Orbital optimization in the density matrix renormalization group, with applications to polyenes and beta-carotene

In previous work we have shown that the density matrix renormalization group (DMRG) enables near-exact calculations in active spaces much larger than are possible with traditional complete active space algorithms. Here, we implement orbital optimization with the DMRG to further allow the self-consistent improvement of the active orbitals, as is done in the complete active space self-consistent field (CASSCF) method. We use our resulting DMRG-CASSCF method to study the low-lying excited states of the all-trans polyenes up to C24H26 as well as beta-carotene, correlating with near-exact accuracy the optimized complete pi-valence space with up to 24 active electrons and orbitals, and analyze our results in the light of the recent discovery from resonance Raman experiments of new optically dark states in the spectrum.     

Extremely localized molecular orbitals: theory and applications

Orbitals that are extremely localized on molecular fragments represent a powerful tool for a number of purposes: to cite a few examples, they allow to reduce strongly the complexity of calculations on large systems and are easily transferable from one molecule to another, providing a suitable and efficient way to build up the electronic structure of large molecules. Recently, we have developed efficient algorithms to determine extremely localized molecular orbitals (ELMOs), which will be reviewed in this paper. As a rigorous localization is strictly connected to a reduction in the number of variational parameters, which reflects into an increased value of the associated energy with respect to the Hartree Fock value, we have developed a number of strategies to relax the wavefunction built up using transferred localized orbitals. The extreme localization has also been exploited in connection with the “Divide and Conquer” technique to determine the electron densities of large polypeptides assembled from orbitals computed on small model molecules. Moreover, we will discuss the recent application of the ELMOs in the framework of the hybrid QM/MM methods to describe the frontier region. We will also show that the ELMOs can be used to extract chemical interpretations from numerical results. A variety of applications will be presented.     

Optimal virtual orbitals to relax wave functions built up with transferred extremely localized molecular orbitals

Extremely localized molecular orbitals (ELMOs), namely orbitals strictly localized on molecular fragments, are easily transferable from one molecule to another one. Hence, they provide a natural way to set up the electronic structure of large molecules using a data base of orbitals obtained from model molecules. However, this procedure obviously increases the energy with respect to a traditional MO calculation. To gain accuracy, it is important to introduce a partial electron delocalization. This can be carried out by defining proper optimal virtual orbitals that supply an efficient set for nonorthogonal configurations to be employed in VB-like expansions.     

Symmetry operators of generalized wreath products and their applications to chemical physics

Symmetry operators of generalized wreath product groups are formulated. Several applications of these operators to nonrigid molecular problems in chemical physics are outlined.     

Coupling routes to hexafluoro-1,3-butadiene, substituted-1,3-fluorine-containing butadienes and fluorinated polyenes

Hexafluoro-1,3-butadiene was readily prepared via a variety of self-coupling processes, such as Cu(0) mediated self-coupling of iodotrifluoroethene, Pd(0) catalyzed coupling of iodotrifluoroethene with the trifluorovinylzinc reagent, and CuBr₂ mediated coupling of the trifluorovinylzinc reagent. Perfluoro-2,3-dimethyl-1,3-butadiene was readily synthesized by the reaction of pentafluoropropenyl-2-zinc reagent with either CuBr₂ or FeCl₃. Alternatively, perfluoro-2,3-dimethyl-1,3-butadiene was prepared by oxidation of the pentafluoropropenyl-2-copper reagent with dioxygen. Cu(0) mediated coupling of an (E)-substituted ,β-difluoro-β-iodostyrene provided the first useful route to a (Z)(Z)-1,4-diaryl-1,3-tetrafluorobutadiene. Extension of the Cu(0) mediated coupling methodology to a perfluorodienyl iodide demonstrated a useful stereospecific route to perfluoropolyenes.     

The scaling approach to multicenter molecular integrals with Slater-type orbitals

A scaling approach to multicenter molecular integrals with Slater-type orbitals (STOs) is presented. The result is significant in that it shows (1) the existence of a simple relationship between multicenter integrals and (2) an implied computational savings. Operation count estimates indicate that the significant savings would occur for a system having large numbers of STOs on each atom.     

Virtual orbital transformation prior to configuration interaction with localized orbitals

A virtual orbital transformation is proposed involving pairing of localized occupied orbitals with virtual orbitals. The virtual orbitals are transformed so that each virtual orbital is as “close” as possible to its occupied counterpart, where closeness is the inverse of the particular definition of localization. The appropriate transformation is derived for the special case of Foster–Boys localization, and an illustrative CNDO /2 calculation on HNO is presented. INDO CI results on the series N₂, CO, BF indicate that use of this transformation may reduce the number of energetically significant configurations.     

Perturbative configuration interaction using localized orbitals in the INDO hypothesis. I. Theory and applications to energetical problems

A complete development of the PCILO method is presented for the INDO approximation. The introduction of exchange terms in the theoretical treatment is discussed and a detailed analysis of the successive contributions to the total energy is given. The method is applied to various conformational problems (geometry optimization, rotation, and inversion barriers). Its ability to deal with biological systems is tested on the acetylcholine neurotransmitter. Inversion barriers are highly improved with respect to the corresponding CNDO algorithm. A better account of σ-π exchange phenomena can be expected with the presented PCILO -INDO development.     

An efficient method for calculating maxima of homogeneous functions of orthogonal matrices: applications to localized occupied orbitals

We present here three new algorithms (one purely iterative and two DIIS-like [Direct Inversion in the Iteractive Subspace]) to compute maxima of homogeneous functions of orthogonal matrices. These algorithms revolve around the mathematical lemma that, given an invertible matrix A, the function f(U)=Tr(AU) has exactly one local (and global) maximum for U special orthogonal [i.e., UU(T)=1 and det(U)=1]. This is proved in the Appendix. One application of these algorithms is the computation of localized orbitals, including, for example, Boys and Edmiston-Ruedenberg (ER) orbitals. The Boys orbitals are defined as the set of orthonormal orbitals which, for a given vector space of orbitals, maximize the sum of the distances between orbital centers. The ER orbitals maximize total self-interaction energy. The algorithm presented here computes Boys orbitals roughly as fast as the traditional method (Jacobi sweeps), while, for large systems, it finds ER orbitals potentially much more quickly than traditional Jacobi sweeps. In fact, the required time for convergence of our algorithm scales quadratically in the region of a few hundred basis functions (though cubicly asymptotically), while Jacobi sweeps for the ER orbitals traditionally scale as the number of occupied orbitals to the fifth power. As an example of the utility of the method, we provide below the ER orbitals of nitrated and nitrosated benzene, and we discuss the chemical implications.     

Direct synthesis of soluble,end-functionalized polyenes and polyacetylene block copolymers

The ring-opening metathesis polymerization (ROMP) of 1,3,5,7-cyclooctatetraene (COT) in the presence of a chain transfer agent (CTA) with a highly active ruthenium olefin metathesis catalyst resulted in the formation of soluble polyenes. Small molecule CTAs containing an internal olefin and a variety of functional groups resulted in soluble telechelic polyenes with up to 20 double bonds. Use of polymeric CTAs with an olefin terminus resulted in polyacetylene block copolymers. These materials were subjected to a variety of solution and solid phase characterization techniques including (1)H NMR, UV/vis, and FT-IR spectroscopies, as well as MALDI-TOF MS and AFM.     

Non-linear polamzabilities of conjugated chains: regular polyenes,solitons, and polarons

The hyperpolarizabilities of linear polyenes are shown to be extremely sensitive to confonnational changes and to the presence of charges along the conjugated chain. Geometries corresponding to those of regular polyenes and soliton and polaron defects are examined.     

Complete active-space configuration interaction with optimized orbitals: Application to Li2

We performed CAS –CI calculations on Li₂ using a set of molecular orbitals (MO ) optimized with a procedure that, in the case of highly symmetric molecules, permits extraction of a small set of MO out of a large set of atomic orbitals (AO ). The dimension of the CAS –CI space was of about 12 million symmetry-adapted determinants. We determined some spectroscopic constants of Li₂ with three different atomic basis sets of increasing quality. The values obtained with the largest atomic basis set are very close to the experimental results. © 1995 John Wiley & Sons, Inc.