**LARGE ORDERS OF THE 1/ n-EXPANSION IN ATOMIC PHYSICS**

POPOV V.S., __SERGEEV A.V.__

*Institute of Theoretical and Experimental Physics, Bol'shaya
Cheriomushkinskaya 25, Moscow,*

* 117259 Russian Federation,*

* S.I.Vavilov State Optical Institute,*

* Tuchkov per. 1, St.Petersburg, 199034 Russian Federation,
e-mail sergeev@soi.spb.su*

*Introduction to the 1/n-expansion*

An asymptotic expansion for the energy of atoms is developed in powers
of 1/*n*, where *n* is the principal quantum number. We are considering
the Rydberg states with large orbital momentum *l* corresponding at
the limit . . to the circular orbits of an electron. So we assume that
*n*=*l*+*nr*+1 while *nr*<<*n*. In an essence,
such an approach is equivalent to recently developed well-known dimensional
expansion, see for example: Goodson D.Z., Lopez-Cabrera M. et al, J.Chem.Phys.,
**97**, 8481, 1992. This method is well-suited to the atoms in strong
fields and nonseparable problems such as two-electron atoms.

The large *n* limit reduces to an exactly solvable classical electrostatic
problem with an effective potential containing an additional centrifugal
term. In this simplifying limit, the problem reduces to minization of the
effective potential. For example, for a hydrogen atom the effective potential,

*V*eff(*r*) = -1/*r* + 1/2*r*2,

(we use atomic units, = 1) has a minimum at *r*=*r*0=1, =*V*eff(*r*0)=1/2:

At finite *n* an electron undergoes small oscillations about fixed
position in an effective potential. The scaled energy = *n*2*Enl*
is expanded in 1/*n*:

The 1/*n*-expansion is similar to the methods of molecular vibration
analysis. It reduces to the Rayleigh - Schrodinger perturbation theory
for anharmonic oscillator. The expansion coefficients can be calculated
exactly and to high order, using recursive relations.

*The divergence of large orders of 1/n- expansion*

The results obtained by simply summing sequential terms in the expansion
are quite poor. The reason is the divergence of the expansion, which renders
convential summation methods ineffective beyond the lowest orders. As an
illustrative example, we show the behaviour of ...... for the electronic
energy of H2+ ion with fixed protons. *R* is the distance between
two protons, and *r*=*n*-2*R* is a scaled distance.

The radius of convergence of the 1/*n*-expansion is zero, because
tends to infinity.

It was confirmed by direct numerical calculation of expansion coefficients that the coefficients grow as factorials,

or

with

In this case, the energy can be accurately evaluated by means of Pade
- Borel summation. The method consists of using Pade approximants for the
Borel function *F*,

and than integrating it with the exponent,

The factorial increase of the coefficients leads to the singularity
in the Borel function at *z*=*a*-1. So, by taking into account
the divergent large-order behaviour of the expansion one can localize the
nearest singularity to the origin in the Borel function and take full advantage
of the Pade - Borel summation technique.

In an earlier paper (Popov V.S., Sergeev A.V., Phys.Lett.A, **172**,
193, 1993) we examine parameter of the asymptotics *a* for the simpliest
case of spherically-symmetric or separable problems, such as screened Coulomb
potentials and Stark effect in a hydrogen atom.

We assume the effective potential to be of the form shown in the following figure.

All states are quasistationary because of the tunneling through the
potential barrier between two turning points, *r*0 and *r*1.
It leads to imaginary part of the energy, Im =-â/2, where â
is the width of the level (from quasiclassical formula for decay rate),

is the classical action and is a constant. Supposing analiticity in
the variable =1/*n* and using the dispersion relations in (which connect
with the integral from the imaginary part of the energy), we obtain the
parameters of the asymptotics: . and *a*=(2*S*)-1. For bound
states, there are no more real turning points *r*1, so the large-order
asymptotics contains only complex-conjugate terms.

__Treating nonspherically-symmetric and two-electron atoms__

Our aim is to extend the results for the behaviour of large orders of on multidimensional effective potentials.

As an example which has all essential features of general problem, we investigate in detail a hydrogen atom in parallel electric ( ) and magnetic ( ) fields. In this case, we deal with a two-dimensional quantum decay problem in an effective potential

where , *z* are cylindrical coordinates, *F*=*n*4 ,
*B*=*n*3 are scaled strengths of electric and magnetic fields,
correspondingly. The 1/*n*-expansion is constructed around the classical
circular orbit with the radius *r*0 which is the root of algebraic
equation *r*(1-*F*2*r*4)2(1+*B*2*r*3/4)=1. The
effective potential has a minimum only for small values of electric field,
*F*<*F**(*B*), where *F**(*B*) is a "classical
ionization threshold" at which a local minimum of *V*eff vanishes.
Note, that if *F*>*F**(*B*), the radius *r*0 and
the coefficients become complex. This solution has no means in classical
mechanics, but in quantum mechanics it describes both the position and
the width of quasistationary state. As an illustration, we present here
the effective potential for *B*=0.5 and *F*=0.2<*F**(*B*)
by means of isolevels.

The central problem (in order to calculate parameter *a* of the
asymptotics) is the determination of the most probable escape path wich
minimizes the classical action. We used two approaches. The first one is
based on the method of characteristics (see: A.Schmid, Ann.Phys.(N.Y.)
**170**, 333, 1986). The classical trajectories in an inverted effective
potential are calculated, a trajectory is chosen which terminates at a
stopping point and which represents the most probable escape path. The
parameter *a* equals to the reciprocal of the action along this trajectory.
In the alternative approach, the action is expanded as a perturbation series
around the minimum of the effective potential.

Our results of calculating the parameter *a* for magnetic fields
*B*=0, 0.5 and 1.0 are presented in the figure (*F**(*B*)=0.2081,
0.2532 and 0.3449, correspondingly). We found, that *a* while *F
F** by the typical law *a* (*F*-*F**)-5/4.

For pure magnetic field (Zeeman effect) the parameter *a* becomes
complex. The dependence of *a* on *B* is represented on the figure.

Also, we examine the parameter *a* for H2+ molecular ion and helium
isoelectronic sequence. For example, for H2+ ion we found:

*a*=(*z*-Arth *z*)-1, (*)

where , *t*=*r*2(1-*t*)4, *r*=n- 2*R*<1.299,
0<*t*<1/3.

(3/2)*r* *a*-1

0.8 -0.47480062*

-0.474*795*L

1.0 -0.31384119*

-0.313841*21*L

1.2 -0.19751662152*

-0.19751661*876*L

* - Exact, formula (*)

L - results of M.Lopez-Cabrera et al, Phys.Rev.Lett., **68**, 1992
(1992).

For two-electron atoms, we found the second Borel singularity = 1/*a*1
(the nearest to the origin Borel singularity =1/*a* remains to be
calculated by our method):

*Z*

2 0.8849 **

0.3 3.7G

3 0.6327 **

0.3 3.5G

10 0.2343 **

-0.05 3.3G

** Exact results, obtained by method of characteristics

G D.Z.Goodson et al, J.Chem.Phys. 1992, **97**, 8481

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Designed by A. Sergeev.