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## Convergence of the Wavefunction

A consequence of the variational theorem is that as the energy E of an approximate variational wavefunction approaches the exact energy , the approximate wavefunction approaches the exact one . This follows from equation (3.12), which shows that as the energy E is minimized (or, equivalently, as is minimized), then is minimized; that is, the sum of squares of the absolute values of the coefficients of excited states with weight factors is minimized. It is apparent that these weight factors might not be optimal if we want the which gives the best value for a property other than the energy, such as dipole moment. However, in the limit that E is minimized with a sufficiently large basis so that , then , or c0 = 1, implying that . Once we have the exact wavefunction, then of course all properties can be computed exactly (again, within the Born-Oppernheimer approximation and neglecting relativistic effects). To summarize, variational improvements in the energy give improvements in the approximate wavefunction, which in turn improves the values of all other properties; however, these other properties will not necessarily converge as fast as the energy with respect to N-electron basis set improvement.

Next: Variational Theorem Bounds on Up: The Variational Theorem Previous: Application of the Variational
C. David Sherrill
2000-04-18