474
N. C. Bacalis
From Fig. 13.3 it is shown that the HUM optimization tries to correct the 2nd
root as much as possible, via other orbitals (in the same small basis) as static (just
descriptive) correlation corrections. Thus, although the energy may approach the
correct eigenvalue E 1 , the normalized wave function remains orthogonal to the
deteriorated 0 , and far from
+
1 , and much farther from the saddle point ψ 1 .
Similar results are found for Li 2 S and 4 S low-lying excited states (cf. Table
13.1).
Since the optimization of higher HUM roots fails for the smallest atoms
He and Li, it does not provide any guarantee for larger systems, where, most
probably, small basis sets are needed.
Presently, the direct minimization of the functional F n has been tested by varying
the parameters of the trial normalized expansion n (while the lower normalized
approximants i < n must be kept unvaried), but this procedure is hard to use and
is time-consuming. It is desirable, by nullifying (setting equal to 0) the derivatives
of F n with respect to the parameters, to reduce the minimization to an eigenvalue
problem, so that the lowest eigenvalue would be used. This would be immediately
accomplishable if the differentiation yielded a set of linear equations.
However, because of the form of F n in the denominator, the Rayleigh-Ritz
quotient yields a nonlinear system of equations of higher degree. Thus, there should
be a plan to solve the problem self-consistently, i.e., to consider n in certain
locations in the formula of the Rayleigh-Ritz quotient, except one location, as
constant (initial guess), so as, by nullifying the (varied) derivatives, to use the lowest
root as a new guess n in the other (artificially un-varied) locations, hoping to
converge. A choice that has not been yet exhaustively tested, but seems to work
efficiently, is to consider as initial guess (ig) all locations of the 2nd term of the
F n formula, except inside the quadratic term (because this term always converges
quadratically), i.e., specifically,
F n
ig , , n
≡ E [ n ] + 2
i
ig
| n 2
E
ig
− E [ i ]
1 −
i
2
−1
which is of the form
F n
ig , , n
≡
n | H | n
n | n
+ 2Z
ig
i
ig
n /
√
n | n
2
E
ig
− E [ i ]
,
where n is yet unnormalized, and which reduces to a standard generalized
eigenvalue problem:
F n
ig , , n
n | n ≡ n | H | n + 2Z
ig
i
ig
| n 2
E
ig
− E [ i ]
.
N. C. Bacalis
From Fig. 13.3 it is shown that the HUM optimization tries to correct the 2nd
root as much as possible, via other orbitals (in the same small basis) as static (just
descriptive) correlation corrections. Thus, although the energy may approach the
correct eigenvalue E 1 , the normalized wave function remains orthogonal to the
deteriorated 0 , and far from
+
1 , and much farther from the saddle point ψ 1 .
Similar results are found for Li 2 S and 4 S low-lying excited states (cf. Table
13.1).
Since the optimization of higher HUM roots fails for the smallest atoms
He and Li, it does not provide any guarantee for larger systems, where, most
probably, small basis sets are needed.
Presently, the direct minimization of the functional F n has been tested by varying
the parameters of the trial normalized expansion n (while the lower normalized
approximants i < n must be kept unvaried), but this procedure is hard to use and
is time-consuming. It is desirable, by nullifying (setting equal to 0) the derivatives
of F n with respect to the parameters, to reduce the minimization to an eigenvalue
problem, so that the lowest eigenvalue would be used. This would be immediately
accomplishable if the differentiation yielded a set of linear equations.
However, because of the form of F n in the denominator, the Rayleigh-Ritz
quotient yields a nonlinear system of equations of higher degree. Thus, there should
be a plan to solve the problem self-consistently, i.e., to consider n in certain
locations in the formula of the Rayleigh-Ritz quotient, except one location, as
constant (initial guess), so as, by nullifying the (varied) derivatives, to use the lowest
root as a new guess n in the other (artificially un-varied) locations, hoping to
converge. A choice that has not been yet exhaustively tested, but seems to work
efficiently, is to consider as initial guess (ig) all locations of the 2nd term of the
F n formula, except inside the quadratic term (because this term always converges
quadratically), i.e., specifically,
F n
ig , , n
≡ E [ n ] + 2
i
ig
| n 2
E
ig
− E [ i ]
1 −
i
2
−1
which is of the form
F n
ig , , n
≡
n | H | n
n | n
+ 2Z
ig
i
ig
n /
√
n | n
2
E
ig
− E [ i ]
,
where n is yet unnormalized, and which reduces to a standard generalized
eigenvalue problem:
F n
ig , , n
n | n ≡ n | H | n + 2Z
ig
i
ig
| n 2
E
ig
− E [ i ]
.