Hydrogenic eigenfunctions

To normalize radial hydrogenic eigenfunctions it will be necessary to evalute the integral... 

Fig. 1. Radial dependence of hydrogen eigenfunctions. (From Atomic Spectra and Atomic Structaret by Gerhard Herzberg, trans. by J. W. T. Spinks, reprinted through permission by Dover Publications, Inc., New York 14, X. Y.) (a) Normalized,...

Fig. 3. An so(4) subtower representing the n2 scaled hydrogenic eigenfunctions which form...

This result could be used to obtain explicit forms algebraically for all bound-state hydrogenic eigenfunctions. 

Fig. 4. The collection of so(4) subtowers for n = 1,2, 3,... forms an so(4, 2) tower of scaled hydrogenic eigenfunctions such that all eigenfunctions belong to a single unitary irreducible representation of so(4, 2). The top state in any so(4) subtower can be reached by successive application of the so(2, 1) operator T+ to the ground state. See text for details.

We shall not consider the general representation theory of so(4, 2) (Barut and Bohm, 1970). It is clear from our construction of the hydrogenic realization that we have all bound-state hydrogenic eigenfunctions within a single unirrep of so(4, 2), since this is true for the subalgebra so(4, 1). We only note that there are three independent Casimir operators for so(4, 2) given by the rather complicated expressions (see, e.g., Barut, 1971, p. 45)... 

In this section we shall discuss in some detail the formalism needed to apply the so(4, 2) algebraic methods to problems whose unperturbed Hamiltonian is hydrogenic. First a scaling transformation is applied to obtain a new Hamiltonian whose unperturbed part is just the so(2, 1) generator T3, which has a purely discrete spectrum. Next we use the scaled hydrogenic eigenfunctions of T3 as a basis for the expansion of the exact wave function. This discrete basis is complete with respect to the expansion of bound-state wave functions whereas the usual bound-state eigenfunctions do not form a complete set continuum functions must also be included to ensure completeness (cf. Section VI,A)-... 

Hybrids constructed from hydrogenic eigenfunctions are examined in their momentum-space representation. It is shown that the absence of certain cross-terms that cause the breaking of symmetry in position space, cause inversion symmetry in the complementary momentum representation. Analytical expressions for some simple hybrids in the momentum representation are given, and their nodal and extremal structure is examined. Some rather unusual features are demonstrated by graphical representations. Finally, special attention is paid to the topology at the momentum-space origin and to the explicit form of the moments of the electron density in both spaces. 

The radial behavior of the hydrogenic eigenfunctions in position and momentum space is exponential and Lorentzian , respectively, and their nodal structure depends on the associated Laguerre and Gegenbauer polynomials, respectively. 

In Fig.(l), we display the momentum density contributions of commonly encountered hybrid orbitals, obtained from hydrogenic eigenfunctions with Z = 1. The figure shows surface plots of the densities for in the 2-plane for a = 1, 2 and 3. It may be seen that, while the sp hybrid exhibits a maximum at p = 0, greater p-contributions flatten this maximum out, leading to a plateau for sp2, and finally a saddle point for sp3. 

Where published values of F2, F4, Fq, and were not available, initial assignments were based on levels calculated from the parameters F and f which were obtained by extrapolation from neighboring members of the series (4). Both F2 and can be assumed to be approximately linear functions of Z (atomic number) within the series, and by using the ratios F4/F2 and Fq/F2 calculated for a 4/ or 5f hydrogenic eigenfunction (15), all three electrostatic parameters can be evaluated from F2 alone. 

Based on your knowledge of the first few hydrogenic eigenfunctions, deduce general formulas, in terms of n and , for (i) the number of radial nodes in an atomic orbital, (ii) the number of angular nodes, (iii) the total number of nodes. 

Fig. 2.1 Radial part of the hydrogen eigenfunctions for n = 1.2. 3. (From Herzberg. G. Atomic Spectra and Atomic Structure, Dover New York, 1944. Reproduced with permission.]...

The behavior of a hydrogenic eigenfunction provides a typical example of such cusp behavior. At electron-electron coalescences of electrons with antiparallel spins (so that the wavefunction does not vanish because of the Pauli exclusion principle), fiij = 1/2 and qtqj = 1 is positive, so that the wavefunction displays an inverted cusp behavior, as illustrated in the graph below ... 

To understand the origin of this power of 3 in a more fundamental way, let us consider the generalisation of our analysis to a hydrogenic wavefunction of angular momentum / in I spatial dimensions. For simplicity, we take the principal quantum number n = / + 1. Then such a hydrogenic eigenfunction has in configuration space the form... 

Qualitative conclusions on the relative strengths of the various spin-orbit interactions can be deduced from the expression obtained for through the use of the Coulomb potential energy, C7 = -Zcge lr, and hydrogenic eigenfunctions ... 

Since, on assuming a Coulomb potential and hydrogenic eigenfunctions, the electrostatic parameters are proportional to Zeg, whereas, the spin-orbit parameters are proportional to Ztg, the increase of the spin-orbit parameters, on moving to the right hand side of the group is much greater than the increase of the electrostatic parameters. 

On assuming that the E increase as Zes, C increases as Z and not as ZU, as in the case of the Coulomb potential energy and hydrogenic eigenfunctions. It is interesting also to calculate the amount of increase of starting with Lalll 4f and ending with Tmlll 4f ... 

The hydrogen eigenfunctions are commonly called orbitals. The orbitals for the hydrogen atom are classified according to their angular distribution, or / value. Each different I value is assigned a... 

However, the result of 24.41 V, given in the Note added in proof , which I obtained with the help of mixed helium and hydrogen eigenfunctions, was wrong because of a small computational error. Indeed we do not progress in this way, because the slow convergence of the calculations is due to the expansion of l/ri2 in terms of spherical functions... 

If electron 1 is at nucleus a then the well-known, corresponding hydrogen eigenfunction is... 

For convenience we wiD first consider a system with only two electrons the nuclear charge s Ze Z > 1). The coordinates of the two electrons are r, I , quantum numbers of their oscillations in the initial state (excited state ) n, /, m, respectively N, L, M. As a first approximation one can represent the oscillations of the system through the hydrogen eigenfunctions fnim and fNLM ... 

Here we have to view ipk as the series of hydrogen eigenfunctions at the place of one atom, and as the corresponding set of eigenfunctions, but at the place of the other atom. The wave equation therefore reads... 

The eigenfunctions of the approximate Hamiltonian 9-13 are known as atomic orbitals. Since we have chosen a potential function which is a function of r only, they will be similar in many ways to the hydrogen eigenfunctions. The factors of the orbital depending on B and

radial factor is similar to the radial... 

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