**Author****Sergeev A. V.**, Sherstyouk A. I.**Title**- Higher orders and structure of perturbation theory series for screened Coulomb potential
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.82, no.4, April 1982, pp.1070-8. USSR
**Abstract**- An effective method is developed for calculating the higher orders in perturbation theory for potentials of the polynomial type. It is based on the transition to the problem of perturbation theory for the Fock operator with a purely discrete spectrum. Allowance for dynamic symmetry of the unperturbed operator permits one in this case to express the corrections of any order in perturbation theory as polynomials with rational coefficients. The method is employed to calculate the bound states of the Schroedinger equation with a screened Coulomb potential. The analytical properties of the solutions are investigated. Exact values of the energy in a broad range of variation of the screening parameter is obtained by the Pade technique. The real and imaginary parts of the energies of the quasi-stationary states are calculated.

**Author****Sergeev A. V.**, Sherstyuk A. I.**Institution**- S.I. Vavilov State Opt. Inst., USSR
**Title**- Higher-order perturbation theory for the bound states of the Dirac equation with Yukawa-type potential
**Source**- Yadernaya Fizika, vol.39, no.5, May 1984, pp.1158-64. USSR
**Abstract**- The method of calculating higher orders of perturbation theory based on perturbation of the Fock operator with a purely discrete spectrum is generalized to the case of the Dirac equation with a potential of the Yukawa type. Corrections to any order of perturbation theory in the energy of an arbitrary bound state are given as finite polynomials which are determined through recurrence relations derived from the dynamical symmetry of the unperturbed problem. The authors propose a modified Pade approximant method which can be used to transform a divergent perturbation series into a rapidly convergent sequence for all bound and quasistationary states of the system.

**Author**- Vainberg V. M., Mur V. D., Popov V. S.,
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., USSR
**Title**- Strong-field Stark effect
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.44, no.1, 10 July 1986, pp.9-12. USSR
**Abstract**- The Stark shifts and the widths of the ground and excited states of the
hydrogen atom are calculated. Two independent calculation methods are used: a
summation of a divergent perturbation-theory series and a 1/
*n*expansion.

**Author**- Vainberg V. M., Mur V. D., Popov V. S.,
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, USSR
**Title**- Stark effect for the Rydberg states of the hydrogen atom
**Source**- Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, Pis'Ma V Redaktsiyu, vol.46, no.5, 10 Sept. 1987, pp.178-82. USSR
**Abstract**- The Stark shifts and the widths of highly excited (
*n*=15-30) states of a hydrogen atom in a strong electric field, which is higher than the classical ionization threshold of an atom, are calculated. The results of the calculations are in agreement with the experiment.

**Author**- Vainberg V. M., Mur V. D., Popov V. S.,
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., USSR
**Title**- The hydrogen atom in a strong electric field
**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.93, no.2, Aug. 1987, pp.450-62. USSR
**Abstract**- A calculation is reported of Stark shifts and widths of the ground and first
excited states of the hydrogen atom in a strong electric field
*E*up to values of*E*comparable with the field strength on an atomic orbit. Two independent methods are used, namely, summation of divergent perturbation-theory series and the 1/*n*expansion. The two methods have a wide region of overlap (in the parameters*n*_{1},*n*_{2},*m*, and*E*). The calculations are confirmed by considering the Rydberg limit as n to infinity .

**Author**- Vainberg V. M., Mur V. D., Popov V. S.,
**Sergeev A. V.**, Shcheblykin A. V. **Institution**- Inst. of Theor. & Exp. Phys., USSR
**Title**- The 1/
*n*expansion in quantum mechanics **Source**- Teoreticheskaya i Matematicheskaya Fizika, vol.74, no.3, March 1988, pp.399-411. USSR
**Abstract**- The classical approximation (
*l*=*n*-1 to infinity ) for the energy in*E*^{(0)}and the semiclassical expansion in quantum mechanics are discussed. A recursive method is proposed for calculating the quantum corrections of arbitrary order to in*E*^{(0)}, this being valid for both bound and quasistationary states. The generalization of the method to states with an arbitrary number of nodes and the possibility of a more general choice of the parameter of the semiclassical expansion are considered. The method is illustrated by the example of the Yukawa and 'funnel' potentials and for the Stark effect in the hydrogen atom. These examples demonstrate the rapid convergence of the 1/*n*expansion even for small quantum numbers.

**Author**- Mur V. D.
Popov V. S.
**Sergeev A. V.**Shcheblykin A. V. **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- Stark resonances and scaling in Rydberg atoms.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.96, no.1, July 1989,

pp.91-106. USSR.

**Abstract**- The Stark shifts and the widths of highly excited states are calculated in a
strong electric field (n/sup 4/E approximately 1, n[right angle
bracket][right angle bracket]1) for an arbitrary atom (the difference between
the atomic field and the Coulomstrong field is taken into account by introducing
quantum defects in a parabolic basis). A correction to the Bohr-Sommerfeld
quantization rule, necessitated by the finite barrier penetration factor, is
introduced. The possibility of analytic continuation into the above-barrier
region is discussed. Scaling relations are obtained via a 1/
*n*-expansion for the near-threshold Stark resonances with quantum numbers n/sub 1/[right angle bracket][right angle bracket]1, n/sub 2/, and m approximately 1. The theoretical results are compared with the experimental data on the photoionization of atoms in the presence of a constant electric field E.

**Author**-
**Sergeev A. V.** **Institution**- S.I. Vavilov State Opt. Inst., USSR.
**Title**- 1/
*N*expansion for the three-body problem.

**Source**- Yadernaya Fizika, vol.50, no.4, Oct. 1989, pp.945-50. USSR.

**Abstract**- An expansion in powers of 1/
*N*, where*N*is the dimension of space, is used for the calculation of the energy of a quantum-mechanical three-body system. It is proved that the limit*N*to infinity corresponds to the classical motion of a rigid configuration of particles in four-dimensional space. The results of the summation of the series in 1/*N*are discussed in the case of an anharmonic oscillator the mu mu alpha muonic atom, and screened helium.

**Author**- Mur V. D.
Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Experimental Phys., Moscow Eng.-Phys. Inst., USSR.
**Title**- The 1/
*N*expansion in quantum mechanics.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.97, no.1, Jan. 1990,

pp.32-46. USSR.

**Abstract**- The authors consider the application of the 1/
*N*expansion to the calculation of energies (of both discrete levels and quasi-stationary states) and wave functions, as well as to the problem of two Coulomb centers and the three-body problem. They show that in many cases (for example, for the cc charmonium states) the method yields high accuracy even for low quantum numbers, including the ground state. The relation of these results to the properties of minimizing-uncertainty coherent states is discussed.

**Author**- Krepostnov PI.
**Sergeev A. V.**Utkin AB. **Title**- Optical properties of a dense xenon plasma in flashlamps.

**Source**- Optika i Spektroskopiya, vol.67, no.3, Sept. 1989, pp.538-41. USSR.

**Abstract**- Theoretical calculations of the continuous and line absorption coefficients of a xenon flashlamp at T=8000-20000 K are carried out. The results are compared to experimental data and possible causes of the discrepancy between calculations and these data are discussed.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.**Weinberg V. M. **Institution**- Int. Centre for Theor. Phys., Trieste, Italy.
**Title**- Strong-field Stark effect: perturbation theory and 1/
*n*expansion.

**Source**- Physics Letters A, vol.149, no.9, 15 Oct. 1990, pp.418-24. Netherlands.

**Abstract**- The Stark shifts and the widths of the ground and excited states of a
hydrogen atom are calculated. Two independent calculation methods are used: a
summation of divergent perturbation theory series and 1/
*n*expansion. The results of the calculations for the Rydberg (n[right angle bracket][right angle bracket]1) states are in agreement with the experiment.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.** **Institution**- Inst. Centre for Theor. Phys., Trieste, Italy.
**Title**- 1/
*n*-expansion and scaling for the Stark effect in Rydberg atoms.

**Source**- Physics Letters A, vol.149, no.9, 15 Oct. 1990, pp.425-30. Netherlands.

**Abstract**- The Stark shifts and widths of atomic states in an intense electric field are
calculated with the help of the 1/
*n*-expansion. A previous theory is generalized to highly excited states of an arbitrary atom. The scaling relations for near-threshold (E approximately=0) resonances are obtained which are in good agreement with experiment. The modification of the Bohr-Sommerfeld quantization rule due to the finite barrier penetrability is also considered.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exper. Phys., Moscow, USSR.
**Title**- Quantization rules with barrier penetrability.

**Source**- Journal of the Moscow Physical Society, vol.1, no.1, Feb. 1991, pp.15-31.

UK.

**Abstract**- A correction to the quasiclassical quantization rules that allows for the presence of a barrier factor is found. The equation obtained in the process determines both the position of the quasistationary level E/sub r/ and the level width Gamma . The results are compared with numerical solutions of the Schrodinger equation and with those obtained from an exactly solvable model. Finally, a generalization of the Gamow formula for systems with separable variables is derived.

**Author**- Vainberg V. M.
Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., USSR.
**Title**- The 1/
*n*expansion for a hydrogen atom in an external field.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.98, no.3, Sept. 1990,

pp.847-60. USSR.

**Abstract**- The authors construct a 1/
*n*expansion for a hydrogen atom in constant electric and magnetic fields. The summation of the 1/*n*expansion enables them with a large degree of accuracy to find the position and the width of atomic levels in a strong field. The method is generalized to resonances in a repulsive Coulomb center and to two-electron atoms.

**Author**- Popov V. S.
Mur V. D.
**Sergev**A. V. **Institution**- Inst. of Theoret. & Exp. Phys., Moscow, USSR.
**Title**- Generalization of the Gamow formula to the multidimensional case.

**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.53, no.9, May

1991, pp.433-6. USSR.

**Abstract**- The Gamow formula for a level width is generalized to the case of multidimensional systems with separable variables. A condition for the applicability of this approximation is found.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Experimental Phys., Moscow, USSR.
**Title**- Quantization rules for quasistationary states.

**Source**- Physics Letters A, vol.157, no.4-5, 29 July 1991, pp.185-91. Netherlands.

**Abstract**- The modification of the Bohr-Sommerfeld quantization rules, which is due to the barrier penetrability, is found. The equation obtained is valid for an arbitrary analytical potential U(x), obeying the quasiclassical conditions. It determines both the position E/sub r/ and the width Gamma of the quasistationary state. A generalization of the Gamow formula for multidimensional systems with separable coordinates is derived. A comparison with exactly solvable models as well as with numerical solutions of the Schrodinger equation for the Stark problem is performed.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.** **Institution**- Moscow Eng. Phys. Inst., USSR.
**Title**- Quantization rules with allowance for barrier penetration.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.100, no.1, July 1991,

pp.20-44. USSR.

**Abstract**- The semiclassical quantization rules are corrected with allowance for barrier penetration. The resulting equation determines both the position E/sub r/ and the width Gamma of the quasistationary level. A comparison is made with numerical solutions of the Schrodinger equation and with exactly solvable models. The generalization of the Gamov formula for systems with separating variables is found. The possibility of generalizing the results obtained to the relativistic case is discussed.

**Author**- Mur V. D.
Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Generalization of the Gamow formula to the multidimensional case.

**Source**- Yadernaya Fizika, vol.54, no.4, Oct. 1991, pp.950-61. USSR.

**Abstract**- A generalization of the Gamow formula for the width Gamma of a quasistationary level (with energy E=E/sub r/-i Gamma /2) is given for the case of multidimensional systems with separable variables. The condition for applicability of this approximation is obtained, and some examples are considered.

**Author**-
**Sergeev A. V.** **Institution**- State Opt. Inst., Russia.
**Title**- Positron-nucleus resonances in electric and magnetic fields.

**Source**- Yadernaya Fizika, vol.54, no.5, Nov. 1991, pp.1225-31. Russia.

**Abstract**- Resonance states of a positron in the Coulomb field of a nucleus and in
parallel homogeneous electric and magnetic fields are studied. In the
classical description, the positron undergoes oscillatory motion along the
symmetry axis, being reflecting consecutively from the nucleus and the anode.
The magnetic field does not permit the positron to go around the nucleus and
move to the cathode, stabilizing the resonances. The nucleus is regarded as
infinitely heavy, and the problem reduces to a one-particle Schrodinger
equation. The energies and widths of the resonances are determined by means
of versions of perturbation theory in weak electric and b magnetic
fields, the 1/
*n*expansion, and the quasiclassical approximation. Different methods are in good agreement with one another.

**Author**- Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Large orders of the 1/
*n*expansion in quantum mechanics.

**Source**- Physics Letters A, vol.172, no.4, 4 Jan. 1993, pp.193-8. Netherlands.

**Abstract**- The asymptotics of large orders of the 1/
*n*expansion in quantum mechanics has been found. It is shown that the coefficients epsilon /sup (k)/ grow as k]a/sup k/ with k to infinity , and the dependence of the parameter a on the coupling constant is investigated.

**Author**- Popov V. S.
**Sergeev A. V.**Shcheblykin A. V. **Institution**- Inst. of Theor. & Exp. Phys., Russia.
**Title**- Structure of higher orders of the 1/
*n*expansion.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.102, no.5, Nov. 1992,

pp.1453-68. Russia.

**Abstract**- The asymptote of higher orders of the 1/
*n*expansion in quantum mechanical problems is found. It is shown that the coefficients epsilon /sup (k)/ of the 1/*n*expansion grow factorially like k] a/sup k/ as k to infinity . The parameter a as a function of the coupling constant is studied. The derived analytic expressions agree with numerical calculations. As examples, the Yukawa, Hulthen and V funnel potentials are considered, as well as the Stark effect in the hydrogen atom and the molecular ion H/sub 2//sup +/.

**Author**- Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Russian Acad. of Sci., Moscow, Russia.
**Title**- Asymptotic form of higher orders of the 1/
*n*expansion.

**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.57, no.5, 10

March 1993, pp.273-6. Russia.

**Abstract**- The asymptotic form of higher orders of the 1/
*n*expansion in quantum mechanics is factorial. The Yukawa potential and the hydrogen atom in electric and magnetic fields are discussed.

**Author**- Mur V. D.
Popov V. S.
**Sergeev A. V.** **Institution**- Moscow Eng. Phys. Inst., Russia.
**Title**- The 1/
*n*-expansion in quantum mechanics and quasi-stationary states.

**Source**- Journal of the Moscow Physical Society, vol.2, no.3, Aug. 1992, pp.189-217.

UK.

**Abstract**- Applications of the 1/
*n*-expansion to energy and wavefunction calculations, especially for quasi-stationary states are considered. As an illustration the authors have examined the funnel and Yukawa potentials, the Stark effect in a strong field and a hydrogen atom in electric and magnetic fields. They have found that in many cases the method ensures high accuracy, even for small quantum number, n approximately 1. The connection between these results and the properties of coherent states is briefly discussed. Finally, the behaviour of the large-order terms in the 1/*n*-expansion is investigated.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Critical fields and above-barrier Stark resonances.

**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.59, no.3, 10

Feb. 1994, pp.150-3. Russia.

**Abstract**- Exact values of the critical field C/sub c/ are calculated for a hydrogen atom, including the case of the ground state. The widths Gamma /sub n/ of Stark resonances at C=C/sub c/, are also calculated. In the above-barrier region (C[right angle bracket]C/sub c/ the widths Gamma /sub n/(C) are essentially linear functions of the electric field strength.

**Author**- Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Asymptotic form of higher orders of the 1/
*n*expansion for multidimensional

problems.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.105, no.3, March 1994,

pp.568-91. Russia.

**Abstract**- The asymptotic form of higher orders of the 1/
*n*expansion is analyzed for multidimensional problems of quantum mechanics and atomic physics. Problems in which variables can and cannot be separated are discussed. Examples of the former are the molecular hydrogen atom and the Stark effect; an example of the latter is the hydrogen atom in electric and magnetic fields. The asymptotic form is always factorial. Its parameters can be calculated through a calculation of electron tunneling trajectories by the "imaginary time" method. Possible types of singularities of the asymptotic parameter alpha at the collision point of the classical solutions are determined.

**Author**- Popov V. S.
**Sergeev A. V.**Mur V. D. Shcheblykin A. V. **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- On the asymptotics of high-order terms of the 1/
*n*expansion.

**Source**- Yadernaya Fizika, vol.57, no.6, June 1994, pp.1119-28. Russia.

**Abstract**- Analytic and numerical methods for determining the asymptotics of high-order
terms of the 1/
*n*expansion in quantum-mechanical problems are developed. It is shown that this asymptotics is always of the factorial type. The dependence of parameters of the asymptotics on the form of the potential and on the coupling constant is especially analyzed in the vicinity of the point of collision of classical solutions.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Critical electric fields and Stark resonances in the hydrogen atom.

**Source**- Physics Letters A, vol.193, no.2, 26 Sept. 1994, pp.159-64. Netherlands.

**Abstract**- The exact values of critical electric fields E/sub c/ for different states of a hydrogen atom, including its ground state, as well as the widths Gamma /sub n/ of the Stark resonances at E=E/sub c/, are calculated (at E=E/sub c/ (n/sub 1/, n/sub 2/, m) the potential barrier for an electron in the (n/sub 1/, n/sub 2/, m) state disappears). Using the modified quantization condition with barrier penetrability included, we explain the dependence of Gamma /sub n/ on the principal quantum number n.

**Author**- Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Large orders of 1/
*n*-expansion for multidimensional problems.

**Source**- Physics Letters A, vol.193, no.2, 26 Sept. 1994, pp.165-72. Netherlands.

**Abstract**- The asymptotics of large orders of the 1/
*n*-expansion is investigated for multidimensional problems of quantum mechanics and atomic physics, including those with separable variables (the hydrogen molecular ion H/sub 2//sup +/), and those where separation of variables is impossible (a hydrogen atom in electric and magnetic fields). It is shown that the parameters of the asymptotics can be found by means of calculating sub-barrier trajectories with the help of the "imaginary time" method, as well as by solution of the eikonal equation.

**Author**- Popov V. S.
Mur V. D.
**Sergeev A. V.** **Institution**- Int. Inst. of Phys., Moscow, Russia.
**Title**- Theory of the Stark effect in a strong field: critical fields, above-barrier

resonances, dependence on dimensionality.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.106, no.4, Oct. 1994,

pp.1001-30. Russia.

**Abstract**- We have calculated accurate values of the critical fields E/sub c/ for
various states (n/sub 1/, n/sub 2/, m) of a hydrogen atom, including the
ground state. Using modified quantization rules that take into account the
finite permeability of the barrier, we determine how the widths Gamma /sub n/
of the Stark resonances depend on n for E=E/sub c/. Using the 1/
*n*expansion, along with several model problems as examples, we show that in the above-barrier (E[right angle bracket]E/sub c/) region the Stark widths Gamma /sub n/(E) depend almost linearly on the electric field. We determine the dependence of the ground-state Stark shift of a d-dimensional "hydrogen atom" on the dimensionality D of the space.

**Author**-
**Sergeev A. V.** **Institution**- S.I. Vavilov State Optical Inst., St. Petersburg, Russia.
**Title**- Summation of the eigenvalue perturbation series by multi-valued Pade

approximants: application to resonance problems and double wells.

**Source**- Journal of Physics A-Mathematical & General, vol.28, no.14, 21 July 1995,

pp.4157-62. UK.

**Abstract**- Quadratic Pade approximants are used to obtain energy levels both for the anharmonic oscillator x/sup 2//2- lambda x/sup 4/ and for the double well -x/sup 2//2+ lambda x/sup 4/. In the first case, the complex-valued energy of the resonances is reproduced by summation of the real terms of the perturbation series. The second case is treated formally as an anharmonic oscillator with a purely imaginary frequency. We use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function. Then, we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants. In this way we can reconstruct the real energy by summation of the complex series. Such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory (around the minima of the double well potential) incorrect.

**Author**- Mur V. D.
Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Coulomb corrections to nuclear scattering lengths and effective ranges for

weakly bound systems.

**Source**- Yadernaya Fizika, vol.59, no.1, Jan. 1996, pp.68-77. Publisher: MAIK

Nauka/Interperiodica Publishing, Russia.

**Abstract**- A procedure is considered for extracting the purely nuclear scattering length a/sub s/ and effective range r/sub s/ (which correspond to a strong-interaction potential V/sub s/ with disregarded Coulomb interaction) from the experimentally determined nuclear quantities a/sub cs/ and r/sub cs/ which are modified by Coulomb interaction. The Coulomb renormalization of a/sub s/ and r/sub s/ is especially strong if the system under study involves a level with energy close to zero (on the nuclear scale). This applies to formulas that determine the Coulomb renormalization of the low energy parameters of s scattering (l=0). Detailed numerical calculations are performed for coefficients appearing in the equations that determine Coulomb corrections for various models of the potential V/sub s/(r). This makes it possible to draw qualitative conclusions about the dependence of Coulomb corrections on the form of the strong-interaction potential and, in particular, on its small-distance behavior. A considerable enhancement of Coulomb corrections to the effective range r/sub s/ is found for potentials with a barrier.

**Author**- Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Effect of a magnetic field on the ionization of atoms.

**Source**- Pis'Ma V Zhurnal Eksperimental'Noi i Teoreticheskoi Fiziki, vol.63, no.6, 25

March 1996, pp.398-402. Publisher: AIP, Russia.

**Abstract**- The ionization probability of an atomic s state under the action of static electric and magnetic fields is calculated taking into account the Coulomb interaction between the escaping electron and the atomic core. The structure of the perturbation series for the energy of the level is investigated and the asymptotic behavior of the higher orders of the perturbation theory is found.

**Author**- Mur V. D.
Popov V. S.
**Sergeev A. V.** **Institution**- Eng. Phys. Inst., Acad. of Sci., Moscow, Russia.
**Title**- Coulomb corrections to scattering lengths and effective ranges for l not=0.

**Source**- Yadernaya Fizika, vol.59, no.11, Nov. 1996, pp.1958-65. Publisher: MAIK

Nauka/Interperiodica Publishing, Russia.

**Abstract**- Coulomb corrections to the low-energy scattering parameters (scattering lengths and effective ranges) of states with nonzero orbital angular momentum are studied. The greatest attention is paid to the case of the p wave. Resonance scattering is considered for a system with a shallow nuclear bound or quasistationary (resonance) p level. Coulomb corrections are calculated numerically, and their dependence on the shape of the strong-interaction potential-in particular, on its short-distance behavior-is studied in detail.

**Author**-
**Sergeev A. V.**Goodson DZ. **Institution**- Dept. of Chem., Southern Methodist Univ., Dallas, TX, USA.
**Title**- Self-consistent field perturbation theory of molecular vibrations.

**Source**- Molecular Physics, vol.93, no.3, 20 Feb. 1998, pp.477-84. Publisher: Taylor &

Francis, UK.

**Abstract**- Perturbation theory is used to perform non-iterative calculations of energy eigenvalues of the coupled ordinary differential equations that result from imposing separability assumptions in terms of normal coordinates on vibrational wavefunctions. Various model Hamiltonians with 2 or 3 coupled normal modes are studied and the increase of computational cost with the number of degrees of freedom is analysed. Quadratic Pade approximants of the perturbation expansions are rapidly convergent, and directly yield complex numbers for resonance eigenvalues. For a 3-mode system, results are obtained within partial separability assumptions, with a pair of modes left coupled. Large-order perturbation theory with partial separability is suggested as an alternative to low-order exact perturbation theory.

**Author**-
**Sergeev A. V.**Goodson DZ. **Institution**- Dept. of Chem., Southern Methodist Univ., Dallas, TX, USA.
**Title**- Summation of asymptotic expansions of multiple-valued functions using

algebraic approximants: application to anharmonic oscillators.

**Source**- Journal of Physics A-Mathematical & General, vol.31, no.18, 8 May 1998,

pp.4301-17. Publisher: IOP Publishing, UK.

**Abstract**- The divergent Rayleigh-Schrodinger perturbation expansions for energy eigenvalues of cubic, quartic, sextic and octic oscillators are summed using algebraic approximants. These approximants are generalized Pade approximants that are obtained from an algebraic equation of arbitrary degree. Numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree. Different branches of the approximants converge to different branches of the function. The success of the high-degree approximants is attributed to their ability to model the function on multiple sheets of the Riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter. An efficient recursive algorithm for computing the diagonal approximant sequence is presented.

**Author**- Popov V. S.
**Sergeev A. V.** **Institution**- Inst. of Theor. & Exp. Phys., Moscow, Russia.
**Title**- Ionization of atoms in weak fields and the asymptotic behavior of

higher-order perturbation theory.

**Source**- Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki, vol.113, no.6, June 1998,

pp.2047-55. Publisher: AIP, Russia.

**Abstract**- Using the imaginary time method, we study the structure of the perturbation series for the hydrogen atom in electric E and magnetic H fields. It is shown that there is a "critical" value of the ratio gamma =H/E at which the perturbation series for the ground state changes from having a fixed sign (for gamma [left angle bracket] gamma /sub c/) to having a variable sign (for gamma [right angle bracket] gamma /sub c/). This conclusion is confirmed by direct higher-order perturbation calculations. The change in the asymptotic regime is explained by competition among the contributions of the various complex trajectories that describe the subbarrier motion of the electrons. Here the parameter gamma /sub c/ depends on the angle theta between the electric and magnetic fields.

**Author**- Elout MO.
Goodson DZ.
Elliston CD.
Shi-Wei Huang.
**Sergeev A. V.**Watson DK. **Institution**- Dept. of Chem., Southern Methodist Univ., Dallas, TX, USA.
**Title**- Improving the convergence and estimating the accuracy of summation

approximants of 1/D expansions for Coulombic systems.

**Source**- Journal of Mathematical Physics, vol.39, no.10, Oct. 1998, pp.5112-22.

Publisher: AIP, USA.

**Abstract**- The convergence of large-order expansions in delta =1/D, where D is the dimensionality of coordinate space, for energies E( delta ) of Coulomb systems is strongly affected by singularities at delta =1 and delta =0. Pade-Borel approximants with modifications that completely remove the singularities at delta =1 and remove the dominant singularity at delta =0 are demonstrated. A renormalization of the interelectron repulsion is found to move the dominant singularity of the Borel function F( delta )= Sigma jE/sub j/'/j], where E/sub j//sup '/ are the the expansion coefficients of the energy with singularity structure removed at delta =1, farther from the origin and thereby accelerate summation convergence. The ground-state energies of He and H/sub 2//sup +/ are used as test cases. The new methods give significant improvement over previous summation methods. Shifted Borel summation using F/sub m/( delta )= Sigma jE/sub j/'/ Gamma (j+1-m) is considered. The standard deviation of results calculated with different values of the shift parameter m is proposed as a measure of summation accuracy.

**Author**-
**Sergeev A. V.**Goodson DZ. **Institution**- Dept. of Chem., Southern Methodist Univ., Dallas, TX, USA.
**Title**- Semiclassical self-consistent field perturbation theory for the hydrogen atom

in a magnetic field.

**Source**- International Journal of Quantum Chemistry, vol.69, no.2, 5 Aug. 1998,

pp.183-92. Publisher: Wiley, USA.

**Abstract**- A recently developed perturbation theory for solving self-consistent field equations is applied to the hydrogen atom in a strong magnetic field. This system has been extensively studied using other methods and is therefore a good test case for the new method. The perturbation theory yields summable large-order expansions. The accuracy of the self-consistent field approximation varies according to field strength and quantum state but is often higher than the accuracy from adiabatic approximations. A new derivation is presented for the asymptotic adiabatic approximation, the most useful of the adiabatic approaches. This derivation uses semiclassical perturbation theory without invoking an adiabatic hypothesis.

**Author**- Goodson DZ.
**Sergeev A. V.** **Institution**- Dept. of Chem., Southern Methodist Univ., Dallas, TX, USA.
**Title**- On the use of algebraic approximants to sum divergent series for Fermi

resonances in vibrational spectroscopy.

**Source**- Journal of Chemical Physics, vol.110, no.16, 22 April 1999, pp.8205-6.

Publisher: AIP, USA.

**Abstract**- Pade summation of large-order perturbation theory can often yield highly accurate energy eigenvalues for molecular vibrations. However, for eigenstates involved in Fermi resonances the convergence of the Pade approximants can be very slow. This is because the energy is a multivalued function of the perturbation parameter while Pade approximants are single valued, and Fermi resonances occur when a branch point lies close to the physical value of the parameter. Algebraic approximants are multivalued generalizations of Pade approximants. Using the (200) state of H/sub 2/S and the (400) state of H/sub 2/O as examples of Fermi resonances, it is demonstrated here that algebraic approximants greatly improve the summation convergence.

**Author**-
**Sergeev A. V.**Kais S. **Institution**- Dept. of Chem., Purdue Univ., West Lafayette, IN, USA.
**Title**- Variational principle for critical parameters of quantum systems.

**Source**- Journal of Physics A-Mathematical & General, vol.32, no.39, 1 Oct. 1999,

pp.6891-6. Publisher: IOP Publishing, UK.

**Abstract**- The variational principle for eigenvalue problems with a nonidentity weight operator is used to establish upper or lower bounds on critical parameters of quantum systems. Three problems from atomic physics are considered as examples. Critical screening parameters for the exponentially screened Coulomb potential are found using a trial function with one nonlinear variational parameter. The critical charge for the helium isoelectronic series is found using a Hylleraas-type trial function. Finally, critical charges for the same system subjected to a magnetic field are found using a product of two hydrogen-like basis sets.

**Author**-
**Sergeev A. V.**Kais S. **Institution**- Dept. of Chem., Purdue Univ., West Lafayette, IN, USA.
**Title**- Critical nuclear charges for N-electron atoms.

**Source**- Wiley. International Journal of Quantum Chemistry, vol.75, no.4-5, 15 Nov.-5

Dec. 1999, pp.533-42. USA.

**Conference Information**- International Symposium on Atomic, Molecular, and Condensed Matter Theory. St

Augustine, FL, USA. 27 Feb.-5 March 1999.

**Abstract**- One-particle model with a spherically-symmetric screened Coulomb potential is proposed to describe the motion of a loosely bound electron in a multielectron atom when the nuclear charge, which is treated as a continuous parameter, approaches its critical value. The critical nuclear charge, Z/sub c/, is the minimum charge necessary to bind N electrons. Parameters of the model are chosen to meet known binding energies of the neutral atom and the isoelectronic negative ion. This model correctly describes the asymptotic behavior of the binding energy in the vicinity of the critical charge, gives accurate estimation of the critical charges in comparison with ab initio calculations for small atoms, and is in full agreement with the prediction of the statistical theory of large atoms. Our results rule out the stability of doubly charged atomic negative ions in the gas phase. Moreover, the critical charge obeys the proposed inequality, N-2[left angle bracket]or=Z/sub c/[left angle bracket]or=N-1. We show that in the presence of a strong magnetic field many atomic dianions become stable.

**Author**-
**Sergeev A. V.**Kais S. **Institution**- Dept. of Chem., Purdue Univ., West Lafayette, IN, USA.
**Title**- Resonance states of atomic anions.

**Source**- International Journal of Quantum Chemistry, vol.82, no.5, 20 April 2001,

pp.255-61. Publisher: Wiley, USA.

**Abstract**- We study destabilization of an atom in its ground state with decrease of its
nuclear charge. By analytic continuation from bound to resonance states, we
obtain complex energies of unstable atomic anions with nuclear charge that is
less than the minimum "critical" charge necessary to bind N electrons. We use
an extrapolating scheme with a simple model potential for the electron, which
is loosely bound outside the atomic core. Results for O
^{2-}and S^{2-}are in good agreement with earlier estimates. Alternatively, we use the Hylleraas basis variational technique with three complex nonlinear parameters to find accurately the energy of two-electron atoms as the nuclear charge decreases. Results are used to check the less accurate one-electron model.

**Author**-
**Sergeev A. V.**Segev B. **Institution**- Dept. of Chem., Ben-Gurion Univ. of the Negev, Beer-Sheva, Israel.
**Title**- Most probable path in phase space for a radiationless transition in a molecule.

**Source**- Journal of Physics A-Mathematical & General, vol.35, no.7, 22 Feb. 2002, pp.1769-89.

**Abstract**- We study a radiationless transition in a polyatomic molecule when the electronic energy of an excited electronic state is transferred to the vibrational degrees of freedom of the nuclei, and when some nuclear coordinates change abruptly. This jump between the donor energy surface and the acceptor one gives the initial conditions for the subsequent dynamics on the acceptor surface, and the partition of energy between competing accepting modes. In the Wigner representation, the physical problem of recognizing the accepting modes for a radiationless vibronic relaxation reduces to the mathematical problem of finding the maximum of a function of many variables under a constraint. The function is the initial Wigner function of the nuclei and the constraint is energy conservation. In a harmonic approximation for the potential surfaces, the problem is equivalent to finding the distance from a given point to a multidimensional ellipsoid. This geometrical problem is solved in closed form. For nonharmonic potentials, the optimization problem is solved perturbatively.

**Author**- Kallush S., Segev B.,
**Sergeev A. V.** **Institution**- Dept. of Chem., Ben-Gurion Univ. of the Negev, Beer-Sheva, Israel.
**Title**- Surface jumping: Franck-Condon factors and Condon points in phase space.

**Source**- Journal of Physical Chemistry, vol.106, no.25, 27 June 2002, pp.6006-16.

**Abstract**- We generalize the concept of Tully-Preston surface hopping to include larger jumps in the case that the surfaces do not cross. Instead of identifying a complex hopping point, we specify a jump between two locales in phase space. This concept is used to find propensity rules for the accepting vibrational mode(s) in a radiationless vibronic relaxation transition. A model inspired by the S/sub 2/ to S/sub 0/ vibronic relaxation transition of the benzene molecule in which 30 modes of vibration compete for the electronic energy is studied within this approach. For this model, we show that almost all of the energy must go to a single C-H local stretching. The initial conditions for vibrations of this mode are a coordinate jump of the hydrogen atom toward the ring. All of the other modes undergo an almost vertical transition, in which the energy that they take is determined by their equilibrium displacement between the two surfaces. We observe that for a large energy gap the masses and frequencies become the defining parameters for choosing the accepting mode. Anharmonicities are very important when a competition between degenerate modes occurs. These conclusions are demonstrated by the specific model considered here but apply in general to any weak internal conversion process.

**Author**- Heller EJ, Segev B.,
**Sergeev A. V.** **Institution**- Departments of Chem. & Phys., Harvard Univ., Cambridge, MA, USA.
**Title**- Hopping and jumping between potential energy surfaces.

**Source**- Journal of Physical Chemistry B, vol.106, no.33, 22 Aug. 2002, pp.8471-8.

**Abstract**- Novel approaches to surface hopping (in the case in which surfaces cross in classically allowed regions) and surface "jumping" (in cases in which they never cross or they cross in classically forbidden regions) are discussed. Classically forbidden transitions necessarily involve discontinuous "jumps" in position or momentum or both (but so as to preserve energy). In general, the jumps are discontinuous changes in nuclear positions or momenta on the time scale of the electronic transition. After reviewing various approaches in one dimension, a phase-space approach is applied to multidimensional systems with large energy gaps, in which the traditional semiclassical approaches are more difficult to apply. The concept of jumps extends the spirit of surface hopping into new regimes.

**Author**-
**Sergeev A. V.**Segev B. **Institution**- Dept. of Chem., Ben-Gurion Univ. of the Negev, Beer-Sheva, Israel.
**Title**- Semiclassical estimation of franck-Condon factors and transition rates for vertical and nonvertical
transitions.

**Source**- Journal of Chemical Physics, vol.118, no.13, 1 April 2003, pp.5852-61.

**Abstract**- We develop a systematic way for estimating multidimensional Franck-Condon factors and transition rates for vertical and nonvertical transitions. By analyzing the phase-space overlap integral, we find the most probable positions and momenta of the nuclei immediately after the electronic transition. We find the transition rate by treating the dominant region in phase space as a funnel for the transition and by calculating the flow of probability through this funnel. We use the Wigner representation and its semiclassical limit and find that the transition occurs through a point(s) on the final surface of constant energy where the initial Wigner function is maximal. This dominant contribution is estimated analytically. Results are illustrated for Harmonic, Morse and Poeschl-Teller oscillators.

**Author**- Segev B.
**Sergeev A. V.** **Institution**- Dept. of Chem., Ben-Gurion Univ. of the Negev, Beer-Sheva, Israel.
**Title**- Dominant channels of vibronic transitions in molecules with several identical modes.

**Source**- Chemical Physics Letters, vol.367, no.3-4, 6 Jan. 2003, pp.382-9.

**Abstract**- Weak-coupling radiationless transitions (internal conversion or inter system crossing) are studied assuming separability and symmetry over N identical modes. Franck-Condon factors control the branching ratios between exciting just one of the equivalent modes, or equally distributing the available energy. The dominant process can be predicted by an exact quantum mechanical solution if the wavefunctions are known (Gaussian initial distributions and accepting Morse or Poeschl-Teller oscillators, for example); or more generally by a Wigner phase space surface-jumping analysis based on a classical limit of the Wigner function, using only the donor distribution and the acceptor potential surface.