Gauge Invariance and Finite Basis Sets

Before moving on to magnetic properties, we need to reexamine the role of gauge invariance for molecular calculations. As shown in chapter 3, the introduction of the 

The Dirac equation for an electron in a vector potential A may be written 

A gauge transformation A- A = A- -VA would be expected to cause a change in the wave function, say ilr lr. We assume that this may be written as 

Because 1 is a solution for the system without the gauge term, (13.15), this expression reduces to 

the wave function must incorporate the gauge term exponentially in order for the energy to remain invariant under gauge transformations. We should add that the same factor must be included also for the nonrelativistic treatment of the magnetic fields. For a more general derivation for the nonrelativistic case, the reader should consult the book by Sakurai (1967). 

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