Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings.

The density-matrix renormalization group algorithm has emerged as a promising new method in ab initio quantum chemistry. However, many problems still need to be solved before this method can be applied routinely. At the start of such a calculation, the orbitals originating from a preceding quantum chemical calculation must be placed in a specific order on a one-dimensional lattice. This ordering affects the convergence of the density-matrix renormalization group iterations significantly. In this paper, we present two approaches to obtain optimized orderings of the orbitals. First, we use a genetic algorithm to optimize the ordering with respect to a low total electronic energy obtained at a predefined stage of the density-matrix renormalization group algorithm with a given number of total states kept. In addition to that, we derive orderings from the one- and two-electron integrals of our test system. This test molecule is the chromium dimer, which is known to possess a complicated electronic structure. For this molecule, we have carried out calculations for the various orbital orderings obtained. The convergence behavior of the density-matrix renormalization group iterations is discussed in detail.

[1]  A. D. McLean,et al.  f-Type functions in the orbital basis for calculating molecular interactions involving d electrons Cr2 and Mo2 , 1983 .

[2]  Hans W. Horn,et al.  Fully optimized contracted Gaussian basis sets for atoms Li to Kr , 1992 .

[3]  Karen Hallberg.  Density Matrix Renormalization , 1999 .

[4]  White,et al.  Density matrix formulation for quantum renormalization groups. , 1992, Physical review letters.

[5]  M. Ratner.  Molecular electronic-structure theory , 2000 .

[6]  Hermann Stoll,et al.  The Cr2 potential curve: a multireference pair functional treatment , 1996 .

[7]  Garnet Kin-Lic Chan,et al.  An algorithm for large scale density matrix renormalization group calculations. , 2004, The Journal of chemical physics.

[8]  Guido Fano,et al.  Quantum chemistry using the density matrix renormalization group , 2001 .

[9]  David E. Goldberg,et al.  Genetic Algorithms in Search Optimization and Machine Learning , 1988 .

[10]  K. Andersson,et al.  THE CR2 POTENTIAL ENERGY CURVE STUDIED WITH MULTICONFIGURATIONAL SECOND-ORDER PERTURBATION THEORY , 1994 .

[11]  P. Palmieri,et al.  Epstein-Nesbet second-order perturbation treatment of dynamical electron correlation and ground state potential energy curve of Cr2 , 1997 .

[12]  J. Sólyom,et al.  Optimizing the density-matrix renormalization group method using quantum information entropy , 2003 .

[13]  Richard L. Martin,et al.  Ab initio quantum chemistry using the density matrix renormalization group , 1998 .

[14]  B. A. Hess,et al.  QC-DMRG study of the ionic-neutral curve crossing of LiF , 2002 .

[15]  M. Head‐Gordon,et al.  Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group , 2002 .

[16]  A. H. Sherman,et al.  Comparative Analysis of the Cuthill–McKee and the Reverse Cuthill–McKee Ordering Algorithms for Sparse Matrices , 1976 .

[17]  Garnet Kin-Lic Chan,et al.  Exact solution (within a triple-zeta, double polarization basis set) of the electronic Schrödinger equation for water , 2003 .

[18]  B. A. Hess,et al.  Controlling the accuracy of the density-matrix renormalization-group method: The dynamical block state selection approach , 2002, cond-mat/0204602.

[19]  K. Wilson.  The renormalization group: Critical phenomena and the Kondo problem , 1975 .

[20]  C. Daul,et al.  Full‐CI quantum chemistry using the density matrix renormalization group , 1999 .

[21]  Guido Fano,et al.  Quantum chemistry using the density matrix renormalization group II , 2003 .

[22]  White,et al.  Density-matrix algorithms for quantum renormalization groups. , 1993, Physical review. B, Condensed matter.

[23]  E. Davidson.  The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , 1975 .