502
N. C. Bacalis
13.6.2 Demonstration of Identifying a “Flipped Root”
Let the ground and 1st excited state wave functions of a hydrogen-like ion be
parametrized as
ψ 0 (z 0 ; r) = a 0 (z 0 ) e
−Z r z 0 , ψ 1 (z 1 , g; r) = a 1 (z 1 , g) e
−z 1 Z r/2 (1 − g Z r/2)
where Z is the nuclear charge, z 0 , z 1 , g are variational parameters, and a 0 (z 0 ),
a 1 (z 1 , g) are normalization constants. These functions are not orthonormal, unless
z 0 = 1, z 1 = 1, g = 1, when they form eigenfunctions of the Hamiltonian H ψ j (r) =
−ψ
j (r)/2 − ψ
j (r)/r − Zψ j (r)/r.
In their 2 × 2 subspace create an orthonormal basis
0 (r) = ψ 0 (r), , 1 (r) = (ψ 1 (r) − ψ 0 (r) ψ 0 |ψ 1 ) /
1 − ψ 0 |ψ 1 2 ,
whose overlap matrix is δ i, j . In diagonalizing their Hamiltonian matrix H i,j =
∞
0
4πr 2 i (r)H H j (r)dr, let the two normalized eigenfunctions be 1r (r), 2r (r),
with their eigenvalues (“roots”) depending on z 0 , z 1 , g.
Now, around z 0 ≈ 2 a root crossing occurs for a wide range around z 1 ≈ 1
and g ≈ 1. Near and “before” the crossing the continuation of ψ 0 (z 0 ; r) is 1r (r)
and the continuation of ψ 1 (z 1 , g; r) is 2r (r), whereas “beyond” the crossing the
continuation of ψ 1 (z 1 , g; r) is 1r (r) and the continuation of ψ 0 (z 0 ; r) is 2r (r). The
question is to decide, via F 1 , whether a given value (“point”) of (z 0 , z 1 , g), near the
crossing, is “before” or “beyond” the crossing, in order to use the continuation, ε, of
(always) E[ψ 1 (z 1 , g; r)] in an optimization algorithm. (In the present demonstration
Newton-Raphson (NR) is used: ε
:= (∂ε/∂z 0 , ∂ε/∂z 1 , ∂ε/∂g) = 0 is solved by
proceeding iteratively to a new point p + δp = p − J −1 · ε
– or less if the method
diverges – having started at some point p = (z 0 , z 1 , g), where J is the Jacobian
matrix, Hessian of ε [85].)
Thus, consider
F 1
φ 0 ;
nr
= E
nr
+ 2
(φ 0 | H | nr − E [ nr ] φ 0 | nr )
2
(E [ nr ] − E [φ 0 ])
1 − φ 0 | nr 2 , n = 1, 2
where the fixed predetermined (deliberately not very accurate) ground state approximant is φ 0
z 0
0 ; r
= a 0
z 0
0
e
−Z r z 0
0 with z 0
0 = 1–0.05. (By direct minimization,
F 1 [φ 0 ; 2r ] is minimized to F 1 → E[ 2r (r)] = E[ψ 1 (1, 1; r)] = − 0.125 at z 1 = 1,
g = 1, (z 0 ≈ 1).) As explained above in the theory (cf. Eq. 13.15), among the two
“roots,” the continuation of the excited state, near the crossing, is the one with the
lowest F 1 . Indeed, for Z = 1, using (first traditionally) blindly the “2nd root,” i.e.,
keeping ε to E[ 2r (r)], i.e., regardless of which n (n th root) the lowest F 1 suggests,
“root flipping” occurs: The sequence of Table 13.4 is obtained, even by using half
NR-step.
N. C. Bacalis
13.6.2 Demonstration of Identifying a “Flipped Root”
Let the ground and 1st excited state wave functions of a hydrogen-like ion be
parametrized as
ψ 0 (z 0 ; r) = a 0 (z 0 ) e
−Z r z 0 , ψ 1 (z 1 , g; r) = a 1 (z 1 , g) e
−z 1 Z r/2 (1 − g Z r/2)
where Z is the nuclear charge, z 0 , z 1 , g are variational parameters, and a 0 (z 0 ),
a 1 (z 1 , g) are normalization constants. These functions are not orthonormal, unless
z 0 = 1, z 1 = 1, g = 1, when they form eigenfunctions of the Hamiltonian H ψ j (r) =
−ψ
j (r)/2 − ψ
j (r)/r − Zψ j (r)/r.
In their 2 × 2 subspace create an orthonormal basis
0 (r) = ψ 0 (r), , 1 (r) = (ψ 1 (r) − ψ 0 (r) ψ 0 |ψ 1 ) /
1 − ψ 0 |ψ 1 2 ,
whose overlap matrix is δ i, j . In diagonalizing their Hamiltonian matrix H i,j =
∞
0
4πr 2 i (r)H H j (r)dr, let the two normalized eigenfunctions be 1r (r), 2r (r),
with their eigenvalues (“roots”) depending on z 0 , z 1 , g.
Now, around z 0 ≈ 2 a root crossing occurs for a wide range around z 1 ≈ 1
and g ≈ 1. Near and “before” the crossing the continuation of ψ 0 (z 0 ; r) is 1r (r)
and the continuation of ψ 1 (z 1 , g; r) is 2r (r), whereas “beyond” the crossing the
continuation of ψ 1 (z 1 , g; r) is 1r (r) and the continuation of ψ 0 (z 0 ; r) is 2r (r). The
question is to decide, via F 1 , whether a given value (“point”) of (z 0 , z 1 , g), near the
crossing, is “before” or “beyond” the crossing, in order to use the continuation, ε, of
(always) E[ψ 1 (z 1 , g; r)] in an optimization algorithm. (In the present demonstration
Newton-Raphson (NR) is used: ε
:= (∂ε/∂z 0 , ∂ε/∂z 1 , ∂ε/∂g) = 0 is solved by
proceeding iteratively to a new point p + δp = p − J −1 · ε
– or less if the method
diverges – having started at some point p = (z 0 , z 1 , g), where J is the Jacobian
matrix, Hessian of ε [85].)
Thus, consider
F 1
φ 0 ;
nr
= E
nr
+ 2
(φ 0 | H | nr − E [ nr ] φ 0 | nr )
2
(E [ nr ] − E [φ 0 ])
1 − φ 0 | nr 2 , n = 1, 2
where the fixed predetermined (deliberately not very accurate) ground state approximant is φ 0
z 0
0 ; r
= a 0
z 0
0
e
−Z r z 0
0 with z 0
0 = 1–0.05. (By direct minimization,
F 1 [φ 0 ; 2r ] is minimized to F 1 → E[ 2r (r)] = E[ψ 1 (1, 1; r)] = − 0.125 at z 1 = 1,
g = 1, (z 0 ≈ 1).) As explained above in the theory (cf. Eq. 13.15), among the two
“roots,” the continuation of the excited state, near the crossing, is the one with the
lowest F 1 . Indeed, for Z = 1, using (first traditionally) blindly the “2nd root,” i.e.,
keeping ε to E[ 2r (r)], i.e., regardless of which n (n th root) the lowest F 1 suggests,
“root flipping” occurs: The sequence of Table 13.4 is obtained, even by using half
NR-step.