The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DMRG/SCHOLLWOCK=RMP_dmrg.pdf

The density-matrix renormalization group

Spectra of porphyrins

A general discussion of approximate methods for obtaining self‐consistent molecular orbitals for all valence electrons of large molecules is presented. It is shown that the procedure of neglecting differential overlap in electron‐interaction integrals (familiar in π‐electron theory) without further adjustment may lead to results which are not invariant to simple transformations of the atomic orbital basis set such as rotation of axes or replacement of s, p orbitals by hybrids. The behavior of approximate methods in this context is examined in detail and two schemes are found which are invariant to transformations among atomic orbitals on a given atom. One of these (the simpler but more approximate) involves the complete neglect of differential overlap (CNDO) in all basis sets connected by such transformations. The other involves the neglect of diatomic differential overlap (NDDO) only, that is only products of orbitals on different atoms being neglected in the electron‐repulsion integrals.

Approximate Self-Consistent Molecular Orbital Theory. I. Invariant Procedures

The CNDO method has been modified by substitution of semiempirical Coulomb integrals similar to those used in the Pariser‐Parr‐Pople method, and by the introduction of a new empirical parameter κ to differentiate resonance integrals between σ orbitals from those between π orbitals. The CNDO method with this change in parameterization is extended to the calculation of electronic spectra and applied to the isoelectronic compounds benzene, pyridine, pyridazine, pyrimidine, and pyrazine. The results obtained were refined by a limited CI calculation and compared with the best available experimental data. It was found that the agreement was quite satisfactory for both n→π* and π→π* singlet‐singlet transitions. The relative energies of the pi and lone‐pair orbitals in pyridine and the diazines are compared and an explanation proposed for the observed orders. Also, the nature of the “lone pairs” in these compounds is discussed.

Use of the CNDO Method in Spectroscopy. I. Benzene, Pyridine, and the Diazines

A previously given theory of the electronic spectra and structure of complex unsaturated molecules is further elucidated by listing specific formulas for the charge density, bond order, transition moment, and configuration interaction matrix elements.The theory is then applied to give a number of simple rules for alternant hydrocarbons. These refer to the energies of the singly excited states and the effects of configuration interaction, the degeneracy between certain singlet and triplet states, the prohibition of certain electronic transitions which would be allowed from a group theoretical point of view, and the charge density distribution of the ground and singly excited states. A mechanism is proposed to explain some of the anomalously rapid singlet‐triplet radiationless transitions which are observed in connection with studies of the lower triplet states of complex molecules.Quantitative application of the theory is made to the spectra of the polyacenes, benzene through pentacene, and to the bond ord...

J. A. Pople

Papers

Electron interaction in unsaturated hydrocarbons