Electronic wave function radial

BohrPT Bohr Model of the Atom - Four Quantum Numbers -Electron Configuration of the Atom - Electron Shells - Shapes of Orbitals - Wave Nature of the Electron - Wave Functions, Radial and... 

The first step beyond the statistical model was due to Hartree who derived a wave function for each electron in the average field of the nucleus and all other electrons. This field is continually updated by replacing the initial one-electron wave functions by improved functions as they become available. At each pass the wave functions are optimized by the variation method, until self-consistency is achieved. The angle-dependence of the resulting wave functions are assumed to be the same as for hydrogenic functions and only the radial function (u) needs to be calculated. 

For H2, accurate theoretical calculations3 of the vibration-rotation energy levels have been done by solving the radial differential equation (4.11) using numerical integration. The potential-energy function used is that found from a 100-term variational electronic wave function. 

According to the general relationship (5.9), rotations in isospin space transform the electron creation operators by the D-matrix of rank 1/2. If we go over from these operators to the one-electron wave functions they produce, then we shall have the unitary transformation of radial orbitals... 

There are three restrictions that are normally incorporated into Hartree-Fock calculations, and a fourth often appears when the Hartree-Fock formalism is used to parametrize the experimental results. (1) The spacial part of a one-electron wave function pi is assumed to be separable into a radial and an angular part, so that = r lUi(r)Si(e,)Si(a) where Si(a) is a spin function with spin... 

The electronic wave function can be represented as a product of a radial and an angular component ... 

The differential equation which governs the radial part of the one-electron wave functions inside a single atomic or muffin-tin sphere is the radial Schrodinger equation (1.17). This we write symbolically in the form... 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

We recognize, in this expression of the two-electron wave function, the numerical radial functions of Section 1.2. As is usual, we can require that the individual radial functions be normalized, so that... 

Fig. 10.17. Schematic picture of a metal with 4f-hybridized conduction-electron wave-functions. Haloes indicate the probability maximum of the radial part of the 4f function at a distance of roughly 0.4 A from the nucleus. The picture probably applies to La metal and Ce at high pressure.

Fig. 1. Perspective plot of a two-dimensional slice of a potential energy surface near a conical intersection. The degeneracy point is located at the origin in the uv plane. The radial distance from the intersection is denoted by r and the azimuthal angle around the intersection denoted by g. The adiabatic ground state real electronic wave function changes sign for any closed path in uv space which encircles the origin (such as the dashed curve C).

H H2, we know that it is synunetric across the symmetry hne which extends radially outward from the origin at = 0 (i.e. to the right of the conical intersection). It is also symmetric across the two symmetry lines which extend radially outward from the origin at = 120°. The geometric phase alters the symmetry of the real electronic wave function for H3, so that it is also antisymmetric across the symmetry line which extends radially outward from the origin at = 7t (i.e. to the left of the conical intersection). By symmetry, it is also antisymmetric across the two symmetry lines which extend radially outward from the origin at = 60°. The antisynunetric behavior is a direct consequence of the wave function s double-valuedness. In order to satisfy Fermi statistics for aU nuclear geometries, the product of the nuclear motion wave function and nuclear spin wave function must also be double-valued and be antisynunetric across the synunetry hnes at (f) = 0, (f> = 120°, and = —120°. The product must also be synunetric across the symmetry lines at = tt, = 60°, and 4> = —60°. 

Because the one-electron operators are identical in form to the one-electron operator in hydrogen-like systems, we use for the independent particle model of Eq. (8.96) for the basis of the many-electron wave function a product consisting of N such hydrogen-like spinors. This ansatz allows us to treat the nonradial part analytically. The radial functions remain unknown. In principle, they may be expanded into a set of known basis functions, but we focus in this chapter on numerical methods, which can be conveniently employed for the one-dimensional radial problem that arises after integration of all angular and spin degrees of freedom. 

In this last section, we shall present some examples of results of numerical atomic structure calculations to demonstrate properties of radial functions and the magnitude of specific effects like the choice of the finite nucleus model or the inclusion of the Breit operator. It should be emphasized that the reliability of numerical calculations is solely governed by the affordable length of the Cl expansion of the many-electron wave function since the numerical solution techniques allow us to determine spinors with almost arbitrary accuracy. Expansions with many tens of thousands of CSFs can be routinely handled (with the basis-set techniques of chapter 10, expansions of billions of CSFs are feasible via subspace iteration techniques [372]). 

In eq. (23), Fq is the average potential in the interstitial region of space. In eq. (24), f(fis) and g(Uj) denote the small and large components of the electron wave function at the sphere radius a, respectively, which depend on E. They are obtained by solving the radial Kohn-Sham-Dirac equations for a given E,... 

As to spherical parameters, at least the Slater integrals and the spin-orbit coupling constant, it should be possible to evaluate them with precision from the radial part of the f-electron wave function obtained by solving the Hartree or Hartree-Fock equations. Unfortunately, the F calculated this way are 30-50% greater than the fitted ones. It seems that the lanthanides are almost alone (even if there are fourteen elements ) in the Periodic Table in displaying such pronounced misbehaviour. Elements to help understand this may be found in a paper by Rajnak and Wyboume (1963) who showed that the effects of configuration interactions in configurations may be represented by... 

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