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THE ENERGY LEVELS OF A CONFIGURATION 37 Fig. 1 gives E a and Eb as a function of ( E u — E22). It is a hyperbola with the straight lines E u and E22 as asymptotes and the distance 2 E 12 between the two levels for E n — E 22 = 0, agreeing with eqn. (18). The stabilization of the lowest state by intermixing is given by the value of χ from eqn. (19) x = En - ED = _ Eb = E 12 = - E 1 2 tg φ (26) Thus, the result of the intermixing of a small part, a ^ , of a new wave function into Ψλ is exactly the opposite of that expected when E 12 is neglected.
However, eqn. (8) is only valid for molecules, containing more ORBITALS IN ATOMIC S P E C T R O S C O P Y 29 nuclei, if the corrections for variation of Ε with the internuclear distances r are introduced : Ek in = -E - dE r— and dE E p ot = 2E + r ^ (9) Eqn. (9) reduces to eqn. (8) in the cases r = oo and r = r 0, the equilibrium internuclear distance. Eqn. (5) shows that if two isomorphous radial functions are related to each other by a scaling parameter η, having the dimension of a reciprocal radius, then the kinetic energy 2 will be proportional to η and the potential energy (having the opposite sign for stationary states) will be be proportional to η.
In most text-books, the variables of A are chosen as the angles φ and Θ of a polar co-ordinate system, while the variable R is the distance r from the nucleus. , are rather complicated and do not give a clear mental picture, and since we shall be especially interested in octahedral complexes later on, we shall write A as a linear combination of homogeneous polynomials A p of Ith degree in the Cartesian co-ordinates x, y and z, viz. : a b c _ i 2 2 2 2 Ap = x y z r , where a + b + c = / and r = x + y + z (3) We shall mention here that the orthogonalization condition for two wave functions Ψ1 and Ψ2 to be independent is that the integral of ΨχΨ2 over all its variables is zero, while the normalization refers to 2 2 the similar integrals of Ψχ and Ψ2 being 1.