Self-consistent field definition

Scanning tunneling microscopy, STM ordered adlattices, 264 oxygen adlattices, 261 platinum, 261 sodium adlattices, 262 spillover-backspillover, 259 Self-consistent field, 269 Selectivity definition, 17... 

Let us briefly mention some formal aspects of the above-introduced formalism, which have been discussed in detail by Blaizot and Marshalek [218]. First, it is noted that the both the Schwinger and the Holstein-Primakoff representations are not unitary transformations in the usual sense. Nevertheless, a transformation may be defined in terms of a formal mapping operator acting in the fermionic-bosonic product Hilbert space. Furthermore, the interrelation of the Schwinger representation and the Holstein-Primakoff representation has been investigated in the context of quantization of time-dependent self-consistent fields. It has been shown that the representations are related to each other by a nonunitary transformation. This lack of unitarity is a consequence of the nonexistence of a unitary polar decomposition of the creation and annihilation operators a and at [221] and the resulting difficulties in the definition of a proper phase operator in quantum optics [222]. 

The self-consistent field (SCF) orbital approximation method developed by Hartree is especially well suited for applications in chemistry. Hartree s method generates a set of approximate one-electron orbitals, associated energy levels, s , reminiscent of those for the H atom. The subscript a represents the appropriate set of quantum numbers (see later in this chapter for a definition). The electronic structure of an atom with atomic number Z is then built up by placing Z electrons into these orbitals in accordance with certain rules (see later in this chapter for descriptions of these rules). 

The Definition of the Multiconfiguration Self-consistent Field Method... 

The PPP method is the first semiempirical method presented here where the Fock matrix does depend on the MO coefficients C [via the density matrix elements P, see Eq. (11)]. Therefore, the Roothaan equations (by definition due to the ZDO approximation) in the orthogonal basis, Eq. (13), have to be solved in an iterative process until self-convergence is achieved [self-consistent field (SCF) procedure]. As starting coefficients C°, usually the orbitals of an HMO calculation are used. 

Equation (75) expresses the partition function of the many-polymer system in terms of the partition functions of single polymers subjected to external fluctuating fields. The self-consistent field theory approximates this functional integral over the fields by the value of the integrand evaluated at those values of the fields, and Wb, that minimize the functional F[h, 4, vb]. From the definition of F it follows that these functions satisfy the self-consistent equations... 

Two terms in the sum, Jj(i) and /fj(i), correspond to the Coulomb interaction between electron i and all other electrons j, and the exchange interaction between electrons with the same spins, respectively. The calculations of the HF potential energy for electron i require prior knowledge of all other occupied MOs. We have to guess the initial form of the MOs, use them in the definition of the Coulomb and exchange operators, and solve the HF equations iteratively in a process known as the self-consistent field (SCF) procedure. 

For one of the coupled pair of self-consistent-field equations we may write the matrix A and the inhomogeneity vector g with the following definition of matrix and vector elements... 

Electronic excitation is usually connected with an unpairing of electrons, which, as a rule of thumb, contributes 1 eV correlation energy change per pair. Furthermore, even the usual definition of the correlation energy (Ecorr = Eexact — Ehf) IS not unambiguous for excited states because a HE self-consistent field (SCF) description (which is used as uncorrelated reference) is rarely possible. Because the degree of sophistication of the theoretical treatment that can be performed is usually limited, it is important to know which main factors influence the magnitude of the electron-correlation contributions. 

Basis Sets Correlation Consistent Sets Circular Dichro-ism Electronic Complete Active Space Self-consistent Field (CASSCF) Second-order Perturbation Theory (CASPT2) Configuration Interaction Density Functional Applications Density Functional Theory (DFT), Hartree-Fock (HF), and the Self-consistent Field Electronic Diabatic States Definition, Computation, and Applications ESR Hyperfine Calculations Magnetic Circular Dichroism of rt Systems Non-adiabatic Derivative Couplings Relativistic Theory and Applications Structure Determination by Computer-based Spectrum Interpretation Valence Bond Curve Crossing Models. 

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