The localized-density-matrix method (Yokojima, S.; Chen, G. H. Chem. Phys. Lett. 1998, 292, 379) is employed to simulate the optical responses of very large carbon nanotubes and polyacetylene oligomers containing 10 000 carbon atoms. The Pariser-Parr-Pople Hamiltonian is used to describe the

The linear scaling localized-density-matrix (LDM) method is generalized to calculate the nonlinear optical responses of large polyacetylene oligomers. The ground state reduced single-electron density matrix is initialized by the divide-and-conquer approach and is obtained subsequently using the LDM method. The third-harmonic generation spectra of large oligomers are calculated. The saturation of off-resonant second hyperpolarizability x (3) (0) has been reinvestigated, and the value of x (3) (0) is found to depend mainly on the optical gap and the number of double bonds.

A generalized linear scaling localized-density-matrix (LDM) method is developed to adopt the nonorthonormal basis set and retain full Coulomb differential overlap integrals. To examine its validity, the method is employed to evaluate the absorption spectra of polyacetylene oligomers containing up to 500 carbon atoms. The semiempirical Hamiltonian for the p electrons includes explicitly the overlap integrals among the p orbitals, and is determined from the ab initio Hartree– Fock reduced single-electron density matrix. Implementation of the generalized LDM method at the ab initio molecular orbital calculation level is discussed.

A variety of chemical systems exhibit multiple reaction pathways that adjoin to a common reactant state. In fact, any reaction producing side products or proceeding via a stable intermediate involves a species possessing at least two reaction pathways. Despite improvements in ab initio transition-state search algorithms it remains difficult to detect multiple reaction pathways. Typically, multiple reaction pathways can only be detected by intuitively varying the initial point in the transition-state search trajectory. This reliance on intuition limits the ability to discover new and unexpected chemistry using ab initio methods. This paper proposes a systematic and intuition-free method for biasing a transition-state search to identify multiple reaction pathways originating from a common reactant state. The method allows the successive location of transition states, with each successful search contributing to a cumulative bias potential for the following search. The method is applicable to all psuedo-Newton-Raphson-type transition-state searches. The procedure was tested for a model potential energy surface and for the thermal rearrangement of trans-1,4-dimethylcyclobutene. In the latter case, four reaction pathways were discovered: two exothermic conrotatory ring openings leading to hexadienes, an endothermic methyl migration pathway leading to a carbene, and an exothermic rearrangement leading to 3-methyl-1,4-pentadiene. In accordance with experiment, the calculations predict that the conrotatory pathway leading to trans,trans-2,4-hexadiene is the kinetically dominant pathway. The methodology was also used to compute selectivities for competitive pathways producing trans and cis triflouropentadiene products in the thermal rearrangement of 3-triflouromethyl-cyclobutene. Again, results were in accord with experimental observations.

Dual basis sets are employed as an economical way to approximate self-consistent field (SCF) calculations, such as Kohn-Sham density functional theory (DFT), in large basis sets. First, an SCF calculation is performed in a small subset of the full set of basis functions. The density matrix in this small basis is used to construct the effective Hamiltonian operator in the large basis, from which a correction for basis set extension is obtained for the energy. This correction is equivalent to a single Roothaan step (diagonalization) in the large basis. We present second order nonlinear equations that permit this step to be obtained without explicit diagonalization. Numerical tests on part of the Gaussian-2 dataset, using the B3LYP density functional, show that large-basis results can be accurately approximated with this procedure, subject to some limitations on the smallness of the small basis. Computational savings are approximately an order of magnitude relative to a self-consistent DFT calculation in the large basis.