Extending the Wave Function Calculation

You can extend the calculation of the Hartree-Eock semi-empirical wave function by choosing Configuration Interaction (Cl) in the 

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). 

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

There are two types of electron correlations static and dynamic. Static correlation refers to a near degeneracy of a given state a dynamic correlation refers to the instantaneous avoidance of electrons with each other. 

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If a covalent bond is broken, as in the simple case of dissociation of the hydrogen molecule into atoms, then theRHFwave function without the Configuration Interaction option (see Extending the Wave Function Calculation on page 37) is inappropriate. This is because the doubly occupied RHFmolecular orbital includes spurious terms that place both electrons on the same hydrogen atom, even when they are separated by an infinite distance. 

See also Extending the Wave Function Calculation on page 37. 

The non-BO wave functions of different excited states have to differ from each other by the number of nodes along the internuclear distance, which in the case of basis (49) is r. To accurately describe the nodal structure in aU 15 states considered in our calculations, a wide range of powers, m, had to be used. While in the calculations of the H2 ground state [119], the power range was 0 0, in the present calculations it was extended to 0-250 in order to allow pseudoparticle 1 density (i.e., nuclear density) peaks to be more localized and sharp if needed. We should notice that if one aims for highly accurate results for the energy, then the wave function of each of the excited states must be obtained in a separate calculation. Thus, the optimization of nonlinear parameters is done independently for each state considered. 

Heather, R.W. and Metiu, H. (1987). An efficient procedure for calculating the evolution of the wave function by fast Fourier transform methods for systems with spatially extended wave function and localized potential, J. Chem. Phys. 86, 5009-5017. 

In many places elsewhere in this book we describe the analysis of spectra, the definition and determination of molecular parameters from the spectra, and the relationships between these parameters and the wave functions for the molecules in question. Later in this chapter we will outline the principles and practice of calculating accurate wave functions for diatomic molecules. Before we can do that, however, we must discuss the calculation of atomic wave functions the methods originally developed for atoms were subsequently extended to deal with molecules. This is not the book for an exhaustive discussion of these topics, and so many accounts exist elsewhere that such a discussion is not necessary. Nevertheless we must pay some attention to this topic because the interpretation of spectroscopic data in terms of molecular wave functions is one of the primary motivations for obtaining the data in the first place. 

In this way, once the results for any given LS state have been obtained via the operator equivalent technique, those for any of the J sub-levels thereof may easily be derived. This procedure could of course be extended to evaluate matrix elements between states of different J, arising from the same LS state, but for calculations, whether in the LSMlMs) or the LSJMj) basis, in which it is required to incorporate mixing of different LS states the resulting cross-product matrix elements cannot be found by the operator equivalent method but must be determined directly from the wave functions. 

As we have suggested recently [68] the technique involving separation of the CM motion and representation of the wave function in terms of explicitly correlated gaussians is not only limited to non- adiabatic systems with coulombic interactions, but can also also extended to study assembles of particles interacting with different types of two- and multi-body potentials. In particular, with this approach one can calculate the vibration-rotation structure of molecules and clusters. In all these cases the wave function will be expanded as symmetry projected linear combinations of the explicitly correlated fa of eqn.(29) multiplied by an angular term, Y M. 

In calculating a theoretical photoelectron spectmm, the atomic ionization cross section a. is usually taken so far from the theoretical values calculated for a neutral free atom in the ground state. However, the MO calculation by DV-Xa method is carried out self consistently and provides Q. by Mulliken population analysis using the SCF MO wave function calculated. In the present calculations, the atomic orbital Xj used for the basis function flexibly expands or contracts according to reorganization of the charge density on the atom in molecule in the self-consistent field. Furthermore, excited state atomic orbitals are sometimes added to extend the basis set. In such a case, the estimation of peak intensity of the photoionization using the data of Oj previously published is not adequate. Thus a calculation of the photoionization cross section is required for the atomic orbital used in the SCF calculation in order... 

It appears from the above discussion that the most satisfactory approach to the quantum mechanical calculation of lattice energies is that developed by Yamashita, in which the parameters of the outer wave functions of the ions are adjusted by a variational method to minimize the total energy of the crystal. Orthogonalization of the simple free ion wave functions seems to produce a result rather worse than that achieved by ignoring the correction. No doubt with the availability of electronic computers Yamashita s method will be extended to crystals in addition to LiF, where it may be necessary to adjust the wave functions of both the ions by a variational method, to allow for the effect of the crystal field. This will produce an exceedingly tedious calculation. Yamashita (1S2) has also used the method described above to show that the 0 ion is stable in the MgO crystal, though not in the gas phase. 

Fig. 5. Schematic representation (after [13]) of the numerical calculation of the spatial part of the matrix element Mspace in the p + p—> d c u, reaction. The top part shows the potential well of depth Vo and nuclear radius R of deuterium with binding energy of —2.22 MeV. The next part shows the radius dependence of the deuterium radial wave function Xd(r)- The wave-function extends far outside the nuclear radius with appreciable amplitude due to the loose binding of deuterium ground state. The p-p wave-function XppM which comprise the U = 0 initial state has small amplitude inside the final nuclear radius. The radial part of the integrand entering into the calculation of Mspac is a convolution of both Xd and Xpp in the second and third panels and is given with the hatched shading in the bottom panel. It has the major contribution far outside the nuclear radius...

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