102
S.O. Adamson et al.
Taking into account that all the listed problems of the UCS-stabilization method
will be solved only through systematic investigations of the well studied molecular
systems, the aim of this paper is to assess the influence of the UCS-potential parameters on the accuracy of the resonance energy and width calculation. It is expected
that the choice of the UCS-potential parameters (charge and radius of the uniformly
charged sphere) should also influence the accuracy of the estimated resonance parameters by analogy with the optical potential method [21, 22]. Two lowest 1 S resonances of H − below the threshold n = 2 were chosen as the computational examples
because both theoretical calculations performed by different methods [1, 23–34] and
measurements [35] are known for these resonances. The experiments give the energy and width (E res = 0.35092 ± 0.00048 and Γ = 0.00232 ± 0.0003 a.u.) for the
lowest resonance and prove the existence of one more resonance (E res ≈ 0.37403
a.u.) [35]. The energies and widths for the six 1 S resonances below the threshold n = 2 were estimated in the later theoretical works and for the first and second resonances E res = 0.35122025, Γ = 0.00173870 and E res = 0.373979753,
Γ = 9.0934 × 10 −5 a.u. [1, 33]. The energies and widths given in other mentioned works were calculated less accurately and differ from the above ones by
about 10 −4 a.u. [24–32, 34].
The theoretical background is described in the next section. Following parts contain the details of the Hamiltonian matrix construction and method used for the
resonance parameters calculation. In conclusion the obtained results and their discussion are presented.
5.2 Theory
Within the limit of the infinite nuclear mass approximation the Hamiltonian of the
H − system has the form
ˆ
H s =
2
i=1
ˆ
H s (i) +
e 2
r 12
,
(5.1)
where ˆ
H s (i) = − 2 Δ i /2m − e 2 /r i + V s (r i ), m is electron mass, e is elementary
charge, r i is distance between the nucleus and i-th electron, and r 12 is distance
between electrons. The external (stabilization) potential is given by the formula
V s (r i ) =
0,
r i < R,
V 0 − Z 0 e 2 /r i , r i ≥ R,
(5.2)
where V 0 and R are independent parameters, and Z 0 is determined by the condition
Z 0 = RV 0 .
Consider the application of the variational method for the calculation of the stationary eigenfunctions Ψ
LS with eigenvalues E lying in the continuum spectrum of
the system, which consists of a target and an electron. The brief notation LS is used
for the full set of the quantum numbers {α, L, S, L z , S z }, where L, S—are the total
S.O. Adamson et al.
Taking into account that all the listed problems of the UCS-stabilization method
will be solved only through systematic investigations of the well studied molecular
systems, the aim of this paper is to assess the influence of the UCS-potential parameters on the accuracy of the resonance energy and width calculation. It is expected
that the choice of the UCS-potential parameters (charge and radius of the uniformly
charged sphere) should also influence the accuracy of the estimated resonance parameters by analogy with the optical potential method [21, 22]. Two lowest 1 S resonances of H − below the threshold n = 2 were chosen as the computational examples
because both theoretical calculations performed by different methods [1, 23–34] and
measurements [35] are known for these resonances. The experiments give the energy and width (E res = 0.35092 ± 0.00048 and Γ = 0.00232 ± 0.0003 a.u.) for the
lowest resonance and prove the existence of one more resonance (E res ≈ 0.37403
a.u.) [35]. The energies and widths for the six 1 S resonances below the threshold n = 2 were estimated in the later theoretical works and for the first and second resonances E res = 0.35122025, Γ = 0.00173870 and E res = 0.373979753,
Γ = 9.0934 × 10 −5 a.u. [1, 33]. The energies and widths given in other mentioned works were calculated less accurately and differ from the above ones by
about 10 −4 a.u. [24–32, 34].
The theoretical background is described in the next section. Following parts contain the details of the Hamiltonian matrix construction and method used for the
resonance parameters calculation. In conclusion the obtained results and their discussion are presented.
5.2 Theory
Within the limit of the infinite nuclear mass approximation the Hamiltonian of the
H − system has the form
ˆ
H s =
2
i=1
ˆ
H s (i) +
e 2
r 12
,
(5.1)
where ˆ
H s (i) = − 2 Δ i /2m − e 2 /r i + V s (r i ), m is electron mass, e is elementary
charge, r i is distance between the nucleus and i-th electron, and r 12 is distance
between electrons. The external (stabilization) potential is given by the formula
V s (r i ) =
0,
r i < R,
V 0 − Z 0 e 2 /r i , r i ≥ R,
(5.2)
where V 0 and R are independent parameters, and Z 0 is determined by the condition
Z 0 = RV 0 .
Consider the application of the variational method for the calculation of the stationary eigenfunctions Ψ
LS with eigenvalues E lying in the continuum spectrum of
the system, which consists of a target and an electron. The brief notation LS is used
for the full set of the quantum numbers {α, L, S, L z , S z }, where L, S—are the total