The radial equation

The operator 2 in equation (5.32) commutes with the Hamiltonian operator H in (6.14) because 1 commutes with itself and does not involve the variable r. Likewise, the operator Lz in equation (5.31c) commutes with H because it commutes with U- as shown in (5.15a) and also does not involve the variable r. Thus, we have 

The simultaneous eigenfunctions of 1 and 2 are the spherical harmonics Yim(0, cp) given by equations (5.50) and (5.59). Since neither nor involve the variable r, any specific spherical harmonic may be multiplied by an arbitrary function of r and the result is still an eigenfunction. Thus, we may write (r, 0, cp) as 

Our next task is to solve the radial equation (6.17) to obtain the radial function R(r) and the energy E. The many solutions of the differential equation (6.17) depend not only on the value of I, but also on the value of E. Therefore, the solutions are designated as Ri.iU Y Since the potential energy /.e ir is always negative, we are interested in solutions with negative total energy, i.e., where E 0. It is customary to require that the functions Ri i(r) be normal- 

Through an explicit integration by parts, we can show that 

the operator H/ is hermitian and the radial functions REi(r) constitute an orthonormal set with a weighting function w(r) equal to r  

We next make the following conventional change of variables 

To complete the solution of the Schrodinger equation we return to the radial equation in atomic units. Equation (A9.9), which can be rearranged to give 

we have introduced the final quantum number we need for this three-dimensional problem. This is the principal quantum numbCT and is usually written as n. In this form we have what looks like a normal Schrodinger equation the first term on the left being the radial part of the kinetic energy operator (see Equations (A9.6) and (A9.2)), the term in brackets a potential energy operator and on the right-hand side we have the state energy multiplying the wavefunction R.  

The potential includes the Coulomb interaction, which has been written explicitly in terms of the effective nuclear charge. But it also includes a term that depends on the angular quantum number Z. This tells us that the effective potential in which the electron moves depends on the magnitude of its angular momentum. This observation is analogous to the centrifugal effect experienced by a macroscopic rotating body. 

In the classical description we showed how the centripetal force is generated by the Coulomb interaction and keeps the electron on its orbit. If we could instantaneously remove the electron-nuclear attraction the electron would continue along the tangential direction, flying away from the nucleus. This means that the electron motion is continually acting to separate the particles but this is counteracted by the nuclear attraction. This aspect of the angular momentum is often called the centrifugal effect. 

In the quantum mechanics model, if the electron has angular momentum (Z 0), then the additional potential represents the centrifugal effect and becomes increasingly positive as the electron approaches the nucleus. At very short separations it overrides the attractive Coulomb potential, and so the wavefunction for the electron tends to zero as it approaches the nucleus. 

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Let us now turn our attention to the radial equation, whieh is the only plaee that the explieit form of the potential appears. Using our derived results and speeifying V(r) to be the eoulomb potential appropriate for an eleetron in the field of a nueleus of eharge +Ze, yields ... 

The general solution to the radial equation is then taken to be of the form ... 

In order to retain the orbital model for a many-electron atom, Hartree assumed that each electron came under the influence of the nuclear charge and an average potential due to the remaining electrons. He therefore retained the form of the radial equation for a one-electron atom, equation 12.2, but assumed that the mutual potential energy U was the sum of... 

Sir William Hartree developed ingenious ways of solving the radial equation, and they are documented in Douglas R. Hartree s book (1957). By the time this book was published, the SCF method had been well developed, and its connection with the variation principle was finally understood. It is interesting to note that Chapter 2 of Douglas R. Hartree s book deals with the variation principle. 

We briefly recall here a few basic features of the radial equation for hydrogen-like atoms. Then we discuss the energy dependence of the regular solution of the radial equation near the origin in the case of hydrogen-like as well as polyelectronic atoms. This dependence will turn out to be the most significant aspect of the radial equation for the description of the optimum orbitals in molecules. 

We present here numerical results illustrating that the solutions of the radial equations (eq.(5) for the hydrogen-like case and eq.(14) for polyelectronic atoms) are weakly dependent of e in a finite volume. 

The radial equations was then solved using the Runge-Kutta method (7). 

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... 

The first normalized solution to the radial equation with = 0 is then found from the condition... 

Ideally, i], is obtained from integration of the radial equation for a complex potential. This resolves (II.5) into a pair of coupled differential equations. For... 

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... 

If the radial equation is solved by using the more conventional power series solution,2 the two mathematically allowed solutions for R(r) have leading terms... 

In addition to transforming the radial equation to the Numerov form, the transformation also changes the scale to one more nearly matching the frequency of the oscillations of the radial function. For bound functions, the wavelength of the wavefunction increases with r, and the substitution x = n(r) keeps the number of points per lobe closer to being constant. 

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... 

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