116
S.O. Adamson et al.
level for the basis functions with max(n
1 , n 1 , n
2 , n 2 ) ≤ 10. The check of the twoindexes recursion, which is analogous to the (5.22) [51], showed that this one provides a good precision only for integrals with max(n
1 , n 1 , n
2 , n 2 ) ≤ 13 when 128th
bit representation of real numbers is used. This is in accord with the remark in [51]
about the possibility of using this recursion for integral calculation with a good
accuracy max(n
1 , n 1 , n
2 , n 2 ) ≤ 14. As it was not known beforehand if in constructing a wave function one can select polynomials with n ≤ 13 only, the combined
scheme was used. Calculations have been done with the formulas (5.36)–(5.38), if
max(n
1 , n 1 , n
2 , n 2 ) < 11 and the quadrature integration being used in the opposite
case. Using the replacement of u = x and v = y − x, similar to [57], the integral
(5.35) can be represented as
R L =
∞
0
duf 1 (u)
∞
0
dvf 2 (u + v)
u L
(u + v) L+1
+
∞
0
duf 2 (u)
∞
0
dvf 1 (u + v)
u L
(u + v) L+1 .
(5.39)
Thus, it is possible to use Gauss-Laguerre integration for calculation (5.39). Note,
that the relation as
n
m=0 L α
m (x)L
β
n−m (y) = L
α+β+1
n
(x + y) [41] allows to factorize
the two-dimensional integral completely.
References
1. Nicolaides CA (2010) Theory and state-specific methods for the analysis and computation
of field-free and field-induced unstable states in atoms and molecules. In: Nicolaides CA,
Brändas E (eds) Unstable states in the continuous spectra, Part I: analysis, concepts, methods,
and results. Advances in quantum chemistry, vol 60. Elsevier, Amsterdam, pp 163–267, and
references therein
2. Moiseyev N (2011) Non-Hermitian quantum mechanics. Cambridge University Press, Cambridge
3. Nicolaides CA, Brändas E (eds) (2012) Unstable states in the continuous spectra, Part II:
interpretation, theory and applications. Advances in quantum chemistry, vol 63. Elsevier, Amsterdam
4. Sabelli NH, Gislason EA (1984) SCF study of the lowest 2 Σ +
u resonance of H
−
2 . J Chem Phys
81:4002–4007
5. DeRose E, Gislason EA, Sabelli NH (1985) A new method for computing properties of negative ion resonances with application to 2 Σ +
u states of H
−
2 . J Chem Phys 82:4577–4584
6. Chao JS-Y, Falcetta MF, Jordan KD (1990) Application of the stabilization method to the
N
−
2 (1 2 Π g ) and Mg − (1 2 Π) temporary anion states. J Chem Phys 93:1125–1135
7. Izmaylov AF, Adamson SO, Zaitsevskii A (2004) Multipartitioning many-body perturbation
theory calculations on temporary anions: applications to N
−
2 and CO − . J Phys B, Atom Mol
Phys 37:2321–2329
8. Izmaylov AF, Shchegoleva LN, Scuseria GE, Zaitsevskii A (2005) Ab initio study of temporary anions of benzene and fluorobenzenes using the multipartitioning many-body perturbation
theory. Phys Chem Chem Phys 7:3933–3937
9. Adamson S et al (2007) Multiscale multiphysics non-empirical approach to calculation of
light emission properties of chemically active non-equilibrium plasma: application to Ga-I3
system. J Phys D, Appl Phys 40:3857–3881
S.O. Adamson et al.
level for the basis functions with max(n
1 , n 1 , n
2 , n 2 ) ≤ 10. The check of the twoindexes recursion, which is analogous to the (5.22) [51], showed that this one provides a good precision only for integrals with max(n
1 , n 1 , n
2 , n 2 ) ≤ 13 when 128th
bit representation of real numbers is used. This is in accord with the remark in [51]
about the possibility of using this recursion for integral calculation with a good
accuracy max(n
1 , n 1 , n
2 , n 2 ) ≤ 14. As it was not known beforehand if in constructing a wave function one can select polynomials with n ≤ 13 only, the combined
scheme was used. Calculations have been done with the formulas (5.36)–(5.38), if
max(n
1 , n 1 , n
2 , n 2 ) < 11 and the quadrature integration being used in the opposite
case. Using the replacement of u = x and v = y − x, similar to [57], the integral
(5.35) can be represented as
R L =
∞
0
duf 1 (u)
∞
0
dvf 2 (u + v)
u L
(u + v) L+1
+
∞
0
duf 2 (u)
∞
0
dvf 1 (u + v)
u L
(u + v) L+1 .
(5.39)
Thus, it is possible to use Gauss-Laguerre integration for calculation (5.39). Note,
that the relation as
n
m=0 L α
m (x)L
β
n−m (y) = L
α+β+1
n
(x + y) [41] allows to factorize
the two-dimensional integral completely.
References
1. Nicolaides CA (2010) Theory and state-specific methods for the analysis and computation
of field-free and field-induced unstable states in atoms and molecules. In: Nicolaides CA,
Brändas E (eds) Unstable states in the continuous spectra, Part I: analysis, concepts, methods,
and results. Advances in quantum chemistry, vol 60. Elsevier, Amsterdam, pp 163–267, and
references therein
2. Moiseyev N (2011) Non-Hermitian quantum mechanics. Cambridge University Press, Cambridge
3. Nicolaides CA, Brändas E (eds) (2012) Unstable states in the continuous spectra, Part II:
interpretation, theory and applications. Advances in quantum chemistry, vol 63. Elsevier, Amsterdam
4. Sabelli NH, Gislason EA (1984) SCF study of the lowest 2 Σ +
u resonance of H
−
2 . J Chem Phys
81:4002–4007
5. DeRose E, Gislason EA, Sabelli NH (1985) A new method for computing properties of negative ion resonances with application to 2 Σ +
u states of H
−
2 . J Chem Phys 82:4577–4584
6. Chao JS-Y, Falcetta MF, Jordan KD (1990) Application of the stabilization method to the
N
−
2 (1 2 Π g ) and Mg − (1 2 Π) temporary anion states. J Chem Phys 93:1125–1135
7. Izmaylov AF, Adamson SO, Zaitsevskii A (2004) Multipartitioning many-body perturbation
theory calculations on temporary anions: applications to N
−
2 and CO − . J Phys B, Atom Mol
Phys 37:2321–2329
8. Izmaylov AF, Shchegoleva LN, Scuseria GE, Zaitsevskii A (2005) Ab initio study of temporary anions of benzene and fluorobenzenes using the multipartitioning many-body perturbation
theory. Phys Chem Chem Phys 7:3933–3937
9. Adamson S et al (2007) Multiscale multiphysics non-empirical approach to calculation of
light emission properties of chemically active non-equilibrium plasma: application to Ga-I3
system. J Phys D, Appl Phys 40:3857–3881