476
N. C. Bacalis
After normalization, the normalized n is used as a new ig ← n , until
convergence. Note that, this way, we need only the extra computation of i | n
and i |H| n , which is standard feature in most standard electronic structure
codes, with n expanded in the chosen (presumably small) basis.
In other words, this would extend the known MCSCF (multi-configuration selfconsistent field) methodology with a Rayleigh-Ritz quotient of not only the energy
(of the 1st term of F n ) under the normalizing condition of the trial function but of
the whole functional F n . Note that now, having used the i < n lower, rather crude,
approximants, the desired root of the resulting secular equation is the lowest root
above the (n th HUM root – δE) where δE should be a reasonably small quantity.
(This precaution should be taken especially for the initial guess.)
Since the above self-consistency is presently still being tested, in the following
demonstrations, for the linear part of the process, the optimization of the expansion
coefficients minimizing F n is performed on the 1st term of F n while checking
whether the 2nd term vanishes (being essentially Schrodinger’s equation “dotted”
on the unvaried i < n ). Thus, the lowest (not the n th ) HUM root, if it is above all
lower-lying (crude) energies and if it does not nullify the 2nd term and is not a
saddle point, is rejected. But, in this way (by checking every time), the correct local
minimum may be missed, that is why a targeted but self-consistent nullification of
the derivatives of the Rayleigh-Ritz quotient from F n itself is rather needed.
If the method succeeds for larger atoms, then it could be extended to molecules
and larger systems using CI expansions in standard Gaussian bases, so that it may
be proven useful for large systems demanding small bases, appropriate for both time
saving and easy understanding of the (small and comprehensible) wave functions.
These, however, are future plans; here we shall see a presentation of F n and some
demonstrations of its applications.
13.3 The Construction of F n
The central idea is to invert the sign of L in Eq. 13.1, in order to have all parabolas
upward and then to make a continuation from the unknown exact set of {ψ i } to
known approximants φ i < n .
Consider the energy (Eq. 13.1) of a normalized approximant φ n of the n th excited
state, expanded in the exact eigenfunctions {ψ i }
E [φ n ] = E n − L + U,
where L and U are the downward and upward paraboloids defined in Eq. 13.2.
Invert the sign of L to introduce the functional
F [φ n ] = E n + L + U ;
now all parabolas are upward and F[φ n ] has a local minimum at ψ n : F[φ n ] = E n .
N. C. Bacalis
After normalization, the normalized n is used as a new ig ← n , until
convergence. Note that, this way, we need only the extra computation of i | n
and i |H| n , which is standard feature in most standard electronic structure
codes, with n expanded in the chosen (presumably small) basis.
In other words, this would extend the known MCSCF (multi-configuration selfconsistent field) methodology with a Rayleigh-Ritz quotient of not only the energy
(of the 1st term of F n ) under the normalizing condition of the trial function but of
the whole functional F n . Note that now, having used the i < n lower, rather crude,
approximants, the desired root of the resulting secular equation is the lowest root
above the (n th HUM root – δE) where δE should be a reasonably small quantity.
(This precaution should be taken especially for the initial guess.)
Since the above self-consistency is presently still being tested, in the following
demonstrations, for the linear part of the process, the optimization of the expansion
coefficients minimizing F n is performed on the 1st term of F n while checking
whether the 2nd term vanishes (being essentially Schrodinger’s equation “dotted”
on the unvaried i < n ). Thus, the lowest (not the n th ) HUM root, if it is above all
lower-lying (crude) energies and if it does not nullify the 2nd term and is not a
saddle point, is rejected. But, in this way (by checking every time), the correct local
minimum may be missed, that is why a targeted but self-consistent nullification of
the derivatives of the Rayleigh-Ritz quotient from F n itself is rather needed.
If the method succeeds for larger atoms, then it could be extended to molecules
and larger systems using CI expansions in standard Gaussian bases, so that it may
be proven useful for large systems demanding small bases, appropriate for both time
saving and easy understanding of the (small and comprehensible) wave functions.
These, however, are future plans; here we shall see a presentation of F n and some
demonstrations of its applications.
13.3 The Construction of F n
The central idea is to invert the sign of L in Eq. 13.1, in order to have all parabolas
upward and then to make a continuation from the unknown exact set of {ψ i } to
known approximants φ i < n .
Consider the energy (Eq. 13.1) of a normalized approximant φ n of the n th excited
state, expanded in the exact eigenfunctions {ψ i }
E [φ n ] = E n − L + U,
where L and U are the downward and upward paraboloids defined in Eq. 13.2.
Invert the sign of L to introduce the functional
F [φ n ] = E n + L + U ;
now all parabolas are upward and F[φ n ] has a local minimum at ψ n : F[φ n ] = E n .