Nonlinear Variation The Helium Atom

We mentioned in Chapter 5 that the ground-state wavefunction ls(l)ls(2) for helium was much too contracted if the Is functions were taken from the He+ ion without modification. Physically, this arises because, in He+, the single electron sees only a 

For He+, Z = 2, but we have just seen that this gives a function that is too contracted. Smaller values of Z would cause the function to die away more slowly with r. Therefore, it is reasonable to replace the atomic number Z with a variable parameter f and find the value of f that gives the lowest average energy. Hence, we let 

Since 77(1,2) contains no spin operators at our level of approximation, the integral separates into an integral over the space coordinates of both electrons and an integral over the spin coordinates of both electrons. The integration over spins gives a factor of unity. There remains 

Since the operator in the integrand operates only on coordinates of electron 1, we can separate this into a product of two integrals  

Varying a parameter in the argument of an exponential produces a nonlinear change in the function, and so calculations of the type described above are referred to as nonlinear variation calculations. Such calculations tend to become mathematically complicated and are not frequently used except for fairly simple systems. The fact 

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