13. If Truncated Wave Functions of Excited State Energy Saddle Points Are Computed as Energy Minima, Where Is the Saddle Point?

Theoretical computations tend to compute electronic properties of increasingly larger systems. To understand the properties, we should rather need small truncated but concise and comprehensible wave functions. For electronic processes, in particular charge transfer, which occur in excited states, we need both the energy and the wave function in order to draw and predict correct conclusions. But the excited states are saddle points in the Hilbert space, and, as shown here, the standard methods for excited states, based on the Hylleraas-Undheim and MacDonald (HUM) theorem, compute indeed the correct energy but may give misleadingly incorrect truncated wave functions, because they search for an energy minimum, not a saddle point (many functions can have the correct energy). Then, where is the saddle point? We shall see the use of a functional F _n of the wave function that has a local minimum at the excited state saddle point, without using orthogonality to approximants of lower-lying states, provided these approximants are reasonable, even if they are crude. Therefore F _n finds a correct, albeit small and concise, thus comprehensible truncated wave function, approximant of the desired excited state saddle point, allowing correct predictions for the electronic process. This could also lead to computational developments of more appropriate (to excited state) truncated basis sets. It is further shown that, via a correct approximant of the 1st excited state, we can improve the ground state. Finally it is shown that, in iterative computations, in cases of “root flipping” (which would deflect the computation), we can use F _n to identify the flipped root. For all the above, demonstrations are given for excited states of He and Li. The grand apophthegm is that HUM finds an energy minimum which, only if the expansion is increased, can approach the excited state saddle point, whereas F _n has local minimum at the saddle point, so it finds it independently of the size of the expansion.