Chapter 13
If Truncated Wave Functions of Excited
State Energy Saddle Points Are
Computed as Energy Minima, Where Is
the Saddle Point?
N.C. Bacalis
Abstract 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.
N. C. Bacalis ()
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens,
Greece
e-mail: nbacalis@eie.gr
© Springer Nature Singapore Pte Ltd. 2020
T. Onishi (ed.), Theoretical Chemistry for Advanced Nanomaterials,
https://doi.org/10.1007/978-981-15-0006-0_13
465
If Truncated Wave Functions of Excited
State Energy Saddle Points Are
Computed as Energy Minima, Where Is
the Saddle Point?
N.C. Bacalis
Abstract 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.
N. C. Bacalis ()
Theoretical and Physical Chemistry Institute, National Hellenic Research Foundation, Athens,
Greece
e-mail: nbacalis@eie.gr
© Springer Nature Singapore Pte Ltd. 2020
T. Onishi (ed.), Theoretical Chemistry for Advanced Nanomaterials,
https://doi.org/10.1007/978-981-15-0006-0_13
465