Further Development of the Polyelectronic Model

The next step involves solving the Schrodinger equation for electron 1, which is moving in a potential field created by the nucleus and the electrons in orbitals 1/ 2, tfis,, 4 n- The repulsions between electron 1 and the other electrons are computed at each point in space from the sum of the average electron densities (probabilities) corresponding to 0 213, in volume element dv around that point. With the aid of a computer, the problem is solved to yield the wave function for electron 1, which we will label 4 i- 

The next step is to do the same type of calculation to obtain a new wave function for electron 2 moving in a field of electrons described by the wave functions tf/ i, tf/, if/4,. . . , i/ n- This step leads to a new function 2 for electron 2. Now the process is carried out for electron 3 interacting with electrons described by the wave functions i/ i, tif x, 4,. . ., ilfN to produce a new function This procedure continues until all electrons have been covered to yield the wave functions i/ 2, / a, , iAn- Then the entire process starts again with electron 1 and continues through electron N to give the new functions i/ 2, 1 3, , N- The procedure is diagramed in Fig. 12.32. When a given cycle produces a set of wave functions that are virtually identical to the previous set, a self-consistent field is achieved and the procedure is terminated. 

Finally, although it is not precisely correct to assume that the N electrons in an atom occupy N independent one-electron orbitals, this remains a very useful idea for understanding many atomic properties, including the organization 

A schematic of the SCF method for obtaining the orbitals of a polyelectronic atom. 

Repeat cycle until new set of functions is virtually the same as the previous set 

Radium, element 88, is in Period 7 (and Group 2A), as shown in Fig. 12.31. Thus radium has two electrons in the 7s orbital, and the configuration is 

In contrast to the Schrodinger equation for the hydrogen atom, the Schrodinger equation for a polyelectronic atom cannot be solved exactly. For example, although the hydrogen and helium atoms are similar in many respects, the mathematical descriptions of these atoms are fundamentally 

The orbitals that arise from the SCF method are quite similar to hydrogen orbitals. They have the same angular characteristics (same type of bound- 

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