N.6 Is the variational approximation best?

Clearly, best is a subjective term. If you are looking for the wave function within a definite set that has the most accurate expectation value of energy, then minimizing the expectation value of energy will do it. This function will also approximate the true eigenfunction shape the best, in some technical sense {A.7}. (There are many ways the best approximation of a function can be defined; you can demand that the maximum error is as small as possible, or that the average magnitude of the error is as small as possible, or that a root-mean-square error is, etcetera. In each case, the best answer will be different, though there may not be much of a practical difference.)

But given a set of approximate wave functions like those used in finite element methods, it may well be possible to get much better results using additional mathematical techniques like Richardson extrapolation. In effect you are then deducing what happens for wave functions that are beyond the approximate ones you are using.