Wavefunction density analysis

QUANTUM DYNAMICS OF THE THREE-DIMENSIONAL F + H2 REACTION WAVEFUNCTION DENSITY ANALYSIS... 

In this chapter, we have presented an analysis of the scattering wavefunction for the three-dimensional F + H2 reaction, for total angular momentum J = 0, at the resonance energy, 0.36 eV. The results were mainly presented in terms of a series of contour maps of the weighted the weighted scattering wavefunction density. 

For over a decade, the topological analysis of the ELF has been extensively used for the analysis of chemical bonding and chemical reactivity. Indeed, the Lewis pair concept can be interpreted using the Pauli Exclusion Principle which introduces an effective repulsion between same spin electrons in the wavefunction. Consequently, bonds and lone pairs correspond to area of space where the electron density generated by valence electrons is associated to a weak Pauli repulsion. Such a property was noticed by Becke and Edgecombe [28] who proposed an expression of ELF based on the laplacian of conditional probability of finding one electron of spin a at t2, knowing that another reference same spin electron is present at ri. Such a function... 

The impurity interacts with the band structure of the host crystal, modifying it, and often introducing new levels. An analysis of the band structure provides information about the electronic states of the system. Charge densities, and spin densities in the case of spin-polarized calculations, provide additional insight into the electronic structure of the defect, bonding mechansims, the degree of localization, etc. Spin densities also provide a direct link with quantities measured in EPR or pSR, which probe the interaction between electronic wavefunctions and nuclear spins. First-principles spin-density-functional calculations have recently been shown to yield reliable values for isotropic and anisotropic hyperfine parameters for hydrogen or muonium in Si (Van de Walle, 1990) results will be discussed in Section IV.2. 

For the analysis of the wavefunctions we computed the expectation values of a number of operators, using formulas in the works already cited. The quantities (5(ri)) and ( (ri2)) give probability densities for pairwise particle coincidences (5(ri)5(ri2)) gives the same data for the triple coincidence. The quantity Vi is... 

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. 

Knowledge of the molecular wavefunction enables us to determine the electron density at any given point in space. Here we inquire about the amount of electronic charge that can be associated in a meaningful way with each individual atom of a A -electron system. Our analysis covers Mulliken s celebrated population analysis [31], as well as a similar, closely related method. 

We conclude that KS orbitals seem to be just as suitable, if not better, for qualitative MO theoretical considerations than other orbitals, e.g., HF orbitals. The KS orbitals offer the advantage, in particular over semiempirical orbitals, but also over HF, that they are connected in an interesting way with the exact wavefunction and with exact energetics. So the MO-theoretical analysis put forward in the next section deals with energetic contributions that sum up to the exact or, with the present state of the art in density functionals, at least accurate interaction energy. The KS model offers an MO-theoretical universe of discourse in which molecular energetics can be interpreted in terms of considerations that until now were necessarily inaccurate and qualitative. Is this MO-... 

As chemists we can pose a simple, focussed question how do the Woodward-Hoffmann rules (WHR) [18] arise from a purely electron density formulation of chemistry The WHR for pericyclic reactions were expressed in terms of orbital symmetries particularly transparent is their expression in terms of the symmetries of frontier orbitals. Since the electron density function lacks the symmetry properties arising from nodes (it lacks phases), it appears at first sight to be incapable of accounting for the stereochemistry and allowedness of pericyclic reactions. In fact, however, Ayers et al. [19] have outlined how the WHR can be reformulated in terms of a mathematical function they call the dual descriptor , which encapsulates the fact that nucleophilic and electrophile regions of molecules are mutually friendly. They do concede that with DFT some processes are harder to describe than others and reassure us that Orbitals certainly have a role to play in the conceptual analysis of molecules . The wavefunction formulation of the WHR can be pictorial and simple, while DFT requires the definition of and calculations with some nonintuitive ( ) density function concepts. But we are still left uncertain whether the successes of wavefunctions arises from their physical reality (do they exist out there ) or whether this successes is merely because their mathematical form reflects an underlying reality - are they merely the shadows in Plato s cave [20]. 

Analysis of the radial pair distribution function for the electron centroid and solvent center-of-mass computed at different densities reveals some very interesting features. At high densities, the essentially localized electron is surrounded by the solvent resembling the solvation of a classical anion such as Cr or Br. At low densities, however, the electron is sufficiently extended (delocalized) such that its wavefunction tunnels through several neighboring water or ammonia molecules (Figure 16-9). 

When a population analysis is performed for the various wavefunctions used above, for the hydrogen molecule, it is found that the overlap populations are always positive (indicating a flow of electron density into the overlap region between the nuclei) but that the predicted enhancement of the density in the bond region depends on which function is used for example, the MO and VB functions give bond populations... 

The method that is used in most of the work described in this chapter is the distributed multipole analysis (DMA) of Stone,which is implemented in the CADPAC ab initio suite. DMA is based on the density matrix p,y of the ab initio wavefunction of the molecule, expressed in terms of the Gaussian primitives q that comprise the atomic orbital basis set ... 

The partial density of states (PDOS) provides a convenient analysis of wavefunction composition versus energy, making use of the Mulliken population analysis of individual orbitals. Here PDOS is defined as... 

The use of the exponential operator MCSCF formalism, or more specifically the use of optimization methods that require only the density matrix instead of the coupling coefficients over the CSF expansion terms (or even worse, over the single excitation expansion terms), has allowed relatively large CSF expansion lengths to be used in MCSCF wavefunction optimization. These larger expansion lengths allow CSFs to be included based on formal analysis or computational facility with little or no penalty in those cases where some of... 

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