J. Phys. Chem. 1994, 98, 10054-10062
10054
Combining Molecular Mechanics with Quantum Treatments for Large Conjugated Hydrocarbons. 1. A Geometry-Dependent Hiickel Hamiltonian Gabin Treboux I.R.S.A.M.C., Laboratoire de Physique Quantique, Unit6 associke au C.N.R.S. (U.A. 505), Universitk Paul Sabatier, 31 062 Toulouse Cedex, France Received: April 18, 1994; In Final Form: July 12, 1994@
A geometry-dependent Huckel Hamiltonian is defined for conjugated hydrocarbons. The model leads to n electron energies that can be coupled with the classical additive u potential. The use of analytical expressions for the first and second derivatives of the energy allows us to address the potential hypersurfaces and vibrational properties of large conjugated hydrocarbons with reduced computational effort. The method is tested on various conjugated systems ranging from linear to polycyclic hydrocarbons. The geometries and vibrational spectra are in good agreement with experimental data, which opens the field of interpretation and anticipation of vibrational spectroscopy of very large conjugated systems such as polycondensed aromatic hydrocarbons.
I. Introduction: General Strategy and Possible Options Molecular mechanics (MM) has proved to be an outstanding instrument for the study of the geometrical structures, energies, and vibrational properties of rather large molecules (see for instance ref 1). Its success is based on both its computational efficiency and its accuracy. The low computation cost is due to the simplicity of the potential energy expression, as sum of two-body (and eventually a few three-body) potentials. Since these potentials are analytical, the first and second derivatives are easily calculated, facilitating the research of equilibrium geometries and vibrational frequencies. The MM accuracy is limited to the use of empirically fitted two-body potentials, which incorporate the experimental information regarding bond distances, bond angles, and force constants. Reaching such a precision from ab-initio calculations, despite significant progress on the algorithms and in computers (for a striking illustration of the improved efficiency of ab-initio calculations, see for instance the performances of the TURBOMOLE package*), is hopeless for moderately large molecules since it would require both large basis sets and huge configuration interactions calculations. The theoretical foundation of molecular mechanics, which assumes additivity of bond energies, rest of the localizability of the wave function into localized electron pairs located in either bond orbitals or lone pairs. The localizability is known to be good for nonconjugated and ionic or covalent crystals, but it will fail when at least a subset of electrons is intrinsically strongly delocalized, as occurs in metals and in conjugated molecules for n electrons. This problem is well-known by practitioners of MM, who have proposed to modify the effective potential V, between two bonded conjugated atoms r and s as a function of the n electron density on that bond; more precisely, the V, potential becomes a function of the corresponding bond index P,:
where P , is obtained from a self-consistent quantum mechanical calculation restricted to the n system. This is, in particular, the solution adopted by Allinger et a L 3 s 4 g 5 in their chain of MM, (n I3) programs. Of course when P , = 1, for instance, the @
Abstract published in Advance ACS Abstracts, September 1, 1994.
0022-365419412098-10054$04.50/0
V,, potential is the one of ethylene; it becomes that of a single bond when P , = 0. The use of a self-consistent procedure to calculate P , leads to two difficulties; it may face well-known symmetry-breakingphenomena of the self-consistentfield (SCF) solution (as occurs for instance in ClsHlg6) and makes laborious the energy calculation and the geometry optimization due to the iterative scheme of the SCF algorithm. To speed up the process, some authors' have replaced R, in eq 1 by the Mulliken overlap population calculated from an extended Hiickel calculation. This solution is somewhat paradoxical since it forgets the u electron localizability and introduces the u contribution to the overlap population. One may wonder why simple n-Hiickel bond indexes have not been employed in eq 1. This strategy is limited to the ground state; it can treat neither ionized molecular states nor excited states since the ionization or excitation of a conjugated system is intrinsically delocalized. This is a strong limitation and forbids the use of this approach for spectroscopic purposes. Another direction has been proposed by Karplus et al.,8 who suggested benefiting from the well-known u-n separability and writing
E = E, iE,
(2)
where Eo is additive (according to the localizability of the u electrons),
E, =
Vij ij
-
and E, is calculated through a n-only Hamiltonian. Since the n* electronic target of these authors essentially was the n excitations, they used Pariser-Parr-Pople (PPP) type Hamiltonians and SCF truncated configuration interactions (CI). These calculations are, of course, quite heavy and can only concern the spectroscopy of small conjugated molecules. In this series of works, of which the present paper is the first member, the Karplus strategy will be adopted, namely, a partition of the total energy into an additive u part and a quantum mechanically calculated n part, using the simplest evaluation of the n energy from the exact solutions of explicit Hamiltonians. We would like, therefore, to avoid self-consistent procedures such as SCF steps. The driving ideas are here both computational and physical, namely, (1) an appropriate geometry0 1994 American Chemical Society
J. Phys. Chem., Vol. 98, No. 40, 1994 10055
Huckel Hamiltonian for Conjugated Hydrocarbons dependent Hamiltonian for each electronic situation (neutral fundamental state, Huckel (HU); neutral (in the VB sense) excited state, Heisenberg (HEI)) and (2) classic modeling for the relocalizable part of the wave function which is common to all Hamiltonians (MM). In the present paper we shall study the ground state of neutral molecules using a geometry-dependent Huckel Hamiltonian, which is defined in the next section. We will try to show the pertinence of the Huckel molecular mechanics scheme (the corresponding results will be labeled hereafter as HUMM) and the accuracy of its results for both potential energy surface studiesgJOand R or Raman spectra. A forthcoming paper will give some results on monocation systems concerning ionization potentials, Jahn-Teller distortions, and relaxation energies, which surprisingly were found to be very accurate.
+
E, =
nici i
hf#li= €if$i (ni is the occupation number of the molecular orbital r#~iin the considered state). The Huckel-type Hamiltonian h is a tight-binding Hamiltonian, which only introduces interactions between chemically bonded atoms. In this sense, it reflects the topology, but the amplitude of the interaction will depend on the distance between adjacent atoms and the torsion around conjugated bonds. This Huckel Hamiltonian, expressed in second quantization formalism, has the following form:
where Naphtalcne
Anthracene
trans-Stilbene
d
The off-diagonal term Pls depends on the rotation wrs around the bond (more precisely o, is the angle between the two ?G atomic orbitals) as cis-Stilbene
% trans-Butadiene
Biphenyl
Azulenc
w trans-1,3,5, Hexatricne
and trans-1,3,5,7, Octatretraene
(5) Coronene
The HUMM model would be useless for the study of the excited states of neutral systems since it does not discriminate between triplet and singlet states. For these problems, and more precisely for the neutral states (in the VB sense) of conjugated molecules, we can use the nonempirical magnetic (or Heisenberg) Hamiltonian, derived some years ago by Said et al.," that proved to be efficient in the calculation of structural and spectroscopicproperties. This model, which uses eigensolutions of an analytical Hamiltonian, is well suited for ground states, triplet states, and some low-lying singlet states. The ionic states (in the VB sense), for which the transition from the ground state is dipolarly allowed, are not accessible from the present Hamiltonian, and excitonic models should be preferred in this case. At a further level of sophistication the corresponding cations will be addressed through an effective Hamiltonian which generalizes the Heisenberg Hamiltonians in order to treat the delocalization of a hole in a half-filled band. This Hamiltonian has been popularized under the name of the t-J model in the field of high-Tc superconductors. It has been proposed in an r-independent scheme by Gadea et al.,12 and a geometry-dependent version should be easy to derive, as well.
11. Geometry-Dependent Hiickel Hamiltonian and the Modification of the u Potential Let us start from eq 2, in which En is defined as a sum of one-electron energies obtained as the eigenvalues of a oneelectron Hamiltonian, h:
with 1, being the corresponding bond length and lo the typical CC single-bond length. The quadratic term reflects the the extent of bond alternation; then lo' is kept at a nonaltemated mean distance of 1.399 A. The six parameters COO, p, p, lo, lo', and KO) may be determined by fitting the following properties: (i) C=C bond lengths in ethylene, benzene, and butadiene; (ii) vibrational frequency AI, (breathing) of benzene; (iii) C=C torsion vibrational frequency in ethylene. Without interactions between nonbonded atoms, our formalism would make possible a direct analytical deduction of these properties, leading to a unique set of these six parameters. However, we are taking into account the interations between nonbonded atoms, and consequently one has more parameters and the system is no longer exactly solvable. We solved this difficulty by taking the interactions between nonbonded atoms (repulsion dispersion) as identical to those proposed by Allinger et ai. in the MM2-875 version of his algorithm and by subsequently modifying our previously obtained six parameters in order to get the same agreement with the experimental values. Because our target is related to spectroscopic properties, the u part of the force field was defined from available spectroscopy data. In this way, the stretch potential of C-H VCH= &(I - ro)2 is determined by a best of the C-H bond length of ethylene and the C-H vibrational frequency (A,) of that molecule. The bending potentials have the following form:
V,, = Krs&12(1
+ K,AO4)
KCCHis extracted from CCH bending (Azu) of benzene since
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10056 J. Phys. Chem., VoI. 98, No. 40, 1994 this vibrational Frequency is independent of CCC bending potential. KCW is next determined by a best fit of the B,. CH2 bending vibrational frequency of ethylene, and Kccc is extracted from the CCC bending (E$ of benzene. Let us now address the crucial problem of the energy change occumng upon rotation around a multiple CC bond. The loss of conjugation will be described by the dependence as cos w of the ,!?integral. This is why the u-related parameters are weaker here than in the other non quantum formalism programs. Three rotational barriers are required to fit the potentials which take into account the u energy during the twisting of any C% double bond. These potentials have the following form:
Vikji= Kv&2 cos2 w
TABLE 1: HUMM Pnnuoetcr Set for Conjugated Hydrocarbons a Parameters swetching parameters
1.265 0.58
C-H
bending parameters Kim
CCC CCH HCH
- I)
where w is the dihedral angle ijkl (structure I).
1.5215 1.101
(hameelmi’)
(deg)
0.0219
120.0 120.0, 120.5 119.0
0.01IO 0.0088
torsion and dihedralization parameters Kdj
Krm (hameelmol)
(hameel.&’)
cccc
o.ooo50 o.ooo10 O.WO45
HCCC HCCH HorC
1
K H Cis~fitted from the ethylene rotational barrier. KCCCCis fitted from the 1-2 biphenyl rotational barrier (structure 2a). K m H is next determined by fitting the 1-2 rotational barrier of butadiene (structure 2b).
W
E
2a
0.14
n Parameters -0.315 h W -0.0338 h W 0.201586 hameel.& 0.0155 hartree/A2
a0 Bo
B’
Y
If one accepts the third definition of cos w , the n energy will only contribute to the A. mode, which physically leads to the nullification of the following variations of the n overlap (structures 5a and 5b).
2b
Concerning the definition of the dihedral angle for torsions, three definitions are possible, namely,
cosw=
(7,
*
A
(7,
rabr, r, sin e,
A
sin
5.
7,)
e,,
The second definition of cos w implies a weak influence of
9, variation on the BI. mode and a zero effect for the Bzg mode
often used, for instance, in MM. algorithms, casu=-
-n i n-z
n1n2 where iil is the normal vector to AEC plane and i i 2 the normal vector to the BDF plane, and
--
-
where ki = i i i
A
-BC.
casu=-
5b
kik2 klk2
(Le. nullifies the variation of t h e n overlap shown in Structure 5b). The first definition of cos w implies a strong contribution ! lvariation on the Bzg mode and a zero effect on the BI. of , mode (i.e. nullifies the variation of the n overlap shown in structure 5a). In the HUMM package, the second definition has been selected, so that we must add a penalty potential in order to calculate the Bzg distortion (structure 5b) in a realistic way. In chemical language, we take into account the dihedralization effect (structure 6).
C
6 These three definitions are not equivalent, as can be seen for the three out of plane vibrational modes of ethylene.
4.
B,, CH, bending
4b
B,&CH
c
4, C% torsion
This penalty potential is defined as Vdi = &As, where Ar is the distance between C and the ABD plane and Kdi is chosen as a best fit of the out of plane vibrational frequencies of ethylene. All parameters of the HUMM package are summarized in Table 1.
m.
Computational Method
Geometry Optimization. The geometry optimization is performed according to a steepest descent procedure making
J. Phys. Chem., Vol. 98, No. 40, 1994 10057
Huckel Hamiltonian for Conjugated Hydrocarbons use of the analytical derivatives. The derivation of the Q energy is straightforward. The derivative of the n energy with respect to an 1, bond length or an angle w is given by
TABLE 2: CC Bond Lengths and Bond Angled of Conjugated Hydrocarbons comwund ethylene4 benzeneb naphthaleneC
where P, = &niCi,Ci, is the classical bond index for the bond rs. The second derivative of the x energy are calculated through a second-order perturbative development. The displacement along X of atom i may be associated to a local perturbation operator V,i, implying the modification of a few p integrals, and
a2E,
axiavi=
C
anthracened
Ek
d e
trans-stilbene'
d e f g
h
abc cis-stilbenef
c=c,,
a b ab Ph-b 1-1314-7 biphenyl twistedg a b C
biphenyl plan&
IV. Results
d cdc' a b C
A. Ground State Geometries. Table 2 gives a series of results conceming linear and polycyclic molecules, altemant or nonaltemant, together with experimental values and the results of the MM3 calculation. For the CC bond length in altemant molecules, the standard deviation with respect to the experimental electron diffraction (ED) values happens to be quite comparable in the present work and in MM35 (namely, 0.004 8, for a series of 13 different bonds). Comparison with X-ray data is not appropriate since constraints of crystal packing are not included in our model. Nevertheless for altemant molecules, one can check that our simple quantum treatment for n energy does not include larger errors than the SCF treatment does in the MM3 method. On the other hand, for nonaltemant systems (see for instance azulene) our results are poorer, certainly due to the appearance of net charges and electrostatic effects, neglected in our model. Table 3 shows that calculation of bond alternation in linear polyenes is difficult to obtain whatever the model. For instance, both the C1C2 bond length in rrans-hexatriene and the CC bond length of ethylene are 1.337 A, which is even beyond a DCI level. Nevertheless, bond alternation in large polyenes is qualitatively reproduced by our model (for instance, minimal bond alternation in Cle is Ar = 0.006 8, in comparison to 0.007 for the same system and experimental data for infinite polyene Ar = 0.008 8,). Table 4 shows the evaluation of the CC bond length in polyaromatic species. The analytical form of our model enables us to calculate the CC bond length in graphite. In this case the total energy can be approximated for one unit cell by E = 3K0(r - rJ2
a b C
- El*
Constrained Minima. Routines of geometry optimization under constraint (for instance N fixed values of dihedral angles) are also available and have been used in electronic molecular studie~.~J~ Vibrational Spectrum. The vibrational spectrum is calculated in the frame of the harmonic approximation proposed by Wilson, Decius, and Cross.13 A Williams projection14has been used in order to eliminate translation and rotation contributions in the Cartesian Hessian matrix (which may perturb significantly the value of the lower vibrational frequencies in large molecule~).~~ The code has been entirely home written, but the implementation of our Huckel n energy in standard molecular mechanics programs would certainly be of interest as well.
d a b C
(a)klVX,la)l*)(Q)l*Iv~ila),d
CkCl*
bond a a a b
+ 2(a + p) 4- 1.1498
referring to a set of two n electrons, three o CC bonds, and the
d azulene' b C
d e f
HUMM
MMI
exDtl
1.337 1.399 1.420 1.385 1.424 1.416 1.426 1.379 1.432 1.429 1.410 1.365 1.470 1.413 1.397 1.400 1.399 1.397 1.410 10.8 1.400 1.492 1.355 123.3 40.3 5.8
1.337 1.397 1.421 1.374 1.429 1.412 1.433 1.367 1.441 1.424 1.404 1.355 1.477 1.410 1.395 1.397 1.394 1.397 1.407 1.7 1.400 1.479 1.350 125.9 37.3 6.7
1.337 F 0.001 1.399 'f 0.001 1.417 F 0.004 1.381 F 0.002 1.422 T 0.003 1.412 T 0.008 1.421 1.370 1.440 1.428 1.405 1.338 1.473 1.406 1.393 1.394 1.391 1.390 1.402 5.2 1.397 T 0.003 1.488 'f 0.002 1.338 F 0.006 129.5 'f 0.3 43.9 T 1.3
method
1.399 1.398 1.407 1.494 44.1 1.396 1.398 1.415 1.489 1.399 1.408 1.409 1.400 1.409 1.446
1.396 1.396 1.403 1.488 46.2 1.393 1.396 1.412 1.497 1.401 1.409 1.395 1.396 1.399 1.472
1.396 T 0.008 ED(r,) 1.395 'f 0.006 1.403 T 0.004 1.503 T 0.004 44.2 7 1.2 1.385 X-ray, room 1.388 temp 1.397 1.496 T 0.003 1.399 F 0.009 ED(r,) 1.418 'f 0.010 1.383 F 0.008 1.406 F 0.016 1.403 70.014 1.501 T 0.005
Reference 16. Reference 17. Reference 18. Reference 19. e Reference 20. f Reference 21. Reference 22. * Reference 23. Reference 24. j Bond lengths in angstroms and angles in degrees. corresponding delocalization energy. The condition 6El6r = 0 leads to r = 1.4223 8, for graphite, which is in good agreement with experimental data. In conclusion, our model happens to give sufficiently correct geometries for altemant conjugated hydrocarbons (linear, branched, or polycyclic) with a standard deviation lower than 0.01 A for CC bond lengths. The low CPU requirement of our package allows us to address systems that are much larger (by 1 or 2 orders of magnitude) than those treated by current abinitio calculations. Although our purpose is not to give, here, a quantitative comparison between HUMM and ab-initio calculations or between HUMM and MM3 calculations, we would indicate that, in the typical case of coronene, both the ground state geometry and vibraitonal properties are obtained within 40 s of CPU time on a HP720 CRX workstation (17.9 MFlops). B. Out of Plane Deformations, Isomerization Energies, and Rotational Barriers. In our model, the problem of o-n interaction during out of plane deformation is taken into account by three parameters only: KCCCC, KCCCH, and KHCCH.In spite of such simplicity, the results are quite satisfactory. In biphenyl, for instance, KCCCCwas fitted to the experimental value of Emin - E900 and not to the @-@ dihedral angle (@ labels the
10058 J. Phys. Chem., Vol. 98, No. 40, 1994
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TABLE 3: CC Bond Lengths of Polyenes PPP compound butadiene
bond a b a b
DCIa HUMM 1.346 1.349 1.466 1.463 truns-l,3,5-hexatriene 1.347 1.352 1.462 1.456 C 1.359 1.364 trans-l,3,5,7-octatetraene a 1.347 1.353 b 1.461 1.454 C 1.361 1.368 d 1.457 1.448 Reference 8. Reference 5. Reference 26. Reference 27.e Reference 28.
TABLE 4: Structure of Benzene, Coronene, and Graphite compound bond HUMM exptl method benzene a 1.399 1.399f 0.001 ED(r,)" coronene a 1.418 1.438 ED(rgIb 1.381 b 1.402 C 1.437 1.444 d 1.392 1.382 a 1.430 X-rayc 1.430 b C 1.415 1.385 d graphite a 1.422 X-ray 1.421 Reference 17. Reference 29. Reference 30.
TABLE 5: Relative Energies of Conformations (kcaVmo1) compound conformation HUMM other calc exptl biphenyl twist 0 0" ob planar 2.8 2.52 2.0 1.o 1.1 1.o 90' stilbene trans 0 0' od cis 3.7 5.7 3.0 Reference 5. Reference 31. Reference 32. Experimental estimation, ref 33. TABLE 6: Rotational Barriers around Double and Single Bonds p,p+l of Even Polyenes ( k d m o l ) compound bond HUMM other calc" ethylene c1c2 62.5 62.5 52.2 butadiene c1c2 52.2 6.0 5.4 c2c3 47.7 48.0 hexatriene ClCZ 6.8 6.0 cZc3 41.4 42.5 c3c4 45.3 45.7 octatetraene ClC2 7.0 6.2 cZc3 37.1 38.5 c3c4 7.8 6.8 c4c5 Reference 1 1. benzyl group), and a good value was obtained for both this angle (see Table 2) and (E- - EO.) (see Table 5 ) . This efficiency is seems to extend further on since, for stilbene, correctly evaluated as well as the Q-C-C dihedral angle in the cis configuration (see Table 5). On the other hand, the Q-C=C dihedral angle in the trans configuration is too weak regarding its experimental value, but the latter has recently been q~estioned.~~ Concerning the rotational barrier around double bonds in general, our results are close to those calculated from a Heisenberg Hamiltonian some years ago" (see Table 6). These quantities are correctly predicted with our model, which encouraged us to study various molecular switches built from extended conjugated systems.10 C. Ground State Vibrational Spectroscopy. Tables 8- 13 compare our results with experimental data and other calcula-
exutl 1.3447 0.001 1.467F 0.001 1.337F 0.002 1.458'f 0.002 1.368F 0.004 1.336 1.451 1.327 1.451
MM3b 1.344 1.468 1.345 1.466 1.353
exutl method ED(reY ED(rdd X-ray'
TABLE 7: Relativ Corrections Due to CC-CC Interactions (mdyd ) symmetry classic formalism FI1 = ma If Elu C 0.0 Eze 3c 0.81 Bzu 4c 3.66 Reference 34.* Reference 35,
1
F;.P exptl
0.257 0.927 3.66
TABLE 8: Fundamental Frequencies for Benzene (in cm-l) 5 QCFF/ MO/ SCF symmetry exptl" paramb PIc 8STd SQMe HUMM MMY In Plane Frequencies 3073 3030 3091 3058 3095 3065 3058 993 794 1046 997 983 976 911 1350 1283 1389 1352 1365 1330 1368 3056 3037 3089 3055 3061 3052 3043 1599 1571 1614 1605 1607 1595 1637 1178 1048 1149 1168 1183 1178 1185 606 646 665 604 607 596 611 3057 3042 3093 3057 3051 3051 3038 1010 1123 1068 968 997 999 947 1309 1520 1460 1306 1297 1336 1657 1146 1078 1158 1183 1162 1048 1249 3064 3031 3087 3055 3080 3059 3051 1482 1402 1502 1467 1482 1468 1482 1037 890 1046 1026 1036 982 999 Out of Plane Frequencies 990 996 967 1119 707 701 639 581 846 843 852 773 667 673 729 569 967 969 965 1030 361 402 398 375 Reference 41. Reference 42.' Reference 43. Reference 34. e Reference 37.f Reference 5. tions for the vibrational spectroscopy. The most sophisticated methods such as ab-initio CI, ab-initio scale factor, and even PPP DCI rapidly become CPU time consuming so that coronene, for instance, is hardly manageable. The only methods which actually permit one to address the spectroscopy of extended systems are molecular mechanics methods such as MM3. We would mention especially Ohno's package,34which obtains excellent results but deals with in plane spectroscopy only. The formalism used in the quantum part of our modeling happens to be quite close to the quantum part of Ohno's method, except that our bond indexes are now geometry dependent.
+
+
E, = 2 C P g i I
J. Phys. Chem., Vol. 98, No. 40, 1994 10059
Huckel Hamiltonian for Conjugated Hydrocarbons
TABLE 9: Fundamental Frequencies for Naphthalene (in cm-I) symmetry exptlb SCF SQM” HUMM In Plane Frequencies 3085 3063 3060 sn, 3055 sd, 3056 3052 3031 sn, 3004 sd, 3025 m 1590 1581 1578 sn 1458 1455 1460 sn 1385 1342 1380g 1170 1195 1144 sd, 1163 sd 1023 1025 1020 g 757 783 761 g 5 14 505 514 g 3056 3067 3092sn,3055sd 3050 3047 3006 sd, 3060 sn, 2980 m, 3055 sd 1631 1644 1624 av 1468 1458 1436 av, 1445 sd, 1458 sd 1259 1255 1240 av 1117 1156 1099 g, 1145 sn, 1158 sn, 1168 sd 938 940 939 g 542 512 508 g 3058 3070 3065g, 3056 sn 3050 3058 g, 3055 sd, 3029 sn 3049 1584 1595 1595 g 1383 1389 g 1391 1271 1272 1265 g 1136 1137 1125 g 810 748 g, 810 sd 792 436 354 359 g 3062 3083 3090 g, 3056 sn 305 1 3052 3027 sn, 3005 sn, 3029 sn 1538 1515 1509g 1360 1341 1361g 1193 1204 1209 g 1021 1158 1144 sd, 1163 sn 995 1003 1008 g 671 619 g 626 Out of Plane Frequencies 981 955 825 853 540 622 163 188 964 969 955 g 797 777 780 g 459 480 474 g 172 166 g 172 939 951 sd 952 710 705 620 g, 717 sn 352 385 g 387 983 sd 966 987 884 879 875 av 764 773 465 g, 772 sd 47 1 442 395 g, 465 sd, 470 sd Reference 45. Reference 45, see therein. For experimental values, sd, sn, m, and g indicate solid, solution, melt, and gas; av indicates a phase-averaged value.
(7) occ un
=2 ~ ~ ( c & + c c&$la)[(ckccld & k
+ c&lc)/(Ek
-
l
the bond indices i a n d j refer to atomic pairs a-b and c-d, respectively. Pi and ?lijare the JT bond order and the bondbond polarizability. The main difference between our model and the MM3 algorithm is that in the latter the quantum information is directly included in the K,, constant. Consequently, first and second
TABLE 10: Fundamental Frequencies for Coronene (in cm-’1 symmetry exptl‘ QCFFPI” M0/8STb HUMM In Plane Frequencies 3059 3057 1590 1600 1603, 1604 1607 1299 1364 1432 1354,1370 1234 1199 1216 1090 1050 1080 511 45 1 513 470,488 3059 3057 305 1 3055 1608 1625 1622,1636 1695 1442 1473 1439,1457 1498 1446 1431,1444 1437 1486 1397 1401,1410 1452 1384 1203 1218 1224, 1236 1256 1134 1149 1154, 1166 1159 1022 992,998 1062 997 698 715 667 534 586 484 568 354 384 363,371 371 3058 3054 3057 3049 3045 3054 1620 1610 1612 1644 1519 1525 1500 1518 1385 1410 1494 1449 1321 1314 1390 1320 1205 1209 1192 1133 1160 1141 1125 1154 832 811 867 811 814 843 774 769,772 423 387 379 406 1577 1541 1555 1237 1287 1240 955 996 937 717 747 677 3049 3055 1586 1563 1636 1426 1435 1387 1217 1282 1199 707 659 681 600 598 56 1 3050 3056 1481 1478 1591 1167 1274 1175 1013 1150 1126 507 520 483 Out of Plane Frequencies 958 810 805 594 220 145 941 865 541 450 124 947 851 621 408 285 955 823 738 513 284 76 Reference 48. Reference 34. Reference 34, see therein. derivatives are totally defined in a classical formalism. Quite in contrast, our model makes use of both a classical and a quantum formalism for the successive derivatives. Let us
10060 J. Phys. Chem., Vol. 98, No. 40, 1994
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TABLE 11: 1,3-trans-Butadiene(CU)Vibrational Frequencies (in cm-l) symmetry exptl" PPPDCIb MM3' HUMM In Plane Frequencies 3102 3080 3108 Ag 3101 3043 3060 3014 3063 3014 2987 3010 2994 1643 1701 1596 1631 1474 1448 1442 1451 1279 1303 1213 1187 1166 1209 1185 1205 890 873 883 899 513 532 507 574 Bu 3102 3094 3108 3101 3056 3063 3037 3049 3009 2994 3010 2987 1557 1593 1569 1599 1385 1394 1396 1352 1296 1289 1253 1250 981 1033 1028 991 364 397 301 343 Out of Plane Frequencies Au 1013 1057 1101 1015 908 947 955 922 524 535 508 542 163 181 164 157 Bg 967 980 1090 963 911 931 956 927 753 664 661 707 Reference 46. Reference 8. Reference 5. exemplify this fundamental difference on a cyclic chain of six vibrators (this may be related to the C=C stretches in benzene). The general expression of the Hessian matrix F is imposed by the symmetry
d
o
m
p
m
o
o
d
o
m
p
m
m
o
d
o
m
p
p
m
o
d
o
m
m
p
m
o
d
o
o m p m o d with d = a - (20 2m p ) and o = -Fv ortho, m = -Fij metal, and p = -FQ para. In the classical formalism, and within the tight binding approximation, a and o should account for the force constants of the vibrators and the coupling constant between linked vibrators, so that the matrix is written
+
a - 20 0
+
0
0
a-20 0 o a-20 0
0
a-20 0
0
a-20
eigenvalues is necessarily AI, < El,, < Ezg < Bz,. This happens to be in contradiction with the benzene C=C stretch spectrum: AI, < B2,, < El, < Ezg. This is why M M 3 leads to an unrealistic value for the B2, mode, as illustrated in Table 8. Note that, beyond the tight binding approximation, to reproduce the exact spectrum of benzene, the third neighbor should be given a coupling larger than that of the first neighbor, which is incompatible with the chemical assignment of various potentials in classical force fields. In the HUMM model, due to 0-n separation, and because of the very weak value of pl, parameters (see Table l),we can take the quantum part of Fo as equal to bond-bond polarizations nij (see eq 7 ) . It is interesting to study the relation between nv and F p The values of ni, vary according to the method, but for benzene they must always verify the following equation:
0
a-20 where the eigenvalues are respectively a (AI&, a - o (El,,), a - 30 (Ezg), and a - 40 (Bz,,). In the quantum model, the parameters 0,m, and p depend on the method used, but even within the tight binding approximation, all terms 0,m and p are defined according to eq 6, and the correspondingeigenvalues are a (Alg), a - o - 3m - 2p (El,,), a - 30 - 3m (E2& and u - 40 - 2p (Bzu). In the classical model, one may fit a and o to experimental 0
TABLE 12: Fundamental Frequencies of Hexatriene (in cm-') symmetry exptla PPP DCIb HUMM In Plane Frequencies A, 3085 3091 3101 3039 3078 3064 3039 3063 3048 2989 2986 2994 1582 1623 1619 1547 1573 1577 1394 1413 1388 1280 1350 1267 1238 1296 1193 1187 1193 1108 963 897 934 444 457 489 422 347 385 3091 3098 3101 B" 3040 3082 3056 3012 3063 3041 2994 3012 2986 1623 1618 1575 1429 1443 1452 1294 1303 1317 1255 1272 1207 1130 1180 1163 941 973 947 635 540 578 209 174 Out of Plane Frequencies 1017 1011 1060 941 965 945 899 923 923 658 619 646 243 210 106 88 990 1021 985 928 946 924 897 843 855 594 601 228 223 a References 47, 48. Reference 8.
0
modes such as AI, and El, in order to fix the C-C stretch spectrum. However, whatever the fitting the ordering of the
3n,
+ no+ 2np = 0
The physical significance of this equation is that the El, eigenvalues are insensitive to bond-bond polarization. Fd current experimental values leads to 3nm no 2np = 0.257 for the correction of the El, eigenvalues. Most importantly, the intensity of corrections for the different symmetries now makes possible the evaluation of the nij-Fij overlap (see Table 7). Bond-bond polarization seems to be the main quantity to explain the range of the different stretch C=C modes. In benzene each IR band appears once in the n system, and one can find the molecular orbital directly from the symmetry.
+ +
J. Phys. Chem., Vol. 98, No. 40, 1994 10061
Huckel Hamiltonian for Conjugated Hydrocarbons
TABLE 13: Fundamental Frequencies of Octatetraene (in cm-l) symmetry A,
B U
a
exptl" PPP DCIb In Plane Frequencies 3090 3095 3005 3083 3005 3078 3063 3005 2986 3005 1612,1613 1617 1604 1608,1613 1439 1432, 1423 1299 1367 1313 1304,1291 1291 1285,1281 1187, 1179 1190 1183 1138, 1136 951 958, 956 583 546, 538 337 353, 343 242 257 3070,3091 3099 3009,3030 3090 2988 3080 2955,3009 3063 2920,2967 2986 1631, 1632 1625 1558 1570 1405,1405 1423 1303 1335 1279,1280 1298 1249 1229,1229 1138, 1139 1188 940 931 565, 565 594 390 438 104 108, 86 Out of Plane Frequencies 1007, 1011 1097 954, 960 1023 897, 900 967 839, 840 856 627, 629 615 245 250 181 189 64 1069 984 905, 896 925 883, 877 889 632 353, 343 339 164 150
7
2WO.00
E
HUMM 3101 3065 3054 3042 2994 1586 1545 1456 1350 1234 1176 1170 1055 974 627 357 296 3101 3060 3048 3040 2993 1581 1529 1407 1304 1272 1200 1152 959 639 488 132 997 964 932 869 628 24 1 148 46 1018 923 923 832 617 307 137
Reference 49. Reference 8.
becomes a function of the energy gap between occupied and unoccupied levels only. More precisely, one can demonstrate from symmetry considerationsthat the Bzu mode correction from J C is ~ only a function of AE23, whereas that for Ezg is a function Of AE23 AE14.
+
B 1
1
0
0.00
-1m.00
-
-
-2W0.00 -0.22
-0.17
-0.12
-0.07
-0.02
Figure 1. Dependence of the calculated Bzu frequency on the HOMOLUMO gap in benzene according to our model introducing bond polarizabilities.
on both the Bzu and E2g experimental stretches. It should be emphasized that the Hiickel method contains the implicit function AE23 = (1/2)AE14. In addition, in the present model, to get a precise fitting of nij, a coupling between CC stretch and CCH bending should be further required, as is done in Ohno's Actually, the second Bzu vibration of benzene, analyzed as the CH trigonal bend, is calculated with an error of 100 cm-', which is due to the lack of such coupling. The corresponding improvement will be included in the subsequent version of our program. Let us focus on the value of B2u as a function of AE23. The AIg frequency will govern the Fii value, and one can take the approximation Fv = ~ t i when j /3" is sufficiently small. When A E 2 3 is varied continuously, we get the curve in Figure 1. Two zones clearly appear in the curve. When the Bzu frequency is positive, the stable form of benzene has D6h symmetry, while it has D3h symmetry when the Bzu frequency is negative. Both the u system and that part of the JC system which is proportional to P , stabilize the D6h form, while the nij part destabilizes it to the benefit of the D3h form. This result is quite in line with that obtained from Shaik's Through this example, we have illustrated the crucial role played by the quantum part in the spectra of conjugated hydrocarbons. In summary, quantum terms due to JC electron conjugation effects need to be included explicitly in the potential functions for conjugated system^.^^.^^
V. Concluding Remarks
The well-known bad evaluation of h E 2 3 from the Hiickel method ~ AnG", is avoided in our model by a parameter A such that J C = where zoois calculated from our parameterization and A is fitted
The present package is mainly devoted to the study of conjugated hydrocarbons. Emergence of a new area of research such as molecular electronics (ME) or interstellar chemical physics (IPC) has made this molecular family still of current interest. Such topics demand data for systems with large number of carbon atoms. Such data-not attainable from abinitio calculations-in turn require the development of models that are potentially cost effective in regard to the CPU time. Only the simple and homogeneous character of this unique
10062 J. Phys. Chem., Vol. 98, No. 40,I994 chemical family has made possible such a simple but efficient method. On the other hand, the questions raised by such compounds may cover a very broad range of observations on a broad range of electronic configurations (excited states, cationic or anionic states, ...). This diversity of electronic situations to modelize has induced us to develop an entirely new package.
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Ohno,K. J . Chem. Phys. 1991, 95, 5524. Ozabak, A. G . ; Goodman, L. J . Chem. Phys. 1987,87, 2564. Albertazzi, E.; Zerbetto, F. Chem. Phys. 1987, 164, 91. Pulay, P.; Fogarasi, G.; Boggs, J. E. J. Chem. Phys. 1981,74,3999. Shaik. S . S.: Hilbertv. P. C.: Lefour. J. M.: Ohanessian. G. J . Am. Chem.'Soc. 1987, 109, 363.' (39) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R. Soc., Ser. A 1948, 193, 456. (40) Ohno,K. J . Mol. Spectrosc. 1978, 72, 238. (41) Brodersen, S.; Langseth, A. K. Dan. Vidensk. Selsk. Mat. Fys. SK.1 1956, No. 1. (42) Whitmer, J. C.; Cyvin, S. J.; Cyvin, B. N. Z. Narurforsch. Teil A 1978, 33, 45. (43) Warshel, A.; Karplus, M. J . Am. Chem. SOC.1972, 94, 5612. (44) Orlandi, G.; Zerbetto, F. Chem. Phys. 1988, 123, 175. (45) Sellers, H.; Pulay, P.; Boggs, J. E. J . Am. Chem. SOC.1985, 107, 6487. (46) Popov, E. M.; Kogan, G . A. Opt. Spectrosc. 1964, 17, 362. (47) Lippincott, E. R.; White, C. E.; Sibilia, J. P. J. Am. Chem. Soc. 1958,80, 2926. (48) Lippincott, E. R.; Kenney, T. W. J. Am. Chem. SOC.1962,84,3641. (49) Hamilton, T. P.; Pulay, P. J . Phys Chem. 1989,93,2341. Yoshida, H.; Tasumi, M. J . Chem. Phys. 1988, 89, 2803.
