Radial equation

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. 

Inserting this form of p into the eq.(l) gives the equation to be satisfied by / (r), the so called radial equation ... 

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). 

Preg and pirreg are the regular and irregular solutions of the two radial equations corresponding to the internal and external zones (assuming Ccfj and e to be known). 

This description is completed by describing what happens outside the molecule the partial waves of the optimum orbital are there proportional to the irregular solution of a radial equation involving the actual energy of the orbital. ... 

The solution of these six ordinary differential equations is readily obtained by any ODE software. Note that the rate equation could be any function of C without complicating the solution procedure unduly. When higher axial derivatives are present, each radial equation must be reduced to a set of first order ones. 

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... 

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