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Lattice-dynamical calculations for fluorene, carbazole, dibenzofuran and dibenzothiophen
Volume
79. number
3
LAmICE-DYNAMICAL FOR FLUORENE, Giuseppe
DIBENZOFURAN
FILIPPINI, Carlo M. GFWMACCIOLI
3 December
1 hlay 1981
CALCULATIONS
CARBAZOLE,
Ishtuto dr Chrmwz Fisraz e Centro CNR, Received
PHYSICS LETTERS
CHEhlICAL
1980;
AND DIBENZOTHIOPHEN
and Massimo SIMONETTA
Uiznwsztci dr Mhano. 20133 Mlan, Ital,
m fmal form 28 January
1981
For crystalbrie fluorene, carbazole, dibenzofuran and dibenzothlophen several propertles have been calculated usingsemiemplrlcal atom-atom potentrals Calculated geometrxal parameters. Raman-actlvc htt:ce frcquencles, cnthalples of subhmaUon and crystallographic temperature factors agree with observed values However. except for dlberuothlophen, mfraredactive httrce frequencies are not consistent ~th any potential so far tried
1. Introduction Calculation of external vlbratlonal modes of molecules m sohds has become a widely accepted procedure which brings about the posslbllity of prelctmg their structural, thermodynamic and spectroscopic properties [l-6] . While for hydrocarbons there 1s a choice of rehable atom-atom sem:-empmcal potential functions [S-7], for other molecules the sltuatlon IS more compiex, at least if vibratlonal propertles are taken mto account, besides the packmg energy. In the literature, examples of this tid are reported for a relatively limited number of substances, such as, for mstance. N, [8], 02 [9], fluorinated [IO] and chlonnated [I I] compounds. In order to test the vah&ty of these potentials, and in view of their unprovement and generahzation, we Table 1 Crystallographx
data of the selected Compound
considered a series of slmllar rlgtd aromatic molecbies wth different kmds of heteroatoms and no hydrogen bonds, for whch spectroscopic and crystallographic data are currently available. On this basis, we selected fIuorene, carbazole, dlbenzofuran and dlbenzottiophen as possible candidates. Although a general objection about testmg potentials wth such comphcated molecules might be raised, nevertheless these molecules do not differ considerably from the corresponchng hydrocarbons, for which good to excellent agreement between observed propertles and lattlce-dynamlcal results IS obtained. On the other hand, an efficient screemng of the potentials for the heteroatoms reported m ihe hterature IS achieved, to be used as a reference for future work m the field The first three compounds are practically Isomorphous, space group Pnma = Dig, whereas the fourth crystal-
substances
a
b
C
b W%)
Space
Z
Ref.
group
a)
fluorene
849
18 97
5 72
90
Pnma
4
[12]
carbazole dlbenzofuran
7 77 7.702
19.18 19 185
5 72 5 825
90 90
Pnma Pnma
4 4
1131 [14j=)
dibenzothiophen
867
6 00
W,
4
1151
18.70
113.9
/C
a)
The expenmental data are reported m terms of another unit cell (Pnam) where b and c are Interchanged
470
0 009-26
14/8 I /0000-0000/S
02.50
0 North-Holland F’ubhshmg Ccmparly
Volume 79,
number 3
CHEMICAL
PHYSICS
ties m a space group of lower symmetry P21/c = C5,. CrystaUographrc data are summarized r.n table 1_ For C-C, C-H and H-H non-bonded mteractrons, the potentrals grven by W~ihams [ 161 and Kitargorodskri [ 171 have been used: since no substantral drfferences have been observed between the frequencies calculated from these two sets, results are reported here for the former set only. For N-N the function given by Luty and Pawley [ 181 and for S-S the one given by Pawley and co-workers [ 191 were chosen_ For O-O four ddferent functrons have been euammed, namely the ones proposed by Grgho et al [20]. by Parsonage and , Pemberton [2 I ] , by Laufer and Leror [22] and by Krtargorodskn et al [23] . A set of calculatrons was also performed using the coefficients of the C-C potentials for the N-N, O-O and S-S contacts. Except for the case in whrch the whole set of Kitargorodskri functrons (KMN) was used, for mrxed mteractrons, C-X and H-X, coefficrents were always assumed to be the geometrrc or arithmetrc mean of the coefficrents relative to equal atoms, accordmg to the usual procedure. The C-H bond length was assumed to be 1 09 A and the N-H bond 1.03 A. Potential functions are collected m table 2.
LEI-I’ERS
1 May 1981
2 Method of calculation
The packmg energy and its derivatives with respect to molecular coordinates are considered to be a sum of independent terms, each one of them due to a van der Waals mteractron between two non-bonded atoms III drfferent molecules. The lattice sums are extended to a maxrmum distance of 15 A for the calculation of the packing energy and of 8 a for the calculation of lattree frequencies and of the temperature factors: this ensures convergence [S] . A first step involves mimrmzation of packing energy as a function of molecular coordmates for this purpose, a Raphson-Newton procedure 1s used. From thrs stage, we obtam a value of the packing energy and the drfference between experimental and calculated molecular orientation and posrtron m the urut cell: these data already grve an idea about the intrmsic validity of the assumed potentral function and of the proposed treatment A second step mvolves a Born-von K&man dynamreal treatment, which leads to the ergenvalue equation.
where o IS the frequency of a normal mode in crystal, M(q) IS the “mass adjusted” dynamrcal and t(q) are the “mass adjusted” displacement For 4 = 0, thrs treatment leads to mfrared- and
the matrix vectors. Raman-
Table 2 Potential functions Poten-
Contact
Energy (kcal/mole)
\vlllXIlns (WV.4)
C-C
E =
C-H
E=
8766 exp(-3
H-H
E=
2634 exp(-3.74r)-27.3/r6
N-N
E =
s-s
E = 199900 eup(-3
Luty-Pawley Rmall-Pawlay Cl&o
(LP) (RP)
et aL (GLM)
Parsonage-Pemberton
(PP)
83630 eup(-3 6@)-568/r6
26270 eup(-3
67r)-125/r6 40r)-452/r6 49r)-2149/r6
o-o
E = 186400 exp(-4Sr)-200/r6
o-o
E = 259000/r’*
- 358/r6 - 328/r6
Laufer-Lerol
(LL)
o-o
E = 233433/r’*
Kltagorodsku
et al. (KhIN)
c-c
E =
42000 exp(-3.5&)-358/r6
C-H
E=
42000 exp(-4_12r)-ls4/r6
H-H
E =
42000 exp(4_86r)-57/r’
o-o
E =
78000 exp(-4
l&)-259/@
c-o
E =
57000 exp(-3
85r)-313/r6
H-O
E =
49000 exp(4
SOr)-106/r6
471
CHEMICAL PHYSICSLETTERS
Volume 79. number 3
active lattice vlbratlon frequencies; thermodynamic functions and crystallographc temperature factors are obtained after an appropriate sampling of the Bnllouin zone [24].
3.Results and discussion For all considered compounds the &fference between the experrmental and the calculated molecular coordmates in the crystal is reasonably small, the maximum values being 0.12 A and 3” for translation and rotation, respectively. These values are as good as for hydrocarbons [7], and from this point of view all the potential functions here adopted appear to be equally satisfactory (see table 3).
1 May 1981
Srm~lar conclusions can be drawn wth respect to enthalpy of subhmation, at least m the two cases where measurements are avalable (see table 4). Expenmental and calculated lattice vlbratlon frequencles are reported m table 5. Smce the functions proposed by Laufer and Leroi and by Parsonage and Pemberton gwe unaginary frequencies even for CJ= 0, no results relatwe to these functions are reported m table 5, and they are omltted from further connderatlon. In order to help the reader m vlsuahzmg correspondence between these data, the percentage mutual agreement IS gven as R = 100 C Ivobs-vdl/Qobs It must be stressed, however, that smce a rlBd model has been used in these calculations, the highest calculated frequencies should III general exceed the correspondmg expermental values.
Table 3 Evaluation
of agreement
between
crystallographic
data and results
fluorene
carbazole dtbenzofuran
potential
functions -__
Transhnonal and rotaaonal displacements
(A) (deg)
RB b,
WWA
1
0001
0 005
- d 22
a)
WIVA + LP
2
0 11
0 34
all WIVA
1
0.12
0 17
19
WNA
+ GLM
3
0 06
2 05
WNA
+ PP
4
0 10
2 17
47 _ d)
WNA
+ LL
5
0 10
2 17
6
0 05
3 09
43
all WNA
1
0 06
240
22
WNAIRP
7
0 09
0.45
25
alI WNA
1
0 07
1 03 --
47
KhlN dlbenzothiophen
from various
Set
Potentlal functton
Compound
derived
_ d)
a) For explanation of the symbols see table 2 b)RB= 1OOxIB r,(obs) - ~~(ca~c)l/Zl~Ij(obs)I iS the perCentage agreement between calculated and experuncntal amsotropic temperature factors. ‘) Expenmental values are not reported III ref_ [ 121. d, Existence of lmatiary frequencies even for 4 = 0 does not allow calculations of temperature factors
Table 4 Compartson
between
the packmg
energy
Compound
a) AU Wdhams
472
NA
potentA
and the enthalpy -Epack
of subhmatlon
(kcal/mole)
-_
ms
fluorene
19 5 a)
18.3
carbazole
236a)
225 [261d),21.9 [271d),20_2 [281d,
functions.
b, !6 l-300%
[131 b), 19.8 [251 c,
c) 33-5o”c.
d, 244-352OC
crystallographic
2
(127)
98
70
bju
78“ I08
94
89
98
35 76 90 23 38 63
31 44 66 24 38 71
39 84
34
1
I301 2
I
21
I4
39 84
123 21
100
70 20
54
[31] a)
104
33
I10
84
73
53
96
75
60
69
58
55
85 _ b)
42
obs a) -_-___-___
Dlbenzofuran
3
22
16
us
14 32
98
12
22
77
48
26
99
1
23
I4
84
I7 35
104
23 83
75
49
21
107
35 48
34
41
46 111
41
10
104
83
25 27 86
cillc.
41
34
104
45
40
5
98
88 12
I9 27
CA.
---
6
26
I7
93
34
13
110
75
25
94
57
25
119
55
35
47
125
40 44
10
Ill
97 75
23 25
talc --_
bg bg bg bg bg bg
a6
“6
“g “6 “g a6
bU
bU
bU
bU
311
“U
313
QU
nU
-_____
1321
81 104
63
33
I05
79
49
31
171) 104
53
109
89
721
obs.
1
6
7
102
87
61
53
41
32
108
88
77
46
41
21
6
99
16
62
36
110
65 90
32 45
CIIC.
Dlbenzothlophcn --
I
14
12
85
75
60
26 41 43
92
14
40 67
19 37
15
80
64
31 58
94
15
21 45 64
cnlc.
a) The labelhng hns been changed accordmg to the new umt cell (see table I). b, In addltlon values of 101 ond 130 cm-’ for fluorcne, 103 and 129 cm? for carbazolc and 71 cm-’ for dlbcnzofuran arc rcportcd as bz,, lattlcc frcquenaes. these do not mntch our wlculatlons.
set of potentrals
[29] a)
21
24
R,,,(Z)
ref.
I4
26
103
28
28
122
70 20
58
88 I07
54
36
36 78
I38
137
52
54
40
38
I41
54
47
53
40
55
43
52?
28 57
88
37
53 141
RRaman(%)
b3E
b3is
bag
b2s
b2g
b2E
k
btg
“6 bU
“6
“6
R]R(%)
20
69
61
46
bxi
48
_b)
52
bzu
I7
_ b)
I23 17
blu bzu
122
90 18
100 _ b)
blu
100
98
49 104 - 1))
74
88
88
27 33
CZllC
56
CA.
28 33
obs.
25 54
Wk.
Carbazole
80
QU
“U
PU
obs a)
lIuorcne
Observedand culculatcd lattlcc frequenaes
5 _
i Q
i! i.z
z
3 !I
G 5? E ::
z>
.z
2
< L
Volume 79, number 3
CHEMICAL
wbratlon
The agreement is essentially good m the case of Raman spectra for aI.I molecules and also for IR measurements in &benzottiophen. For the IR spectra of the other compounds the agreement is less satsfactory, especially d the b,, lattice Table 6
1 May 1981
PHYSICS LETTERS
modes are considered
(see footnote
b of tab:e
5). Although such a sltuatlon might Involve a wrtual madequacy of these potentials to account for the vlbratlonal properties of these sohds, there are good reasons not to discard the possible
vahdlty
of our results
and
and calculated crystallographrc anlsotroplc temperature factors (all values are multlplled by 104)
Expenmentd
B 22
Bil
obs
CalC
obs
B 12
833
CalC
obs
CalC.
GllC.
obs
B23
f313
-
obs.
Gllc
obs.
talc
5
-11
carbazole a)
144 195
39
3s
232
297
C2 c3
134 141
45
37
217
221
34
32
c4
208
222
38
31
C.5
149
177
45
32
C6
105
131
38
30
N
179
162
47
41
290 455 373 242 229 231
404 499 458 337 277 288
339 488 647 586 404 316 318
Cl
-4
14
5
-14
-11
6
13
-29
-29
-13
-6
56
50
-11
-18
6
18
29
-4
5
2
-4
0
-4
0
-58
2
6
0
23
-3 9
10 8
5 -6
-22
-1 0
0
dtbenzofuran b,
C6
125
175
38
44
322 347 600 517 359 311
0
234
239
46
64
319
Cl
153
203
41
53
c2
196
281
52
58
c3
214
314
3s
48
c4
207
257
39
44
C5
165
236
4s
45
-17
-7
20
10
-1s
-20
1
22
-33
-47
-16
-16
86
73
-2s
-34
0
54
51
32
7
3
6
5
8
10
2
5 -1 -6
-1
0
0
28 -54
14
11
-9
-2s
-2 0
0
dibenzothlophen ‘) Cl
193
206
331
420
24
37
-28
-53
41
27
-11
C2
162
191
350
508
27
-20
-36
34
19
:
c3
184
179
330
457
34
37 43
51
24
14
c4
178
172
233
341
32
41
cs
183
181
250
311
27
37
C6
170
192
282
404
29
c7
157
166
370
431
31
C8
170
162
285
356
32
42
29
Cl0
179
174
256
306
24
35
-1
32 13
24
13
53
32
-11
47
3s
-10
-16
37
-54
-30
44
30
-16
-23
38
-26
-13
45
24
9
48
29
15
-29
44
32
18
1
7 25
-10
-3
Cl1
137
152
217
277
25
33
6
-11
43
33
Cl2
137
149
213
262
22
33
24
-13
36
34
Cl3
145
155
224
276
33
29
6
34
46
-1
S
197
208
246
294
33
46
5
48
37
-12
-13 18
a) Experimental values from ref [ 131, calculated from WIVA+i_.P potenti b, Experimenti values from ref. [ 141, calculated from all WIVA potentials. ‘1 Expenmental values from ref. [ 15 1, calculated from WIVA+RP potenmls.
474
21
-17
6 -3
-1 6 -9 4 -1 6 -20
Volume 79, number 3
CHEMICAL
PHYSICS
to consider instead the opportumty of reexammmg the mterpretation of the experrmer,tal IX data. In fact, It is difficult to image an essential failure III reproducing a whole series of IR-active vlbratlonal modes and in the same tune a substantial success m reproducmg the whole seyes of Raman modes. Moreover, these drfficul-
ties arise even for fluorene, I.e. 2 compound ~nth no heteroatoms, m contrast with the general agreement for hydrocarbons between calculated and experimental lattice frequencies [7] . Even for IR frequencies a good agreement is achieved for hbenzothiophen, whose crystal structure IS different from the ones of the other three compounds Therefore, some parallel msmterpretation of spectra might have occurred. Comparison of anisotroplc crystallographuz temperature factors with experimental values, as given by X-ray diffraction, shows an overall satisfactory agreement for all the substances (see table 6). for some potentials the R mdex (see table 3) 1s ~20% and this 1s w~thm the actual accuracy of the experimental data. Furthermore, it must be pomted out that calculabon of temperature factors was possible since no imagmary frequencies appeared throughout the Brillouin zone: a test of ths kind has always proved to be quite important for venfymg the valilty of any potenti function [33 1. It seems that better results are obtained for dbenzofuran if the potential proper to the heteroatom is replaced by a “common” C-C potentialthis might mean that the van der Waals interaction of the oxygen atom enclosed in these rings is tiferent from the corresponding mteraction in the substances from which these potentials were derived. In conclusion, our calculations show that an overall good agreement with experimental data 1s obtained, except for some [R-active vlbratlonal modes: however the calculated frequencies are at the limit or beyond the range of observation and future experunental work seems to be advisable.
References I1 J C-S. Pawley. Phys. Stat. Sol 20 (1967) 347. [21 G.S Pawley, Phys. Stat. Sol. B49 (1972) 475. [3] S. Llfson and A. Warshel. J. Chem whys. 49 (1968) 5116.
1 hiay 1981
LE’l-fERS
141 A Warshel and S. Llfson, J Chem. Phys 53 (1970) 582. 151 G Fdippuu, CM Gramaccloh. M. Sunonetta and G B. Suffrlttl,Chem. Phys 8 (1975) 136. hf Slmonetta and G-B J6J G Fd~ppuu, C_M Gramaccroli. Suffrith. Chem. Phys. Letters 35 (1975) 17. hf. Sunonetta and G B. 171 G. Fibppim. CM. Gramaccloli, Suffntti, J. Chem. Phys. 59 (1973) 5088 J81 G Fti~ppuu. CM Gramacciolr. hl Sunonetta and G B Suffritu, hfol. Phys. 35 (1978) 1659. and references
‘herein
hl L. Klem and V. Chandrasekharan, J Chem. phys 71 (1979) 843. hf. Slmonetta. G-B Suf1101 G Fd~ppin~,C hf. Cramaccloh, J91 K. Kobashi,
fn’nttland 0. Sala, Chem. Phys. Letters 39 (1976) 14. [I11 L -Y Hsu and D E. Wdbarns. Acta Cryst. A36 (1980) 277. Jl21 D.hf Bums and J. Iball. Proc. Roy. Sot. 227A (1955) 200. 1131 hf. Kurahashl, M Fukuo, A. Shunada. A Furusakr and I. Nitta, Bull. Chem. Sot. Japan 42 (1969) 2174. J14J 0 Dldeberg, L DuPont and J-M. AnclrC, Acta Cryst. 828 (1972) 1002. 1151 R hf Schaffrm and 3. Trotter, J. Chem Sot. A (1970) 1563_ 1161 D E. Wilhams, J. Chem Phys 47 (1967) 4680 J. Chun. Phys. 63 (1966) 6. Jl71 A I fitayorodsku, iI81 T Luty and G S. Pawley. Chem Phys Letters 28 (1974) 593. 1191 R P. Rinald~ and G.S. Pawley. Nuovo Cuncnto 16B (1973) 55; J Kunttu and G S. Pawiey. ActaCryst A29 (1973) 615. 1201 E. Gigbo. A hf. LIquorI and L. hfazzarella, Lettere Nuovo Cunento 1 (1969) 135 [21 J N.G. Parsonage and R C Pemberton. Trans Faraday Sot 63 (1967) 311 [22J J C Laufer and G E Lero!, J Chem. Phys. 55 (1971) 993 [23] A.1 Kltigorodsku, K.V. Muskaya and V.V. Nauchitel’, Knstallografiya 14 (1969) 900. [24 J G. Fihppuu, CM. Gramaccioh, M. Sunonetta and G.B. Suffrltti. Acta Cryst. A32 (1976) 259. [251 R.S Bradley and T-G Cleasby, J. Chem Sot (1953) 1690 126 J International Cntical Tables 3 (1928) 226,5 (1929) 134. [27J F.S. Mortimer and D.V. Murphy, Ind. Eng. them. 15 (1923) 1140. 1281 A Aihara, J. Cbem Sot. Japan 76 (1955) 492. 1291 A. Bree and R. Zwarich. J. Chem. Phys 51 (1969) 912 [30] A. Bree and R. Zwanch, J. Chem Phys. 49 (1968) 3344. 131 J A- Bree. V-V B. VI&OS and R. Zwanch, J. Mol. Spectiy. 48 (1973) 124.
1321 A. Bree and R. Zwarich, Spectrochun. Acta 27~ (1971) 599. J33J S. Ram&~. GM. Parkmson, J.M. Thomas, C M. Gramacciofi, G. Fllippmi, M. Sunonetta and MJ. Cormge, Nature
284 (1980)
153
475
79. number
3
LAmICE-DYNAMICAL FOR FLUORENE, Giuseppe
DIBENZOFURAN
FILIPPINI, Carlo M. GFWMACCIOLI
3 December
1 hlay 1981
CALCULATIONS
CARBAZOLE,
Ishtuto dr Chrmwz Fisraz e Centro CNR, Received
PHYSICS LETTERS
CHEhlICAL
1980;
AND DIBENZOTHIOPHEN
and Massimo SIMONETTA
Uiznwsztci dr Mhano. 20133 Mlan, Ital,
m fmal form 28 January
1981
For crystalbrie fluorene, carbazole, dibenzofuran and dibenzothlophen several propertles have been calculated usingsemiemplrlcal atom-atom potentrals Calculated geometrxal parameters. Raman-actlvc htt:ce frcquencles, cnthalples of subhmaUon and crystallographic temperature factors agree with observed values However. except for dlberuothlophen, mfraredactive httrce frequencies are not consistent ~th any potential so far tried
1. Introduction Calculation of external vlbratlonal modes of molecules m sohds has become a widely accepted procedure which brings about the posslbllity of prelctmg their structural, thermodynamic and spectroscopic properties [l-6] . While for hydrocarbons there 1s a choice of rehable atom-atom sem:-empmcal potential functions [S-7], for other molecules the sltuatlon IS more compiex, at least if vibratlonal propertles are taken mto account, besides the packmg energy. In the literature, examples of this tid are reported for a relatively limited number of substances, such as, for mstance. N, [8], 02 [9], fluorinated [IO] and chlonnated [I I] compounds. In order to test the vah&ty of these potentials, and in view of their unprovement and generahzation, we Table 1 Crystallographx
data of the selected Compound
considered a series of slmllar rlgtd aromatic molecbies wth different kmds of heteroatoms and no hydrogen bonds, for whch spectroscopic and crystallographic data are currently available. On this basis, we selected fIuorene, carbazole, dlbenzofuran and dlbenzottiophen as possible candidates. Although a general objection about testmg potentials wth such comphcated molecules might be raised, nevertheless these molecules do not differ considerably from the corresponchng hydrocarbons, for which good to excellent agreement between observed propertles and lattlce-dynamlcal results IS obtained. On the other hand, an efficient screemng of the potentials for the heteroatoms reported m ihe hterature IS achieved, to be used as a reference for future work m the field The first three compounds are practically Isomorphous, space group Pnma = Dig, whereas the fourth crystal-
substances
a
b
C
b W%)
Space
Z
Ref.
group
a)
fluorene
849
18 97
5 72
90
Pnma
4
[12]
carbazole dlbenzofuran
7 77 7.702
19.18 19 185
5 72 5 825
90 90
Pnma Pnma
4 4
1131 [14j=)
dibenzothiophen
867
6 00
W,
4
1151
18.70
113.9
/C
a)
The expenmental data are reported m terms of another unit cell (Pnam) where b and c are Interchanged
470
0 009-26
14/8 I /0000-0000/S
02.50
0 North-Holland F’ubhshmg Ccmparly
Volume 79,
number 3
CHEMICAL
PHYSICS
ties m a space group of lower symmetry P21/c = C5,. CrystaUographrc data are summarized r.n table 1_ For C-C, C-H and H-H non-bonded mteractrons, the potentrals grven by W~ihams [ 161 and Kitargorodskri [ 171 have been used: since no substantral drfferences have been observed between the frequencies calculated from these two sets, results are reported here for the former set only. For N-N the function given by Luty and Pawley [ 181 and for S-S the one given by Pawley and co-workers [ 191 were chosen_ For O-O four ddferent functrons have been euammed, namely the ones proposed by Grgho et al [20]. by Parsonage and , Pemberton [2 I ] , by Laufer and Leror [22] and by Krtargorodskn et al [23] . A set of calculatrons was also performed using the coefficients of the C-C potentials for the N-N, O-O and S-S contacts. Except for the case in whrch the whole set of Kitargorodskri functrons (KMN) was used, for mrxed mteractrons, C-X and H-X, coefficrents were always assumed to be the geometrrc or arithmetrc mean of the coefficrents relative to equal atoms, accordmg to the usual procedure. The C-H bond length was assumed to be 1 09 A and the N-H bond 1.03 A. Potential functions are collected m table 2.
LEI-I’ERS
1 May 1981
2 Method of calculation
The packmg energy and its derivatives with respect to molecular coordinates are considered to be a sum of independent terms, each one of them due to a van der Waals mteractron between two non-bonded atoms III drfferent molecules. The lattice sums are extended to a maxrmum distance of 15 A for the calculation of the packing energy and of 8 a for the calculation of lattree frequencies and of the temperature factors: this ensures convergence [S] . A first step involves mimrmzation of packing energy as a function of molecular coordmates for this purpose, a Raphson-Newton procedure 1s used. From thrs stage, we obtam a value of the packing energy and the drfference between experimental and calculated molecular orientation and posrtron m the urut cell: these data already grve an idea about the intrmsic validity of the assumed potentral function and of the proposed treatment A second step mvolves a Born-von K&man dynamreal treatment, which leads to the ergenvalue equation.
where o IS the frequency of a normal mode in crystal, M(q) IS the “mass adjusted” dynamrcal and t(q) are the “mass adjusted” displacement For 4 = 0, thrs treatment leads to mfrared- and
the matrix vectors. Raman-
Table 2 Potential functions Poten-
Contact
Energy (kcal/mole)
\vlllXIlns (WV.4)
C-C
E =
C-H
E=
8766 exp(-3
H-H
E=
2634 exp(-3.74r)-27.3/r6
N-N
E =
s-s
E = 199900 eup(-3
Luty-Pawley Rmall-Pawlay Cl&o
(LP) (RP)
et aL (GLM)
Parsonage-Pemberton
(PP)
83630 eup(-3 6@)-568/r6
26270 eup(-3
67r)-125/r6 40r)-452/r6 49r)-2149/r6
o-o
E = 186400 exp(-4Sr)-200/r6
o-o
E = 259000/r’*
- 358/r6 - 328/r6
Laufer-Lerol
(LL)
o-o
E = 233433/r’*
Kltagorodsku
et al. (KhIN)
c-c
E =
42000 exp(-3.5&)-358/r6
C-H
E=
42000 exp(-4_12r)-ls4/r6
H-H
E =
42000 exp(4_86r)-57/r’
o-o
E =
78000 exp(-4
l&)-259/@
c-o
E =
57000 exp(-3
85r)-313/r6
H-O
E =
49000 exp(4
SOr)-106/r6
471
CHEMICAL PHYSICSLETTERS
Volume 79. number 3
active lattice vlbratlon frequencies; thermodynamic functions and crystallographc temperature factors are obtained after an appropriate sampling of the Bnllouin zone [24].
3.Results and discussion For all considered compounds the &fference between the experrmental and the calculated molecular coordmates in the crystal is reasonably small, the maximum values being 0.12 A and 3” for translation and rotation, respectively. These values are as good as for hydrocarbons [7], and from this point of view all the potential functions here adopted appear to be equally satisfactory (see table 3).
1 May 1981
Srm~lar conclusions can be drawn wth respect to enthalpy of subhmation, at least m the two cases where measurements are avalable (see table 4). Expenmental and calculated lattice vlbratlon frequencles are reported m table 5. Smce the functions proposed by Laufer and Leroi and by Parsonage and Pemberton gwe unaginary frequencies even for CJ= 0, no results relatwe to these functions are reported m table 5, and they are omltted from further connderatlon. In order to help the reader m vlsuahzmg correspondence between these data, the percentage mutual agreement IS gven as R = 100 C Ivobs-vdl/Qobs It must be stressed, however, that smce a rlBd model has been used in these calculations, the highest calculated frequencies should III general exceed the correspondmg expermental values.
Table 3 Evaluation
of agreement
between
crystallographic
data and results
fluorene
carbazole dtbenzofuran
potential
functions -__
Transhnonal and rotaaonal displacements
(A) (deg)
RB b,
WWA
1
0001
0 005
- d 22
a)
WIVA + LP
2
0 11
0 34
all WIVA
1
0.12
0 17
19
WNA
+ GLM
3
0 06
2 05
WNA
+ PP
4
0 10
2 17
47 _ d)
WNA
+ LL
5
0 10
2 17
6
0 05
3 09
43
all WNA
1
0 06
240
22
WNAIRP
7
0 09
0.45
25
alI WNA
1
0 07
1 03 --
47
KhlN dlbenzothiophen
from various
Set
Potentlal functton
Compound
derived
_ d)
a) For explanation of the symbols see table 2 b)RB= 1OOxIB r,(obs) - ~~(ca~c)l/Zl~Ij(obs)I iS the perCentage agreement between calculated and experuncntal amsotropic temperature factors. ‘) Expenmental values are not reported III ref_ [ 121. d, Existence of lmatiary frequencies even for 4 = 0 does not allow calculations of temperature factors
Table 4 Compartson
between
the packmg
energy
Compound
a) AU Wdhams
472
NA
potentA
and the enthalpy -Epack
of subhmatlon
(kcal/mole)
-_
ms
fluorene
19 5 a)
18.3
carbazole
236a)
225 [261d),21.9 [271d),20_2 [281d,
functions.
b, !6 l-300%
[131 b), 19.8 [251 c,
c) 33-5o”c.
d, 244-352OC
crystallographic
2
(127)
98
70
bju
78“ I08
94
89
98
35 76 90 23 38 63
31 44 66 24 38 71
39 84
34
1
I301 2
I
21
I4
39 84
123 21
100
70 20
54
[31] a)
104
33
I10
84
73
53
96
75
60
69
58
55
85 _ b)
42
obs a) -_-___-___
Dlbenzofuran
3
22
16
us
14 32
98
12
22
77
48
26
99
1
23
I4
84
I7 35
104
23 83
75
49
21
107
35 48
34
41
46 111
41
10
104
83
25 27 86
cillc.
41
34
104
45
40
5
98
88 12
I9 27
CA.
---
6
26
I7
93
34
13
110
75
25
94
57
25
119
55
35
47
125
40 44
10
Ill
97 75
23 25
talc --_
bg bg bg bg bg bg
a6
“6
“g “6 “g a6
bU
bU
bU
bU
311
“U
313
QU
nU
-_____
1321
81 104
63
33
I05
79
49
31
171) 104
53
109
89
721
obs.
1
6
7
102
87
61
53
41
32
108
88
77
46
41
21
6
99
16
62
36
110
65 90
32 45
CIIC.
Dlbenzothlophcn --
I
14
12
85
75
60
26 41 43
92
14
40 67
19 37
15
80
64
31 58
94
15
21 45 64
cnlc.
a) The labelhng hns been changed accordmg to the new umt cell (see table I). b, In addltlon values of 101 ond 130 cm-’ for fluorcne, 103 and 129 cm? for carbazolc and 71 cm-’ for dlbcnzofuran arc rcportcd as bz,, lattlcc frcquenaes. these do not mntch our wlculatlons.
set of potentrals
[29] a)
21
24
R,,,(Z)
ref.
I4
26
103
28
28
122
70 20
58
88 I07
54
36
36 78
I38
137
52
54
40
38
I41
54
47
53
40
55
43
52?
28 57
88
37
53 141
RRaman(%)
b3E
b3is
bag
b2s
b2g
b2E
k
btg
“6 bU
“6
“6
R]R(%)
20
69
61
46
bxi
48
_b)
52
bzu
I7
_ b)
I23 17
blu bzu
122
90 18
100 _ b)
blu
100
98
49 104 - 1))
74
88
88
27 33
CZllC
56
CA.
28 33
obs.
25 54
Wk.
Carbazole
80
QU
“U
PU
obs a)
lIuorcne
Observedand culculatcd lattlcc frequenaes
5 _
i Q
i! i.z
z
3 !I
G 5? E ::
z>
.z
2
< L
Volume 79, number 3
CHEMICAL
wbratlon
The agreement is essentially good m the case of Raman spectra for aI.I molecules and also for IR measurements in &benzottiophen. For the IR spectra of the other compounds the agreement is less satsfactory, especially d the b,, lattice Table 6
1 May 1981
PHYSICS LETTERS
modes are considered
(see footnote
b of tab:e
5). Although such a sltuatlon might Involve a wrtual madequacy of these potentials to account for the vlbratlonal properties of these sohds, there are good reasons not to discard the possible
vahdlty
of our results
and
and calculated crystallographrc anlsotroplc temperature factors (all values are multlplled by 104)
Expenmentd
B 22
Bil
obs
CalC
obs
B 12
833
CalC
obs
CalC.
GllC.
obs
B23
f313
-
obs.
Gllc
obs.
talc
5
-11
carbazole a)
144 195
39
3s
232
297
C2 c3
134 141
45
37
217
221
34
32
c4
208
222
38
31
C.5
149
177
45
32
C6
105
131
38
30
N
179
162
47
41
290 455 373 242 229 231
404 499 458 337 277 288
339 488 647 586 404 316 318
Cl
-4
14
5
-14
-11
6
13
-29
-29
-13
-6
56
50
-11
-18
6
18
29
-4
5
2
-4
0
-4
0
-58
2
6
0
23
-3 9
10 8
5 -6
-22
-1 0
0
dtbenzofuran b,
C6
125
175
38
44
322 347 600 517 359 311
0
234
239
46
64
319
Cl
153
203
41
53
c2
196
281
52
58
c3
214
314
3s
48
c4
207
257
39
44
C5
165
236
4s
45
-17
-7
20
10
-1s
-20
1
22
-33
-47
-16
-16
86
73
-2s
-34
0
54
51
32
7
3
6
5
8
10
2
5 -1 -6
-1
0
0
28 -54
14
11
-9
-2s
-2 0
0
dibenzothlophen ‘) Cl
193
206
331
420
24
37
-28
-53
41
27
-11
C2
162
191
350
508
27
-20
-36
34
19
:
c3
184
179
330
457
34
37 43
51
24
14
c4
178
172
233
341
32
41
cs
183
181
250
311
27
37
C6
170
192
282
404
29
c7
157
166
370
431
31
C8
170
162
285
356
32
42
29
Cl0
179
174
256
306
24
35
-1
32 13
24
13
53
32
-11
47
3s
-10
-16
37
-54
-30
44
30
-16
-23
38
-26
-13
45
24
9
48
29
15
-29
44
32
18
1
7 25
-10
-3
Cl1
137
152
217
277
25
33
6
-11
43
33
Cl2
137
149
213
262
22
33
24
-13
36
34
Cl3
145
155
224
276
33
29
6
34
46
-1
S
197
208
246
294
33
46
5
48
37
-12
-13 18
a) Experimental values from ref [ 131, calculated from WIVA+i_.P potenti b, Experimenti values from ref. [ 141, calculated from all WIVA potentials. ‘1 Expenmental values from ref. [ 15 1, calculated from WIVA+RP potenmls.
474
21
-17
6 -3
-1 6 -9 4 -1 6 -20
Volume 79, number 3
CHEMICAL
PHYSICS
to consider instead the opportumty of reexammmg the mterpretation of the experrmer,tal IX data. In fact, It is difficult to image an essential failure III reproducing a whole series of IR-active vlbratlonal modes and in the same tune a substantial success m reproducmg the whole seyes of Raman modes. Moreover, these drfficul-
ties arise even for fluorene, I.e. 2 compound ~nth no heteroatoms, m contrast with the general agreement for hydrocarbons between calculated and experimental lattice frequencies [7] . Even for IR frequencies a good agreement is achieved for hbenzothiophen, whose crystal structure IS different from the ones of the other three compounds Therefore, some parallel msmterpretation of spectra might have occurred. Comparison of anisotroplc crystallographuz temperature factors with experimental values, as given by X-ray diffraction, shows an overall satisfactory agreement for all the substances (see table 6). for some potentials the R mdex (see table 3) 1s ~20% and this 1s w~thm the actual accuracy of the experimental data. Furthermore, it must be pomted out that calculabon of temperature factors was possible since no imagmary frequencies appeared throughout the Brillouin zone: a test of ths kind has always proved to be quite important for venfymg the valilty of any potenti function [33 1. It seems that better results are obtained for dbenzofuran if the potential proper to the heteroatom is replaced by a “common” C-C potentialthis might mean that the van der Waals interaction of the oxygen atom enclosed in these rings is tiferent from the corresponding mteraction in the substances from which these potentials were derived. In conclusion, our calculations show that an overall good agreement with experimental data 1s obtained, except for some [R-active vlbratlonal modes: however the calculated frequencies are at the limit or beyond the range of observation and future experunental work seems to be advisable.
References I1 J C-S. Pawley. Phys. Stat. Sol 20 (1967) 347. [21 G.S Pawley, Phys. Stat. Sol. B49 (1972) 475. [3] S. Llfson and A. Warshel. J. Chem whys. 49 (1968) 5116.
1 hiay 1981
LE’l-fERS
141 A Warshel and S. Llfson, J Chem. Phys 53 (1970) 582. 151 G Fdippuu, CM Gramaccloh. M. Sunonetta and G B. Suffrlttl,Chem. Phys 8 (1975) 136. hf Slmonetta and G-B J6J G Fd~ppuu, C_M Gramaccroli. Suffrith. Chem. Phys. Letters 35 (1975) 17. hf. Sunonetta and G B. 171 G. Fibppim. CM. Gramaccloli, Suffntti, J. Chem. Phys. 59 (1973) 5088 J81 G Fti~ppuu. CM Gramacciolr. hl Sunonetta and G B Suffritu, hfol. Phys. 35 (1978) 1659. and references
‘herein
hl L. Klem and V. Chandrasekharan, J Chem. phys 71 (1979) 843. hf. Slmonetta. G-B Suf1101 G Fd~ppin~,C hf. Cramaccloh, J91 K. Kobashi,
fn’nttland 0. Sala, Chem. Phys. Letters 39 (1976) 14. [I11 L -Y Hsu and D E. Wdbarns. Acta Cryst. A36 (1980) 277. Jl21 D.hf Bums and J. Iball. Proc. Roy. Sot. 227A (1955) 200. 1131 hf. Kurahashl, M Fukuo, A. Shunada. A Furusakr and I. Nitta, Bull. Chem. Sot. Japan 42 (1969) 2174. J14J 0 Dldeberg, L DuPont and J-M. AnclrC, Acta Cryst. 828 (1972) 1002. 1151 R hf Schaffrm and 3. Trotter, J. Chem Sot. A (1970) 1563_ 1161 D E. Wilhams, J. Chem Phys 47 (1967) 4680 J. Chun. Phys. 63 (1966) 6. Jl71 A I fitayorodsku, iI81 T Luty and G S. Pawley. Chem Phys Letters 28 (1974) 593. 1191 R P. Rinald~ and G.S. Pawley. Nuovo Cuncnto 16B (1973) 55; J Kunttu and G S. Pawiey. ActaCryst A29 (1973) 615. 1201 E. Gigbo. A hf. LIquorI and L. hfazzarella, Lettere Nuovo Cunento 1 (1969) 135 [21 J N.G. Parsonage and R C Pemberton. Trans Faraday Sot 63 (1967) 311 [22J J C Laufer and G E Lero!, J Chem. Phys. 55 (1971) 993 [23] A.1 Kltigorodsku, K.V. Muskaya and V.V. Nauchitel’, Knstallografiya 14 (1969) 900. [24 J G. Fihppuu, CM. Gramaccioh, M. Sunonetta and G.B. Suffrltti. Acta Cryst. A32 (1976) 259. [251 R.S Bradley and T-G Cleasby, J. Chem Sot (1953) 1690 126 J International Cntical Tables 3 (1928) 226,5 (1929) 134. [27J F.S. Mortimer and D.V. Murphy, Ind. Eng. them. 15 (1923) 1140. 1281 A Aihara, J. Cbem Sot. Japan 76 (1955) 492. 1291 A. Bree and R. Zwarich. J. Chem. Phys 51 (1969) 912 [30] A. Bree and R. Zwanch, J. Chem Phys. 49 (1968) 3344. 131 J A- Bree. V-V B. VI&OS and R. Zwanch, J. Mol. Spectiy. 48 (1973) 124.
1321 A. Bree and R. Zwarich, Spectrochun. Acta 27~ (1971) 599. J33J S. Ram&~. GM. Parkmson, J.M. Thomas, C M. Gramacciofi, G. Fllippmi, M. Sunonetta and MJ. Cormge, Nature
284 (1980)
153
475