Density Matrix Search Research Articles

Obtaining Hartree–Fock and density functional theory doubly excited states with Car–Parrinello density matrix search

The calculation of doubly excited states is one of the major problems plaguing the modern day excited state workhorse methodology of linear response time dependent Hartree-Fock (TDHF) and density function theory (TDDFT). We have previously shown that the use of a resonantly tuned field within real-time TDHF and TDDFT is able to simultaneously excite both the alpha and beta electrons to achieve the two-electron excited states of minimal basis H(2) and HeH(+) [C. M. Isborn and X. Li, J. Chem. Phys. 129, 204107 (2008)]. We now extend this method to many electron systems with the use of our Car-Parrinello density matrix search (CP-DMS) with a first-principles fictitious mass method for wave function optimization [X. Li, C. L. Moss, W. Liang, and Y. Feng, J. Chem. Phys. 130, 234115 (2009)]. Real-time TDHF/TDDFT is used during the application of the laser field perturbation, driving the electron density toward the doubly excited state. The CP-DMS method then converges the density to the nearest stationary state. We present these stationary state doubly excited state energies and properties at the HF and DFT levels for H(2), HeH(+), lithium hydride, ethylene, and butadiene.

Car–Parrinello density matrix search with a first principles fictitious electron mass method for electronic wave function optimization

In spite of its success in molecular dynamics and the advantage of being a first order propagation technique, the Car-Parrinello method and its variations have not been successful in self-consistent-field (SCF) wave function optimization due to slow convergence. In this article, we introduce a first principles fictitious mass scheme to weigh each individual density element differently and instantaneously. As an alternative to diagonalization in SCF, the Car-Parrinello scheme is implemented as a density matrix search method. Not only does the fictitious mass scheme developed herein allow a very fast SCF convergence, but also the Car-Parrinello density matrix search (CP-DMS) exhibits linear scaling with respect to the system size for alanine helical chain test molecules. The excellent performance of CP-DMS holds even for very challenging compact three-dimensional quantum particles. While the conventional diagonalization based SCF method has difficulties optimizing electronic wave functions for CdSe quantum dots, CP-DMS shows both smooth and faster convergence.

Density matrix search using direct inversion in the iterative subspace as a linear scaling alternative to diagonalization in electronic structure calculations

For electronic structure calculations on large systems, solving the self-consistent-field (SCF) equations is one of the key bottlenecks. Density matrix search methods provide an efficient linear scaling approach for circumventing the traditional O(N3) diagonalization procedure for solving the SCF equations. The conjugate gradient density matrix search (CG-DMS) method is a successful implementation of this approach. An alternative density matrix search method, QN–DMS, employs direct inversion in the iterative subspace using a quasi-Newton (QN) step to estimate the error vector. For linear polyglycine chains of 10–100 residues, the present approach scales linearly and is 30% faster than CG-DMS. For clusters of up to 300 water molecules, this method shows smoother and efficient convergence, and displays nearly linear scaling.

Polarized atomic orbitals for linear scaling methods

We present a modified version of the polarized atomic orbital (PAO) method [M. S. Lee and M. Head-Gordon, J. Chem. Phys. 107, 9085 (1997)] to construct minimal basis sets optimized in the molecular environment. The minimal basis set derives its flexibility from the fact that it is formed as a linear combination of a larger set of atomic orbitals. This approach significantly reduces the number of independent variables to be determined during a calculation, while retaining most of the essential chemistry resulting from the admixture of higher angular momentum functions. Furthermore, we combine the PAO method with linear scaling algorithms. We use the Chebyshev polynomial expansion method, the conjugate gradient density matrix search, and the canonical purification of the density matrix. The combined scheme overcomes one of the major drawbacks of standard approaches for large nonorthogonal basis sets, namely numerical instabilities resulting from ill-conditioned overlap matrices. We find that the condition number of the PAO overlap matrix is independent from the condition number of the underlying extended basis set, and consequently no numerical instabilities are encountered. Various applications are shown to confirm this conclusion and to compare the performance of the PAO method with extended basis-set calculations.

Linear scaling density matrix search based on sign matrices

This paper presents a new approach to the linear scaling evaluation of density matrices in electronic structure theory. The new approach is based on the iterative computation of a special matrix function, the sign of the matrix and its performance is compared to that of some other methods developed for similar purpose. One particular variant of the sign approach turned out to be very competitive with other linear scaling density matrix evaluation algorithms, in terms of computational time and accuracy. It is also shown that a special damping technique greatly improves the stability of self-consistent field (SCF) calculations when using density matrix purification and sign methods.

Geometry optimization of Kringle 1 of plasminogen using the PM3 semiempirical method

The results of a geometry optimization on the 1226 atom Kringle 1 of plasminogen are presented. The energy and gradients were calculated using a linear- scaling PM3 semiempirical method with a conjugate gradient density matrix search replacing the diagonalization step. The geometry was optimized with the rational function optimization technique combined with a modified version of the direct inversion in the iterative subspace procedure. The optimization required 362 geometry update steps to reach a local minimum. An analysis is given of the optimization and timing results using a single processor on the SGI Origin2000. Q 2000 John Wiley & Sons, Inc. Int J

Converging difficult SCF cases with conjugate gradient density matrix search

The possibility of converging difficult self constent field (SCF) cases with the conjugate gradient density matrix search (CGDMS) method is investigated. Density functional theory calculations using various functionals and basis sets were performed on CrC, Cr2, UO2(OH)4 and UF4 which we deem representative of molecules which have SCF convergence problems. From calculations on these molecules we found that CGDMS has better SCF convergence properties than diagonalization alone, while it is comparable with diagonalization in combination with an adequately chosen level shift.

What is the best alternative to diagonalization of the Hamiltonian in large scale semiempirical calculations?

Recently, several linear scaling approaches have been introduced which replace the time dominating diagonalization step in semiempirical methods, enabling practical calculations to be performed on very large molecules. This paper compares the accuracy and performance of pseudodiagonalization (PD), conjugate gradient density matrix search (CG-DMS), the Chebyshev polynomial expansion method (CEM), and purification of the density matrix (PDM) as linear scaling substitutions for diagonalization. The scaling, speed, and reliability of these methods are compared for AM1 single point energy calculations on polyglycine chains (up to 20 000 atoms), water clusters (up to 12 300 atoms), and nucleic acids (up to 6300 atoms).

Comparison of conjugate gradient density matrix search and Chebyshev expansion methods for avoiding diagonalization in large-scale electronic structure calculations

We report a comparison of two linear-scaling methods which avoid the diagonalization bottleneck of traditional electronic structure algorithms. The Chebyshev expansion method (CEM) is implemented for carbon tight-binding calculations of large systems and its memory and timing requirements compared to those of our previously implemented conjugate gradient density matrix search (CG-DMS). Benchmark calculations are carried out on icosahedral fullerenes from C60 to C8640 and the linear scaling memory and CPU requirements of the CEM demonstrated. We show that the CPU requisites of the CEM and CG-DMS are similar for calculations with comparable accuracy.

Semiempirical methods with conjugate gradient density matrix search to replace diagonalization for molecular systems containing thousands of atoms

Conventional semiempirical methods using diagonalization are not practical for calculations on molecular systems containing more than a few hundred atoms because of O(N3) time and O(N2) memory requirements, where N is the number of atoms. Currently, the time dominating step is diagonalization of the Fock matrix. This paper demonstrates how O(N3) diagonalization and O(N2) memory requirements are eliminated by using a conjugate gradient search for the density matrix with sparse matrix techniques. Our method makes high accuracy energy calculations on molecules containing thousands of atoms possible on the typical workstation. Benchmark examples are presented on polyglycine chains (20000 atoms), water clusters (up to 1800 atoms), and nucleic acids (up to 6304 atoms).

Linear scaling conjugate gradient density matrix search as an alternative to diagonalization for first principles electronic structure calculations

Advances in the computation of the Coulomb, exchange, and correlation contributions to Gaussian-based Hartree–Fock and density functional theory Hamiltonians have demonstrated near-linear scaling with molecular size for these steps. These advances leave the O(N3) diagonalization bottleneck as the rate determining step for very large systems. In this work, a conjugate gradient density matrix search (CG-DMS) method has been successfully extended and computationally implemented for use with first principles calculations. A Cholesky decomposition of the overlap matrix and its inverse is used to transform to and back from an orthonormal basis, which can be formed in near-linear time for sparse systems. Linear scaling of CPU time for the density matrix search and crossover of CPU time with diagonalization is demonstrated for polyglycine chains containing up to 493 atoms and water clusters up to 900 atoms.

An O(N) tight-binding study of carbon clusters up to C8640: the geometrical shape of the giant icosahedral fullerenes

We report an efficient implementation of the Li-Nunes-Vanderbilt conjugate gradient density matrix search method that exploits sparse matrix algorithms for storage and matrix operations. The scheme avoids the N3 diagonalization bottleneck and scales linearly with the system's size. Benchmarks on large carbon systems indicate that the method becomes faster than diagonalization for small number of atoms (⩽250). Clusters containing thousands of atoms are studied with modest workstation computation resources. We also report geometry optimizations of very large icosahedral fullerenes (up to C8640) which clearly indicate that these clusters have a faceted (polyhedral) geometrical shape.