Applications of the Classical Two-Coulomb-Center Systems to Atomic/Molecular Physics
by
Nikolay Kryukov
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
December 14, 2013
Keywords: two-Coulomb-center systems, diatomic Rydberg quasimolecules, crossings of energy 
terms, effects of external fields and plasma screening, continuum lowering, Runge-Lenz vector
Copyright 2013 by Nikolay Kryukov
Approved by
Eugene Oks, Chair, Professor of Physics
Michael S. Pindzola, Professor of Physics
Joe D. Perez, Professor of Physics
Abstract
We advanced the classical studies of the two-Coulomb-center (TCC) systems consisting 
of two nuclei of charges Z and Z' and an electron in a circular or a helical state centered on the 
internuclear axis. These systems represent diatomic Rydberg quasimolecules encountered, e.g., in 
plasmas containing more than one kind of multicharged ions. Diatomic Rydberg quasimolecules 
are one of the most fundamental theoretical playgrounds for studying charge exchange. Charge 
exchange and crossings of corresponding energy levels that enhance charge exchange are 
strongly connected with problems of energy losses and of diagnostics in high temperature 
plasmas; besides, charge exchange is one of the most effective mechanisms for population 
inversion in the soft x-ray and VUV ranges. The classical approach is well-suited for Rydberg 
quasimolecules. First, we considered diatomic Rydberg quasimolecules subjected to a static 
electric field parallel to the internuclear axis. We found the appearance of an additional (fourth) 
term, which was absent at the zero field, and which had a V-type crossing with the lowest term. 
We also found X-type crossings (absent at the zero field) which significantly enhance charge 
exchange. Second, we studied effects of the electron screening in plasmas on diatomic Rydberg 
quasimolecules. We found that the screening stabilizes the nuclear motion for Z = 1 and 
destabilizes it for Z > 1. We also found that a so-called continuum lowering in plasmas was 
impeded by the screening, creating the effect similar to that of the magnetic field and opposite to 
that of the electric field. The continuum lowering plays a key role in calculations of the equation 
ii
of state, partition function, bound-free opacities, and other collisional atomic transitions in 
plasmas. Third, we considered diatomic Rydberg quasimolecules in a laser field. For the situation 
where the laser field is linearly-polarized along the internuclear axis, we found an analytical 
solution for the stable helical motion of the electron valid for wide ranges of the laser field 
strength and frequency. We also found resonances, corresponding to a laser-induced unstable 
motion of the electron, that result in the destruction of the helical states. For the case of a 
circularly-polarized field, polarization plane being perpendicular to the internuclear axis, we 
found an analytical solution for circular Rydberg states valid for wide ranges of the laser field 
strength and frequency. For this case we demonstrated also that there is a red shift of the primary 
spectral component. We showed that both under the linearly-polarized laser field and under the 
circularly-polarized laser field, in the electron radiation spectrum in the addition to the primary 
spectral component at (or near) the unperturbed revolution frequency of the electron, there 
appear satellites. Under a laser field of a known strength, in the case of the linear polarization the 
observation of the satellites would be the confirmation of the helical electronic motion in the 
Rydberg quasimolecule, while in the case of the circular polarization the observation of the red 
shift of the primary spectral component would be the confirmation of the specific type of the 
phase modulation of the electronic motion. Conversely, if the laser field strength is unknown, 
both the relative intensities of the satellites and the red shift of the primary spectral component 
could be used for measuring the laser field strength. Fourth, we considered TCC systems 
consisting of a proton, muon and an electron. We found that a muonic hydrogen atom can attach 
an electron, with the muon and electron being in circular states. The technique of the separation 
of rapid and slow subsystems was used, where the muon represented the rapid subsystem and the 
iii
electron the slow subsystem. We showed that the spectral lines emitted by the muon experience a 
red shift compared to the corresponding spectral lines in a muonic hydrogen atom. Observing 
this red shift should be one of the ways to detect the formation of such muonic-electronic 
negative hydrogen ions. Studies of muonic atoms and molecules, where one of the electrons is 
substituted by the heavier lepton ??, have several applications, such as muon-catalyzed fusion, a 
laser-control of nuclear processes, and a search for strongly interacting massive particles 
proposed as dark matter candidates and as candidates for the lightest supersymmetric particle. 
Fifth, we studied fundamental algebraic symmetry of the TCC systems leading to an additional 
conserved quantity: the projection of a super-generalized Runge-Lenz vector on the internuclear 
axis. We derived the correct super-generalized Runge-Lenz vector, whose projection on the 
internuclear axis is conserved, and showed that the corresponding expressions by other authors 
did not correspond to a conserved quantity and thus were incorrect. The correct super-
generalized Runge-Lenz vector for the TCC systems that we derived should be of a general 
theoretical interest since the TCC systems represent one of the most fundamental problems in 
physics. It can also have practical applications: e.g., it can be used as a necessary tool while 
applying to the TCC systems the robust perturbation theory for degenerate states based on the 
integrals of the motion.
iv
Dedicated to my beloved grandmother,
Nina Stepanovna Lipatova
v
Acknowledgments
   First, I would like to thank my academic advisor, Prof. Dr. Eugene A. Oks, for directing 
my steps towards completing this work. His outstanding experience and knowledge guided me 
throughout the whole process of thinking, finding out, deriving, calculating and writing. I am 
also obliged to him for leading me into and supporting my graduate study. 
   Accomplishing this work would not have been possible without necessary preparation in 
graduate-level physics. Besides Dr. Oks, my student's gratitude goes to Profs. J. D. Perez, M. S. 
Pindzola, A.-B. Chen, J. D. Hanson, S. Hinata, J. J. Dong, who taught me the graduate physics 
courses. Unfortunately, Dr. Chen is no longer with us, and, besides his help with my application 
for the Ph.D. program, I will forever remember the interest he inspired in me in his class, and 
that one homework I did for twenty-two hours nonstop. Since mathematical preparation is 
indispensable for physics research, it was also a pleasure to attend the classes of Profs. M. Liao 
and D. Glotov, which I took at the mathematics department. 
   I owe my admission to physics graduate school to the education I received in Russia. Russian 
traditional physics education and teaching is considered to be one of the best in the world, and I 
was certainly privileged, perhaps not always aware of this, to take the courses at the University 
of Nizhny Novgorod and the Institute of Applied Physics of the Russian Academy of Sciences 
led by Profs. M. I. Bakunov, S. B. Biragov, B. V. Lisin, O. N. Repin, A. M. Feygin, A. V. 
Abrosimov, G. M. Zhislin, A. G. Demekhov, I. A. Shereshevsky and other professors and 
vi
teachers. Profs. Bakunov and Biragov, in addition to teaching university students, organized 
preparatory physics courses and physics olympiads for secondary-school students, which not 
only stimulated the spirit of healthy competition, but also eased the transition from school to 
university. Here, I also acknowledge the support of my tutor Dr. A. S. Garevsky, who prepared 
me for the entrance exams to the University of Nizhny Novgorod. 
   Going back to the time when physics was shown to me as a tool to satisfy my curiosity about 
the world around me, I have to mention the master who introduced me to it: my secondary-
school physics teacher V. Yu. Kovalev. His ability to ignite curiosity, encourage analysis and 
discussion, together with his sense of humor, planted the seed of interest in me, then a seventh-
grader, which budded and, after nineteen years, fed by many, troubled at times, but always 
moving upwards, has grown into the present work. 
   Finally, returning to here and now, I would like to thank Dr. F. Robicheaux for an important 
discussion which resulted into a notable contribution to this dissertation. The discrepancy, which 
was noticed while preparing Chapter 6, helped discover errors in earlier publications of other 
authors and obtain new, correct results.
vii
Table of Contents
Abstract.........................................................................................................................................ii
Acknowledgments........................................................................................................................vi
List of Figures..............................................................................................................................ix
List of Abbreviations...................................................................................................................xii
Chapter 1. Introduction: Two-Coulomb Center Systems Representing Diatomic Rydberg
Quasimolecules .................................................................................................................1
Chapter 2. Enhancement of Charge Exchange and of Ionization by a Static Electric Field  .....10
Chapter 3. Effect of Plasma Screening on Circular States of Diatomic Rydberg Quasimolecules
and their Application to Continuum Lowering in Plasmas ............................................30
Chapter 4. Helical and Circular States of Diatomic Rydberg Quasimolecules in a Laser Field 65
Chapter 5. Attachment of an Electron by Muonic Hydrogen Atoms: Circular States ................84
Chapter 6. Super-generalized Runge-Lenz Vector ...................................................................109
Chapter 7. Conclusions  ............................................................................................................117
References  ...............................................................................................................................121
viii
List of Figures
Figure 1.1 ......................................................................................................................................4
Figure 1.2 ......................................................................................................................................7
Figure 2.1 ....................................................................................................................................15
Figure 2.2 ....................................................................................................................................16
Figure 2.3 ....................................................................................................................................18
Figure 2.4 ....................................................................................................................................19
Figure 2.5 ....................................................................................................................................20
Figure 2.6 ....................................................................................................................................20
Figure 2.7 ....................................................................................................................................21
Figure 2.8 ....................................................................................................................................23
Figure 2.9 ....................................................................................................................................23
Figure 3.1 ....................................................................................................................................35
Figure 3.2 ....................................................................................................................................36
Figure 3.3 ....................................................................................................................................37
Figure 3.4 ....................................................................................................................................38
Figure 3.5 ....................................................................................................................................38
Figure 3.6 ....................................................................................................................................39
Figure 3.7 ....................................................................................................................................39
ix
Figure 3.8 ....................................................................................................................................40
Figure 3.9 ....................................................................................................................................40
Figure 3.10 ..................................................................................................................................41
Figure 3.11 ..................................................................................................................................43
Figure 3.12 ..................................................................................................................................44
Figure 3.13 ..................................................................................................................................44
Figure 3.14 ..................................................................................................................................45
Figure 3.15 ..................................................................................................................................46
Figure 3.16 ..................................................................................................................................47
Figure 3.17 ..................................................................................................................................47
Figure 3.18 ..................................................................................................................................49
Figure 3.19 ..................................................................................................................................49
Figure 3.20 ..................................................................................................................................50
Figure 3.21 ..................................................................................................................................51
Figure 3.22 ..................................................................................................................................52
Figure 3.23 ..................................................................................................................................53
Figure 3.24 ..................................................................................................................................53
Figure 3.25 ..................................................................................................................................53
Figure 3.26 ..................................................................................................................................54
Figure 3.27 ..................................................................................................................................55
Figure 3.B.1 ................................................................................................................................60
Figure 3.B.2 ................................................................................................................................60
x
Figure 3.B.3 ................................................................................................................................61
Figure 3.C.1 ................................................................................................................................62
Figure 3.C.2 ................................................................................................................................63
Figure 3.C.3 ................................................................................................................................63
Figure 3.C.4 ................................................................................................................................64
Figure 3.C.5 ................................................................................................................................64
Figure 4.1 ....................................................................................................................................71
Figure 4.2 ....................................................................................................................................73
Figure 4.3 ....................................................................................................................................73
Figure 4.4 ....................................................................................................................................75
Figure 4.5 ....................................................................................................................................78
Figure 4.6 ....................................................................................................................................81
Figure 4.7 ....................................................................................................................................82
Figure 5.1 ....................................................................................................................................87
Figure 5.2 ....................................................................................................................................91
Figure 5.3 ....................................................................................................................................93
Figure 5.4 ....................................................................................................................................98
Figure 5.5 ..................................................................................................................................101
Figure 5.6 ..................................................................................................................................102
Figure 5.7 ..................................................................................................................................103
Figure 5.A.1 ..............................................................................................................................108
xi
List of Abbreviations
CL Continuum lowering
CRS Circular Rydberg states 
OCC One Coulomb center
TCC Two Coulomb centers
TCP Two-centers problem
xii
Chapter 1. Introduction: Two-Coulomb Center Systems Representing Diatomic Rydberg 
Quasimolecules
Charge exchange and crossings of corresponding energy levels that enhance charge exchange are 
strongly connected with problems of energy losses and of diagnostics in high temperature 
plasmas ? see, e.g., [1.1, 1.2] and references therein. Besides, charge exchange was proposed as 
one of the most effective mechanisms for population inversion in the soft x-ray and VUV ranges 
[1.3 ? 1.6]. One of the most fundamental theoretical playgrounds for studying charge exchange is 
the problem of electron terms in the field of two stationary Coulomb centers (TCC) of charges Z 
and Z' separated by a distance R. It presents fascinating atomic physics: the terms can have 
crossings and quasicrossings.
   The crossings are due to the fact that that the well-known Neumann-Wigner general theorem 
on the impossibility of crossing of terms of the same symmetry [1.7] is not valid for the TCC 
problem of Z' ? Z ? as shown in paper [1.8]. Physically it is here a consequence of the fact that 
the TCC problem allows a separation of variables in the elliptic coordinates [1.8]. As for the 
quasicrossings, they occur when two wells, corresponding to separated Z- and Z'-centers, have 
states ? and ?', characterized by the same energies E = E', by the same magnetic quantum 
numbers m = m', and by the same radial elliptical quantum numbers k = k' [1.9 ? 1.11]. In this 
situation, the electron has a much larger probability of tunneling from one well to the other (i.e., 
of charge exchange) as compared to the absence of such degeneracy.
   These rich features of the TCC problem also manifest in a different area of physics such as 
1
plasma spectroscopy as follows. A quasicrossing of the TCC terms, by enhancing charge 
exchange, can result in unusual structures (dips) in the spectral line profile emitted by a Z-ion 
from a plasma consisting of both Z- and Z'-ions, as was shown theoretically and experimentally 
[1.12 ? 1.17]. From the experimental width of these dips it is possible to determine rate 
coefficients of charge exchange between multicharged ions, which is a fundamental reference 
data virtually inaccessible by other experimental methods [1.17].
   Before year 2000, the paradigm was that the above sophisticated features of the TCC problem 
and its flourishing applications were inherently quantum phenomena. But then in year 2000, 
papers [1.18, 1.19] were published presenting a purely classical description of both the crossings 
of energy levels in the TCC problem and the dips in the corresponding spectral line profiles 
caused by the crossing (via enhanced charge exchange). These classical results were obtained 
analytically based on first principles without using any model assumptions. 
   In the classical studies the TCC systems represent diatomic Rydberg quasimolecules 
encountered, e.g., in plasmas containing more than one kind of multicharged ions. Naturally, the 
classical approach is well-suited for Rydberg quasimolecules.
   Later applications of the results from [1.18, 1.19] included a magnetic stabilization of Rydberg 
quasi-molecules [1.20], a problem of continuum lowering in plasmas (which plays a key role in 
calculations of the equation of state, partition function, bound-free opacities, and other 
collisional atomic transitions in plasmas) [1.21], and the study of the classical Stark effect for 
Rydberg quasi-molecules ([1.22] and Chapter 2).
   In these studies a particular attention was given to circular Rydberg states. Circular states of 
atomic and molecular systems are an important subject in its own right. They have been 
2
extensively studied both theoretically and experimentally for several reasons (see, e.g., [1.18 ? 
1.20, 1.24 ? 1.37] and references therein): (a) they have long radiative lifetimes and highly 
anisotropic collision cross sections, thereby enabling experiments on inhibited spontaneous 
emission and cold Rydberg gases, (b) these classical states correspond to quantal coherent states, 
objects of fundamental importance, (c) a classical description of these states is the primary term 
in the quantal method based on the 1/n-expansion, and (d) they can be used in developing atom 
chips.
   As examples of experimental studies of Rydberg states, we refer to paper [1.23] where such 
studies made in the last three decades have been enumerated [1.23]. In particular, Day and Ebel 
[1.38] predicted theoretically in 1979 that probability of a wake electron being captured by fast-
moving ions traversing a solid to a state with large principal (n) and angular momentum (l) 
quantum numbers is quite high and much of the time the electron is captured into circular 
Rydberg states (l = n ? 1) distributed over a narrow band near nmax. Note here that l = n ? 1 
defines circular orbits, whereas the full qualification of circular Rydberg states (CRS) requires 
| ml | = l = n ? 1. Day and Ebel proposed existence of an "optical window" in ion velocity as a 
possible explanation for non-observability of the CRS in beam-foil spectroscopy work. Also, the 
CRS are both long lived with respect to radiative transitions and short lived with respect to 
collisions, hence their observation requires a wide aperture and very good vacuum. Pegg et al. in 
1977 [1.39] observed a strong cascade tails in the decay curves of Cu18+ in a beam-foil interaction 
and attributed it to the successive decay of long-lived CRS or "yrast states". Note that CRS can 
radiate only to the next lower state, which leads to a chain of successive yrast transitions till they 
reach to the ground state. Recently, from the study of the time-resolved beam-foil X-ray spectra 
3
of projectile or projectile-like ions of 2p,2s ? 1s transitions in H-like Fe, Ni, Cu, and Zn at 
different delay times (in the range 250?1600 ps). Nandi identified, in each case, a single circular 
Rydberg and/or an elliptic Rydberg state cascading successively to the 2p or 2s level.
   Coming back to the ground-breaking theoretical papers [1.18, 1.19], it should be emphasized 
that the analysis there was not confined to circular orbits of the electron. Paper [1.19] presented a 
detailed study of helical Rydberg states of these diatomic Rydberg quasimolecules. For the stable 
motion, the electron trajectory was found to be a helix on the surface of a cone, with the axis 
coinciding with the internuclear axis. In this helical state, the electron, while spiraling on the 
surface of the cone, oscillates between two end-circles which result from cutting the cone by two 
parallel planes perpendicular to its axis (Fig. 1.1).
Fig. 1.1. Sketch of the helical motion of the electron in the ZeZ'-system at the absence of the magnetic 
field. We stretched the trajectory along the internuclear axis to make its details better visible.
   Let us now reiterate how classical energy terms of diatomic Rydberg quasimolecules were 
obtained in paper [1.19]. (The meaning of ?classical energy terms? is clarified below.) They were 
obtained by considering the Hamiltonian of the particle in a circular state in the cylindrical 
4
coordinates:
H=12 ? p?2? p?
2
?2 ?pz
2?? Z
?z2??2?
Z'
??R?z?2??2
 (1.1)
where (?, ?, z) are the cylindrical coordinates with the z-axis being the internuclear axis and 
(p?, p?, pz) are the corresponding canonical momenta, Z and Z' are the charges at z = 0 and z = R. 
Since ? is cyclic, p? = const = L and in the circular state pz = p? = 0, so (1.1), which is the particle 
energy, can be written as
E= L
2
2?2 ?
Z
?z2??2?
Z'
??R?z?2??2   (1.2)
which, using the scaled quantities defined below,
w= zR, v= ?R, ?= L?ZR, b=Z'Z , ?=?ERZ  (1.3)
takes the form
?= 1?w2?v2? b?
?1?w?2?v2
? ?
2
2v2   (1.4)
In equilibrium, the derivatives of ? by both scaled coordinates (w, v) must vanish, so by taking 
the partial derivatives of (1.4) by each coordinate we obtain two more equations. The partial 
derivative of (1.4) by w set equal to zero yields
5
v2=w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3   (1.5)
and the partial derivative of (1.4) by v set equal to zero yields
?2=v4? 1?w2?v2?3/2 ? b??1?w?2?v2?3/2?   (1.6)
Equation (1.5) determines the points (w, v) where the equilibrium is located. For b > 1, the 
equilibrium value of v exists for 0 < w < w1 and for w3 < w < 1; for b < 1, it exists for 0 < w < w3 
and w1 < w < 1. Here we introduced the quantities w1 = 1/(1 + b1/2) and w3 = b/(1 + b). For 
definiteness, we shall consider the cases of b > 1 (or Z' > Z). There are no equilibrium points for 
w < 0 or w > 1 (i.e., for z < 0 or z > R): the z-component of the total Coulomb attraction force of 
the two centers has no balancing force at those points. 
   Solving (1.5) for v and substituting it into (1.6) and (1.4) and then solving (1.6) for ? and 
substituting it into (1.4), we obtain ?(w, b) ? the scaled energy depending on the scaled 
internuclear coordinate w for a given ratio of the charges of the nuclei b.
   If we scale the internuclear distance R as r = (Z/L2)R, and given ? = ?ER/Z from (1.3), the 
energy of the electron can be represented in the form E = ? (Z2/L2)?1, where we define ?1 = ?/r. 
The scaling of E to ?1 includes no more R and includes L, just like the scaling of R to r. Next, 
from the scaled quantities (1.3) we have ?2 = L2/(ZR) and from earlier in this paragraph, 
r = ZR/L2, therefore, r = 1/?2; with ?2 taken from (1.6) and with v substituted from (1.5) we obtain 
r(w, b). Thus, for any L > 0 and any b > 0, the dependence ?1(r), which represents the classical 
energy terms for this system, can be presented in a parametric form ?(w, b), r(w, b) via the 
6
parameter w for a given b. Here we give the explicit form of this dependence:
?1=p2? 1?w2?p?3/2 ? b??1?w?2?p?3/2 ?? w
2?p/2
?w2?p?3/2 ?
b??1?w?2?p/ 2?
??1?w?2?p?3/2 ?   (1.7)
r=p?2? 1?w2?p?3/2? b??1?w?2?p?3/2?
?1
  (1.8)
where we have defined the quantity p
p=v2=w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3   (1.9)
The plot of the energy terms given by (1.7) and (1.8) for b = 3 is given below. The same plot, 
with different ranges on both axes, is given in paper [1.19].
Fig. 1.2. Classical energy terms: the dependence of the scaled classical energy ?1 on the scaled 
internuclear distance r given by eq. (1.7) and (1.8) for b = 3.
7
   At this point it might be useful to clarify the relation between the classical energy terms ??1(r) 
and the energy E. The former is a scaled quantity related to the energy as specified above: 
E = ? (Z/L)2 ?1 . The projection L of the angular momentum on the internuclear axis is a 
continuous variable. The energy E depends on both ?1 and L. Therefore, while the scaled quantity 
?1 takes a discrete set of values, the energy E takes a continuous set of values (as it should be in 
classical physics).
   It turns out that the form of the parametric dependence can be significantly simplified and 
some of the properties of it can be found analytically by introducing a new parameter
?=? 1w?1?
1/3
  (1.10)
In this case, w = 0 will correspond to ? = +? and w = 1 will correspond to ? = 0, thus ? > 0 in the 
allowed regions. The points w1 = 1/(1 + b1/2) and w3 = b/(1 + b) defining the allowed regions 
0 < w < w1, w3 < w < 1 (here we assume b > 1) will correspond to ?1 = b1/6 and ?3 = 1/b1/3 (notice 
that 0 < w < w1 corresponds to +? > ? > ?1 and w3 < w < 1 corresponds to ?3 > ? > 0). The energy 
terms ?1(r) will take the following parametric form:
?1??,b?=?b
2/3??4?2????3?2??b2/3?2?3?1??
2??3?1?2??6?1?   (1.11)
r??,b?=?b
2/3 ?2?1??6?1?3/2
??b2/3??4?2   (1.12)
8
The parametric plot of (1.11) and (1.12) with the parameter ? varied from 0 to 1/b1/3 and from b1/6 
to +? for b = 3 will yield the same graph as in Fig. 1.2.
   The crossing of the top two terms corresponds to the point where r(?, b) has a minimum or 
?1(?, b) has a maximum for a given b. Thus, taking the derivative of either function by ? and 
setting it equal to zero will yield a solution for the ? on the interval ? > 1 corresponding to the 
crossing. The equation for ? obtained from differentiating r(?) is a 6th-power polynomial and 
cannot be solved analytically; however, the equation for ? obtained from differentiating ?1(?) can 
be solved analytically for ?. Below is the critical value ?0 corresponding to the crossing.
?0=?b1/3??b?1?
1/3
b1/6 ???b?1?
1/3???b?1?1/3?   (1.13)
Substituting (1.13) into (1.12), we can obtain analytically the value of r corresponding to the 
crossing.
   In the following chapters we consider the same system subjected to external potentials of 
various nature, as well as other configurations, such as a quasimolecule consisting of a proton, an 
electron and a muon, and examine the integrals of motion in the TCC system. Atomic units 
(? = e = me = 1) are used throughout the whole study.
9
Chapter 2. Enhancement of Charge Exchange and of Ionization by a Static Electric Field
2.1. Introduction
Charge exchange is one of the most important atomic processes in plasmas. Charge exchange 
and crossings of corresponding energy levels that enhance charge exchange are strongly 
connected with problems of energy losses and of diagnostics in high temperature plasmas ? see, 
e.g., [1.1, 1.2] and references therein. Charge exchange was proposed as an effective mechanism 
for population inversion in the soft x-ray and VUV ranges [1.3 ? 1.6]. One of the most 
fundamental theoretical domains for studying charge exchange is the problem of electron terms 
in the field of two stationary Coulomb centers (TCC) of charges Z and Z' separated by a distance 
R. It presents an intriguing atomic physics: the terms can have crossings and quasicrossings. 
   The crossings are due to the fact that that the well-known Neumann-Wigner general theorem 
on the impossibility of crossing of terms of the same symmetry [1.7] is not valid for the TCC 
problem of Z' ? Z ? as shown in paper [1.8]. Physically it is here a consequence of the fact that 
the TCC problem allows a separation of variables in the elliptic coordinates [1.8]. As for the 
quasicrossings, they occur when two wells, corresponding to separated Z- and Z'-centers, have 
states ? and ?', characterized by the same energies E = E', by the same magnetic quantum 
numbers m = m', and by the same radial elliptical quantum numbers k = k' [1.9 ? 1.11]. In this 
situation, the electron has a much larger probability of tunneling from one well to the other (i.e., 
of charge exchange) as compared to the absence of such degeneracy.
   These intrinsic features of the TCC problem also manifest in a different area of physics such as 
10
plasma spectroscopy as follows. A quasicrossing of the TCC terms, by enhancing charge 
exchange, can result in unusual structures (dips) in the spectral line profile emitted by a Z-ion 
from a plasma consisting of both Z- and Z'-ions, as was shown theoretically and experimentally 
[1.12 ? 1.17].
   Before year 2000, the paradigm was that the above sophisticated features of the TCC problem 
and its flourishing applications were inherently quantum phenomena. But then in year 2000 
papers [1.18, 1.19] were published presenting a purely classical description of both the crossings 
of energy levels in the TCC problem and the dips in the corresponding spectral line profiles 
caused by the crossing (via enhanced charge exchange). These classical results were obtained 
analytically based on first principles without using any model assumptions. Later applications of 
these results included a magnetic stabilization of Rydberg quasi-molecules [1.20] and a problem 
of continuum lowering in plasmas [1.21].
   In papers [1.18, 1.20, 1.21] the study was focused at Circular Rydberg States (CRS) of the TCC 
system (the analysis in paper [1.19] went beyond CRS). CRS of atomic and molecular systems, 
with only one electron, correspond to |m| = (n ? 1) >>1, where n and m are the principal and 
magnetic electronic quantum numbers, respectively. They have been extensively studied [2.1 ? 
2.4] both theoretically and experimentally for several reasons: (a) CRS have long radiative 
lifetimes and highly anisotropic collision cross sections, thereby enabling experiments on 
inhibited spontaneous emission and cold Rydberg gases [1.36, 2.5], (b) classical CRS correspond 
to quantal coherent states, objects of fundamental importance, and (c) a classical description of 
CRS is the primary term in the quantal method based on the 1/n-expansion (see, e.g. [2.6] and 
references therein).
11
   While the authors of paper [1.20] studied analytically the effect of a magnetic field (along the 
internuclear axis) on CRS of the TCC system, in the present chapter we study the effect of an 
electric field (along the internuclear axis) on CRS of the TCC system. We provide analytical 
results for strong fields, as well as numerical results for moderate fields. We show that the 
electric field leads to the following consequences.
   First, it leads to the appearance of an extra energy term: the fourth classical energy term ? in 
addition to the three classical energy terms at zero field. Second, but more importantly, the 
electric field creates additional crossings of these energy terms. We show that some of these 
crossings enhance charge exchange while other crossings enhance the ionization of the Rydberg 
quasi-molecule. 
2.2. Calculations of the classical Stark effect 
for a Rydberg quasi-molecule in a circular state
We consider a TCC system, where the charge Z is at the origin and the Oz axis is directed to the 
charge Z', which is at z = R. A uniform electric field F is applied along the internuclear axis ? in 
the negative direction of Oz axis. We study CRS where the electron moves around a circle in the 
plane perpendicular to the internuclear axis, the circle being centered at this axis.
   Two quantities, the energy E and the projection L of the angular momentum on the internuclear 
axis are conserved in this configuration. We use cylindrical coordinates to write the equations for 
both. 
12
E=12 ? ??2??2 ??2??z2??Zr ?Z'r' ?F z  (2.1)
L=?2 ??  (2.2)
where ? is the distance of the electron from the internuclear axis, ? is its azimuthal angle, z is the 
projection of the radius-vector of the electron on the internuclear axis, r and r' are the distances 
of the electron from the particle to Z and Z', respectively.
   The circular motion implies that d?/dt = 0; as the motion occurs in the plane perpendicular to 
the z-axis, dz/dt = 0. Further, expressing r and r' through ? and z, and taking d?/dt from (2.2), we 
have:
E= L
2
2?2 ?
Z
??2?z2?
Z'
??2??R?z?2 ?Fz
 (2.3)
With the scaled quantities
w= zR, v= ?R, b=Z'Z , ?=?ERZ , ?= L?Z R, f =F R
2
Z , r=
Z R
L2
 (2.4)
our energy equation takes the form below:
?= 1?w2?v2? b?
?1?w?2?v2
? f w? ?
2
2v2
 (2.5)
We can seek the equilibrium points by finding partial derivatives of ? by the scaled coordinates 
w, v and setting them equal to zero. This will give the following two equations.
f ? b?1?w???1?w?2?v2?3/2= w?w2?v2?3/2  (2.6)
13
?2=v4? 1?w2?v2?3/2 ? b??1?w?2?v2?3/2?  (2.7)
From the definitions of the scaled quantities (4), ?2 = 1/r and E = ? (Z/R) ?. Since r = ZR/L2, 
E = ? (Z/L)2 ?/r, where r = 1/?2 can be obtained by solving (2.7) for ?. Substituting ? into the 
energy equation, we get the three master equations for this configuration.
?1=p2? 1?w2?p?3/2 ? b??1?w?2?p?3/2 ?? w
2?p/2
?w2?p?3/2 ?
b??1?w?2?p/2?
??1?w?2?p?3/2 ? f w?  (2.8)
r=p?2? 1?w2?p?3/2 ? b??1?w?2?p?3/2 ?
?1
 (2.9)
f ? b?1?w???1?w?2?p?3/2= w?w2?p?3/2  (2.10)
where E = ? (Z/L)2 ?1 and p = v2. Thus, ?1 is the ?true? scaled energy, whose equation for E does 
not include R. The scaled energy ?1 and internuclear distance r in (2.8) and (2.9) now explicitly 
depend only on the coordinates w and p (besides the constants b and f). Therefore, if we solve 
(2.10) for p and substitute it into (2.8) and (2.9), we will have the parametric solution ?1(r) with 
the parameter w. 
   Our focus is at crossings of energy terms of the same symmetry. In the quantum TCC problem, 
?terms of the same symmetry? means terms of the same magnetic quantum number m [1.8 ? 
1.11]. Therefore, in our classical TCC problem, we fixed the angular momentum projection L 
14
and study the behavior of the classical energy at L = const ? 0 (the results for L and ?L are 
physically the same). 
   Equation (2.10) does not allow an exact analytical solution for p. Therefore, we will use an 
approximate analytical method.
   Figure 2.1 shows a contour plot of (2.10) for a relatively weak field f = 0.3 at b = 3, with w on 
the horizontal axis and p on the vertical. The plot has two branches. The left branch spans from 
w = 0 to w = w1. The right one actually has a small two-valued region between some w = w3 and 
1 (w3 < 1). Indeed, at w = 1, there are two values of p: p = 0 and p = f ?2/3 ? 1. Thus, the two-
valued region exists only for f < 1. 
Fig. 2.1. Contour plot of (2.10) for a relatively weak field f = FR2/Z = 0.3 at b = Z'/Z = 3.
15
The right branch touches the abscissa at w = 1 and at some w = w2. Analytical expressions for w1 
and w2 are given in Appendix A. The quantity w3 is a solution of the equation
f 2/5?2w3?1?3/5=w32/5?b2/5?1?w3?2/5  (2.11)
The method, by which w3 was found from (2.11), is presented in Appendix B.
   Figure 2 shows a contour plot of (2.10) for a relatively strong field f = 20 at b = 3. It is seen 
that there is no two-valued region. 
Fig. 2.2. The same as in Fig. 2.1, but for a relatively strong field f = FR2/Z = 20.
   From now on we consider the situation where the radius of the electronic orbit is relatively 
small, meaning that p << 1. Physically this corresponds to strong fields f > fmin ~ 10.
16
   Solving (2.10) in the small-p approximation, we obtain 
p=? w
f ? b?1?w?2 ?
2/3
?w2  (2.12)
for the left branch (0 < w < w1) and
p=?b?1?w?1
w2? f ?
2/3
??1?w?2  (2.13)
for the right branch (1 < w < w2). Substituting these results into (2.8) and (2.9), we get 
approximate solutions for energy terms ??1(r) in both regions in a parametric form, w being the 
parameter. Now we plot classical energy terms ??1(r) by varying the parameter w over both 
regions, using the appropriate formula for each one. 
   Figure 2.3 shows classical energy terms at b = 3 for f = 20. Figure 2.4 presents classical energy 
terms at b = 3 for f = 5. 
17
Fig. 2.3. Classical energy terms at b = 3 for f = 20.
18
Fig. 2.4. Classical energy terms at b = 3 for f = 5.
We also solved the same problem numerically. By comparison we found that the approximate 
analytical solution is accurate for fields f = 5 and above. 
   Figures 2.5 and 2.6 show the numerically obtained classical energy terms at b = 3 for f = 2 and 
f = 0.1, respectively. For comparison, Fig. 2.7 shows the classical energy terms at b = 3 at the 
absence of the electric field (it had been previously presented in papers [1.18, 1.19] and is the 
same as Fig. 1.2).
19
Fig. 2.5. Classical energy terms at b = 3 for f = 2.
Fig. 2.6. Classical energy terms at b = 3 for f = 0.1.
20
Fig. 2.7. Classical energy terms at b = 3 at the absence of the electric field.
   The electric field causes several important new features compared to the zero-field case. While 
at f = 0 there are three classical energy terms, the electric field brings up the fourth classical 
energy term. Indeed, let us take as an example the case of f = 5 at b = 3 presented in Fig. 2.4. 
There are four energy terms that we label as follows:
#1 ? the lowest term;
#2 ? the next term up (which has a V-type crossing with term 1);
#3 ? the next term up;
#4 ? the highest term (which has a V-type crossing with term 3).
We will use this labeling also while discussing all other plots (except the plot in Fig. 2.7 for 
f = 0): terms 1 and 2 will be those having the V-type crossing at the lower energy, terms 3 and 4 
21
will be those having the V-type crossing at the higher energy,
   At f = 0 term 2 is absent, but it appears at any non-zero value of f ? no matter how small. 
Actually, as f approaches zero, this term behaves like ?f/r, which is why it disappears at f = 0.
   The existence of this additional term can be explained physically as follows. When f = 0, 
equilibrium of the orbital plane to the right of Z' (i. e., for w > 1) does not exist, so that the values 
of w1 and w3 reduce to the ones presented in papers [1.18, 1.19] and the right branch of p(w) 
asymptotically goes to infinity when w goes down to w3. When an infinitesimal field f appears, 
the right branch flips over positive infinity and ends up on the abscissa at w2 ? ?, thus enabling 
the whole region w > 1 for equilibrium. As the field grows, w2 decreases. Physically, the force 
from the electric field at w > 1 balances out the Coulomb attraction of the Z?Z' system on the left 
? the situation not possible for f = 0. This term is obtained by varying the parameter w from 1 to 
w2.
   We emphasize that the above examples presented for Z'/Z = 3 represent a typical situation. In 
fact, for any pair of Z and Z' ? Z, at the presence of the electric field, there are four classical 
energy terms of the same symmetry for CRS.
   Another important new feature caused by the electric field is X-type crossings of the classical 
energy terms. This kind of crossings and their physical consequences are discussed in the next 
section.
2.3. X-type crossings of classical energy terms and their physical consequences
Figure 2.8 shows a magnified version of the energy terms 2, 3, and 4 at b = 3 for f = 2. Figure 2.9 
22
shows a further magnified version of the energy terms 2 and 4 at b = 3 for f = 2. Compared to 
Fig. 2.5 for the same b and f, in Figs. 2.8 and 2.9 we decreased the exhibited energy range, but 
increased the exhibited range of the internuclear distances r.
Fig. 2.8. Magnified plot of the classical energy terms 2, 3, and 4 at b = 3 for f = 2.
Fig. 2.9. Further magnified plot of the classical energy terms 2 and 4 at b = 3 for f = 2.
23
   It is seen that term 2 has the X-type crossing with term 3 at r = 7.8 and the X-type crossing 
with term 4 at r = 32. The situation where there are two X-type crossings exists in a limited range 
of the electric fields. For example, for b = 3: 
- two X-type crossings exist at 1.31 < f < 2.4;
- there are no X-type crossings at f < 1.31;
- there is one X-type crossing at f > 2.4 (the crossing of terms 2 and 3).
   To reveal physical consequences of the X-type crossings, let us first discuss the origin of all 
four classical energy terms for arbitrary Z'/Z ? 1. At r ? ?, term 3 corresponds to the energy of a 
hydrogenlike ion of the nuclear charge Zmin = min(Z', Z), slightly perturbed by the charge 
Zmax = max(Z', Z), as shown in [1.18, 1.19]. 
   At r ? ?, term 4 corresponds to a near-zero-energy state (where the electron is almost free), as 
shown in [1.18, 1.19]. If the ratio Z'/Z is of the order of (but not equal to) unity, this term at 
r ? ? can be described only in the terminology of elliptical coordinates (rather than parabolic or 
spherical coordinates), meaning that even at r ? ? the electron is shared between the Z- and Z'-
centers. However, in the case of Z' >> Z, this term can be asymptotically considered as the Z'-
term, as shown in [1.18, 1.19]. It has the V-type crossing with term 3, which asymptotically is the 
Z-term (since Zmin = Z for Z' > Z). Likewise, in the case of Z' << Z, term 4 can be asymptotically 
considered as the Z-term, as shown in [1.18, 1.19]. It has the V-type crossing with term 3, which 
asymptotically is the Z'-term (since Zmin = Z' for Z' < Z).
   At r ? ?, term 1 corresponds to the energy of a hydrogenlike ion of the nuclear charge Zmax 
slightly perturbed by the charge Zmin , as shown in [1.18, 1.19]. As for the term 2, at r ? ? it has 
properties similar to term 4, but with the interchange of Zmax and Zmin . In particular, in the case of 
24
Z' >> Z, this term can be asymptotically considered as the Z-term, having the V-type crossing 
with term 1, which asymptotically is the Z'-term (since Zmax = Z' for Z' > Z). In the case of 
Z' << Z, term 2 can be asymptotically considered as the Z'-term, having the V-type crossing with 
term 1, which asymptotically is the Z-term (since Zmax = Z for Z' < Z).
   Thus, when Z and Z' differ significantly from each other, the V-type crossings occur between 
two classical energy terms that can be asymptotically labeled as Z- and Z'-terms. This situation 
classically depicts charge exchange, as explained in papers [1.18, 1.19]. Indeed, say, initially at 
r ? ?, the electron was a part of the hydrogenlike ion of the nuclear charge Zmin . As the charges 
Z and Z' come relatively close to each other, the two terms undergo a V-type crossing and the 
electron is shared between the Z- and Z'-centers. Finally, as the charges Z and Z' go away from 
each other, the electron ends up as a part of the hydrogenlike ion of the nuclear charge Zmax .
   So, the first distinction caused by the electric field is an additional, second V-type crossing 
leading to charge exchange ? compared to the zero-field case where there was only one such 
crossing. However, the second V-type crossing (the crossing of terms 1 and 2) occurs at the 
internuclear distance rV2 << rV1, where rV1 is the internuclear distance of the first V-type crossing 
(the crossing of terms 3 and 4). Therefore the cross-section of the charge exchange 
corresponding to the second V-type crossing is much smaller than the corresponding cross-
section for the first V-type crossing.
   Now let us discuss the X-type crossing from the same point of view. When Z and Z' differ 
significantly from each other, the X-type crossing of terms 2 and 4 is the crossing of terms that 
can be asymptotically labeled as Z- and Z'-terms. Thus, this situation again classically depicts 
charge exchange. The most important is that this crossing occurs at the internuclear distance 
25
rX1 >> rV1 >> rV2 . Therefore, the cross-section of charge exchange due to this X-type crossing is 
much greater than the corresponding cross-sections of for the V-type crossings. This is the most 
fundamental physical consequence caused by the electric field: a significant enhancement of 
charge exchange.
   When Z and Z' differ significantly from each other, the X-type crossing of terms 2 and 3 is the 
crossing of terms having the same asymptotic labeling: either both of them are Z-terms or both of 
them are Z'-terms. Therefore this second X-type crossing (at r = rX2) does not correspond to 
charge exchange ? rather it represents an additional ionization channel. Indeed, say, initially at 
r ? ?, the electron resided on term 3 of the hydrogenlike ion of the nuclear charge Z. As the 
distance between the charges Z and Z' decreases to r = rX2, the electron can switch to term 2, 
which asymptotically corresponds to a near-zero-energy state (of the same hydrogenlike ion of 
the nuclear charge Z) where the electron would be almost free. So, as the charges Z and Z' go 
away from each other, the system undergoes the ionization. Thus, another physical consequence 
caused by the electric field is the appearance of the additional ionization channel. This should 
have been expected since the electric field promotes the ionization of atomic and molecular 
systems.
2.4. Conclusions
We studied the effect of an electric field (along the internuclear axis) on circular Rydberg states 
of the two Coulomb centers system. We provided analytical results for strong fields, as well as 
numerical results for moderate fields. We showed that the electric field had the following effects.
26
   The first effect is the appearance of an extra energy term: the fourth classical energy term ? in 
addition to the three classical energy terms at zero field. This term exhibits a V-type crossing 
with the lowest energy term. The two highest energy terms continue having a V-type crossing 
like at the zero field. In the situation where the charges Z' differ significantly from each other, 
both V-type crossings correspond to charge exchange.
   The second effect is the appearance of a new type of crossings ? X-type crossings. One of the 
X-type crossings (existing in a limited range of the electric field strength) corresponds to charge 
exchange at a much larger internuclear distance compared to the V-type crossings. Therefore the 
cross-section of charge exchange due to this X-type crossing is much greater than the cross-
section of charge exchange due to V-type crossings. Thus, the electric field can significantly 
enhance charge exchange. We believe that this is the most important result of the present chapter.
   The other X-type crossing does not correspond to charge exchange. Instead, it represents an 
additional ionization channel.
Appendix A. The limits w1 and w2 on the graph of p(w) in Eq. (2.10).
The analytical results for the quantities w1 and w2, obtained using the software Mathematica, 
have the following form.
   For w1:
27
)6)1(4)1(
3)1(2)1(
)1(36
3)1(2)1(3(63
1
1
2
1
1
2
1
1
2
1
+?+???+?
+?+?+?+
++
++?+?+?+?=
f
fb
f
a
fa
fb
f
fb
f
a
fa
fbf
b
f
fb
f
a
fa
fbw
where
3 3233
1 ))7(1(3)1()1(336)1(45 +++?+?++?++= ffbffbbfbfbfba
   For w2:
)2)1(4)1(
1)1(2)1(
)1(36
1)1(2)1(3(63
2
2
2
2
2
2
2
2
2
2
+++?+??
+++++?
?+
++++++?+=
f
b
f
a
fa
fb
f
b
f
a
fa
fbf
b
f
b
f
a
fa
fbw
where 
3 323323
2 )1(3))7(1(3)1(36))61(1(3)1(3)1( fbfbbfbfbffbbfbfa ?++?+?++++???+?=
Appendix B. Finding the lower limit w3 of the two-valued region 
on the graph of p(w) in Eq. (2.10).
Defining a function
F?p,w?= f ? b?1?w???1?w?2?p?3/2 ? w?w2?p?3/2 , (2.B.1)
28
we can rewrite (2.10) as F(p, w) = 0. From the graph it is seen that at w3, dw/dp = 0. Since 
F(p, w) = 0, dF/dp = 0 as well. On the other hand, F(w, p) = F(w(p), p) = 0 and 
dF
dp=
?F
?w
dw
dp ?
?F
? p =0  (2.B.2)
from where we get 
dw
dp =?
?F/? p
?F/?w  (2.B.3)
Setting the right side of (2.B.1) and (2.B.3) to zero, we obtain the system of two equations, 
solving which for w will give us the point on the contour plot of F(p, w) = 0 where the derivative 
dw/dp vanishes, i. e., the desired point. Excluding p from the system, we reduce the equation to
f 2/5?2w3?1?3/5=w32/5?b2/5?1?w3?2/5  (2.B.4)
where w was renamed to w3 for clarity. This is (2.11) of this chapter.
29
Chapter 3. Effect of Plasma Screening on Circular States of Diatomic Rydberg 
Quasimolecules and their Application to Continuum Lowering in Plasmas
3.1. Introduction
Circular Rydberg States (CRS) of atomic and molecular systems containing only one electron 
have been extensively studied both theoretically and experimentally for several reasons (see, e.g., 
[1.18 ? 1.20, 1.24 ? 1.37, 3.1] and references therein). First, they have long radiative lifetimes 
and highly anisotropic collision cross sections, thereby enabling experiments on inhibited 
spontaneous emission and cold Rydberg gases. Second, these classical states correspond to 
quantal coherent states, i.e., objects of fundamental importance. Third, a classical description of 
these states is the primary term in the quantal method based on the 1/n-expansion (n is the 
principal quantum number). Fourth, they can be used in developing atom chips.
   In the previous chapters we studied analytically CRS of the two-Coulomb-center system, the 
system (denoted as ZeZ') consisting of two nuclei of charges Z and Z', separated by a distance R, 
and one electron ? see also [1.18 ? 1.20, 1.24, 1.25, 1.34, 3.1]. Energy terms of these Rydberg 
quasimolecules were obtained for a field-free case, as well as under an electric field or under a 
magnetic field.
   The Rydberg quasimolecules of this type naturally occur in high density plasmas of several 
types of ions, where a fully-stripped ion of the charge Z' is in the proximity of a hydrogenlike ion 
of the nuclear charge Z (or where a fully-stripped ion of the charge Z is in the proximity of a 
30
hydrogenlike ion of the nuclear charge Z'). Therefore in the present chapter we study the effects 
of plasma screening on CRS of these Rydberg quasimolecules ? the effects not taken into 
account in the previous works. We provide analytical results for weak screening, as well as 
numerical results for moderate and strong screening. We show that the screening leads to the 
following consequences.
   The screening causes an additional energy term to appear ? compared to the absence of the 
screening. This new term has a V-type crossing with the lowest energy term. The internuclear 
potential is also affected by the screening, destabilizing the nuclear motion for Z > 1 and 
stabilizing it for Z = 1. 
   We also study the effect of plasma screening on continuum lowering (CL) in the ionization 
channel. CL was studied for over 50 years ? see, e.g., books/reviews [3.2 ? 3.6] and references 
therein. Calculations of CL evolved from ion sphere models to dicenter models of the plasma 
state [3.4, 3.7 ? 3.12]. One of such theories ? a percolation theory [3.4, 3.9] ? calculated CL 
defined as an absolute value of energy at which an electron becomes bound to a macroscopic 
portion of plasma ions (a quasi-ionization). In 2001 the value of CL in the ionization channel was 
derived analytically, which was disregarded in the percolation theory: a quasimolecule, 
consisting of the two ion centers plus an electron, can get ionized in a true sense of this word 
before the electron would be shared by more than two ions [1.21]. It was also shown in [1.21] 
that, whether the electron is bound primarily by the smaller or by the larger out of two positive 
charges Z and Z', makes a dramatic qualitative and quantitative difference for this ionization 
channel. The results in [1.21] were obtained for circular states of the corresponding Rydberg 
quasimolecules.
31
   In the present chapter we show that the screening decreases CL in the ionization channel, 
making CL vanish as the screening factor increases.
3.2. Calculation of the effect of plasma screening and classical energy terms for a Rydberg 
quasimolecule in a circular state
Plasma screening of a test charge is a well-known phenomenon. For a hydrogen atom or a 
hydrogen-like ion (an H-atom, for short), it is effected by replacing the pure Coulomb potential 
by a screened Coulomb potential which contains a physical parameter ? the screening length a. 
For example, the Debye-H?ckel (or Debye) interaction of an electron with the electronic shielded 
field of an ion of charge Z is U(R) = ?(Ze2/R)exp(?R/a), where 
a = (kT / (4?e2Ne))1/2 ? 1.304 ? 104(1010/Ne)1/2T1/2a0 , where Ne (cm?3) and T (K) are the electron 
density and temperature, respectively.
   We study a two-Coulomb center (TCC) system with the charge Z placed at the origin, and the 
Oz axis is directed at the charge Z', which is at z = R, the system being submerged in a plasma of 
a screening length a. We consider the circular orbits of the electron which are perpendicular to 
the internuclear axis and centered on the axis.
   Two quantities, the energy E and the projection L of the angular momentum on the internuclear 
axis are conserved in this configuration. We use cylindrical coordinates to write the equations for 
both:
E=12 ? ??2??2 ??2??z2??Zr e?r/a?Z'r' e?r'/a  (3.1)
32
L=?2 ??  (3.2)
where r and r' are distances from the electron to Z and Z'. The circular motion implies that 
d?/dt = 0; as the motion occurs in the plane perpendicular to the z-axis, dz/dt = 0. Further, 
expressing r and r' through ? and z, and taking d?/dt from (3.2), we have:
E= L
2
2?2?
Z
??2?z2 e
???2?z2/a? Z'
??2??R?z?2 e
???2??R?z?2/a  (3.3)
With the scaled quantities
w= zR, p=??R?
2
, b=Z'Z , ?=?ERZ , ?= L?ZR , ?=Ra , r=ZRL2  (3.4)
our energy equation takes the form below:
?=e
???w2? p
?w2?p?
be????1?w?2?p
??1?w?2?p?
?2
2 p
 (3.5)
   We can seek the equilibrium points by finding partial derivatives of ? by the scaled coordinates 
w, p and setting them equal to zero. This will give the following two equations.
we???w2?p
w2?p ?
1
?w2?p???=
b?1?w?e????1?w?2?p
?1?w?2?p ?
1
??1?w?2?p???
 (3.6)
?2
p2=
e???w2?p
w2?p ?
1
?w2?p????
be????1?w?2?p
?1?w?2?p ?
1
??1?w?2?p???
 (3.7)
From the definitions of the scaled quantities (3.4), ?2 = 1/r and E = ? (Z/R) ?. Since r = ZR/L2, 
33
E = ? (Z/L)2 ?/r, where r = 1/?2 can be obtained by solving (3.7) for ?. Thus, the scaled energy 
without explicit dependence on R is ?/r, which we shall denote ?1. Using this, equations (3.5), 
(3.6) and (3.7) can be transformed into the following three master equations for this 
configuration.
?1=? p?1???w
2?p?e???w2? p
?1?w??w2?p?3/2 ?
2??1?w??w2?p?
1???w2?p
? w??1?w?
2?p?
1????1?w?2?p
? p2 ?  (3.8)
r=?1?w??w
2?p?3/2e??w2? p
p2?1???w2?p?
 (3.9)
w?1???w2?p?e???w2?p
?w2?p?3/2 =
b?1?w??1????1?w?2?p?e????1?w?2?p
??1?w?2?p?3/2
 (3.10)
The quantities ?1 and r now depend only on the coordinates w and p (besides the constant ?). 
Therefore, if we resolve (3.10) for p and substitute it to (3.8) and (3.9), we obtain the parametric 
solution for the energy terms ?1(r) with the parameter w for the given b and ?.
   Equation (3.10) does not allow an exact analytical solution for p. Therefore, we will use an 
approximate analytical method.
   Below is the contour plot of this equation for b = 3 and ? = 0.1.
34
Fig. 3.1. Contour plot of equation (3.10) for b = 3 and ? = 0.1.
As in [1.19] and Chapter 2, the plot has two branches, the left one spanning from w = 0 to 
w = w1, and the right one from the asymptote w = w3 to w = 1. w1 is a solution of the equation
?1?w1?2 ?1??w1?e??1?2w1?=bw12?1???1?2w1??  (3.11)
in the interval 0 < w1 < 1, and w3 does not depend on ? and equals b/(1 + b) ? the same as in 
[1.19] for ? = 0 ? the "default" case described in Chapter 1. As ? increases, w1 and the p-
coordinate of the maximum of the left branch increase, but the general shape of both curves is 
preserved. Below is the plot for a relatively strong ? = 2.
35
Fig. 3.2. Contour plot of equation (3.10) for b = 3 and ? = 2.
   An approximation was made for small values of ?. Approximating (3.10) in the first power of 
?, we obtain the expression involving only the second and higher powers of ?. Therefore, an 
attempt was made using the value of p(w) for ? = 0 presented in [1.19], which we shall denote as 
p0; it is the same as the quantity in (1.5). Further, taking the higher powers of ? into account, we 
obtained the next-order approximation for p(w):
p?w?=p0??
2
6 ?1?2w??1??1?2w??
w2/3?b2/3?1?w?2/3
w2/3?b2/3?1?w?2/3 ?
2?  
(3.12)
where
p0=w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3  (3.13)
36
? the zero-? value as in (1.5).
   Equation (3.11) can be approximated by substituting 1 + ?(1 ? 2w1) in place of exp(?(1 ? 2w1)), 
which will render it a 4th-degree polynomial in w1. The analytical expression for it is given in 
Appendix A.
   Substituting (3.12) into (3.8) and (3.9), we obtain the approximate parametric solution for the 
energy terms ? ?1(r) by running the parameter w on 0 < w < w1 and w3 < w < 1. Empirically, by 
comparison with the numerical results, it was found that using the value of p from (3.13) on the 
0 < w < w1 range and from (3.12) on the w3 < w < 1 range gives the best approximate results. 
Below are the approximate terms for b = 3 and different values of ?.
Fig. 3.3. Approximate classical energy terms for b = 3 at ? = 0.1.
37
Fig. 3.4. Approximate classical energy terms for b = 3 at ? = 0.2.
Fig. 3.5. Approximate classical energy terms for b = 3 at ? = 0.3.
38
   A numerical solution has also been made. It confirmed that the analytical solution was a good 
approximation for ? < 0.3. Below are the terms plotted for selected arbitrary values of ?.
Fig. 3.6. Numerical classical energy terms for b = 3 at ? = 0.2.
Fig. 3.7. Numerical classical energy terms for b = 3 at ? = 0.5.
39
Fig. 3.8. Numerical classical energy terms for b = 3 at ? = 1.
Fig. 3.9. Numerical classical energy terms for b = 3 at ? = 2.
40
Fig. 3.10. Numerical classical energy terms for b = 3 at ? = 3.
   The following reminder should be made. The above plots represent ?classical energy terms? of 
the same symmetry. (In physics of diatomic molecules, the terminology "energy terms of the 
same symmetry" means the energy terms of the same projection of the angular momentum on the 
internuclear axis.) For a given R and L, the classical energy E takes only several discrete values. 
However, as L varies over a continuous set of values, so does the classical energy E (as it should 
be in classical physics).
3.3. Crossings of the energy terms
Several properties of these energy terms have been studied. We note that in case of small or 
41
moderate ?, we observe four terms, both pairs of which have a V-type crossing. As an example, 
we shall take the plot of the terms for ? = 0.2 and number the lowest term 1 and the highest term 
2; the remaining terms will be numbered 3 and 4, from the lower one to the higher one. 
Therefore, terms 1 and 2 and terms 3 and 4 undergo V-type crossings, to which we shall refer as 
V12 and V34. Using a small-? approximation by choosing (3.13) as the p(w) solution for the 
parametric energy terms (essentially, a zero-? approximation), we can substitute (3.13) into (3.9), 
which will give it the form below.
r=
?1?2w?3/2 ?b2/3?? w1?w?
2/3
w3?b2/3?? 1?ww ?
4/3?2  (3.14)
For a given b, the terms 3 and 4 are produced by varying w between 0 and w1. The V34 crossing 
occurs at the value of w where r(w) has a minimum [1.19]. Therefore, setting the derivative 
dr/dw to zero, we obtain the equation whose solution for w in the range 0 < w < w1 gives us the 
point on the parametric axis which produces the V34 crossing.
9w4/3?1?w?4/3?w4/3?b4/3?1?w?4/3?=b2/3?1?4w?22w2?36w3?18w4?  (3.15)
This equation has no dependence on ? and is therefore equivalent to the Coulomb-potential case 
described in Chapter 1. Therefore, an analytical solution exists for (3.15), which is shown in 
(1.13). Going back to the w-parametrization, we obtain the analytical solution of (3.15):
42
wV34= 1
1??b1/3??b?1?
1/3
b1/6 ???b?1?
1/3???b?1?1/3??
3/2  (3.16)
Substituting it into (3.9) and using the numerical solution for p of (3.10), we obtain the semi-
analytical dependence rV34(?) for a given b. Below is the plot for b = 3.
Fig. 3.11. Semi-analytical plot of rV34(?) for b = 3.
   Since the plot in Fig. 3.11 was obtained using a zero-? approximation for the point of the V34 
crossing, we also graphed this dependence numerically point by point. The graph below for the 
same b shows that in relation to terms 3 and 4, this approximation works well even for moderate 
values of ?.
43
Fig. 3.12. Numerical plot of rV34(?) for b = 3.
   The energy of the V34 crossing can be obtained semi-analytically by substituting the numerical 
solution for p of (3.10) into (3.8), and by further substituting (3.16) into the resulting formula. It 
could be seen that as ? grows, the energy of the crossing grows and at a relatively large ? 
becomes positive. A numerical graph can also be made in a fashion similar to Fig. 3.12. A visual 
comparison shows a good similarity between the two. Below, a numerical graph is given for 
b = 3.
Fig. 3.13. Numerical plot of ??V34(?) for b = 3.
44
   As we see, the energy of the V34 crossing becomes positive after ? = 2.96, has a maximum, 
and later asymptotically approaches zero. For b = 4/3, the V34 crossing reaches zero energy at 
? = 2.13.
   The shape of the terms 3 and 4 is also affected by the screening. Term 3, whose energy 
increases as r increases, becomes nearly horizontal at energy ?0.5 at a certain value of ?; at 
further ?, its energy decreases with r. For b = 3, this value of ? is about 1.1; for b = 4/3, it is 
about 0.7. The plots are shown below.
Fig. 3.14. Classical energy terms for b = 3 at ? = 1.1; 
term 3 is nearly constant at energy ??1 = ?0.5.
45
Fig. 3.15. The same as Fig. 3.14 for b = 4/3 at ? = 0.7.
   For V12 crossing, the small-? approximation is not applicable since this crossing is not 
observed at ? = 0. Therefore, only numerical methods were used. A situation of particular interest 
is the behavior of term 1 at very small r, because as r ? 0 it corresponds to the energy of the 
hydrogenic ion of the nuclear charge Z + Z' [1.25]. The point with the smallest r is the V12 
crossing. A comparison was made of the dependence of the electronic energy on the screening 
parameter ? between [1.25] and the limiting case r ? 0 in our situation. Since in the paper 
mentioned above, the calculation was performed for a single Coulomb center Z, we had to re-
scale the quantities to make a valid comparison. The electronic energies are related as 
?1(TCC) = (1 + b)2?1(OCC), where OCC stands for "one Coulomb center". Since the scaling for the 
screening parameter in the OCC case did not include R (the internuclear distance), the scaling 
factor between the screening parameter includes r: ?(TCC) = r(1 + b)?(OCC). Taking this into 
46
account, we can plot the energy dependence on ? for the limiting case r ? 0.
Fig. 3.16. Plot of the energy of the electron versus the scaled screening factor for b = 3 in the limit r ? 0.
Below is the dependence obtained in [1.25] for OCC:
Fig. 3.17. Plot of the energy of the electron in a OCC system versus the scaled screening factor.
47
3.4. The effect of the plasma screening on the internuclear potential
Another aspect of this problem worth studying is the internuclear potential. Previously its 
properties were studied for the same system with ? = 0 and a magnetic field parallel to the 
internuclear axis [1.20]. Particularly, the magnetic field created a deep minimum in the 
internuclear potential, which stabilized the nuclear motion and transformed a Rydberg quasi-
molecule into a real molecule. Here we shall investigate the effect of the screening on the 
internuclear potential. Its form in atomic units is
Uint=ZZ'R ?E  (3.17)
where E is the electronic energy. Using the scaled quantities from (3.4), we have the scaled 
internuclear potential
uint=bZr ??1  (3.18)
where Uint = (Z/L)2uint. By plotting its dependence on r, we found out that in cases of Z > 1 the 
screening tends to flatten the minimum, producing the effect opposite to the one of the magnetic 
field. Compare the plots of uint(r) in the case of Z = 2, b = 2 for ? = 0 and ? = 0.3.
48
Fig. 3.18. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 2, 
Z' = 4, ? = 0.
Fig. 3.19. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 2, 
Z' = 4, ? = 0.3.
49
   The screening increases the potential of the point of intersection of the two branches; the upper 
branch, which has a very shallow minimum at ? = 0, loses it as ? increases.
   A completely different behavior was observed for Z = 1. A small ? creates a deep minimum in 
the upper branch of the potential. For comparison, we present the plots of the potential in the 
case of Z = 1, b = 2 for ? = 0 and ? = 0.3.
Fig. 3.20. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 1, 
Z' = 2, ? = 0.
50
Fig. 3.21. The plot of the scaled internuclear potential versus the scaled internuclear distance for Z = 1, 
Z' = 2, ? = 0.3.
   The figures above reveal the case of the screening stabilization of the nuclear motion for the 
case of Z = 1 and destabilization for Z > 1.
3.5. The effect of the plasma screening on the continuum lowering
Our analysis of the stability of the electronic motion shows results similar to those obtained 
previously in [1.18, 1.19]. Namely, term 3 corresponds to a stable motion while term 4 ? to an 
unstable motion. So, the crossing point of terms 3 and 4 corresponds to the transition from the 
stable motion to the unstable motion, leading the electron to the zero energy (i.e., to the free 
motion) along term 4, which constitutes the ionization of the molecule.
51
   Therefore, we arrive at the following. For the ionization of the hydrogenlike ion of the 
nuclear charge Zmin perturbed by the charge Zmax, it is sufficient to reach the scaled energy 
?c(b) = ?(wV34(b), b) < 0. At that point, the electron switches to the unstable motion and the 
radius of its orbit increases without a limit. This constitutes CL by the amount of 
Z<1/R> | ?(wV34(b), b) |, where <1/R> is the value of the inverse distance of the nearest 
neighbor ion from the radiating ion averaged over the ensemble of perturbing ions.
   Thus, obtaining CL in the ionization channel requires calculations of the scaled energy ? at the 
crossing point wV34 of terms 3 and 4.
   CL for the "default" (? = 0) TCC system was studied in [1.21]. Particularly, the scaled CL 
energy ?c(b) = ?(wV34(b), b) = ?E/(Z<1/R>) was graphed on a double-logarithmic scale, where ? 
is defined in (1.3) and wV34 is given by (3.16). The graph is given below; "lg x" stands for 
"log10 x".
Fig. 3.22. The plot of the CL energy versus b on a double logarithmic scale for ? = 0.
   We have made several plots of ?c(b) for several values of ?. A numerical value for wV34 was 
taken to increase precision.
52
Fig. 3.23. The plot of the CL energy versus b on a double logarithmic scale for ? = 0.1.
Fig. 3.24. The plot of the CL energy versus b on a double logarithmic scale for ? = 0.5.
Fig. 3.25. The plot of the CL energy versus b on a double logarithmic scale for ? = 2.
53
From the figures above we can see that the plasma screening decreases the value of CL in the 
ionization channel. Also, starting from about ? = 1.7, we observe the "cutoff" value of b > 1, 
below which ?c becomes negative, i.e., the electron energy at wV34 becomes positive. This means 
that there is no more CL in this ionization channel ? instead, the continuum becomes higher than 
for the isolated hydrogen-like ion of the nuclear charge Z. This effect cannot be observed in the 
logarithmic graphs above because the cutoff value of energy (zero) corresponds to lg ?c = ??. 
Below we made the standard, non-logarithmic plots of ?c(b) for selected values of ? at which this 
effect is observed.
Fig. 3.26. The plot of the CL energy versus b for ? = 2.
54
Fig. 3.27. The plot of the CL energy versus b for ? = 3.
In Fig. 3.27 we can see that there is no CL for b = 2 and b = 3 at ? = 3.
   In Appendices B and C we present the effects of the electric and magnetic fields on CL. The 
effect of the magnetic field was observed to decrease the value of CL, similar to the case above, 
while the electric field increases the value of CL, promoting ionization.
3.6. Conclusions
We studied the effects of the plasma screening on the classical energy terms of the electron in the 
field of two Coulomb centers. We provided analytical results for the small values of the 
screening factor and numerical results for the medium values.
   We found that the plasma screening leads to the appearance of the fourth energy term ? in 
addition to the three classical energy terms with no screening. This term exhibits a V-type 
crossing with the lowest energy term. The two highest energy terms continue having a V-type 
55
crossing like at the zero field. 
   We studied the effect of the screening on the internuclear potential. We found that the nuclear 
motion was stabilized by screening for Z = 1 and destabilized for Z > 1.
   The effect of the screening on the continuum lowering was studied as well. The plasma 
screening decreases the value of CL in the ionization channel, similar to the effect of the 
magnetic field [1.20].
Appendix A. The analytical expression for the limit w1 in Eq. (3.11) in the small-? 
approximation.
56
57
58
Appendix B. The effect of the electric field on continuum lowering
Using the value of the scaled energy of the electron in the TCC system given in (2.5) with the 
substitution of the numeric or approximate solution for p from (2.10) into (2.5) and (2.7) and the 
further substitution of ? from (2.7) into (2.5), we obtain the dependence of the scaled energy on 
the scaled coordinate w in the situation considered in Chapter 2, where the electric field was 
parallel to the internuclear axis. Then we numerically find the point on the w-axis corresponding 
to the V34 crossing for a given value of the scaled electric field f and substitute it into the 
59
formula for the scaled energy, obtaining the critical energy, which is the value of CL.
   Below are a few logarithmic plots (lg ?c versus lg b) made for selected values of f.
Fig. 3.B.1. The plot of the CL energy versus b for f = 0.1.
Fig. 3.B.2. The plot of the CL energy versus b for f = 1.
60
Fig. 3.B.3. The plot of the CL energy versus b for f = 10.
   CL increases as the electric field increases. This is expected because the electric field promotes 
ionization.
Appendix C. The effect of the magnetic field on continuum lowering
In the case of the magnetic field B parallel to the internuclear axis, the default energy given in 
(1.2) will acquire an additional term
?L??
2?2
2  (3.C.1)
where ? = B/(2c) is the Larmor frequency. We apply the same method as we used in the 
beginning of each chapter to find the energy dependent on one spatial parameter. Substituting the 
scaled quantities as defined in (3.4) and defining ? = ?L3/Z2, taking the derivatives by w and p 
and setting them equal to zero, solving for p and ?, substituting them back into the formula for 
61
energy, and further substituting the parameter ? as given in (1.10), we arrive at the following 
expressions of the scaled energy ? = ?ER/Z: and the scaled internuclear distance r:
?=??
4?2??b2/3?2?3?1?????3?1??b2/3?2?1?
2 ???3?1?3/2 ?
? ?
2?b2/3??4?
??3?1?2?b2/3 ?2?1???????
2???
3?1?5/2?b2/3?2?1?3/2
?3??3?1?3/2 ?
 (3.C.2)
r= ??
3?1?4?b2/3?2?1?2
?4?b2/3??4?2??2???
3?1?5/2 ?b2/3?2?1?3/2
?3??3?1?3/2 ?
 (3.C.3)
   To find the point of the V34 crossing, we take the derivative of r by ? and set it equal to zero. 
The numerical solution for this equation determines the value of ? corresponding to the minimum 
of r(?) for given b and ?, which corresponds to the crossing. Substituting it to the expression for 
the energy in (3.C.2), we obtain ?c(b, ?) ? the dependence of CL on b for a given ?.
   Below we present several double-logarithmic plots, similar to those in Section 3.5 and 
Appendix B, for selected values of ?. 
Fig. 3.C.1. The plot of the CL energy versus b on a double-logarithmic scale for ? = 0.5.
62
Fig. 3.C.2. The plot of the CL energy versus b on a double-logarithmic scale for ? = 1.
Fig. 3.C.3. The plot of the CL energy versus b on a double-logarithmic scale for ? = 2.
   From the graphs in Figs. 3.C.1 ? 3.C.3 it is seen that the effect of the magnetic field on CL is 
similar to the effect of the plasma screening ? it decreases CL. The "cutoff" values of b, below 
which there is no CL for a given ?, are also observed as ? becomes large. For example, for 
? = 2.8 CL at b = 2 vanishes, so the values of b corresponding to CL start at b > 2. At ? = 4.3, 
CL starts at b > 3. (See Figs. 3.C.4 and 3.C.5 below.)
63
Fig. 3.C.4. The plot of the CL energy versus b for ? = 2.8.
Fig. 3.C.5. The plot of the CL energy versus b for ? = 4.3.
64
Chapter 4. Helical and Circular States of Diatomic Rydberg Quasimolecules in a Laser 
Field
4.1. Introduction
Circular states of atomic and molecular systems in general, as well as circular Rydberg states 
(CRS) in particular, have been extensively studied both theoretically and experimentally for 
several reasons (see, e.g., [1.18 ? 1.20, 1.24 ? 1.37] and references therein). Namely: (a) they 
have long radiative lifetimes and highly anisotropic collision cross sections, thereby enabling 
experiments on inhibited spontaneous emission and cold Rydberg gases, (b) these classical states 
correspond to quantal coherent states, objects of fundamental importance, (c) a classical 
description of these states is the primary term in the quantal method based on the 1/n-expansion, 
and (d) they can be used in developing atom chips.
   In the previous works [1.18 ? 1.20, 1.24, 1.25, 1.28, 1.34] and the previous chapters, analytical 
studies of circular Rydberg states of two-Coulomb-center systems consisting of two nuclei of 
charges Z and Z', separated by a distance R, and one electron were carried out. Energy terms of 
these Rydberg quasimolecules for a field-free case [1.18, 1.19] were obtained, as well as under a 
static electric field ([1.24] and Chapter 2) or under a static magnetic field [1.20], and crossings of 
the energy terms were studied ? the crossings that enhance charge exchange in these systems.
   The analysis was not confined to circular orbits of the electron. For example, paper [1.19] 
studied in detail helical Rydberg states of these Rydberg quasimolecules. In order to make those 
results more transparent, we briefly outline here the scheme of that analysis. In cylindrical 
65
coordinates (z, ?, ?) with the z-axis along the internuclear axis, using the axial symmetry of the 
problem, the z- and ?-motions can be separated from the ?-motion. The ?-motion can be then 
determined from the calculated ?-motion. Equilibrium points of the two-dimensional motion in 
the z?-space were studied and a condition distinguishing between two physically different cases, 
where the effective potential energy either has a two-dimensional minimum in the z?-space or 
has a saddle point in the z?-space, was explicitly derived. In particular, it turned out that the 
boundary between these two cases corresponds to the point of crossing of the upper and middle 
energy terms (out of the three energy terms in this system). For the stable motion, the trajectory 
was found to be a helix on the surface of a cone, with axis coinciding with the internuclear axis. 
In this helical state, the electron, while spiraling on the surface of the cone, oscillates between 
two end-circles which result from cutting the cone by two parallel planes perpendicular to its 
axis (Fig. 1.1).
   In the present chapter we study such Rydberg quasimolecules under a laser field. For the 
situation where the laser field is linearly-polarized along the internuclear axis, we found an 
analytical solution for the stable helical motion of the electron valid for wide ranges of the laser 
field strength and frequency. We also found resonances, corresponding to a laser-induced 
unstable motion of the electron, that result in the destruction of the helical states. For the 
situation where such Rydberg quasimolecules are under a circularly-polarized field, polarization 
plane being perpendicular to the internuclear axis, we found an analytical solution for circular 
Rydberg states valid for wide ranges of the laser field strength and frequency. We showed that 
both under the linearly-polarized laser field and under the circularly-polarized laser field, in the 
electron radiation spectrum in the addition to the primary spectral component at (or near) the 
66
unperturbed revolution frequency of the electron, there appear satellites. We found that for the 
case of the linearly-polarized laser field, the intensities of the satellites are proportional to the 
squares of the Bessel functions Jq2(s), (q = 1, 2, 3, ? ), where s is proportional to the laser field 
strength. As for the case of the circularly-polarized field, we demonstrated that there is a red shift 
of the primary spectral component ? the shift linearly proportional to the laser field strength.
4.2. Analytical solution for the case of a linearly-polarized laser field
We consider the case where the laser is polarized parallel to the internuclear axis and oscillates 
sinusoidally with the frequency ?. The angular momentum L is conserved here due to ?-
symmetry. The corresponding Hamiltonian is
H= p?
2?p
?
2
2 ?
L2
2?2?
Z
??2?z2?
Z'
??2??R?z?2 ?zFcos ?t
 
(4.1)
Below we scale all frequencies using the factor (R3/Z)1/2 : for example, the scaled laser frequency 
is ? = ?(R3/Z)1/2 . We also use scaled coordinates as in papers [1.18, 1.19] and the other chapters
w= zR, v= ?R  (4.2)
where R is the internuclear distance. The origin is at the location of charge Z.
   Without the electric field, in the vicinity of the equilibrium the motion in z?-space corresponds 
to a two-dimensional harmonic oscillator [1.19]. Its scaled eigen-frequencies are
67
??= 1?w2?v2?3/4 ? 11?w? 3w?
?w2?v2???1?w?2?v2?
 (4.3)
where the equilibrium value of v connected to w as follows ? as in (1.5) and in [1.18, 1.19]:
v?w,b?=?w
2/3?1?w?4/3?b2/3w2
b2/3?w2/3?1?w??2/3  (4.4)
The motion occurs on the axes (w', v'), which are the original axes (w, v) rotated by an angle ? 
given in [1.19]. The dependence of the angle ? on the scaled coordinate w can be expressed in 
the most compact form by introducing the notation as in (1.15):
?=? 1w?1?
1/3
 (4.5)
 In the ?-representation it has the form
?=12 arctg ??b
2/3 ?2?1???4?b2/3?
??b2/3???  (4.6)
The scaled eigen-frequencies ?? and ?+ are the scaled frequencies of small oscillations about the 
equilibrium along the coordinates w', v' accordingly.
   As we introduce the oscillating electric field, these oscillations become forced, with the forces 
F cos ? cos ?t on w' and F sin ? cos ?t on v'. Therefore, the deviations from equilibrium on 
(w', v') are (see, e.g., textbooks [4.1, 4.2])
68
?w'= f cos??
?
2??2 cos??, ?v'=
f sin?
??2??2 cos??  (4.7)
where ? = ?(R3/Z)1/2 and ? = t(Z/R3)1/2. Now we revert to the original coordinates (w, v) and 
obtain the equations of motion in the linearly-polarized oscillatory electric field in the vicinity of 
the equilibrium: the electron follows the circular path corresponding to the case with no electric 
field with deviations from equilibrium depending on the scaled time ?:
?w=f ? cos
2?
??2??2?
sin2?
??2??2?cos??,
?v= f sin?cos?? 1?
?
2??2?
1
??2??2?cos ??
 (4.8)
   From (4.8) it is seen that the strength and frequency of the laser field affect the amplitudes of 
the forced oscillations on w- and v-axes; in fact, these amplitudes are proportional to the field 
strength f. The frequencies of the forced oscillations on the axes are equal to that of the laser 
field, instead of ?? and ?+.
   Since the Hamiltonian from (4.1) does not depend on ?, the corresponding momentum is 
conserved:
p?=?2 d?dt ?L=const  (4.9)
We can re-write (4.9) in the scaled notation as
d?
d?=
?
v2???  (4.10)
69
where ? = L/(ZR)1/2 is the scaled angular momentum. Substituting in (4.10) v(?) = v0 + ?v(?), 
where v0(w) is the equilibrium value of the scaled radius v of the electron orbit from (4.4) and 
?v(?) is given by (4.8), we obtain
d?
dt ?
?
v02 ?
2?
v03 ?v???  (4.11)
which after the integration with respect to time yields:
??t?? ?v
0
2 ??
2?
?v03 f sin?cos??
1
??2??2?
1
??2??2 ?sin??  (4.12)
From (4.12) it is seen that the ?-motion is a rotation about the internuclear axis with the scaled 
frequency ?/v02, slightly modulated by oscillations of the scaled radius of the orbit v at the laser 
frequency ? (i.e., at the laser frequency ? in the usual notation).
   Thus, from (4.8) and (4.12) it is clear that the electron is bound to a conical surface which 
incorporates the original circular orbit. In Fig. 4.1 below the three-dimensional actual trajectory 
is plotted for b = 3, f = 1, ? = 1 at w = 0.2.
70
Fig. 4.1. The actual trajectory of the electron in the linearly-polarized laser
field for b = 3, f = 1, ? = 1 at w = 0.2. The z-axis is along the internuclear axis.
   The expression for ?(?) from (4.12), i.e., ?(t(Z/R3)1/2), enters the following Fourier-transform 
that determines the amplitude of the power spectrum of the electron radiation
Al???= 1??
0
?
dtcos??t???t? ZR3 ??  (4.13)
where ? is the radiation frequency measured, e.g., by a spectrometer. The sinusoidal modulation 
of the phase ? is analogous to the situation where hydrogen spectral lines are modified by an 
external monochromatic field at the frequency ?, the latter problem being solved analytically by 
Blochinzew as early as in 1933 [4.3] (a further study can be found, e.g., in book [4.4]).
   From Blochinzew?s results it follows for our case in the electron radiation spectrum, this 
71
helical motion should manifest as follows. The most intense emission would be at the frequency 
? = d?/dt of the rapid ?-motion. In addition, there will be satellites at the frequencies ? ? q?, 
where q = 1, 2, 3, ? , whose relative intensities Iq are controlled by the Bessel functions Jq(s):
Iq=Jq2 ?s?, s= 2??v
0
3 f sin?cos??
1
??2??2?
1
??2??2 ?  (4.14)
   The oscillatory motion of the electron in the z?-space with the laser frequency ? should lead 
also to the radiation at this frequency. However, since ? << ?, this spectral component would be 
far away from the primary spectral line and its satellites.
   From (4.8) it is also seen that there are resonances when the laser frequency is equal to one of 
the eigen-frequencies of the motion in the z?-space, i.e., when either ? = ?+ or ? = ?? . It turns 
out that these conditions yield three resonance points on the w-axis for the laser field frequency ? 
below a certain critical value ?c, or five resonance points for ? > ?c ? see the figures below.
72
Fig. 4.2. Eigen-frequencies of the motion in the z?-space ?+ (solid curves) and ?? (dashed curves) versus 
w, i.e., versus the scaled z-coordinate of original circular Rydberg state. The scaled laser frequency ? is 
shown by the horizontal straight line. The plot is for b = 3 and ? = 4. Three resonant points are seen.
Fig. 4.3. Same as in Fig. 4.2, but for b = 3 and ? = 9. Five resonant points are seen.
73
For instance, in the case of b = 3, for ? = 8, we observe resonances at the following five values of 
w: 0.02883, 0.1106, 0.2497, 0.9852, 0.9878. The critical value corresponds to the minimum of 
?+ (w) for a given b in the interval 0 < w < w1 at the equilibrium point (the equilibrium scaled 
coordinate v being expressed via w by (4.4)). Calculating the derivative of ?+ with respect to w 
and setting it equal to zero, we find the point of the minimum. The value of ?+ at this point will 
be equal to the critical value of the scaled laser frequency ?c. For example, for b = 3 at 
w = 0.17642 (the minimum of ?+ in Figs. 4.2 and 4.3) this critical value is ?c = 7.5944. As the 
ratio of nuclear charges b increases, so does also the critical value ?c of the scaled laser 
frequency.
   These resonances correspond to a laser-induced unstable motion of the electron that result in 
the destruction of the helical states. This is illustrated in Fig. 4.4 showing the three-dimensional 
actual trajectory of the electron (for various directions of its initial velocity) for a resonance case 
where b = 3, f = 1, ? = 8, and w = 0.111 (w = 0.111 is one of the three values of w, at which the 
laser frequency ? coincides with the eigen-frequency ?+). A striking difference is seen compared 
to the stable helical motion depicted in Fig. 4.1: the resonance destroyed the helical state.
74
Fig. 4.4. The actual trajectory of the electron (for various directions of its initial velocity) in the linearly-
polarized laser field for a resonance case where b = 3, f = 1, ? = 8, and w = 0.111. The z-axis is along the 
internuclear axis.
4.3. Analytical solution for the case of a circularly-polarized laser field
Now we consider the case of a circular polarization of the laser field, polarization plane being 
perpendicular to the internuclear axis. The laser field varies as
75
F=F?ex cos ?t?ey sin ?t?  (4.15)
where ex and ey are the unit vectors along the x- and y-axes, F is the amplitude and ? is the 
frequency. The Hamiltonian for the electron in this configuration will take the following form.
H=12 ? p?2?pz2? p?
2
?2 ??
Z
??2?z2 ?
Z'
??2??R?z?2?F?cos????0?
 (4.16)
where we introduced ?0 = ?t. As in [1.19], we consider ?-motion to be the rapid subsystem, i.e. 
d?/dt is much greater than the laser frequency ? and the frequencies of z- and ?-motion. The 
canonical equations for the ?-motion obtained from (4.16) are
d?
dt =
?H
? p?=
p?
?2
 (4.17)
d p?
dt =?
?H
?? =F ?sin????0?
 (4.18)
Combining (4.17) and (4.18), we get
d2?
dt2 =
F
? sin????0?  (4.19)
After a substitution ? ? ?0 = ? + ?, (4.19) becomes
d2?
dt2 =?
F
? sin?  (4.20)
76
which is the equation of motion of a mathematical pendulum of length ? in gravity F. Its two 
possible modes are libration and rotation; since ? is the rapid coordinate, we have the case of 
rotation. The solution for ?(t) is well-known and can be expressed in terms of Jacobi amplitude:
??t?=2am??t2 , 4F??2?  (4.21)
Here we denoted d?/dt at t = 0 as ?. For rapid rotations, the change in the angular speed on ? is 
insignificant compared to the initial speed and d?/dt ? ?.
   The expression for ?(t) enters the following Fourier-transform that determines the amplitude of 
the power spectrum of the electron radiation:
Ac??, 4F??2?= 1??
0
?
dt cos??t???t , 4F??2 ??  (4.22)
Figure 4.5 shows as an example the power spectrum of the electron radiation spectrum (i.e., Ac2) 
versus the dimensionless radiation frequency ?/? for the case where 4F/(??) = 0.1. It is seen 
that the most intense component in the spectrum is at the frequency ? approximately equal to, but 
slightly less than ?. It is also seen that the laser modulation of the primary frequency of the 
electron rotation results in a series of relatively small satellites of the primary spectral 
component.
77
Fig. 4.5. The power spectrum of the electron radiation P (in arbitrary units) versus the dimensionless 
radiation frequency ?/? for the case where 4F/(??) = 0.1. Here ? is the frequency of the electron 
radiation at the absence of the laser field. A certain width is assigned to all spectral components to display 
a continuous spectral line profile.
   The red shift of the primary spectral component can be calculated analytically as follows. Since 
?-motion is rapid, we can average the Hamiltonian in (4.16) with respect to time. Integrating 
(4.20) with the initial condition d?/dt = ?, we get
?2??d?dt ?
2
=4F? sin2 ?2  (4.23)
By averaging this equation with respect to time, we obtain
78
?2???d?dt ?
2?
=2F?  (4.24)
Thus, the ?-momentum term in the Hamiltonian (4.16) becomes
? p?2?2 ?=?2 ??d?dt ?2?=?2?2?1? 2F??2?  (4.25)
The last term in the Hamiltonian from (4.16) vanishes after the time averaging so that the time-
averaged Hamiltonian depends only on ?- and z-coordinates and their corresponding momenta. 
The result is the following quasi-stationary Hamiltonian with no explicit time dependence:
H=12 ? p?2?pz2?? Z?
?2?z2
? Z'?
?2??R?z?2
?12 ?2 ?2??F  (4.26)
Introducing the scaled quantities
w= zR, v= ?R, f =FR
2
Z , ?=??
R3
Z
 (4.27)
and using the Hamiltonian equations, we obtain the following two differential equations of 
motion:
? ?w= w?w2?v2?3/2? b?1?w???1?w?2?v2?3/2  (4.28)
??v=v? 1?w2?v2?3/2? b??1?w?2?v2?3/2??2??f  (4.29)
79
(the dot above the letter indicates the differentiation by the scaled time ? = t(Z/R3)1/2).
   In this section we consider these Rydberg quasimolecules in circular (not helical) states, so that 
the plane of the electron orbit has a stationary position on the internuclear axis. Therefore, the 
right-hand side of (4.28) vanishes and the relationship between w and v becomes the same as 
given by (4.4). This makes the scaled radius of the orbit v a constant as well.
   Since the angular momentum is L = ??2 for a stationary circular orbit, the averaging of the ?-
momentum in (4.25) is equivalent to changing L for L(1 ? F?3/L2). Using scaled units and the 
relationship L = ??2, we find out that the case of the circularly-polarized laser field is equivalent 
to a field-free case, but with an effective frequency ? given by the substitution:
????1????? f ?  (4.30)
where
????= ?
6??3?1?3/2??4?b2/3?3/2
??3?1?11/2?b2/3 ?2?1?3  (4.31)
 
   The quantity ? (?)? f is the red shift of the primary spectral component. This result is valid as 
long as the relative correction (?)? f to the unperturbed angular frequency ? of the electron 
remains relatively small. Figures 4.6 and 4.7 illustrate the situation for the case where the ratio of 
the nuclear charges is b = 2. On the horizontal axis is the scaled coordinate w, i.e., the scaled 
coordinate along the internuclear axis of the Rydberg quasimolecule. The solid curve, having two 
branches, shows the unperturbed angular frequency ? of the electron. The dashed curve shows 
80
the correction ? (?)? f . It is seen that the correction remains relatively small for the entire left 
branch of ? and for a significant part of the right branch of ?. (Figures 4.6 and 4.7 differ only by 
the range of the vertical scale, so that Fig. 4.6 allows to see more clearly the region where the 
solid and dashed curves intersect and the region of validity of the results for the right branch of 
?.) Physically, the left branch corresponds to the situation where the electron is primarily bound 
by the charge Z. The region of the right branch, where the correction is relatively small, 
physically corresponds to the situation where the electron is primarily bound by the charge Z'.
Fig. 4.6. Dependence of the unperturbed angular frequency ? of the electron (solid curve, two branches) 
and of the correction ? (?)? f for f = 1 (dashed curve) on the scaled coordinate w along the internuclear 
axis of the Rydberg quasimolecule.
81
Fig. 4.7. The same as in Fig. 4.6, but with better visible details in the region of the right branch of ?(w).
4.4. Conclusions
   While studying diatomic Rydberg quasimolecules under a laser field that is linearly-polarized 
along the internuclear axis, we found an analytical solution for the stable helical motion of the 
electron valid for wide ranges of the laser field strength and frequency. Namely, the linearly-
polarized laser field makes the motion in the z?-space to be forced oscillations at the frequency 
of the laser field. We also found resonances, corresponding to a laser-induced unstable motion of 
the electron, that result in the destruction of the helical states. For the situation where such 
Rydberg quasimolecules are under a circularly-polarized field, polarization plane being 
perpendicular to the internuclear axis, we found an analytical solution for circular Rydberg states 
valid for wide ranges of the laser field strength and frequency.
   We showed that both under the linearly-polarized laser field and under the circularly-polarized 
laser field, in the electron radiation spectrum in the addition to the primary spectral component at 
82
(or near) the unperturbed revolution frequency of the electron, there appear satellites. We found 
that for the case of the linearly-polarized laser field, the intensities of the satellites are 
proportional to the squares of the Bessel functions Jq2(s), (q = 1, 2, 3, ? ), where s is 
proportional to the laser field strength. As for the case of the circularly-polarized field, we 
demonstrated that there is a red shift of the primary spectral component ? the shift linearly 
proportional to the laser field strength.
   Under a laser field of a known strength, in the case of the linear polarization the observation of 
the satellites would be the confirmation of the helical electronic motion in the Rydberg 
quasimolecule, while in the case of the circular polarization the observation of the red shift of the 
primary spectral component would be the confirmation of the specific type of the phase 
modulation of the electronic motion described by (4.21). Conversely, if the laser field strength is 
unknown, both the relative intensities of the satellites and the red shift of the primary spectral 
component could be used for measuring the laser field strength.
83
Chapter 5. Attachment of an Electron by Muonic Hydrogen Atoms: Circular States 
5.1. Introduction
Studies of muonic atoms and molecules, where one of the electrons is substituted by the heavier 
lepton ??, have several applications. The first one is muon-catalyzed fusion (see, e.g., [5.1 ? 5.3] 
and references therein). When a muon replaces the electron either in the dde-molecule (D2+), 
which becomes the dd?-molecule, or in the dte-molecule, which becomes the dt?-molecule, the 
equilibrium internuclear distance becomes about 200 times smaller. At such small internuclear 
distances, the fusion can occur with a significant probability, which has been observed in dd? or 
even with a higher rate in dt? [5.1 ? 5.3]. The second application is a laser-control of nuclear 
processes. This has been discussed in the context of the interaction of muonic molecules with 
superintense laser fields [5.4]. Another application is a search for strongly interacting massive 
particles (SIMPs) proposed as dark matter candidates and as candidates for the lightest 
supersymmetric particle (see, e.g., [5.5] and references therein). SIMPs could bind to the nuclei 
of atoms, and would manifest themselves as anomalously heavy isotopes of known elements. By 
greatly increasing the nuclear mass, the presence of a SIMP in the nucleus effectively eliminates 
the well-known reduced mass correction in a hydrogenic atom. Muonic atoms are better 
candidates (than electronic atoms) for observing this effect because the muon's much larger mass 
(compared to the electron) amplifies the reduced mass correction [5.5]. This may be detectable in 
astrophysical objects [5.5]. 
84
   Another line of research is studies of the negative ion of hydrogen H?, which can be also 
denoted as epe-system (electron-proton-electron), constitute an important line of research in 
atomic physics and astrophysics. It has only one bound state ? the ground state having a 
relatively small bound energy of approximately 0.75 eV. This epe-system exhibits rich physics. 
Correlations between the two electrons are strong already in the ground state. With long-range 
Coulomb interactions between all three pairs of particles, the dynamics is particularly subtle in a 
range of energies 2 ? 3 eV on either side of the threshold for break-up into proton + electron + 
electron at infinity [5.6]. There are strong correlations in energy, angle, and spin degrees of 
freedom, so that perturbation theory and other similar methods fail [5.6]. Experimental studies of 
H? provided a testing ground for the theory of correlated multielectron systems. Compared to the 
helium atom, the structure of H? is even more strongly influenced by interelectron repulsion 
because the nuclear attraction is smaller for this system [5.7]. In addition to the above 
fundamental importance, the rich physics of H? is also important in studies of the ionosphere's D-
layer of the Earth atmosphere, the atmosphere of the Sun and other stars, and in development of 
particle accelerators [5.6]. 
   In this chapter we combine the above two lines of research: studies of muonic atoms/molecules 
and studies of negative hydrogen ion. Namely, we consider whether a muonic hydrogen atom can 
attach an electron and become a muonic negative hydrogen ion, i.e. ?pe-system. Specifically, we 
study a possibility of circular states in such system. We show that the muonic motion can 
represent a rapid subsystem, while the electronic motion can represent a slow subsystem ? the 
result that might seem counterintuitive. 
   First, we find analytically classical energy terms for the rapid subsystem at the frozen slow 
85
subsystem, i.e., for the quasimolecule where the muon rotates around the axis connecting the 
immobile proton and the immobile electron. The meaning of classical energy terms is explained 
below. We demonstrate that the muonic motion is stable. We also conduct the analytical 
relativistic treatment of the muonic motion. 
   Then we unfreeze the slow subsystem and analyse a slow revolution of the axis connecting the 
proton and electron. We derive the condition required for the validity of the separation into the 
rapid and slow subsystems. 
   Finally we show that the spectral lines, emitted by the muon in the quasimolecule ?pe, 
experience a red shift compared to the corresponding spectral lines that would have been emitted 
by the muon in a muonic hydrogen atom (in the ?p-subsystem). Observing this red shift should 
be one of the ways to detect the formation of such muonic negative hydrogen ions. 
   As for physical processes leading to the formation of muonic-electronic negative hydrogen 
ions, one of the processes could be the following:
e + ?p ? ?pe
(which sometimes might be followed by the decay ?pe ? ? + pe). Such formation of the ?pe-
systems was discussed, e.g., in paper [5.8], where these systems were called resonances. The 
theoretical approach based on the separation of rapid and slow subsystems requires in this case 
the muon to be in a state of a high angular momentum. Luckily, the experimental methods to 
create muonic hydrogen atoms ?p (necessary for the above reaction) lead to the muon being in a 
highly-excited state (see, e.g., review [5.9] and paper [5.10]). We also mention paper [5.11] 
where it has been shown, in particular, that the distribution of the muon principal quantum 
number in muonic hydrogen atoms peaks at larger and larger values with the increase of the 
86
energy of the muon incident on electronic hydrogen atoms.
5.2. Analytical solution for classical energy terms of the rapid subsystem
We consider a quasimolecule where a muon rotates in a circle perpendicular to and centered at 
the axis connecting a proton and an electron ? see Fig. 5.1. As we show below, in this 
configuration the muon may be considered the rapid subsystem while the proton and electron 
will be the slow subsystem, which essentially reduces the problem to the two stationary Coulomb 
center problem, where the effective stationary ?nuclei? will be the proton and electron. The 
straight line connecting the proton and electron will be called here ?internuclear? axis. We use 
the atomic units in this study.
Fig. 5.1. A muon rotating in a circle perpendicular to and centered at the axis connecting the proton and 
the electron. 
   Because of the difference of muon and electron masses, the muon-proton separation is much 
87
smaller than the electron-proton separation. Therefore, it should be expected that the spectral 
lines, emitted by this system, would be relatively close to the spectral lines emitted by muonic 
hydrogen atoms. In other words, the presence of the electron should result in a relatively small 
shift of the spectral lines (compared to muonic hydrogen atoms); however, this shift would be an 
important manifestation of the formation of the quasimolecule ?pe.
   A detailed classical analytical solution of the two stationary Coulomb center problem, where an 
electron revolves around nuclei of charges Z and Z', has been presented in papers [1.18, 1.19]. 
We base our results in part on the results obtained therein.
   The Hamiltonian of the rotating muon is
H= 12m? pz2?p?2? p?
2
?2 ??
Z
?z2??2?
Z'
??R?z?2??2
 (5.1)
where m is the mass of the muon (in atomic units m = 206.7682746), Z and Z' are the charges of 
the effective nuclei (in our case, Z = 1 and Z' = ?1), R is the distance between the effective 
nuclei, (?, ?, z) are the cylindrical coordinates, in which Z is at the origin and Z' is ar z = R, and 
(p?, p?, pz) are the corresponding momenta of the muon. 
   Since ? is a cyclic coordinate, the corresponding momentum is conserved:
?p??=const=L  (5.2)
With this substituted into (5.1), we obtain the Hamiltonian for the z- and ?-motions
Hz?= pz
2?p
?
2
2 ?Ueff ?z,??
 (5.3)
where an effective potential energy is
88
Ueff ?z,??= L
2
2m?2?
Z
?z2??2?
Z'
??R?z?2??2
 (5.4)
Because in a circular state pz = p? = 0, the total energy E(z, ?) = Ueff(z, ?).
   With Z = 1, Z' = ?1 and the scaled quantities
w= zR, v= ?R, ?=?ER, ?= L?mR, r=mRL2  (5.5)
we obtain the scaled energy ? of the muon:
?= 1?w2?v2? 1?
?1?w?2?v2
? ?
2
2v2
 (5.6)
The equilibrium condition with respect to the scaled coordinate w is ??/?w = 0; the result can be 
brought to the form:
??1?w?2?v2?3/2
?w2?v2?3/2 =
w?1
w
 (5.7)
Since the left-hand side of (5.7) is positive, the right-hand side must also be positive: (w ? 1)/w > 
0. Consequently, the allowed ranges of w here are ?? < w < 0 and 1 < w < +?. This means that 
equilibrium positions of the center of the muon orbit could exist (judging only by the equilibrium 
with respect to w) either beyond the proton or beyond the electron, but there are no equilibrium 
positions between the proton and electron.
   Solving (5.7) for v2 and denoting v2 = p, we obtain:
p?w?=w2/3?w?1?2/3?w2/3??w?1?2/3?  (5.8)
   The equilibrium condition with respect to the scaled coordinate v is ??/?v = 0, which yields
89
?2=p2? 1?w2?p?3/2? 1??1?w?2?p?3/2?  (5.9)
Since the left-hand side of (5.9) is positive, the right-hand side must be also positive. This entails 
the relation w2 + p < (1 ? w)2 + p, which simplifies to 2w ? 1 < 0, which requires w < 1/2.
   Thus, the equilibrium with respect to both w and v is possible only in the range ?? < w < 0, 
while in the second range, 1 < w < +? (derived from the equilibrium with respect to w only) 
there is no equilibrium with respect to v.
   From the last two relations in (5.5), we find r = 1/?2; thus
r=p?2? 1?w2?p?3/2 ? 1??1?w?2?p?3/2 ?
?1  
(5.10)
where p is given by (5.8). Therefore, the quantity r in (5.10) is the scaled "internuclear" distance 
dependent on the scaled internuclear coordinate w. 
   Now we substitute the value of ? from (5.9), as well as the value of p from (5.8) into (5.6), 
obtaining ?(w) ? the scaled energy of the muon dependent on the scaled internuclear coordinate 
w. Since E = ? ?/R and R = rL2/m, then E = ? (m/L2)?1 where ?1 = ?/r. The parametric dependence 
?1(r) will yield the energy terms. 
   The form of the parametric dependence ?1(r) can be significantly simplified by introducing a 
new parameter ? = (1 ? 1/w)1/3, as was shown in Chapter 1 starting from (1.10). The region 
? ? < w < 0 corresponds to 1 < ? < ?. The parametric dependence will then have the following 
form:
?1???= ?1???
4?1??2?2
2?1????2?2?1??2??4?
 (5.11)
90
r???=?1??
2??4?3/2
??1??2?2
 (5.12)
Classical energy terms given by the parametric dependence of the scaled energy ?1 = (L2/m)E on 
the scaled internuclear distance r = (m/L2)R are presented in Fig. 5.2.
Fig. 5.2. Classical energy terms: the scaled energy ??1 = (L2/m)E versus the 
scaled internuclear distance r = (m/L2)R.
   Fig. 5.2 actually contains two coinciding energy terms: there is a double degeneracy with 
respect to the sign of the projection of the muon angular momentum on the internuclear axis. We 
remind the readers that L is the absolute value of this projection ? in accordance to its definition 
in (5.2).
   The minimum value of R, corresponding to the point where the term starts, can be found from 
(5.12). The term starts at w = ? ?, which corresponds to ? = 1; taking the value of (5.12) at this 
point, we find 
91
Rmin=3
3/2
4
L2
m
 (5.13)
With the value of m = 206.7682746, (5.13) yields R = 0.00628258 L2.
   The following note might be useful once again. The plot in Fig. 5.1 represents two degenerate 
classical energy terms of "the same symmetry". (In physics of diatomic molecules, the 
terminology "energy terms of the same symmetry" means the energy terms of the same 
projection of the angular momentum on the internuclear axis.) For a given R and L, the classical 
energy E takes only one discrete value. However, as L varies over a continuous set of values, so 
does the classical energy E (as it should be in classical physics).
   The revolution frequency of the muon ? is
?= Lm?2 = LmR2v2 = LmR2 p  (5.14)
in accordance with the previously introduced notation p = v2 = (?/R)2. Since R = L2r/m (see 
(5.5)), then (5.14) becomes ? = (m /L3)f, where f = 1/(pr2). Using (5.12) for r(?) and (5.8) for 
p(w) with the substitution w = 1/(1 ? ?3), where ? > 1, we finally obtain:
?= mL3 f ???, f ???=?1??
2?3?1??3?2
?1??2??4?3
 (5.15)
where f(?) is the scaled muon revolution frequency. Fig. 5.3 shows the scaled muon revolution 
frequency f = (L3/m)? versus the scaled internuclear distance r = (m/L2)R.
92
Fig. 5.3. The scaled muon revolution frequency f = (L3/m)? versus 
the scaled internuclear distance r = (m/L2)R.
   It is seen that for almost all values of the scaled internuclear distance r = (m/L2)R, the scaled 
muon revolution frequency f = (L3/m)? is very close to its maximum value fmax = 1, 
corresponding to large values of R. (The quantity fmax can be easily found from (15) given that 
large values of R correspond to ? >> 1 and that this limit yields fmax = 1.) In other words, for 
almost all values of R, the muon revolution frequency ? is very close to its maximum value 
?max=mL3  (5.16)
In Sect. 5.3, we will compare the muon revolution frequency with the corresponding frequency 
of the electronic motion and derive the condition of validity of the separation into rapid and slow 
subsystems.
   To analyse the stability of the muon motion, corresponding to the degenerate classical energy 
terms, while considering a classical circular motion of a charged particle (which was the electron 
93
in [1.19]) in the field of two stationary Coulomb centers, using the same notation as in this 
chapter, it was shown [1.19] that the frequencies of small oscillations of the scaled coordinates w 
and v of the circular orbit around its equilibrium position are given by
??= 1?w2?p?3/4 ? 11?w?3wQ  (5.17)
where
Q=??w2?p???1?w?2?p?  (5.18)
These oscillations are in the directions (w', v') obtained by rotating the (w, v) coordinates by the 
angle ?:
?w'=?wcos???vsin?, ?v'=??wsin???vcos ?  (5.19)
where the "?" symbol stands for the small deviation from equilibrium. The angle ? is determined 
by the following relation:
?=12 arctg ?1?2w??pw?1?w??p  (5.20)
The quantity Q in (5.18) is always positive since it contains the squares of the coordinates. From 
(5.17) it is seen that the condition for both frequencies to be real is
1
1?w?
3w
Q  (5.21)
   For the frequency ?? to be real, (5.17) requires Q ? 3w(1 ? w). For any w < 0 (which is the 
allowed range of w), this inequality is satisfied: the left-hand side is always positive while the 
right-hand side is always negative.
94
   For the frequency ?+ to be real, the following function F(w) must be positive (in accordance 
with (5.17) and (5.18)):
F?w?=?w2?p???1?w?2?p??9w2?1?w?2  (5.22)
After replacing w by ? = (1 ? 1/w)1/3, (5.22) becomes
F???=?
2??2?1?2?1?4?2??4?
??3?1?4
 (5.23)
Since the allowed range of w < 0 corresponds to ? > 1, it is seen that F(?) is always positive.
   Thus, the corresponding classical energy terms corresponds to the stable motion.
5.3. Electronic motion and the validity of the scenario
Now we unfreeze the slow subsystem and analyse a slow revolution of the axis connecting the 
proton and electron, the electron executing a circular orbit. In accordance with the concept of 
separating rapid and slow subsystems, the rapid subsystem (the revolving muon) follows the 
adiabatic evolution of the slow subsystem. This means that the slow subsystem can be treated as 
a modified ?rigid rotator? consisting of the electron, the proton, and the ring, over which the 
muon charge is uniformly distributed, all distances within the system being fixed (see Fig. 5.1).
   The potential energy of the electron in atomic units (with the angular-momentum term) is
Ee= M
2
2R2?
1
R?
1
??2??R?z?2
 (5.24)
where M is the electronic angular momentum. Its derivative by R must vanish at equilibrium, 
95
which yields
dEe
dR =?
M2
R3 ?
1
R2?
R?z
??2??R?z?2?3/2=0
 (5.25)
which gives us the value of the scaled angular momentum
?e= M?R  (5.26)
corresponding to the equilibrium:
?e2=1? 1?w??1?w?2?p?3/2  (5.27)
where the scaled quantities w, p of the muon coordinates are defined in (5.5). Using the muon 
equilibrium condition from (5.7) with v2 denoted as p, we can represent (5.27) in the form
?e2=1? w?w2?p?3/2  (5.28)
After replacing w by ? = (1 ? 1/w)1/3, we obtain
?e???=?1??1???
2?1????2
?1????2?3/2
 (5.29)
   The electron revolution frequency is ? = M/R2 = ?e(?)/R3/2 given that M = ?e(?)R1/2 in 
accordance with (5.26). Since R = L2r(?)/m (see (5.5)) with r(?) given by (5.12), then from 
? = ?e(?)/R3/2 we obtain
?=m
3/2?
e???
L3r3/2???
 (5.30)
96
   From (5.15) and (5.30) we find the following ratio of the muon and electron revolution 
frequencies:
?
?=
1
?m
f ???r3/2???
?e???
 (5.31)
where f(?) is given in (5.15).
   In addition to the above relation R = L2r(?)/m, the same quantity R can be expressed from 
(5.26) as R = M2/?e2(?). Equating the right-hand sides of these two expressions, we obtain the 
equality L2r(?)/m = M2/?e2(?), from which it follows:
L
M=
?m
?e????r???
 (5.32)
   The combination of (5.31) and (5.32) represents an analytical dependence of the ratio of the 
muon and electron revolution frequencies ?/? versus the ratio of the muon and electron angular 
momenta L/M via the parameter ? as the latter varies from 1 to ?. This dependence is presented 
in Fig. 5.4.
97
Fig. 5.4. The ratio of the muon and electron revolution frequencies 
?/? versus the ratio of the muon and electron angular momenta L/M.
   For the separation into the rapid and slow subsystems to be valid, the ratio of frequencies ?/? 
should be significantly greater than unity. From Fig. 5.4 it is seen that this requires the ratio of 
angular momenta L/M to be noticeably greater than 20.
98
   There is another validity condition to be checked for this scenario. Namely, the revolution 
frequency ? of the muon must also be much greater than the inverse lifetime of the muon 1/Tlife, 
where Tlife = 2.2 ?s = 0.91 ? 1011 a.u.: ?Tlife >> 1. Since for almost all values of R, the muon 
revolution frequency ? is very close to its maximum value ?max = m/L3, as shown in Sect. 5.2, 
then the second validity condition can be estimated as (m/L3)Tlife >> 1, from which it follows
L?Lmax=?mTlife?1/3=26600  (5.33)
(we remind that m = 206.7682746 in atomic units). So, the second validity condition is fulfilled 
for any practically feasible value of the muon angular momentum L.
   Thus, for the ratio of angular momenta L/M noticeably greater than 20, we deal here with a 
muonic quasimolecule where the muon rapidly rotates about the axis connecting the proton and 
electron following a relatively slow rotation of this axis. 
5.4. Red shift of spectral lines compared to muonic hydrogen atoms
The muon, rotating in a circular orbit at the frequency ?(R), should emit a spectral line at this 
frequency. The maximum value ?max = m/L3 corresponds to the frequency of spectral lines 
emitted by the muonic hydrogen atom (by the ?p-subsystem). For the equilibrium value of the 
proton-electron separation ? just as for almost all values of R ? the frequency ? is slightly 
smaller than ?max . Therefore, the spectral lines, emitted by the muon in the quasimolecule ?pe, 
experience a red shift compared to the corresponding spectral lines that would have been emitted 
by the muon in a muonic hydrogen atom. The relative red shift ? is defined as follows
99
?=???0?
0
=?max???  (5.34)
where ? and ?0 are the wavelength of the spectral lines for the quasimolecule ?pe and the muonic 
hydrogen atom, respectively. Using (5.15), the relative red shift can be represented in the form
????= 1f ????1  (5.35)
where f(?) is given in (5.15).
   The combination of (5.35) and (5.32) represents an analytical dependence of the relative red 
shift ? on the ratio of the muon and electron angular momenta L/M via the parameter ? as the 
latter varies from 1 to ?. Figure 5.5 presents the dependence of ? on L/(m1/2M). In this form the 
dependence is ?universal?, i.e., valid for different values of the mass m: for example, it is valid 
also for the quasimolecule ?pe where there is a pion instead of the muon. Figure 5.6 presents the 
dependence of ? on L/M specifically for the quasimolecule ?pe.
100
Fig. 5.5. Universal dependence of the relative red shift ? of the spectral lines of the quasimolecule ?pe (or 
?pe) on L/(m1/2M), which is the ratio of the muon and electron angular momenta L/M divided by the 
square root of the mass m of the muon or pion.
101
Fig. 5.6. Dependence of the relative red shift ? of the spectral lines of the 
quasimolecule ?pe on the ratio of the muon and electron angular momenta L/M. 
   It is seen that is the relative red shift of the spectral lines is well within the spectral resolution 
??res/? of available spectrometers: ??res/? ~ (10?4 ? 10?5) as long as the ratio of the muon and 
electron angular momenta L/M < 80. Thus, this red shift can be observed and this would be one 
of the ways to detect the formation of such muonic negative hydrogen ions.
   Figure 5.7 presents the dependence of the relative red shift ? on the ratio of the muon and 
electron revolution frequencies ?/?. It is seen that the relative red shift decreases as the ratio of 
the muon and electron revolution frequencies increases, but it remains well within the spectral 
102
resolution ??res/? of available spectrometers.
Fig. 5.7. Dependence of the relative red shift ? on the ratio of 
the muon and electron revolution frequencies ?/?.
5.5. Conclusions
We studied whether a muonic hydrogen atom can attach an electron, the muon and the electron 
being in circular states. We showed that it is indeed possible for a muonic hydrogen atom to 
attach an electron and to become a muonic negative hydrogen ion. We demonstrated that in this 
103
case, the muonic motion can represent a rapid subsystem while the electronic motion can 
represent a slow subsystem ? the result that might seem counterintuitive. In other words, the 
muon rapidly revolves in a circular orbit about the axis connecting the proton and electron while 
this axis slowly rotates following a relatively slow electronic motion. 
   We used a classical analytical description to find the energy terms of such a system, i.e., 
dependence of the energy of the muon on the distance between the proton and electron. We 
found that there is a double-degenerate energy term. We demonstrated that it corresponds to a 
stable motion. We also conducted the analytical relativistic treatment of the muonic motion, 
which is presented in Appendix. It was found that the relativistic corrections are relatively small. 
Their relative value is ~ 1/(cL)2 ~ 0.5 ? 10?4/L2 (we remind the readers that here c = 137.036 is 
the speed of light in atomic units). 
   Then we unfroze the slow subsystem and analysed a slow revolution of the axis connecting the 
proton and electron. The slow subsystem can be treated as a modified ?rigid rotator? consisting 
of the electron, the proton, and the ring, over which the muon charge is uniformly distributed, all 
distances within the system being fixed. We derived the condition required for the validity of the 
separation into the rapid and slow subsystems. 
   Finally we showed that the spectral lines, emitted by the muon in the quasimolecule ?pe, 
experience a red shift compared to the corresponding spectral lines that would have been emitted 
by the muon in a muonic hydrogen atom (in the ?p-subsystem). The relative values of this red 
shift, which is a ?molecular? effect, are significantly greater than the resolution of available 
spectrometers and thus can be observed. Observing this red shift should be one of the ways to 
detect the formation of such muonic negative hydrogen ions. 
104
Appendix. Relativistic treatment of the muonic motion
The Hamiltonian of the rotating muon is
H=c?m2c2?pz2?p?2? p?
2
?2 ?
Z
?z2??2?
Z'
??R?z?2??2?mc
2  (5.A.1)
Since ? is a cyclic coordinate, the corresponding momentum is conserved:
?p??=const=L  (5.A.2)
With this substituted into (5.A.1) and taking into account that in a circular state, pz = p? = 0, we 
obtain the energy of the muon in a circular state
E=c?m2c2?L
2
?2?
Z
?z2??2?
Z'
??R?z?2??2 ?mc
2  (5.A.3)
   With Z = 1, Z' = ?1 and the scaled quantities
w= zR, v= ?R, ?=?ER, ?= LmcR , r=RL  (5.A.4)
we obtain the scaled energy ? of the muon:
105
?= 1?
w2?v2
? 1?
?1?w?2?v2
?mc2 R?1??1??
2
v2 ?  (5.A.5)
The equilibrium condition with respect to the scaled coordinate w is ??/?w = 0, which yields
p?w?=w2/3?w?1?2/3?w2/3??w?1?2/3?  (5.A.6)
where p = v2.
   The equilibrium condition with respect to the scaled coordinate v is ??/?v = 0, which yields:
?2= p
2
mc2 R ?1?
?2
p ?
1
?w2?p?3/2?
1
??1?w?2?p?3/2?  (5.A.7)
From the relation before last in (5.A.4), we find R = L/(mc?). Substituting this in (5.A.7), we can 
solve it for ? and obtain:
?=?c
2 L2
p4
1
? 1?w2?p?3/2? 1??1?w?2?p?3/2 ?2 ?
1
p?
?1/2
 (5.A.8)
From the last two relations in (5.A.4), we find r = 1/(mc?); thus
r= 1mc ?c
2 L2
p4
1
? 1?w2?p?3/2? 1??1?w?2?p?3/2?2?
1
p  (5.A.9)
where p is given by (5.A.6). Therefore, the quantity r in (5.A.9) is the scaled "internuclear" 
distance dependent on the scaled internuclear coordinate w for a given absolute value of the 
angular momentum projection on the internuclear axis L. 
   Now we substitute R = L/(mc?) and the value of ? from (5.A.8), as well as the value of p from 
106
(5.A.6) into (5.A.5), obtaining ?(w, L) ? the scaled energy of the muon dependent on the scaled 
internuclear coordinate w for a given value of the angular momentum L. Since E = ? ?/R and 
R = rL, then E = ? ?1/L where ?1 = ?/r. The parametric dependence E(R), where E = ? ?1/L and 
R = Lr will yield the energy terms for a given value of L. 
   After introducing the parameter ? = (1 ? 1/w)1/3, the parametric dependence takes the following 
form:
E??,L?=?mc2?1? 1?
?6???
?
??1????2
??3??  (5.A.10)
R??,L?=L
2
m
??6??
??1??2  (5.A.11)
where quantities ? and ? are defined as follows:
?=?1??
2??4
1??2 , ?=?
1??3
cL ?
2
 (5.A.12)
The revolution frequency of the muon is
?=mc
2
L
?
?6?? ?1?
?
?6??  (5.A.13)
Let us check the degree of the relativity of the muon motion. Figure 5.A.1 shows the ratio ? of 
the muon velocity to the speed of light versus the "internuclear" distance R for L = 1 and L = 3. It 
is seen that for all values of R ? n2 (n = 1, 2, 3, ...), this ratio is practically equal to some constant 
value ?max. It is easy to find that ?max = 1/(cL) = 1/(137.036 L) ? since c = 137.036 in a.u.
107
Fig. 5.A.1. The ratio ? of the muon velocity to the speed of light versus the ?internuclear? 
distance R (a.u.) for L = 1 (the upper curve) and L = 3 (the lower curve).
   It is interesting to compare the above ?max with the corresponding average value of ?e for the 
electron motion in hydrogen atoms: ?e = 1/(cn). So, ?max for the muon motion differs from ?e for 
the electron motion in hydrogen atoms only by the substitution of the principal quantum number 
n of the electron by the angular momentum quantum number L of the muon.
   Thus, even for L = 1 (for which ?max is the highest), the muon motion is only weakly-
relativistic. The relativistic correction to the average frequency of the muon radiation is ~ 1/(cL)2 
(a.u.), where c = 137.036. Thus, the relative correction is insignificant even for L ~ 1 and it 
rapidly diminishes as L grows: for example, it is ~ 10?5 for L = 3 and ~ 10?7 for L = 15.
108
Chapter 6. Super-generalized Runge-Lenz Vector
6.1. Introduction
A so-called Two Centers Problem (TCP) has the following two mathematically equivalent, but 
physically different embodiments. The first one is the motion of an electron in the field of two 
stationary Coulomb centers of charges Z and Z' separated by a distance R, which is one of the 
most fundamental problems in quantum mechanics (see, e.g., [1.10, 1.11]). The second one is the 
motion of a planet in the gravitational field of two stationary stars of generally different masses, 
which is one of the most fundamental problems in celestial mechanics (see, e.g., [6.1, 6.2]). The 
geometrical symmetry of the TCP dictates the conservation of the energy and of the projection of 
the angular momentum on the internuclear (or interstellar) axis. It is also well-known that the 
TCP possesses a higher than geometrical symmetry and that there should be an additional 
integral of the motion. The existence of the additional integral of the motion is intimately 
connected with the fact that the TCP allows the separation of variables in the elliptical 
coordinates ? the fact shown as early as in 1760 by Euler [6.3] (see also [6.4, 6.5]).
   In the limit of large R, the problem of two Coulomb centers reduces to the problem of a 
hydrogenic ion of the nuclear charge Z in the uniform electric field F = Z'/R2, which is another 
fundamental problem in quantum mechanics (the corresponding celestial problem reduces to the 
problem of the one-center Kepler system in the uniform gravitational field). This simpler quantal 
problem also possesses higher than geometrical symmetry (connected with the fact that this 
109
problem allows the separation of variables in the parabolic coordinates). The corresponding 
integral of the motion is known as a projection of a generalized Runge-Lenz vector on the 
internuclear axis. The generalized Runge-Lenz vector A for this simpler problem, introduced by 
Redmond in 1964 [6.6], and its projection Az on the axis Oz||F can be represented in the forms, 
respectively 
A=A0??r?F??r2 , Az=A0z??x
2?y2?F
2
 (6.1)
(atomic units are used throughout the chapter). Here A0 is the well-known Runge-Lenz vector for 
one isolated Coulomb center*:
A0=p?L?Z rr  (6.2)
We note that the corresponding expressions in Redmond?s paper [6.6] differed from (6.1), (6.2) 
by factor ? 1/Z.
   After Redmond introduced the generalized Runge-Lenz vector for the asymptotic case of the 
TCP, the challenge was to find out whether a super-generalization of the Runge-Lenz vector is 
possible for the general (non-asymptotic) TCP. At least, two groups of authors claimed that they 
accomplished this task. Namely, Krivchenkov and Liberman in 1968 [6.11] and Gurarie in 1992 
[6.12] presented expressions for a super-generalized Runge-Lenz vector and claimed that its 
* Vector A0 is also called Laplace-Runge-Lenz vector. It is interesting to note that none of these three 
scientists was the first to introduce this vector. Historically, it was introduced as early as in 1710 by 
Hermann [6.7, 6.8] and Bernoulli [6.9] and therefore is also called Hermann-Bernoulli vector or 
Ermanno-Bernoulli vector (different just by the spelling of the first author?s name, who is the same 
person in both cases) ? more details on this history can be found in [6.10].
110
projection on the internuclear (or interstellar) axis is conserved.
   In the present chapter, first, we show that their claims are incorrect: the projection of their 
super-generalized Runge-Lenz vector is not conserved (despite the fact that in the limit of large 
R, their expressions reduce to Redmond?s result). Second, we derive a correct super-generalized 
Runge-Lenz vector whose projection on the internuclear/interstellar axis does conserve. Third, 
since in the literature there are several expressions for the separation constant for the TCP ? the 
expressions not having the form of a projection of any vector on the internuclear/interstellar axis 
? we provide relations between those expressions and our result.
6.2. Super-generalized Runge-Lenz vector
Krivchenkov and Liberman [6.11] considered the TCP described by the Hamiltonian (or the 
Hamilton function):
H= p
2
2 ?
Z1
r1 ?
Z2
r2 ?
Z2
R
 (6.3)
The charge Z1 was placed at the origin and Z2 at z = R. They presented the following expression 
for a super-generalized Runge-Lenz vector:
A?KL?=p?L?Z1 r1r
1
?Z2 r2r
2
?Z2  (6.4)
In the limit of large R, it reduces to Redmond?s result from (6.1).
   We calculated the Poisson bracket [A(KL), H] of the projection of this vector on the internuclear 
axis with the Hamiltonian from [6.11]. Surprisingly, the result was not zero:
111
[A?KL?,H]=?2Z1r
1
3 r?L?ez?
2Z2
r23 R?x px?y py?
 (6.5)
Thus, the projection of Krivchenkov-Liberman?s vector on the internuclear axis does not 
conserve ? contrary to their claim.
   Gurarie [6.12] considered the TCP described by the Hamilton function:
H= p
2
2 ?
Z1
r1 ?
Z2
r2
 (6.6)
where Z1 and Z2 are placed into z = a and z = ?a, respectively (atomic units are also used here). 
He presented the following expression for a super-generalized Runge-Lenz vector:
A?G?=p?L?Z1 r?aez?r?ae
z?
?Z2 r?aez?r?ae
z?
 (6.7)
In the limit of large R, it reduces to Redmond?s result from (6.1).
   We calculated the Poisson bracket [A(G), H] of the projection of this vector on the interstellar 
axis with the Hamiltonian from [6.12]. Surprisingly, again the result was not zero:
[A?G?,H]=?2a?x px?y py?? Z1?r?ae
z?
3 ?
Z2
?r?aez?3 ?
 
(6.8)
Thus, the projection of Gurarie?s vector on the interstellar axis does not conserve ? contrary to 
his claim.
   To derive a correct expression for the super-generalized Runge-Lenz vector, we started from 
the Hamiltonian (or Hamilton function) given by (6.3) and followed the first few steps from 
Krivchenkov-Liberman?s paper [6.11], arriving at the same expression in the elliptical 
112
coordinates as they did, for the additional conserved quantity:
A= 1R 1w2?v2 ??w2?1??1?v2??pw2 ?pv2???w
2?1
1?v2 ?
1?v2
w2?1? p?
2??Z 1?wv
w?v ?
?Z' ?1?v??w?1?w?v
 
(6.9)
Then, using the relation between the elliptical coordinates and the Cartesian coordinates
x=R2 ??w2?1??1?v2?cos?, y=R2 ??w2?1??1?v2?sin?, z=R2 wv  (6.10)
we obtained from (6.9) the following: the super-generalized Runge-Lenz vector, whose 
projection on the internuclear axis is really conserved, has the form
A=p?L?L
2
R ez?Z
r
r?Z'
R?r
?R?r??Z'ez, ez=
R
R
 (6.11)
The Poisson bracket of the projection of the vector from (6.11) on the internuclear axis
Az=p?L?ez?L
2
R ?Z
z
r ?Z'
R?z
?R?r??Z'
 (6.12)
with the Hamilton function from (6.3) vanishes indeed.
   In the limit of large R, the expression from (6.12) reduces to the following asymptotic form
?Az?as=p?L?ez?L
2
R ?Z
z
r?Z'
x2?y2
2R2
 (6.13)
Here the direction of the z-axis is chosen from the charge Z to the charge Z' . Compared to 
Redmond?s result from (6.1), there is an extra term in (6.13): ?L2/R. At the first glance, this might 
seem puzzling. However, the Poisson bracket of ?L2/R with the Hamilton function from (6.3) 
yields a term of the order of 1/R3. Thus, the Poisson bracket of Aas from (6.13) with the Hamilton 
113
function from (6.3), being calculated up to (including) terms of the order of 1/R2, vanishes. This 
resolves what might have seemed as the puzzle.
   In the literature there are several expressions for the separation constant for the TCP ? the 
expressions not having the form of a projection of any vector on the internuclear/interstellar axis. 
Below we provide relations between those expressions and our result.
   In 1949, Erikson and Hill [6.13] presented the following expression for the separation constant:
?=L??L??2a?Z1cos ?1?Z2 cos?2?  (6.14)
where L' and L'' are the angular momenta of the electron with respect to Z1 and Z2, ?1 and ?2 are 
the angles between the radii-vectors r1 and r2 (directed from the nuclei to the electron) and the 
positive direction of the z-axis (from Z1 to Z2), and 2a is the internuclear distance.
   We found that it is related to our result from (6.12) as follows:
?=R?Z'?Az?  (6.15)
   In Landau-Lifshitz?s book of 1960 [6.5], there is the following expression for the separation 
constant:
?=?2? p?2? p?
2
?2 ??M
2?2m???
1 cos?1??2cos ?2?
 (6.16)
where ?1 and ?2 are the nuclear charges with the sign opposite to ours, ? and ? are the cylindrical 
coordinates corresponding to the z-axis which is the internuclear axis, p? and p? are the canonical 
momenta corresponding to these coordinates, m is the mass of the particle (in our units m = 1), M 
is the angular momentum of the electron with respect to the origin (which is at half the 
internuclear distance), 2? is the internuclear distance, and ?1 and ?2 are the angles between the 
114
radii-vectors r1 and r2 and the internuclear axis measured inside the triangle made by r1, r2 and 
2?.
   We found that it is related to our result from (6.12) as follows:
?=R?Az?Z'?  (6.17)
   In 1967, Coulson and Joseph [6.14] presented the following expression for the separation 
constant:
B=L2?a2 pz2?2aq1zr
1
? 2aq2zr
2
 (6.18)
where q1 and q2 are the nuclear charges with the sign opposite to ours, L is the angular 
momentum of the particle with respect to the origin (which is at half the internuclear distance), 
2a is the internuclear distance, r1 and r2 are the distances from the particle to the nuclei, z is the 
coordinate along the internuclear axis and pz is its corresponding canonical momentum.
   We found that it is related to our result from (6.12) as follows:
B=R?Z'?Az??R
2H
2
 (6.19)
where H is the Hamiltonian.
   In the book by Komarov, Ponomarev, and Slavyanov published in 1976 [1.11], the following 
expression for the separation constant was presented:
?=L2?R
2 p
z
2
4 ?
RZ1 z
r1 ?
RZ2 z
r2 ?
R2H
2
 (6.20)
   We found that it is related to our result from (6.12) as follows:
115
?=R?Z'?Az?  (6.21)
6.3. Conclusions
We showed that the expressions for a super-generalized Runge-Lenz vector presented for the 
TCP by Krivchenkov and Liberman [6.11] and by Gurarie [6.12] are incorrect. Its projection on 
the internuclear (or interstellar) axis is not conserved ? contrary to their claims.
   We derived a correct super-generalized Runge-Lenz vector, whose projection on the 
internuclear/interstellar axis does conserve. We also analysed the asymptotic form of the 
projection in the limit of large R.
   Finally, since in the literature there are several expressions for the separation constant for the 
TCP ? the expressions not having the form of a projection of any vector on the 
internuclear/interstellar axis ? we provided relations between those expressions and our result.
   The correct super-generalized Runge-Lenz vector for the TCP that we derived should be of a 
general theoretical interest since the TCP is one of the most fundamental problems in physics. It 
can also have practical applications: for example, it can be used as a necessary tool while 
applying to the TCP the robust perturbation theory for degenerate states (based on the integrals 
of the motion) developed by Oks and Uzer [6.15].
116
Chapter 7. Conclusions
In this work we studied classically two-Coulomb-center systems consisting of two nuclei of 
charges Z and Z' and an electron in the field of these nuclei in a circular or a helical state . These 
systems represent diatomic Rydberg quasimolecules encountered, e.g., in plasmas containing 
more than one kind of multicharged ions. Diatomic Rydberg quasimolecules are one of the most 
fundamental theoretical playgrounds for studying charge exchange. Charge exchange and 
crossings of corresponding energy levels that enhance charge exchange are strongly connected 
with problems of energy losses and of diagnostics in high temperature plasmas; besides, charge 
exchange is one of the most effective mechanisms for population inversion in the soft x-ray and 
VUV ranges. The classical approach is well-suited for Rydberg quasimolecules.
   Chapter 1 presented a history of classical studies of diatomic Rydberg quasimolecules. In these 
studies one of the goals was to find the energy terms of this system ? the dependence of the 
energy of the electron on the internuclear distance. There were three energy terms found, two of 
them crossing; the crossing had a V-shape.
   In Chapter 2, we considered diatomic Rydberg quasimolecules subjected to a static electric 
field parallel to the internuclear axis. First, it led to the appearance of the fourth energy term, 
which was absent at the zero field. This term had a V-crossing with the lowest energy term. 
Second ? more importantly ? the electric field caused additional crossings and these crossings 
had an X-shape. The X-crossings occurred at much larger internuclear distances than the V-
crossings, and one of them was found to significantly enhance charge exchange.
117
   In Chapter 3, diatomic Rydberg quasimolecules were considered in a plasma of a given 
screening length, which altered the original potential of either nucleus by an exponential factor 
(see (3.1)). As in the previously mentioned case, the screening led to the appearance of the fourth 
energy term having a V-type crossing with the lowest energy term. More importantly, it was 
found that the screening stabilizes the nuclear motion for Z = 1 and destabilizes it for Z > 1. We 
also found that a so-called continuum lowering in plasmas was impeded by the screening, 
creating the effect similar to that of the magnetic field and opposite to that of the electric field. 
The continuum lowering plays a key role in calculations of the equation of state, partition 
function, bound-free opacities, and other collisional atomic transitions in plasmas.
   In Chapter 4, diatomic Rydberg quasimolecules were studied in a laser field. For the situation 
where the laser field is linearly-polarized along the internuclear axis, we found an analytical 
solution for the stable helical motion of the electron valid for wide ranges of the laser field 
strength and frequency. We also found resonances, corresponding to a laser-induced unstable 
motion of the electron, that result in the destruction of the helical states. For the case of a 
circularly-polarized field, polarization plane being perpendicular to the internuclear axis, we 
found an analytical solution for circular Rydberg states valid for wide ranges of the laser field 
strength and frequency. For this case we demonstrated also that there is a red shift of the primary 
spectral component. We showed that both under the linearly-polarized laser field and under the 
circularly-polarized laser field, in the electron radiation spectrum in the addition to the primary 
spectral component at (or near) the unperturbed revolution frequency of the electron, there 
appear satellites. Under a laser field of a known strength, in the case of the linear polarization the 
observation of the satellites would be the confirmation of the helical electronic motion in the 
118
Rydberg quasimolecule, while in the case of the circular polarization the observation of the red 
shift of the primary spectral component would be the confirmation of the specific type of the 
phase modulation of the electronic motion. Conversely, if the laser field strength is unknown, 
both the relative intensities of the satellites and the red shift of the primary spectral component 
could be used for measuring the laser field strength.
   In Chapter 5, a system consisting of a proton, muon and an electron was studied. It was found 
that a muonic hydrogen atom can attach an electron, with the muon and electron being in circular 
states. The technique of the separation of rapid and slow subsystems was used, where the muon 
represented the rapid subsystem and the electron the slow subsystem. The energy terms of the 
rapid muon in the field of the two slow (quasi-static) proton and electron were found ? a double-
degenerate energy term, which corresponds to stable motion. By analysing the slow subsystem, 
we derived the validity conditions for separation of rapid and slow subsystems. Finally we 
showed that the spectral lines emitted by the muon experience a red shift compared to the 
corresponding spectral lines in a muonic hydrogen atom. Observing this red shift should be one 
of the ways to detect the formation of such muonic-electronic negative hydrogen ions. Studies of 
muonic atoms and molecules, where one of the electrons is substituted by the heavier lepton ??, 
have several applications, such as muon-catalyzed fusion, a laser-control of nuclear processes, 
and a search for strongly interacting massive particles proposed as dark matter candidates and as 
candidates for the lightest supersymmetric particle. Additionally, analytical relativistic treatment 
of the muonic motion was conducted, which showed that the relativistic corrections are relatively 
small.
   Chapter 6 presented our study of fundamental algebraic symmetry of the TCC systems leading 
119
to an additional conserved quantity: the projection of a super-generalized Runge-Lenz vector on 
the internuclear axis. We derived the correct super-generalized Runge-Lenz vector, whose 
projection on the internuclear axis is conserved, and showed that the corresponding expressions 
by other authors did not correspond to a conserved quantity and thus were incorrect. The correct 
super-generalized Runge-Lenz vector for the TCC systems that we derived should be of a 
general theoretical interest since the TCC systems represent one of the most fundamental 
problems in physics. It can also have practical applications: e.g., it can be used as a necessary 
tool while applying to the TCC systems the robust perturbation theory for degenerate states 
based on the integrals of the motion.
120
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