Short distance relativistic atom-atom forces

                                              J. F. Babb

                        Institute for Theoretical Atomic and Molecular Physics,

                            Harvard-Smithsonian Center for Astrophysics,

                          60 Garden Street, Cambridge, Massachusetts 02138

                                            (March 1, 1999)

      About fifty years ago two important papers appeared describing novel interactions. One,

  by Casimir, discussed the case of two interacting walls [1] and the other, by Casimir and

  Polder, considered the interactions between an atom and a wall and between two atoms [2].

  The history and some of the many interesting aspects of these interactions, their derivations,

  and their importance in field theory and atomic and molecular physics are summarized

  elsewhere [3- 5].  Indeed, recent experiments give strong evidence of the reality of both the

  atom-wall [6] and the wall-wall interactions [7] predicted in those two papers.

      Here I will focus on the connection between the QED result of Casimir and Polder and

  other results for relativistic atom-atom interactions at short distances of the order of, say,

  20 a0.  The interaction between an electron and an ion will also be considered.  For typical

  atomic systems these relativistic effects are very small corrections to the non-relativistic

  potentials arising from the van der Waals and Coulomb interactions for, respectively, the

  atom-atom interaction and the electron-ion interaction.

      Casimir and Polder used QED and old-fashioned perturbation theory and their result

  was subsequently duplicated by other authors with different methods,  cf. [3].  One way

  to write their result for the interaction potential V (R) between two spherically symmetric

  atoms separated by a distance R is as a one-dimensional integral

                                 1   Z 1                         2
                      V (R) = - _____    d! exp (-2ff!R)[ffd(i!)] P (!ffR);                        *
                                ssR6  0

  with P (x) = x4 + 2x3 + 5x2 + 6x + 3 and ff the fine structure constant.  Atomic units

h   = m = e = 1 are used throughout and in these units c = 1=ff. The function ffd(i!) is the

  dynamic electric dipole polarizability at imaginary frequency,

                                ffd(i!) =    fu=[(Eu - E0)2 + !2];                                 *

  where fu is the oscillator strength of state u and Eu - E0 is the transition frequency between

  the states u and 0, with 0 denoting the ground state of the atom, and the summation in (2)


   *Contribution for third issue of the Bulletin of the Asia Pacific Center for Theoretical Physics,

  Seoul, Korea.


includes an integration over continuum states. The function ffd(i!) is a smooth function of

! with no singularities. The limit of V (R) for asymptotically large separations of the atoms

can be obtained from the Casimir-Polder integral (1) yielding

                                       23 [ffd(0)]2
                             V (R) ! - ____________;      R ! 1:                                 (3)
                                       4ss  ffR7

    What about the limit for small R? The result is

                                  3   Z 1             2
                       V (R) ! - _____    d! [ffd(i!)] ;      R  137;                            (4)
                                 ssR6  0

and upon integration (4) yields

                                       V (R) ! -C6=R6;                                           (5)

where C6 is the van der Waals constant expressed as a double sum over oscillator strengths

and the correct form of the atom-atom interaction at short distances (say 20 a0) is re-

produced.  The van der Waals constant is of vast importance for all sorts of molecular

spectroscopic  and  atomic  collision  problems,  of  course.   The  result  for  two  H  atoms  is

C6 = 6:499 026 705::: and for studies of atomic collisions at ultracold temperatures C6 plays

a crucial role in characterizing the interactions [8]. So it is nice to see QED connect nicely

with non-relativistic molecular quantum mechanics. What is the next correction?

    If one more term is retained in the small R expansion then [9]

                                       C6     2 W4        3  3
                             V (R) = - ___+ ff  ____+ O(ff =R )                                  (6)
                                       R6       R4


                                       1 Z  1     2         2
                                  W4 = __     d! ! [ffd(i!)] :                                   (7)
                                       ss  0

By integrating (7),  the coefficient W4 can be expressed as a double sum over oscillator

strengths and it was evaluated for a small number of diatomic systems using various approx-

imations, both semi-empirical [10] and computational [11,12]. The result for two H atoms is

W4 = 0:462 807.  The derivations above assume that the two atoms are well-separated and

accordingly do not include considerations involving electron exchange.

    How do the results above connect with results from the Breit-Pauli approximation to

the Dirac equation?  The van der Waals potential was shown above to be the short range

limit of the QED result;  yet it is also the long-range limit of the molecular  interaction

potential.   The full power of quantum-chemical methods (recognized in the 1998 Nobel

Prize in Chemistry) enables, at least in principle, calculation of the molecular potential

by solution of the nonrelativistic Schr"odinger equation. Relativistic effects are treated using

perturbation theory on the terms in the Breit-Pauli Hamiltonian (or for molecules containing


high-Z atoms by solution of the Dirac equation.) The connection to (6) was given by Power

and Zienau [13,9] who showed using perturbation theory that the matrix element of the

orbit-orbit interaction Hoo reproduces the second term in (6) as R increases,

                               <"0|Hoo|"0> ! ff2 ____;      R ~ R0;                              (8)

where R0 is of order, say, 10 to 20 a0 and |"0> is the molecular ground electronic state wave

function. Therefore there is a smooth connection between the relativistic and Casimir-Polder


    This relativistic R-4  term might be studied by incorporating it into theoretical calcu-

lations of collisions of ultra-cold atoms, particularly for H-H, H-Li, and Li-Li where high

precision determinations of the molecular potentials are possible.  There are of course ad-

ditional subtle effects to be accounted for such as deviations from the Born-Oppenheimer

approximation through isotope effects and nonadiabatic terms (nonlocal terms arising from

the action of the nuclear kinetic energy operator on the electronic wave function) and addi-

tional relativistic terms like the p4 and Darwin terms, for example, but these are unrelated

to the Casimir-Polder result.

    The Casimir-Polder-type interaction between an electron and an ion is closely related

to that of the atom-atom interaction (3).  Kelsey and Spruch [14] exhibited the result for

asymptotic separations,

                                      11  ffffd(0)
                              U (R) ! ___ ________;      R ! 1;                                  (9)
                                      4ss    R5

where R now denotes the electron-ion distance.  They obtained (9) using QED and old-

fashioned perturbation theory and they considered the possibility of measurement of this

potential through spectroscopy of the Rydberg states of atoms. Later, the integral form of

U (R), analogous to (1), was obtained [15,16] yielding an expression not particularly more

complicated than (1) and which can be obtained essentially by replacing the polarizability

ffd(i!) of one of the atoms by the quantity 1=!2, which is an excellent approximation to the

polarizability of the weakly bound electron [17]. (Some care is required, however, due to the

additional Coulomb interaction present for the ion-electron case, see [16] for details.)  The

limit of the electron-ion "Casimir-Polder" potential for small R for an electron interacting

with an ion (of net charge Z - 1) is [18]

                               ff2 1        3  3                2
                       U (R) = ___ ___+ O(ff =R );      R  137=Z :                              (10)
                               Z2  R4

Similarly to the atom-atom case, the relativistic R-4  term in the ion-electron interaction

was derived alternatively using Hoo with perturbation theory on the non-relativistic wave

function of the Rydberg atom [19] providing a connection to the QED result (10).  This


term is a small correction to the much larger Coulomb interaction between the two charged

particles, but nevertheless, through much theoretical work by Drachman, Drake, and oth-

ers [20], there is definitive evidence for the first term of (10) from a long series of careful

measurements of energies of Rydberg states of the helium atom by Lundeen and collabo-

rators [20,21].  At present the asymptotic part of U (R), (9), has not been measured and

Hessels and collaborators [22] conclude from their measurements that there is, in fact, no

experimental evidence for deviations from (10). Additional experiments are in progress [23]

and it will be interesting to see if the ion-electron Casimir effect will be verified.  From a

theoretical point of view there are interesting connections at short [24] and long distance

between the order O(ff3=R3) QED corrections in (6) and (10).

    This work was supported in part by the National Science Foundation through a grant for

the Institute for Theoretical Atomic and Molecular Physics at the Smithsonian Astrophysical

Observatory and Harvard University.



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