13 If Truncated Wave Functions of Excited State Energy Saddle Points Are. . .
509
He 3 S
He 3 S 1s3s (1st Excited State φ 1 )
The lowest state of He 3 S is 1s2s, obtainable by simply minimizing the energy. For
the 1st excited state φ 1 , as seen in Fig. 13.3 (a), again it is demonstrated that “large”
functions, HUM and F, are practically equivalent. The F “small” function (b) has
the same main orbitals, 1s and 3s, as both the “large” functions F (a) and HUM (d),
where the 2s just adds some static correlation correction near the nucleus (as well
as all higher orbitals of the “large” expansion (c)). But the HUM “small” expansion
(e), orthogonal to a deteriorated 1st root 3 S “1s2s” (f), has main orbitals 1s2s (with
opposite sign), while the 3s orbital acts as a static correlation correction of the total
wave function, to approach the correct energy. Thus, the HUM solution incorrectly
predicts, as “HOMO” orbital, the 2s instead of the 3s; therefore, it is misleading.
Of course, blindly considering as “LUMO” the 1st higher “unoccupied” orbital, i.e.,
the HUM 3s, is completely out of question.
The above conclusions are clearly confirmed by Fig. 13.4, where all HUM
functions have minimum, whereas all F functions, either “large” or “small,”
have energy saddle point. (Of course, the larger the expansion, the better is the
approximation, but for larger systems this is not feasible; rather “small” but reliable
functions are needed.)
Li
Similar results are found for Li: Several low-lying excited states of Li 2 S and 4 S
are shown in Table 13.1, where the main configuration, the energy E and the F n
value are displayed, along with the RMS extent of the main orbitals. The low-lying
HUM “small” functions were misleading. However, by appropriately changing the
basis, guided by the F n functional, much better HUM “small” functions were found,
used for final comparisons with the F n functions. This, as well as the corresponding
aforementioned results for He, suggests that small expansions for excited states
would be more correctly described in terms of more appropriate basis functions.
The last (9th) column shows the number of configurations along with the overlap
HUM| F n of the corresponding functions. Clearly, the larger the expansion, the
better the approximation; nevertheless, the 2nd derivatives in the eighth column
assert that the HUM functions always have energy minimum and therefore are not
the excited state saddle points, whereas the F n functions are saddle points.
13.6.4 Immediate Improvement of a Lowest State Approximant
A very accurate φ 1 approximant of ψ 1 was used for He 1 S 1s2s, in terms of 19
AVOLTOs up to 5 g with unrestricted s orbitals up to 4s, 53 CI terms, 263 SDs,
509
He 3 S
He 3 S 1s3s (1st Excited State φ 1 )
The lowest state of He 3 S is 1s2s, obtainable by simply minimizing the energy. For
the 1st excited state φ 1 , as seen in Fig. 13.3 (a), again it is demonstrated that “large”
functions, HUM and F, are practically equivalent. The F “small” function (b) has
the same main orbitals, 1s and 3s, as both the “large” functions F (a) and HUM (d),
where the 2s just adds some static correlation correction near the nucleus (as well
as all higher orbitals of the “large” expansion (c)). But the HUM “small” expansion
(e), orthogonal to a deteriorated 1st root 3 S “1s2s” (f), has main orbitals 1s2s (with
opposite sign), while the 3s orbital acts as a static correlation correction of the total
wave function, to approach the correct energy. Thus, the HUM solution incorrectly
predicts, as “HOMO” orbital, the 2s instead of the 3s; therefore, it is misleading.
Of course, blindly considering as “LUMO” the 1st higher “unoccupied” orbital, i.e.,
the HUM 3s, is completely out of question.
The above conclusions are clearly confirmed by Fig. 13.4, where all HUM
functions have minimum, whereas all F functions, either “large” or “small,”
have energy saddle point. (Of course, the larger the expansion, the better is the
approximation, but for larger systems this is not feasible; rather “small” but reliable
functions are needed.)
Li
Similar results are found for Li: Several low-lying excited states of Li 2 S and 4 S
are shown in Table 13.1, where the main configuration, the energy E and the F n
value are displayed, along with the RMS extent of the main orbitals. The low-lying
HUM “small” functions were misleading. However, by appropriately changing the
basis, guided by the F n functional, much better HUM “small” functions were found,
used for final comparisons with the F n functions. This, as well as the corresponding
aforementioned results for He, suggests that small expansions for excited states
would be more correctly described in terms of more appropriate basis functions.
The last (9th) column shows the number of configurations along with the overlap
HUM| F n of the corresponding functions. Clearly, the larger the expansion, the
better the approximation; nevertheless, the 2nd derivatives in the eighth column
assert that the HUM functions always have energy minimum and therefore are not
the excited state saddle points, whereas the F n functions are saddle points.
13.6.4 Immediate Improvement of a Lowest State Approximant
A very accurate φ 1 approximant of ψ 1 was used for He 1 S 1s2s, in terms of 19
AVOLTOs up to 5 g with unrestricted s orbitals up to 4s, 53 CI terms, 263 SDs,