The Virial Theorem for Atoms and Diatomic Molecules

We introduce a scale factor ri into i/r. This factor affects the lengths of the vectors r, but not their directions. That is, 

is more contracted in 3k-dimensional space than For 1, is more diffuse. 

We must check to see if our scaled function is normalized. We know that 

We could make the integral equal to V if we could get the scale factor into all the r terms in the operator and also into dv. The volume element dv requires, which is already present in Eq. A8-13 from the normalization constants. To get n into the operator, we need to multiply the operator by Multiplying Eq. A8-13 by ] gives, then 

The general result is that, for any quantum-mechanical system where 

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