Wave approximate construction

The approximations defining minimal END, that is, direct nonadiabatic dynamics with classical nuclei and quantum electrons described by a single complex determinantal wave function constructed from nonoithogonal spin... 

At this point we are sufficiently equipped to consider briefly the methods used to approximate the wave functions constructed in the restricted subspace of orbitals. So far the only approximation was to restrict the orbital basis set. It is convenient to establish something that might be considered to be the exact solution of the electronic structure problem in this setting. This is the full configuration interaction (FCI) solution. In order to find one it is necessary to construct all possible Slater determinants for N electrons allowed in the basis of 2M spin-orbitals. In this context each Slater determinant bears the name of a basis configuration and constructing them all means that we have their full set. Then the matrix representation of the Hamiltonian in the basis of the configurations ( >K is constructed ... 

For very small field amplitudes, the multiphoton resonances can be treated by time-dependent perturbation theory combined with the rotating wave approximation (RWA) [10]. In a strong field, all types of resonances can be treated by the concept of the rotating wave transformation, combined with an additional stationary perturbation theory (such as the KAM techniques explained above). It will allow us to construct an effective Hamiltonian in a subspace spanned by the resonant dressed states, degenerate at zero field. 

Eq. (2.22) is much simpler than the original many-electron Schro-dinger equation yet it cannot be solved in closed form and approximation methods must be used. It is customary to choose a finite set of one-electron basis functions % and approximate the Hartree-Fock orbitals

total energy given by a wave function constructed from orbitals of this form, one gets a homogeneous set.of linear equations ... 

Theoretically, Vineyard described GIXD with a distorted-wave approximation in the kinematical theory of x-ray diffraction [4]. In terms of the ordinary dynamical theory of Ewald [5] and Lane [6], Afanas ev and Melkonyan [7] worked out a formulation for the dynamical diffraction of x-rays under specular reflection conditions and Aleksandrov, Afanas ev, and Stepanov [8] extended this formalism to include the diffraction geometry of thin surface layers. Subsequently, the properties of wave Adds constructed during specularly diffracted reflections have been discussed in more detail by Cowan [9] and Sakata and Hashizume [10]. Meanwhile, a geometrical interpretation of GIXD based on a three-dimensional dispersion surface has been proposed by Hoche, Briimmer, and Nieber [11]. 

Non-relativistic quantum theory of atoms and molecules is built upon wave-functions constructed from antisymmetrized products of single particle wave-functions. The same scheme has been adopted for relativistic theories, the main difference now being that the single particle functions are 4-component spinors (bispinors). The finite matrix method approximates such 4-spinors by writing... 

Atomic Valence States. The valence state of an atom for a given molecular electronic state is the state in which the atom exists in the molecule. Since individual atoms do not really exist in molecules, the valence-state concept is an approximate one. The VB approximation constructs molecular wave functions from wave functions of the individual atoms. We use the VB wave function to define the valence state of an atom as the wave function obtained on removing all other atoms to infinity, while keeping the form of the molecular wave function invariant. This process is purely hypothetical, and the valence state is not in general a stationary atomic state. 

The expression (4.53) expanded to third or forth order is well suited for this purpose, ii) With these wave-functions construct an approximation to H t + At) and to U t- -At/2,t + At). iii) Apply (4.55). This procedure leads to a very stable propagation. 

Any wave function constructed in this way is inevitably an approximate one. The coefficients have to be chosen to give the best wave functions possible within the limits of the approximation. The variation principle tells us that the energy corresponding to an approximate wave function is always 16... 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT 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). 

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