THE GENERAL THEORY OF MOLECULAR FORCES.

By F. London (Paris).

Received 31st July, 1936.

Following Van der Waals, we have learnt to think of the molecules as centres of forces and to consider these so-called Molecular Forces as the common cause for various phenomena: The deviations of the gas equation from that of an ideal gas, which, as one knows, indicate the identity of the molecular forces in the liquid with those in the gaseous state; the phenomena of capillarity and of adsorption; the sublimation heat of molecular lattices; certain effects of broadening of spectral lines, etc. It has already been possible roughly to determine these forces in a fairly consistent quantitative way, using their measurable effects as basis.

In these semi-empirical calculations, for reasons of simplicity, one imagined the molecular forces simply as rigid, additive central forces, in general cohesion, like gravitation; this presumption actually implied a very suggestive and simple explanation of the parallelism observed in the different effects of these forces. When, however, one began to try to explain the molecular forces by the general conceptions of the electric structure of the molecules it seemed hopeless to obtain such a simple result.

[section] 1. Orientation Effect.

Since molecules as a whole are usually uncharged the dipole moment μ was regarded as the most important constant for the forces between molecules. The interaction between two such dipoles μI and μII depends upon their relative orientation. The interaction energy is well known to be given to a first approximation by

U = - μIμII/R3 (2 cos θI cosθII - sin θI sin θII cos(φI - φII (I)

where θI, πI; φI; θII, πII are polar co-ordinates giving the orientation of the dipoles, the polar axis being represented by the line joining the two centres, R = their distance. We obtain attraction as well as repulsion, corresponding to the different orientations. If all orientations were equally often realised the average of ITLμITL would be zero.

But according to Boltzmann statistics the orientations of lower energy are statistically preferred, the more preferred the lower the temperature. Keesom, averaging over all positions, found as a result of this preference:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

For low temperatures or small distances ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) this expression does not hold. It is obvious that the molecules cannot have a more favourable orientation than parallel to each other along the line joining the two molecules, in which case one would obtain as interaction energy (see (1)):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

which gives in any case a lower limit for this energy. (2) and (3) represent an attractive force, the so-called orientation effect, by which Keesom tried to interpret the Van der Waals attraction.

[secdtion] 2. Induction Effect.

Debye remarked that these forces cannot be the only ones. According to (2) they give an attraction which vanishes with increasing temperature. But experience shows that the empirical Van der Waals corrections do not vanish equally rapidly with high temperatures, and Debye therefore concluded that there must be, in addition, an interaction energy independent of temperature. In this respect it would not help to consider the actual charge distribution of the molecules more in detail, e.g. by introducing the quadrupole and higher moments. The average of these interactions also would vanish for high temperatures.

But by its charge distribution alone a molecule is, of course, still very roughly characterised. Actually, the charge distribution will be changed under the influence of another molecule. This property of a molecule can very simply be described by introducing a further constant, the polarisability [varies]. In an external electric field of the strength F a molecule of polarisability [varies] shows an induced moment

M = α x F.... (4)

(in addition to a possible permanent dipole moment) and its energy in the field F is given by

U = 1/2 [varies] x F2.... (5)

Now the molecule I may produce near the molecule II an electric field of the strength

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

This field polarises the molecule II and gives rise to an additional interaction energy according to (5)

U = -1/2 [varies]IIF2 = [varies]II/2 μI2/R6 (I + 3 cos2 θI) .. (7)

which is always negative (attraction) and therefore its average, even for infinitely high temperatures, is also negative. Since cos2 θ = 1/3 we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A corresponding amount would result for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e. for the action of μII upon [varies]I. As total interaction of the two molecules we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... (8)

If the two molecules are of the same kind (μI = II = μ and [varies]i = [varies]I = [varies]II = [varies]) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].... (8')

This is the so-called induction effect.

In such a way Debye and Falckenhagen believed it possible to explain the Van der Waals equation. But many molecules have certainly no permanent dipole moment (rare gases, H2, N2, CH4, etc.). There they assumed the existence of quadrupole moments τ, which would of course also give rise to a similar interaction by inducing dipoles in each other. Instead of (8) this would give:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].... (9)

Since no other method of measuring these quadrupoles was known, the Van der Waals corrections (second Virial coefficient) were used in order to determine backwards ITLτITL which, after μ and [varies], has been regarded as the most fundamental molecular constant.

[section] 3. Criticism of the Static Models for Molecular Forces.

The most obvious objection to all these conceptions is that they do not explain the above mentioned parallelism in the different manifestations of the molecular forces. One cannot understand why, for example, in the liquid and in the solid state between all neighbours simultaneously practically the same forces should act as between the occasional pairs of molecules in the gaseous state. All these models are very far from simply representing a general additive cohesion:

Suppose that two molecules I and II have such orientations of their permanent dipoles that they are attracted by a third one ; then between the two former molecules very different forces are usually operative, mostly repulsive forces. Or, if the forces are due to polarisation, the acting field will usually be greatly lowered, when many molecules from different sides superimpose their polarising fields. One should expect, therefore, that in the liquid and in the solid state the forces caused by induced or permanent dipoles or multipoles should at least be greatly diminished, if not by reasons of symmetry completely cancelled.

The situation seemed to be still worse when wave mechanics showed that the rare gases are exactly spherically symmetrical, that they have neither a permanent dipole nor quadrupole nor any other multipole. They showed none of the mentioned interactions. It is true, that for H2, N2, etc., wave mechanics, too, gives at least quadrupoles. But for H2 we are now able to calculate the value of the quadrupole moment numerically by wave mechanics. One gets only about 1/100 of the Van der Waals forces that were attributed hitherto to suitably chosen quadrupoles.

On the other hand, wave mechanics has provided us with a completely new aspect of the interaction between neutral atomic systems.

[section] 4. Dispersion Effect; a Simplified Model.

Let us take two spherically symmetrical systems, each with a polarisability [varies], say two three-dimensional isotropic harmonic oscillators with no permanent moment in their rest position. If the charges e of these oscillators are artificially displaced from their rest positions by the displacements

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

respectively, we obtain for the potential energy:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].... (10)

Classically the two systems in their equilibrium position

(xI = XII = … = zII = 0)

would not act upon each other and, when brought into finite distance (R >3 [square root of [2 [varies], remain in their rest position. They could not influence a momentum in each other.

However, in quantum mechanics, as is well known, a particle cannot lie absolutely at rest on a certain point. That would contradict the uncertainty relation. According to quantum mechanics our isotropic oscillators, even in their lowest states, make a so-called zero-point motion which one can only describe statistically, for example, by a probability function which defines the probability with which any configuration occurs; whilst one cannot describe the way in which the different configurations follow each other. For the isotropic oscillators these probability functions give a spherically symmetric distribution of configurations round the rest position. (The rare gases, too, have such a spherically symmetrical distribution for the electrons around the nucleus.) We need not know much quantum mechanics in order to discuss our simple model. We only need to know that in quantum mechanics the lowest state of a harmonic oscillator of the proper frequency v has the energy

E0 = 1/2 hv..... (11)

the so-called zero-point energy. If we introduce the following co-ordinates ("normal"-co-ordinates):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

the potential energy (10) can be written as a sum of squares like the potential energy of six independent oscillators (while the kinetic energy would not change its form):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10')

The frequencies of these six oscillators are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Here v0 = e/[square root of] [m[varies]] is the proper frequency of the two elastic systems, if isolated from each other (R [right arrow] ∞), and m is their reduced mass. Assuming [varies] [much less than] R3, we have developed the square roots in into powers of ([varies]/R3.

The lowest state of this system of six oscillators will therefore be given, according to (II), by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first term 3hv0] is, of course, simply the internal zero-point energy of the two isolated elastic systems. The second term, however,

U = 3/4 hv0[varies]2/R6.... (13)

depends upon the distance R and is to be considered as an interaction energy which, being negative, characterises an attractive force. We shall presume that this type of force, which is not conditioned by the existence of a permanent dipole or any higher multipole, will be responsible for the Van der Waals attraction of the rare gases and also of the simple molecules H2, N2, etc. For reasons which will be explained presently these forces are called the dispersion effect.

[section] 5. Dispersion Effect; General Formula.

Though it is of course not possible to describe this interaction mechanism in terms of our customary classical mechanics, we may still illustrate it in a kind of semi-classical language.

If one were to take an instantaneous photograph of a molecule at any time, one would find various configurations of nuclei and electrons, showing in general dipole moments. In a spherically symmetrical rare gas molecule, as well as in our isotropic oscillators, the average over very many of such snapshots would of course give no preference for any direction. These very quickly varying dipoles, represented by the zero-point motion of a molecule, produce an electric field and act upon the polarisability of the other molecule and produce there induced dipoles, which are in phase and in interaction with the instantaneous dipoles producing them. The zero-point motion is, so to speak, accompanied by synchronised electric alternating field, but not by a radiation field: The energy of the zero-point motion cannot be dissipated by radiation.