488
N. C. Bacalis
and
| = 2π
2
∞
0
s
0
u
0
1
p=0
(−1)
p s
2(1−p) t
2p u u u dt du ds
The integrals, of the form J (z s , z t ; n s , n t , n u ) =
∞
0
s
0
u
0 e −z s s−z t t s n s t n t
u n u dt du ds, are reported in [81].
13.6.1.1 F n Minimization Procedure
To minimize F n , the fixed lower approximants {φ i , i < n} (that can be crude, but
reasonable, needed to compute F n ) have been computed (independently of each
other, and not necessarily orthogonal to each other, since, in F n , Schrödinger’s
equation is “dotted” into them). For each (varied) value of the parameters z n and
a n, k , the critical points of F n (saddle points of E n ) are found, and F n is minimized
(without demanding orthogonality to the lower approximants .{φ i , i < n} Orthogonality, among the functions that minimize each F i , i < n, should be an outcome). In
finding the saddle points of E n , for certain trial values of the parameters z n and a n, k ,
the linear part, ∂ φ n | H |φ n /∂c i s ,2i t ,i u = 0, can be solved either by direct variation
of the c-coefficients or, preferably, by reducing to a generalized eigenvalue problem
(requiring φ n | φ n for normalization). Thus, the lowest eigenvalue (1st root) above
the known (highest) E[φ n−1 ] and the c-coefficients of its eigenvector are substituted
in Eq. 13.4 of F n , and improved values of z n and a n, k are sought until minimization
of F n .
13.6.1.2 Establishing “Exact” Wave Functions ψ 0 , ψ 1 and Truncated
Approximants φ 0 , φ 1
First a reliable wave function basis |ψ 0 ∼ = |0, |ψ 1 ∼ = |1 is established
by taking 27 terms, i.e., up to s 2 (t 2 ) 2 u 2 , (n s , n t , n u ) = (2, 2, 2), adequate to
achieve coincidence of the HUM and F 1 minima: (E[φ 0 ∼ψ 0 ] = −2.90371 a.u.,
z 0 = 1.9549), (E
ψ HUM
1
= −2.14584, z 1,1 = 1.8348, z 1,2 = 1.9745, a 1,2,1 = 0;
or E
ψ
F 1
1
= −2.14577, z 1,1 = 1.930501, z 1,2 = 1.827298, a 1,2,1 = 0.799760;
ψ
F 1
1 |ψ HUM
1
= 0.99996) – thus, going up to (n s , n t , n u ) = (2, 2, 2) is sufficient.
Then, in order to exhibit the aforesaid demonstrations, truncations up to s 1 (t 2 ) 1 u 1 ,
(n s , n t , n u ) = (1, 1, 1), i.e., 8 terms, will be used.
In F 1 , a fixed 1-term normalized φ 0, i.e., up to s 0 (t 2 ) 0 u 0 , (n s , n t , n u ) = (0, 0, 0)
(z 0 = 1.6875, E[φ 0 ] = −2.84766 a.u.), is used.
N. C. Bacalis
and
| = 2π
2
∞
0
s
0
u
0
1
p=0
(−1)
p s
2(1−p) t
2p u u u dt du ds
The integrals, of the form J (z s , z t ; n s , n t , n u ) =
∞
0
s
0
u
0 e −z s s−z t t s n s t n t
u n u dt du ds, are reported in [81].
13.6.1.1 F n Minimization Procedure
To minimize F n , the fixed lower approximants {φ i , i < n} (that can be crude, but
reasonable, needed to compute F n ) have been computed (independently of each
other, and not necessarily orthogonal to each other, since, in F n , Schrödinger’s
equation is “dotted” into them). For each (varied) value of the parameters z n and
a n, k , the critical points of F n (saddle points of E n ) are found, and F n is minimized
(without demanding orthogonality to the lower approximants .{φ i , i < n} Orthogonality, among the functions that minimize each F i , i < n, should be an outcome). In
finding the saddle points of E n , for certain trial values of the parameters z n and a n, k ,
the linear part, ∂ φ n | H |φ n /∂c i s ,2i t ,i u = 0, can be solved either by direct variation
of the c-coefficients or, preferably, by reducing to a generalized eigenvalue problem
(requiring φ n | φ n for normalization). Thus, the lowest eigenvalue (1st root) above
the known (highest) E[φ n−1 ] and the c-coefficients of its eigenvector are substituted
in Eq. 13.4 of F n , and improved values of z n and a n, k are sought until minimization
of F n .
13.6.1.2 Establishing “Exact” Wave Functions ψ 0 , ψ 1 and Truncated
Approximants φ 0 , φ 1
First a reliable wave function basis |ψ 0 ∼ = |0, |ψ 1 ∼ = |1 is established
by taking 27 terms, i.e., up to s 2 (t 2 ) 2 u 2 , (n s , n t , n u ) = (2, 2, 2), adequate to
achieve coincidence of the HUM and F 1 minima: (E[φ 0 ∼ψ 0 ] = −2.90371 a.u.,
z 0 = 1.9549), (E
ψ HUM
1
= −2.14584, z 1,1 = 1.8348, z 1,2 = 1.9745, a 1,2,1 = 0;
or E
ψ
F 1
1
= −2.14577, z 1,1 = 1.930501, z 1,2 = 1.827298, a 1,2,1 = 0.799760;
ψ
F 1
1 |ψ HUM
1
= 0.99996) – thus, going up to (n s , n t , n u ) = (2, 2, 2) is sufficient.
Then, in order to exhibit the aforesaid demonstrations, truncations up to s 1 (t 2 ) 1 u 1 ,
(n s , n t , n u ) = (1, 1, 1), i.e., 8 terms, will be used.
In F 1 , a fixed 1-term normalized φ 0, i.e., up to s 0 (t 2 ) 0 u 0 , (n s , n t , n u ) = (0, 0, 0)
(z 0 = 1.6875, E[φ 0 ] = −2.84766 a.u.), is used.