Radial wave equation

In polar coordinates the radial part of V2 reduces to + 2 A and the wave equation... 

Multiplet structures are ignored completely and electronic configurations are defined only in terms of the occupation numbers of the various orbitals. Accordingly, the radial HFS wave equations for a free atom or ion are written in the form... 

Besides the modules of maximum values of ig-function radial part were compared with Po-parameter values, and the line dependence between these values was found. Using some properties of wave function for P-parameter, the wave equation of P-parameter was obtained. 

Example 5. The diatomic molecule.6 The radial wave equation is... 

This is called the radial wave equation. Apart from the term involving l, it is the same as the one-dimensional time-independent Schrodinger equation, a fact that will be useful in its solution. The last term is referred to as the centrifugal potential, that is, a potential whose first derivative with respect to r gives the centrifugal force. 

The Xi and %2 wavefunctions describe the motion of particles of total energy W/4 in the potentials F( )/4 or V rj)l4. It is also interesting to note that for the case E=0 the wave equations of Eqs. (6.22a) and (6.22b) are similar to the radial equation for the coulomb potential in spherical coordinates. Explicitly, making the substitutions... 

What is of further interest here, as a model of the hydrogen atom and its angular momentum, is the vibration of a three-dimensional fluid sphere in a central field. As in 2D the wave equation separates into radial and angular parts, the latter of which determines the angular momentum and is identical with the angular part of Laplace s equation. 

This equation can be solved by separation of variables, provided the potential is either a constant or a pure radial function, which requires that the Lapla-cian operator be specified in spherical polar coordinates. This transformation and solution of Laplace s equation, V2 / = 0, are well-known mathematical procedures, closely followed in solution of the wave equation. The details will not be repeated here, but serious students of quantum theory should familiarize themselves with the procedures [15]. 

The radial wave function has (n — l+l) nodes, where n and l are the quantum numbers. To solve the radial atomic wave equation above, the Herman-Skillman method [12] is usually used. The equation above may be rewritten in a logarithmic coordinate of radius. The radial wave equation is first expressed in terms of low-power polynomials near the origin at the nucleus [13]. With the help of the derived polynomials near the origin, the equation is then numerically solved step by step outward from the origin to satisfy the required node number. At the same time, the radial wave equation is solved numerically from a point far away from the origin, where the radial wave function decays exponentially. The inner and outer solutions are required to be connected smoothly including derivative at a connecting point. 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

Solution of the wave equation for these conditions, would give three expressions for the wave function /, and we could again plot radial probability distributions These are not shown, but in all cases, the probability is zero at the origin, rises to a maximum value and decreases as r becomes large. We may again construct surfaces which will enclose nearly all the probability of finding an electron with the above values of the quantum numbers. 

Solving this equation will not concern us, although it is useful to note that it is advantageous to work in spherical polar coordinates (Figure 1.4). When we look at the results obtained from the Schrodinger wave equation, we talk in terms of the radial and angular parts of the wavefunction,... 

Orbitals that use the usual angular factors obtained by solving the wave equation for hydrogen together with these simple radial functions are called Slater-type orbitals (STO s). 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

Equations (12.4)-(12.9) describe an outgoing transverse spherical wave propagating radially with the phase velocity v = cojk and having mutually perpendicular complex electric and magnetic field vectors. The wave is homogeneous in that the real and imaginary parts of the complex wave vector kx are parallel. The surfaces of constant phase coincide with the surfaces of constant amplitude and are spherical. Obviously,... 

For systems of chemical interest the amplitude function ip that occurs as a solution of (4.19) is postulated to give a complete description, provided the potential energy V, is correctly specified. In reality, the only chemically significant problem that has been solved is of an electron associated with an isolated stationary proton, with potential energy V = jr, in atomic units. The differential wave equation is separable in spherical polar coordinates. Separate solutions, as functions of radial (r) and angular 9, ip) coordinates, describe the quantized energy and angular momentum of the electron as ... 

The discrete variational (DV) method numerically calculates the basis atomic orbitals using the following wave equation for the radial atomic orbital function Rja r) in spherical coordinates... 

The radial wave equation for a particle moving with orbital angular momentum I in a Coulomb field is... 

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) ... 

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