13 If Truncated Wave Functions of Excited State Energy Saddle Points Are. . .
479
(For clarity in the above expressions, care has been taken to be expressed in terms
of positive quantities.) In these expressions, if the lower approximants φ i < n are
accurate enough, the coefficients inside the “2[square brackets]” are normally small
(squared: even smaller), and the factor of “1,” before them, normally dominates
over them, so that A k
> 0, A n
n > 0. Therefore, if the lower approximants
φ i < n , without being very accurate, fulfill the conditions A k
> 0, A n
n > 0, the
functional F[φ n ] has a local minimum at φ n = ψ n . (In practice, if these conditions
are not fulfilled, F[φ n ] drops down to −∞).
It is important to note that no orthogonality to the lower approximants φ i < n
has been assumed, whatsoever; it is not needed, the functional F[φ n ] has a local
minimum at φ n = ψ n for any reasonable φ i < n (that would satisfy Eqs. 13.9 and
13.10). Also, as mentioned before, it is not necessary to compute and diagonalize
the Hessian and count its negative eigenvalues to identify the saddleness at φ n = ψ n .
However, a simple criterion of being at the saddle point is to compute the 2nd
derivative of the energy E[φ n ] with respect to the parameters: There should be
n down-parabolas. Of course, the corresponding 2nd derivatives of F n would be
positive.
13.4 Improving a Ground State Approximant φ 0 Via
an Accurate φ 1
The central idea is that: if our approximant φ 1 is more accurate than an approximant
φ 0 , then diagonalizing the Hamiltonian between the two, opens their energy gap
and improves φ 0 more than φ 1 (since φ 1 is more accurate than φ 0 ). Then repeat
the process by working in the space orthogonal to φ 1 , by diagonalizing between the
new φ 0 and any function orthogonal to both φ 1 and the new φ 0 , and so on, until no
further improvement.
13.4.1 Improving φ 0 Orthogonally to the Exact ψ 1 (Analysis)
First, if we had the exact ψ 1 , then we could immediately improve an approximant
φ 0 orthogonally to ψ 1 [2]: In the subspace of {φ 0 , ψ 1 ,}, the highest Hamiltonian
eigenvector, + , is ψ 1 itself:
+
= ψ 1 .
(13.11)
The lowest, − , is orthogonal to ψ 1 ,
−
= φ
+
0 ≡
φ 0 − ψ 1 ψ 1 |φ 0
1 − ψ 1 |φ 0 2
,
(13.12)
479
(For clarity in the above expressions, care has been taken to be expressed in terms
of positive quantities.) In these expressions, if the lower approximants φ i < n are
accurate enough, the coefficients inside the “2[square brackets]” are normally small
(squared: even smaller), and the factor of “1,” before them, normally dominates
over them, so that A k
> 0, A n
n > 0. Therefore, if the lower approximants
φ i < n , without being very accurate, fulfill the conditions A k
> 0, A n
n > 0, the
functional F[φ n ] has a local minimum at φ n = ψ n . (In practice, if these conditions
are not fulfilled, F[φ n ] drops down to −∞).
It is important to note that no orthogonality to the lower approximants φ i < n
has been assumed, whatsoever; it is not needed, the functional F[φ n ] has a local
minimum at φ n = ψ n for any reasonable φ i < n (that would satisfy Eqs. 13.9 and
13.10). Also, as mentioned before, it is not necessary to compute and diagonalize
the Hessian and count its negative eigenvalues to identify the saddleness at φ n = ψ n .
However, a simple criterion of being at the saddle point is to compute the 2nd
derivative of the energy E[φ n ] with respect to the parameters: There should be
n down-parabolas. Of course, the corresponding 2nd derivatives of F n would be
positive.
13.4 Improving a Ground State Approximant φ 0 Via
an Accurate φ 1
The central idea is that: if our approximant φ 1 is more accurate than an approximant
φ 0 , then diagonalizing the Hamiltonian between the two, opens their energy gap
and improves φ 0 more than φ 1 (since φ 1 is more accurate than φ 0 ). Then repeat
the process by working in the space orthogonal to φ 1 , by diagonalizing between the
new φ 0 and any function orthogonal to both φ 1 and the new φ 0 , and so on, until no
further improvement.
13.4.1 Improving φ 0 Orthogonally to the Exact ψ 1 (Analysis)
First, if we had the exact ψ 1 , then we could immediately improve an approximant
φ 0 orthogonally to ψ 1 [2]: In the subspace of {φ 0 , ψ 1 ,}, the highest Hamiltonian
eigenvector, + , is ψ 1 itself:
+
= ψ 1 .
(13.11)
The lowest, − , is orthogonal to ψ 1 ,
−
= φ
+
0 ≡
φ 0 − ψ 1 ψ 1 |φ 0
1 − ψ 1 |φ 0 2
,
(13.12)