Hydrogen atom radial equation solution

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence will turn out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules. 

We present here numerical results illustrating that the solutions of the radial equations (eq.(5) for the hydrogen-like case and eq.(14) for polyelectronic atoms) are weakly dependent of e in a finite volume. 

In general, the Slater function is not an exact solution of any Schrodinger equation (except the Is- wavefunction, which is the exact solution for the hydrogen-atom problem). Nevertheless, asymptotically, the orbital exponent C is directly related to the energy eigenvalue of that state. Actually, at large distances from the center of the atom, the potential is zero. Schrodinger s equation for the radial function R(r) is... 

Equation (9.38), if restricted to two particles, is identical in form to the radial component of the electronic Schrodinger equation for the hydrogen atom expressed in polar coordinates about the system s center of mass. In the case of the hydrogen atom, solution of the equation is facilitated by the simplicity of the two-particle system. In rotational spectroscopy of polyatomic molecules, the kinetic energy operator is considerably more complex in its construction. For purposes of discussion, we will confine ourselves to two examples that are relatively simple, presented without derivation, and then offer some generalizations therefrom. More advanced treatises on rotational spectroscopy are available to readers hungering for more. 

Show that the Is orbital givien in eqn 4.22 is a solution of Schrodinger s equation for the hydrogen atom with the correct ground-state energy, either by substitution into the radial equation (eqn 4.19), or, if you are feeling brave, by substitution into the full equation (eqn 4.17). You will find the latter method distinctly harder, and will need to use the result, applicable to any function /which depends on r only,... 

Very recently, Lavenda devised an interesting method of solution of the Kramers problem in the extreme low-friction limit. He was able to show that it could be reduced to a formal Schrddinger equation for the radial part of the hydrogen atom and thus be solved exactly. One particular form of the long-time behavior of the rigorous rate equation coincides with that obtained by Kramers with the quasi-stationary hypothesis and may thus clarify the implications of this hypothesis. The method of Lavenda is reminiscent of that used by van Kampen but applied to a Smoluchowski equation for the diffusion of the energy. 

The natures of the Lanczos functions and various properties of Stleltjes orbitals are best demonstrated by specific example. In Figure 3 are shown the first ten radial Lanczos functions for the ls->kp spectrum of a hydrogen atom (18). These are obtained from solution of Equations (4) for j=l to 10 employing the appropriate hydrogenic Hamiltonian in the operator A(H) and the Is ground state orbital in the test function. In this case. Equation (4b) can be solved for the v. in terms of Laguerre functions with constant expo-... 

A second linearly independent solution of Kummer s equation, conventionally denoted U(a, b, x), is not regular at the origin and can often be excluded on physical grounds. However, as we have shown recently [2-4], when a normal hydrogen atom is confined inside a finite sphere or spherical shell, its radial eigenfunctions may involve U(a,b, x) or a combination of two linearly independent solutions of Kummer s equation, chosen so as to satisfy the boundary conditions which reflect the particular confinement considered. We have felt it necessary to address this choice in Section 2 of the present work. 

In the Darwin solution of the Dirac equation for a hydrogenic atom, the radial functions g(r) (great) and /(r) (fine) satisfy the system... 

Equation 35-34 is closely related to the radial equation 18-37 of the hydrogen atom and may be solved in exactly the same manner. If this is done, it is found that it is necessary to restrict v to the values 0, 1, 2, in order to obtain a polynomial solution.1 If we solve for W by means of Equations 35-35 and the definitions of Equations 35-33, 35-32, and 35-28, we obtain the equation... 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

The exact solutions to the separate equations, which result from this coordinate transformation of the Schrddinger equation for the hydrogen atom, are the sets of functions known as the associated Laguerre polynomials, for the radial equation, and the spherical harmonics, for the angular equation. The quantum numbers, n,l and m arise naturally in the solution of Schrddinger s equation, and so the symbolic form, for the eigenfunction solutions to the H-atom problem, known as atomic orbitals, is... 

Table 1.1 The radial and angular components of the hydrogenic atomic orbitals with distinct normalization constants for the radial and angular functions. The parameter, p = extends the application of the functions in the table entries for non-hydrogen one-electron atomic species. Remember that the solutions to the angular equation in are exp(-f / — im0) and the real forms given are obtained by taking the sums and differences of the expansions of the complex exponentials and then applying equations 1.1 to 1.3 to these results. The column headed -I-/- indicates the particular choices of sum when relevant. ...

This contains two constants A and A2, as required for a solution to a second-order differential equation, and it is indeed the complete solution. The two integration constants Ai and A2 must be assigned based on physical arguments. For the radial part of the hydrogen atom, for example, the A constant is zero since the wave function must be finite for all values of x, and A2 becomes a normalization constant. 

In the continuum limit a K) —> 0) eqn (E.3) is identical to the effective one-dimensional equation for the radial part of the three-dimensional hydrogen atom wavefunction, u r) = r (r), for the case of zero angular momentum, where 4> r) is the radial wavefunction (see Cohen-Tannoudji et al. (1977, p. 792)). This equation was studied in detail by Loudon (1959). It is useful to treat the even and odd parity solutions separately. 

The basis functions used in constructing MOs are the AOs based on the hydrogen atom solutions of the Schrddinger equation discussed in Appendix 9, with the proviso that accurate energies will require flexibility in the radial decay constants. Before moving on to molecules more complex than H2, it is worth looking at the shapes of the AOs relevant for the first row of the periodic table. We have already used the shapes of s-, p- and d-functions to discuss the symmetry of particular AOs (e.g. the d-orbitals of the central metal atom in transition metal complexes were covered in Section 5.8). These shapes are based on the... 

A third form of the Laguerre equation, important in the wave-mechanical solution of the radial part of the hydrogen atom, is the equation... 

At first sight, we are foreed to solve this equation numerically, but its overall form allows a qualitative insight into the number of solutions and their approximate values. For example, one easily see that S represents a sum of two identical quasi-atomic (onedimensional) functions each centered on the corresponding hydrogen nucleus. The functions are quite similar to 2pz Gaussian functions, but they differ by their one-dimensionality and by a different radial dependence. Indeed, instead of the usual... 

There are also solutions to the radial differential equation (6.17) for positive values of the energy E, which correspond to the ionization of the hydrogen-like atom. In the limit r oo, equations (6.17) and (6.18) for positive E become... 

Figure 6.2 The radial behavior of various basis functions in atom-centered coordinates. The bold solid line at top is the STO (f = 1) for the hydrogen Is function for the one-electron H system, it is also the exact solution of the Schrodinger equation. Nearest it is the contracted STO-3G Is function...

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