A simple way of understanding the nonadditivity of van der Waals
dispersion forces
C. Farina,a) F. C. Santos,b) and A. C. Tortc)
Instituto de Fı́sica, Universidade Federal do Rio de Janeiro, Cidade Universitária, Ilha do Fundão,
Caixa Postal 68528, 21945-970 Rio de Janeiro, Brazil
~Received 1 April 1998; accepted 22 September 1998!
Using a semiclassical model for three interacting fluctuating dipoles we introduce a simple scenario
in which the nonadditivity of the van der Waals dispersion forces arises in a very transparent way.
For simplicity, we illustrate our model in the case of nonretarded dispersion forces. The argument
can be straightforwardly generalized to the case of N interacting fluctuating dipoles. For pedagogical
reasons, we also give a brief review of some basic points concerning van der Waals forces. © 1999
American Association of Physics Teachers.
I. INTRODUCTION
The name of van der Waals is inextricably related to the
equation of state for real gases, proposed by him in his
thesis1 in 1873. Written for 1 mol of gas this equation reads
S
P1
D
a
~ V2b ! 5RT,
V2
~1!
where P, V, and T are the pressure, volume, and the absolute
temperature, respectively, and R is the universal gas constant. The parameters a and b are known as van der Waals
constants and their respective values can be obtained by fitting this equation of state to the experimental data. The presence of the constant b in ~1! takes into account the finite size
of the molecules and tells us that one individual gas molecule does not have access to the total volume V. As to the
parameter a, this constant was associated by van der Waals
with an attractive force between two molecules. A consequence of such intermolecular ~attractive! forces is to diminish the pressure at the walls. Although van der Waals suggested explicit forms for these intermolecular potentials, he
was not able to explain the precise origin of the forces responsible for the appearance of parameter a.
When we talk about van der Waals attraction, it is tacitly
assumed that we are dealing with molecules ~atoms! whose
separation is large enough to exclude the overlapping of
electronic orbitals. In this case, we can distinguish three
types of van der Waals forces, to wit: orientation, induction,
and dispersion van der Waals forces, which we now describe
briefly ~a more detailed discussion can be found in Refs. 2, 3,
and 4!.
The orientation van der Waals force corresponds to the
force between two polar molecules ~polar here means that
the molecule has a permanent electric dipole moment, for
instance a water molecule!. W. H. Keesom, in the early
twenties, was the first to compute the thermal average of the
force between two polar molecules and found a temperaturedependent interaction energy, namely, (22/3)( p 21 p 22 /
4 p e 0 r 6 )(1/k B T), ~for k B T@( p 1 p 2 /4p e 0 r 3 )), where p 1 and
p 2 are the magnitudes of the electric dipole moments of the
two molecules, respectively, r is the distance between the
molecules, and k B is the Boltzmann constant. The attractive
nature of the orientation force is not difficult to understand:
Although the number of attractive orientations is exactly the
same as the number of repulsive ones, the attractive orientations are statistically favored over the repulsive ones. This is
344
Am. J. Phys. 67 ~4!, April 1999
so because the Boltzmann weight is e 2E/k B T ~and hence it
diminishes with increasing energy E!, and the smaller energies correspond to attractive orientations. Notice also that the
orientation force decreases with increasing temperature. This
is quite natural since as the temperature increases repulsive
orientations tend to be as accessible as attractive orientations.
Approximately at the same time, P. Debye and others recognized that even if one molecule had a permanent dipole
moment ~or even a higher multipole moment, as for example,
a permanent quadrupole moment! and the other had not, the
first molecule would induce a dipole moment in the second
one and a resulting dipole–dipole ~quadrupole–dipole,...! attractive force would appear. That the induction van der
Waals force is also attractive is not difficult to understand: It
suffices to remember a very common experience, namely,
that neutral tiny pieces of dielectric material such as small
pieces of paper are always attracted by a charged object ~like
a small bar of glass charged by friction!. This is so because
the induced dipoles point in the same direction as the electric
field that induced them, and these relative orientations give
rise to an attractive resultant force. Moreover, due to this
correlation between the induced dipole and the dipole of the
polar molecule, the induction force does not vanish with increasing temperature. There is still another important feature
of induction forces: its dependence on the distance r between
the two molecules is proportional to 1/r 7 , which can be
qualitatively understood as follows: Recall that the magnitude of the induced dipole on the second molecule ( p 2 ) is
proportional to the magnitude of the electric field created by
the ~permanent! dipole moment of the first molecule (E 1 ),
which in turn is of the form E 1 ' p 1 /r 3 . The electrostatic
energy between the molecules is then
U ~ r ! ' p 2 E 1 ~ r ! ' a E 21 ~ r ! '
a p 21
,
r6
~2!
where a is the molecular polarizability. Since U(r) is proportional to 1/r 6 , the force has the 1/r 7 behavior mentioned
above.
However, neither the orientation forces nor the induction
ones can explain the attraction between molecules ~atoms! in
all cases, as for example the attractive interaction between
two noble gas atoms, since these types of forces still require
at least one molecule ~atom! to exhibit a permanent dipole
~quadrupole! moment.
© 1999 American Association of Physics Teachers
344
The explanation of the true origin of such general forces
had to wait for the advent of quantum mechanics. In 1930, F.
London5 using perturbation theory showed that even if neither molecule posessed a permanent dipole moment, they
still would attract each other. It was only necessary that dipole moments could be induced in both molecules. In other
words the molecules ought to be polarizable ~the molecular
polarizability a must be different from zero!. Since the polarizability is related to the refractive index and dispersion,
this type of force is known as dispersion force. At this point
it is quite natural to ask ourselves how these induced dipoles
arise in both molecules ~remember that none of them had
initially a permanent dipole moment!. Here, classical physics
fails completely as dispersion forces have a genuine quantum
origin. Modern quantum field theory, considered as one of
the most important scientific achievements of the twentieth
century, tells us that even when the ‘‘medium’’ is the
vacuum, there is a residual ~quantized! electromagnetic field
E(r,t) and B„r,t) whose vacuum expectation value is zero,
but whose fluctuations given by ^ E2 & and ^ B2 & do not average to zero. The vacuum field will induce an instantaneous
dipole moment in one molecule, which in turn, together with
the vacuum electromagnetic field, will also induce in the
second molecule another instantaneous dipole moment. We
can say then that the vacuum electromagnetic field induces
fluctuating dipoles in both atoms and the dipole–dipole interaction of these zero-mean but correlated dipoles corresponds to the van der Waals dispersive interaction ~see
Milonni’s book4 for a detailed discussion on the connection
between dispersion forces and the vacuum electromagnetic
field!.
There are two types of dispersion forces, to wit: nonretarded and retarded dispersion forces. Nonretarded forces occur when the speed of light is taken to be infinite, so that the
interaction between two molecules is instantaneous. Retarded dispersion forces occur when we take into account the
finiteness of the speed of light. In this case, the dipole field
of the first molecule, which is induced by the vacuum fluctuations, will reach the second one after the time interval
r/c. Consequently, the reaction field of the second molecule
will be delayed at the first one by the time interval 2r/c.
Nonretarded forces are then good approximations when the
distance between the molecules is small. As the distance between the two molecules increases, retardation effects become more and more important. London’s work5 refers to
nonretarded dispersion forces and shows that for two identical polarizable atoms the interaction potential energy is
given by
U ~ r ! 52
S D
1
4pe0
2
3\ v 0 a 2
,
4r 6
~3!
where in the above formula a means the static polarization
of each atom and v 0 is the transition frequency between the
ground state and the first excited one. Almost two decades
after London’s result Casimir and Polder6 obtained for the
first time, after an involved perturbative quantum electrodynamics calculation, the expression for the retarded dispersion
van der Waals forces between two polarizable atoms. Basically, retardation effects change the 1/r 6 behavior of the interaction energy to 1/r 7 . However, this change in the power
law for the dispersion force with increasing distance was not
easy to verify experimentally and only in 1968 Tabor and
Winterton7 were able to achieve this goal.
345
Am. J. Phys., Vol. 67, No. 4, April 1999
To conclude this brief review concerning van der Waals
forces, it is worth emphasizing at this point that van der
Waals forces ~orientation, induction, and dispersion! have a
great variety of theoretical and practical applications as for
example: in condensation and crystallization, in structural
and energetic effects in colloid chemistry and biology, in
surface physics and chemistry, in the large field of adhesion
and its implications in washing ~good detergents must diminish van der Waals attraction between dirt and clothing!, in its
connection with the so-called Casimir effect8 ~retarded dispersion force!, among others.
Here we shall be concerned with dispersion forces, in particular with a very important property of these forces,
namely, their nonadditivity. It is a well-known fact that the
retarded and unretarded dispersion van der Waals forces are
nonadditive, that is, the resultant force which acts on a given
fluctuating dipole ~polarizable molecule! due to the presence
of the other fluctuating dipoles ~other polarizable molecules!
is not simply given by the sum of the individual forces that
each one of them would exert if all the others were absent. In
other words, the force between two fluctuating dipoles ~polarizable molecules! is altered in a nontrivial way by the
presence of a third one, see, for example, Refs. 2, 3, and 4,
and references therein. The degree of nonadditivity, however, may depend on the density of the medium. In fact, for
rarefied media it is possible to suppose that the forces are
additive. For two semi-infinite dielectric slabs this leads, for
instance, to an attractive macroscopic force which is inversely proportional to the cube of the distance between the
slabs in the nonretarded case. If retardation effects are taken
into account then one obtains a force which is inversely proportional to the fourth power of the distance between the
slabs. Although the functional dependence of the force with
the distance between the slabs is correct, the theoretical values computed taking into account only pair interaction do not
fit experimental data. It was precisely this discrepancy between experimental results and microscopic theories that assume additive van der Waals forces that led to a consideration of their nonadditivity.9
The main purpose of this article is to show in a simple and
transparent approach how the nonadditivity of these forces
arises allowing in this way students with a very elementary
knowledge of quantum mechanics to understand such an important issue. For simplicity, we shall work here only with
nonretarded van der Waals forces. For pedagogical reasons
this article is organized as follows: In order to introduce the
notation and basic ideas of the model to be employed, Sect.
II is devoted to a rederivation of London’s result; in Sect. III
we generalize the model of Sect. II to the case of three polarizable molecules which suffices to make the first ‘‘nonadditive’’ term ~in the expression for the energy of the system!
appear. Section IV is left for the conclusions and final remarks.
II. LONDON’S RESULT: A SEMICLASSICAL
APPROACH
In order to introduce the basic ideas as well as to establish
our notation, we shall review London’s result5 using an elementary approach for the dispersive nonretarded van der
Waals forces between two neutral but polarizable atoms. In
this model the instantaneous dipole moments exhibited by
each atom will be simulated by a point charge e oscillating
harmonically with angular frequency v 0 around a fixed point
Farina, Santos, and Tort
345
~‘‘the electronic cloud around a fixed nucleus’’!. The restoring harmonic force linking the electron to the center of force
simulates its binding to the nucleus ~which remains fixed by
assumption!. It is well known that for many purposes an
atom in its ground state level can be represented by a
charged harmonic oscillator.4 Hence, this assumption means
that we shall obtain the interaction energy between two
ground state atoms. To neglect retardation effects in the
propagation of the electromagnetic field means essentially
that the time interval that a light signal takes to propagate
between the two molecules, which is given by DT5r/c, is
much smaller than the characteristic time associated with the
system, which is given by D t '1/v 0 , where v 0 5(E 1
2E 0 )/\'(mc 2 /\) a 2fsc. Here, a fsc is the fine structure constant ( a fscªe 2 /(4 p e 0 \c)'1/137). Since the Bohr radius
a 0 can be written as a 0 5\c/ a fscmc 2 , it follows that the
nonretarded regime holds if DT!D t , or r!137a 0 .
For our purposes, it suffices to consider these oscillations
in fixed directions ~although with the relative orientations of
the two instantaneous dipoles completely arbitrary!. We
know that the force between induced dipoles must be attractive, so it is desirable that this model reproduces this peculiarity for arbitrary directions. This is not obvious and we
will come to this point later. Besides, we shall neglect the
radiation emitted by these oscillating dipoles and consequently in the following discussion there will be no radiation
reaction terms. This fact should cause no surprise: Recall
that we are trying to obtain the interaction energy between
two atoms in their ground state. In order to get an expression
for the nonretarded van der Waals interaction energy between these atoms in such a way that its connection with
zero-point energy becomes clear, we must proceed as follows: First, we write down the ~coupled! classical equations
of motion for both oscillating charges, taking into account
the electric force that one exerts on the other; then, we decouple these equations and compute the frequencies of the
corresponding normal modes. Finally, since dispersion
forces are genuinely quantum in nature, we must quantize
the system of the two independent oscillators ~normal
modes!, compute the associated ground state energy level,
and identify the desired interaction energy as the difference
between this energy and the ground state energy of the decoupled system of oscillating charges. This is equivalent to
taking as the interaction energy the expression containing
only the terms involving the distance r between the atoms.
Denoting by x i the displacement of the electron attached
to atom i from its equilibrium position and assuming that u x i u
is much smaller than the distance r between the atoms, the
electric field created, for instance by the first oscillating
charge at the position of the second one, can be considered as
the field of an electric dipole pW 1 (t)5ex 1 (t) m̂ 1 , where m̂ 1 is
a unit vector which determines the fixed ~but otherwise arbitrary! direction of oscillation of the first instantaneous dipole
~recall that there is no monopole term because the atom is
neutral and the oscillating charge is just a way of simulating
its fluctuating dipole!. Denoting by w the angular frequency
of the oscillating dipole, this field is given by,10
W 1 ~ P 2 ,t ! 5
E
H
S
D
J
346
Am. J. Phys., Vol. 67, No. 4, April 1999
W 1 ~ P 2 ,t ! • m̂ 2 5
e 2E
~4!
e 2x 1~ t !
@ 3 ~ r̂• m̂ 1 !~ r̂• m̂ 2 ! 2 m̂ 1 • m̂ 2 # .
4 p e 0r 3
~5!
Hence, the equations of motion for these oscillating
charges are simply
ẍ 1 ~ t ! 1 v 20 x 1 ~ t ! 5K 12x 2 ~ t ! ,
~6!
ẍ 2 ~ t ! 1 v 20 x 2 ~ t ! 5K 12x 1 ~ t ! ,
where for convenience we defined
K 125
S D
e2
Q 12
,
4 p e 0r 3 m
~7!
Q 125 @ 3 ~ r̂• m̂ 1 !~ r̂• m̂ 2 ! 2 m̂ 1 • m̂ 2 # ,
where m is the mass of the electron. The factor Q 12 is the
spatial orientation factor. The above equations describe a
coupled system with two vibrational degrees of freedom,
which can be decoupled in the usual way by introducing
normal coordinates, say h 1 and h 2 , as follows:
h 1 ~ t ! 5x 1 ~ t ! 1x 2 ~ t ! ,
~8!
h 2 ~ t ! 5x 1 ~ t ! 2x 2 ~ t ! .
~9!
The resulting ~decoupled! equations are simply
ḧ 1 1 v 21 h 1 50,
~10!
ḧ 2 1 v 22 h 2 50,
~11!
where the normal frequencies are given by
v 6 5 ~ v 20 7K 12! 1/2
S
5 v 0 17
K 12
K 212
K 312
5K 412
2v0
8v0
16v 0
128v 0
22
47
62
8 1O
S DD
K 512
v 10
0
,
~12!
K 12! v 20
where we assumed here that
~see the discussion in
Sec. IV!. The quantization of this system is immediate since
it consists of two independent harmonic oscillators with frequencies v 1 and v 2 , respectively. The corresponding
ground state energy is then
E5
\
~ v 11 v 2 !
2
H
5\ v 0 12
e
x ~ t2r/c ! @ 3 ~ r̂• m̂ 1 ! r̂2 m̂ 1 #
4pe0 1
1 ik
k 2 ~ r̂3 m̂ 1 ! 3r̂
3 32 2 1
,
r
r
r
where k5 v /c. In the above equation, we identify clearly the
near field electrostatic contribution ~terms proportional to
1/r 3 ! and the radiation field ~the term proportional to 1/r!.
Since we are interested in nonretarded dispersion forces,
which means small distances between the atoms, the dominant term turns out to be the near field contribution. Also,
x 1 (t2r/c) can be approximated by x 1 (t).
The component of the relevant electric force exerted by
the dipole e 1 x 1 m̂ 1 upon e 2 along the allowed direction of
oscillations of e 2 ~determined by m̂ 2 ! is then (e 1 5e 2 5e)
K 212
8 v 40
2
5K 412
128v 80
1O
S DJ
K 612
v 12
0
.
~13!
Substituting ~7! into ~13! and identifying U(r) with the
r-dependent terms we obtain to lowest order in powers of
K 12 / v 20 ,
Farina, Santos, and Tort
346
U~ r ! 5E223 21 \ v 0
52
S D
1
4pe0
2
~14!
Q 212\ v 0 a 2
8r 6
,
~15!
where in the previous equation e 2 /m v 20 was identified with
the classical static polarizability a of an atom ~considered as
a charged harmonic oscillator!.
Now, we can understand why this model ~with fixed directions! works: We know that the total quantum mechanical
energy of two independent oscillators always diminishes
when coupling terms between them of the form given by Eq.
~6! are introduced. This fact becomes very clear if we look at
Eq. ~12! ~notice that the first corrections to v 1 and v 2 have
opposite signs, but the second-order corrections have the
same sign and are both negative!.
Of course the semiclassical model presented here could
not yield London’s result with all numerical factors. However, it must be clear that our purpose in this section was
simply to introduce the fluctuating oscillator model. Anyway, London’s result can be obtained if we think of a as the
quantum static polarization and v 0 as the angular frequency
associated with the transition between the ground state and
the first excited one and change Q 212 by its corresponding
quantum mechanical evaluation which gives the factor 2
33, see for instance Refs. 11 and 12. ~Quantum mechanically, a is given by (2/3\)( u dW u / v 0 ), where u dW u is the dipole
moment amplitude between the ground state and the first
excited one.!
III. NONADDITIVITY OF VAN DER WAALS
DISPERSION FORCES
The standard discussions of the nonadditivity of dispersion
van der Waals forces are in general rather involved for students with an elementary knowledge of quantum theory ~the
interested reader can find a detailed discussion on this subject in the excellent book by Langbein2!. It is also possible to
introduce the nonadditive effects in connection with the
~quantized! vacuum electromagnetic field. For instance, in
order to discuss the nonadditivity of van der Waals forces,
Milonni4 considered the interaction energy of an ~induced!
oscillating dipole pW with the quantized source-free electromagnetic field ~vacuum field!. In the quantization of the
field, or in other words, in computing the mode functions for
the field, the presence of all the other atoms ~considered also
as induced oscillating dipoles! must be taken into account.
After a rather involved calculation it is shown that nonadditive terms contribute to the expectation value of the energy
of the induced dipole pW i at point rW i in the quantized sourcefree electric field E(ri ,t), namely, ^ Ei & 52 21 ^ pi •E(ri ,t) & .
These terms are three-body contributions of third order ~associated with the triplet i,j,l! of the form4
^ Ei jl & }
a3
,
r 3i j r 3jl r 3il
~16!
where a is the static polarizability of the atoms, considered
identical in the formula above. This kind of term had already
been derived before through standard perturbation theory and
studies concerning its magnitude and sign date from the early
forties.13
It is precisely this kind of term that we want to derive here
using an elementary approach, namely, a simple generaliza347
Am. J. Phys., Vol. 67, No. 4, April 1999
tion of the model used in Sec. II. Hence, in order to discuss
the nonadditivity of nonretarded van der Waals dispersion
forces in a simple way we shall introduce a third fluctating
dipole ~polarizable atom! interacting with the other two. Using the same assumptions as in Sec. II, the equations of motion now read
ẍ 1 ~ t ! 1 v 20 x 1 ~ t ! 5K 12x 2 ~ t ! 1K 13x 3 ~ t ! ,
ẍ 2 ~ t ! 1 v 20 x 2 ~ t ! 5K 12x 1 ~ t ! 1K 23x 3 ~ t ! ,
~17!
ẍ 3 ~ t ! 1 v 20 x 3 ~ t ! 5K 13x 1 ~ t ! 1K 23x 2 ~ t ! ,
where in an obvious notation the symmetrical coefficients
K i j define the coupling constants between the three oscillating charges and are given by
Ki j5
Qij
4 p e 0 r 3i j
with
S D
e2
m
Q i j 53 ~ m̂ i •r̂ i j !~ m̂ j •r̂ i j ! 2 m̂ i • m̂ j .
~18!
As before, the above coupled equations describe a system of
three ~vibrational! degrees of freedom that can also be decoupled by introducing normal coordinates. Since we need
only compute the normal frequencies and recalling that when
the system vibrates in one of its normal modes all particles
oscillate with the same frequency, we can substitute into ~17!
x i 5C i exp$ivt% which yields
S
v 20 2 v 2
2K 12
2K 13
2K 12
v 20 2 v 2
2K 23
2K 23
v 20 2 v 2
2K 13
DS D S D
x1
0
x2 5 0 .
x3
0
~19!
In order to avoid trivial solutions we must require that the
above transformation have no inverse, which is equivalent to
setting the determinant of the above matrix to zero. This
assumption simply gives
~ v 20 2 v 2 ! 3 2 ~ K 2121K 2131K 223!~ v 20 2 v 2 ! 22K 12K 13K 2350.
~20!
The desired eigenfrequencies v i , i51,2,3, are then the positive roots of this equation. After quantization the ground
state energy of the three fluctuating dipoles is given by
E5
\
~ v 11 v 21 v 3 !.
2
~21!
Defining b ª v 20 2 v 2 , we rewrite Eq. ~20! in the form
b 3 2 ~ K 2121K 2131K 223! b 22K 12K 13K 2350
and Eq. ~21! in the form
E5
\v0
2
HS
12
b1
v 20
D S
1/2
1 12
b2
v 20
D S
1/2
1 12
b3
v 20
~22!
DJ
1/2
,
~23!
where b i ª v 20 2 v 2i , i51,2,3 are the roots of Eq. ~22!. Next,
making a Taylor expansion of each term in Eq. ~23! in powers of b i / v 20 , we get
E5
H
\v0
1
32
~ b 1 b 21 b 3 !
2
2 v 20 1
2
1
8 v 40
~ b 21 1 b 22 1 b 23 ! 2
J
~24!
Farina, Santos, and Tort
347
1
16v 60
~ b 31 1 b 32 1 b 33 ! .
It is interesting to observe that all the terms on the righthand side ~RHS! of Eq. ~24! are symmetrical functions of the
roots of Eq. ~22! and consequently, it will be not necessary to
determine explicitly each root b i , since these functions can
be expressed directly as functions of the coefficients of the
algebraic equation. For a general algebraic equation of the
third degree, namely,
a 0 x 3 1a 1 x 2 1a 2 x1a 3 50,
x 1 x 2 x 3 52
which implies an interaction energy given by
U~ r 12 ,r 13 ,r 23! 5E2
1
a2
,
a0
~26!
a3
.
a0
Comparing Eqs. ~22!, ~25!, and ~26!, we can write
b 1 1 b 2 1 b 3 50,
~27!
b 1 b 2 1 b 1 b 3 1 b 2 b 3 52 ~ K 2121K 2131K 223! ,
~28!
b 1 b 2 b 3 52K 12K 13K 23 .
~29!
What we must do now is to use the above equations in
order to obtain the expressions for the sum of the squares of
the roots ( b 21 1 b 22 1 b 23 ) as well as the sum of the cubes of
the roots ( b 31 1 b 32 1 b 33 ) as required by Eq. ~24!.
Taking then the square of ~27! we get
b 21 1 b 22 1 b 23 522 ~ b 1 b 2 1 b 1 b 3 1 b 2 b 3 !
52 ~ K 2121K 2131K 223! ,
~30!
where we used Eq. ~28!.
For the sum of the cubes of the roots, we take as a starting
point the identity
b 31 1 b 32 1 b 33 1 b 21 b 2 1 b 22 b 1 1 b 21 b 3 1 b 23 b 1 1 b 22 b 3 1 b 23 b 2
50,
~31!
which follows immediately from Eq. ~27! if we multiply
both sides by ( b 21 1 b 22 1 b 23 ).
On the other hand, using Eq. ~27! again, we can write
b 1 b 2 ~ b 1 1 b 2 1 b 3 ! 50→ b 21 b 2 1 b 22 b 1 52 b 1 b 2 b 3 ,
b 1 b 3 ~ b 1 1 b 2 1 b 3 ! 50→ b 21 b 3 1 b 23 b 1 52 b 1 b 2 b 3 ,
~32!
b 2 b 3 ~ b 1 1 b 2 1 b 3 ! 50→ b 22 b 3 1 b 23 b 2 52 b 1 b 2 b 3 ,
so that
b 21 b 2 1 b 22 b 1 1 b 21 b 3 1 b 23 b 1 1 b 22 b 3 1 b 23 b 2 523 b 1 b 2 b 3 .
~33!
Substituting the previous equation into ~31! and also using
Eq. ~29!, we obtain
b 31 1 b 32 1 b 33 56K 12K 13K 23 .
~34!
Therefore, substituting Eqs. ~27!, ~30!, and ~34! into Eq.
~24! we finally obtain
348
J
'2
a1
,
a0
x 1 x 2 1x 1 x 3 1x 2 x 3 51
H
1
3
\v0
2
2
2
32
K K K ,
4 ~ K 121K 131K 23 ! 2
2
4v0
8 v 60 12 13 23
~35!
~25!
we write below the basic combinations of its roots as
x 1 1x 2 1x 3 52
E'
Am. J. Phys., Vol. 67, No. 4, April 1999
3
\v0
2
\v0
8
HS D S
a
4pe0
S D
a
3
2 4pe0
3
2
Q 212
r 612
1
Q 12Q 13Q 23
r 312r 313r 323
Q 213
J
r 613
1
.
Q 223
r 623
D
~36!
The first three terms on the RHS of ~36! represent pairwise
additive contributions to the interaction potential between the
three oscillators. The last term on the RHS of ~36! is a threebody contribution to the interaction and therefore it spoils the
additivity. This can be seen as follows: If the force that acts
on one of the dipoles, say number one, is computed from
~36!, then the result will be different from the sum FW 21
1FW 31 , where FW 21(FW 31) is the force that dipole number two
~three! would exert on dipole number one in the absence of
dipole number three ~two!. In order to connect the threebody contribution appearing in ~36! with that written in Eq.
~16!, it suffices to set a 5e 2 /m v 20 and identify this constant
with the polarizability at the transition frequency v 0 5(E 1
2E 0 )/\ between the ground and the first excited state,
which is the principal transition for this system. After this we
will have the kind of term appearing as the lowest order
nonadditive contribution to van der Waals forces in the case
of N interacting fluctuating dipoles in Milonni’s calculation.4
Notice also that this term, depending on the geometrical arrangement of the oscillators, may yield an attractive or a
repulsive contribution.13
IV. FINAL REMARKS AND CONCLUSIONS
In this paper we applied the fluctuating dipole model to
the case of three ~polarizable! atoms in order to explain on
elementary grounds how the nonadditivity of dispersion van
der Waals forces arises. The application of this model to the
case of N interacting fluctuating dipoles is straightforward
and yields correctly many interesting results as for example:
~i! There will not be any linear term in K i j in the expression
for U in analogy with what happens for N52 @see Eq. ~14!#
and for N53 @see Eq. ~36!#. ~ii! Although the cubic term in
K 12 is absent from the expression for U in the N52 case @see
Eq. ~14!#, cubic terms in K i j will appear for N>3, giving
rise to the lowest order nonadditive terms. These terms correspond to the three-body contribution to the interaction van
der Waals energy. ~iii! Higher order nonadditive terms also
appear, namely, four-body contributions,..., N-body contributions. The N-body term is of the form ( a /4p e 0 r 3 ) N , a and r
being a typical atomic polarizability and a typical interatomic distance, respectively. Consequently, the N-body
term is a /4p e 0 r 3 times smaller than the (N21), term.
Hence, in order to make sense the model must satisfy the
condition a /4p e 0 r 3 !1. In fact, substituting typical values
@recall that a 5e 2 /m v 20 and that v 0 can be thought of as the
transition frequency between the ground state and first excited one, that is, v 0 is a few eV ~s! per \# we obtain for an
Farina, Santos, and Tort
348
intermolecular distance r'10 Å that a /4p e 0 r 3 '1023 !1. It
is worth emphasizing that this value for r is compatible with
an earlier assumption, namely, r!137a 0 .
We would also like to emphasize the importance of the
nonadditive terms in the evaluation of the force between
macroscopic objects like spheres, cylinders, or slabs. Although the dependence of this force on the relevant distance
involved in each case is correct even if we take into account
only pair interactions, numerical values will fit correctly the
experimental data only when nonadditivity is not neglected.
Finally, we cannot conclude this article without mentioning that retarded dispersion forces between macroscopic objects ~with all nonadditive contributions! correspond to the
so-called electromagnetic Casimir force between them,
which since Casimir’s seminal paper8 can be computed directly from the fluctuations of the confined ~vacuum! electromagnetic field subjected to the appropriate boundary conditions. The Casimir effect was verified experimentally for
the first time by Sparnaay14 with a poor accuracy and recently by Lamoreaux,15 and Mohideen and Roy16 with an
excellent agreement between theory and experimental data,
but that is another story.
ACKNOWLEDGMENTS
The authors are indebted to M. V. Cougo-Pinto, A. N.
Vaidya, V. Mostepanenko, and G. L. Klimchitskaya for enlightning discussions. One of us ~CF! wishes to acknowledge
the partial financial support of CNPq ~the National Research
Council of Brazil!.
a!
Electronic mail: farina@if.ufrj.br
Electronic mail: filadelf@if.ufrj.br
c!
Electronic mail: tort@if.ufrj.br
1
J. D. van der Waals, ‘‘Over de continuiteit van den gas-en vloeistoftoestand,’’ Dissertation, Leiden, 1873.
2
Dieter Langbein, Theory of Van der Waals Attraction, Springer Tracts in
Modern Physics, Vol. 72 ~Springer-Verlag, Berlin, 1974!.
3
H. Margenau and N. R. Kestner, Theory of Intermolecular Forces ~Pergamon, New York, 1969!.
4
P. W. Milonni, The Quantum Vaccum: An Introduction to Quantum Electrodynamics ~Academic, New York, 1994!.
5
F. London, ‘‘Zur Theorie und Systematik der Molecularkrafte,’’ Z. Phys.
63, 245 ~1930!.
6
H. B. G. Casimir and D. Polder, ‘‘The Influence of Retardation on the
London-van der Waals Forces,’’ Phys. Rev. 73, 360–372 ~1948!.
7
D. Tabor and R. H. S. Winterton, ‘‘Direct measurement of normal and
retarded van der Waals forces,’’ Nature ~London! 219, 1120–1121 ~1968!.
8
H. B. G. Casimir, ‘‘On the Attraction Between Two Perfectly Conducting
Planes,’’ Proc. K. Ned. Akad. Wet. 51, 793–795 ~1948!.
9
E. M. Lifshitz, ‘‘The Theory of Molecular Attractive Forces between Solids,’’ Sov. Phys. JETP 2, 73–83 ~1956!.
10
J. D. Jackson, Classical Electrodynamics ~Wiley, New York, 1975!, 2nd
ed., p. 395.
11
P. W. Milonni and P. L. Knight, ‘‘Retardation in the resonant interaction
of two identical atoms,’’ Phys. Rev. A 10, 1096–1108 ~1974!; ‘‘Retarded
interaction of two nonidentical atoms’’ ibid., 11, 1090–1092 ~1975!.
12
P. W. Milonni, ‘‘Semiclassical and Quantum-Electrodynamical Approach
in NonRelativistic Theory,’’ Phys. Rep., Phys. Lett. 25C, 1–81 ~1976!.
13
B. M. Axilrod and E. Teller, ‘‘Interaction of the van der Waals Type
Between Three Atoms,’’ J. Chem. Phys. 11, 299–300 ~1943!.
14
M. J. Sparnaay, ‘‘Measurements of Attractive Forces between Flat
Plates,’’ Physica ~Amsterdam! 24, 751–764 ~1958!.
15
S. K. Lamoreaux, ‘‘Demonstration of the Casimir Force in the 0.6 to 6 mm
Range,’’ Phys. Rev. Lett. 78, 5–8 ~1997!.
16
U. Mohideen and A. Roy, ‘‘A precision measurement of the Casimir force
from 0, 1 to 0, 9 mm,’’ hep-ph/9805038.
b!
PROPER USE OF THE SEMICOLON
With the war ended, Purcell could hardly suppress thoughts of returning to academia. The
problem was, he had agreed to stay on and help grind out the Rad Lab Series. While never
doubting the task’s importance, he nevertheless found it hard to concentrate on writing when the
long-suppressed world of research beckoned. It would have been bad enough just writing. But the
stultifying bureaucracy associated with the series heightened his angst. Drafts had to be fed to a
cadre of nitpicky English teachers who hammered away at the physicists’ syntax and structure.
They never let up. As the writing slipped further and further behind schedule, a flood of memos
arrived about editorial principles. One day Purcell received a two-page, single-spaced bulletin on
the proper use of the semicolon... .
Robert Buderi, The Invention that Changed the World ~Simon and Schuster, New York, 1996!, p. 258.
349
Am. J. Phys., Vol. 67, No. 4, April 1999
Farina, Santos, and Tort
349