Wavefunctions of discrete orbitals

The wavefunctions of discrete orbitals in a central potential factorize according to 

For convenience, the lowest functions are reproduced here using atomic units  

The lowest spherical harmonics are given by (phase convention of [CSh35]) 

Special cases are the Legendre polynomials P/cos 9), which follow from 

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The wavefunctions hydrogenic wavefunctions. In the present context such wavefunctions for discrete orbitals and for the continuum describing the emission of a photo- or Auger electron are of interest. 

The wavefunction which fits the equation and leads to discrete values of V is the eigenfunction. The search for such eigenfunctions and eigenvalues can be a most demanding mathematical excercise, and need not be considered here. Let us note however that the solutions of the Schrodinger equation lead to the definitions of the orbital quantum numbers n, l and m. The quantum numbers of rotational and vibrational levels are also derived from the Schrodinger equation. 

In the periodic cluster approach, the size of the supercell influences the number of orbitals and thus the discrete representation of the band structure. Using this model to describe localized defects such as vacancies introduces an additional difficulty because now the spatial extent of the vacancy wavefunction must be related to the size of the supercell. A 64-atom supercell was found to be too small to adequately describe the vacancy wavefunction, the amplitude of the wave function of a monovacancy at the cell boundary is still V4 of the maximum value. This means that the wavefunction has a considerable overlap with the vacancy wavefunctions of the neighboring cells. This artificial defect-defect interaction leads to a dispersion of about 0.8 eV for the vacancy states. When a 216-atom supercell is considered, the wavefunction amplitude at the cell boundary is only Vs of the maximum value and the dispersion is less than 0.2 A supercell of at... 

These wavefunctions are qualitatively similar to the Bohr electron orbits. The energies and wavefunctions are discrete functions of an integer n, and each wavefunction has — 1 nodes, the points at which the wavefunction crosses zero. We emphasize that the wavefunction crosses zero at the nodes to distinguish nodes from the regions outside the box, where the wavefunction stays at zero. Because the probability density is given by 1, a node in the wavefunction corresponds to a region where the probability density also reaches zero. The mass, the charge, all the measurable quantities we associate with the particle—all of these are zero at a node. 

One particular problem that massively increases the cost of correlated calculations is the slow convergence of the correlation energy with the size of the basis set that is used for the discretization of the equations. This problem comes about because all commonly used correlation methods try to expand the wavefunction in a linear combination of antisymmetrized orbital products, i.e. Slater determinants. This ansatz, however, cannot correctly describe short-range correlation effects, i.e. the shape of the wavefunction when two electron approach each other closely, and very large expansions augmented with extrapolation techniques are needed to get a sufficiently converged correlation energy. 

The only method found so far which is flexible enough to yield ground and excited state wavefunctions, transition rates and other properties is based on expanding all wavefunctions and operators in a finite discrete set of basis functions. That is, a set of one-particle spin-orbitals < >. s-x are selected and the wavefunction is expanded in Slater determinants based on these orbitals. A direct expansion would require writing F as... 

We restrict our attention in this chapter to the simple but widely used Hiickel MO (HMO) method for calculating orbitals for rc-electron and aromatic molecules [2], The HMO scheme assumes that a conjugated n-electron molecule consists of a network of sp2-hybridized carbon atoms lying in a plane and each participating atom i has a 2p electron in an atomic orbital, < ) , perpendicular to this plane. Linear combinations of these atomic orbitals (LCAO) result in the molecular n wavefunctions, q/j, each of which has a discrete energy, Ej. In terms of the parameters used in HMO computations,... 

Other Related Methods.—Baerends and Ros have developed a method suitable for large molecules in which the LCAO form of the wavefunction is combined with the use of the Xa approximation for the exchange potential. The method makes use of the discrete variational method originally proposed by Ellis and Painter.138 The one-electron orbitals are expanded in the usual LCAO form and the mean error function is minimized. 

The first-principles calculations for theoretical XANES spectra consist of three procedures, that is obtaining the self-consistent charge density, the discretized continua and the X-ray absorption spectra. The self-consistent charge densities for the chemical species were calculated with software called SCAT which implemented the DV-Xa molecular orbital method (15). For calculations of the continua and X-ray absorption spectra, the method was extended within the framework of square-integrable (L ) discretized wavefunction method (9-11). 

The basis sets used for the spectra of the Ce02 and CeO 75 clusters consisted of orbitals from 1a- to 9p for Ce and l.s to 4f for O, including/ wavefunctions and the atomic continua from 0 Hr to +1.6 Hr with a step of 0.2 Hr, containing a, p, d and / wavefunctions. The present basis sets produced 11 discretized states per eV for the continua. 

In the method, the differential equation for the continua is resolved in the same way as for the molecular orbital method. Sufficient basis sets represent the continuum wavefunctions in the molecular region and then my depends only on the density of the discretized states. Therefore, the density curves derived from smoothing the oscillator strengths of the states correspond to the photoabsorption spectra (10, 21). In the present work, the Lorentzian curve with the peak width (FWHM) of 3.0cV was used as a combined function of <5 and the smoothing. Though the discretized states depend on the choice of basis set, the spectrum obtained is practically independent of this, fhis was confirmed by using different sizes of the basis sets. 

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