170
A.V. Glushkov
of the state becomes complex: E = E r − iΓ /2. The total problem is thus reduced
to the diagonalization of the energy complex matrix for a many-electron atom (see
details in Refs. [12, 59–63, 71–76]). The matrix of interaction operator should be
then calculated [6, 61, 75]:
E
(i−j)
= Re E
(i−j)
+ i Im E
(i−j) .
(9.25)
Further it is possible to limit the calculation accuracy to the relationship
Im E/ Re E (the justification for this approach is set out in detail, for example,
in [6, 75]). The components of the eigenvectors of states can be obtained as a result
of the diagonalization of the real part of the energy matrix. The imaginary part of
the energy matrix is as follows:
Im E ik =
n,m
ˆ
C
∗
in Im E
i−j
nm ˆ
C mk ,
(9.26)
where C is the matrix of eigenvectors. The widths of the resonances are determined
by the corresponding imaginary parts. As a result, this approach allows in principle
to calculate the characteristics of the Stark resonances in the spectrum of an arbitrary
multi-electron atom in a strong external electric field, which is a great interest to a
wide range of applications in modern atomic, molecular and laser physics, quantum
electronics, plasma physics and chemistry etc.
9.3 Calculation Results and Discussion
9.3.1 The Stark Resonances Energies and Widths of Hydrogen
Atom
The calculation results for the Stark resonances energies and widths of the ground
state hydrogen atom in the DC electric field with the strength ε = 0.04, 0.08, 0.10,
0.80 a.u. are presented in Tables 9.1 and 9.2. The comparison with earlier similar
results, obtained within the generalized WKB approximation, summation of divergent PT series, the numerical solution of the differential equations following from
expansion of the wave function over finite basis, a complex scaling plus B-spline
calculation [15–51] shows quite acceptable agreement.
The calculation results of the Stark resonances parameters for the excited state
H atom (n = 2, 5, 15) for different strength values are listed in Table 9.3. The comparison with earlier similar results, obtained within the summation of divergent PT
series, the numerical solution of the differential equations with using the finite basis
expansion of the wave function again shows acceptable agreement. It is important to
compare the theoretical values of the resonance energy and width for the H atom in
the field ε = 16.8 kV/cm with experimental data [4]. There is quite good agreement
between theory and experiment. Note that our results are obtained in the first PT
A.V. Glushkov
of the state becomes complex: E = E r − iΓ /2. The total problem is thus reduced
to the diagonalization of the energy complex matrix for a many-electron atom (see
details in Refs. [12, 59–63, 71–76]). The matrix of interaction operator should be
then calculated [6, 61, 75]:
E
(i−j)
= Re E
(i−j)
+ i Im E
(i−j) .
(9.25)
Further it is possible to limit the calculation accuracy to the relationship
Im E/ Re E (the justification for this approach is set out in detail, for example,
in [6, 75]). The components of the eigenvectors of states can be obtained as a result
of the diagonalization of the real part of the energy matrix. The imaginary part of
the energy matrix is as follows:
Im E ik =
n,m
ˆ
C
∗
in Im E
i−j
nm ˆ
C mk ,
(9.26)
where C is the matrix of eigenvectors. The widths of the resonances are determined
by the corresponding imaginary parts. As a result, this approach allows in principle
to calculate the characteristics of the Stark resonances in the spectrum of an arbitrary
multi-electron atom in a strong external electric field, which is a great interest to a
wide range of applications in modern atomic, molecular and laser physics, quantum
electronics, plasma physics and chemistry etc.
9.3 Calculation Results and Discussion
9.3.1 The Stark Resonances Energies and Widths of Hydrogen
Atom
The calculation results for the Stark resonances energies and widths of the ground
state hydrogen atom in the DC electric field with the strength ε = 0.04, 0.08, 0.10,
0.80 a.u. are presented in Tables 9.1 and 9.2. The comparison with earlier similar
results, obtained within the generalized WKB approximation, summation of divergent PT series, the numerical solution of the differential equations following from
expansion of the wave function over finite basis, a complex scaling plus B-spline
calculation [15–51] shows quite acceptable agreement.
The calculation results of the Stark resonances parameters for the excited state
H atom (n = 2, 5, 15) for different strength values are listed in Table 9.3. The comparison with earlier similar results, obtained within the summation of divergent PT
series, the numerical solution of the differential equations with using the finite basis
expansion of the wave function again shows acceptable agreement. It is important to
compare the theoretical values of the resonance energy and width for the H atom in
the field ε = 16.8 kV/cm with experimental data [4]. There is quite good agreement
between theory and experiment. Note that our results are obtained in the first PT