Configuration-interaction expansions

HyperChem uses single determinant rather than spin-adapted wave functions to form a basis set for the wave functions in a configuration interaction expansion. That is, HyperChem expands a Cl wave function, in a linear combination of single Slater determ in ants... 

The advantage over the HF scheme is that whereas in conventional ah initio theory we must resort to costly perturbation theory or configuration interaction expansions, in DFT electron correlation is already included explicitly in the exchange-correlation functional. The key problem is instead to find an appropriate expression for xc. As stated above, when we have the correct functional we should be able to extract the exact energy, the exact ground state density, and all properties for our system. 

We take the initial wavefunctions to be very limited configuration-interaction expansions, containing only s orbitals. Thus, we take as the first wavefunc-tion in the one formed by all single and double excitations generated from the basic configuration [Is ] by substitutions comprising the 2s orbital for the second wavefunction, we span over the 2s and 3s orbitals and for the third one, we use orbitals Is, 2s and 3s. Hence, are all these wavefunctions are full Cl wavefunctions for such bases. Explicitly, these wavefunctions are... 

Table 2. Configuration interaction expansion coefficients for the wavefunctions...

M. Venuti, P. Decleva, Convergent multichannel continuum states by a general configuration interaction expansion in a B-spline basis Application to H photodetachment, J. Phys. B At. Mol. Opt. Phys. 30 (1997) 4839. 

Applying this projector on each of the terms of a rotational configuration interaction expansion, built up on the basis of the double free rotor solutions, the symmetry eigenvectors are obtained [21,22,34]. 

The coordinate—spin representation of the states n/jm) of an IV-electron atom or ion can be expanded in an M-dimensional linear combination of single-determinant configurations pk) (5.1). This is the configuration-interaction expansion. The orbitals a) forming the determinants are represented as orthonormal square-integrable functions aix). 

We denote the atomic eigenstate n jm) by i) and the configuration-interaction expansion by... 

Fig. 11.6. The 1200 eV noncoplanar-symmetric momentum profiles for the ground-state (n = 1) and summed n = 2 transitions in helium (Cook et al., 1984). Curves indicated DWIA, distorted-wave impulse approximation PWIA, plane-wave impulse approximation. The curves are calculated using a converged configuration-interaction expansion (McCarthy and Mitroy, 1986) for the helium ground state. The long-dashed curve is the distorted-wave impulse approximation for the Hartree—Fock ground state. Experimental data are normalised to the Is curve at low momentum. From McCarthy and Weigold (1991).

We have indicated the ability of spin-coupled valence bond (SCVB) calculations to provide compact, though accurate, nonorthogonal configuration interaction expansions for ground and excited states. These wavefunctions can be made even more compact using the SCVB method, which starts from direct... 

An ADT and its associated production rule set for a simple configuration interaction expansion problem... 

Configuration interaction expansion coefficients for the approximate ground-state wavefunctions [ (l-s) of the beryllium atom for orbits [i] = 1,..., 4 and densities... 

Linear response theory expression Alternatively, the spin-spin coupling constant can be expressed using the linear response theory formalism. Let us write the electronic energy of the system perturbed by the nuclear magnetic dipole moments M/f in the form E = E(Mjf, A), where A are the variational parameters of the wave function. A may represent orbital rotation parameters for the SCF wave function, or orbital rotation parameters and coefficients of the configuration interaction expansion for the MCSCF... 

In Section 2 we discuss the Lie algebra structure of Fermi-Fock space, which is the direct sum of all of the state spaces W, in Section 3 we will describe the possible decompositions of the Lie algebra u 2s) of U 2s), the one-particle group associated with, and thus the factorizations of U (25) that lead to manifolds of CSs in that can be based on U 2s). In Section 4 we study in detail the forms of AGP CSs. We also show how the CS construction applied to the unitary group of A-electron space can reproduce the familiar configurational interaction expansion. We conclude with a summary and discussion in Section 5. 

Wave function (4) is the standard configuration interaction expansion in mo theory. The parameter can take values from -1 to -1-1. Putting /r = 0 gives the pure molecular orbital description first considered by Coulson [49] in 1937. Table 1 summarizes the behaviour of the approximate wave functions as a function of the parameters k and /r. in the Coulson-Fischer analysis. (This table is taken from the work of Coulson and Luz [50].)... 

We may object that our conclusions seem quite naive. Indeed, there is something to worry about. We have assumed that, independent of the reaction stage, the ground-state wave function represents a single Slater determinant I o, whereas we should use a configuration interaction expansion. In such an expansion, besides the dominant contribution of I o, double excitations would be the most important (p. 653), whieh in our simple approximation of the three (p orbitals means a leading role for 2d and i sd-... 

The AMl/FCI formalism, as adopted for geometry optimization, but considering a larger configuration interaction expansion (including up to 16 molecular orbitals) to insure convergence of the spectroscopic properties. 

However, this approach, which exploits the variation theorem to determine the correlated wavefunction for the non-relativistic problem, was soon thwarted by the slow convergence of the configuration interaction expansion. 

I. Shavitt, B. J. Rosenberg, and S. Palalikit, Comparison of Configuration Interaction Expansions Based on Different Orbital Transitions, Int. J. Quantum Chem. Symp. 10, 33 6 (1976) erratum, ibid. 11, 651 (1977). 

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