Approximate solution for electron correlation through the use of Schwinger probes

Computation of the Fock matrix is currently the limiting factor in the application of Hartree-Fock and hybrid Hartree-Fock/density functional theories to larger systems. Computation of the Fock matrix is dominated by calculation of the Coulomb and exchange matrices. With conventional Gaussian-based methods, computation of the Fock matrix typically scales as ∼N2.7, where N is the number of basis functions. A hierarchical multipole method is developed for fast computation of the Coulomb matrix. This method, together with a recently described approach to computing the Hartree-Fock exchange matrix of insulators [J. Chem. Phys. 105, 2726 (1900)], leads to a linear scaling algorithm for calculation of the Fock matrix. Linear scaling computation the Fock matrix is demonstrated for a sequence of water clusters at the restricted Hartree-Fock/3-21G level of theory, and corresponding accuracies in converged total energies are shown to be comparable with those obtained from standard quantum chemistry programs. Restri...

Linear scaling computation of the Fock matrix

A simplified version of the Li, Nunes and Vanderbilt [Phys Rev B 47, 10891 (1993)] and Daw [Phys Rev B 47, 10895 (1993)] density matrix minimization is introduced that requires four fewer matrix multiplies per minimization step relative to previous formulations The simplified method also exhibits superior convergence properties, such that the bulk of the work may be shifted to the quadratically convergent McWeeny purification, which brings the density matrix to idempotency Both orthogonal and nonorthogonal versions are derived The AINV algorithm of Benzi, Meyer, and Tůma [SIAM J Sci Comp 17, 1135 (1996)] is introduced to linear scaling electronic structure theory, and found to be essential in transformations between orthogonal and nonorthogonal representations These methods have been developed with an atom-blocked sparse matrix algebra that achieves sustained megafloating point operations per second rates as high as 50% of theoretical, and implemented in the MondoSCF suite of linear scaling SCF programs For the first time, linear scaling Hartree–Fock theory is demonstrated with three-dimensional systems, including water clusters and estane polymers The nonorthogonal minimization is shown to be uncompetitive with minimization in an orthonormal representation An early onset of linear scaling is found for both minimal and double zeta basis sets, and crossovers with a highly optimized eigensolver are achieved Calculations with up to 6000 basis functions are reported The scaling of errors with system size is investigated for various levels of approximation

A simplified density matrix minimization for linear scaling self-consistent field theory

Publisher Summary This chapter discusses basic modeling. The chapter illustrates that quantum chemistry aims at understanding the properties of matter through the modeling of its behavior at a subatomic scale, where matter is described as an assembly of nuclei and electrons. At this scale, the equation that rules the interactions between these constitutive elements is the Schrodinger equation. It can be considered as a universal model for at least three reasons. First, it contains all the physical information of the system under consideration so that any of the properties of this system can be deduced in theory from the Schrodinger equation associated to it. Second, the Schrodinger equation does not involve any empirical parameter, except some fundamental constants of Physics; it can thus be written for any kind of molecular system provided its chemical composition, in terms of natures of nuclei and number of electrons, is known. Third, this model enjoys remarkable predictive capabilities, as confirmed by comparisons with a large amount of experimental data of various types.

Computational quantum chemistry: A primer

Two-electron reduced density matrix as the basic variable in many-electron quantum chemistry and physics.

Starting with Schr\"odinger's time-independent equation for $N$ indistinguishable particles, an exact equation is derived for the reduced density matrix of order $s$, ${\ensuremath{\rho}}_{s}$. The equation contains terms dependent on ${\ensuremath{\rho}}_{s+1}$ and ${\ensuremath{\rho}}_{s+2}$. The Hartree-Fock equation is derived, without recourse to the variational method, by assuming that ${\ensuremath{\rho}}_{2}$ and ${\ensuremath{\rho}}_{3}$ are particular functionals of ${\ensuremath{\rho}}_{1}$.

Hierarchy equations for reduced density matrices

An equation for the Hartree–Fock density matrix is discussed and the possibility of solving this equation directly for the density matrix instead of solving the Hartree–Fock equation for orbitals is considered. Toward that end the density matrix is expanded in a finite basis to obtain the matrix representative equation. The closed shell case is considered. Two numerical schemes are developed and applied to a number of examples. One example is given where the standard orbital method does not converge while the method presented here does.

C. Frishberg

Papers

Hierarchy equations for reduced density matrices

Hartree-Fock density matrix equation.