, in Version I, this non-local interaction is also approximated by the local potential function of Gaussian type. This non-local to local approximation for the exchange operator shares the same concept with Slater s Xa density functional model [69]. Under such an approximation, the one-electron hamiltonian for the valence space in an atom (Eq. 8.5) is rewritten as... 

The Dewar-type theories treat the molecule as a collection of valence electrons and atomic cores, where each core consists of an atomic nucleus and the inner-shell (core) electrons. For example, the core of a carbon atom consists of the nncleus and the two Is electrons. The simplest approach would be to take Vcc = 2b>a2aVcc,ab = 2B>A2ACACB/f AB. where Ca and Cb are the core charges of cores A and B. For example, for a carbon atom, Ca = 6 — 2 = 4, the number of valence electtons. Although this form of V c is used in CNDO and INDO, it is more consistent with the approximations used to evaluate the electron-core interaction integrals in Dewar-type theories to take... 

R. C. Raffenetti, /. Chem. Phys., 58,4452 (1973). General Contraction of Gaussian Atomic Orbitals Core, Valence, Polarization and Dif se Basis Sets Molecular Integral Evaluation. 

Due to their localized nature, core electrons can only be adequately described with G - vectors of very high frequency, which would necessitate the use of prohibitively large basis sets in a standard plane wave scheme. Consequently, only valence electrons are treated explicitly and the effect of the ionic cores is integrated out using a pseudopotential formalism. Consistent with the first-principles character of Car-Parrinello simulations, the pseudopotentials used for this purpose are ab initio pseudopotentials (AIPPs). AIPPs are derived directly fixnn atomic all-electron calculations and different schemes exist for their construction. One of the general... 

In these expressions, x lM is the /ith atomic orbital on the Mth atom epAf(9Af) is the ionization potential of the /tth valence orbital on the Mth atom with charge qM SnM,vN and T m,vN are the overlap and kinetic energy integrals of the /tth and tdh orbitals on the Mth and A th atom, respectively (XpAf XiiM) is the core-attraction integral. The effective charge qm is calculated using the Mulliken population analysis, i.e. qm = Pfi where the AO population is defined by (6.17). 

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