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.