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Effect of non-central interactions on energy and lattice dynamics of molecular crystals
EFFb OF NON-mti ON ENERGY AND LA-I-IKE
k-ERACI-IONS DYNAhilCS OF MOhULAR
LA_ REh4IzOV. V-G_ PODOPRIGORA,
r.
: CRYSTALi -
AN_ BOh’ICH.
Atom-atom, multipoIe-multipole and three-body potentials have b&n used to calculate the energy and lsttice dynimics of ¶dichIorobenzene. phenanthrme and mchIoronitrobenzcne Noncentral interactions are shokn to &z a signikant contniution to the intermokcuIar interaction energy but have littIe effect on the phonon
1,Introduction
2. Crystal potential function
In a number of papers [l-4] atom-atom potentials of the 6-exp type were employed to describe the dynamics and the structure of molecular crystal lattices However, when investigating strongly.anisotropic crystals the parameters of the atom-atom interaction curve have to be varied for various substanks_ Besides, the application of the potential is limited since some specific, eg non-additive and electrostatic, interactions, anisotropic by- nature are neglected here_ Recently a number of papers have demonstrated the effect of multipo!e-muItipoIe interactions on ihe dynamics and energy of the molecular cjstal lattice. However the results appear to be extremely conflicting (see, for example, refs_ [5,6Jj_ The effects of many-body non-additive interactions have been studied in refs_ ]7,8] for solid inert gases They were found to play a significant role for the crystal structure and the lattice dy&mics_ For more complicated mokcufar crystals such data. are not yet avaifable in the literature_ The present paper aims at investigating the infhrence of three-body and multipoIe~muhipole interactions on energy and dynamical properties of moiecular crystals of different space groups: ¶dichlorobenzene, phenanthrene and m-chloronitrobenzen~-
We present the crystal lattice energy as the sum of the potentials
w = y-1 + y-
+
y(3)*
where V.’ is a central atom-atom pair potential, Vz denotes an anisotropic pair potential of muhipole-multipole interaction, Yes is a non-additive three-body potentiai. The first term on the right-hand side of the above expression is chosen in the 6-exp form (modified Buckingham potential): Y, = c [ - CiiR,7p -t- Bij exp( -pi&)] 8
_
(0
The coefficients C, B and p have been obtained from structural and optical eqeriments employing the lattice energy data from refs. [4,9] (Rji is the distance between two atoms i and i j_ The interaction of constant multipoles of ttio neutral mokcuks Q and fi, according to ref. [6], is -defined by 00
a =<= (-1)*+‘2~+s!r!/(2s)!(2t)!.
0301_0104/85/S03_30 Q EIsevier Science Publishers B-V. (North-Holland Physics Publishing Division)
~_ _
-.
164
LA.
Q**’
is an
*s+=r
= - ,=+,(
(3 i
t)th
rank
RZiJ_
Rem&z
interaction
CI
aL ,* .Von -ccmraJ
tensor (3)
K&h’)is the electric multipole moment tensor of sth order for the ath molecule and R4 is the distance between the centers of mass of molecules Q and j3_ When s = 0, 1,2._ __ we deal with a charge. dipole. quadrupoIe, ettc, respectively_ A three-body potential consists of short- and long-range parts. the latter containing the potentiafs of inductive and dispersive interactions_ The former potential, being non-additive by nature includes terms referring to simultaneous two- and three-body interactions_ Non-pairwise terms apRear in the second order of the perturbation theor)= i_c in the same approximation as van der Waals pair forces_ Restricting ourseXves to the dipole appro_ximation the inductive potential can be. as well as (2). expressed in tensor te_rms[lo]
where M(*’ is the dipole moment and A, is the generahz&i polarizability tensor of the ath molecule_ The dispersive potential cf three interacting asymmetric moIecuIes has the following form
(5) where G,,, is given by the coefficient C$’ [lo] at R$ in a two-body attractive potential derived in second-order perturbation theory_ This coefficient must speciahy be chosen and here it is the main difficulty in employing eq_ (5) For three spherical bodies_ eq_ (5) transforms into
~(R&x$s,)-t-
(6)
as su_ggested in ref. [ll]_ v-. TV. or are the internal art&s of the dipolar triangle with sides Rag, R,,, RBr_ The description of the three-body interaction by means of eq_ (6) is more exact the closer the molecular shape is to spherical_ For the first time we ha\= treated a three-body energy as the sum of triple interactions of non-vaIent atoms to describe
imcramicws in
nwidar
qnnz&
complicated molecular crystals_ The magnitude of G ,,s, in (6) appears to be determired exactly by the atom-atom potential. The short-range part of the three-body potential takes into account the overlap of electronic clouds_ For inert gases this interaction was satisfactorily described by the potential [12] V’3’=cDi;k
expi-((~~~R;~fp;~R,~)].
(7)
where for solid inert gases pij = pir = p (p is the Buckingham potential parameter oi eq_ (1)) and Di,k = D_ While D is related witk 3 from the Buckingham potential_ its value must be obtained from additional experiments_ such as cohesive energy and elastic constant measut-emtnfs [13]. So here we tty to obtain D from atom-atom potential parameters_ The ¶dichlorobenzene crystal u-as taken for the caiculaticus because of its simple structure with one molecule per unit cell, so that there are only three lines in its Raman specsra We obtain D;,k = 3 X 103( BiiBjjBkr-)‘f3 from inverse spectral problem solution with C_ B and p from (1). So three-body interaction parameters may be calculated using pairvise ones. To check the obtained D,.+. value. phonon spectra for some molecuhxr crystaIs have been calculated_ The results are analysed below_ To calculate the effective pokuizability A of a molecule, the method of induced dipoles has been used to take into account the effecl of the lattice environment on a molecuhxr electronic cloud [9j_ The component A, of the effective polarizability tensor is given by
where 4 are the elements of the free molecule pohuizabihty tensor, a, and ct,. refer to the atomic polarizabilities of different R,@pazed moiecules. 11, is the direction cosine of the bond i-t’_ S, is the Kronecker symboL This model provides A derivatives not only with respezt to the hbrating external coordinates but also with respect to the trans1ationa.l ones which results in a. change of the force constant values for three-body dispersion and induction interactions_ To estimate the interaction tensor T (lattice
sum) describing the structural crystal anisotropy, the EwaId method has been applied, invoking summation over the lattice sites both in direct &d reciprocal spaces 1141. Previous papers, however, used &e point-molecule apprcximation disregarding the real size and mutual molecular orientations in the ctyscal lattice. when determining the inceraction contribution co the lattice dynamics of molecular crystak In that case the lattice sum 7 depends Tn the external translational molecular coordinates only, which brings about an inaccuracy in the estimations of the oriencaciond phonon parameters Other well-known approaches are either to divide a mokcule into force centers (atom groups) 1151 or co dea1 with atomic sublacciccs [16,17]. The latter enables one to obtain T derivatives not only with respect to translational but also with respect co 0riencationaI molecuk displacements from equilibrium_ Thus for complicated
mokcular vibraiions in a unit cell, second derivatives of the crystal energy obtained within the present model may differ appreciably from those found for point-molecule and oriented-gas approximations- The calculating ckchnique of crystai force constants typical of pair, electrostatic and three-body interactions is treated in refs- [4,6,lS], respectively_
3_ Results and discussion RI_ p-dichlorobetcene
and phenanthrene
The p-dichlorobenzene crystai belongs to the Pi space group with one molecule per unit ceII Iocated at the inversion center [19]. According to the selection ruIes, three Iibrational modes of A, species are active in its Raman spectrum_ Phenanthrene crystaIIizes in the space group P2, with two molecules per unit ceU [19]_ Zero wave vector lattice vibrations in terms of irreducible representations of its factor group are ~classified as follows: five modes of A species and six modes of B species_ For ¶dichIorobenzene the first non-zero mu!tipole moment is the quadrupole one_ Molecular symmetry aIIows three components of the quadrupole moment: ML?, M,‘:),~ MC& two of
them being independent: M$!i = - (ME) 4 Mi!‘); where u, u; w are the long, medium and normal axes, respectively_ The char& on the atoms fat. &C,H,CI, have been determined in ref. [20]. I%ng these estimations, the diagonal components of IW censor. have been evaluated to give Mzi’ = 62 x lo-* C m’, Mz: = Miz) = -31 X lo-j0 C m’_ The dipole moment of phenanthrene lying in. the plane of a benzene ring cquaIs 0.2 D 1215 The components of the quadrupole moment have the following magnitudes [21]: Miz = 27-9 x lOma C m’, M:z’ = 32-7 X lo-C m’. Dipole and quadrupole Iattice sums of p-dichlorobenzene and phenanthrene were calculated on the atomic sublattices with a subsequent averaging per molecule. Effective polarizability tensors are presented in table l_ Lattice energges and phonon frequencies of & paradichlorobenzene and phenanthrene derived in different appro.ximaticns and those obtained from polarized Raman scattering experiments are given in table 2. The atom-atom potential parameters are taken from refs. [3,4]. As seen from tabIe 2, the absolute values of the short-range three-body repulsion and induction energies are small_ The dispersive non-additive energy gives a large& positive contribution to the cocaI energy and, being cakulated by (5) in a point-mokcule approximatiou. is slowly converggng_ Varying the summation radius from 37 to 40 A, the energy of @-paradichlorobenzene changes by more than 5%_ The calculation of T and A over atomic sublattices provides a much better convergence_ Thus, in the range from 24 to 28 A the energy changes by less than 0.001 kcaI/moIe_ The tensor potential of dispersive forces is nor applicable for the phenanthrene crqrstal because of the difficulty to obtain its attractive coefficient [lo].
lS3
-02 14.4
0 0.1 10.8
20.3
32 22%
2x5s 02s 314
=~~YW=W=JW I 11. fzupelX&(cm-~) 1
I?
‘1 w2
UwdeA
2 e =s =a as
modtB
(JI == w3 ma
(1)
(2)
-14-O --13-I
-18-9
-1x1
-16.01
-15[22]
-31.0
- 29-4
-2539
-22[2q
ss
87
s2
$1
50
51
41 106 79 55 47 26 95 74 47 29
41 166 Sl 5s 50 25 96 75 51 31
59 40 109 SS 63 54 30 102 73 56 u
SS 41 110 90 63 53 32 103 72 54 36
The three-body dispersion energy of phenanthrene has been derived from direct summa tion of isotropic potentiak over eve- three non-vaknt atoms [Is]_ The contribution of the dispersive non-pair interaction to the energy of both ctystaIs yields = 15% cf the total lattice energy_ 32. m-&Ioror~itrok~ene The f73-chloronitrobenzene cn_stal belongs to the Pbn2, space group with four polar molecules (Mt” = 4 D) per unit ceil [19]_ In terms of the irreducibIe representation of the factor group. zero wave vector lattice vibrations are ciassified as folIows: 5At + dA,+ 23, f .S&. all of them beins active in Raman scattering- proceeding from the avaiIab!e charge distribution in an m-chIoronitrc~ benzene mokcule i3). quadntpole and octupde moIecuIar moments have been found by the usuaI
Qwdmpok Ilmhuhr IO=225
(@Q.
ocwpok
(Wn)
mommas
and effaxive
,ExperimalI
(4)
(3)
82(18]
56 l&31 97 61 47 32 99 60 36 32
118(24] 89 62 33 100 60 31
technique [6]_ The vahxs obtained and the effective polarizabihty are summarizd in table 3_ The lattice sums were caIcuIated over atomic sublattices with a subsequent averaging per moIecuIe_ Phonon spectra and Iattice enagies resulting from various approximations are given in table 4_ As can be seen, the electrostatic interaction causes a displacement of the lattice vibration frequencies by l-3 cm-’ (except for the frequencies &A,) and a_(Bt) where the shift is equal to 7 cm-‘). ix_ by no more than lo!& whiIe the contribution to the crystal energy appears to be significant and achieves 25%. The addition of the threebody interaction to the lattice dynamics does not resttic in an appreciabIe redistribution of the frequencies, though for some frequencies (e-gwS(A,). u,(B,). w,(A,)) it improves the agreement with experiment_ But the contribution to the Iattice energy appears to be fairIy large (= 15%)
nwkcuhr
poIaizabili:y
tensor (A)
of m-chloronitrobauax
coordixu~e s)-jCem)
.we (C
ox=)
lo-
-4,t.W
- 246
0
-4-76
0 2-61
X9-75
0.69
I.45
1836
-OS4 10.09
A$:‘:
-157(meu)
(C
m’) -324(umc)
Lo7(no)
--0_4s (UC@) 1.17 (CIW) -Lo5 &Iaw)
(in the
.-
ZA. Ronira, Table 4 Energy and
h-c!qmm-esof tions.T-2soK=)
IaItia
viilations
.-
et aL /
Ncyz -coral
of m4zbIoronluvbalzale
-219
0=WmoIe)
rrcqucncics (cm-‘) modeA,
01 W_ &Z 3 *, es 06 01 *2 =J 04 *5
mode &_
=: 02 cs, a,
(3)
- 23.7
-252
potemial;
0 110 90 75
58 36 0 107 91 75
55 46 34 95 78
56 48 32 95 71
58 50 32 93 70
60 45 28 0 97 66 58 46
59 45 30 0 94 68 58 45 24 0
63 46 33 0 92 70 62 45 28 0
(2): (I)+multipole-multipoic
porcntiak
4_ Coudusion CaIcuIations of phonon spectra of crystals belonging to different space groups show no significant influence of the long-range interactions on the lattice vibrational frequencies. Thus, taking into account the ekctrostatic interaction can change the cakutated frequencies of Raman spec: tra by at most 10%. even for the m-chloronitrobenzene crystal with strongly polar molecuies_ Dispersive and inductive potentials give a still smaller contribution to the force co&ants. A short-range interaction taking into account rep&ion of three bodies at short distances has a greater effect on the Iattice dynamics. But in this case an appreciabfe cfiange is observed only for a limited number of frequencies (see tables 2 and 4) However. the I
and caI&rcd
I.- 167.:; __I -_
:
.i .. ._.-.... in different approxim+
Expcrimult :
107 104~ 78 61 45 0 106 94 79
24 0
us a’ (1): atom-atom
(3
107 101 74
68 57 32
es
mode 9,
.
108 loo
03 rz,
~;
cQmzLs
obtained cxpaimcntaUy
-. -1 cz
modeA=
in -
I
(1) =agy
inr-&
(3): <2)+-lh_cr-body
-24[22]
,-_’ ..
107 [25] -94 _..
_.:..:.-
--
76 59 50 107 93 75 58 49 30 92 68 44 32 92 67 60 44 26
in:cracrions.
contribution of long-range forces to the lattice energy reaches a sufficient value to make it necessay to take into account these interactions-when evaluating subhmation energies and equilibrium configurations of crystals.
References [lJ AL Kitaigomdsky, Molecular cqsds (Nauka, Moscour. 1971). [2] G. Taddei. H. Ekmadco. M_P_ hfanoahi and S C&fan% J_ Chem_ Ph_ys_58 (1973) 966. [3] A_ Girard, M. Sanqucr. J.-C Messger and I ~Ueinxl_ Mol. Cryst Liquid Cryh 28 (1974) 249. (41 V-F_ Sbabanov. V-G_ Podoprigora and V_F_ Spiridonov. Opt. i Sp&roskopi~~ 39 (1975) 1%. 1193[S] H. Gamba and H. Bonadco. J. Ghan Phys 75. (1981)
-v)59.
-_
IA.
168 [6] N. New. chmr m
R
Pays
N_P_ Gupta
Ri@ni. 29 (l978) and
ZZcntikw ff aL / RnwxmnZ
S Cdifano
2nd S-H.
Wahmky,
Card_
SoIid
State Commun
IS
(1976) 799_ [S] JX Barker. .Sf_Y_ Robatic and ML Ktdp Ph>s Lettess 34A (19TT) 415; phys. Rev_ B5 (1972) 3185. [9] V_F_ Sbabmxw and V-G_ Podoprigom Opt_ i S+kkopiya 41 (1976) 235: 45 (1978) 493_ [lo] D-E srrogr)n MoL Pfqx 22 (1981) 8X_ fl!] B_M_ Axilxxl and EJ_ TeiIe. I Chau Ph>-% 11 0949) 39. [12] D. Roy and AX_ Bay. Pbys. Star SOL 38 (1970) 367. [13] AX_ Sadcar and S Scqupta. Ph_ys_ State SoL 36 (1369) 35L [l:] P-G_ Gmmins. PA_ Dunmur_ RW_ hiunn and RJ_ Se-.=ham. Aaa C,rt A32 (1976) 847s flS] T_ Lucy. Chum_ wx_ Letters 44 (1976) 335 [X6] V-G_ PcdoprIgors. A_N_ Botvich V_l=_ Shabamx and V-P_ Etmakov. CrjmIIogctf-tya
26 <1981) 6Z’_
in nwkdar
i-qnaLs
Mum. Cham Phj% 5; (1981) 26% V-G_. Podoprigorri V_F- Shabanov add. IA_ Appt Spauy_ USSR 38 (1983) 993-
[17] RW_ [18]
167.
Px-
inwracIiaw
Ranizcw-J_
119) RW_G. Wyzkoff. CtystaI structures. VoL 6 (Wiley-Intasciara. New York. 1%9)_ [20] K_ Morokwna. S Otmishi. T. Muas& and K F&t& BuIL CfemSocJapcm360%3)1~ 1211 A Chabkx D_W_I_ G&kshak A Hinchwfe and RW. .u,mn. Ghan phys_ Letters 7s (1981) 424. fu~YlrApc&in.Physicsd~-of~~mpounds in Ihe solid state (Nauka Mosanv. l%r, [23] I_ coiombo. Cham Phys Lcttas 1241 A_ BID= F-G_ Sohen and V-V-B. [rs]
48 (w77) 166Vihs. J. ,UoL Spccq_
44 (1972) 298_ V-G. Podopti3ora. AN_ Botvidt. N-P. &stakov Shabanov. Opt. i Spcktroskopiya SO (1581) Un.
and V-F-
k-ERACI-IONS DYNAhilCS OF MOhULAR
LA_ REh4IzOV. V-G_ PODOPRIGORA,
r.
: CRYSTALi -
AN_ BOh’ICH.
Atom-atom, multipoIe-multipole and three-body potentials have b&n used to calculate the energy and lsttice dynimics of ¶dichIorobenzene. phenanthrme and mchIoronitrobenzcne Noncentral interactions are shokn to &z a signikant contniution to the intermokcuIar interaction energy but have littIe effect on the phonon
1,Introduction
2. Crystal potential function
In a number of papers [l-4] atom-atom potentials of the 6-exp type were employed to describe the dynamics and the structure of molecular crystal lattices However, when investigating strongly.anisotropic crystals the parameters of the atom-atom interaction curve have to be varied for various substanks_ Besides, the application of the potential is limited since some specific, eg non-additive and electrostatic, interactions, anisotropic by- nature are neglected here_ Recently a number of papers have demonstrated the effect of multipo!e-muItipoIe interactions on ihe dynamics and energy of the molecular cjstal lattice. However the results appear to be extremely conflicting (see, for example, refs_ [5,6Jj_ The effects of many-body non-additive interactions have been studied in refs_ ]7,8] for solid inert gases They were found to play a significant role for the crystal structure and the lattice dy&mics_ For more complicated mokcufar crystals such data. are not yet avaifable in the literature_ The present paper aims at investigating the infhrence of three-body and multipoIe~muhipole interactions on energy and dynamical properties of moiecular crystals of different space groups: ¶dichlorobenzene, phenanthrene and m-chloronitrobenzen~-
We present the crystal lattice energy as the sum of the potentials
w = y-1 + y-
+
y(3)*
where V.’ is a central atom-atom pair potential, Vz denotes an anisotropic pair potential of muhipole-multipole interaction, Yes is a non-additive three-body potentiai. The first term on the right-hand side of the above expression is chosen in the 6-exp form (modified Buckingham potential): Y, = c [ - CiiR,7p -t- Bij exp( -pi&)] 8
_
(0
The coefficients C, B and p have been obtained from structural and optical eqeriments employing the lattice energy data from refs. [4,9] (Rji is the distance between two atoms i and i j_ The interaction of constant multipoles of ttio neutral mokcuks Q and fi, according to ref. [6], is -defined by 00
a =<= (-1)*+‘2~+s!r!/(2s)!(2t)!.
0301_0104/85/S03_30 Q EIsevier Science Publishers B-V. (North-Holland Physics Publishing Division)
~_ _
-.
164
LA.
Q**’
is an
*s+=r
= - ,=+,(
(3 i
t)th
rank
RZiJ_
Rem&z
interaction
CI
aL ,* .Von -ccmraJ
tensor (3)
K&h’)is the electric multipole moment tensor of sth order for the ath molecule and R4 is the distance between the centers of mass of molecules Q and j3_ When s = 0, 1,2._ __ we deal with a charge. dipole. quadrupoIe, ettc, respectively_ A three-body potential consists of short- and long-range parts. the latter containing the potentiafs of inductive and dispersive interactions_ The former potential, being non-additive by nature includes terms referring to simultaneous two- and three-body interactions_ Non-pairwise terms apRear in the second order of the perturbation theor)= i_c in the same approximation as van der Waals pair forces_ Restricting ourseXves to the dipole appro_ximation the inductive potential can be. as well as (2). expressed in tensor te_rms[lo]
where M(*’ is the dipole moment and A, is the generahz&i polarizability tensor of the ath molecule_ The dispersive potential cf three interacting asymmetric moIecuIes has the following form
(5) where G,,, is given by the coefficient C$’ [lo] at R$ in a two-body attractive potential derived in second-order perturbation theory_ This coefficient must speciahy be chosen and here it is the main difficulty in employing eq_ (5) For three spherical bodies_ eq_ (5) transforms into
~(R&x$s,)-t-
(6)
as su_ggested in ref. [ll]_ v-. TV. or are the internal art&s of the dipolar triangle with sides Rag, R,,, RBr_ The description of the three-body interaction by means of eq_ (6) is more exact the closer the molecular shape is to spherical_ For the first time we ha\= treated a three-body energy as the sum of triple interactions of non-vaIent atoms to describe
imcramicws in
nwidar
qnnz&
complicated molecular crystals_ The magnitude of G ,,s, in (6) appears to be determired exactly by the atom-atom potential. The short-range part of the three-body potential takes into account the overlap of electronic clouds_ For inert gases this interaction was satisfactorily described by the potential [12] V’3’=cDi;k
expi-((~~~R;~fp;~R,~)].
(7)
where for solid inert gases pij = pir = p (p is the Buckingham potential parameter oi eq_ (1)) and Di,k = D_ While D is related witk 3 from the Buckingham potential_ its value must be obtained from additional experiments_ such as cohesive energy and elastic constant measut-emtnfs [13]. So here we tty to obtain D from atom-atom potential parameters_ The ¶dichlorobenzene crystal u-as taken for the caiculaticus because of its simple structure with one molecule per unit cell, so that there are only three lines in its Raman specsra We obtain D;,k = 3 X 103( BiiBjjBkr-)‘f3 from inverse spectral problem solution with C_ B and p from (1). So three-body interaction parameters may be calculated using pairvise ones. To check the obtained D,.+. value. phonon spectra for some molecuhxr crystaIs have been calculated_ The results are analysed below_ To calculate the effective pokuizability A of a molecule, the method of induced dipoles has been used to take into account the effecl of the lattice environment on a molecuhxr electronic cloud [9j_ The component A, of the effective polarizability tensor is given by
where 4 are the elements of the free molecule pohuizabihty tensor, a, and ct,. refer to the atomic polarizabilities of different R,@pazed moiecules. 11, is the direction cosine of the bond i-t’_ S, is the Kronecker symboL This model provides A derivatives not only with respezt to the hbrating external coordinates but also with respect to the trans1ationa.l ones which results in a. change of the force constant values for three-body dispersion and induction interactions_ To estimate the interaction tensor T (lattice
sum) describing the structural crystal anisotropy, the EwaId method has been applied, invoking summation over the lattice sites both in direct &d reciprocal spaces 1141. Previous papers, however, used &e point-molecule apprcximation disregarding the real size and mutual molecular orientations in the ctyscal lattice. when determining the inceraction contribution co the lattice dynamics of molecular crystak In that case the lattice sum 7 depends Tn the external translational molecular coordinates only, which brings about an inaccuracy in the estimations of the oriencaciond phonon parameters Other well-known approaches are either to divide a mokcule into force centers (atom groups) 1151 or co dea1 with atomic sublacciccs [16,17]. The latter enables one to obtain T derivatives not only with respect to translational but also with respect co 0riencationaI molecuk displacements from equilibrium_ Thus for complicated
mokcular vibraiions in a unit cell, second derivatives of the crystal energy obtained within the present model may differ appreciably from those found for point-molecule and oriented-gas approximations- The calculating ckchnique of crystai force constants typical of pair, electrostatic and three-body interactions is treated in refs- [4,6,lS], respectively_
3_ Results and discussion RI_ p-dichlorobetcene
and phenanthrene
The p-dichlorobenzene crystai belongs to the Pi space group with one molecule per unit ceII Iocated at the inversion center [19]. According to the selection ruIes, three Iibrational modes of A, species are active in its Raman spectrum_ Phenanthrene crystaIIizes in the space group P2, with two molecules per unit ceU [19]_ Zero wave vector lattice vibrations in terms of irreducible representations of its factor group are ~classified as follows: five modes of A species and six modes of B species_ For ¶dichIorobenzene the first non-zero mu!tipole moment is the quadrupole one_ Molecular symmetry aIIows three components of the quadrupole moment: ML?, M,‘:),~ MC& two of
them being independent: M$!i = - (ME) 4 Mi!‘); where u, u; w are the long, medium and normal axes, respectively_ The char& on the atoms fat. &C,H,CI, have been determined in ref. [20]. I%ng these estimations, the diagonal components of IW censor. have been evaluated to give Mzi’ = 62 x lo-* C m’, Mz: = Miz) = -31 X lo-j0 C m’_ The dipole moment of phenanthrene lying in. the plane of a benzene ring cquaIs 0.2 D 1215 The components of the quadrupole moment have the following magnitudes [21]: Miz = 27-9 x lOma C m’, M:z’ = 32-7 X lo-C m’. Dipole and quadrupole Iattice sums of p-dichlorobenzene and phenanthrene were calculated on the atomic sublattices with a subsequent averaging per molecule. Effective polarizability tensors are presented in table l_ Lattice energges and phonon frequencies of & paradichlorobenzene and phenanthrene derived in different appro.ximaticns and those obtained from polarized Raman scattering experiments are given in table 2. The atom-atom potential parameters are taken from refs. [3,4]. As seen from tabIe 2, the absolute values of the short-range three-body repulsion and induction energies are small_ The dispersive non-additive energy gives a large& positive contribution to the cocaI energy and, being cakulated by (5) in a point-mokcule approximatiou. is slowly converggng_ Varying the summation radius from 37 to 40 A, the energy of @-paradichlorobenzene changes by more than 5%_ The calculation of T and A over atomic sublattices provides a much better convergence_ Thus, in the range from 24 to 28 A the energy changes by less than 0.001 kcaI/moIe_ The tensor potential of dispersive forces is nor applicable for the phenanthrene crqrstal because of the difficulty to obtain its attractive coefficient [lo].
lS3
-02 14.4
0 0.1 10.8
20.3
32 22%
2x5s 02s 314
=~~YW=W=JW I 11. fzupelX&(cm-~) 1
I?
‘1 w2
UwdeA
2 e =s =a as
modtB
(JI == w3 ma
(1)
(2)
-14-O --13-I
-18-9
-1x1
-16.01
-15[22]
-31.0
- 29-4
-2539
-22[2q
ss
87
s2
$1
50
51
41 106 79 55 47 26 95 74 47 29
41 166 Sl 5s 50 25 96 75 51 31
59 40 109 SS 63 54 30 102 73 56 u
SS 41 110 90 63 53 32 103 72 54 36
The three-body dispersion energy of phenanthrene has been derived from direct summa tion of isotropic potentiak over eve- three non-vaknt atoms [Is]_ The contribution of the dispersive non-pair interaction to the energy of both ctystaIs yields = 15% cf the total lattice energy_ 32. m-&Ioror~itrok~ene The f73-chloronitrobenzene cn_stal belongs to the Pbn2, space group with four polar molecules (Mt” = 4 D) per unit ceil [19]_ In terms of the irreducibIe representation of the factor group. zero wave vector lattice vibrations are ciassified as folIows: 5At + dA,+ 23, f .S&. all of them beins active in Raman scattering- proceeding from the avaiIab!e charge distribution in an m-chIoronitrc~ benzene mokcule i3). quadntpole and octupde moIecuIar moments have been found by the usuaI
Qwdmpok Ilmhuhr IO=225
(@Q.
ocwpok
(Wn)
mommas
and effaxive
,ExperimalI
(4)
(3)
82(18]
56 l&31 97 61 47 32 99 60 36 32
118(24] 89 62 33 100 60 31
technique [6]_ The vahxs obtained and the effective polarizabihty are summarizd in table 3_ The lattice sums were caIcuIated over atomic sublattices with a subsequent averaging per moIecuIe_ Phonon spectra and Iattice enagies resulting from various approximations are given in table 4_ As can be seen, the electrostatic interaction causes a displacement of the lattice vibration frequencies by l-3 cm-’ (except for the frequencies &A,) and a_(Bt) where the shift is equal to 7 cm-‘). ix_ by no more than lo!& whiIe the contribution to the crystal energy appears to be significant and achieves 25%. The addition of the threebody interaction to the lattice dynamics does not resttic in an appreciabIe redistribution of the frequencies, though for some frequencies (e-gwS(A,). u,(B,). w,(A,)) it improves the agreement with experiment_ But the contribution to the Iattice energy appears to be fairIy large (= 15%)
nwkcuhr
poIaizabili:y
tensor (A)
of m-chloronitrobauax
coordixu~e s)-jCem)
.we (C
ox=)
lo-
-4,t.W
- 246
0
-4-76
0 2-61
X9-75
0.69
I.45
1836
-OS4 10.09
A$:‘:
-157(meu)
(C
m’) -324(umc)
Lo7(no)
--0_4s (UC@) 1.17 (CIW) -Lo5 &Iaw)
(in the
.-
ZA. Ronira, Table 4 Energy and
h-c!qmm-esof tions.T-2soK=)
IaItia
viilations
.-
et aL /
Ncyz -coral
of m4zbIoronluvbalzale
-219
0=WmoIe)
rrcqucncics (cm-‘) modeA,
01 W_ &Z 3 *, es 06 01 *2 =J 04 *5
mode &_
=: 02 cs, a,
(3)
- 23.7
-252
potemial;
0 110 90 75
58 36 0 107 91 75
55 46 34 95 78
56 48 32 95 71
58 50 32 93 70
60 45 28 0 97 66 58 46
59 45 30 0 94 68 58 45 24 0
63 46 33 0 92 70 62 45 28 0
(2): (I)+multipole-multipoic
porcntiak
4_ Coudusion CaIcuIations of phonon spectra of crystals belonging to different space groups show no significant influence of the long-range interactions on the lattice vibrational frequencies. Thus, taking into account the ekctrostatic interaction can change the cakutated frequencies of Raman spec: tra by at most 10%. even for the m-chloronitrobenzene crystal with strongly polar molecuies_ Dispersive and inductive potentials give a still smaller contribution to the force co&ants. A short-range interaction taking into account rep&ion of three bodies at short distances has a greater effect on the Iattice dynamics. But in this case an appreciabfe cfiange is observed only for a limited number of frequencies (see tables 2 and 4) However. the I
and caI&rcd
I.- 167.:; __I -_
:
.i .. ._.-.... in different approxim+
Expcrimult :
107 104~ 78 61 45 0 106 94 79
24 0
us a’ (1): atom-atom
(3
107 101 74
68 57 32
es
mode 9,
.
108 loo
03 rz,
~;
cQmzLs
obtained cxpaimcntaUy
-. -1 cz
modeA=
in -
I
(1) =agy
inr-&
(3): <2)+-lh_cr-body
-24[22]
,-_’ ..
107 [25] -94 _..
_.:..:.-
--
76 59 50 107 93 75 58 49 30 92 68 44 32 92 67 60 44 26
in:cracrions.
contribution of long-range forces to the lattice energy reaches a sufficient value to make it necessay to take into account these interactions-when evaluating subhmation energies and equilibrium configurations of crystals.
References [lJ AL Kitaigomdsky, Molecular cqsds (Nauka, Moscour. 1971). [2] G. Taddei. H. Ekmadco. M_P_ hfanoahi and S C&fan% J_ Chem_ Ph_ys_58 (1973) 966. [3] A_ Girard, M. Sanqucr. J.-C Messger and I ~Ueinxl_ Mol. Cryst Liquid Cryh 28 (1974) 249. (41 V-F_ Sbabanov. V-G_ Podoprigora and V_F_ Spiridonov. Opt. i Sp&roskopi~~ 39 (1975) 1%. 1193[S] H. Gamba and H. Bonadco. J. Ghan Phys 75. (1981)
-v)59.
-_
IA.
168 [6] N. New. chmr m
R
Pays
N_P_ Gupta
Ri@ni. 29 (l978) and
ZZcntikw ff aL / RnwxmnZ
S Cdifano
2nd S-H.
Wahmky,
Card_
SoIid
State Commun
IS
(1976) 799_ [S] JX Barker. .Sf_Y_ Robatic and ML Ktdp Ph>s Lettess 34A (19TT) 415; phys. Rev_ B5 (1972) 3185. [9] V_F_ Sbabmxw and V-G_ Podoprigom Opt_ i S+kkopiya 41 (1976) 235: 45 (1978) 493_ [lo] D-E srrogr)n MoL Pfqx 22 (1981) 8X_ fl!] B_M_ Axilxxl and EJ_ TeiIe. I Chau Ph>-% 11 0949) 39. [12] D. Roy and AX_ Bay. Pbys. Star SOL 38 (1970) 367. [13] AX_ Sadcar and S Scqupta. Ph_ys_ State SoL 36 (1369) 35L [l:] P-G_ Gmmins. PA_ Dunmur_ RW_ hiunn and RJ_ Se-.=ham. Aaa C,rt A32 (1976) 847s flS] T_ Lucy. Chum_ wx_ Letters 44 (1976) 335 [X6] V-G_ PcdoprIgors. A_N_ Botvich V_l=_ Shabamx and V-P_ Etmakov. CrjmIIogctf-tya
26 <1981) 6Z’_
in nwkdar
i-qnaLs
Mum. Cham Phj% 5; (1981) 26% V-G_. Podoprigorri V_F- Shabanov add. IA_ Appt Spauy_ USSR 38 (1983) 993-
[17] RW_ [18]
167.
Px-
inwracIiaw
Ranizcw-J_
119) RW_G. Wyzkoff. CtystaI structures. VoL 6 (Wiley-Intasciara. New York. 1%9)_ [20] K_ Morokwna. S Otmishi. T. Muas& and K F&t& BuIL CfemSocJapcm360%3)1~ 1211 A Chabkx D_W_I_ G&kshak A Hinchwfe and RW. .u,mn. Ghan phys_ Letters 7s (1981) 424. fu~YlrApc&in.Physicsd~-of~~mpounds in Ihe solid state (Nauka Mosanv. l%r, [23] I_ coiombo. Cham Phys Lcttas 1241 A_ BID= F-G_ Sohen and V-V-B. [rs]
48 (w77) 166Vihs. J. ,UoL Spccq_
44 (1972) 298_ V-G. Podopti3ora. AN_ Botvidt. N-P. &stakov Shabanov. Opt. i Spcktroskopiya SO (1581) Un.
and V-F-