Angular equation

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

For = 0, Y = constant is easily seen to be a solution of the angular equation. Normalizing this solution to one over the solid angle sin 6 de d0, we get Y = V1/471. Looking now at the radial equation for large values of r, (1-16) reduces to the asymptotic equation... 

We now turn to the possible solutions of the Schrodinger equation for it 0. For = 1 the angular equation gives three solutions which are orthogonal to each other. These are the so-called p... 

For l = 2 we obtain the following five linearly independent solutions to the angular equation they are called the d orbitals ... 

The angular equation requires X = ((( + 1) where is a positive integer. Using this value for X the radial equation of Eq. (2.5) can be written as... 

We now further assume that the angular solution may be factored into the form, Y 9, (/>) = P 9)u ( )). Substituting this solution into the angular equation results once again in a decoupling of the 9 and dependence and a second separation constant, namely. 

The great merit of this approximation is that we need solve the angular equation only once in order to establish the general solution. Thereafter, all bound electrons in all atoms possess angular functions... 

The fact that the stability rules of negative ions differ so much from those for neutral atoms is, again, a consequence of their radial properties. Binding by a polarisation potential is completely different from binding by a Coulomb well, even if the angular equations are identical. 

In the simplified summary given here, only the radial part of the problem has been mentioned. Clearly, there are also angular equations and angular correlation quantum numbers to consider. Another, related, approach is to use group theoretical methods to classify doubly-excited states, and this has been pursued mainly by Herrick [328]. 

The exact solutions to the separate equations, which result from this coordinate transformation of the Schrddinger equation for the hydrogen atom, are the sets of functions known as the associated Laguerre polynomials, for the radial equation, and the spherical harmonics, for the angular equation. The quantum numbers, n,l and m arise naturally in the solution of Schrddinger s equation, and so the symbolic form, for the eigenfunction solutions to the H-atom problem, known as atomic orbitals, is... 

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. ...

While the angular equation (12) can be solved numerically (cf. [12]) or by expansion techniques (cf. [ll]) the set of coupled radial equations (13) has been treated by propagating quantum wavefunctions (cf. [11,20,21]) or, equivalently, S-matrices (cf. [12]) or R-matrices (cf. [22,23]) along the hyperspherical radius r. Due to the potential boundaries at cp =0 and cp = cp the angular wavefunctions I (r,cp ) form a discrete and complete set for the entire range of collision energies E, even in the domain above the dissociation limit... 

The spherical polar expression for the Laplacian operator appears much more foreboding than the Cartesian coordinate version. However, this is not really the case, since now the Schrddinger equation can be split into radial and angular equations that can be solved separately. To see this, we first write the wavefunction as a product of a function that only depends on r and a function that only depends on the angles 9 and < ... 

We have chosen to write the constant that ensures Equation (A9.8) is obeyed as /(/ +1), because this preempts the solution of the angular equation in which / will be shown to be the angular momentum quantum number. The solutions to Equations (A9.9) and (A9.10) will depend on the particular value of I, and so this has been added as a subscript to the Rpi and Yip functions. These two equations can now be solved separately. 

The angular equation can be rearranged to give a form that looks very like a Schrodinger equation with only a kinetic term ... 

The angular equation can be further subdivided into two differential equations, one... 

Table A9.1 The solutions for the angular equation, Equation (A9.11), for angular momentum quantum number I from 0 to 2. For mj 0 the linear combinations can be taken to give real functions whose Cartesian forms follow the familiar orbital labels. 

As an example demonstration that spherical harmonics are working solutions of the angular part of the Schrddinger equation, we will substitute Ti i back into the angular equation, Equation (A9.11), and show that Equation (A9.22) is valid in this case. Using the function from Table A9.1, the left-hand side of Equation (A9.12) becomes... 

Concentrating on the mathematical solutions of the Schrodinger equation, we can easily lose touch with the physical problem that is being considered. We will now recap the classical picture of the electron motion in an orbit around the nucleus so that we can try to relate the solutions found for the angular equation to this more tangible model. This will also allow the differences between the classical and quantum pictures of matter at the atomic scale to be highlighted. 

The radial functions in Table A9.2 are already real functions and are common factors for the orbitals from which we will take linear combinations, and so we will concentrate on the spherical harmonic solutions of the angular equation in this section. 

Any of the allowed solutions given in the first column of Table A9.1 satisfy the angular equation (Equation (A9.11)). This also means that any linear combination of the solutions will also be a valid solution. For example, taking the sum Tn + Ti i ... 

Euler s problem is very close to the equation for the function of

separating variables for the angular equation ... 

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