13 If Truncated Wave Functions of Excited State Energy Saddle Points Are. . .
497
to the “low 2 × 2 root” φ 0 = (0.999 0− 0.048 ψ 1 ), with (indeed lower than
E[ 0 ] = −2.84766) energy: E[φ 0 ] = −2.84926 a.u. differing from the energy
of the exact orthogonal to ψ 1 (in the { 0 , ψ 1 } subspace [cf. Eq. 13.13]) by
−1.57 × 10 −9 a.u. (which means that the “low 2 × 2 root” φ 0 is actually orthogonal
to ψ 1 ). This verifies that ψ 1 is indeed very close to the exact excited saddle point
eigenfunction and that φ 0 , in order to further approach ψ 0 , needs be rotated only
orthogonally to ψ 1 .
Similarly, a 2 × 2 diagonalization between the F 1 27-term Φ 1 and the 1-term
0 yields: φ 1 = (ψ 1 –1.4 × 10 −4 0 ), E[φ 1 ] – E[ 1 ] = −1.34 × 10 −8 a.u.,
φ 0 = (0.999 0− 0.048 ψ 1 ), with lower energy: E[φ 0 ] = −2.84934 a.u. differing
from the energy of the exact orthogonal to 1 (in the { 0 , 1 } subspace [cf. Eq.
13.13]) by −1.34 × 10 −8 a.u., which verifies that the F 1 27-term 1 is also very
close to the exact saddle point, as expected.
Now, to check F 1 , a 2 × 2 diagonalization between the 8-term Φ 1 and the
1-term 0 indeed leaves 1 also almost unaffected, as “high root” φ 1 = (ψ 1 –
6.2 × 10 −4 0 ), E[φ 1 ] – E[ 1 ] = −2.69 × 10 −7 a.u., and, by opening their “gap,”
improves 0 to the “low root” φ 0 = (0.999 0− 0.052 ψ 1 ), with lower energy:
E[φ 0 ] = −2.84957 a.u. differing from the energy of the exact orthogonal to 1 (in
the { 0 , 1 } subspace [cf. Eq. 13.13]) by −2.69 × 10 −7 a.u., which means that
the “low 2 × 2 root” φ 0 is practically orthogonal to 1 and verifies that 1 1s2s is
indeed very close to the exact excited saddle point eigenfunction; φ 0 could still be
further improved by rotating it orthogonally to 1 .
Repeating the check with the 8-term HUM 2nd root 1s1s and the 1-term 0 ,
yields: φ 1 = (0.999 ψ 1 –1.2 × 10 −3 0 ), E[φ 1 ] – E[ 1 ] = −1.02 × 10 −6 a.u.,
φ 0 = (0.999 0− 0.049 ψ 1 ), with lower energy: E[φ 0 ] = −2.84937 a.u. differing
from the energy of the exact orthogonal to 1 (in the { 0 , 1 } subspace [cf. Eq.
13.13]) by −1.02 × 10 −6 a.u., which verifies that 8-term HUM 2nd root 1s1s is
indeed inferior than the F 1 8-term 1s2s function.
Secondly, as shown in Fig. 13.8, F 1 has local minimum, always above the
energy, while (thirdly) the energy E[ 1 ] is a saddle point at the minimum of F 1 ,
showing that it has approached the exact excited state 1s2s. For some parameters (z,
c, etc.), the energy has local minimum, while for others (a 1,2 ) it has local maximum.
13.6.1.6 Quick Check of Reasonableness via the Main Orbitals and via
2nd Derivatives
Figure 13.5 shows the main orbitals of both the 27-term and the 8-term optimized
wave functions of the HUM 2nd root and also of φ LE
1 (the lowest orthogonal to
φ 0 ) and of F 1 excited state solutions, compared to the ground state. Observe that
the HUM answer is 1s1s , instead of 1s2s (because a 1,2 ≈ 0, as mentioned above),
and the HUM 8-term function deviates from the 27-term ψ 1 , as anticipated in the
introduction (also the HUM 1s orbital is, unexpectedly, slightly more diffuse than
1s 2 ); therefore the HUM function is veered away from the exact, whereas the F 1
answer is indeed 1s2s, the F 1 8-term function already almost coincides with the
497
to the “low 2 × 2 root” φ 0 = (0.999 0− 0.048 ψ 1 ), with (indeed lower than
E[ 0 ] = −2.84766) energy: E[φ 0 ] = −2.84926 a.u. differing from the energy
of the exact orthogonal to ψ 1 (in the { 0 , ψ 1 } subspace [cf. Eq. 13.13]) by
−1.57 × 10 −9 a.u. (which means that the “low 2 × 2 root” φ 0 is actually orthogonal
to ψ 1 ). This verifies that ψ 1 is indeed very close to the exact excited saddle point
eigenfunction and that φ 0 , in order to further approach ψ 0 , needs be rotated only
orthogonally to ψ 1 .
Similarly, a 2 × 2 diagonalization between the F 1 27-term Φ 1 and the 1-term
0 yields: φ 1 = (ψ 1 –1.4 × 10 −4 0 ), E[φ 1 ] – E[ 1 ] = −1.34 × 10 −8 a.u.,
φ 0 = (0.999 0− 0.048 ψ 1 ), with lower energy: E[φ 0 ] = −2.84934 a.u. differing
from the energy of the exact orthogonal to 1 (in the { 0 , 1 } subspace [cf. Eq.
13.13]) by −1.34 × 10 −8 a.u., which verifies that the F 1 27-term 1 is also very
close to the exact saddle point, as expected.
Now, to check F 1 , a 2 × 2 diagonalization between the 8-term Φ 1 and the
1-term 0 indeed leaves 1 also almost unaffected, as “high root” φ 1 = (ψ 1 –
6.2 × 10 −4 0 ), E[φ 1 ] – E[ 1 ] = −2.69 × 10 −7 a.u., and, by opening their “gap,”
improves 0 to the “low root” φ 0 = (0.999 0− 0.052 ψ 1 ), with lower energy:
E[φ 0 ] = −2.84957 a.u. differing from the energy of the exact orthogonal to 1 (in
the { 0 , 1 } subspace [cf. Eq. 13.13]) by −2.69 × 10 −7 a.u., which means that
the “low 2 × 2 root” φ 0 is practically orthogonal to 1 and verifies that 1 1s2s is
indeed very close to the exact excited saddle point eigenfunction; φ 0 could still be
further improved by rotating it orthogonally to 1 .
Repeating the check with the 8-term HUM 2nd root 1s1s and the 1-term 0 ,
yields: φ 1 = (0.999 ψ 1 –1.2 × 10 −3 0 ), E[φ 1 ] – E[ 1 ] = −1.02 × 10 −6 a.u.,
φ 0 = (0.999 0− 0.049 ψ 1 ), with lower energy: E[φ 0 ] = −2.84937 a.u. differing
from the energy of the exact orthogonal to 1 (in the { 0 , 1 } subspace [cf. Eq.
13.13]) by −1.02 × 10 −6 a.u., which verifies that 8-term HUM 2nd root 1s1s is
indeed inferior than the F 1 8-term 1s2s function.
Secondly, as shown in Fig. 13.8, F 1 has local minimum, always above the
energy, while (thirdly) the energy E[ 1 ] is a saddle point at the minimum of F 1 ,
showing that it has approached the exact excited state 1s2s. For some parameters (z,
c, etc.), the energy has local minimum, while for others (a 1,2 ) it has local maximum.
13.6.1.6 Quick Check of Reasonableness via the Main Orbitals and via
2nd Derivatives
Figure 13.5 shows the main orbitals of both the 27-term and the 8-term optimized
wave functions of the HUM 2nd root and also of φ LE
1 (the lowest orthogonal to
φ 0 ) and of F 1 excited state solutions, compared to the ground state. Observe that
the HUM answer is 1s1s , instead of 1s2s (because a 1,2 ≈ 0, as mentioned above),
and the HUM 8-term function deviates from the 27-term ψ 1 , as anticipated in the
introduction (also the HUM 1s orbital is, unexpectedly, slightly more diffuse than
1s 2 ); therefore the HUM function is veered away from the exact, whereas the F 1
answer is indeed 1s2s, the F 1 8-term function already almost coincides with the