Hydrogen orbitals radial functions

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. 

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... 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

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... 

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). 

Figure 2.22 A plot of the radial function for a 3s hydrogen atomic orbital...

The radial function of the 3s atomic orbital for the hydrogen atom has the form given in equation (2.44). 

This is the most stable orbital of a hydrogen-like atom—that is, the orbital with the lowest energy. Since a Is orbital has no angular dependency, the probability density 2 is spherically symmetrical. Furthermore, this is true for all s orbitals. We depict the boundary surface for an electron in an s orbital as a sphere (Figure 1-2). The radial function ensures that the probability for finding the particle goes to zero for r — °°. 

By way of example, Table 18.3 provides the squares of the largest weight coefficients a ax, / ax of the wave functions that are obtained using radial hydrogen orbitals after the matrix of the electrostatic interaction operator has been diagonalized in a conventional basis... 

The explicit forms of the radial functions of hydrogenic orbitals Is through 4f are listed in Table 2.1.3, where Z is the nuclear charge of the atom and ao is the Bohr radius ... 

Table 2.1.3. The radial functions for hydrogenic orbitals with n = 1-4...

The radial probability function Dni or rRni 2, drawn to the same scale, of the first six hydrogenic orbitals. 

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... 

The radial functions for the first three orbitals in the hydrogen diom are... 

The sizes and shapes of the hydrogen atom orbitals are important in chemistry because they provide the foundations for the quantum description of chemical bonding and the molecular shapes to which it leads. Sizes and shapes of the orbitals are revealed by graphical analysis of the wave functions, of which the first few are given in Table 5.2. Note that the radial functions are written in terms of the dimensionless variable a, which is the ratio of Zr to ao- For Z = 1, a- = 1 at the radius of the first Bohr orbit of the hydrogen atom. 

The most apparent feature of the radial wave functions is that they all represent an exponential decay , and that for n = 2 the decay is slower than for n = I. This may be generalized for all radial functions They decay as eFor this reason, the radius of the various orbitals (actually, the most probable radius) increases with increasing n. A second feature is the presence of a node in the 2s radial function. At r = 2uo/Z, R = 0 and the value of the radial function changes from positive to negative. Again, this may be generalized s orbitals have n — 1 nodes, p orbitals have n — 2 nodes, etc. The radial functions for the hydrogen Is, 2s, and 2p orbitals are shown in Fig. 2.1. 

Figure 10.2. The radial probability distribution functions for hydrogen orbitals Is (gray) 2s (black) and 3s (heavy black).

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