13 If Truncated Wave Functions of Excited State Energy Saddle Points Are. . .
481
and as a lower eigenvector φ
+
0 itself as improved φ 0 . This, by just a 2 × 2
diagonalization, is another simple criterion of being at the saddle point.
Incidentally, all other (small) components are less relevant, so that the opposite
procedure of optimizing φ 1 orthogonally to φ 0 can lead to φ MIN
1
, unpredictably
far from ψ 1 , even orthogonal to ψ 1 , with still E
φ MIN
1
<
∼
E [ψ 1 ], as shown
(as analysis) in the following counterexample: Even in the subspace {ψ 0 , ψ 1 ,
ψ 2 } (supposedly known for the present counterexample), the orthonormal trial
functions φ 0 = aψ 0 + 0ψ 1 + bψ 2 and φ 1 = bψ 0 − aψ 2 with a =
(E 1 −ε)−E 0
E 2 −E 0
,
b =
E 2 −(E 1 −ε)
E 2 −E 0
, (small ε), where, by construction, φ 0 is orthogonal to ψ 1 and
φ 1 is orthogonal to both ψ 1 and φ 0 , have energies φ 0 |H|φ 0 = a 2 E 0 + b 2 E 2 =,
E[φ 0 ] = E 0 + E 2 − (E 1 − ε) > E 0 , and E[φ 1 ] = E 1 − ε, while φ 0 may approximate
ψ 0 . (To make the counterexample numerical, consider He 1 S, for which it is known
that, in a.u., E 0 = − 2.903, E 1 = − 2.146, E 2 = − 2.06. In terms of {ψ 0 (=1s 2 ),
ψ 1 (=1s2s), ψ 2 (=1s3s)} (supposedly known for the present counterexample), the
function φ 0 = 0.9476 ψ 0 + 0.3194 ψ 2 would have E[φ 0 ] = − 2.832, and the
function φ 1 = 0.3194 ψ 0 − 0.9476 ψ 2 chosen orthogonal to both ψ 1 and φ 0
would have E[φ 1 ] = − 2.146 = E 1 (i.e., it would be a function with the energy of
ψ 1 , although orthogonal to ψ 1 ), so that, any function orthogonal to the same φ 0 ,
between this φ 1 and φ 1
+ = ψ 1 (if φ 0 is orthogonal to ψ 1 , the “closest” to ψ 1 is
ψ 1 itself), could be a minimization result, φ MIN
1
, with arbitrary
ψ 1 |φ MIN
1
and with
E 1 − ε ≤ E
φ MIN
1
≤ E 1 . In this counterexample, using F 1 this danger would
not exist: Minimization of F 1 , for the same, not particularly accurate, unvaried φ 0
(0.3194 is not very small), by varying φ 1 = c ψ 0 +d ψ 2 +ψ 1
√
1 − c 2 − d 2 , yields:
c < tol(=10 −8 ) and d < tol, i.e., φ 1 = ψ 1 , with E[φ 1 ] = − 2.146 [so that, from Eq.
13.14, φ
+
0 = φ 0 ]. Below we shall see an actual demonstration.
13.4.3 Further Improvement of φ 0
If E
φ
+
0
≤ E [φ 0 ], then, by rotating around φ 1 , as described above [after Eq.
13.13], since the Hamiltonian always opens the energy gap between mutually
orthogonal functions (all orthogonal to φ 1 ), φ
+
0 can be further improved (until
ψ 0 | φ 1 2 > ψ 1 | φ 0 2 ), by always computing the 2 × 2 eigenfunctions (both
orthogonal to φ 1 ) and taking the lowest current eigenfunction − as: φ
(m−)
0
= − .
(At any step, φ
(m−)
0
could be used as a new φ 0 to improve φ 1 via F 1 of Eq.
13.4.) In the above example (using the above He values), rotating φ 0 around φ 1
immediately gives φ
(1−)
0
= − = ψ 0 (and + = ψ 2 ). Below we shall see an
actual demonstration, performing a few “rotating” cycles.
481
and as a lower eigenvector φ
+
0 itself as improved φ 0 . This, by just a 2 × 2
diagonalization, is another simple criterion of being at the saddle point.
Incidentally, all other (small) components are less relevant, so that the opposite
procedure of optimizing φ 1 orthogonally to φ 0 can lead to φ MIN
1
, unpredictably
far from ψ 1 , even orthogonal to ψ 1 , with still E
φ MIN
1
<
∼
E [ψ 1 ], as shown
(as analysis) in the following counterexample: Even in the subspace {ψ 0 , ψ 1 ,
ψ 2 } (supposedly known for the present counterexample), the orthonormal trial
functions φ 0 = aψ 0 + 0ψ 1 + bψ 2 and φ 1 = bψ 0 − aψ 2 with a =
(E 1 −ε)−E 0
E 2 −E 0
,
b =
E 2 −(E 1 −ε)
E 2 −E 0
, (small ε), where, by construction, φ 0 is orthogonal to ψ 1 and
φ 1 is orthogonal to both ψ 1 and φ 0 , have energies φ 0 |H|φ 0 = a 2 E 0 + b 2 E 2 =,
E[φ 0 ] = E 0 + E 2 − (E 1 − ε) > E 0 , and E[φ 1 ] = E 1 − ε, while φ 0 may approximate
ψ 0 . (To make the counterexample numerical, consider He 1 S, for which it is known
that, in a.u., E 0 = − 2.903, E 1 = − 2.146, E 2 = − 2.06. In terms of {ψ 0 (=1s 2 ),
ψ 1 (=1s2s), ψ 2 (=1s3s)} (supposedly known for the present counterexample), the
function φ 0 = 0.9476 ψ 0 + 0.3194 ψ 2 would have E[φ 0 ] = − 2.832, and the
function φ 1 = 0.3194 ψ 0 − 0.9476 ψ 2 chosen orthogonal to both ψ 1 and φ 0
would have E[φ 1 ] = − 2.146 = E 1 (i.e., it would be a function with the energy of
ψ 1 , although orthogonal to ψ 1 ), so that, any function orthogonal to the same φ 0 ,
between this φ 1 and φ 1
+ = ψ 1 (if φ 0 is orthogonal to ψ 1 , the “closest” to ψ 1 is
ψ 1 itself), could be a minimization result, φ MIN
1
, with arbitrary
ψ 1 |φ MIN
1
and with
E 1 − ε ≤ E
φ MIN
1
≤ E 1 . In this counterexample, using F 1 this danger would
not exist: Minimization of F 1 , for the same, not particularly accurate, unvaried φ 0
(0.3194 is not very small), by varying φ 1 = c ψ 0 +d ψ 2 +ψ 1
√
1 − c 2 − d 2 , yields:
c < tol(=10 −8 ) and d < tol, i.e., φ 1 = ψ 1 , with E[φ 1 ] = − 2.146 [so that, from Eq.
13.14, φ
+
0 = φ 0 ]. Below we shall see an actual demonstration.
13.4.3 Further Improvement of φ 0
If E
φ
+
0
≤ E [φ 0 ], then, by rotating around φ 1 , as described above [after Eq.
13.13], since the Hamiltonian always opens the energy gap between mutually
orthogonal functions (all orthogonal to φ 1 ), φ
+
0 can be further improved (until
ψ 0 | φ 1 2 > ψ 1 | φ 0 2 ), by always computing the 2 × 2 eigenfunctions (both
orthogonal to φ 1 ) and taking the lowest current eigenfunction − as: φ
(m−)
0
= − .
(At any step, φ
(m−)
0
could be used as a new φ 0 to improve φ 1 via F 1 of Eq.
13.4.) In the above example (using the above He values), rotating φ 0 around φ 1
immediately gives φ
(1−)
0
= − = ψ 0 (and + = ψ 2 ). Below we shall see an
actual demonstration, performing a few “rotating” cycles.