Hydrogenic radial function

However, for practical purposes, even the hydrogen radial functions are represented as linear combinations of other functions, which can be applied more easily in molecular orbital calculations. Nowadays, these functions and the approximations to the numerical radial functions for the other atoms of the elements of the Periodic Table derived from the hydrogenic ones are the basis sets of modem molecular orbital theory. 

Table 1.5 The Gaussian basis sets proposed by Reeves to represent the hydrogenic radial functions. The table entries, for each basis set, are the exponents, a, of the primitive Gaussians and then in the second columns the complete normalized coefficients, d, of the linear combinations.

We here examine the radial extent of the hydrogenic radial functions R i and the Laguerre functions In terms of the associated Laguerre polynomials L (x), these functions may be expressed as... 

Table Al.l Normalized radial functions / ,/(r) for hydrogen-like atoms...

Figure Al.l Radial functions for a hydrogen atom. (Note that the horizontal scale is the same in each graph but the vertical scale varies by as much as a factor of 100. The Bohr radius Oq = 52.9 pm.)...

Table 6.1. Radial functions R i for the hydrogen-like atom for n = to 6. The variable p is given by p = 2 Zrj na ...

Figure 6.4 The radial functions for the hydrogen-like atom.

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. 

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and multipole formalism, the density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

The deformation functions, however, must also describe density accumulation in the bond regions, which in the one-center formalism is represented by the atom-centered terms. They must be more diffuse, with a different radial dependence. Since the electron density is a sum over the products of atomic orbitals, an argument can be made for using a radial dependence derived from the atomic orbital functions. The radial dependence is based on that of hydrogenic orbitals, which are valid for the one-electron atom. They have Slater-type radial functions, equal to exponentials multiplied by r1 times a polynomial of degree n — l — 1 in the radial coordinate r. As an example, the 2s and 2p hydrogenic orbitals are given by... 

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... 

As an example we apply the variational principle to the evaluation of the ground state energy of a hydrogen-like atom using a minimum basis set of two-component radial functions ... 

The Schrodinger equation can be solved exactly for the case of the hydrogen atom (see, for example, Chapter 12 of Gasiorowicz (1974)). If distances are measured in atomic units, then the first few radial functions take the form... 

Fig. 2.12 The radial function, / n/ (dashed lines) and the probability density. Pm (solid lines) as a function of r for the 1s, 2s and 2p states of hydrogen.

The radial functions for the Is, 2s, and 2p states of the hydrogen atom are given by... 

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...

The signs have been chosen to make the orbitals antibonding for positive values of the constants. This choice is based on the assumption of nodeless radial functions for all orbitals. If hydrogen-like radial functions are used, not all constants will be positive. 

The radial functions f (r) will be different for different atoms. Only for the hydrogen atom is the exact analytical form of the i2((r) s known. For other atoms the f (r) s will be approximate and their form will depend on the method used to find them. They might be analytical functions (e.g. Slater orbitals) or tabulated sets of numbers (e.g. numerical Hartree-Fock orbitals). 

The wave functions for the electron in the hydrogen atom are all products of two functions. First there is the radial function R(n, r), which depends on the principal quantum number n and the coordinate r. Then there is the... 

The wave function F for a single electron, in a hydrogen atom for example, may be written as a product of four factors. These are the radial function R(r), which is dependent only on the radial distance rfrom the nucleus two angular functions 0(0) and ( ), which depend only on the angles 0 and (j>... 

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