- Email: [email protected]
Quadrupole moment calculations for some aromatic hydrocarbons
Volume
78, number
CHEMICAL
3
QUADRUPOLE A. CHABLO, Department Recewed
MOMENT CALCULATIONS
D.W.J. CRUICKSHANK,
of ClremarrJ:
24 November
UMST, Mm&ester
1980:
PHYSICS
15 March 1981
LETTERS
FOR SOME AROMATIC
A. HiNCHLIFFE
HYDROCARBONS
and R.W. MUNN
Al60 I QD. UK
m final form 18 December
1980
Quadrupole tensor calculatmns are reported for benzene, naphthalene, anthrdcenc, phenanthrene, biphenyl (planar and twisted), p> rene, [ 18]-annulene dnd azulene. Double-zeta and STOJ4G basis sets give similar ratios between naphthalene and benzene components. ST0/4G calculations _giveout-of-phne components proportional to the number of valence elec-
trons for nll molecules except azulene.
l_ Introduction
Crystals of the aromatrc hydrocarbons have been of the class of molecular crystals. The crystals can be obtained m reasonably pure, perfect and stable forms, while the constituent molecules are amenable to simple molecular orbital treatments. Detailed i-rformatron is avarlable on the structure, vibrations, spectroscopic and electrical properties of the crystals, and there has been considerable progress in interpreting these properties in terms of molecular properties and interactions. One important interaction is the classical electrostatic interaction between the molecular charge distributions. The leading contribution to this interaction is given by the lowesr non-vanishing permanent electric multipole moment of the molecule_ The most intensively studied aromatic hydrocarbons, notably naphthalene and anthracene, have a symmetry which precludes a permanent dipole moment, so that their quadrupole moments are of particular interest_ Lattice-energy calculations for benzene [1] and naphthalene [2] show that the point quadrupole-quadrupole interaction is significant but not dominant in determining the structure. However, the addition of point quadrupole-quadrupole interactions to atom-atom potentials in calculations of the lattice dynamics of these two crystals significantly improves the agreement between theory and experiment, giving for the first time the correct ordering of the highest 4 and widely studied as represcntatrves
424
B, modes in naphthalene [3]. Similar evidence for the importance of the quadrupole moment is discussed elsewhere in relation to structure [4] and dynamics [s] _ A different quadrupole contribution arises in the interaction with an excess charge in a crystal which helps to determine earner conductton and trapping energy levels [6].
However, experimental information on the quadrupole moments is scanty. Benzene is the most favourable aromatic hydrocarbon because of its volatility and its high symmetry, which means there is only one independent non-zero component of the traceless quadrupole moment tensor in the molecular axes. Values of this component derived form various experiments [3,4,7-121 are shown in table 1. As can be seen, the values vary by a factor of four, but most are deduced rather indirectly from experiment_ The recent direct measurements from electric-field-gradient induced birefringence [4,12] now seem to have largely resolved these inconsistencies_ (The earlier fieldgradient birefringence measurement [8] omitted certam corrections [4] _) The two measurements differ by slightly more than their combined estimated uncertainties, and for present purposes we prefer the gasphase result [ 121, which has a smaiIer uncertainty and fewer problems in ehminating the effects of intermolecular interactions than the result from solution [4]. There appear to be no experimental values of the quadrupole moment for polycyclic aromatic hydrocarbons, except for a recent value for the outof-plane
0 OOP-2614/81/0000-0000/$02.50
0 North-Holland
Publishing Company
Table 1 Axial component ONA~ (in 104’ C m’) of the quadrupole tensor for benzene. For comparison, the atomic unit e=g (ao is the Bohr radius) is 4.48659 X 104’ C m*, and the electrostatic unit 1O-26 esu cm* (1 buckimham) is 3.33564 X 1040 Cm* Ref.
Value
-10.4 -42*4
[7]
240 -19k9.5 -21*11‘ 1 -28+11 -27.0 -33.3c2.1 -29.oc1.7
-30.7 -32.8 -34.9
[ll] I:; [I21
[141 1
-29.7 -13.6 -32.2
-30.9 -27.2 -31.5 I -28.6 -13.5 -31.9 -29.5
component gradient
181 191 1101
I151 I161 1171 1181 this work
Experrmental microwave line broadening field+radient biref-ence second virial coefficient comparison with CsFs microwave line broadenin3 (different methods) fit to lattice vibrations fieldsradient birefrmsence field+radrent blrefringencc
obtained
from
in solution
@MM~@LL _____
-51.0 -52.4 -48.0 -20.7 -48.9 -4525 (solution) i&s)
Theoretrcal semi-empirical amsotropic ST0/4G basis similar extended basis polarized lame basis isotropic ST0/4G basis anisotropic STO/4G basis isotropic DZ basis amsotropic DZ basrs polarized DZ bans many body isotropic ST0/4G basis isotropic DZ basis 4-21G basis
birefringence
Table 2 Quadrupole tensor components 0~ for naphthalene calculated in the molecular axes: the out-of-plane component 0~1 the m-plane ratio @mf/@~~, and the ratio R = ONN(naphth lene)/ONN(benzene) in each method (see table 1) WIV (IPa C m*)
Method
in naphthalene, induced
15 March 1981
CHEMICAL PHYSICS LETTERS
Volume 78, number 3
field[ 131.
Various theoretical mformation on the quadrupole moments is available. For benzene there have been semr-empirical MO calculations [ 141 and various ab initio studies [ I5-181. These results are quoted in table 1, where it can be seen that all except the simple ST0/4G basis set calculations agree well with the experimental value [4]_ For naphthalene the quadrupole tensor has been calculated by the same semi-empirical method [14] as for benzene; from the point-charge distribution given by floating spherical gaussian orbitals (FSGOs) deduced from ab initio molecular fragment calculations [3];and from a fit to lattice vibrational frequencies assuming the FSGO anisotropy [3].
a) Assumed
1 1.242 (1.242) 1.169 1.123 -
R
-~
a)
Method
1.667 1.778 1.531 1.532 1.4 -e 0.2
semi-empirical [ 143 FSGO [3] fit to lattice vibrations [ ;] STO/4G (this work) DZ (this work) fieldaradient birefrmgence (solution) [ 131
from FSGO calculatron.
quadrupole moment tensor for benzene, naphthalene, anthracene, phenanthrene, biphenyl (planar and twisted), pyrene, [ 18]-annulene and azulene. Moleculat wavefunctions are taken from ab initio SCF MO calculations using both a double-zeta (DZ) quahty basis set [ 191 (for benzene and naphthalene [20] only and an isotropic ST0/4G minimal basis set widely use1 for ab initio calculations on large molecules [21] for all the molecules). The DZ basis set should yield an accurate quadrupole moment, as table 1 indicates, while the smaller more compact isotropic ST0/4G basis set underestimates the quad,upole moment. However, the relative magnitudes of the quadrupole moments for different molecules and the anisotropy of the quadrupole moment tensor for each molecule should depend less strongly on the basis set. By performing calculations with the two basis sets for benzene and naphthalene we are able to test these expectations. It appears that the less time-consuming STO/IG calculations are adequate for predicting the relativevalues of quadrupole moments of aromatic hydrocarbons_ In combination with an experimental value for benzene, such calculations then predict absolute values.
2. Method
These results are summarized in table 2 with the ex-
perimental value 1131. We have now calculated the components
of the
All geometries were idealized, the rings treated as regular with R(C-C) = 140 pm and R(C-II) = 109 pm 425
CHEMICAL PHYSICS LETI’ERS
Volume 78, number 3
In twisted biphenyl the angle of twist was taken as 8 = 4.P. The components of the quadrupole tensor were calcuIated as
- (3rArB - z 26,1
,
(1)
1 where A and B denote Cartesian components_ Here e is the proton charge, Z,,e is the charge on nucleus IZ at position r(rz), and the last term is an average over the electron distribution. All the molecules have symmetry high enough to fu the principal axes of the quadrupole moment ten-
sor along the molecular axes. For the planar molecules, the longer m-plane axis is labelled L, the shorter
(medium) in-plane axis M, and the normal axis AT.In twisted biphenyl, the twist axis is labelled L and the other axis in the plane bisecting *Ahesmaller dihedral angle is labelled M. Except for phenanthrene and azulene, the molecules have no permanent dipole moment, so that the quadrupole tensor is independent of the choice of origin. For phenanthrene and azulene the origin is chosen as the centre of dipole, midway between the centres of positive and negative charge in each molecule. The centre of dipole for phenanthrene lies 209 pm from the midpoint of the C~--C,O bond, in the direction towards the opposite side of the central ring; that for azulene lies 55 pm from the midpoint of the shared bond, in the seven-membered ring. The dipole moments calculated from our STO/ 4G results are 0.13 X lo-30 C m (0.04 D) for phenanthrene and 7.62 X IO-30 C m (2.28 D) for azulene, so that the origin dependence is very weak for phenantb_rene but very strong for azulene. However, the C, symmetry of these molecules makes the combmation - ONN for @LL - ONN for phenanthrene and O,,, azulene independent of origin.
15 March 1981
quite with that in the gas phase [ 12]_ Table 1 also shows our value of ONN calculated for benzene with a 4-21 G basis set [22] of intermediate size. As expected, the greater flexibility in the valence orbitals of this basis set compared with the ST0/4G basis set brings the calculated value of ONN much closer to the expenmental one, if not quite into agreement with it. However, since the 4-21G basis set offers neither the economy in computing time of the STO/ 4G basis set nor the flexibility and accuracy of the DZ basis set, we have not used it for quadrupole moment calculations on the larger molecules. Our results for naphthalene are gwen in table 2. The two basis sets yield similar ratios OM,,~/@~~ and almost exactly the same value for R, the value of
O,, relative to that for benzene calculated from the same basis set. There is also broad agreement with the FSCO results [3] _The DZ value of OA-N agrees with the experimental one [13]. It appears that the STO/4G calculations give about the right anisotropy, and will give the correct magnitude if scaled to the experimental value for benzene. The results obtained with the STOJ4G basis for naphthalene, anthracene, phenanthrene, biphenyl, [18]-annulene and azulene are shown in table 3. Each component is scaled by the factor which makes the value of ONN calculated for benzene with this basis set agree with the gas-phase experimental value 1121, as suggested by the naphthalene results. For naphthalene, ONN then agrees with the solution value [I 31. The variation in the components from molecule to
Table 3 Quadrupole tensor components 0~ calculated III the molecular a_?,eswith the ST0/4G basis. Each value is scaled by the factor 2.15 = O~fl(exp)/ONN(cPc) for benzene 111this basis Molecule
om --
3. Results and discussion The axial component ONN of the quadrupole tensor for benzene obtained from our two basis sets is given in table l_ The values agree with those obtamed from comparable basis sets, and the DZ value agrees with the experimental value from solution [4], if not 426
naphthalene antbracene phenanthrene biphenyl (0 = 0”) (e = 45”) pyrene [ 18 ]-annulene azulene
(10 4o C m’)
LL
MM
NN
20.5 26.5 27.9 20.5 22.1 33.1 39.4 -4.7
23.9 34.7 32.7 33.1 19.6 35.4 39.4 40.5
-44.4 -61.1 -60.6 -53.6 -41.7 -68.5 -78.7 -35.8
15 hiarch 1981
CHEMICAL PHYSICS LETTERS
Volume 78, number 3
molecule clearly reflects the molecular geometry and the number of atoms. For example, in going from naphthalene to anthracene the magnitude and anisotropy of the components both increase, while in going from anthracene to phenanthrene the anisotropy decreases with the magnitude hardly changed_ Twrsting biphenyl reduces the magnitude and anisotropy of the quadrupole moment tensor as the deviation from spherical symmetry is reduced. in [ 181 -anuuIene, %L and %fif are not required to be equal by symmetry, but differ only in the fifth significant figure. The results for azulene are less readily interpreted and are in any case very sensitive to the position of the origin, as noted above, which in turn depends on the caIculated nuclear and electronic contributions to the dipole moment _ For a charge distribution made up of a superposition of spherical neutral atomic distributions, the quadrupole moment is zero, since a sphere contnbutes zero relative to rts nucleus as origin and transformation to the molecular origin contributes only through the atomic dipole, which is again zero. The components of the quadrupole tensor thus reflect the rather subtle changes in charge distribution caused by bonding. For example, the negative sign of O,, for benzene can be ascribed to a net movement of electron density towards the centre of the ring [4]. Similar redistribution makes O,, negative in all the other molecules studied here. Redistribution in the carbon core electrons is small, so these changes affect mairJy the valence electrons_ Table 4 shows the ratio of @NATto the number of valence electrons V for the planar molecules. As can be seen, -ONA,/ vlies in the range (0.92 f 0.05) X 10e40 C m?- for all the molecules except azulene. The Table 4 Out-of-plane components O,vlv, scaled as m table 3, dtvided by the number of valence electrons V Molecule
--OAq#
benzene naphthalene anthracene phenanthrene biphenyl(0 = 0) pyrene [ 18]-annulene azulene
0.968 0.926 0.927 0.917 0.923 0.925 0.874 0.746
C m*)
deviation for azulene may be partly attributable to uncertainty in the location of the centre of dipole, but azulene is also the only non-altemant hydrocarbon in the set, and calculations show a large transfer of negative charge into the five-membered ring [23] (hence the large dipole moment). Such more extensive charge redistribution is consistent with a different pattern of components. in any case, the large dipole moment means that uncertainties in the quadrupole moment are of little prectical significance. We conclude that for planar alternant aromatic hydrocarbons, Of,r~ should be given within -10% by -0.92 VX lo-40 C rnz. Given an estimate of the anisotropy in the molecular plane, this conclusion would suffice to provide reasonable estimates of the quadrupole tensors for such molecules_ Our calculations provide a coherent picture of the quadrupole moments in these aromatic hydrocarbon molecules_ The results will be of use in electrostatic theories for charge-carrier transport in the crystals [S] and perhaps in explaining the phase transitions in phenanthrene [24] and biphenyl [25] _ Such calculations can more generally assist in deriving reliable potential functions for crystals, as already shown in benzene and naphthalene [3] and in heteroaromatic crystals [s] .
Acknowledgement We are grateful to Dr. S.H. Walmsley for helpful discussicns and correspondence.
References
t13 D.P. Craig, R. Mason, P. Patding and D.P. Santry, Proc.
Roy. Sot. A286 (1965) 98. D.P. Craig, P.A. Dobosh, R. Mason and D.P. Santry, Discussions Faraday Sot. 40 (1965) 121. 131 S. Cal&no, R. Rrghini and S-H. Walmsley, Chem. Phys. Letters 64 (1979) 491. 143 J. Vrbancich and G.L.D. Ritchie, J. Chem. Sot. Faraday II 76 (1980) 648. 151 Z. Gamba and H_ Bonadeo, Chem. Phys. Letters 69
PI
(1980)
525.
161 F. Gutmann and L.E. Lyons, Organic semiconductors
(Wiley, New York, 1967); R.W. hlunn, Mot Cryst. Liquid Cryst. 3 1 ( 1975) 105; P.J. Bounds and R.W. Munn, 9th Molecular Crystal Symposium, Kleinwakertal, Austria (1980).
427
Volume [7] [S] [9] [lo] [ 1 I] [12]
[ 131 [ 141 [15] [ 161
428
78, number
3
CHEMICAL
PHYSICS
R.M. Hrll and W.V. Smith, Phks. Rev. 82 (1951) 451; W-V. Smith, J. Chem. Phys. 25 (1956) 510. S. Golub, Ph.D. Thesis, Columbia University (1968). T.H. Spurhng and A.G. de Rocco, J. Chem. Phys. 49 (1968) 2867. R.L. Shoemaker and W.H. Flygare, J. Chem. Phys. 51 (1969) 2988. G.K. Johri, V. Prakash and S.L. Srivastava, Indian J. Pure Appl. Phys. 14 (1976) 417. J.H. Williams, Ph.D. Thesis, Cambridge University (1980), M.R Battaglia, A.D. Buckingham and H.J. Williams, Chem. Phys. Letters 000 (1981) 421. R.L. Calvert and G.L.D. Ritchie, J. Chem. Sot. Faraday II 76 (1980) 1249. A. Schweig, Mol. Phys. 14 (1968) 533. J.M. Schulman, C.J. Homback and J.W. Moskowrtz, Chem. Phys. Letters 8 (1971) 361. J. Almlof, B. Roos, U. Wahlgren and H. Johansen, J. Electron Spectry_ 2 (1973) 51.
LETTERS
15 March
1981
[17]
‘WC. Ermler and C.W. Kern, J. Chem. Phys. 58 (1973) 3458. [ 181 \V. von Niessen, L.S. Cederbaum and 1V.P. Kraemer, 3. Chem. Phys. 6.5 (1976) 1378. [ 191 L.C. Snyder and H. Basch, Molecular wavefunctions and [20] [21] [22] [23] [24] [25]
properties (Wiley-Interscience, New York, 1972). A. Hinchliffe, J. Chem. Sot. Faraday II 73 (1977) 1627. W.J. Hehre, R Ditchfield, R.F. Stewart and J.A. Pople, J. Chem. Phys. 51 (1969) 1657. P. Pulay, G. Fogarasi, F. Pang and J.E. Boggs, J. Am. Chem. Sot. lOl(l979) 2550. R.J. Buenker and SD. Peyerimhoff, Chem. Phys. Letters 3 (1969) 37. L.J. Soltzberg, P.A. Piliero and M.R. Shea, hlol. Cryst. Liquid Cryst. 29 (1974) 151. H. Cailleau, J.L. Baudour, J. hleinnel, A. Dworkm, f. Moussa and C.M.E. Zeyen, Faraday Discussions Chem. Sot. 69 (19801, to be published.
78, number
CHEMICAL
3
QUADRUPOLE A. CHABLO, Department Recewed
MOMENT CALCULATIONS
D.W.J. CRUICKSHANK,
of ClremarrJ:
24 November
UMST, Mm&ester
1980:
PHYSICS
15 March 1981
LETTERS
FOR SOME AROMATIC
A. HiNCHLIFFE
HYDROCARBONS
and R.W. MUNN
Al60 I QD. UK
m final form 18 December
1980
Quadrupole tensor calculatmns are reported for benzene, naphthalene, anthrdcenc, phenanthrene, biphenyl (planar and twisted), p> rene, [ 18]-annulene dnd azulene. Double-zeta and STOJ4G basis sets give similar ratios between naphthalene and benzene components. ST0/4G calculations _giveout-of-phne components proportional to the number of valence elec-
trons for nll molecules except azulene.
l_ Introduction
Crystals of the aromatrc hydrocarbons have been of the class of molecular crystals. The crystals can be obtained m reasonably pure, perfect and stable forms, while the constituent molecules are amenable to simple molecular orbital treatments. Detailed i-rformatron is avarlable on the structure, vibrations, spectroscopic and electrical properties of the crystals, and there has been considerable progress in interpreting these properties in terms of molecular properties and interactions. One important interaction is the classical electrostatic interaction between the molecular charge distributions. The leading contribution to this interaction is given by the lowesr non-vanishing permanent electric multipole moment of the molecule_ The most intensively studied aromatic hydrocarbons, notably naphthalene and anthracene, have a symmetry which precludes a permanent dipole moment, so that their quadrupole moments are of particular interest_ Lattice-energy calculations for benzene [1] and naphthalene [2] show that the point quadrupole-quadrupole interaction is significant but not dominant in determining the structure. However, the addition of point quadrupole-quadrupole interactions to atom-atom potentials in calculations of the lattice dynamics of these two crystals significantly improves the agreement between theory and experiment, giving for the first time the correct ordering of the highest 4 and widely studied as represcntatrves
424
B, modes in naphthalene [3]. Similar evidence for the importance of the quadrupole moment is discussed elsewhere in relation to structure [4] and dynamics [s] _ A different quadrupole contribution arises in the interaction with an excess charge in a crystal which helps to determine earner conductton and trapping energy levels [6].
However, experimental information on the quadrupole moments is scanty. Benzene is the most favourable aromatic hydrocarbon because of its volatility and its high symmetry, which means there is only one independent non-zero component of the traceless quadrupole moment tensor in the molecular axes. Values of this component derived form various experiments [3,4,7-121 are shown in table 1. As can be seen, the values vary by a factor of four, but most are deduced rather indirectly from experiment_ The recent direct measurements from electric-field-gradient induced birefringence [4,12] now seem to have largely resolved these inconsistencies_ (The earlier fieldgradient birefringence measurement [8] omitted certam corrections [4] _) The two measurements differ by slightly more than their combined estimated uncertainties, and for present purposes we prefer the gasphase result [ 121, which has a smaiIer uncertainty and fewer problems in ehminating the effects of intermolecular interactions than the result from solution [4]. There appear to be no experimental values of the quadrupole moment for polycyclic aromatic hydrocarbons, except for a recent value for the outof-plane
0 OOP-2614/81/0000-0000/$02.50
0 North-Holland
Publishing Company
Table 1 Axial component ONA~ (in 104’ C m’) of the quadrupole tensor for benzene. For comparison, the atomic unit e=g (ao is the Bohr radius) is 4.48659 X 104’ C m*, and the electrostatic unit 1O-26 esu cm* (1 buckimham) is 3.33564 X 1040 Cm* Ref.
Value
-10.4 -42*4
[7]
240 -19k9.5 -21*11‘ 1 -28+11 -27.0 -33.3c2.1 -29.oc1.7
-30.7 -32.8 -34.9
[ll] I:; [I21
[141 1
-29.7 -13.6 -32.2
-30.9 -27.2 -31.5 I -28.6 -13.5 -31.9 -29.5
component gradient
181 191 1101
I151 I161 1171 1181 this work
Experrmental microwave line broadening field+radient biref-ence second virial coefficient comparison with CsFs microwave line broadenin3 (different methods) fit to lattice vibrations fieldsradient birefrmsence field+radrent blrefringencc
obtained
from
in solution
@MM~@LL _____
-51.0 -52.4 -48.0 -20.7 -48.9 -4525 (solution) i&s)
Theoretrcal semi-empirical amsotropic ST0/4G basis similar extended basis polarized lame basis isotropic ST0/4G basis anisotropic STO/4G basis isotropic DZ basis amsotropic DZ basrs polarized DZ bans many body isotropic ST0/4G basis isotropic DZ basis 4-21G basis
birefringence
Table 2 Quadrupole tensor components 0~ for naphthalene calculated in the molecular axes: the out-of-plane component 0~1 the m-plane ratio @mf/@~~, and the ratio R = ONN(naphth lene)/ONN(benzene) in each method (see table 1) WIV (IPa C m*)
Method
in naphthalene, induced
15 March 1981
CHEMICAL PHYSICS LETTERS
Volume 78, number 3
field[ 131.
Various theoretical mformation on the quadrupole moments is available. For benzene there have been semr-empirical MO calculations [ 141 and various ab initio studies [ I5-181. These results are quoted in table 1, where it can be seen that all except the simple ST0/4G basis set calculations agree well with the experimental value [4]_ For naphthalene the quadrupole tensor has been calculated by the same semi-empirical method [14] as for benzene; from the point-charge distribution given by floating spherical gaussian orbitals (FSGOs) deduced from ab initio molecular fragment calculations [3];and from a fit to lattice vibrational frequencies assuming the FSGO anisotropy [3].
a) Assumed
1 1.242 (1.242) 1.169 1.123 -
R
-~
a)
Method
1.667 1.778 1.531 1.532 1.4 -e 0.2
semi-empirical [ 143 FSGO [3] fit to lattice vibrations [ ;] STO/4G (this work) DZ (this work) fieldaradient birefrmgence (solution) [ 131
from FSGO calculatron.
quadrupole moment tensor for benzene, naphthalene, anthracene, phenanthrene, biphenyl (planar and twisted), pyrene, [ 18]-annulene and azulene. Moleculat wavefunctions are taken from ab initio SCF MO calculations using both a double-zeta (DZ) quahty basis set [ 191 (for benzene and naphthalene [20] only and an isotropic ST0/4G minimal basis set widely use1 for ab initio calculations on large molecules [21] for all the molecules). The DZ basis set should yield an accurate quadrupole moment, as table 1 indicates, while the smaller more compact isotropic ST0/4G basis set underestimates the quad,upole moment. However, the relative magnitudes of the quadrupole moments for different molecules and the anisotropy of the quadrupole moment tensor for each molecule should depend less strongly on the basis set. By performing calculations with the two basis sets for benzene and naphthalene we are able to test these expectations. It appears that the less time-consuming STO/IG calculations are adequate for predicting the relativevalues of quadrupole moments of aromatic hydrocarbons_ In combination with an experimental value for benzene, such calculations then predict absolute values.
2. Method
These results are summarized in table 2 with the ex-
perimental value 1131. We have now calculated the components
of the
All geometries were idealized, the rings treated as regular with R(C-C) = 140 pm and R(C-II) = 109 pm 425
CHEMICAL PHYSICS LETI’ERS
Volume 78, number 3
In twisted biphenyl the angle of twist was taken as 8 = 4.P. The components of the quadrupole tensor were calcuIated as
- (3rArB - z 26,1
,
(1)
1 where A and B denote Cartesian components_ Here e is the proton charge, Z,,e is the charge on nucleus IZ at position r(rz), and the last term is an average over the electron distribution. All the molecules have symmetry high enough to fu the principal axes of the quadrupole moment ten-
sor along the molecular axes. For the planar molecules, the longer m-plane axis is labelled L, the shorter
(medium) in-plane axis M, and the normal axis AT.In twisted biphenyl, the twist axis is labelled L and the other axis in the plane bisecting *Ahesmaller dihedral angle is labelled M. Except for phenanthrene and azulene, the molecules have no permanent dipole moment, so that the quadrupole tensor is independent of the choice of origin. For phenanthrene and azulene the origin is chosen as the centre of dipole, midway between the centres of positive and negative charge in each molecule. The centre of dipole for phenanthrene lies 209 pm from the midpoint of the C~--C,O bond, in the direction towards the opposite side of the central ring; that for azulene lies 55 pm from the midpoint of the shared bond, in the seven-membered ring. The dipole moments calculated from our STO/ 4G results are 0.13 X lo-30 C m (0.04 D) for phenanthrene and 7.62 X IO-30 C m (2.28 D) for azulene, so that the origin dependence is very weak for phenantb_rene but very strong for azulene. However, the C, symmetry of these molecules makes the combmation - ONN for @LL - ONN for phenanthrene and O,,, azulene independent of origin.
15 March 1981
quite with that in the gas phase [ 12]_ Table 1 also shows our value of ONN calculated for benzene with a 4-21 G basis set [22] of intermediate size. As expected, the greater flexibility in the valence orbitals of this basis set compared with the ST0/4G basis set brings the calculated value of ONN much closer to the expenmental one, if not quite into agreement with it. However, since the 4-21G basis set offers neither the economy in computing time of the STO/ 4G basis set nor the flexibility and accuracy of the DZ basis set, we have not used it for quadrupole moment calculations on the larger molecules. Our results for naphthalene are gwen in table 2. The two basis sets yield similar ratios OM,,~/@~~ and almost exactly the same value for R, the value of
O,, relative to that for benzene calculated from the same basis set. There is also broad agreement with the FSCO results [3] _The DZ value of OA-N agrees with the experimental one [13]. It appears that the STO/4G calculations give about the right anisotropy, and will give the correct magnitude if scaled to the experimental value for benzene. The results obtained with the STOJ4G basis for naphthalene, anthracene, phenanthrene, biphenyl, [18]-annulene and azulene are shown in table 3. Each component is scaled by the factor which makes the value of ONN calculated for benzene with this basis set agree with the gas-phase experimental value 1121, as suggested by the naphthalene results. For naphthalene, ONN then agrees with the solution value [I 31. The variation in the components from molecule to
Table 3 Quadrupole tensor components 0~ calculated III the molecular a_?,eswith the ST0/4G basis. Each value is scaled by the factor 2.15 = O~fl(exp)/ONN(cPc) for benzene 111this basis Molecule
om --
3. Results and discussion The axial component ONN of the quadrupole tensor for benzene obtained from our two basis sets is given in table l_ The values agree with those obtamed from comparable basis sets, and the DZ value agrees with the experimental value from solution [4], if not 426
naphthalene antbracene phenanthrene biphenyl (0 = 0”) (e = 45”) pyrene [ 18 ]-annulene azulene
(10 4o C m’)
LL
MM
NN
20.5 26.5 27.9 20.5 22.1 33.1 39.4 -4.7
23.9 34.7 32.7 33.1 19.6 35.4 39.4 40.5
-44.4 -61.1 -60.6 -53.6 -41.7 -68.5 -78.7 -35.8
15 hiarch 1981
CHEMICAL PHYSICS LETTERS
Volume 78, number 3
molecule clearly reflects the molecular geometry and the number of atoms. For example, in going from naphthalene to anthracene the magnitude and anisotropy of the components both increase, while in going from anthracene to phenanthrene the anisotropy decreases with the magnitude hardly changed_ Twrsting biphenyl reduces the magnitude and anisotropy of the quadrupole moment tensor as the deviation from spherical symmetry is reduced. in [ 181 -anuuIene, %L and %fif are not required to be equal by symmetry, but differ only in the fifth significant figure. The results for azulene are less readily interpreted and are in any case very sensitive to the position of the origin, as noted above, which in turn depends on the caIculated nuclear and electronic contributions to the dipole moment _ For a charge distribution made up of a superposition of spherical neutral atomic distributions, the quadrupole moment is zero, since a sphere contnbutes zero relative to rts nucleus as origin and transformation to the molecular origin contributes only through the atomic dipole, which is again zero. The components of the quadrupole tensor thus reflect the rather subtle changes in charge distribution caused by bonding. For example, the negative sign of O,, for benzene can be ascribed to a net movement of electron density towards the centre of the ring [4]. Similar redistribution makes O,, negative in all the other molecules studied here. Redistribution in the carbon core electrons is small, so these changes affect mairJy the valence electrons_ Table 4 shows the ratio of @NATto the number of valence electrons V for the planar molecules. As can be seen, -ONA,/ vlies in the range (0.92 f 0.05) X 10e40 C m?- for all the molecules except azulene. The Table 4 Out-of-plane components O,vlv, scaled as m table 3, dtvided by the number of valence electrons V Molecule
--OAq#
benzene naphthalene anthracene phenanthrene biphenyl(0 = 0) pyrene [ 18]-annulene azulene
0.968 0.926 0.927 0.917 0.923 0.925 0.874 0.746
C m*)
deviation for azulene may be partly attributable to uncertainty in the location of the centre of dipole, but azulene is also the only non-altemant hydrocarbon in the set, and calculations show a large transfer of negative charge into the five-membered ring [23] (hence the large dipole moment). Such more extensive charge redistribution is consistent with a different pattern of components. in any case, the large dipole moment means that uncertainties in the quadrupole moment are of little prectical significance. We conclude that for planar alternant aromatic hydrocarbons, Of,r~ should be given within -10% by -0.92 VX lo-40 C rnz. Given an estimate of the anisotropy in the molecular plane, this conclusion would suffice to provide reasonable estimates of the quadrupole tensors for such molecules_ Our calculations provide a coherent picture of the quadrupole moments in these aromatic hydrocarbon molecules_ The results will be of use in electrostatic theories for charge-carrier transport in the crystals [S] and perhaps in explaining the phase transitions in phenanthrene [24] and biphenyl [25] _ Such calculations can more generally assist in deriving reliable potential functions for crystals, as already shown in benzene and naphthalene [3] and in heteroaromatic crystals [s] .
Acknowledgement We are grateful to Dr. S.H. Walmsley for helpful discussicns and correspondence.
References
t13 D.P. Craig, R. Mason, P. Patding and D.P. Santry, Proc.
Roy. Sot. A286 (1965) 98. D.P. Craig, P.A. Dobosh, R. Mason and D.P. Santry, Discussions Faraday Sot. 40 (1965) 121. 131 S. Cal&no, R. Rrghini and S-H. Walmsley, Chem. Phys. Letters 64 (1979) 491. 143 J. Vrbancich and G.L.D. Ritchie, J. Chem. Sot. Faraday II 76 (1980) 648. 151 Z. Gamba and H_ Bonadeo, Chem. Phys. Letters 69
PI
(1980)
525.
161 F. Gutmann and L.E. Lyons, Organic semiconductors
(Wiley, New York, 1967); R.W. hlunn, Mot Cryst. Liquid Cryst. 3 1 ( 1975) 105; P.J. Bounds and R.W. Munn, 9th Molecular Crystal Symposium, Kleinwakertal, Austria (1980).
427
Volume [7] [S] [9] [lo] [ 1 I] [12]
[ 131 [ 141 [15] [ 161
428
78, number
3
CHEMICAL
PHYSICS
R.M. Hrll and W.V. Smith, Phks. Rev. 82 (1951) 451; W-V. Smith, J. Chem. Phys. 25 (1956) 510. S. Golub, Ph.D. Thesis, Columbia University (1968). T.H. Spurhng and A.G. de Rocco, J. Chem. Phys. 49 (1968) 2867. R.L. Shoemaker and W.H. Flygare, J. Chem. Phys. 51 (1969) 2988. G.K. Johri, V. Prakash and S.L. Srivastava, Indian J. Pure Appl. Phys. 14 (1976) 417. J.H. Williams, Ph.D. Thesis, Cambridge University (1980), M.R Battaglia, A.D. Buckingham and H.J. Williams, Chem. Phys. Letters 000 (1981) 421. R.L. Calvert and G.L.D. Ritchie, J. Chem. Sot. Faraday II 76 (1980) 1249. A. Schweig, Mol. Phys. 14 (1968) 533. J.M. Schulman, C.J. Homback and J.W. Moskowrtz, Chem. Phys. Letters 8 (1971) 361. J. Almlof, B. Roos, U. Wahlgren and H. Johansen, J. Electron Spectry_ 2 (1973) 51.
LETTERS
15 March
1981
[17]
‘WC. Ermler and C.W. Kern, J. Chem. Phys. 58 (1973) 3458. [ 181 \V. von Niessen, L.S. Cederbaum and 1V.P. Kraemer, 3. Chem. Phys. 6.5 (1976) 1378. [ 191 L.C. Snyder and H. Basch, Molecular wavefunctions and [20] [21] [22] [23] [24] [25]
properties (Wiley-Interscience, New York, 1972). A. Hinchliffe, J. Chem. Sot. Faraday II 73 (1977) 1627. W.J. Hehre, R Ditchfield, R.F. Stewart and J.A. Pople, J. Chem. Phys. 51 (1969) 1657. P. Pulay, G. Fogarasi, F. Pang and J.E. Boggs, J. Am. Chem. Sot. lOl(l979) 2550. R.J. Buenker and SD. Peyerimhoff, Chem. Phys. Letters 3 (1969) 37. L.J. Soltzberg, P.A. Piliero and M.R. Shea, hlol. Cryst. Liquid Cryst. 29 (1974) 151. H. Cailleau, J.L. Baudour, J. hleinnel, A. Dworkm, f. Moussa and C.M.E. Zeyen, Faraday Discussions Chem. Sot. 69 (19801, to be published.