Hydrogenic radial function normalization

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.

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

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. 

Figure 1.4 The normalized radial function for the 2s orbital in the hydrogen atom.

Figure 2.2 The Simpson s rule integration procedure, on an EXCEL spreadsheet, to determine the normalization constant, N, for the Is radial function in hydrogen defined in Table 1.1. The constant N in cell F 7 is the inverse of the square root of the value of the integral in cell F 6.

For the hydrogen Is radial function the normalization constant is exactly 2.0. As you see, in Figure 2.2, the normalization constant has been calculated to be 2.00000 to five decimal places for the integration mesh of 3200 radial points in the range 0 < r < 10.00. You should change the integration range and redo this calculation to determine the extent of the tail of the wave function. 

Since the interaction (4.304) is central, the associate wave equation may be separated in spherical polar coordinates to produce the normalized radial function. For the bound states hydrogenic atoms in the case of an infinitely heavy nucleus it looks like (Bransden Joachain, 1983) ... 

Now we are in position to calculate the normalization constant of the Hydrogenic wave-function, as well some of the radial mean values in direct manner. For instance, for the normalization constants we have ... 

A is a normalization constant and T/.m are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. 

The quantity p2 as a function of the coordinates is interpreted as the probability of the corresponding microscopic state of the system in this case the probability that the electron occupies a certain position relative to the nucleus. It is seen from equation 6 that in the normal state the hydrogen atom is spherically symmetrical, for p1M is a function of r alone. The atom is furthermore not bounded, but extends to infinity the major portion is, however, within a radius of about 2a0 or lA. In figure 3 are represented the eigenfunction pm, the average electron density p = p]m and the radial electron distribution D = 4ir r p for the normal state of the hydrogen atom. 

Fig. 1-2.—The wave function u, its square, and the radial probability distribution function 47rrVi fo the normal hydrogen atom.

Here n denotes "effective quantum number", exponent 5C, is an arbitrary positive number, r, t, y) are polar coordinates for a point with respect to the origin A in which the function (2,3) is centered. Apart from the first two terms that represent a normalizing factor, the function (2,3) is closely related to hydrogen-like orbitals. For the hydrogen Is orbital the function I q q 0 identical with Q q, if we assume Z Z/n, However, it should be recalled that in contrast to hydrogen-like orbitals STO s are not mutually orthogonal. Another essential difference is in the number of nodes. Hydrogen functions have (n -i - 1) nodes, whereas STO s are nodeless in their radial part. Alternatively, the STO may be expressed by means of Cartesian coordinates as follows... 

Orbitals (GTO). Slater type orbitals have the functional form e, if) = NYi, d, e- -- (5.1) is a normalization constant and T are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. centre of a bond. 5.2 Classification of Basis Sets Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

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. ...

Figure 1.3 The normalized [see Chapter 3] radial wave function, Ri(r), for the Is atomic orbital in the hydrogen atom [Z = 1 ] presented as an EXCEL graph, constructed as described in the text. Note, that since Yqo = 1 for s-orbitals, the only difference between this function and the total Is atomic orbital is the factor (l/4jrf- due to the normalization constant over the angular coordinates.

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