Wave functions radial

To calculate the matrix elements for H2 in the minimal basis set, we approximate the Slater Is orbital with a Gaussian function. That is, we replace the Is radial wave function... 

Spherically symmetric (radial) wave functions depend only on the radial distance r between the nucleus and the election. They are the Is, 2s, 3s. .. orbitals... 

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

Figure 1.7 Plots of (a) the radial wave function (b) the radial probability distribution function and (c) the radial charge density function 4nr Rl( against p...

The metal cluster will be modeled as an infinitely deep spherical potential well with the represented by an infinitely high spherical barrier. Let us place this barrier in the center of the spherical cluster to simplify the calculations. The simple Schrodinger equation, containing only the interaction of the electrons with the static potential and the kinetic energy term and neglecting any electron-electron interaction, can then be solved analytically, the solutions for the radial wave functions being linear combinations of spherical Bessel and Neumann functions. 

Consider now the solutions of the spherical potential well with a barrier at the center. Figure 14 shows how the energies of the subshells vary as a function of the ratio between the radius of the C o barrier Rc and the outer radius of the metal layer R ui- The subshells are labeled with n and /, where n is the principal quantum number used in nuclear physics denoting the number of extrema in the radial wave function, and / is the angular momentum quantum number. 

The radial wave functions used are thus the hydrogen-like 2p and 3d functions, J ai(r) and J 32-(r), for all orbitals of the L and M shells, respectively the symbols pts, and i 3j, piP, p3d represent these multiplied by the angular parts 1 (for s), /3 cos 8 (for p), and /5/4 (3 cos2 0-1) (for d), rather than the usual hydrogen-like orbitals. The 2-axis for each atom points along the internuclear axis toward the other atom. 

The effective atomic numbers in the radial wave functions cannot be evaluated by minimizing the energy integral, because of neglect of inner shells. In all the calculations reported the effective atomic numbers were given the value 1. 

The Relation between the Shell Model and Layers of Spherons.—In the customary nomenclature for nucleon orbitals the principal quantum number n is taken to be nr + 1, where nr> the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nT + l + 1.) The nucleon distribution function for n = 1 corresponds to a single shell (for Is a ball) about the origin. For n = 2 the wave function has a small negative value inside the nodal surface, that is, in the region where the wave function for n = 1 and the same value of l is large, and a large value in the region just beyond this surface. 

The nature of the radial wave functions thus leads us to the following interpretation1 of the subshells of the shell model ... 

The Slater—Condon integrals Ft(ff), Ft(fd), and Gj-(fd), which represent the static electron correlation within the 4f" and 4f 15d1 configurations. They are obtained from the radial wave functions R, of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23,31... 

Figure 3.6 Radial wave functions R for atomic orbitals with n = 1,2, and 3. (Reproduced with permission from J. E. Huheey, E. A. Keiter, and R. L. Keiter, Inorganic Chemistry, 4th ed., Harper Collins, 1993, Fig. 2.2 and G. Herzberg, Atomic Spectra and Atomic Structure, 1944, Dover, New Y ork, 1944.)...

Substituting from the table of associated Laguerre polynomials (1.17) the first few normalized radial wave functions are ... 

In order to obtain one-electron radial wave functions from the energy expression by the variational method, it is assumed that a set of coefficients i exists such that... 

The Sehriidinger equation governing tile radial wave functions for positive energy slates in n Coulomb field is... 

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends... 

In the rigid rotor approximation, the radial wave function is independent of /. The total wave function is... 

The typical behavior of M0 v is shown in Figure 1.6. One should note that, for the Morse potential, and in lowest approximation, the radial wave functions and thus v are independent of /. This is no longer the case for more general potentials and for the exact solution of the Morse problem. 

In the Clementi and Roetti tables, the radial wave function of all orbitals in each electron subshell j is described as a sum of Slater-type functions ... 

The radial wave functions depend on radius only via the combination p = rrUr-Za and it is convenient to write it explicitly as a function of this dimensionless variable... 

The gamma functions Ak(p) and Bj(pt) may be obtained by the use of recursion formulas an extensive tabulation is due to Flodmark (141). In the case of Slater orbitals of principal quantum number 4 or 6, application of Slater s rules leads to nonintegral powers of r in the radial wave function consequently, changing to spheroidal coordinates introduces A and B functions of nonintegral k values, that is, incomplete gamma functions. These functions can, however, be computed (56, 57) and the overlap... 

Line shapes. Because of the energy normalization of the radial wave-functions, Eq. 5.63, the summations over the free-state energies, Eq. 5.62, become integrations,... 

Let us consider a bond along the z axis. The only orbitals that extend in this direction have the following radial wave functions all others have nodes along the z axis ... 

Because no symmetry operation can alter the value of R(n, r), we need not consider the radial wave functions any further. Symmetry operations do alter the angular wave functions, however, and so we shall now examine them in more detail. It should be noted that, since A(0, 0) does not depend on n, the angular wave functions for all s, all / , all d, and so on, orbitals of a given type are the same regardless of the principal quantum number of the shell to which they belong. Table 8.1 lists the angular wave functions for sy p, d, and / orbitals. 

Oces the presence or one or more nodes and maxima have any chemical effect The answer depends upon the aspect or bonding in which we are interested. We shall see later that covalent bonding depends critically upon the overlap of orbitals. Conceivably, ir one atom had a maximum in its radial wave function overlapping with a region with a node (minimum) in the wave function of a second atom, the overlap would be poor.4 However, in every case in which careful calculations liave been made, it has been found that the nodes lie too close to the nucleus to affect the bonding appreciably. 

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