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A Monte Carlo study of thecondensed phases of biphenylAparna Chakrabarti a , S. Yashonath a & C.N.R. Rao aa Solid State and Structural Chemistry Unit , Indian Instituteof Science , Bangalore, 560012, IndiaPublished online: 22 Aug 2006.

To cite this article: Aparna Chakrabarti , S. Yashonath & C.N.R. Rao (1995) A Monte Carlostudy of the condensed phases of biphenyl, Molecular Physics: An International Journal at theInterface Between Chemistry and Physics, 84:1, 49-68, DOI: 10.1080/00268979500100041

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MOLECULAR PHYSICS, 1995, VOL. 84, No. 1, 49-68

A Monte Carlo study of the condensed phases of biphenylt

By APARNA CHAKRABARTI, S. YASHONATH and C. N. R. RAO

Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India

(Received 12 July 1994; revised version accepted 9 September 1994)

Detailed calculations on the condensed phases of biphenyl have been carried out by the variable shape isothermal-isobaric ensemble Monte Carlo method. The study employs the Williams and the Kitaigorodskii intermolecular poten- tials with several intramolecular potentials available from the literature. Thermo- dynamic and structural properties including the dihedral angle distributions for the solid phase at 300 K and 110 K are reported, in addition to those in the liquid phase. In order to get the correct structure it is necessary to carry out calculations in the isothermal-isobaric ensemble. Overall, the Williams model for the intermolecular potential and Williams and Haigh model for the intramolecular potential yield the most satisfactory results. In contrast to the results reported recently by Baranyai and Welberry, the dihedral angle distribu- tion in the solid state is monomodal or weakly bimodal. There are interesting correlations between the molecular planarity, the density and the intermolecular interaction.

1. Introduction

The geometry of biphenyl in the free state as well as in the different condensed phases has been of considerable interest for several decades. Gas phase electron diffraction studies [1-3] suggest that the phenyl-phenyl torsion angle is --~43 ~ Molecular force-field calculations [4] suggest that the dihedral angle is 32 ~ in solution, but experimental measurements [5, 6] yield a value in the 19-26 ~ range. X-Ray diffraction studies [7, 8] of the solid phase at room temperature, on the other hand, show the molecule to be planar. At room temperature the solid has a monoclinic structure with two molecules per unit cell. The low temperature phase at 110 K also has monoclinic symmetry, with slightly different cell parameters [9]. X-Ray diffraction studies [9] at 110 K and 293 K have shown a large component of the librational motion about the long molecular axis with a mean-square amplitude of 45.7 ~ at 110 K and 105.9 ~ at 293 K, suggesting that the zero dihedral angle between the phenyl rings obtained from X-ray diffraction studies [7, 8] is likely to be an average value, and that there is a considerable deviation from the planar conformation. Apart. from exhibiting large variations in the dihedral angle in the gaseous and condensed phases, biphenyl undergoes several phase transitions at low temperatures. High-resolution elastic neutron scattering studies [ 10] have shown that the transition around 38 K, referred to as the 'twist' transition, involves an incommensurate modulation in both the a* and b* directions in orientational space and the average dihedral angle changes from 0 ~ to 7-10 ~ Another transition, called the 'lock-in' transition, is associated with the disappearance of the modulation in the a* direction below 17-20K. Raman spectroscopic studies [11] confirm the

t Contribution No. 1003 from the Solid State and Structural Chemistry Unit, Bangalore.

0026-8976/95 $10.00 �9 1995 Taylor & Francis Ltd.

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50 A. Chakrabarti et al.

existence of these two transitions. Variable-temperature Raman [12] as well as electron paramagnetic resonance and electron-nuclear double resonance studies [13] also support this observation.

Computational approaches have been used with some success in understand- ing the different properties of biphenyl. Fischer-Hjalmars [14] calculated the dependence of the conjugation energy on dihedral angle 0 using a semi-empirical approach. Busing 1-15] calculated the properties of biphenyl at 22 K, 110 K and 293 K using a temperature-dependent potential. Benkert, Heine and Simmons (BHS) [16] proposed an intramolecular potential which includes both the non-bonded and the conjugation energies. They modelled this system to study the incommensurate phase transition of biphenyl [16]. Baranyai and Welberry [17, 18] have recently reported molecular dynamics calculations at constant volume using the intermolecular potential of Williams and Cox [19] and the BHS intramolecular potential [16]. Their results suggest the presence of two sublattices in the solid state with dihedral angles of 0-2 ~ and 22 ~ . However, on the basis of available experimental results one cannot distinguish between a double minimum in the potential proposed by Charbonneau and Delugeard [9] and the arrangement proposed by Baranyai and Welberry.

We have carried out detailed Monte Carlo (MC) calculations in the isothermal- isobaric ensemble employing the intermolecular potentials of Williams and Cox [19] and Kitaigorodskii [20] along with the intramolecular potentials of Haigh [14], Bartell [21], and BHS [16]. In addition, we have used the potentials of Williams and Kitaigorodskii for the intramolecular contributions. We report thermodynamic properties as well as the crystal and the molecular structures of biphenyl based on these calculations. We also examine the structural aspects of this fascinating molecule in the crystalline (monoclinic) phase at 300 K and 110 K and in the liquid state.

2. Potential models

There are several potentials for hydrocarbons in the literature. One of the most widely used potentials among these is that due to Williams and Cox [19]. This has been derived by fitting to about 30 different aromatic and non-aromatic hydrocarbons. It is an atom-atom potential with 22 sites on each of the C and H atoms and has the following form:

~) jk(r jk ) = - - Ajk/r6k -b B j k exp (-- Cjkrjk ) Jr- q j q k / r j k , j, k = C, H (1)

Kitaigorodskii [20] has proposed several potentials for hydrocarbons, and these are used frequently to model aromatic systems. The parametrization of these potentials has been done using structural and enthalpic data for a large number of hydrocarbons. We have used one of these potentials also for the intermolecular interaction. Both potentials have the 6-exp form for the short range interaction but, unlike the Williams potential, no electrostatic interactions are included in the potential given by Kitaigorodskii. In the Williams model, a charge of + q is placed on hydrogen atoms and - q on the carbon atoms. The potential parameters are listed in table 1.

The intramolecular potential for biphenyl consists of two parts. One is the steric interaction of the atoms at the ortho position modelled in terms of the non-bonded interactions between hydrogens and carbons of the two rings at the ortho position. The second is the variation of the conjugation energy [14]. Several models have been proposed for the former. Of the available potentials, the one proposed by Bartell [21] is the oldest. Most potentials include only the H-H non-bonded interaction, but as

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Monte Carlo study of biphenyl 51

Table 1. Atom-atom interaction parameters for biphenyl. The interaction is assumed to be of the Buckingham form (6-exp).

Model Atom A/kJ mol - 1 A- 6 B/kJ mol - 1 C/A- 1 q/e

WW C 2439.8 369 743.0 3.60 - 0" 153 H 136.4 11 971.0 3.74 0' 153

KK C 1498.6 175 812.0 3-58 0.0 H 238.6 175 812.0 4.86 0.0

Bartell has pointed out the C - H and C-C interactions also play an important role. As we shall see, more important than the inclusion of the C-C, C - H interaction is the overall barrier for the planar conformation. Bartell's potential is of the form:

V(C, C) = D/r 12 - - E / r 6 (2)

V(C, H) = F/r6(G exp ( - r /0"49) - 1) (3)

V(H, H) = H exp ( - r/0.245) - I/r 6 . (4)

This potential yields a barrier height of 21.8 kJ mol - 1 across 0 = 0 ~ which is rather large, rt Electron calculations [22, 23] without geometry optimization have yielded a higher barrier height across the planar conformation (20 kJ mol - 1) than across the perpendicular conformation (8.4 kJ mol-1). Barrier heights for the Bartell potential are close to the values obtained from this calculation. More accurate ab initio [24] calculations incorporating geometry optimization especially of the inter-ring C-C bond suggests that the barrier heights are in fact 5"0 and 18.8 kJ mol - 1 for the planar and the perpendicular forms. All barrier heights reported by us here take into account both the conjugation effect and the ortho-ortho steric interaction. Table 2 lists the values of barrier heights for the various potential models, and also from 7t electron calculations [22, 23] and ab initio [24] studies. Bartell's potential has been used earlier to predict the properties of solid and gaseous biphenyl [14, 25]. The potential due to Haigh [14] models the non-bonded interaction potential in terms of the H - H interaction term alone. It has the same form as that of Bartell, but the values of the parameters are different:

V(H, H) = J exp ( - r /0"234) - K/r 6. (5)

More recently, Benkert, Heine and Simmons [16] have proposed a potential of the form

V(O) = g(L exp ( - N O z) + M sin 2 (0)), (6)

where 0 is the dihedral angle (in radians). This potential includes both the conjugation energy and the energy arising out of the interaction of the H atoms at the ortho position. In contrast, all other potentials take into account only the non-bonded interaction, viz., the steric interaction between the atoms at the ortho positions. Hence the conjugation energy had to be accounted for separately. The intramolecular poten- tial parameters are listed in table 3. The barrier across the planar conformation is of utmost importance. The intramolecular non-bonded inter-ring interaction between the atoms at the ortho positions has also been modelled additionally using the standard Williams and Kitaigorodskii potentials, The parameters are the same as those used in evaluating the intermolecular contribution and all three interactions (H-H, C - H

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52 A. C h a k r a b a r t i et al.

0

0

0

0

nnl

0

I

e ~ <

e.i

0

= 6 o ~

�9 ~ , o .

.~.o

"-~ 0 o o

II II

..~

"~, l ~ , - ~ ~ s : l

,-, o= ,~ u " 6 ~

�9 - ~ ~ = . 0 ~.E

" ~

~ ~ ~ ~ :~

0

e ~

0

0

ID

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e~

.o

~ �9

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~ L x

, - ~ X

o 9

0 t'N

, ~,

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~ < •

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Monte Carlo study of biphenyl 53

Figure 1.

-5

-10 0

~ -15 Q >

-20

-25 J

y

/ i

J

/ t

i t

/ J

/" J

/" i

J

/

/ s

i

/ " t

t

/ t

t s

-30 , ~ ' ' '

0 20 40 60 80

0

Dependence of conjugation energy on the dihedral angle 0. This is used in all models except the BHS to compute the intramolecular contribution.

and C-C) have been taken into account. The Williams and Kitaigorodskii potentials have a barrier lower than the ab initio value for the planar conformation (see table 2).

Fischer-Hjalmars [14] has reported several ways of estimating the conjugation energy. In this work we have used the estimate obtained from semi-empirical quantum chemical calculations [14]. The variation of the conjugation energy as a function of the dihedral angle is shown in figure 1.

Based on the electron diffraction data, Almeningen et al. [26] have proposed an intramolecular potential function with barrier heights 6.0 and 6"5 kJ mol- ~ for the planar and perpendicular conformations, respectively. These values, especially, the latter, are considerably different from the ab initio [24] values and hence we preferred not to employ this potential. Similarly we do not employ the intramolecular potential function obtained from the molecular mechanics calculation by Stolevik and Thingstad [27], which predicts barrier heights of 8.4 and 7"5 kJ mol- 1 for the planar and the perpendicular conformations, respectively. Recent ab initio work of Lenstra et al. [28] using a 4-21G basis set suggest a barrier of 7"9 kJ mol- 1 for the planar conformation which compares well with those of the Haigh and BHS potential models. However, they have not given the full potential or the barrier for the perpendicular conformation.

In total, results from eight models were obtained. Of these, four employ the Williams intermolecular potential with the intramolecular potential of Williams, BHS, Bartell and Haigh. These are designated as WW, WBHS, WB and WH models, respectively. The remaining four models, referred to as KK, KBHS, KB and KH,

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54 A. Chakrabart i et al.

are obtained by replacing the Williams intermolecular interaction with that of Kitaigorodskii.

3. Computational details

In recent molecular dynamics calculations on biphenyl, Baranyai and Welberry [17, 18] have employed the canonical ensemble and carried out simulations of the solid at room temperature using the Williams and the BHS potentials for the intermolecular and the intramolecular interactions. In order to ensure that the results obtained actually correspond to the potential used, we investigated the effect of the cut-off criteria and variable shape simulation on the results. Three sets of calculations were carried out on the Wil l iams-BHS model. Calculations using the a t o m - a t o m cut-offs in the canonical ensemble, and the variable shape isothermal-isobaric ensemble are referred to as sets A and B, respectively. Set C uses the conditions of set B except that centre of mass-centre of mass (com-com) cut-off has been used in place of the a t o m - a t o m cut-off. The unit cell parameters for the three sets are listed in table 4. The c o m - c o m and the C - C radial distribution functions (RDFs) for set B and set C are shown in figure 2. The RDFs for set A are similar to those of set B and hence are not shown. RDFs of set B are sharp and narrow, indicating that the molecules are not performing the thermal motion normally expected of them at room temperature. In contrast, the RDFs of set C are broad and smoothly varying as expected of a room temperature solid. Closer examination revealed that the Coulomb

lO

8

, - . ,6

e~ 4

2

o 4

10

8

~ 6

~ 4

2

o

(a) i n

f

; ,', , ,, - - - Set B

t

,' ", Set C i

i i i ~ COM COM

I

6 r,A

i I h i I h h ; i

' i I I I

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8 10

(b) ,, ' . . . . H ~ l ~ q I

I

i i i

4 6 r, A

Set B

Set C

C-C

1.5

0 10

0.5

Figure 2. (a) The centre of mass-centre of mass radial distribution function (RDF) for set B and set C for the room temperature solid phase. (b) C-C RDF for set B and set C for the room temperature solid phase.

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Monte Carlo study of biphenyl 55

0.12[ 0.1 ..~ ~i

I 't J! j,! t ~176 li' 0.08 I- ," , " ' ' ; / ' / i i

._, ] 2~ :i~~ ~ i

0 04 [

I 0.02 [ ,' ',

~ t _.. ~'""

. . . . . Set B

- - Set B long run

Set C

% ! i i I ~',l

-40 -20 0 20 40

|

Figure 3. Distribution S(O) of the dihedral angle between two phenyl rings from set B, set C, and for a long run of set B. Inset shows S(O) from set B and longer run of set B on an expanded scale. S(O) is the fraction of molecules per degree interval at a given value of 0.

term did not converge in the case of sets A and B due to the use of a t o m - a t o m cut-offs, and that the sum itself was fluctuating widely. The dihedral angle distribution for sets B and C are shown in figure 3. The distributions for the two sets of calculations are completely different. The normal duration of about 10000 MC passes was extended by another 20 000 MC passes for set B, and the results for this are referred to as set B long run (see figure 3). It is seen that for set B there is a peak near 0 = 10 ~ apart from the main peak near 0 ~ This shoulder tended to shift towards higher values of 0 for the longer runs. The distribution is reminiscent of that obtained by Baranyai and Welberry [17, 18]. For set C, which employs the c o m - c o m cut-off, a symmetric bimodal distribution is observed. This suggests that the a t o m - a t o m cut-off should not be employed for the Wil l iams-BHS model. Further, these results suggest that a constant pressure variable shape simulation is essential for obtaining the properties corresponding to the potential being employed.

In view of the non-convergence of the Coulomb contribution when the a t o m - a t o m cut-offs were used, we carried out all subsequent simulations using the com com cut-off. A cut-off of 10/~ has been used in all the runs. The Coulomb contribution was obtained by direct summation. Neglect of the Coulomb interaction beyond 10/~ leads to a ~ 2% error in the calculation of the total intermolecular energy.

All the results discussed in the paper relate to the isothermal-isobaric ensemble on a system of N = 72 biphenyl molecules. The simulation was carried out on

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56 A. Chakrabarti et al.

ID

,4

,.< ~b ob 6-

t"-I r r

d ~ 6 6 ~

G'~ C h ~

~s d ~ 6 ~ 6

~ o o

I I

I t I

II

II

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II

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Monte Carlo study of biphenyl 57

3 x 4 x 3 (36 unit cells x 2 molecules per cell = 72 molecules) crystallographic unit cells. Solid biphenyl was modelled using a variable shape simulation cell. Calculations on solid biphenyl were carried out at a pressure of one atmosphere and at temperatures of 300 and 110 K. The imposition of periodic boundary conditions in the simulation prevent the system from exhibiting incommensurate behaviour. The starting configuration for the room temperature solid was taken from the room temperature crystallographic data of Trotter [7], and for the 110 K run the structure reported by Charbonneau and Delugeard [9] has been used. In modelling liquid biphenyl, a cubic simulation cell was used. Calculations on the liquid are reported at 400 K and atmospheric pressure. For liquid state calculations random starting configurations were used.

One MC pass is defined as N attempted MC steps, once for each of the N molecules. A single MC step consists of an attempted random translational displace- ment of the com, a random rotational displacement around a randomly chosen axis and a random intramolecular rotation around the central C-C bond joining the two rings. In the simulation of the liquid beiphenyl, prior to equilibration at 400 K, the liquid was kept at 900 K for a few hundred MC passes. For all the condensed phases, equilibration was carried out for about 10000 MC passes and the averages were calculated over an equivalent number of MC passes.

4. Results and discussion

4.1. Solid biphenyl

In table 5, we list the thermodynamic properties of solid biphenyl at 300 K for all eight potential models. The heat of vapourization A n v a p w a s obtained from the expression.

AHva p = E~natra - - (Einte r + Eintra) + P ( V gas -- V) + (11o - H ) (7)

gas Eintr a w a s taken to be the Boltzmann average of intramolecular energy at 800 K. The (/40 - H) term represents the Berthelot correction for the deviation from ideal gas behaviour, and V gas is the volume of ideal gas. It is seen that the best agreement for the heat of vapourization is obtained for models employing the Williams inter- molecular interaction potential. For models employing the Kitaigorodskii inter- molecular potential, the heat of vapourization is generally lower by 8-18 kJ mol - 1 compared with the experimental value (81 kJ mo1-1) [29, 30] Among the models employing the Williams intermolecular potential, the best agreement is obtained for the WW model. The intramolecular contribution to the total interaction energy is between - 8 and - 16 kJ mol - ~ for all potential models with the exception of the BHS model which gives a large and positive intramolecular contribution of nearly 57 kJ mo1-1. The densities for the various models listed in table 5, compare favourably with the experimental value of 1.185 x 103 kg m -3 [7]. The best agree- ment is, however, found for the WW model. The density for the K K model is somewhat higher (1.22 x 103 kg m-3).

Table 6 lists the unit cell parameters a, b, c, a, fl and 7. Overall, the models predict the unit cell parameter c better than b, the deviations being 1"4% and 9%, respectively. The deviation in the unit cell parameter a is considerably larger being in the range of 6-16%. All the models with the exception of the K K model, underestimate b and overestimate a. Models employing the Williams intermolecular potential predict the

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,...i

~

& ID

I

e~

•

I

I

O

A

V

I

A

V

7

v

I I I I I I I I I I I

I I I I I I I I I I I I I I I I

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Monte Carlo study of biphenyl

Table 6. Cell parameters of solid biphenyl at 300 K.

59

Model a/A c/A c/A ct/deg fl/deg ~/deg

Expt 8.12 5.63 9-47 90.00 95.40 90.00 WW 8.26 5.62 9.34 90-48 92.75 90-18 WH 8.59 5-57 9.38 90.26 93-59 90.15 WBHS 8-69 5-53 9.42 90.06 97-18 89.39 WB 9.08 5-51 9.39 91.23 95.23 90.18 KK 752 5"95 9"39 86'16 9515 8946 KH 877 540 9"40 92'49 95 10 90-64 KBHS 9.40 5-12 9.48 89.30 99-68 88.17 KB 8.62 5.58 9.60 88.60 100-88 90.64

value of ~ more accurately. A larger deviation is observed in the value of ft. Considering the overall performance, the WW model seems to perform better than others. The value of fl (92.7~ however, is lower than the crystallographic value of 95.4 ~ . This result is in agreement with the work of Busing [-15] on the solid phase of biphenyl; in earlier work on the high pressure monoclinic phase of benzene [31] it was found that the Williams potential underestimated the value of ft. But 7 is predicted satisfactorily by all the models.

Figure 4 shows a plot of the com-com and the C-C RDFs for different potential models. The positions of the com-com peaks from the X-ray crystal structure [7] are indicated by vertical lines. The relative heights of the lines are proportional to their intensities. The RDFs exhibit the following characteristics. All models except the KK and the WW show an absence of fine structure in the com-com as well as the C-C RDFs. As we shall see shortly, this seems to be related to the nature of the dihedral angle distribution. Model K K exhibits a shift by about 0.25 ~ of the first peak towards lower distances; the peak appears at 4.7 ~ for model K K compared with 4.95/~ for the X-ray data. The second peak at 5"5 ]~ shows an outward shift to higher r values for model KK whereas all the other models predict the second peak correctly. The third peak near 8 ~ is not predicted correctly by any of the KX (X = BHS, B, H and K) models. The Williams models show the third peak, or a shoulder at a slightly higher distance, around 8.5 ~. The WW model shows a clear peak around 8.2 A. The principal lacunae in the Kitaigorodskii intermolecular potential appears to be the absence of Coulomb interaction. In particular, recent studies have suggested that a quadrupolar interaction between aromatic rings is of vital importance in modelling any molecule possessing phenyl rings. Thus, models for the benzene molecule not accounting for the quadrupolar term were unsuccessful in predicting the properties of condensed phases of benzene. In particular, the PS model of benzene [31], which underestimated the quadrupolar contribution, exhibited a large inward shift of the first peak in the com-com RDF as compared with the experimental structure; this is similar to the shift of first peak exhibited by the K K model in figure 4. The intramolecular interactions in the KBHS, KB and KH models seems to be responsible for the absence of any such inward shift. Another consequence of the non-inclusion of the quadrupolar interactions is the high density of the K K model. Judging from the relative success of the various models in predicting the heat of vapourization, density, cell parameters and the com com and C-C RDFs, it appears that only WW, WH and WBHS perform satisfactorily.

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60 A. Chakrabart i et al.

t=3ooK I Ii~[',i - - -WBHS I [Jl ii . . . . w 8 I

........ .. W H I - w w /

I/J i , ~ :OMCOM ' / ' ~

i

4 6 8 10

r , , ~

1.5

0.5

0

(c) C-C --T=30OKs- WBH

~ .... WB b , / % , ......... w.

4 6 8 10

r,]~

exl)

Figure 4.

6 (t)

4 -

3 -

J

o ~

e ~

T=300K --- KBHS

,,~'!, .... KB _\~'~, ' ........ KH

COM-COM, /,,;

6 8 10

r,A

1.5

0.5

(d) C-C T=300K ] --- KBHS /

,- . . . . . . KB / "~... ~ ~ , . ......... KH ,.,

I i I i I I

4 4 6 8 10

r , / ~ k

Radial distribution functions for solid biphenyl at 300 K with the Bartell, BHS, and Haigh models and the Williams or Kitaigorodskii intramolecular potential. (a) Centre of mass-centre of mass RDF for models employing intermolecular potential given by Williams and Cox; (b) same as (a) but for models employing the Kitaigorodskii intermolecular potential; (c) C-C RDF for the Williams intermolecular potential; and (d) C-C RDF for the Kitaigorodskii intermolecular potential. Peak positions in the com-com RDF corresponding to the X-ray structure are marked by vertical lines. The lengths of the lines correspond to the relative peak heights.

Dihedral angle distribution functions for the various models are shown in figure 5. Models using the Bartell and BHS intramolecular potential functions show a clear bimodal distribution. The former shows a zero intensity near 0 = 0 ~ The latter shows a small non-zero intensity near 0 = 0 ~ The Haigh potential shows a distribution which may be described as lying somewhere between bimodal and monomodal . Both the WW and K K models show a monomodal function with a maximum near 0 = 0 ~ suggesting the most probable conformation is the planar conformation in the room temperature solid phase. The RDFs for these two models show well defined features which seem to be correlated with the monomodal S(O) exhibited by them.

The average dihedral angles calculated from the S(O) for the various models are listed in table 7. Two different types of average can be calculated. (i) The first is the

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Monte Carlo study of biphenyl 61

0.04

0.03

0.02

0.01

0

-60

�9 T = 3 0 0 K ~ - - - WBHS

(a) I",.. / \ / " , . . . . . wa i .:,. / \ ..f. ,, - - w .

/ , " ".:'" ^ / \ A ,"i ',,', - - w w

i , , '.. _ _ �9 , , ,

.... , . - : : , : ~ ~ . . - : - . . . . . / :,i'-:,, .... -40 -20 0 20 40 60

O

0.04

(b) T=300K --- KBHS

@

0.02 / ' : : , �9 / , , ' , ' . \

0.01 / " " " \ i / / '~ %\.

-60 -40 -20 0 20 40 60

e

Figure 5. Dihedral angle distribution function for solid biphenyl at 300 K for models containing the intermolecular potential given by (a) Williams and Cox, and by (b) Kitaigorodskii.

Table 7. Average dihedral angle calculated as described in the text.

Model O/deg (0)

WW 8.78 - 0-02 WH 14.94 0.02 WBHS 26.20 0.43 WB 31-92 - 0.55 KK 9-39 -0.09 KH 22-70 0"27 KBHS 30.90 0.32 KB 37.10 - 1.94

average angle as obtained by averaging between 0 ~ and 90 ~ and between - 9 0 ~ and 0 ~ separately. The overall average is then obtained by taking the mean of the modulus of the two values�9 This is represented by 0 in table 7. The average angle that is obtained from X-ray crystallography, however, is different from 0. (ii) Ano the r average ( 0 ) has been obtained by calculating the average over the range between - 9 0 ~ and + 90 ~ Such an average would correspond to the X-ray crystallographic average. It is seen that ( 0 ) is nearly zero for all of the models investigated. Hence, it is not possible to rule out any of the models here on the basis of X-ray crystallographic data [7, 8]. However, the X-ray data of Charbonneau and Delugeard [9] have also indicated that the libration tensor L22 along the long axis has a

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62 A. Chakrabarti et al.

significantly large amplitude of 105"9 ~ Ab-initio [24] calculations and X-ray and electron-diffraction studies indicate a torsional amplitude of 10-15 ~ [32]. From figure 5 it is clear that only the WH, WW and KH, KK models have an amplitude in this range. Other models, in particular WBHS, WB, KBHS and KB, have a considerably larger amplitude (30-35~ As has been pointed out, the recent molecular dynamics calculations of Baranyai and Welberry [17, 18] have suggested yet another possible dihedral angle distribution: a main peak near 0 = 0 ~ and a shoulder around 22 ~ which develops into a peak at low temperatures. However, neither experimental nor any of the existing theoretical investigations to date have confirmed such a possibility. Furthermore, as pointed out in the previous section, the dihedral angle distributions obtained by them seems to be an artefact of the possible use of the atom-atom cut off in their calculations. The thermodynamic and unit cell parameters obtained by us also indicate that the constant pressure calculation with variable shape yield properties which are significantly different from the constant volume calculation.

4.2. Structure-potential correlations

Here we analyse possible correlations between the structure and the potential employed. We are able to do this because of the large number of potentials employed in the present study. The densities obtained with the different models based on the Kitaigorodskii intermolecular potential (see table 5) are in the order: KK(1.226) > experiment(I-185) [-7] > KH(1.153) > KBHS(1.138) > KB(I.127). Let us now look at the probability of the biphenyl molecules being planar in the solid phase, by taking the intensity of the dihedral angle distribution at 0 = 0 ~ as a measure of the planarity. We see that the intensity decreases in the order KK > KH > KBHS > KB (see figure 5), suggesting that there is a strong correlation between the planarity of the molecule and the density of the solid. Similar trends are seen in the models based on Williams intermolecular potential models, where the density as well as the planarity vary in the order WW > experiment > WH > WBHS > WB. These trends suggest that the large barrier for rotation around the inter-ring C-C bond near 0 = 0 ~ is responsible for the low density and demonstrate the relationship between intramolecular potential and thermodynamic properties. Since the experimental value of the density lies close to the values given by the WW and WH models, and since the torsional amplitudes for these two models are comparable with the experimental value, it appears that the dihedral angle distribution in the real crystal could be closer to that exhibited by the WW and WH models.

We now compare two potential models with the same intramolecular potential but different intermolecular potentials. For example, comparing the results from KX (X = BHS, Bartell or Haigh) with those from the corresponding WX models indicates that the WX models always have a higher intermolecular contribution to the total energy (see table 5). Figure 5 indicates that, for a given intramolecular potential, the molecules in the solid are more planar if the intermolecular contribution is larger. This finding is in agreement with that of Casalone et al. [25], who found that the planarity of the phenyl rings in the solid phase may be related to the intermolecular energy. The present study demonstrates that there is indeed a one-to-one corres- pondence.

Overall, it appears that the WW and WH models give the best description of the room temperature solid. Both of them, however, underestimate ft. In the present

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Monte Carlo study of biphenyl

Table 8. Cell parameters of solid biphenyl at 110 K.

63

Model a/,~ b/A c/A e/deg fl/deg 7/deg

Expt 7.82 5-58 9.44 90.00 94-62 90.00 WW 8.00 5.61 9.20 89"59 92.24 89.86 WH 8.35 5.51 9-28 89.14 95.00 89.25 WBHS 8.55 5.50 9.36 90.72 96.44 90.29 WB 8-62 5.50 9.41 90.47 97.70 90'35 KK 7.17 5.98 9.32 88.26 94.30 89.14 KH 7.81 5.61 9.34 89-94 95.37 90.00 KBHS 7.98 5.59 9.42 90-61 97.67 90.96 KB 8.08 5.50 9-45 90-38 98.80 90.28

study no dynamical properties have been calculated and hence it is not possible to comment on the performance of the Williams model in predicting dynamical properties accurately. However, it is worthwhile to note that an earlier study of solid benzene using the Williams intermolecular potential found that it leads to a rather tightly packed structure [31].

4.3. The solid at 110 K

Table 8 lists the values of the cell parameters for the various models along with the values obtained from X-ray diffraction measurements [9]. It is seen that the deviations of the cell parameters from the experimental values of Charbonneau and Delugeard [9] are similar to those seen at 300 K. Thus, all of the cell parameters are predicted satisfactorily by the WW and the WH models with the exception of fl and a, respectively. The K K model predicts fl well but the density is too large, a is too small and ct deviates significantly from 90 ~ The dihedral angle distribution functions for different models are shown in figure 6. The general features for the various models are similar to those of the room temperature solid. Models employing the Haigh intramolecular potential show a slight change in the shape of the curve with respect to the room temperature solid; the peaks near _+ 20 ~ appear to be split peaks. At sufficiently low temperatures, the dihedral angle in the solid is non-zero (10 ~ at 22 K from neutron diffraction studies of Cailleau and Baudour [33]).

4.4. The liquid phase

Only the WW and WH models perform satisfactorily in predicting the properties of solid biphenyl, and since the results for these two models are on the whole only slightly different, we choose only the WW model to study the liquid phase of biphenyl. In addition, for purposes of comparison and to understand the effect of neglect of quadrupolar interaction on the thermodynamic and structural properties of liquid biphenyl, we have chosen the K K model. The thermodynamic properties at 400 K are listed in table 9. The calculated heat of vapourization for both models lies within about 4~o of the experimental value [34], the densities being 0.879 and 0.978, respectively, for the two models (the experimental value is 0.866 x 103 kg m - 3 [34]).

In figure 7 we show a plot of the corn-corn and the C C RDFs for the K K and

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64 A. Chakrabarti et al.

I

o c ~ 6 c ~

X

I

0

V

I

~ ~ I ~ =

.5

~ g

i

0

/ - . I I

v ~ t ' l

9

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Monte Carlo study of biphenyl 65

0.06

0.05

0.04

0.03

0.02

0.01

0

T - 1 1 0 K ,,, - - - WBH~ ( a ) ,.., - / ~ ! ! - - - w B

: , , , , / '\ ';,, i - / :"" A / \ A : I - ' :,' ~f':',J\ / \ I \ A i ,,",

/ ' , \

-40 -20 0 20 40

O

0.06 , . , / ~ --- KBHS

0.05 CO) r,, T=110K / \ - - - KB

0.04 / ~ / \ - - KH / \ 8, / '~,', . / \ - - ~ , - i \

i ' ' ' / " / \ A , i ' , "' ~0.03 i " \ ~ \ / ~ / \ / ' t i ' '~

/ , ' , ' \

0.01 , / , ' ' ' \

-40 -20 0 20 40

O

Figure 6. Dihedral angle distribution function for solid biphenyl at l l 0 K for models containing the intermolecular potential given by (a) Williams and Cox, and by (b) Kitaigorodskii.

1.5

0.5

(a) --- ww " .......... ~ .....---~.".--:.:--~,~-,,,,r-.

T = 4 0 0 K ,, '

/ C O M - C O M s"

! / , "

/ . r - I ~ I I

4 6 8 10 r,A

Figure 7.

1.2

1

0.8

0.6

0.4

0.2

0

(b) T = 4 0 0 K --- w w , I _.-- - .~ "" . . . . ~ - - KK

, / , , "r

, /

' / C C , /

,, , /

", I I I

2 4 6 8

r,A 10

Radial distribution function for liquid biphenyl at 400 K for the WW and KK models: (a) com-com; and (b) C-C.

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66 A. Chakrabarti e t a l .

Figure 8.

0.02

|

0.015

O.Ol

0.005

T=4OOK

. . . . W W

......... KK .~ :hA

i ~? : :

i :v , ",. f , : \ f , ~, t " r ~ , i , i',

i , : i~ . , " "..,,

i ", '"' i '

:," i ' , ' # i : / ~,,

: / ', " i ~ ' ,

/ / ! ,

�9 / .. , ," / \ ",

�9 t

t , ~ / , L

. , " . . - ' r ' " - , I , , , , i , , , I " " - . - . . . - ' 2 . - _ . . 7 . 7 " -

-50 0 50

0 Dihedral angle distribution function for liquid biphenyl at 400 K for the WW and

KK models.

WW models. The K K model starts at a significantly lower distance (3 A) compared with the WW model. This distance seems to be too small in view of the fact that two benzene molecules in liquid benzene do not approach closer than 4 A 1-31]. The Kitaigorodskii intermolecular potential seems to give a wrong packing of the nearest neighbours in the liquid as well. Thus, it appears that the absence of quadrupolar interactions leads to the inward shift of the first peak of the com-com RDF and to a significantly higher density. The first peak and the main peak in the com-com # ( r )

appear around 5.2 A and 7"2 A, respectively. The C-C RDF shows a broad peak with a maximum around 6.4 A and a shoulder around 4'2 A. This shoulder is characteristic of all hydrocarbons and corresponds to the peripheral groups between neighbouring molecules [31]. Thus, the shoulder around 4.2 A in liquid biphenyl arises from the nearest CH groups.

The dihedral angle distribution for the liquid is shown in figure 8. Both the WW and the K K models show a bimodal distribution. This is in contrast to the solid where a monomodal distribution was observed for these potential models. The trends are in agreement with experiment where it is known that the rings have a non-zero dihedral angle in the solution phase. The average dihedral angle is found to be 24"6 ~ and 20.4 ~ for WW and K K models, respectively. This may be compared with the experimental value of 19-26 ~ for the solution phase [5, 6]. The distribution has a larger width for the WW model, extending almost over the whole range of 0 ( - 9 0 ~ to 90~

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Monte Carlo study o f biphenyl 67

5. Conclusion

Of the eight models investigated, the overall agreement with the experimental structure of biphenyl could be reproduced most satisfactorily only by the WW and WH models. Structurally, the WW and WH models predict all the unit cell parameters satisfactorily except ft. An inward shift of the nearest neighbours is observed for the room temperature solid phase as well as the liquid phase for the KK model. This is attributed to the absence of quadrupole interaction between the phenyl rings. The dihedral angle distribution and the torsional amplitude of 10-15 ~ observed experimentally are again best reproduced by the WW and WH models. Consequently, S(0) in real crystals may be described as somewhere in between monomodal and weakly bimodal with a significant non-zero intensity near 0 = 0 ~ The results obtained here suggest that the distribution obtained by Baranyai and Welberry [17, 18] where some biphenyls have near zero dihedral angle while others have a value of 0 = 22 ~ may be an artefact due to the possible use of a tom-atom cut-off in their simulations.

Important correlations have been observed in solid biphenyl. The planarity of the molecules in the crystal is higher, the higher the density. In addition, the intermolecular contribution to the total interaction energy is larger, the larger is the planarity. Potentials employing the Williams intermolecular function generally lead to higher planarity. The calculated dihedral angle for the liquid phase is in good agreement with available experimental data for the solution phase. Table 2 and the results obtained here suggest that actual barrier height for the planar conformation are between 2.8 and 8.4 kJ mol- 1, which are the heights for the WW and WH models, respectively. The range of barrier heights is comparable to the ab initio result (5"0 kJ tool- 1). Any proposed potential should improve the prediction of unit cell parameters fl and a.

References

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