The effect of nearest-neighbour correlations on the elastic constants of a nematic liquid crystal

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The effect of nearest-neighbour correlations on the elastic constants of a nematic liquid crystal

S. Sarkar, R.J.A. Tough

To cite this version:

S. Sarkar, R.J.A. Tough. The effect of nearest-neighbour correlations on the elastic con- stants of a nematic liquid crystal. Journal de Physique, 1982, 43 (10), pp.1543-1555.

�10.1051/jphys:0198200430100154300�. �jpa-00209535�

The effect of nearest-neighbour correlations on the elastic constants of a nematic liquid crystal

S. Sarkar and R. J. A. Tough

Royal Signals ^{and Radar} Establishment, St Andrews Road, ^Great Malvern, ^WR14 3PS, U.K.

(Reçu le 26 avril 1982, ^{révisé le} 22 juin, accepté ^le 24 juin 1982)

Résumé. 2014 On étudie un modèle de nématique ^{dont les} constantes élastiques, évaluées par ^une théorie de champ

moyen, varient linéairement ^avec le carré du paramètre d’ordre. Les interactions entre molécules dérivent de

potentiels dispersifs ^et dipolaires ^et les corrélations entre plus proches ^voisins, négligées par la théorie de champ

moyen, ^sont traitées par ^une approximation de Bethe-Peierls. Les calculs reposant ^sur l’énergie libre donnent des résultats différents de ceux utilisant l’énergie interne. Les constantes élastiques déduites de l’énergie ^{libre conservent}

leur dépendance ^linéaire en S22 tandis que celles déduites de l’énergie ^{interne ne montrent} pas ^une dépendance ^aussi simple ^{en loi de} puissance.

Abstract.

The elastic constants of a model nematic, ^{which in a mean field} theory ^{show a linear} dependence

on the square of the order parameter S2, ^are investigated. The constituent molecules of the nematic interact through dispersive ^and dipolar potentials ^and nearest-neighbour correlations neglected ^{in the mean field} theory ^{are des-}

cribed using the Bethe-Peierls cluster model. Calculations based on the free and internal energies of the nematic

are found to give ^different results; the elastic constants derived from the free energy retain their linear dependence

on S22, while those derived from the internal energy show ^no such simple power law dependence.

Classification

Physics ^Abstracts

61. 30C

1. Introduction.

The variation with temperature of k11, k22, k33, the elastic constants of a nematic liquid crystalline material, ^and their relation to the structure of its constituent molecules have been studied inten-

sively ^{in recent} years. While much has been achieved in the refinement of the experimental determinations of these elastic constants (for ^{a review see} the article of Gerber and Schadt [1]) the results so obtained are

frequently interpreted only ^{in terms of mean field}

and molecular field models. Of these the simplest ^{is the}

mean field theory of de Gennes [2] from which each of the elastic constants is found to be proportional ^to s1, ^where S2 ^{is a} member of the family ^{of nematic}

order parameters ^defined by

Here Pi ^{is a} Legendre polynomial, ^{0 is the} angle ^bet-

ween the long ^{axis of a} typical molecule and the local

optic ^{axis or director n of the} liquid crystal and the bar denotes an ensemble _average. No molecular details

are incorporated ^{in this} theory. ^This deficiency ^is

remedied in the molecular field model of Priest [3]

which incorporates ^an arbitrary intermolecular poten- tial. However the anisotropy in molecular orientation reflected in the non-zero values of S21’ ^is imposed only

by ^{a molecular} field, which maintains an orientational

ordering consistent with the long-range order in the nematic but neglects details of shorter-range ^correla-

tions. This allows the elastic constants to be expanded

in terms of the 821. ^{Such series are} typically ^truncated

at terms including S4 :

Here T=-I(kll ⁺ k22 ⁺ k33) ^is proportional to S2,

while d, ^{d’ are constants} characteristic of the ^structure of the constituent molecules of the nematic. Attempts

have been made to incorporate effects of short-range

order into Priest’s theory through ^{a wider} interpreta-

tion of the parameters d, ^{d’ and} (albeit ^{in the} special

case of a fully ordered nematic at absolute zero) by modelling the molecular pair distribution function

[4, 5]. Faber has also discussed the variation of the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198200430100154300

elastic constants with the order parameter S2, basing

his analysis ^{on the} change ^on distortion of the internal energy of the liquid crystal [6]. ^This he expresses in terms of non-integral powers of S2, through ^his spin-

wave type ^continuum theory of molecular orienta- tional correlations in the nematic phase [7]. However,

as this model incorporates short-range ^{order not}

described by the molecular field theory, the result of the calculation cannot be expected ^to be identical with a more correct determination of the elastic constants based on the free _energy of the system [3].

Furthermore, both Priest’s and Faber’s theories contain quantities (for example ^{A and 4 ’ in} (2)) ^which should, ⁱⁿ principle, be determined self-consistently

from the intermolecular potential, ^{but are} merely

identified as empirical parameters.

As the elastic constants and order parameter S2 ^{of a}

nematic can now be determined quite reliably, ^detailed comparisons of these theories with experiment ^{can be}

made [8]. ^The principal findings ^{are that the} simple

mean field result

does not hold over the full nematic _range and that deviations from this behaviour are not well accounted for by ^Priest’s expansion ^{truncated at} S4, ^or by ^Faber’s

model. Furthermore effects ascribed to short-range

order and polar interactions have been observed

experimentally and discussed in terms of the ad-hoc extensions of Priest’s calculations outlined above [9].

To assess the validity ^{of the} interpretation ^{of devia-}

tions from mean field behaviour in terms of a simple nearest-neighbour picture ^of short-range ^{order we}

have calculated the elastic constants of a model nematic, ^{for which mean field} theory predicts ^{a linear}

variation with 51 ^over the entire nematic _range.

Short-range ordering ^and dipolar interactions are

treated using the Bethe-Peierls cluster model [10],

which has been used successfully for Chandrasekhar and Madhusadana to account for their effects on the

thermodynamic and dielectric properties ^{of nematic} liquid crystals [11]. ^It should be stressed that in this

approach , correlations between a molecule and its

nearest-neighbours ^{are not} represented by ^some simple but uncontrolled ad-hoc model but are treated in a detailed and self consistent manner. Consequently

our calculations ^{are rather} involved; this is the price

which must be paid ^{if we are to have a realistic assess-}

ment of the value of the nearest-neighbour picture ⁱⁿ

the discussion of elastic constants. Much of the analysis

involved in our work is relegated ^to appendices, ^the

second of which also contains several new properties ^of

moments of the Maier-Saupe orientation distribution familiar from simple ^mean field theories. As we are

able to evaluate the change ^on distortion in both the free and internal energies of the model we are also able to test the validity ^of calculations, ^{such as that of} Faber, ^{which are based on the} internal, rather than the

free, energy change. The rather surprising ^{result of our}

calculations is the persistence of the linear dependence

on S’ of the elastic constants of the model derived from the change in its free energy; the internal energy calculation yields ^{elastic constants which do not show}

this simple linear behaviour and might unwittingly ^be thought ^to show deviations from mean field behaviour attributable to the effects of short-range order. Thus

we are led to the conclusion that calculations of elastic constants based on changes ^{in the internal} energy of the

liquid crystal ^{can be} quite misleading ^{and that} simple

« nearest-neighbour » ^{models of molecular} correlation may ^not be adequate ^to describe deviations from the

mean field dependence of the elastic constants on the nematic order parameters S21-

2. Description of model.

The nematic liquid crystal is modelled by molecules sited on a simple

cubic lattice which interact through ^a potential containing isotropic dispersive ^and pseudo-dipolar

terms.

where Oii ^{is the angle} between the long ^axes of molecules

on sites i, j, pl, P2 ^are again Legendre polynomials ^and A *, ^{B * are} positive constants. Short-range ^{order is}

treated in the Bethe-Peierls approximation [10],

which has been used successfully by Chandrasekhar and Madhusadana in assessing its effects on the ther-

modynamic and dielectric properties of nematic

liquid crystals [11]. ^{Thus we} describe in detail the interactions of a molecule with the members of the cluster it forms with its nearest-neighbours, ^whose

interactions with the remaining ^{molecules on the}

lattice are modelled by ^a molecular field v(Oi), ^where 9i

is the angle between the long axis of the molecule at site i and the director n. The form of v( fJi) ^{is determined} by the intermolecular potential (3) ^{and a} simple ^self consistency requirement [11]. ^The properties ^{of such a}

cluster and their thermodynamic ^self consistency ^are

discussed in the next section.

By positioning the molecules on a lattice we intro- duce a length ^scale (the ^lattice spacing a) ^{into the}

model which is absent in the work of Chandrasekhar and Madhusadana, ^{but is} necessary in a theory ^of

elastic constants. In the models of Priest and Faber this length ^{scale is} provided by the range of the inter- molecular potential. The confinement of the molecules to lattice sites does, however, disqualify ^{the model}

from describing the contribution to the elastic cons- tants of changes ^on distortion of the translational free _energy [3]. ^This deficiency ^{is of course shared} by ^all

calculations based on the internal rather than the free _energy of the nematic.

For a model of the type considered here it is readily

shown that the elastic constants are equal [5, 12] :

and that if short-range ^{order is} neglected, ^{as in a}

molecular field calculation, ^then

holds over the entire nematic _range [13, 14].

3. Thermodynamic properties ^{of an} undistorted cluster.

The cluster formed by ^a molecule and its six nearest-neighbours plays ^a central role in the ana-

lysis ^of short-range order in the Bethe-Peierls approxi-

mation. So, ^before discussing ^the changes ^{on distor-}

tion in the free and internal energies of the nematic

sample, ^we give expressions for the free and internal

energies ^{of an} undistorted cluster and demonstrate their thermodynamic ^self consistency.

The internal _energy of the undistorted cluster, whose central molecule is at site 0, ^is given ^{in terms of}

the intermolecular potential V(Ooi) and the molecular field v(Oi) by

where the sum is taken over the 6 exterior members of the cluster. The angular brackets ( ) ^{indicate an}

average ^over the Boltzmann distribution

where fl ⁼ (kB T)-’, kB is Boltzmann’s constant, ^T is the absolute temperature ^and Zo ^{is the} partition

function of the undistorted cluster

the integration being ^{taken over the} angular ^coordi-

nates of all members of the cluster.

The factor 2 preceding the ( v(Oi) > compensates for double counting of the intermolecular interactions

represented by ^the molecular field. The entropy ^of the cluster is given by the Boltzmann expression

thus we can write the cluster free _energy as

The expressions (5), (9) ^{should be} thermodynamically

self consistent, satisfying ^the identity

It is straightforward ^{to show that}

and _{so, to} ensure thermodynamic ^self consistency,

the equality

must hold. We will now show that (10) ^is satisfied, pro- vided that the free _energy of the cluster is minimized with respect ^to the form of the molecular field v(0;) acting ^{at the} exterior sites of the cluster. v( OJ ^is expand-

ed as a sum of Legendre polynomials

the coefficients _v, being functions of the moments

characterizing ^the long-range ^{order in} the nematic.

On minimizing the cluster free _energy with respect

to the moments pk, ^we obtain the conditions

As

Similarly

and consequently

the equality (10) being ^{seen to hold} provided the cluster free _energy is minimized with respect ^{to the form} (11)

of the molecular field u(0J.

4. Derivation of the elastic constants.

We now identify the elastic constant K of the model nematic by considering ^the change (to ^order q2) in its bulk free _energy resulting ^{from the} imposition ^{of a} long wavelength

director reorientation about the ^x axis with wave vector q parallel ^{to the} lattice y ^axis, which is manifest in a single

cluster through the rotation of the axes of the molecular fields acting ^on the exterior members positioned ^{on the}

y axis (see Fig. 1). ^As the intermolecular potential (3) ^is isotropic ^{there is no loss in} generality (to ^order q2 ) ⁱⁿ

choosing ^{the axes} of the molecular fields at sites 1, 3, 5, ^{6 to} be unrotated. The partition ^function of the distorted cluster can now be written as (cf. ^Eq. (7))

where

and

Fig. ^1.

A distorted cluster. The axes of the molecular fields v(8J ^{at sites} 1, 3, 5, ^{6 are} unrotated ; ^{those at sites} 2,

4 are rotated through + 0, were 0 = t/Jo qa ⁺ 0(q3).

gd(02), gd(04), the Boltzmann factors corresponding ^{to the} rotated molecular fields are obtained from 9(02), 9(04), using ^{the rotation} operators familiar from the quantum theory ^of angular ^momentum [15]

Here il2x, il4x generate rotations about the x axes at sites 2, ⁴ respectively. ^The validity ^{of this} procedure ^{in cal-} culating changes ^{to order} q2 ^has been discussed by ^Priest [3]. ^The unitarity ^{of the rotation} operators ^{allows us to}

recast this result as

To evaluate this integral ^{to order} tf¡2 ^{we first} expand the rotation operators formally ^as power series. The func- tions f (Ooi) ^are expressed ^{in terms of} Legendre polynomials ^{in cos} 00; ^and then, using ^the spherical ^harmonic

addition theorem, ^{in terms of} spherical harmonics in the polar coordinates Qo, Qi defining ^the directions of the

long ^axes of the molecules at sites 0, ^{i :}

The spherical ^harmonics Ck,, ^are normalized so that

and the Condon-Shortley phase convention has been adopted [15]. ^This decomposition of f (Ooi) ^is particularly

useful. The angular coordinates of the cluster members are immediately decoupled and, ^as spherical ^harmonics

of rank k form a rotationally irreducible set, the effect of the rotation operators ^on /(0of) ^{can be} represented very

simply. Furthermore the orthogonality properties ^{of the} spherical ^{harmonics can be} exploited ^{to reduce the}

14-dimensional integral ⁱⁿ (16) ^{to a} 1-dimensional integral, ^which simplifies computations quite dramatically.

The analysis involved in carrying through this program is given ⁱⁿ Appendix ^{A. Here we note that the} partition

function of the distorted cluster is, ^{to order} tf¡2,

where Pk ^{is an} associated Legendre function and

The convergence of the series in the integrand ⁱⁿ (19) is assured by ^the rapid fall-off with k of the quantities fk, Pk (see (B. .10)). ^{Thus we are able to} calculate the free energy of the distorted cluster, noting ^that

as

In summing up the free energies of the distorted clusters ^on the lattice to obtain the change ⁱⁿ the free energy of the bulk sample ^{we first note} that, ^{to order} q2, all distorted clusters ^are equivalent. Thus, ^on summing (21) ^over

lattice sites the one-site terms Vi >d ^are eliminated to avoid double counting. ^As log Zo ^{cannot be} decomposed

into a sum of one and two site terms a summation to give ^the total free _energy of the bulk sample (distorted ^or undistorted) ^cannot be carried out. However, ^the change ^{in the} partition ^{function is} composed ^{of two site terms} generated by the action of the rotation operators ^{on the two} site functions f(Ooi). Thus it is possible ^{to sum} up the change in the free energy per unit volume of the bulk sample ^on distortion, avoiding overcounting, ^to give

where n is number of lattice sites _per unit volume. From this expression ^we can, following ^Priest [3], ^{deduce the}

value of the elastic constant K

where a is the lattice spacing.

To assess the validity ^of calculations in which entropic contributions to the elastic constants are neglected

we also consider the change ^on distortion in the internal _energy of the system. ^{From this we} deduce an elastic

constant K’, ^{which can be} compared ^{with K.} Details of the analysis, ^which parallels ^that leading ^to (22), ^are given

in Appendix ^A where it is shown that the change in the internal _energy is given by ^the expression

where

Zo ^{is the} partition function of the undistorted cluster,

and

ak ^are coefficients in the expansion

K’ is then given by

5. Calculation and discussion.

To investigate the variation of the elastic constants K, ^{K’ of our model} with order parameter S2 ^{we must} evaluate the expressions (22), (24) numerically. Following Chandrasekhar and Madhusadana [11] ^{we assume} v(Oi), the molecular field acting ^on the exterior members of the cluster, ^{to have}

a simple Maier-Saupe ^form

For a given ^{set of values} B*/kBT, A * /kB T, B/kBT is found from the consistency ^relation [11] ]

which must be solved numerically. ^The properties of the solutions we obtained were those observed by ^Madhusa-

dana and Chandrasekhar. The moments

required in the calculations are most readily obtained from the identity

which is derived and discussed in Appendix ^{B. The} coefhcients fk ^{are evaluated} by ^numerical quadrature (see ^A. 3) ^{and can be used to} generate ^the coefficients ak (see ^A. 10). ^The Legendre ^and associated Legendre

functions can be generated ^from recurrence for- mulae [16] ; the numerical integration ^{over cos} 00 ^can

then be performed quite easily using ^{a standard}

Gaussian quadrature technique. ^{Inclusion of ten terms}

in the expansions (A. 3, ^A. 7) is sufficient to ensure

convergence. The order parameter S2 is evaluated in the form

The results of these calculations are shown in

figures ^{2 and 3. In} figure 2 ^we show the variation of the elastic constants K and K’ with the _square of the order parameter ^{for a} system ^with dispersive interactions between its molecules (i.e. A * is taken to be zero

in (3)). The linear variation of K, derived from the free _energy, with S2 ^{contrasts with that} of K’, ^{the inter-}

nal _energy result. We see that K’ is greater ^{than K} and that the difference between the two results is less marked at higher ^order parameter ^values (corres- ponding ^{to lower} temperatures). Figure 3 shows the variation of K with S2 ^{for a} system ^with only dispersive

interactions and for a system ^{with both} dispersive ^and dipolar couplings (A * ^{= 0.5 B * in} (3)). ^{The linear}

Fig. ^2.

^The variation with S2 of the elastic constants K,

K’ derived from the free and internal energies respectively

of a nematic with purely dispersive intermolecular inter- actions.

Fig. ^3.

The variation with S2 ^of the elastic constants K derived from the free energies of nematics with : (i) purely dispersive ^and (ii) dispersive ^and dipolar intermolecular interactions.

variation is seen to hold even in the _presence of dipolar

interactions.

The calculation we have performed, in which short- range order is treated self-consistently using ^{a nearest-} neighbour construction corresponding quite closely

to the intuitive content of more qualitative ^treat-

ments [9], ^{leads to two} conclusions. Firstly ^{we note}

that Priest [3] ^{has shown} that, ^{to order} q2, ^{there is no} change ^{in the} orientational entropy ^{of a nematic as a} result of a periodic reorientation of its director, ^when

this is calculated using ^{a mean field} theory ^{in which} short-range correlations are neglected. Consequently,

as changes in free and internal _energy are related

through

we should expect ^{the elastic constants of our model} K,

K’ (derived from AF and AE respectively) ^{to be iden-}

tical at the mean field level. Thus the difference between K and K’ seen in our results can be attributed to a significant contribution to the change ^{on distor-}

tion in the free _energy of the system ^{from the} change ⁱⁿ

its entropy, revealed in a calculation which includes effects of short-range ordering. ^{Thus the difference}

found between K and K’ is not unexpected ^{and indi-}

cates that the calculation of elastic constants from AE rather than AF is an unreliable procedure, ^{if effects}

of short-range ordering ^are implicit ^{in the model}

under consideration.

Our second finding is, however, ^more surprising.

We have shown that the physically ^{more correct free}

energy calculation yields ^{an elastic constant K which}

varies linearly ^with S’ ^even when effects of short-

range order are included in the Bethe-Peierls approxi-

mation. The persistence ^{of this mean} field functional behaviour in K should be contrasted with that of K’, whose deviation from mean field behaviour could be ascribed to effects of short-range order, ^{if it were not}

realized that it was calculated from the physically inappropriate change ^{in internal} energy. This diffe-

rence in functional behaviour between K and K’

again highlights ^the misleading ^{nature of} calculations based on changes in the internal _energy of the nematic.

The linear variation of K with S’ found in the results of our numerical calculations is certainly worthy ^of

further comment. The Bethe-Peierls nearest-neigh-

bour model of short-range ^{order has} proved ^{to be} quite

successful in accounting for its effects on the thermo-

dynamic and dielectric properties of nematic liquid crystals [11]. ^{Here we} have found that the inclusion of short-range ^{order effects, modelled} using ^{the same} detailed, self consistent nearest-neighbour ^construc- tion, ^{does not lead to a} change in the functional form of the dependence of the elastic constant on the order parameter. ^{Thus we} have shown that this simple picture of molecular correlations (beyond ^those

accounted for by ^{mean field} theory) appears ^to be an

inadequate basis for the discussion of deviations from mean field behaviour observed in experiment.

Acknowledgments.

The authors wish to thank the referees for several helpful ^{comments and} sugges- tions. @ British Crown Copyright ^1982.

Appendix ^A

Evaluation of integrals ^{over a distorted cluster.}

^The partition function of a distorted cluster can be written as

f (00j) ^{can be expanded in spherical harmonics as}

where

while, ^{to order} tf¡2, the rotation operators ^{can be} expanded ^as

The action of Ij., ^{on the} spherical ^harmonic Ckq(Qj) ^is expressed simply ^{in terms of the} step operators Ij+, Ij - ^{[ 15] :}

where

and

Thus in evaluating (A. 1) ^{to order} tf¡2 ^we require ^the integrals

and

On substituting these results into (A.I), noting ^that

and exploiting ^the orthogonality properties ^{of the} spherical harmonics in performing ^the Qo integration ^{we see}

that terms linear in ql ^vanish identically ^and obtain the result (19), ^{which is} simply ^a 1-dimensional integral ^over

cos 00. Zo, ^the partition function of the undistorted cluster ^can of course be written as

The two site contributions to the change ^{in internal} energy of the cluster can be written, ^to lowest order in tf¡2, ^as

where Zo, ^{6Z are as} given ⁱⁿ ((A. 4), (19)),

and

To evaluate E1 ^{we must now make the} decomposition

the coefficients ak being obtained either by ^direct quadrature (cf. (A. 2, ^A. 3)) ^{or from the} previously ^calcula-

ted fk. ^In the latter case we note that V(00;) ^{is itself a sum of} Legendre polynomials ^{in cos} Ooi; ^use of the results

[16] :

and the definitions (A. 2), (A. 7) ^{lead to}

On substituting ^the expansions (A. 2), (A. 7) ^into (A. 6) ^and carrying through ^the analysis ^{as before we obtain}

the following expression ^for El

On forming (A. 5) ^and summing ^over all clusters we arrive at the expression (24) ^{for the} change ^on distortion in the bulk internal _energy.

Appendix ^B

Moments of the maier-saupe distribution.

In our numerical calculations we require quantities ^{of the} type

where A is as large ^{as 6. While it is in} principle possible ^to determine these moments by ^numerical quadrature,

the oscillations in the Legendre polynomials, amplified by ^the rapidly increasing exponential term, make such ^a

computation either inaccurate or time consuming ^for large ^{k. In this} appendix ^{we describe an} alternative, ^and

more elegant, ^{method of} evaluating ^{these moments} which is both fast and accurate, and investigate ^{their beha-}

viour at large k ^{and so} demonstrate the rapid convergence of our series expansions.

We first note the identity [16]

and form the integrals

An integration by parts ^{and use of the} composition property (A. 8) ^{now lead to the} identity

This is satisfied trivially by the odd order moments

Henceforth we shall assume k to be even and, ^where the ^{resulting formulae are} simplified, put k ^{= 2 n.}

Considerable insight into the behaviour of the Pk(A) ^{can be} gained by studying ^the simplified ^form of (B. 2)

in which terms o(k-1) and smaller have been neglected. ^{As the} P-k(A) ^{are expected to} fall off with increasing k ^we

write

and deduce that

The other solution of the recurrence relation _grows and oscillates with increasing k, ^{and is} generated by

as

Thus the identity (B . 2) ^is numerically stable with respect ^{to backwards} recurrence, which suppresses the solution with behaviour (B. 5) [17]. ^As Po(À) is normalized to unity ^{all moments} P-k(A) ^{of the} Maier-Saupe distribution can

in fact be generated ^without performing ^a single ^numerical integration. Consequently ^the recurrence relation (B. 2)

is both simpler ^{and more} convenient for use in computation than that derived by Luckhurst et al. [ 18] :

where

A more precise analysis of the behaviour of the Pk(A) ^{can be based on the} integral

As integrals ^{of this} type ^{can be} evaluated in closed form but are not included in the standard compilations ^this analysis, ^which yields ^{a result} slightly different from (B. 4), ^{will now} be outlined. Firstly ^{we note that}

which follows from the definition (B. 6) and the result (A. 9). ^The recurrence relation (B. 2), which is also satisfied

by ^the Ik(À), ^{is now used to eliminate} Ik - 2 (A), Ik+ 2(À) successively ^from (B. 7) ^to give ^{the «} step-operator » ^iden- tities

and

These provide ^an Infeld-Hull factorization [19] ^{of the} surprisingly simple differential equation

satisfied by ^the Ik. ^{This can be solved} by standard methods [20] ^to yield

where iFi (a ; b ; z) ^{is a confluent} hypergeometric ^function [21]. ^On making k( = ² n) explicitly ^{even the multi-}

plicative ^constant A2n ^{can be} identified by comparing (B. 8) ^{with the} leading ^{term in the} expansion ^of 12n ⁱⁿ

powers of A :

to give

As

it is straightforward ^{to check that} (B . 9) ^{and the} asymptotic behaviours of 1 F 1 (n ⁺ !; ^{2 n} + 2 ; À) ^at large posi-

tive and negative À [21] give

and

as we would hope. The behaviour of iFi (n ⁺ i ; ^{2 n} + 1; A) ^at large n ^but with fixed argument A [21] implies ^that

from which we see that

Thus we see that the result (B. 4) ^{based on the} simplified recurrence relation (B . 3) ^is substantially, ^{but not enti-} rely, ^correct and that the rapid convergence of the series expansions ^{used in our} numerical work is assured.

The coefficients in the expansion ^of f (Ooi) (see (A. 2)) ^{can be found using (B. 2) in the} special ^{case of a} purely dispersive interaction between the molecules; ^a relation similar to (B. 2) ^can be derived for the quantities

required ^{in the} general expansion. ^{However this is not stable with respect to} either forwards or backwards recur- rence and so is not useful in numerical work. Fortunately the values of A, y characterizing f (Ooi) ^{are smaller} (~ 1-2) than those occurring ⁱⁿ the calculation of Pk, ^{and the} integrals ^{are evaluated} quite conveniently by ^nume-

rical quadrature.

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