Self-consistent field wavefunction

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

One solution to this basis set problem in more recent classical VB-style approaches, such as some types of VBSCF (VB self-consistent-field) wavefunctions and the BOVB ( breathing orbital VB) method,is to use variational hybrid atomic orbitals (HAOs) expanded in terms of the basis functions on a single centre only. 

IV. Configuration State Function Expansion Spaces for Multiconfiguration Self-consistent Field Wavefunctions... 

A. Energy Expressions for Multiconfiguration Self-consistent Field Wavefunction Optimization... 

IV. CONFIGURATION STATE FUNCTION EXPANSION SPACES FOR MULTICONFIGURATION SELF-CONSISTENT FIELD WAVEFUNCTIONS... 

II. Optimization of a Complete Active Space Self-consistent Field Wavefunction.405... 

II. OPTIMIZATION OF A COMPLETE ACTIVE SPACE SELF-CONSISTENT FIELD WAVEFUNCTION... 

Figure 1. Charge density contours for N2 (a) and CO (b) obtained from self-consistent field wavefunctions. The outermost contour corresponds to 0.002 atomic units and contains approximately 95% of the total electronic charge density. The center-of-mass of the CO molecule is marked by a dot and is displaced by about 0.2 A from the center of the outermost charge contour shown. (Adapted from Fig. 1 of Ref. 108.)...

A self-consistent field wavefunction (and thus its energy) can be considered a complicated function of the nuclear coordinates, basis functions and basis function coefficients (and, for a Cl calculation, the coefficients of single determinantal wavefunctions). In order to determine the first, second, etc. derivatives of the energy with respect to the nuclear coordinates [Pulay 1977] it is necessary to consider not only how the energy depends directly on the nuclear coordinates but also whether there is an indirect dependence via other parameters. Indeed, it is only the one-electron part of the Hamiltonian that depends directly upon the nuclear coordinates (H (l), Equation (2.125)), to which is added an intemuclear Coulomb repulsion term. For the other parameters the derivative with respect to the nuclear coordinates is generally determined via the chain rule (for first derivatives). For example, for a generic nuclear coordinate and a generic parameter Xj we can write ... 

An entirely different way to treat the electronic structure of molecules is provided by valence bond theory, which was developed at about the same time as the molecular orbital approach. However, valence bond theory was not so amenable to calculations on large molecules, and molecular orbital theory came to dominate electronic structure theory for such systems. Nevertheless, valence bond theories are often considered to be more appropriate for certain types of problem than molecular orbital theory, especially when dealing with processes that involve bonds being broken and/or formed. Recall from Figure 3.2 that a self-consistent field wavefunction gives a wholly inaccurate picture for the dissociation of H2 by contrast, the correct dissociation behaviour is naturally built into valence bond theories. 

Y. Osamura, Y. Yamaguchi, and H. F. Schaefer, III, Chem. Phys., 103,227 (1986). Second-Order Coupled Perturbed Hartree-Fock Equations for Closed-Shell and Open-Shell Self Consistent Field Wavefunctions. 

In 1926, he began studying radiative transitions in Hj, and in so doing, he examined Heisenberg s ideas of symmetric and antisymmetric two-electron states in helium. When Douglas Hartree introduced the self-consistent field method for the electronic structure of atoms in 1928, Slater saw the connection with Heisenberg s two-electron states. Slater published a major paper the next year. It described a theory of complex spectra, and in it he showed that with a determinantal many-electron wavefunction (the Slater determinant) one could achieve a self-consistent field wavefunction and also have the proper symmetry for electron systems (antisymmetric with respect to particle exchange). 

The Multireference Space. The calculations of the embedded-cluster wavefunctions have a first step where multiconfigurational self-consistent field wavefunctions and energies are calculated using complete and/or restricted active spaces (CASSCF [32-34] and/or RASSCF [35,36]). The lanthanide 4/, 5d, and 6s shells must be included in the active space for the calculation of the 4/, 4f 5d, and manifolds, N being the num-... 

M. Modi, M. Dolg, P. Fulde, H. Stoll, Analysis of large-scale multi-configuration self-consistent field wavefunctions by expectation values of local operators, J. Chem. Phys., 105, 2353—2363 (1996). 

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