This article was downloaded by: [University of Toronto Libraries] On: 22 February 2013, At: 12:10 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Molecular Physics: An International Journal at the Interface Between Chemistry and Physics Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tmph20 Forces in the benzene crystal Kalyan Banerjee a b a Mathematical Institute, Oxford b the Department of Physics, I.I.T. Kanpur, Kanpur Version of record first published: 12 Aug 2006. To cite this article: Kalyan Banerjee (1967): Forces in the benzene crystal, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 12:4, 385-398 To link to this article: http://dx.doi.org/10.1080/00268976700101641 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Transcript
This article was downloaded by: [University of Toronto Libraries]On: 22 February 2013, At: 12:10Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
Molecular Physics: An International Journalat the Interface Between Chemistry andPhysicsPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/tmph20
Forces in the benzene crystalKalyan Banerjee a ba Mathematical Institute, Oxfordb the Department of Physics, I.I.T. Kanpur, KanpurVersion of record first published: 12 Aug 2006.
To cite this article: Kalyan Banerjee (1967): Forces in the benzene crystal, Molecular Physics: AnInternational Journal at the Interface Between Chemistry and Physics, 12:4, 385-398
To link to this article: http://dx.doi.org/10.1080/00268976700101641
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Any substantialor systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, ordistribution in any form to anyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation thatthe contents will be complete or accurate or up to date. The accuracy of any instructions,formulae, and drug doses should be independently verified with primary sources. Thepublisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs ordamages whatsoever or howsoever caused arising directly or indirectly in connection with orarising out of the use of this material.
II. The effect of an intramolecular static distortiont
by KALYAN BANERJEE~
Mathematical Institute, Oxford
(Received 19 December 1966)
The possibility of an intramolecular distortion in benzene crystal is considered. It is shown that a static distortion of E2v symmetry imposed on each molecule in the crystal leads to a slightly enhanced lattice energy. The symmetry and magnitude of the expected distortion is obtained by requiring that the change in the total potential energy of the lattice produced by it is cohesive and the maximum. The results agree well with the actual distortion observed experimentally by Bacon et al. [1].
The change in the potential energy of the crystal lattice is obtained from the component changes in the dispersion energy (AWdisp), internal strain energy (AWinternal) and the repulsion energy (AWrepulsion) of the crystal. AWdisp originates from the distortional changes in the various electronic transition moments of the molecule. It is found that a static distortion modifies both allowed and forbidden transition moments to the same order in contrast with the vibronic changes in which only the forbidden transition moments are modified.
1. INTRODUCTION
The neutron diffraction studies of Bacon et al. [1] show that the individual benzene molecule in the crystal at low temperatures is slightly distorted from its hexagonal symmetry. In this paper we have shown that a static distortion of proper symmetry imposed on each molecule in the crystal leads to a slightly enhanced lattice energy. A permanent though small distortion of the molecule in the crystal may therefore be expected on energetic grounds. We have constructed a static distortion of the benzene framework by superposing certain normal coordinate displacements of the nuclei. Under the field of this distortion, the nuclei shift from their equilibrium positions, and certain electronic states which did not mix with each other previously, now begin to interact due to the distorted symmetry. This effect is significant only for the low lying and relatively closely spaced ~ electronic states of benzene; the mixing between the widely space
states is negligible in comparison. We have therefore treated the mixing of electronic states under the distortion simply as a 7r electronic problem. In the distorted configuration of the molecule, the moments of the forbidden ~r electronic transitions assume non-zero values, leading to an increased dispersion force between two such molecules. A static distortion, therefore, introduces a dispersion energy change AWaisp in the crystal. The change in the molecular geometry
t Part of a thesis submitted by K. Banerjee to the University of Oxford for the requirements of the degree of D.Phil. This research was supported in part by the Commonwealth Scholarship Commission, U.K. and by the National Institute of Health, U.S.A. (grant GM 12343). The first paper in this series (paper I) is reference [9] of this paper.
Now at the Department of Physics, I .I .T. Kanpur, Kanpur.
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due to nuclear displacements also produces an internal strain energy AWinternal, and a AWrevulsion is brought in by the changes in H . . . H, C . . . H and C . . . C contact distances between atoms belonging to two different molecules in the crystal. The total change in the potential energy of the crystal lattice, AWtotal pot, is the sum of the changes in the dispersion, internal and repulsion energies. The new equilibrium under the distortion and hence the new molecular geometry is determined by the condition:
{AWtotal pot} = O, (l) ~q
where q's are the nuclear displacements (extensions) constituting the static distortion. This gives us the stationary values of the q's and also of the energy AWtotal pot. It is gratifying to find that the distortion predicted from these simple considerations has the same symmetry and the same order of magnitude as the observed one (Bacon et al. [1]).
In w we show how the symmetry considerations lead to the form of the static distortion in terms of the nuclear coordinates. In w we evaluate AWdisp and AWinternal and show that AWrepulsion is probably negligible in comparison. Then in w we find the extensions q's using the variation (1) and also compute the total change in potential energy of the crystal lattice under the static distortion. Section 5 is for the conclusions.
2. THE FORM OF THE DISTORTION
The form of the static distortion is suggested by the experimentally observed situation--the benzene molecules are distorted almost exclusively in the molecular plane. The distortion should therefore be constructed in terms of the nuclear coordinates which form a complete set in the plane of the molecule; and our problem is to calculate the total change in potential energy when such a static distortion is imposed on each molecule of the crystal.
In the MO description of the electronic states of benzene, the ground state has the closed shell configuration a2u2elg 4 and the symmetry Ale. The first singly excited configuration a2u2elq3e2u 1 gives rise to the states of symmetry B2u, Blu and Elu, whereas E2g states are produced from the other two singly excited configurations (a2ulelg4e2u 1 or a2ueelgab2g 1) involving the lowest or the highest MO's. Out of the possible transitions, whose moments lie in the plane of the molecule, only 1Alg+lElu is allowed by symmetry. The transitions 1Alg-~IBeu, iAlg~-lBlu and 1Alg-->lEeg are symmetry forbidden.]- In the field of the displaced nuclei, however, the situation is different. The low lying ~ states can interact through extensions of the proper symmetry to produce transition densities of the allowed symmetry Elu and thus weak allowed moments may be created even for the forbidden transitions. The transitions Alg~Beu, Alg-+Blu and Alg-+Elu are affected by the nuclear displacements of symmetry X such that:
Elu e Alg{B2u or Blu or Elu}X,
giving X=E2g. The transition Alg-+E2a is, similarly, affected by the nuclear
]- We shall consider the singlet~singlet transitions only and will therefore drop the notation for the multiplicity of states from now on.
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Forces in the benzene crystal 387
displacement of symmetry Y which ensures that:
Elu e AlgE2gY,
giving Y=Elu. Hence our static distortion 8 should be written in the form:
8 = qE2~ + qEl~, (2)
where q's denote the nuclear displacements (extensions) along the normal coordinates belonging to the symmetries of the subscripts.
2.1. The nuclear coordinates
We will need the nuclear coordinates for the calculation of both AWdis p and AWinternal. T o calculate AWinternal we also need the force constants corres- ponding to particular modes of extensions, which have been evaluated by Whiffen [2] for the internal symmetry coordinates of the benzene molecule. AWdis> can also be calculated easily in terms of the same coordinates. We shall define these coordinates in the following. To simplify the problem we shall concentrate on the stretching and bending of the carbon skeleton only and ignore the extensions which arise mainly from the stretching and bending of the C-H links. Out of the possible four pairs of E2g and two pairs of Elu extensions of the benzene framework only two pairs of E2v and one pair of Elu extensions involve the carbon skeleton.
2.2. The internal symmetry coordinates
Whiffen's [2] force constants for the benzene molecule are in terms of these coordinates. Following him, the Eeg bending modes (S6a and S6b) and the stretching modes (S8a and Ssb) of the carbon framework may be defined as:
where Ro is the length of Cj-C;+I link in benzene and the local cartesians R and U (radial and tangential respectively) are centred on the carbon atoms. The S6o and $80 modes are degenerate with S6a and Ssa respectively. The Elu extensions of the carbon skeleton, $193 degenerate with $19b, are stretching modes given by:
It is important to notice that although the internal symmetry coordinates of different symmetries are orthogonal to one another, those of same symmetry are not mutually orthogonal. For example $6, is not orthogonal to S8a, etc.
The relations between the internal symmetry coordinates and the normal coordinates of the molecule can be worked out from the data provided by Whiffen. Denoting the two E2e normal modes of frequencies very nearly equal to 600 cm 1 and 1600 cm -1 by q6 and q16 respectively, one finds:
S6a= (0"968) q6-- (0"251) q16, (6) (
S8a= (0.781) q6+ (0.624) q16. J
In contrast with the internal symmetry coordinates, q6 and q16 are mutually orthogonal.
3. THE CHANGE IN THE TOTAL POTENTIAL ENERGY OF THE CRYSTAL DUE TO A STATIC DISTORTION
Under the static distortion the potential energy of the crystal lattice changes due to the two intermolecular contributions, AWdisp and AWrepulsion, and the intramolecular contribution AWinternal. In the following sections we shall evaluate them separately.
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Forces in the benzene crystal 389
3.1. AWdisp due to a static distortion
The origin of A Wdisp due to static distortion lies in the coupling of the electronic and nuclear motions of the molecule. The analysis resembles Herzberg and Teller's theory [3, 4] of intensities of the vibronic transitions in many ways, but at an important stage the two problems become distinct. Whereas the average values of the normal coordinate functions, viz:
q--~=O, q~q~--O and q 2 # 0
determine the vibronic intensities, the static values of the same functions characterized by:
q~#O, q~q~#O and q 2 # 0
determine the effects of a static distortion. In the following analysis we shall assume the Born-Oppenheimer principle to be valid and introduce the distortion as a perturbation of the nuclear coordinates of the molecule.
The Born-Oppenheimer approximation enables us to write the complete wave function as a product of an electronic and a nuclear part:
~Fe, n(x, q)= ee(X, q)q~e, n(q), (7)
where x and q are respectively the electronic and nuclear (normal) coordinates and e and n are the quantum numbers describing the electronic and vibration- rotation states respectively. The moment of a general transition in which the labels of ~F change from 00-+k/~ is given by:
Moo-~k~ = l'r Nok(q)r dq, (s) qd
where
M01c(q) = j'r q)M(x)r q) dx (9)
is the moment of the electronic transition 0-+tc for the particular nuclear configuration q, M(x) being the electronic dipole moment operator. M0k(q) is a variable in the q space and it is this variation that determines the coupling of electronic and nuclear motions. Hence, let us expand M0~(q) in powers of q's:
Mok(q) = Mok(0) + 2 q : ( ~ ] o + order (q2). (10) . \ q~ /
Here M01c(0) is the value of the transition moment at the equilibrium configuration of the nuclei (it is zero for the forbidden transitions and non-zero for the allowed ones), and
(0M0k] ~q~ \ ~q: /0
is the moment gained at the equilibrium configuration due to nuclear motions. To obtain AWdisv as a function of the extensions q's we will first have to get the change in polarizability due to the q's and this involves the calculation of A M0e 2. Hence, squaring equation (10) and transposing we have:
AMos2 = MokZ(q)- Mo~Z(O)
7q~ (~Molc] + (~Mo~ t 2. ~ 2 M 0 ~ ( 0 ) �9
\ eq~ ]o!
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The first term on the right hand of equation (11) is interesting in that it contains the first degree terms t in q's. It exists only for the allowed transitions, for which N0k(0)# 0, and may have either sign. The second term is a perfect square and is the only term in the AN 2 of a forbidden transition. Hence, under static extensions, the forbidden transitions always contribute to a gain in polarizability whereas the contribution from allowed transitions may have either sign. We remarked earlier that the change in polarizability due to distortion is expected to be significant for the rr electrons only; its magnitude ~ " is given by the second- order perturbation theory expressionS:
Ac%'=2e 2 ~ ' AM~ (12) k ~k-- E0
where ~'s form a complete set of 7r excited states. We shall show in a later section that AWaisp varies linearly with At%'. So we obtain:
AMok 2 AWalsv ~cA%" ~X' (13)
k Ek- ~0
giving AWdisp in terms of the A Mok2's and the energy differences (ek-Eo)'s. We shall figure out the proportionality constants in a later section. What follows now is essentially a calculation of the AMo~2's. \Vr are taking the energy differences from experiment.
There are two quantities to be determined in equation (11): H0k(0) and (~N0~/~q)0. I t is simple to obtain N0~(0) values for the MO states of benzene. We shall determine (8 N@/bq)0 now.
By definition Mok(q) = <0l M(x) [ k), (14)
in which the integration on the right-hand side is over all the electronic coordinates, making Mok(q) a vector in the q space. Differentiating equation (9) with respect to a particular nuclear coordinate q~, we get:
I M(x) l M(x) l ~q~ >, (15)
since b M(x)/~q= = 0 by definition. The differential coefficients
~q~
can be obtained from the second-order perturbation theory (Byers Brown [5]). Expanding ?r in a complete set as:
eel_ Z Cz, m~b,,, (16)
one obtains :
Cz, m -<llaH/aq~lm> for m#l. (17)
]" This does not however mean that there are first-order changes in the vibronic intensity of the allowed transition. The important quantity in a vibronic intensity calculation is the average of AM0~ 2 for all values of q; and since ~=0, the first degree terms in the q's do not contribute to the vibronic intensity.
The change in the energy difference denominator due to distortion is neglected in writing equation (12). See note added in proof.
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Forces in the benzene crystal 391
Hence equation (15) at the equilibrium nuclear configuration (denoted by the subscript zero) may be put in the form (for references, see Banerjee [6]):
where ]l) and ]m) are states for which the corresponding C's and M's are non- zero. Equation (19) is the central one in the theory of interaction between the electronic and nuclear motions. In effect it is an expansion of (~Mok/~q~)o in a complete set of M's and shows how the states ]l} and I m) are brought in by the extension q~ to interact respectively with the states ]0) and ]k) of the transition 10>~1~>.
The C's of equation (19) may now be determined following Murrell and Pople [3]. Equation (17) defining the C's contains a matrix element of the type <llag/aq~lrn>. They show that the matrix element <lleH/aq~lm> represents simply the interaction energy of the dipoles created by the nuclear displacement Pq~ and the single electron transition density <l[m)'. For example, the interaction energy of the transition density:
2
l t t , g a ~ (l;1 = ~ x - I - z
2
and the dipoles created by one of the E2g extensions S6a, shown in figure 1, is the sum of all the individual charge-dipole interactions]-. We have calculated the various transition densities for the LCAO-MO states of benzene represented by:
r ~ 1 {01+ 02+ 03+ 04+ 05+ 06}, V6
1 r {201+ 02- 03 -204- 05+ 06},
r {02+ 0a- 05- 06}, (20)
1 r = 2 { - 02 + 03- 05 + 06},
r ~--112 {201- 02- 03+204- 05- 06},
r ! {01- 02+ 03- 04+ 05- 06}, ~/6 t In PAlg, z2g(o) the charge densi ty is on the carbon atoms. However , there are other
t ransi t ion charge densities, for example PAlg, E~g(~,) where the charge gets concent ra ted at the centres of the bonds.
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where O's are the 2p~ orbitals centred on the carbon atoms marked by the subscript, and the pairs (~b-~, ~b3) and (~ba, ~bs) are degenerate. Following Dewar and Longuet-Higgins and Goeppert-Mayer and Sklar [7] and Pople [8], the ~r excited states may be written as sums or differences of the degenerate excited configuration Xk z obtained by removing one electron from the orbital ~blc and putting it into ~bl. The ASMO representations for the five lowest excited states of benzene are given in table 1 (after Murrell and Pople [3]). We may now obtain all the single electron transition densities and this allows us to evaluate all the matrix elements listed in table 2.
S y m m e t r y
Rig
B2u
Blu
Elu
E2da)
S y m b o l
I LI.,
Configuration
XO ~
1
1 - - (xz 4 + x-~ 5) V2
1 - - (xaS+ x., 4) ~/2
1 ~/~ (x a4- x,,~)
E n e r g y
39 400 cm -1
49 500 cm -1
54 500 cm -1
1 'I'r ~/2
q .y, 1 V J2
{ ~t'a x/2
tl" a,
(Xl 5 -- X2 6) [
(X14 -- X3 6)
- - (x l 5 + x2 6) ]
1 C2 (X14+ xa6)
8.18w ev
8"89w ev
Oscillator s t r e n g t h
0.0014t
0.094{
o.88{
"~ KLEVENS, H. B., and PLATT, J. R., 1954, Tech. Rep. g~iv. Chicago. { HAMMOND, V. J., and PRICE, W. C., 1955, Trans. Faraday Soc., 51, 605. w KEARNS, D. R., 1962, J. chem. Phys., 36, 1608.
Table 1. Lowest excited states of benzene (after Murrell and Pople [3]).
The C's of equation (17) can be obtained very simply by dividing the matrix elements (of table 2) by the corresponding experimental energy differences given in table 1. Hence, we may now get all the (~M/~q)0's from equation (19) and thus obtain the expressions for AM2's in terms of the extensions (in a.v.), using equation (11):
A M A l g , B 2 u 2 = (12" 01)S6a 2 + (29- 15)$8a 2 + (37" 43)S6aSsa, ]
A MAI~, Blu 2 = (6" 96)$6a 2 + (642" 47)Ssa 2 + (133" 73)S6aSsa, f
A MAlg, E,u ~ = (0" 09)$6a e + (1" 53)Ssa 2 + (0" 76)S6oSsa (21) + (1.621)$6a + (6-537)$sa,
A MAlg, E2g(7) 2 = (16" 60)$19a 2,
A MAIg , E2~,(3) 2 = (41- 80)$19aL
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Forces in the benzene crystal 393
OH S <~--H6a>o = 0"403 12
(0~--Hs~>o: = 4" 174 ~
0 H \ =0'761 1 OSsa/~ B 12
a n \ , s o s ~ / v ~ = o. 570 p
OH S < O - ~ s a ) v / = 5 899 l~
? ~ ~,,r - o . 5 o 5 V
/ o n \ , s \ ~ / ~ = - 3- 617 V
0S1~ o / =0 . 715 l- ~
(S = ~ O101+1 dr = 0" 246 ; l = C-C bond length in benzene = 2- 640 A.U.)
Table 2. The matrix elements of the operator aH/Oq.
The same expressions would be obtained for the extensions $6~, Ssb and $19b, which are degenerate with S6a, Ssa and $19a respectively. Therefore, we shall drop the expressions for the former set f rom now on and shall re-introduce them only at the very end.
To write AWaisp in terms of extensions we need just one more information: Wmsp in terms of ~• (benzene). I f we agree to proportion Wdisp into parts according to their relative contributions to o2 (benzene), we find (paper I, [9])that 41 .6 per cent of Wmsp involves ~• (benzene) (the rest comes f rom %~, ~H ~ and ax~)~. Hence we could write the approximate relation in n.u. :
Waisp = c o n s t a n t - 0- 357 x 10 -a x c% ~ (benzene).
For a change Aax ~ in the 7r electronic polarizability, the corresponding change
]" According to this assumption the contribution of ax ~r (benzene) to Waisp is {~a• This assumes that all the parts have isotropic polarizabilities. For an individual pair interaction, this is definitely not true for the 7r electrons of the molecule which have only axially symmetric polarizabilities. However, the individual pair interaction when averaged over all the orientations of the two molecules--a situation approximately reproduced in the crystal--does yield a spherically symmetric type formula, justifying the proportioning that we have assumed above.
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in the dispersion energy AWdisp is therefore given by:
2~Waisp = - 0-357 x 10 -a x 2 y ' 5lvl~ in A.U., (22) k E k - - r
on putting Ac%~ (benzene) from equation (12). Substituting for the sum in equation (22) from the equations (21) and table 1, we get (in A.U.):
This is the expression from the Hfickel theory. We have pointed out in paper I [9] that Malg, El, 2 is over-estimated by the Hiickel theory and that good agreement with the experimental value can be obtained by including configuration interaction which reduces the Hfickel value for I*lAlg, Elu 2 by a factor of 1/2.97. This factor is also expected to be nearly the same for the other transitionsJ-. Hence, to correct the Hfickel estimates for configuration interaction, we have assumed that:
(i) all the bt0~2's are to be lowered by a factor of 1/2.97; (ii) all the C0k2's (since C0k is proportional to the matrix element of the
one-electron operator aH/Oq between the states involving the same transition density) are also to be lowered by the same factor 1/2-97.
To correct the Hiickel estimate for Wdisp (equation (23)) we notice that the coefficients for the quadratic terms (~ C2bt z) are to be lowered by a factor of 1/(2"97) 2 whereas those of the linear terms (~-Cbt 2) are to be lowered by 1/(2.97) ~v/2.97. This gives:
This is the equation we were seeking in this section. It has negative coefficients for the square terms in extensions showing that the molecule may continually lower the lattice dispersion energy (which is negative) by distorting itself more. A check on this is put by the AWinternal and ,-~tYrespulsion created in the distortion process. In the following we shall show how.
3.2. AWinternal due to a static distortion
As the extensions distort the equilibrium geometry of the molecule, it develops an internal strain. The change in internal energy accompanying the distortion may be expressed as the following quadratic in the extensions Saa, Ssa and S19a:~ :
where Za, Aa, )~3 and 54 are the force constants for which the following values
J- This is so because we are considering singlet~singlet transitions only. And the states of the same multiplicity and nearly the same eigen-energies are expected to be modified similarly on including configuration interaction.
:~ We restrict ourselves to those extensions only which also cause kWaisp. Through the extensions not contained in AWaisp the total potential energy can only increase in the form of AWinternaI and hence a distortion cannot occur along them.
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have been assigned by Whiffen [2]:
A3 = 5. 380 x 105 dynes/cm = 0. 3454 A.V.,
Z3 = 0" 846 x 105 dynes/cm = 0.0543 A.U.,
Xa = -- 0. 180 x 105 dynes/cm = - 0.0116 n.u.,
2Aa = 7. 340 x 105 dynes/cm = 0.4712 n.u.
Substituting the force constants in equation (25), we get in n.u. :
It is important to note here that 2AWinternal measures the internal energy change corresponding to a change AWdisp in the dispersion energy because of a double count involved in obtaining the latter. AWinternal is positive quantity and on its own account it opposes any distortion of the molecule. In fact it drains away most of the negative energy AWdisp gained through the distortion.
3.3. AWrepulsion due to a static distortion
An a priori calculation of AWrepulsion is a very difficult problem. Here, no extensions are forbidden by symmetry, neither is there an easy way of distinguishing the important ones among them. But for very small distortions (as we have in the case of benzene) AWrepulsion turns out to be rather small. This happens because of the following balancing effect: in a distortion process, among other changes, the H . . . H, C . . . H and C . . . C contact distances change in the crystal. But this change, on the whole, is a re-distribution of contact distances rather than a consistent packing or loosening of the contacts. Hence we expect that the gains and losses in Wrevuxsion would tend to balance out statistically and only the unbalanced part would produce the AWrepulsion. Since the distortion is very small, the unbalanced repulsion energy, AWrepulsion, should also be very small. We may mention here that no such balancing occurs in the distortional changes in the dispersion or internal strain energy of the crystal. To show the order of magnitudes involved in the calculation of AWrepulsion we shall estimate its upper limit. I t would come mainly from the changes in the closest contact distances, which are about five in the benzene crystal for a chosen central molecule and all of them are of the H . . . H type with an average contact distance of about 2 .7 s The experimental distortion is about 0. 006 s in the bond lengths. A change of 0.006 s in H . . . H contact distance of 2 .7 s contributes about 0.0008 kc/m to AWrepuxsionJ'. Hence if in a distortion all the five H . . . H contacts moved in phase (this is impossible in the benzene crystal because most of the rings are mutually perpendicular) to make a closer or looser packing, the change in repulsion energy would be 0.004 kc/m. For the same distortion, on the other hand, both AWdisp and AWinternal turn out to be about 1 kc/m each.
I t must be emphasized that the situation might be quite different if the distortion were large. Indeed, AWrepulsion is then expected to be the most significant term because the repulsion energy is so sensitively determined by the changes in the H . . . H contact distances. But for the very small intramolecular distortion that has been observed in the benzene crystal [1] the balancing effect mentioned above leaves a negligibly small AWrepulsion behind. Hence, we shall start out by
]" Estimated from the H . . . H potential described in paper I [9].
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n e g l e c t i n g A W r e p u l s i o n and re-write the variational condition (1) as:
{A Wdisp + 2A Winternal} = 0, (27) bq
for all q's to determine the q's and also the stationary value of AWaisp + 2AlVinternal.
4. THE EXTENSIONS
From equations (24) and (26) we obtain (in A.U.): A Wdisp + 2A Winternal = (0" 0464)$6a 2 + (0" 1015)Ssa 2 - (0" O882)S6aSsa
For a distortion to occur AWdisp+2AWinternal should be negative and hence there would not be a distortion of the Etu symmetry, since any extension of this symmetry necessarily leads to a positive value of AW4isp+2A~Vinternal. The coefficients of Saa e and Ssa 2 in equation (28) are positive showing that with respect to either of these E2g extensions the left-hand side possesses a minimum characterized by (using the variational condition (27)):
where the extensions at the minimum are the solutions of the equations (29),
given by : S6a = 0" 0460 a.u., I
Ssa=0"0381A.U. ) (30)
The other pair of extensions S6b and $8o will be precisely equal in magnitude to S6a and SSa respectively. Substituting the extensions we obtain at the minima:
A W d i s p = - 0 . 8 3 kc/m, ]
2AWinternal= 0 .72kc/m, ! (31)
A W t o t a l pot = A W d i s p + 2 A W i n t e r n a l = - 0" 11 kc/m. j
Equations (30) may be compared with the observed distortion [1] by converting the extensions (Ssa, Ssb and S6a, S6v) into bond angle and bond length changes of the benzene framework, through the relations (4). One finds that for the extensions S6a and Ssa of equations (30) the bond angle and bond length changes
Similarly, for the other pair ($6~ and Ssb) one obtains:
- Ac~2= Ac~3= - Aas= Aa6=0-0150 A.U., / (33)
- - AR12 = AR34 = - AR45 = AR61 = 0. 0104 A.U. j
Now since the extensions S6a and S8a are degenerate with S6b and Ssb respectively, the general distortion would be of the form:
~(S6a + S8a ) -~- ~ / ( l - ~2)(36b + Ssb), (34)
where the condition on A is that I h l ~< 1. So there would be an infinity of distorted forms for the molecule which lead to the same total potential energy of the crystal. The molecule has the choice of distorting along any of these infinity of ways and the actual distortion cannot be specified further from the above considerations.
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Forces in the benzene crystal 397
The observed distortion [1] shows two long and four short bonds in benzene; the longer and shorter bonds differing in length by about 0.006 A. The sym- metry of this distortion is the same as that of the extensions (S6a, Ssa) (see equations (32)) and is different from that of the other pair of extensions (S6b, Ssb). In actual fact, therefore, we have a markedly (S6a, Ssa) type of distortion. The distortion effects would show up experimentally in the appearance of a Raman line of frequency 38.5 cm -1 (0.11 kc/m) in the spectrum of crystalline benzene at - 135~ This line should be absent in the liquid or gas phase; and we expect that the frequency would increase as the temperature is lowered, but only very slowly, following the corresponding change in the crystal structure of benzene.
5. CONCLUSION AND DISCUSSION
(i) For an intramolecular distortion to occur in the crystal two conditions must be satisfied: there must be a minimum in the change in the total potential energy AWtotal pot with respect to the distortion, and at the minima AWtotal pot should be negative. It is clear from equation (28) that the minima in AWtotal pot appears because of the first degree terms in q's. These terms are contributed in the form of AWdisp arising from the effect of the static distortion on the allowed transition Alg-+Elu of the molecule. Energetically also the linear terms are very important for they contribute nearly the biggest terms in making up the AWtotal pot at the distorted equilibrium. The magnitude of AWtotal pot at the minima at -135~ is only - 0 . 1 1 k c / m , and compared to W ~-" ( = - 3 - 5 k c / m ) and Waisp (= - 2 3 . 5 kc/m)J-, this is quite small. But it should be remembered that AWtotal pot is really the difference of the two quantities, AWoisp and 2AWinternal which are individually much bigger. Experimentally [1], the intramolecular distortion in the benzene crystal is just on the border line of significance. Our method is neither precise nor complete enough to talk about the magnitudes of such distortions. The object of this attempt was merely to formulate the intramolecular static distortion problem in a crystal, hoping to get the symmetry and the order of the distortion magnitudes right.
(ii) It is interesting to note that the quadratic terms in extensions in AWtota l pot (see equation (30)) are provided almost exclusively by the forbidden transitions, whereas the linear terms come only from the allowed ones. We know that in a vibronic intensity calculation (in which one deals with an oscillatory distortion) all but the square terms in nuclear displacements are eliminated by symmetry. Hence, the vibronic change in the intensity of the allowed transitions is almost negligible--it is the forbidden transitions that derive intensity from vibronic coupling. But under the static distortion the situation is different in that the intensity of the allowed transition is also significantly altered. In table 3 we have separated the contributions of the various electronic transitions to the quantities connected with the static distortion.
It is clear from this table that in determining the effects of a static distortion the changes in the intensities of the forbidden and allowed transitions are equally important. That is the central point of this calculation.
(iii) We have not discussed the alternative assignment E2e for the excited state of the 49 500 cm 1 band. It would bring in slightly different extensions which can be easily obtained from the data provided in this paper. We have not evaluated these extensions because a choice between the alternative assignments Blu and
Table 3. Contributions of the various transitions of benzene molecule to the AWaisp of the crystal.
E2g is too precise a distinction to be made from a distortion which is only just significant.
(iv) We found that the important quantity in the calculation of AWmsp is A 1,!2 and we obtained the A MZ's by expanding M0k(q) only up to the first-degree terms in q's (see equation (10)). However, since we ultimately need Abt01c 2 which does contain the second-degree terms in q's, to be consistent, we must start with an expansion of M0x(q) up to the quadratic terms in q's. This introduces
an extra term: btok(0) . S" q 2 ( 321v!~ \ ~q2 10,
in the expression (11) for Abt0k. It exists evidently for the allowed transitions only. We have calculated this term and found it to be smaller than the other quadratic terms that we have included in equation (11). But, as we mentioned earlier, the quadratic terms contribute negligibly compared to the linear terms to the A btZ of an allowed transition. Hence the new quadratic term could be safely left out.
The author would like to thank Dr. L. Salem for suggesting to him this problem and for many very interesting sessions. He would like to thank Professor C. A. Coulson, for his suggestions and kind encouragement throughout this work. A part of this paper was written when the author was at the Institute of Mathematical Sciences, Madras. The Visiting Membership offered to him is gratefully acknowledged.
Note added in pro@ The meaning of the i] and J_ directions in the text is the same as in paper 119].
For benzene molecule the [] direction is along the sixfold axis and the L directions are in the plane of the molecule. For the bonds however the il direction is along the bond axis and the L directions are mutually perpendicular and also perpendicular to the bond axis.
REFERENCES
[1] BACON, G. E., Cuaav, N. A., and WILSON, S. A., 1964, Proc. R. Soc. A, 279, 98. [2] WHIFFEN, D. H., 1955, Phil. Trans. R. Soc. A, 248, 131. [3] MUaaELL, J. A., 1956, Proc. phys. Soc. A, 69, 245. [4] HERZBERC, G., and TELLEa, E., 1933, Z. phys. Chem. B, 21, 410. [5] BYEas BROWN, W., 1958, Proc. Camb. phil. Soc. math. phys. Sci., 54, 257. [6] BANERJEE, K., 1965, D. Phil. Thesis, University of Oxford, p. 100. [7] DEWAR, M. J. S., and LONCCET-HIGGINS, H. C., 1954, Proc. phys. Soc. A, 67, 795.
GOEPPERT-MAYER, M., and SKLAR, A. L., 1938, J. chem. Phys., 6, 645. [8] POPLE, J. A., 1955, Proc. phys. Soc. A, 68, 81. [9] BANERJEE, K., and SALEM, L., 1966, Molec. Phys., 11, 405.
