Hydrogenic wave function

The Self-Consistent-Field (SCF) procedure can be initiated with hydrogenic wave functions and Thomas-Fermi potentials. It leads to a set of solutions w(fj), each with k nodes between 0 and oo, with zero nodes for the lowest energy and increasing by one for each higher energy level. The quantum number n can now be defined asn = / + l + A to give rise to Is, 2s, 2p, etc. orbitals. 

As required by (36), the variational parameter k is calculated to vary between k = 2 at R = 0 and k = 1 at R > 5ao- The parameter k is routinely interpreted as either a screening constant or an effective nuclear charge, as if it had real physical meaning. In fact, it is no more than a mathematical artefact, deliberately introduced to remedy the inadequacy of hydrogenic wave functions as descriptors of electrons in molecular environments. No such parameter occurs within the Burrau [84] scheme. 

The Hamiltonian for two electrons in the field of two fixed protons is given by (39). For large values of rab the system reasonably corresponds to two H atoms. The wave functions of the degenerate system are ipi = ui5a(l)ui 6(2) and ip2 = UiSt(l)u1Sa(2), where ul5o(l) is the hydrogenic wave function for electron 1 about nucleus A, etc. For smaller values of rab a linear combination of the two product functions is a reasonable variational trial function, i.e. 1p = 1pl +... 

Fig. 5. Stopping cross section for hydrogen in argon. Calculated from binary theory with shell correction based on Clementi [58] and hydrogenic wave functions. Experimental data from numerous laboratories compiled in Ref. [6].

A common alternative is to synthesize approximate state functions by linear combination of algebraic forms that resemble hydrogenic wave functions. Another strategy is to solve one-particle problems on assuming model potentials parametrically related to molecular size. This approach, known as free-electron simulation, is widely used in solid-state and semiconductor physics. It is the quantum-mechanical extension of the classic (1900) Drude model that pictures a metal as a regular array of cations, immersed in a sea of electrons. Another way to deal with problems of chemical interaction is to describe them as quantum effects, presumably too subtle for the ininitiated to ponder. Two prime examples are, the so-called dispersion interaction that explains van der Waals attraction, and Born repulsion, assumed to occur in ionic crystals. Most chemists are in fact sufficiently intimidated by such claims to consider the problem solved, although not understood. 

Waerden, 1968) and can be used to rederive the Bohr formula for the energy levels. Corresponding to each energy level a unirrep of so(4) is obtained. The basis functions for each such unirrep are just the hydrogenic wave functions belonging to the energy level. 

Using either so(4, l)orso(4, 2) we can find infinite dimensional unirrepsfor which all bound-state scaled hydrogenic wave functions form a basis. The Lie algebra so(4, 2) is more suitable for our purposes since we have the simple expressions r = T3 — Tur = B — A. We can then calculate matrix elements of... 

In order to obtain the so(4,2) representation corresponding to the hydrogenic case we must specify the action of the 15 generators on the scaled bound-state hydrogenic wave functions nZm>. We have already done this for the generators L, A of so(4) [cf. Eqs. (177) and (178)]. For TUT2, T3 it follows... 

The H3+ ion represents the simplest example of a two-electron three-center bond in which a molecular orbital containing two electrons encompasses all three of the atoms. Instead of the approach described earlier, a more satisfactory description of the bonding is provided by constructing a molecular orbital from a combination of three hydrogen wave functions,... 

A much lesser known contribution of Pauling to the chemical knowledge, is his explicit expression for the momentum representation of the hydrogenic wave function [3]. Momentum space concepts are common among scattering physicists, some experimental chemists and a few theoreticians however, they have not won over the bulk of chemists nearly as efficiently as the hybrid concept. The reason is that they are somewhat counter intuitive and molecular structure is expressed in a rather indirect and (in the truest sense of the word) convoluted manner. 

Figure 1. Surface plot of the orbital densities for spa hybrids in momentum space. The hybrids are based on the hydrogenic wave functions. The three plots pertain to a = 1, 2 and 3, respectively. The hybrids point in the 2-direction. A section through the density in the 22-plane is displayed.

The radial part of the hydrogenic wave functions for the 4f, 5d and 6s orbitals of cerium (after H.G. Friedman et at. J. Chem. Educ. 1964,41, 357). Reproduced by permission of the American Chemical Society (c) 1964. 

In standard introductory text books, the quantum mechanics of the hydrogen atom is usually discussed in spherical coordinates. In the spherical description, neglecting the electron spin, the hydrogen states are classified with the help of three quantum numbers, the principal quantum number n, the angular quantum number I and the magnetic quantum number m. The hydrogen wave functions are given by... 

For a hydrogenic wave function (T>l e this gives in atomic units (setting m — 1). 

As a final subtlety, the screened hydrogenic wave function /)fg(ri) /) (r2) exchange is included as an additional independent member of the basis set. Without this term, rather large basis sets are required just to recover the screened hydrogenic energy —2 — Z — l) /(2n ) for Rydberg states. 

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