Minimum total energy calculations are reported for π‐conjugated hydrocarbons including neutral (ground, 1 ^{1}*B*_{u}, 2 ^{1}*A*_{g}) 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 *ab* *initio* 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 *ab* *initio* 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.

## REFERENCES

*Electroresponsive Molecular and Polymeric Systems*, edited by T. A. Skotheim (Marcel Dekker, New York, 1991);

*Conjugated Conducting Polymers*, edited by H. Kiess (Springer, Berlin, 1992), pp. 7–133.

*Excited States*, edited by E. C. Lim (Academic, New York, 1982);

*Polydiacetylenes*, NATO Advanced Study Institute, Series E, edited by D. Bloor and R. R. Chance (Martinus Nijhoff, Dordrecht, 1985), Vol. 102, pp. 93–104.

*ab initio*calculation to the SSH Hamiltonian.

*Spectroscopy of Advanced Materials*, edited by R. J. H. Clark and R. E. Hester (Wiley, New York, 1991), pp. 251–353;

*P*of the full basis set grows exponentially with the number of sites

*N*. For example, for the $S\u2009=\u20090$ subspace in a half-filled system, $P\u2009=\u2009(1/(N+1))(N/2N+1)2$ and $h(\gamma ,\upsilon 0,\upsilon 1)$ is represented by a $P\xd7P$ 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).

*M*th to the $(M+1)th$ state has the same effect on this sum as removing both the electrons present in the

*M*th state.

*M*th 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.

*M*th to the $(M+2)th$ level.

*Methods of Electronic Structure Theory*, edited by H. F. Schaefer III (Plenum, New York, 1977), Chap. 1.

*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.

*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.

*CRC Handbook of Chemistry and Physics*, 75th ed., edited by D. R. Lide (Chemical Rubber, Boca Raton, 1995), Chap. 9.

*W*is given by $t0\u2009=\u2009W/(2\u2212\beta l\u2212\beta s),$ and not by the commonly used (see Refs. 3, 6) $4t0\u2009=\u20092W;$ here $1\u2212\beta l$ and $1\u2212\beta s$ 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 4

*N*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 $t0\u22482.5\u2009eV$ ordinarily used. For example, if γ has the value $\gamma \u2009=\u20090.93$ (see Table III, results of fit 3) and $2W\u2009=\u200910\u2009eV,$ we find $t0\u22481.5\u2009eV;$ then for $R\u2009=\u20097.1$ $\xc5\u22121$ we get $\alpha \u224810.65\u2009eV/\xc5$ and $K\u2248140\u2009eV/\xc52.$ Likewise, the comparison of Ref. 4 between the SSH bond lengths and those found for long polyenes in Ref. 44 gave $\gamma \u2009=\u20090.9,$ $R\u2009=\u20094.525\u2009\xc5\u22121,$ $L0\u2009=\u20091.548\xc5,$ $t0\u22481.47\u2009eV,$ $\alpha \u22486.67\u2009eV/\xc5,$ and $K\u224855\u2009eV/\xc52.$

*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.

*Quantum Theory of Solids*(Oxford University, Oxford, 1955), p. 111.