Minimum total energy calculations are reported for π‐conjugated hydrocarbons including neutral (ground, 1 1Bu, 2 1Ag) and doped (1+ and 2+) chains and rings with up to eight carbon atoms. Two models are considered; first, a semiempirical π‐electron Hamiltonian that includes both electron–electron (Hubbard) and electron–lattice (Longuet‐Higgins–Salem) interactions, and second, an accurate abinitio complete‐active‐space self‐consistent‐field (CASSCF) treatment that includes the π‐electron correlation effects most important in determining the bond geometries. The results of the abinitio calculations can be used to estimate the phenomenological parameters entering the semiempirical Hamiltonian and thus to obtain quantitative predictions of bond geometries from the semiempirical treatment. The two models yield qualitatively the same results for the bond geometries in all states considered, and the changes in bond geometry following excitation from ground to doped or excited states find natural interpretation in terms of short‐chain limiting behaviors of soliton and polaron distortions familiar for longer chains. Further, the absolute values and sensitivities of the phenomenological parameters of the semiempirical model to various fitting schemes provide an indication of the different roles played by electron–lattice and electron–electron interactions in determining the properties of these systems. While electron–lattice interactions are found to be the most important factor in determining bond geometries, particularly in the ground and doped states, electron–electron interactions play an important and subtle role in determining the bond geometries and relative energetic orderings of the excited states.

1.
Electroresponsive Molecular and Polymeric Systems, edited by T. A. Skotheim (Marcel Dekker, New York, 1991);
J. H.
Burroughes
,
D. D. C.
Bradley
,
A. R.
Brown
,
R. N.
Marks
,
K.
MacKay
,
R. H.
Friend
,
P. L.
Burn
, and
A. B.
Holmes
,
Nature
347
,
539
(
1990
);
N. S.
Sariciftci
,
L.
Smilowitz
,
A. J.
Heeger
, and
F.
Wudl
,
Science
258
,
1474
(
1992
);
N. C.
Greenham
,
S. C.
Moratti
,
D. D. C.
Bradley
,
R. H.
Friend
, and
A. B.
Holmes
,
Nature
365
,
628
(
1993
).
2.
R.
Hoffmann
,
C.
Janiak
, and
C.
Kollmar
,
Macromolecules
24
,
3725
(
1991
).
3.
D. Baeriswyl, D. K. Campbell, and S. Mazumbar, in Conjugated Conducting Polymers, edited by H. Kiess (Springer, Berlin, 1992), pp. 7–133.
4.
G.
Rossi
,
J. Chem. Phys.
94
,
4031
(
1991
).
5.
G. W.
Hayden
and
Z. G.
Soos
,
Phys. Rev. B
38
,
6075
(
1988
).
6.
See, for example,
A. J.
Heeger
,
S.
Kivelson
,
J. R.
Schrieffer
, and
W. P.
Su
,
Rev. Mod. Phys.
60
,
781
(
1988
).
7.
H. C.
Longuet-Higgins
and
L.
Salem
,
Proc. R. Soc. London Ser., A
251
,
172
(
1959
).
8.
J. A.
Pople
and
S. H.
Walmsley
,
Mol. Phys.
5
,
15
(
1962
).
9.
W. P.
Su
,
J. R.
Schrieffer
, and
A. J.
Heeger
,
Phys. Rev. Lett.
42
,
1698
(
1979
),
W. P.
Su
,
J. R.
Schrieffer
, and
A. J.
Heeger
,
Phys. Rev B
22
,
2099
(
1980
).
10.
B. S.
Hudson
and
B. E.
Kohler
,
Chem. Phys. Lett.
14
,
299
(
1972
);
K.
Schulten
and
M.
Karplus
,
Chem. Phys. Lett.
14
,
305
(
1972
); ,
Chem. Phys. Lett.
B. S. Hudson, B. E. Kohler, and K. Schulten, in Excited States, edited by E. C. Lim (Academic, New York, 1982);
B. S.
Hudson
and
B. E.
Kohler
,
Synth. Met.
9
,
241
(
1984
);
B. E.
Kohler
,
J. Chem. Phys.
88
,
2788
(
1988
);
B. E.
Kohler
,
Chem. Rev.
93
,
41
(
1993
).
11.
B. E.
Kohler
,
C.
Spangler
, and
C.
Westerfield
,
J. Chem. Phys.
89
,
5422
(
1988
).
12.
W. J.
Buma
,
B. E.
Kohler
, and
C.
Song
,
J. Chem. Phys.
92
,
4622
(
1990
);
W. J.
Buma
,
B. E.
Kohler
, and
C.
Song
,
94
,
4691
(
1991
).,
J. Chem. Phys.
13.
G.
Rossi
,
Phys. Rev. B
50
,
1268
(
1994
).
14.
J.
Tinka Gammel
and
D. K.
Campbell
,
Synth. Met.
57
,
4638
(
1993
).
15.
E. H.
Lieb
and
B.
Nachtergaele
,
Phys. Rev. B
51
,
4777
(
1995
).
16.
R. Silbey, in Polydiacetylenes, NATO Advanced Study Institute, Series E, edited by D. Bloor and R. R. Chance (Martinus Nijhoff, Dordrecht, 1985), Vol. 102, pp. 93–104.
17.
Terms describing longer distance (next to nearest neighbor and beyond) e−e interaction could easily be included in our treatment. We neglect them here to avoid introducing additional phenomenological information. We do not expect the inclusion of these terms to significantly modify our results since even the inclusion of the nearest neighbor term does not change the qualitative form of the results that we find including only on site e−e repulsion. By the same token, we also neglect any dependence of V1 on the bond lengths.
18.
Others quantities describing the molecular geometry, such as in plane and out of plane bond angles, are simply not considered in the semiempirical model of Eq. (3).
19.
S. N.
Dixit
,
D.
Guo
, and
S.
Mazumdar
,
Phys. Rev. B
43
,
6781
(
1991
).
20.
S.
Abe
,
J.
Yu
, and
W. P.
Su
,
Phys. Rev. B
45
,
8264
(
1992
);
S.
Abe
,
M.
Schreiber
,
J.
Yu
, and
W. P.
Su
,
Phys. Rev. B
45
,
9432
(
1992
).,
Phys. Rev. B
21.
D.
Guo
,
S.
Mazumdar
,
S. N.
Dixit
,
F.
Kajzar
,
F.
Jarka
,
Y.
Kawabe
, and
N.
Peyghambarian
,
Phys. Rev. B
48
,
1433
(
1993
).
22.
Z. G.
Soos
,
S.
Ramasesha
, and
D. S.
Galvão
,
Phys. Rev. Lett.
71
,
1609
(
1993
).
23.
S.
Mazumdar
and
D.
Guo
,
J. Chem Phys.
100
,
1665
(
1994
).
24.
F.
Guo
,
D.
Guo
, and
S.
Mazumdar
,
Phys. Rev. B
49
,
10
102
(
1994
).
25.
D.
Mudkhopadhyay
,
G. W.
Hayden
, and
Z. G.
Soos
,
Phys. Rev. B
51
,
9476
(
1995
).
26.
The form of the electron-lattice coupling described by the semiempirical Hamiltonian used by
G.
König
and
G.
Stollhoff
,
Phys. Rev. Lett.
65
,
1239
(
1990
) is also such that polaron or soliton excitations cannot be supported. We disagree with the conclusions reached by these authors regarding the origin of the dimerization in conjugated system; indeed our results provide strong confirmation for the validity of the Peierls mechanism. We believe that the conclusions reached by König and Stollhoff regarding the relative unimportance of electron-phonon effects is due to their failure to independently fit the results of their ab initio calculation to the SSH Hamiltonian.
27.
For recent studies regarding the vibrational structure of polyenes, see M. Gussoni, C. Castiglioni, and G. Zerbi, in Spectroscopy of Advanced Materials, edited by R. J. H. Clark and R. E. Hester (Wiley, New York, 1991), pp. 251–353;
G.
Orlandi
,
F.
Zerbetto
, and
M. Z.
Zgierski
,
Chem. Rev.
91
,
867
(
1991
);
M.
Kofranek
,
H.
Lischka
, and
A.
Karpfen
,
J. Chem. Phys.
96
,
982
(
1992
).
28.
Several rigorous results regarding the optimal set of βm for the ground state of neutral rings are proven in Ref. 13.
29.
S.
Mazumdar
and
Z. G.
Soos
,
Synth. Met.
1
,
77
(
1979
).
30.
Obtaining the required eigenvalues of h(γ,υ01) is by far the most computationally expensive step. This is because the size P of the full basis set grows exponentially with the number of sites N. For example, for the S = 0 subspace in a half-filled system, P = (1/(N+1))(N/2N+1)2 and h(γ,υ01) is represented by a P×P matrix. For N>6, if only the lowest few eigenvalues are sought, the Rutishauser algorithm implemented in the NAG Fortran Library—see NAG Fortran Library Manual Mark 16, (NAG, Oxford, 1993)—is considerably more efficient than Householder reduction to tridiagonal form followed by the QL algorithm; for this procedure, see B. T. Smith, J. M. Boyle, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines-EISPACK Guide (Springer, Berlin, 1976).
31.
J. A.
Nelder
and
R.
Mead
,
Comp. J.
7
,
308
(
1965
).
32.
Similar behavior occurs within other independent electron schemes, see
V.
Bach
,
E. H.
Lieb
,
M.
Loss
, and
J. P.
Solovej
,
Phys. Rev. Lett.
72
,
2981
(
1994
).
33.
In the SSH model the energy spectrum is symmetric around E = 0; in other words one has εM+j,s = −εM−j+1,s here j = 1,…,M = N/2 and εm,s({βm})(m = 1,…,N) are the single particle electronic energy levels. The first term on the right-hand side of Eq. (4) is simply the sum of the energies of the single particle levels which are occupied; namely, Ep({βm},υ0 = 0,υ1 = 0) = Σm,sνm,sεm,s({βn}) where νm,s( = 0,1) are the occupation numbers. Moving one electron from the Mth to the (M+1)th state has the same effect on this sum as removing both the electrons present in the Mth state.
34.
The two soliton levels in the Bu1 state are degenerate so that it does not cost any additional energy to move a second electron from the Mth to the (M+1)th level. For N finite the degeneracy of the two soliton levels is broken and the SSH description predicts that the energy of the 21Ag state is higher than that of the 11Bu state.
35.
Note that in the range of g that we are considering, if one retains the dimerized geometry of the ground state (vertical transition), the 21Ag state (for N = 4, 6, and 8) is the one obtained moving one of the two electrons from the Mth to the (M+2)th level.
36.
Perhaps this result reflects the fact that in the SSH model the 11Bu state contains two charged solitons (Ref. 4) so that excesses of charge are concentrated at the solitons locations rather than being spread over the chain.
37.
At υ0 = υ1 = 0 the same bond geometry found for the lowest singlet excited state is obtained removing two electrons from the N = 6 ring. In fact, we find that the ground state of the doped ring with two electrons removed has this kind of irregular hexagon bond geometry throughout the range 0⩽υ0⩽5,0⩽υ1⩽υ0/5.
38.
R.
Shepard
,
Adv. Chem. Phys.
69
,
63
(
1987
);
B. O.
Roos
,
Adv. Chem. Phys.
69
,
399
(
1987
).,
Adv. Chem. Phys.
39.
A. C.
Lasaga
,
R. J.
Aerni
, and
M.
Karplus
,
J. Chem. Phys.
73
,
5230
(
1980
).
40.
M.
Aoyagi
,
Y.
Osamura
, and
S.
Iwata
,
J. Chem. Phys.
83
,
1140
(
1985
);
M.
Aoyagi
,
I.
Ohmine
, and
B. E.
Kohler
,
J. Phys. Chem.
94
,
3922
(
1990
).
41.
P. G.
Szalay
,
A.
Karpfen
, and
H.
Lischka
,
Chem. Phys.
130
,
219
(
1989
).
42.
F.
Zerbetto
and
M. Z.
Zgierski
,
J. Chem. Phys.
93
,
1235
(
1990
).
43.
(a)
L.
Serrano-Andrés
,
R.
Lindh
,
B. O.
Roos
, and
M.
Merchán
,
J. Phys. Chem.
97
,
9360
(
1993
);
(b)
L.
Serrano-Andrés
,
M.
Merchán
,
I.
Nebot-Gil
,
R.
Lindh
, and
B. O.
Roos
,
J. Chem. Phys.
98
,
3151
(
1993
);
(c)
L.
Serrano-Andrés
,
J.
Sánchez-Marín
, and
I.
Nebot-Gil
,
J. Chem. Phys.
97
,
7499
(
1992
).,
J. Chem. Phys.
44.
H. O.
Villar
,
M.
Dupuis
,
J. D.
Watts
,
G. J. B.
Hurst
, and
E.
Clementi
,
J. Chem. Phys.
88
,
1003
(
1988
);
H. O.
Villar
,
M.
Dupuis
, and
E.
Clementi
,
Phys. Rev. B
37
,
2520
(
1988
);
H. O.
Villar
,
M.
Dupuis
, and
E.
Clementi
,
J. Chem. Phys.
88
,
5251
(
1988
);
C. X.
Cui
,
M.
Kertesz
, and
M.
Dupuis
,
J. Chem. Phys.
93
,
5890
(
1990
).,
J. Chem. Phys.
45.
H. O.
Villar
and
M.
Dupuis
,
Theor. Chim. Acta
83
,
155
(
1992
).
46.
R. J.
Cave
and
M. G.
Perrott
,
J. Chem. Phys.
96
,
3745
(
1992
);
R. J.
Cave
and
J. L.
Johnson
,
J. Phys. Chem.
96
,
5332
(
1992
).
47.
M. W.
Schmidt
,
K. K.
Baldridge
,
J. A.
Boatz
,
J. H.
Jensen
,
S.
Koseki
,
M. S.
Gorgon
,
K. A.
Nguyen
,
T. L.
Windus
, and
S. T.
Elbert
,
QCPE Bull.
10
,
52
(
1990
).
48.
T. H. Dunning and P. J. Hay, in Methods of Electronic Structure Theory, edited by H. F. Schaefer III (Plenum, New York, 1977), Chap. 1.
49.
S.
Huzinaga
,
J. Chem. Phys.
53
,
2823
(
1970
).
50.
ACES II, an ab initio program system, authored by J. F. Stanton, J. Gauss, J. D. Watts, W. J. Lauderdale, and R. J. Bartlett, Quantum Theory Project, University of Florida.
51.
G.
Fogarasi
,
R.
Liu
, and
P.
Pulay
,
J. Phys. Chem.
97
,
4036
(
1993
).
52.
R. J.
Bartlett
,
J. Phys. Chem.
93
,
1697
(
1989
).
53.
R. J.
Cave
,
J. Chem. Phys.
92
,
2450
(
1992
);
R. J.
Cave
and
E. R.
Davidson
,
Chem. Phys. Lett.
148
,
190
(
1988
);
R. J.
Cave
and
E. R.
Davidson
,
J. Phys. Chem.
92
,
2173
,
614
(
1988
);
R. J.
Cave
and
E. R.
Davidson
,
91
,
4481
(
1987
).,
J. Phys. Chem.
54.
C. H.
Martin
and
K. F.
Freed
,
J. Phys. Chem.
99
,
2701
(
1995
);
R. L.
Graham
and
K. F.
Freed
,
J. Chem. Phys.
96
,
1304
(
1992
).
55.
Note that results qualitatively similar to those of Figs. 2, 3, and 4 are found in the whole range 0.6⩽γ⩽1.2,0⩽υ1⩽υ0/5 so that the examples shown in these figures are representative of the predictions of the semiempirical treatment for “realistic“ values of the phenomenological parameters.
56.
A similar discrepancy is found in longer chains between the predictions of the SSH model and the results of the earlier ab initio calculations of Ref. 44. The discrepancy can be viewed as a result of the fact that while both calculations predict solitonic bond geometries, the two solitons are somewhat closer to the center of the chain in the SSH formulation. See Ref. 4 for details.
57.
In the limit γ→∞ all the βm vanish [see Eq. (4)] and all bonds have the same length.
58.
See, CRC Handbook of Chemistry and Physics, 75th ed., edited by D. R. Lide (Chemical Rubber, Boca Raton, 1995), Chap. 9.
59.
M. F.
Granville
,
B. E.
Kohler
, and
J. B.
Snow
,
J. Chem. Phys.
75
,
3765
(
1981
).
60.
M. F.
Granville
,
G. R.
Holtom
, and
B. E.
Kohler
,
J. Chem. Phys.
72
,
4671
(
1980
);
R. M.
Gavin
,
C.
Weisman
,
J. K.
McVey
, and
S. A.
Rice
,
J. Chem. Phys.
68
,
522
(
1978
).,
J. Chem. Phys.
61.
Note that, with the parametrization chosen here (and in Ref. 4) for the Hamiltonian of Eq. (3) with υ0 = υ1 = 0, the relation between t0 and the bandwidth 2W is given by t0 = W/(2−βl−βs), and not by the commonly used (see Refs. 3, 6) 4t0 = 2W; here 1−βl and 1−βs are the rescaled hopping constants for the long and short bond in the central region of a long chain (or of a large ring with 4N carbon atoms). In other words, our t0 and the t0 of Refs. 3, 6 are different quantities; for this reason, our values for t0 are considerably smaller than the value t0≈2.5 eV ordinarily used. For example, if γ has the value γ = 0.93 (see Table III, results of fit 3) and 2W = 10 eV, we find t0≈1.5 eV; then for R = 7.1Å−1 we get α≈10.65 eV and K≈140 eV2. Likewise, the comparison of Ref. 4 between the SSH bond lengths and those found for long polyenes in Ref. 44 gave γ = 0.9,R = 4.525 Å−1,L0 = 1.548Å,t0≈1.47 eV,α≈6.67 eV/Å, and K≈55 eV2.
62.
The sum [right-hand side of Eq. (6)] used to obtain the data in Table II contains each input bond length lm(i) as many times as this bond length occurs in the actual molecules. For example, the term containing the benzene bond length was assigned weight 6 [e.g., it appears six times in the sum of Eq. (6)], the term containing the butadiene outerbond was assigned weight 2 and the term containing the butadiene midbond was assigned weight 1.
63.
It has recently been argued [see
B. Y.
Jin
and
R.
Silbey
,
J. Chem. Phys.
102
,
4261
(
1995
)], on the basis of a number of minimum total energy calculations, that in long polyenes the two solitons predicted for the 11Bu state by the SSH model are bound together in a “self-trapped exciton“ as a result of the presence of e−e interactions. While our chains are so short that a direct comparison with the results of Jin and Silbey is not possible, we note that these authors restrict the minimization procedure within a small subset of all the possible bond geometries and that only a very limited sample of e−e correlations is accounted for in their calculations. In view of these restrictions and in view of the difficulties (discussed in Sec. III) that we encountered in obtaining reliable ab initio predictions for the relaxed bond geometry of the 11Bu state, it seems doubtful that the framework adopted by Jin and Silbey captures correctly the physics of the problem.
64.
By contrast the “optimal“ value for L0 falls within a couple of percents of 1.52 Å in all the cases that we have explored.
65.
R. Peierls, Quantum Theory of Solids (Oxford University, Oxford, 1955), p. 111.
66.
G.
Rossi
,
Synth. Met.
49
,
221
(
1992
).
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