Van der Waals Force between Two Localized Clusters of Bosons

R. K. MISHRA, K. BHAUMIK, A. SRIVASTAVA, AND S . S . CHAUDHARY

Department of Biophysics, All India Institute of Medical Sciences, New Delhi-29, India

Abstract

In traditional quantum electrodynamic derivation of a intermolecular van der Waals force, the molecules are assumed to be localized clusters of electrons. In this paper we have assumed the molecules to be localized clusters of bosons and have shown that the expression of the potential re- mains formally the same. This vindicates, through a microscopic derivation, the established concept that the van der Waals interaction is essentially a macroscopic phenomenon.

1. Introduction

The physics of the van der Waals force between neutral molecules is a well- studied phenomenon. In its quantum electrodynamic version the molecules are approximated [ 11 as localized clusters of electrons, interacting with one another through the exchange of photons (which are emitted or absorbed due to the fluctuation of the states of the electrons). The usual technique is to consider the two-photon exchange diagrams, which give the lowest order contribution to the interaction energy, and then to convert the expression of energy into the well- known Mclachlan relation [2], which gives the interaction potential as the in- tegral over the “generalized susceptibilities” [3] of the two systems under con- sideration. From this generalized susceptibility relation, with due consideration to the geometry of the system one can easily arrive at the famous London dis- persion force [4] or the retardation force of Casimir and Polder [5].

In this paper we shall find the intermolecular van der Waals force by assuming the molecules as localized clusters of bosons instead of considering them clusters of fermions. This study is motivated by the fact that the description of a mac- romolecule in terms of its elementary excitations (like excitons, etc.), which are all bosons at low density, is also an useful method of studying the intermolecular interaction. However the intermolecular potential, as given by the Mclachlan relation, contains a statistical weight factor. It is therefore worthwhile investi- gating whether or not the intermolecular van der Waals potential can depend on the statistics of the cluster. It will be shown in this paper that the final ex- pression of the van der Waals potential between two localized clusters of bosons and the corresponding expression between two localized clusters of electrons comes out formally the same. This once again vindicates the macroscopic nature of the van der Waals force even through the microscopic derivation.

International Journal of Quantum Chemistry, Val. XX, 377-383 (1981) 0 1981 by John Wiley & Sons, Inc. CCC Ol61-3642/81/080377-07$01 .OO

378 MISHRA ET AL

2. Interaction Potential

We write the Hamiltonian for the interacting clusters of bosons (in natural unit)

H = C Qibfbi + C f wq (u! uq + uq u ! ) i 4

+ C [ g + ( q ) a $ + g-(q)aqI blbi, (1) i ,k ,q

where b! (b;) and u$ (aq ) are the creation and annihilation [ 6 ] operators for the localized bosons (hereafter called “Bosons”) and the photons, respectively, g* ( 4 ) are the couplings at the vertex, suffixes i and k run over the “Bosonic” states in both of these clusters, suffix q denotes the photonic states, and the other notation has its usual significance. We have shifted the zero of the energy scale by the zero-point energy of the “Bosons” to keep uniformity with the Hamil- tonian in the electronic picture (for example, Ref. 1). The unperturbed state and energy eigenvalues are given by

$0 = INi, Nk, . . ., nq, n,, . . . )

Eo = CQiNi + C w q ( n q + 4)

(2a)

(2b)

and

where N and n refer to the occupancy of “Bosons” and photons, respectively. Since the process under consideration is fluctuation and not scattering, the second order diagrams contribute mainly to the self-energy of the bosons and the pho- tons. Renormalizing up to second order [6], we get

and

The lowest order contribution to the interaction potential comes from the fourth order diagrams. Denoting the bosonic states by suffixes (I, j , . . .) in the first cluster and by suffixes ( c u , ~ , . . .) in the second cluster, we get the corresponding expression for interaction potential

V = -- C C C C N I ( 1 + N;)N,( l + Np)(nq + $ f &)(nr + 4 f 4) 1 2 * qr I; a p

g* ( r k * ( 4 ) (f wq f wr)(Q; - Q[ f w q )

+ g f ( q ) g * ( 4 )

(Q; - Q I + Q2p - Q,)(Qj - Q I f w q ) X

LOCALIZED CLUSTERS OF BOSONS 379

g'(q)g'(r) + g'(r)g'(q) + (Q, - Q , f wq Q j - Q I f w,

g*(r)g*(q) (Q j - s t l f wq + Q , - Q , f U,)(Qj - Q I f w,)

X

After a little rearrangement with the help of the partial fractions as well as taking advantage of the dummy suffixes we get the potential (up to fourth order)

1 V = -2 C C C C N I (1 + Nj>Na (1 + N B )

f 9r l j a@

Srg*(q)g*(r) + Srg*(q)g*(r) s t j - 01 f w, Qj - Q 1 3 = w,

g'(q)g'(r) + gFF(q)gYr) Q , - Q, 3= wq Q , - Q, f wq

S9 Sr g* (4) g* ( r ) ( Q j - Q I f U,)(Qj - Q[ =I= w,)

-

where S, and S, are the signature factors as given by S,g+(r) = + g + ( r ) , S,g-(r) = - g - ( r ) , and similarly for S,g*(q). The summation 2* reminds us that there will be no net emission or absorption of photons and hence proper combination of g's should be taken [for example, (g+(q) g-(r)) (g-(q) g + ( r ) ) ] . It is clear from Eq. (4) that up to fourth order the total energy is linear in (n, + $) and hence by combining Eqs. (3b) and (4), we can write

(w,> = a 9 - x ( w 9 , +q, -9)

x(w,, +q, fr)x(-w,, 7 7 Tr), ( 5 ) Sr - C C

i r f w q f w r

where

Since "generalized susceptibility" is the Fourier transform of the response function [3], which is actually the correlation function of the fluctuations of the system [7], it is obvious that x's as defined by Eq. (6) are the measures of the susceptibility of the system.

3. Statistics of the Quasiparticles

In order to get the desired susceptibility relation from Eq. (4), one has to take the average of the occupancy Ni and n,, which necessitates a knowledge of the statistics of the quasi-"Bosons" and quasiphotons. To this end we diagonalize the Hamiltonian by introducing B: (Bi) and A: (A,) as the quasi-"Boson" and

380 MISHRA ET AL.

the quasiphoton creation (annibilation) operators, respectively. The transfor- mation relation between the diagonalized and nondiagonalized operators are

g - ( r ) g * ( q ) A ,fB:Bj - 2

+ c i j /mq (Q/ - Q; - u,)(Q/ - Q; - 0, + Q i - Q k f wq)

+... (7b) g - ( r ) g*(q)A, fB:Bk

j l k q (Q/ - Q; - o r ) ( Q / - wr - Qk f wq)

and similarly for b, and a,. Here A ; means A , (annihilation) operator. Sub- stituting relations (7a) and (7b) in Eq. (1 ) we get the diagonalized Hamiltonian upto second order. It can of course be diagonalized upto any desired order but for the present discussion second order diagonalization would suffice our purpose. Thus

With the diagonalized Hamiltonian, we can easily write its energy eigenvalues (upto second order)

E2 = Ei N; + c w q (nq + 4). where

& = ( Q i ) + [ (&Iq) - w,](n, + t). (8b)

We can also evaluate Ei alternatively as follows Ref. 1. Since the total energy (upto fourth order) is linear in (nq + t), we may carry out the summation over photon occupation number nq, n,, . . ., so that the total energy (upto fourth order)

LOCALIZED CLUSTERS OF BOSONS

is now given by

381

E kT N ~ , N , ,..., ng,nr kT

E(Ni,Nj, . . .,nq.nr. . . .) - = C C . . . )

where ( w q ) is given by Eq. (5). Expanding ( w q ) around a,, we may write the partition function

Z = [ 2 sinh (%)I-’ Cexp (s), 4

where

1 2 4 2kT

ki = (Qi) + - C [(w,) - w q ] coth (*) + 0(g4).

Comparing Eqs. (8b) and (9), we observe that the relation

(9 )

satisfies the consistency. It, therefore, shows that the quasiphotons are an in- dependent system of bosons. On the other hand quasi-“Bosons” are not inde- pendent. Since the process under consideration is fluctuation, any change in the occupancy N; is going to associate a concommitant change in N, ( i # j ) . Hence the distribution of such a system would be a Bose distribution with finite chemical potential p, as

N; =, l/(e(Ei-p)/kT - 1 ) .

where Ni is the occupancy in the level of energy Ei and p is determined from the condition that the number of “bosons” is equal to their average number. It is easy to observe the identity

( 1 1 ) E k - E j N i ( N k + l ) + N k ( N i + l )

coth ~ - ( 2kT ) - (Ni)(Nk + 1) - Nk(Ni + 1 ) ‘

Using Eqs. (lo), ( 1 l ) , and (6), we may rewrite Eq. (4) in the form

4. Mclachlan’s Relation

Each one of the integrals, summed implicitly in Eq. (12), have series of poles either on the positive or on the negative real axis (but not both). We can therefore distort the contour towards the imaginary axis [8], so that the entire contour may be divided into two parts; one from -i- to +im along the imaginary axis and the other is along an infinite semicircle. No new singularities would be en-

382 MISHRA ET AL.

countered while distorting the contour, except some branch points on the imaginary axis, which are to be avoided. Obviously the contribution from the infinite semicircle vanishes. We can, therefore, write Eq. (1 2) as

9 (13) XI(P> fq , fr) x 2 ( - p . =v. - r )

W r f P x c

qr W q f P where suffixes 1 and 2 indicate the positions of the clusters. Equation ( I 3) is the generalized Mclachlan relation [2] and is identical in form to the van der Waals potential between two localised clusters of electrons [ 11.

Our work thus has proved that the van der Waals force between two localized clusters of bosons is formally identical to that between two localized clusters of electrons. One must appreciate the fact that the determination of statistics of the quasiparticle in the bosonic case is much easier and neater than the de- termination of statistics of the quasiparticle in the electronic case, since in the later system the quasiparticle states would be a messy mixture of the both bosonic and fermionic naked particle states. Hence, we conclude that as far as van der Waals forces are concerned, the representation of a macromolecule in terms of its elementary excitations is equally good to and even better than the traditional representation in terms of electronic states. We are not, at present, in a position to characterize these bosons. In general these may be understood as a set of “virtual” excitation quanta having frequencies equal to the transition frequency of the molecules. To avoid possible confusion we add a note of caution that in this paper we have not tried to find a one-to-one correspondence between the electronic picture and the bosonic picture. In such an exercise a trilinear Hamiltonian in the former picture may turn out to be a quadratic one in the later picture. Our work gives only a formal equivalence in the two pictures as far as van der Waals forces are concerned. This, therefore, reaffirms through a mi- croscopic derivation that the van der Waals force is essentially a macroscopic phenomenon. Understandably this paper is a small step toward the effect of bridging the gap between quantum chemistry and the physics of collective ex- citation.

Acknowledgments

We are thankful to Professor B. Dutta-Roy (SINP, Calcutta), Dr. D. Bhaumik (Bose Institute, Calcutta), and Dr. A. Lahiri (Vidyasagar College, Calcutta) for illuminating suggestions.

Bibliography

[ I ] D. Langbein, “Van der Waals Attraction,” in Springer Tracts in Modern Physics (Springer,

121 A. D. Mclachlan, Proc. R . Soc. London, Ser. A 274, 80 (1963). [ 3 ] A. D. Mclachlan, Proc. R. Soc. London, Ser. A 271,387 (1963).

Berlin-Heidelberg-New York, 1974), Val. 72.

LOCALIZED CLUSTERS OF BOSONS 383

[4] F. London, Z. Phys. Chem. B 11,222 (1930). [5] H. B. G . Casimir and D. Polder, Phys. Rev. 73,360 (1948). [6] J . M. Ziman, Elements of Advanced Qunntum Theory (Cambridge U.P., Cambridge, England,

[7] R. Kubo and K. Tomita, J. Phys. SOC. Jpn. 9,888 (1954). [8] E. M. Lifshitz, Sov. Phys. JETP 2, 73 (1956).

1969).