Sparnaay, M . J . Physica 25 1959 217-231

ON THE ADDITIVITY OF LONDON- VAN DER WAALS FORCES

AN EXTENSION OF LONDON'S OSCILLATOR MODEL"

by M. J. SPARNAAY

Philips Research Laboratories N.V. Philips' Gloeilampenfabrieken, Eindhoven, Nederland

S y n o p s i s

Calculations are given concerning the additivity of London-Van der Waals forces between two groups of atoms, the atoms being represented as isotropic harmonic oscillators. The results indicate that deviations of 10-30% from additivity can be obtained if only dipole-dipole interaction between oscillators of one group is assumed. The effect can be expected to be relatively large if the symmetry of the arrangements of the oscillators in the group is low and it is dependent upon the relative spatial position of the groups.

1. Introduction. L o n d o n 1) introduced in 1930 the harmonic oscillator model for the s tudy of the interaction of two (neutral) atoms a large distance apart. His quantum mechanical treatment resulted in the following ex- pression for the interaction energy:

E1 -~ - - 3 h v o ~ 2 / 4 R 8 (I)

where R is the distance between the two (three-dimensional isotropic) harmonic oscillators which here are taken identical for convenience. The frequency of oscillation is v(=_o~/2~), the polarizability is a, where a = e2/mo~ 2 (e is the charge, m is the oscillating mass). Furthermore h is Planck's constant. In eq. (1) ground states were considered only, i.e. h~(=--ho~) >~ kT (k is Boltzmann's constant. T is absolute temperature).

I t was pointed out by L o n d o n tha t though the oscillator model is rather an oversimplification, it provides values of E which are probably of the right order of magnitude. Because of its simplicity the model is par- ticularly suitable for studying problems involving a large number of atoms, close together, such as calculations of lattice energies z). D r u d e 3) made extensively use of the oscillator model for the s tudy of optical problems. The model is therefore often called the Drude model. The model may also be important in calculations of the attraction between colloidal particles.

The question arises whether London-van der Waals forces can in these

- - 2 1 7 - -

Physica 25

218 M.J. SPARNAAY

cases still be considered as additive. This question will be studied in this paper with a view in particular to possible applications in colloid chemistry Addit ivi ty will be defined in the following way. Consider two groups each of n identical atoms, to be represented as three-dimensional isotropic har- monic oscillators. The distance between the groups is large compared with the dimensions of one group. Addit ivi ty means that the total interaction energy between the groups is given b y

En = n2E1 (2)

Non-addit ivi ty then may be represented as

E . = n2Et[1 +/(c~j, n)] (3)

where the non-additivity factor /(c,~, n) will be proved to depend on the interaction between the atoms i and 1" of one group and on the number n. If there is no interaction between any pair i and ~', the non-additivity factor will be zero. But some sort of interaction, for example dipole interaction, will certainly be present, at least between nearest neighbours. We will, in our treatment, only consider interactions between nearest neighbours, the interaction between non-nearest neighbours being considered as too small to give an appreciable contribution to /(c,j, n). We therefore restrict our- selves to small values of n. In fact we will deal only with the cases n = 2, 3, 5 and 7 as illustrated in figs. 1, 2, 3 and 4. The interaction parameters c~1 will be considered as due to dipole interaction. Owing to our restriction to the oscillator model the results will have a qualitative character only, but never- theless they will be of some interest. For instance, it can be proved that the interaction between two groups of oscillators depends on the relative position in space of the groups. Thus the interaction between two pairs of oscillators in the position drawn in fig. l b is larger than if the position of fig. l c is occupied. To see this consider the directions in space of the oscil- lating dipoles. All directions in space can be occupied by the oscillating dipoles. We have selected one certain direction, the z-direction, which gives maximal attraction and represented it by means of small arrows, pointing to the right. The neighbouring oscillators in fig. l b tend to maintain this position, and this will ultimately lead to an increased attraction between the two pairs. The neighbouring atoms in fig. l c on the contrary tend to reverse the direction of each others oscillating dipoles and this will lead to a decrease of the total attraction between the two pairs. Similar differences may appear when we compare the interactions for n = 3 (fig. 2a and b) and for n = 5 (fig. 3a and b). If the symmetry, of the arrangerhents of oscillators in a group is high (fig. 4) such a phenomenon probably does not occur.

An application of these results may be found in colloid chemistry. If colloidal particles of low crystal symmetry are close enough together it may not be expected that all positions in space are equally important for London-

ON THE ADDITIVlTY OF LONDON-VAN DER WAALS FORCES 219

Van der Waals attractions. In other words: a "directed" flocculation may take place. Since we may estimate the absolute value of/(qj , n) to be of the order of 0. I-0.3 the effect is not large, but it should not be neglected.

0 ~ R o

b; ~ ~ ~ R ,.. • o

r - ~ xl ~ ~ - i r((R

c; i - o - R ~ o- [

o) o o o R o o o

O U

b) o = R ,. o

fig.2 o o N - -

o;

b;

fie.3

O2 0

~0 01 05 R ~ 0 0 0

o3 0

¢ Q

• 0

-.O--~--.e- R

~'~.~

Fig. l. a) Two oscillators a dis tance R apart . For a t t r ac t ion see eq. 1 ( q u a n t u m limit) and eq. 32 (classical limit).

b) and c) Two pairs of oscillators a dis tance R apart . For a t t rac t ion see eq. 25 ( q u a n t u m limit) and eq. 33 (classical limit). For the explana t ion of the arrows, see text .

Fig. 2 a) and b). Two groups of three oscillators a dis tance R apart . For a t t rac t ion see eq. 46 (classical limit) and eq. 70 (quan tum limit).

Fig. 3 ~) ell b). Two groups of five oscillators a dis tance R apart . For a t t rac t ion see eq. 51 (classical limit) and eq. 71 (quan tum limit) for case a) and eq. 55 (classical limit) and eq. 72 ( q u a n t u m limit) for case b).

Fig. 4. Two groups of seven oscillators a dis tance R apart . Addi t iv i ty of the at- t rac t ion is preserved in this case whereas in the other cases (figs. lb, lc, 2a, 2b, 3a, 3b) there is no addi t iv i ty , the a t t rac t ion also depending on the spat ial posi t ion of the groups considered.

2. Considerations o[ the additivity /or the case n = 2. Quantum limit and classical limit. L o n d o n derived eq. (1) in the following way: Consider (fig. l a ) the Hamiltonian function H of two isotropic oscillators 1 and 2 a distance R apart and include dipole interaction only. The kinetic and the potential energy parts (T and V respectively) of the Hamiltonian function

220 ~. J. SPARNAAY

H = T + V are well known to be:

1 T = (pl 2 + psi)

2m e 2

V = ½mro=(ql z + q22) + ~-~ (XlX2 + ylyg. -- 2zlza)

(4)

The momenta P l and P2 and the coordinates ql and q~ can be separated into the three directions x, y and z in space. We have ql ~ = Xl 2 + yl ~ + z12; p l 9" = pxl 2 + Pvl 2 + pzl 2 and similarly for (/9. and 102.

The linear transformation

xl = ½a/2(s~l + s~2) / x 2 = ½ V 2 ( S z l - - s=,o.) • (6 )

and similarly for the other components of coordinates and momenta leaves T unaltered, but V becomes:

V=½moJl+2(s~12+sv12)+½mcol-~(s~2+Sv22)+½m¢o~2-szlg+½mco29+Sz22 (7)

where

oo1_8 = o o ~ ( 1 - ~ - 3 )

c02-2 = oJ~ ( 1 - - 2 b )

Now V, being of purely quadratic form, gives, together with T the Hamil- tonian function of a set of six linear harmonic oscillators. The zero point energy E of this set of oscillators is:

E = ½~(2o1+ + 2oa_ + o~2÷ + o~2-) (9)

This energy should be compared with the zero point energy E0 of the oscillators at infinite distance R.

E0 = 3~oJ (10)

The difference E1 = E -- E0 can be calculated upon expansion of OJl+ and so on as follows:

(.01+ = CO 1 + 2R 3 8R 6

and similarly for the other frequencies. I t is now easily seen, that R -3 terms cancel and that the resulting expression for E1 is identical with eq. (I). We note that the z-component is preponderant. The contribution due to ½m(co2--2Szl 2 + co2+2Sz2 2) is twice the sum of the contributions of the x and y component.

(8)

(s)

ON THE ADDITIVITY OF LONDON-VAN DER WAALS FORCES 221

We now proceed in the same w a y 4) to obta in the interact ion be tween two pairs of a toms (fig. l b and I c). We have for V~, Vv , and Vz where

V---- V x + V v + Vz (12)

the following expressions. The numbers 1 and 2 indicate oscillators of the first pair, 3 and 4 those of the second pair.

V x = a(xl z + x2 2 + x8 2 + x4 2) + 2b(xlx8 + XlX4 + xzxz + xzx4) +

-~- 2Cx(X.lX 2 -~- X.3X4)

V v = a(y l 9" + y2 2 + y3 z + y4 2) -¢: 2b(ylys + YlY4 + Y2Y3 + Y2y4)+ (13) + 2cv(ylY2 + YaY4)

Vz = a(z l 9" + zg. 2 + z3 ~ + z49J - - 4b(zlz3 + ZlZ4 + z2z3 + zzz4) +

+ 2Cz(fllZ2 -I- ZaZ4) We have wr i t ten

a = ½ m o ~ (14)

2b = e2/R 3 (15)

The symbol s cx, cy and c~ represent the parameters of interact ion be tween a toms of one pair. If c~, c v and cz are entirely determined b y dipole inter- actions be tween the oscillators 1 and 2 and be tween 3 and 4, it is useful to consider the two cases g i v e n b y figs. l b and l c. In case c the directions of the lines joining oscillators I and 2 and oscillators 3 and 4 are normal to the z-direction. We consider this as the y-direction.

One has in the case of fig. lb:

2cz ---- 2c v = eZ/ra; cz = - -e2 /r 3 (fig. lb) (16)

and in the case of fig. l c:

2c~ ---- 2cz = e~/rS; c u = --eZ/r a (fig. lc) (17)

where r is the distance be tween the oscillators 1 and 2 and be tween 3 and 4. We consider weak coupling be tween oscillators o f one pair only, i.e. we consider only

Ic,I, lcvi and Ic=l < a.

I t m a y be no ted tha t th roughout the whole t r ea tmen t we will t ake

Ic/al ~ e2/mo~2rZ ~. 1 and Ib/cl ~ r3/R 3 < 1 (18)

However , the case where c= w and a are of the same order of magni tude will briefly be discussed below. The linear t ransformat ion

x l = ½(s=l + s=2 + s~8 + s~4)

x2 = ½(s~l + s~2 - - s~3 - - s~4) (19) x8 = ½(s .1 - s~2 + s~3 - s~4)

x4 = ½(s~l - s~2 - s~3 + s~4)

222 M . J . SPARNAAY

and similarly for the other components of coordinates and moment s leads again to an unal tered expression for T, b u t now Vx, V v and Vz become

Y x =- a z l S x l 2 ~- 6tz2Sx2 2 27 ax3S:~3 2 27 az4Sx4 2 1

V y = a y l s y l 2 27 ay2sy2 2 27 ay3Sv3 2 27 ay4sy4 2 [ (20) Vz = azlSzl 2 27 az2sz2 2 27 azssz3 2 27 az4sz4 2

where

a = l = ½ m w x l 2 = a + c= + 2b;

a=z = ½mw=2 9' = a + cz - - 2b;

a~3 : ½meox3 2 = a - - c x ;

a x 4 = ½mCOx4 2 = a - - c x ;

a z l = ½ m w z l 2 = a + cz - - 4b;

az~. = ½mwz2 9" = a + cz + 4b;

avl = ½mcovl 2 = a + c v + 2b ay2 = ½mo~vz 2 = a + c v - - 2b ay3 = ½toO)y3 2 = a - - Cy

ay4 = ½roOmy4 2 = a - - Cy

az3 : ½~O)z3 2 = a - - Cz

az4 : ½~(Dz4 2 : a - - Cz

(21)

we obta in for the interact ion energy

he4 ( 1 1 1 ) E 2 = - 4 + +

where

oJ~ = w~/1 + 2 c z / m w 2

COy = oJ%/1 27 2Cy/mCo 2 and wz = wW'l + 2cz/moo 2 (24)

(23)

Thus the frequencies which essentially appear in the final expression are now shifted because of the interact ion be tween the oscillators of one pair. The frequencies cox, O~y and coz m a y be developed, retaining first power terms in c~, Cy and cz only and the result can be wr i t ten as

3~(~2 E 1 E2 = - - 4 - 4 - - - ~ - 1 2mo~ 2 (c~ + Cy + 4cz) (25)

Thus the factor /(c,;, n) represent ing non-addi t iv i ty becomes /(c,j, n ) = = - (1 /2mo,2) (c~ + cv + 4c~).

One has in the case i l lustrated in fig. l b:

3 o~/r 3 (fig. lb) (26) / ( c , j , n ) = -

and in the case, i l lustrated in fig. l c:

[(cia, n) = + }a / r 3 (fig. lc) (27)

Now in order to calculate the interact ion energy we note tha t

(Dx3, 60~4, (by3, O)y4, 09z3 and ~oz4

do not contain b, i.e. do not contain te rms with R. W e therefore only consider the following zero point energy:

E = ½h(Wxl 27 o)x2 + OOyl -~- O)y2 -~- Wzl -~- oJz2) (22)

This m a y be eva lua ted in the same w a y as in the simple London case and

ON T H E A D D I T I V I T Y OF L O N D O N - V A N D E R WAALS FORCES 2 2 3

Thus the total attraction between two pairs of oscillators depends on the position in space o~ these pairs. In the case of fig. l b the attraction is higher compared with additivity, in the case of fig. l c it is lower. This phenomenon was visualized in the Introduction.

The polarizability ~2 of a pair of oscillators may be given by:

~2 = eg/mo~+ 2 + eg /m co-~ (28)

In order to determine the frequencies co+ and co_ we have to specify the direction of the incident light waves necessary for the measurement of ~2 with respect to the direction of the line joining the two oscillators (the z '

direction say) since degeneracy is now part ly lost. If the two directions coincide then co+ and co_ can be found from

V~, = ½mco2(zl '2 + z2 '2) - - 2e~z l ' z2 ' / r 3 (29)

b y means of a linear transformation of the coordinates Zl 1 and z21 similar to that used by London (eq. 6). We ultimately obtain:

~2 = (2eg"/rmog")(1 + o~/r s) (30)

A slightly different expression is easily found if the incidence of the light waves is in the x ' or y ' direction. Expressions similar to eq. (30) have already been found by S i l b e r s t e i n 5). We may estimate o~/r 3 to be ot the order of 0.1. The polarizability thus remains practically unaltered whereas the attraction may vary considerably.

If c~, cv and cz, now no longer supposed to be determined b y simple dipole interaction, are increased, we may ultimately arrive at the situation in which movements of the electric charges in the oscillators 1 and 2 and also in 3 and 4 are always in phase (cz, cv, cz < 0) or always in counterphase (c~, c v, cz > 0). Then one coordinate instead of two describes the move- ments of the electric charges in each pair. We then have a simple example of the "monopole" representation 6) as introduced by L o n d o n in 1942. We then would obtain an equation similar to eq. (1) but now with 4a instead of ~ (or 2e instead of e) if cx, cv, cz < 0. If c~, c v, c~ > 0, i.e. if the charges were always exactly in counterphase, R -e terms would be absent.

We have hitherto considered the quantum limit of the problem (all frequencies >~ k T / h . We will now consider the classical limit. Then

l~co~ ~ k T

where co, are frequencies given above. The free energy F of a classical linear harmonic oscillator with a frequency

co is known to be F = - - k T ln(kT/l~o~) (31)

I t is not difficult to calculate the classical free energy of interaction of two three dimensional oscillators, or of two pairs of such oscillators, since all the

224 M.J. SPARNAAY

frequencies, required by eq. (31) are known. Thus, for the interaction be- tween two three-dimensional oscillators one has:

F I = F - - Fo = --(3=2/RS)kT (32)

where again = = eS/mo~ ~ (this polarizability can be measured with the aid of (nearly) static electric fields only) and where F0 is the free energy at infinite distance. For two pairs of oscillators one obtains:

F~ = - - 4(3=~/R6)kT[1 - - (2/3mo~)(c~ + c v + 4cz)] (33)

The same features with respect to non-additivity properties appear as in the quantum limit. These p~operties are now even more pronounced since w~ -4, o~u-4 and o~z -a appear essentially instead of 0= -3, ¢ov-3 and ~oz -3.

We have used in this calculation the frequencies obtained above. However, the classical limit can be calculated without knowing these frequencies. Neither does one need to know coordinate transformations. An example of such a transformation will nevertheless be given in the Appendix.

E i d i n o f f and A s t o n 1) pointed out, that, given a Hamiltonion H of the type

1 r~ 2 H = 2 m ~i=1 P~ "{- ~ = 1 ~ = 1 A,jq, qj

the partition function Z where Z is defined as

1 z = - - (... ['exp hn J J

is "given by

Z = (z~kr~,,/2-m/h) n X [~/D-~-(A)] -1

(34)

(35)

(36/ where Det (A) is the determinant value of the matrix, formed by the ele- ments A,j. Applying this expression of Z we obtain for the interaction of two agglomerates of n oscillators each:

Fn = F -- Fo = -- k T ln(Z/Zo) = -- ½kT ln(Det(Ao)/Det(A)) (37)

where Det (A0) is the determinant value at infinite distance R of the two agglomerates. The matrices whose determinant value should be calculated according to this method are for the simple London case given by:

Jab a b J f ° r t h e x a n d y d i r e c t i ° n l a - - 2 b - - 2 b f ° r thez-d i rec t i °n } ( 3 8 ) a

The matrices for the interaction between two pairs of oscillators are:

a cz b i zb' Cz a b

b b a b b c~

c b i a cz 2b 2b I c v a b and Cz a --2b --2b (39) b b a --2b --2b a b b c v --2b --2b cz

ON THE A D D I T I V I T Y OF L O N D O N - V A N D E R WAALS FORCES 225

A simple method of finding determinant values, which we will appl3~ later on, runs as follows:

The.linear transformation

= ½v'2(s +

= ½ a / 2 ( s , - (i = 1...n) } (40)

leads to a matrix which, in the case of two pairs of oscillators, takes the form *)

a + b c~+b 0 0 ] c~ + b a + b 0 0 and similarly for the other"

0 0 a -- b c~ -- b components. (41) 0 0 cx- -b a - - b

We denote this determinant value as D ( + b) × D(-- b) and the determinant value at zero value of b (i.e. at infinite distance R) as D a. We generally have s)

O ( + b) = D + b X~=I ~-~L10~hl (42)

where ~nt is an underdeterminant. I t differs from D in that the hl element is replaced by unity.

The determinant value of the matrix given above is:

D( +b)D(--b)=[(a~--c~2)2--4b2(a--c~)~][(ag--cv~)2--4bZ(a--cv) ~] × [(a2--Cz2)2--16b2(a--cz) 2] (43)

Identifying Det (A0) with D z and Det (A) with D(+ b)D(-- b) and inserting in eq. (37) then leads to eq. (33) for the interaction free energy of two pairs of oscillators.

This method of calculating the interaction free energy of the classical limit may conveniently be applied to a number of other cases, illustrated by figs. 2, 3 and 4. In these cases the calculation of the determinant values D and D ( + b) D(-- b) is simple and elementary. The non-additivity properties will be present both in the classical case and in the quantum mechanical case. In fact it will be proved that the non-additivity factor in the quantum limit is 3/4 times that in the classical limit for the cases to be considered. (We have seen this property already in the case n = 2 illustrated by figs. l b and 1 c). First the classical limit will be considered. The quantum limit, essentially requiring knowledge of separate shifted frequencies, will be dealt with afterwards. Einstein functions should be used for intermediate cases, i.e. for oscillators whose frequencies are of the order of kT/~ but these cases will not be considered.

In connection with the classical limit we remark that the energy E = F + +TS, where S is the entropy, is always equal to 6nkT. We are therefore only interested in the [ree energy F, which depends on R.

*) The author is indebted to professor B o u w k a m p for pointing out this to h im.

226 M.J. SPARNAAY

3. Class ical l i m i t ]or n = 3, n = 5 a n d n = 7. The relat ive positions of the oscillators in the groups n = 3, n = 5 and n = 7 are i l lustrated by figs. 2, 3 and 4 respectively. The oscillators of the group n = 3 are in linear position, the line joining them being either in the z direction (fig. 2a) or in the x direction (fig. 2b). The oscillators of the group n = 5, being in a plane normal to the y direction (fig. 3a) or normal to the z direction (fig. 3b) form a rectangular set. The positions of the oscillators of the group n = 7 have cubic symmet ry . In all cases the distance between nearest neighbours will be assumed to be equal. We consider nearest neighbour interact ion only, the interact ion always being given by terms of the type 2c,aq,q1; where q, and qj are coordinates of neighbouring oscillators i and 1" and where c,j is a constant . We will draw special a t ten t ion to the case where c,~ is determined b y dipole interaction. Then c,t has not necessarily the same value for any combinat ion i and ?', the spatial position of the oscillators i and i now being of importance. Thus in the case n = 5, i l lustrated by fig. 3, the interact ion parameters between the central oscillator and the two oscillators s i tua ted in the x direction will differ from the interact ion parameters between the central oscillator and the oscillators s i tuated in the z-direction or y direction. The same holds for n = 7.

We first consider n = 3.' The de terminant value D m a y be calculated from the following matrices:

a c~ c~[ a c v c v a c,. cz

c:~ a O J x cv a 0 x c~ a 0 (44) c~ 0 cy 0 a cz 0 a

The de te rminant values D ( + b) and D( - - b) m a y be calculated according to the me t hod outl ined above and the result is:

O ( + b ) D ( - - = b ) = a2[(a 2 -- 2c~ 2) -- 9b2(a --~cx) 2] X a2[(a z -- 2cv~ ) - - 9b~(a - - ~cv) 9"] × (45) a~[(a 2 - - 2cz2) -- 36b2(a -- Icz) ~]

The free energy of interact ion F becomes:

F z = - - 9 ( 3 a ~ / R e ) k T [1 -- (8/9mco2)(cx + Cy + 4cz) (46)

where, as before, only first powers of cx, c v and cz are re ta ined in the neces- sary developments. The same non-addi t iv i ty properties appear as before (n = 2) and we m a y refer to the considerations on the relative position in space of the two groups of oscillators given there.

For the case n = 5 we introduce different parameters for the interact ion between the central oscillator 1 ~nd oscillators 2 and 3, and between the central oscillator and the oscillators 4 and 5, these parameters being denoted as 2c~2, 2Cy2, 2Cz2 and 2Cx4, 2Cy4, 2Cz4 respectively. The de te rminan t value D is then obtained from the following matrices:

ON THE ADDITIVITY OF LONDON-VAN DER WAALS FORCES 227

C~2 Cx2 Cx2 a 0 .

c~z 0 a

Cx4 0 0

Cx4 0 0

Cx4 Cx4 ] a Cy2 Cy2 Cy 4 CV4 a Cz2 cz2 Cz4 Cz4[

o Ol a o o Ol o o o

I 0 O]×lCvz 0 a 0 O ] × l C ~ 0 a 0 0 (47) a 0 Icy4 0 0 a 0 Ic~4 0 0 a 0 0 a ¢y4 0 0 0 a cz4 0 0 0 a

We now consider first the position given in fig. 3a. Then:

Cm2---- - - 2 C x 4 ; C v 2 = C v 4 = C y ; --2Cz2-----Cz4 (48)

Then D can be wr i t ten as:

D = a a ( a 2 - - lOc:~42)aa(a 2 - - 4cvg")aS(a 2 - - lOcz2 z) (49)

and D ( + b) .D(- - b) calculated according to the method outlined above is

D ( + b ) ' D ( - - b) = aO2(a 2 - 10c,42) - 25b2(a z + ~ac.4 -

x # [ ( a " - 4cv~) 2 - 25b2(a - ~cv)2] x

+ a6[(a 2 -- 10cz22) -- 100b2(a 2 + ~acz2 - - ~c,2~)] (50)

The free energy of interact ion F5 becomes after considering tha t for dipole interact ion we have -- cx4 = -- c v = - - cz2 =- c > 0

F 5 a = - - 2 5 ( 3 o ~ Z / R 6 ) k T [ 1 + 4 c / 1 5 m o J ~ ] (fig. 3a) (51)

Thus the a t t rac t ion for two sets of five oscillators in the position indicated b y fig. 3a, is larger t han expected from addi t iv i ty considerations.

I f positions indicated by fig. 3b are occupied, we have:

cx2 = -- 2cx4; -- 2Cyz = %4 and cz2 ---- cz4 = cz (fig. 3b) (52)

The de te rminant D becomes:

D = a S ( a 2 - - l O c ~ 4 9 ) a S ( a 2 - - l O c y 4 2 ) a S ( a 2 - - 4c~ 2) (53)

and D ( + b) .D(- - b) becomes:

~ ) ~ ] x D ( + b ) . D ( - - b) = a4[a2(a~ - - 4c,42)2 - - 25b2(a2 + ~ac ,4 - - 2 36 2-~ X a 4 [ a g ( a 2 - - 10cv~2) 2 -- 25b~(a ~ + -~acyg. - - ~-cy~ )J x

X a6[(a z -- 4cz2) 2 -- 100bg'(a -- 8c,.)9] (54)

The free energy of interact ion F5 becomes upon considering tha t for dipole interact ion we have: -- c~4 = -- % 2 = - - cz = c > 0

Fsb = - - 25(30,2/R6)kT[1 - - 5 6 c / 1 5 m o J 2] (fig. 3b) (55)

Thus now there is a decrease of the a t t rac t ion compared with addi t ivi ty . This decrease is ra ther more pronounced than the increase found in the case, indicated by fig. 3a.

For n = 7 (fig. 4) it is easy to prove, tha t there is one mat r ix only whose

2 2 8 M . J . S P A R N A A Y

determinant value D should be obtained:

This value is:

a --2c --2c c c c c

- - 2 c a 0 0 0 0 0

- - 2 e 0 a 0 0 0 0

c 0 0 a 0 0 0

c 0 0 0 a 0 0

c 0 0 0 0 a 0

c 0 0 0 0 0 a

(56)

!

a c= c= a cy Cvl a c, c,

c= a ~c, x cv a ~cv [x c, a ~c, (60) c= a c , c . a

and we ultimately arrive at:"

FsT,- --9(3=2/R6)kTr_l ~.'t t c = - - i ~ • + cv + 4cz)/mc°2] (61)

This result differs only slightly from the result given by eq. (46) for Fs.

from

D = aS(a2 - - 12c~) (57)

For the value D ( + b) .D(- - b) one obtains for each direction x and y

D ( + b ) 'D( - - b) = al0(a ~' -- 12c2) 2 -- 49b2as(a 2 - - ~c~) 2 (58)

(For the z-direction one has 4 × 49bz(...) instead of 49b~(...)). In contradistinction with the other cases (n = 2, n = 3), n = 5) terms

with c are absent. This means that we now (n = 7) practically have addi- tivity. This is essentially due to the high symmetry of the arrangement of the oscillators (see fig. 4) combined with the fact that we considered dipole interaction only.

Inspection of the matrix sets leading to the determinant values given in this section shows that addit ivity properties can qualitatively be predicted by considering Y~ c, (the sum of the elements of the first row (or first column) minus a). If Y, c = 0 we have addit ivi ty; if Y, c :/= 0 there is no additivity.

The case in which the same interaction was supposed to be present be- tween all the oscillators of one group, the physical nature of this interaction being left out of account, has been treated previously and is particularly simple. The result was 4) :

F n = - - n2(3o~9"/R6)kT[1 + 3(n-- l )c/mco ~'] (59)

and thus the lack of addit ivi ty will be the more important, the larger n is chosen. This case may considered as an extreme one, the other extremes being represented by the cases treated in this paper. Intermediate cases may be treated along the lines of the method given above. As an example we give the case n = 3 where also the dipole interaction between the next nearest neighbours is included. Its determinant value may be determined

ON THE A D D I T I V I T Y OF LONDON-VAN DER WAALS FORCES 9-9-9

4. Q u a n t u m l i m i t / o r n = 3, n = 5 and n = 7. For the calculation of the quantum limit the shifted frequencies should be known separately. How- ever, only the determinant values D, D ( + b) and D(-- b) are known. The determinant values essentially consist of products of squared frequencies. I t may be seen from the previous section that these products contain a number of unshifted squared frequencies and the product of two or three shifted squared frequencies. Thus in order to calculate the quantum limit, quadratic or cubic equations should be solved. We will deal with the so- lution of a cubic equation, the solution of appropriate quadratic equations then easily being derived. The solution of the cubic equation can be written a s

a a - - ~2ac 9" + nb(a ~ + kac + lc 2) = al+ × a2+ × aa+ (62) where

al+ = al + alnb + flln2b 2 and a l = a = {moo2 (63)

a2+ = a2 + ag.nb + fl2n~b 2 az = a + ~c (64)

aa+ = a3 + aanb + flan~b 2 aa = a - - ?c (65)

Sirnilarly we have a l - , a2- and aa- as the roots of the cubic equation in which -- b appears instead of + b. The numbers ~, k and 1 can be obtained from eqs. (45), (50), (54) and (58) and the constants al, ~2, aa and ill , flz and fla are to be determined in terms of ~, k, l and n.

If b = 0 we simply have the complete solution of the determinant D. This solution aUows calculation of the polarizability of the sets of 3 or 5 or 7 oscillators considered, the calculation being carried out along the same method as followed above for the set of two oscillators. I t will then again appear that the interaction as given by c hardly influences the polarizability, i.e. additivity of the polarizabilities is almost preserved.

For b :# 0 the development is stopped after terms with b 2, because second order phenomena will be considered only. The quantum limit can most easily be obtained by writing down the attractive energy using the symbols introduced in eqs. (63), (64), (65). This will be done both for the classical limit and for the quantum limit. Comparison of the two expressions obtained will lead to the desired expression for the quantum limit. The terms containing b 2 of the development of the following p r o d u c t / / s h o u l d be written down for obtaining the classical limit:

1 / = a l + x a2+ x aa+ x a l - x a~_ x aa - (66)

The terms containing b 2 are:

c__ (~32 _ ~ 2 ) - - ~ x a12a~.2a82 (67) {~1 ~ + ~ + ~8 ~ - 2),c(~3 - ~ ) } + ~r a

(We have used here the approximation ( a + y c ) m = am(1 -t-m?,c/a)). After

230 M.j. SPARNAAY

conversion of ~1, ~z, ~8 and/31,/3z, f18 into terms with y, n, k and l it appears that the term in { } is unity•

Thus it contains no non-additivity properties. Non-addit ivity is given b y the term 27c(~8 ~' -- a2Z)]a. I t can easily be checked that after insertion of 7, n, k and l into (67) we arrive at the free energies of interaction Fs, F5 and F7 given in the previous section.

For the quantum limit terms containing b z of the development of the following sum E should be obtained

x = Vaz+ + Va + + Va.+ + <a , - + Va : + Va - (6B)

The terms containing b z are:

[{o~12+o~2~+o~82--27c(fls--/32)}+ 3~-'/(c/a)(o~89--o~2)~n2b2]aV' a (69)

3 Thus now the non-additivity part {?c/a(as ~ --~29") is ~ times the non- addit ivity part which was obtained in the classical limit. This is a gener- alization of the result obtained in section 2 for the case n = 2. The energies of interaction for the quantum limit therefore read: (compare with the eqs. (33), (46), (51), (55) and (58) respectively)

3t~=~ E 2 E.8 = - - 9 ~ 1 3 m ~ ~ ( c ,~+cu+4c , ) (n=3, fig. 2) (70)

3~o~cd z r 1 -I E s , , = - ~-5 4---R--~--L 1 +5---~-~cj (n = 5, fig. 3a) (71)

3ho e. E 14 -] E56 ---- - - 25 ~ 1 5mco 2 c (n = 5, fig. 3b) (72)

3hw¢¢ 9. E7 = -- 49 4R----- ~ (n -~ 7, fig. 4) (73)

where the signs and absolute values of cx, c u and cz were discussed in the previous sections and where c is positive. Thus we see that the interaction of two groups of oscillators will be a function of the symmetry of the arrangement in one group and of their position in space, the total non addit ivity effect being of the order of 10-30%. The two groups of 7 oscillators (fig. 4) will exert an attraction which is practically independent of the position in space and which, moreover, is practically additive.

For applications in physical phenomena sufficient validity of the oscillator model should be assumed and the interaction parameter c, as used in this paper, should be of the givgn magnitude. I~: was the main purpose of this paper to show that, on the basis of these two assumptions, the attraction between two groups of atoms cannot a priori be considered as built up along the theorem 1) 9) of the additivity of London-Van der Waals forces.

ON THE ADDITIVITY OF LONDON-VAN D E R WAALS FORCES 231

A P P E N D I X

Knowledge of general ized coordinates is as a rule not needed for the so- lu t ion of the problem of f inding energy values. I t is sufficient to know tha t a l inear t r ans format ion of the coordinates, leading to generalized coordi- nates , is always possible for problems as t r ea t ed here i0). Fo r convenience however , we give the coordina te t r ans fo rmat ion for n = 3. The t ransfor- ma t ion for n = 5 and n = 7 are more e laborate to work out and ~i l l not be considered.

Af ter car ry ing out the t r ans format ion (40) we obta in for n - - 3 an ex- pression o f ' t he form:

W+ = (a + b)(sl ~ + s ~ + s8 ~) + 2(b + c)sl(s2 + ss) + 2bsg.s3 (74)

This expression can be conver t ed into

W+ = a l+ r l 2 4- ag.+ rg. 9" 4- a3+ r39 (75)

b y means of the following t rans format ion :

Sl = sin X rg. - - cos X r8 ]

s2 = ½1/2r l 4- ½1/2 cos ;~ r2 4- ½1/2 sin Z rz J (76) s8 = - - ½1/2r l 4- ½1/2 cos • r2 4- ½~2 sin Z r8

where the t r igonometr ic funct ions are de te rmined by :

(b + c)/b = ¼1/2 tg 2;~ (77) I t is found t h a t

a l+ = a (78)

as+ = a 4- b 4- (b 4- c)1/2 sin 2 z 4- b cos 2;~ (79)

az+ = a 4- b - - (b 4- c)~/2 sin 2;~ 4- b sing Z (80)

Since we of ten have b ~ c, bo th cos ;~ and sin Z will be near to ½1/2. The t r ans fo rma t ion (76) is a funct ion of b, which is not the case for the coordina te t r ans format ions (6) and (19) for n = 1 and n = 2.

Received 13-11-58

R E F E R E N C E S

1) L o n d o n , F., Z. physik. Chem. B 11 (1930) 222.

2) Bade , W. L., J. chem. Phys. 27 1280 (1957). 3) D r u d e , P. K. L., The theory of Optics, Longmans, Green, London, 1933. 4) O v e r b e e k , J. Th. G. and S p a r n a a y , M. J., Disc. Faraday Society, 18 (1954) 19; J e h l e , J. ,

J . chem. Phys. 18 (1950) 1150. 5) S i l b e r s t e i n , L., Phil. Mag. :18 (1917) 92, 215, 521.

C a b a n n e s , J. , La Diffusion mol~eulaire de la Lumi~re, Parijs, 1929. 6) L o n d o n , F., J. phys. Chem. 411 (1942) 305. 7) E i d i n o f f , M. L. and A s t o n , J. G., J. chem. Phys. 8 (1935) 379. 8) See for example P. B. Fischer, Determinanten, Sammlung, GSschen no. 402. 9) M a r g e n a u , H., Rev. rood. Phys. 11 (1939) 1.

10) See textbooks such as M. B o r n and P. J o r d a n , Elementare Quantenmechanik, Springer, Berlin, 1930. Ch. II and III . E. T. W h i t t a k e r , Analytical Dynamics 4th ed. Dover, 1944, Ch. VII.