Electron correlation ground state

Here, Oo) represents the HF WF, 5/ are symmetry adapted excitation operators and Cl are variable coefficients. The excited state WFs f calculated from the electron correlated ground state WF, f ... 

The ADC method overcomes these drawbacks by using a more complicated basis for the expansion of the (N — l)-electron wavefunctions. The idea is to apply the physical excitation operators, Q, to the perturbation-theoretically corrected, or "correlated", ground state of the neutral system,... 

The short-range form of the correlation potential has been considered in DFT primarily in terms of density-gradient corrections to the LDA [6,7]. The correlation energy functional given by Eq.(7) here is not expressed explicitly as a functional of the electronic density, but is a formally exact expression which could be used to calibrate proposed functionals in particular cases where an accurate Cl expansion of a correlated ground-state wave function is known. When occupied orbital functions are expressed in a localized representation,... 

To assess the effectiveness of the PPP-VB method in describing the correlated ground states of various w-electron systems, we have examined a set of 70 relatively small molecules... 

A discussion is given of electron correlations in d- and f-electron systems. In the former case we concentrate on transition metals for which the correlated ground-state wave function can be calculated when a model Hamiltonian is used, i.e. a five-band Hubbard Hamiltonian. Various correlation effects are discussed. In f-electron systems a singlet ground-state forms due to the strong correlations. It is pointed out how quasiparticle excitations can be computed for Ce systems. 

Fig. 1. Schematic representation of the potential energy surface for the electronic (el) ground state of a molecule existing in two tautomeric forms, A and B. Superscripts exp, HF, CNDO/2, MINDO/3 indicate that energy differences 8 a,b calculated for potential energy surfaces determined either experimentally (exp) or calculated by means of ab initio method in the Hartree-Fock (HF) approximation or by semiempirical methods (CNDO/2, MINDO/3). The symbol eq stands for the geometrical equilibrium of both tautomers, while 2a and Qb indicate nonequilibrium geometries of tautomers A and B, respectively. Note that the theoretical potential surface calculated by sophisticated quantum-mechanical methods ( exact solution of electronic Schrbdinger equation includes electron correlation with geometry optimization) should be the same (or very similar) as that determined experimentally [in this case i>eor) ei

Iwata and co-workers" found that configurations obtained from the SCF determinant by removal of a valence electron together with a double excitation to the Iw orbitals provide very important contributions to the higher shake-up states of Nj. These significant Cl configurations are present in our EOM calculations, since the EOM basis operators act on every determinant in an approximately correlated ground-state wave function. However, the coefficients of these important Cl configurations are not... 

In both molecules and solids the electron-correlation effects may be strong or weak. Let us consider an example of a H2 molecule. For the two electrons involved, Heitler and London [150] suggested the following correlated ground-state wavefunction... 

In order to calculate the correlated ground state jiZ o) and its energy o, we split the many-electron Hamiltonian JI into... 

In the calculation of electronic structures, the presence of correlations thus always represents a difficulty. Perturbation expansions can account for the two extreme cases the delocalized limit in which the effective repulsion U is low compared to the band width, and the quasi-atomic limit where the electron delocalization modifies only slightly the correlated ground state (Anderson, 1959). Some variational techniques (Hubbard, 1964 Gutzwiller, 1965) allow a treatment of systems with U of the order of jS, but they are difficult to use. New methods have recently been developed for adding a part of the Hubbard Hamiltonian to the LDA (local density approximation) ground state (Czyzyk and Sawatzky, 1994). 

Figure Al.6.14. Schematic diagram showing the promotion of the initial wavepacket to the excited electronic state, followed by free evolution. Cross-correlation fiinctions with the excited vibrational states of the ground-state surface (shown in the inset) detennine the resonance Raman amplitude to those final states (adapted from [14].

The electron alfinity (FA) and ionization potential (IP) can be computed as the difference between the total energies for the ground state of a molecule and for the ground state of the appropriate ion. The difference between two calculations such as this is often much more accurate than either of the calculations since systematic errors will cancel. Differences of energies from correlated quantum mechanical techniques give very accurate results, often more accurate than might be obtained by experimental methods. 

Configuration Interaction (or electron correlation) adds to the single determinant of the Hartree-Fock wave function a linear combination of determinants that play the role of atomic orbitals. This is similar to constructing a molecular orbital as a linear combination of atomic orbitals. Like the LCAO approximation. Cl calculations determine the weighting of each determinant to produce the lowest energy ground state (see SCFTechnique on page 43). 

When the orbitals have been classified with respect to symmetry, they can be arranged according to energy and the correlation lines can be drawn as in Fig. 11.10. From the orbital correlation diagram, it can be concluded that the thermal concerted cycloadditon reaction between butadiene and ethylene is allowed. All bonding levels of the reactants correlate with product ground-state orbitals. Extension of orbital correlation analysis to cycloaddition reactions involving other numbers of n electrons leads to the conclusion that the suprafacial-suprafacial addition is allowed for systems with 4n + 2 n electrons but forbidden for systems with 4n 7t electrons. 

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