510
N. C. Bacalis
with E[φ 1 ] = −2.14593739906107 a.u., F 1 = −2.14593739906075, and, as φ 0 ,
a crude approximant 1s 2 , restricted s, with E[φ 0 ] = −2.847656 a.u. In their 2D
space their 2 × 2 overlap matrix, s ij , and Hamiltonian matrix, h ij , are s 11 = 1.0;
s 12 = s 21 = −0.047857; s 22 = 1.0 and h 11 = −2.847656; h 12 = h 21 = 0.102699;
h 22 = −2.145937, as expected, since φ 1 is very close to ψ 1 and (s 12 h 22 –
h 12 ) = 5 × 10 −8 (i.e., in other words: φ 0 |H|φ 1 = E 1 φ 0 | φ 1 ).
The (orthonormal) eigenvectors are indeed + = 1.2 10 −6 φ 0 + 1.00 φ 1 = ψ 1 ,
E + = E 1 = −2.145937 a.u. (unchanged, since it was already essentially a
Hamiltonian eigenfunction) and the improved − = 1.0011471 φ 0 + 0.047910626
φ 1 , E − = −2.8492667 a.u. This is equal to [also computed independently via
Eq. (13.14)] φ
+
0 exactly orthogonal to φ 1 = ψ 1 . By using a partially optimized
wave function f 2 = 2s3s, a function φ
(2+)
0
orthogonal to both {φ
+
0 , ψ 1 } was
obtained, then, in the 2 × 2 subspace of {φ
+
0 , φ
(2+)
0
} (both orthogonal to ψ 1 ),
the lowest Hamiltonian eigenvector − ≡ φ
−
0 = 0.09305114 φ 1 + 0.29149129
φ 0 + 0.715803 f 2 has energy E
φ
−
0
= − 2.8673305 a.u. By further introducing an
individually optimized wave function f 3 = 4s 2 , then − ≡ φ
2−
0 = 0.093057191
φ 1 + 0.29149218 φ 0 + 0.71580519 f 2 –0.0025808019 f 3 was obtained with
E 0
−2 = −2.8673481 a.u. By repeating with an individually optimized wave function
f 4 = 2p3p, then − ≡ φ
3−
0 = 0.09284577 φ 1 + 0.29145852 φ 0 + 0.71572254 f 2 –
0.0025805039 f 3 – 0.013914576 f 4 was obtained with E 0
3− = −2.8677801 a.u., and
so on. All these improvements were done immediately within negligible time.
13.7 Final Remark
The excited states are energy saddle points in the Hilbert space of wave functions.
To compute excited states with truncated wave functions, most standard methods
are based on HUM theorem, which says that, optimizing the desired higher root
of the secular equation approaches the correct energy from above; it does not say
anything about the quality of the wave function. However, the wave function is
needed if we want to understand the behavior of the electrons and to compute
other properties besides the energy. For large systems we need rather small and
comprehensible (truncated) wave functions. We have seen that using small wave
function expansions, HUM theorem, allows us to approach the energy (from above,
i.e., by minimization), but it prevents us from approaching the exact excited
state saddle point wave function. Thus, based on HUM theorem, we must use
large expansions (the larger, the better), which is impracticable for large systems.
We have also seen that the functional F n (Eq. 13.4) has local minimum at the
excited state saddle point, regardless of the size of the expansion. Therefore,
it could be used for large systems with small wave functions. We saw that it has
been tested for small atoms and demonstrated the above statements. So, it is worth
trying it for larger systems. The implementation needs the computation, beyond
the energy φ n |H|φ n , of simple matrix elements φ i < n | φ n and φ i < n |H|φ n , that
N. C. Bacalis
with E[φ 1 ] = −2.14593739906107 a.u., F 1 = −2.14593739906075, and, as φ 0 ,
a crude approximant 1s 2 , restricted s, with E[φ 0 ] = −2.847656 a.u. In their 2D
space their 2 × 2 overlap matrix, s ij , and Hamiltonian matrix, h ij , are s 11 = 1.0;
s 12 = s 21 = −0.047857; s 22 = 1.0 and h 11 = −2.847656; h 12 = h 21 = 0.102699;
h 22 = −2.145937, as expected, since φ 1 is very close to ψ 1 and (s 12 h 22 –
h 12 ) = 5 × 10 −8 (i.e., in other words: φ 0 |H|φ 1 = E 1 φ 0 | φ 1 ).
The (orthonormal) eigenvectors are indeed + = 1.2 10 −6 φ 0 + 1.00 φ 1 = ψ 1 ,
E + = E 1 = −2.145937 a.u. (unchanged, since it was already essentially a
Hamiltonian eigenfunction) and the improved − = 1.0011471 φ 0 + 0.047910626
φ 1 , E − = −2.8492667 a.u. This is equal to [also computed independently via
Eq. (13.14)] φ
+
0 exactly orthogonal to φ 1 = ψ 1 . By using a partially optimized
wave function f 2 = 2s3s, a function φ
(2+)
0
orthogonal to both {φ
+
0 , ψ 1 } was
obtained, then, in the 2 × 2 subspace of {φ
+
0 , φ
(2+)
0
} (both orthogonal to ψ 1 ),
the lowest Hamiltonian eigenvector − ≡ φ
−
0 = 0.09305114 φ 1 + 0.29149129
φ 0 + 0.715803 f 2 has energy E
φ
−
0
= − 2.8673305 a.u. By further introducing an
individually optimized wave function f 3 = 4s 2 , then − ≡ φ
2−
0 = 0.093057191
φ 1 + 0.29149218 φ 0 + 0.71580519 f 2 –0.0025808019 f 3 was obtained with
E 0
−2 = −2.8673481 a.u. By repeating with an individually optimized wave function
f 4 = 2p3p, then − ≡ φ
3−
0 = 0.09284577 φ 1 + 0.29145852 φ 0 + 0.71572254 f 2 –
0.0025805039 f 3 – 0.013914576 f 4 was obtained with E 0
3− = −2.8677801 a.u., and
so on. All these improvements were done immediately within negligible time.
13.7 Final Remark
The excited states are energy saddle points in the Hilbert space of wave functions.
To compute excited states with truncated wave functions, most standard methods
are based on HUM theorem, which says that, optimizing the desired higher root
of the secular equation approaches the correct energy from above; it does not say
anything about the quality of the wave function. However, the wave function is
needed if we want to understand the behavior of the electrons and to compute
other properties besides the energy. For large systems we need rather small and
comprehensible (truncated) wave functions. We have seen that using small wave
function expansions, HUM theorem, allows us to approach the energy (from above,
i.e., by minimization), but it prevents us from approaching the exact excited
state saddle point wave function. Thus, based on HUM theorem, we must use
large expansions (the larger, the better), which is impracticable for large systems.
We have also seen that the functional F n (Eq. 13.4) has local minimum at the
excited state saddle point, regardless of the size of the expansion. Therefore,
it could be used for large systems with small wave functions. We saw that it has
been tested for small atoms and demonstrated the above statements. So, it is worth
trying it for larger systems. The implementation needs the computation, beyond
the energy φ n |H|φ n , of simple matrix elements φ i < n | φ n and φ i < n |H|φ n , that