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 email@example.com
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,
Veff(r) = -1/r + 1/2r2,
(we use atomic units, = 1) has a minimum at r=r0=1, =Veff(r0)=1/2:
At finite n an electron undergoes small oscillations about fixed position in an effective potential. The scaled energy = n2Enl 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-2R 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,
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, r0 and r1. 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=(2S)-1. For bound states, there are no more real turning points r1, 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=n4 , B=n3 are scaled strengths of electric and magnetic fields, correspondingly. The 1/n-expansion is constructed around the classical circular orbit with the radius r0 which is the root of algebraic equation r(1-F2r4)2(1+B2r3/4)=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 Veff vanishes. Note, that if F>F*(B), the radius r0 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=r2(1-t)4, r=n- 2R<1.299, 0<t<1/3.
* - 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/a1 (the nearest to the origin Borel singularity =1/a remains to be calculated by our method):
2 0.8849 **
3 0.6327 **
10 0.2343 **
** 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.