Electronic wave function many-electron atoms

Pauling showed that the quantum mechanical wave functions for s and p atomic orbitals derived from the Schrodinger wave equation (Section 5.7) can be mathematically combined to form a new set of equivalent wave functions called hybrid atomic orbitals. When one s orbital combines with three p orbitals, as occurs in an excited-state carbon atom, four equivalent hybrid orbitals, called sp3 hybrids, result. (The superscript 3 in the name sp3 tells how many p atomic orbitals are combined to construct the hybrid orbitals, not how many electrons occupy each orbital.)... 

Almost all approaches to many-electron wave functions, for both atoms and molecules, involve their formulation as products of one-electron orbitals. If the interelectronic repulsion term in (6.23) is small compared with the other terms, the Hamiltonian is approximately separable into independent operators for each electron and the two-electron wave function, f (1, 2), can be written as a simple product of one-electron functions,... 

In the most commonly utilized approximation, the many-electron wave functions are written in terms of products of one-electron wave functions similar to the solutions obtained for the hydrogen atom. These one-electron functions used to construct the many-electron wave function are called atomic orbitals. They are also called hydrogen-like orbitals since they are one-electron orbitals and also because their shape is similar to that of the hydrogen atom orbitals. 

We found the contraction of the f-wave functions to an atomic like character. This has an important consequense, because we see the onset of a cooperative effect, the orbital magnetic moment to spin conduction band electrons interaction and other correlations will generate an effective mass for the f-electrons which, in turn, will, by contraction of the f-wave function, enhance the atomic like character and then enhance the full process, hence the process will not be smooth or show linear behaviour the process will tend to present a sudden change of the f-electrons from a band character to an almost atomic like behaviour. This many-body effect is responsible for further scattering between the conduction electrons and the rare earth or actinide f electron states,... 

The wave function for an atom simultaneously depends on (describes) all the electrons in the atom. The Schrodinger equation is much more complicated for atoms with more than one electron than for a one-electron species such as a hydrogen atom, and an explicit solution to this equation is not possible even for helium, let alone for more complicated atoms. We must therefore rely on approximations to solutions of the many-electron Schrodinger equation. One of the most common and useful of these is the orbital approximation. In this approximation, the electron cloud of an atom is assumed to be the superposition of charge clouds, or orbitals, arising from the individual electrons these orbitals resemble the atomic orbitals of hydrogen (for which exact solutions are known), which we described in some detail in the previous section. Each electron is described by the same allowed combinations of quantum numbers (w, m(, and /,)... 

Jones et al. [144,214] used direct dynamics with semiempirical electronic wave functions to study electron transfer in cyclic polyene radical cations. Semiempirical methods have the advantage that they are cheap, and so a number of trajectories can be run for up to 50 atoms. Accuracy is of course sacrificed in comparison to CASSCF techniques, but for many organic molecules semiempirical methods are known to perform adequately. 

Many molecular properties can be related directly to the wave function or total electron density. Some examples are dipole moments, polarizability, the electrostatic potential, and charges on atoms. 

Section 1 1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron con figurations of atoms Neutral atoms have as many electrons as the num ber of protons m the nucleus These electrons occupy orbitals m order of increasing energy with no more than two electrons m any one orbital The most frequently encountered atomic orbitals m this text are s orbitals (spherically symmetrical) and p orbitals ( dumbbell shaped)... 

The Schrodinger equation can be solved approximately for atoms with two or more electrons. There are many solutions for the wave function, ij/, each associated with a set of numbers called quantum numbers. Three such numbers are given the symbols n, , and mi. A wave function corresponding to a particular set of three quantum numbers (e.g., n = 2, = 1, mi = 0) is associated with an electron occupying an atomic orbital. From the expression for ij/y we can deduce the relative energy of that orbital, its shape, and its orientation in space. 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

Boys, S. F., Proc. Roy. Soc. London) A207, 181, Electronic wave functions. IV. Some general theorems for the calculation of Schrodinger integrals between complicated vector-coupled functions for many-electron atoms."... 

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