Bringing about matrix sparsity in linear-scaling electronic structure calculations

The performance of linear-scaling electronic structure calculations depends critically on matrix sparsity. This article gives an overview of different strategies for removal of small matrix elements, with emphasis on schemes that allow for rigorous control of errors. In particular, a novel scheme is proposed that has significantly smaller computational overhead compared with the Euclidean norm-based truncation scheme of Rubensson et al. (J Comput Chem 2009, 30, 974) while still achieving the desired asymptotic behavior required for linear scaling. Small matrix elements are removed while ensuring that the Euclidean norm of the error matrix stays below a desired value, so that the resulting error in the occupied subspace can be controlled. The efficiency of the new scheme is investigated in benchmark calculations for water clusters including up to 6523 water molecules. Furthermore, the foundation of matrix sparsity is investigated. This includes a study of the decay of matrix element magnitude with distance between basis function centers for different molecular systems and different methods. The studied methods include Hartree–Fock and density functional theory using both pure and hybrid functionals. The relation between band gap and decay properties of the density matrix is also discussed.     

Prescreening of two-electron integral derivatives in SCF gradient and Hessian calculations

The definition and implementation of a rigorous two-electron integral bound based on Schwarz' inequality both for gradient and hessian calculations is presented. Tests demonstrate the advantages of this easily implemented and effective bound.     

The two-electron integral transformation and two-body density matrix transformation

Ann ⁵ algorithm for the transformation of quantum-mechanical four centre functions is presented in a form best suited for computers having a virtual memory capability. Part of the work to be submitted for the degree of Ph. D. in the University of Newcastle-upon-Tyne.     

Analytic derivatives for the Cholesky representation of the two-electron integrals

We propose a formalism for calculating analytic derivatives of the electronic energy with respect to nuclear coordinates using Cholesky decomposition of the two-electron integrals. The formalism is derived by exploiting the equivalence of Cholesky decomposition and density fitting when a suitable auxiliary basis set is used for expanding atomic orbital product densities in the latter. An implementation of gradients at the nonhybrid density functional theory level is presented, and sample calculations demonstrate that the errors in equilibrium geometries due to the Cholesky representation of the integrals can be controlled by adjusting the decomposition threshold.     

Systematic sparse matrix error control for linear scaling electronic structure calculations

Efficient truncation criteria used in multiatom blocked sparse matrix operations for ab initio calculations are proposed. As system size increases, so does the need to stay on top of errors and still achieve high performance. A variant of a blocked sparse matrix algebra to achieve strict error control with good performance is proposed. The presented idea is that the condition to drop a certain submatrix should depend not only on the magnitude of that particular submatrix, but also on which other submatrices that are dropped. The decision to remove a certain submatrix is based on the contribution the removal would cause to the error in the chosen norm. We study the effect of an accumulated truncation error in iterative algorithms like trace correcting density matrix purification. One way to reduce the initial exponential growth of this error is presented. The presented error control for a sparse blocked matrix toolbox allows for achieving optimal performance by performing only necessary operations needed to maintain the requested level of accuracy.     

Computation of interior eigenvalues in electronic structure calculations facilitated by density matrix purification

Density matrix purification, is in this work, used to facilitate the computation of eigenpairs around the highest occupied and the lowest unoccupied molecular orbitals (HOMO and LUMO, respectively) in electronic structure calculations. The ability of purification to give large separation between eigenvalues close to the HOMO-LUMO gap is used to accelerate convergence of the Lanczos method. Illustrations indicate that a new eigenpair is found more often than every second Lanczos iteration when the proposed methods are used.     

Wavelets for electronic structure calculations

Molecular electronic structure calculations have a multi‐scale character through the presence of a set of singularities corresponding to atomic nuclei, and thus there exists a potential to improve the efficiency of these calculations using fast wavelet transform techniques. We report on the development of a one dimensional prototype benchmark problem of sufficient complexity to capture the features of 3‐D problems that are being solved today in quantum electronics calculations. Theoretical estimates of decay across scales and spatial distribution of wavelet coefficients for the solutions of the 1‐D and 3‐D problems are derived and verified experimentally. Equivalence in a multi‐resolution context of the solutions of the 1‐D prototype and the 3‐D problem is established. This revised version was published online in July 2006 with corrections to the Cover Date.     

On the Beebe-Linderberg two-electron integral approximation

The Beebe-Linderberg two-electron integral approximation, which is generated by a Cholesky decomposition of the two-electron integral matrix ([μν|λσ]), is slightly modified. On the basis of test calculations, two key questions concerning this approximation are discussed: The numerical rank of the two-electron integral matrix and the relationship between the integral threshold and electronic properties. The numerical results presented in this work suggest that the modified Beebe-Linderberg approximation might be considered as an alternative to effective core potential methods.     

Optimization of molecular electronic structure calculations. 2. Calculation of symmetry-reduced matrix elements

A suggested formalism of the local symmetricized orbitals in conjunction with the selection technique for independent blocks of integrals in an original basis is used for a construction of multielectron Hamiltonian matrix elements in the symmetry orbital basis. The optimal molecular electronic structure calculation algorithm with the Hartree–Fock–Roothaan method in the symmetricized basis was obtained as a result. The minimal number of fundamentally distinguished (symmetry attributed) elements both in original and in symmetricized basis is used in the calculations.     

Spectral differences in real-space electronic structure calculations

Real-space grids for electronic structure calculations are efficient because the potential is diagonal while the second derivative in the kinetic energy may be sparsely evaluated with finite differences or finite elements. In applications to vibrational problems in chemical physics a family of methods known as spectral differences has improved finite differences by several orders of magnitude. In this paper the use of spectral differences for electronic structure is studied. Spectral differences are implemented in two electronic structure programs PARSEC and HARES which currently employ finite differences. Applications to silicon clusters and lattices indicate that spectral differences achieve the same accuracy as finite differences with less computational work.     

The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions

The variational approach for electronic structure based on the two-body reduced density matrix is studied, incorporating two representability conditions beyond the previously used P, Q, and G conditions. The additional conditions (called T1 and T2 here) are implicit in the work of Erdahl [Int. J. Quantum Chem. 13, 697 (1978)] and extend the well-known three-index diagonal conditions also known as the Weinhold-Wilson inequalities. The resulting optimization problem is a semidefinite program, a convex optimization problem for which computational methods have greatly advanced during the past decade. Formulating the reduced density matrix computation using the standard dual formulation of semidefinite programming, as opposed to the primal one, results in substantial computational savings and makes it possible to study larger systems than was done previously. Calculations of the ground state energy and the dipole moment are reported for 47 different systems, in each case using an STO-6G basis set and comparing with Hartree-Fock, singly and doubly substituted configuration interaction, Brueckner doubles (with triples), coupled cluster singles and doubles with perturbational treatment of triples, and full configuration interaction calculations. It is found that the use of the T1 and T2 conditions gives a significant improvement over just the P, Q, and G conditions, and provides in all cases that we have studied more accurate results than the other mentioned approximations.     

Spectral-product methods for electronic structure calculations

Progress is reported in development, implementation, and application of a spectral method for ab initio studies of the electronic structure of matter. In this approach, antisymmetry restrictions are enforced subsequent to construction of the many-electron Hamiltonian matrix in a complete orthonormal spectral-product basis. Transformation to a permutation-symmetry representation obtained from the eigenstates of the aggregate electron antisymmetrizer is seen to enforce the requirements of the Pauli principle ex post facto, and to eliminate the unphysical (non-Pauli) states spanned by the product representation. Results identical with conventional use of prior antisymmetrization of configurational state functions are obtained in applications to many-electron atoms. The development provides certain advantages over conventional methods for polyatomic molecules, and, in particular, facilitates incorporation of fragment information in the form of Hermitian matrix representatives of atomic and diatomic operators which include the non-local effects of overall electron antisymmetry. An exact atomic-pair expression is obtained in this way for polyatomic Hamiltonian matrices which avoids the ambiguities of previously described semi-empirical fragment-based methods for electronic structure calculations. Illustrative applications to the well-known low-lying doublet states of the H₃ molecule in a minimal-basis-set demonstrate that the eigensurfaces of the antisymmetrizer can anticipate the structures of the more familiar energy surfaces, including the seams of intersection common in high-symmetry molecular geometries. The calculated H₃ energy surfaces are found to be in good agreement with corresponding valence-bond results which include all three-center terms, and are in general accord with accurate values obtained employing conventional high-level computational-chemistry procedures. By avoiding the repeated evaluations of the many-centered one- and two-electron integrals required in construction of polyatomic Hamiltonian matrices in the antisymmetric basis states commonly employed in conventional calculations, and by performing the required atomic and atomic-pair calculations once and for all, the spectral-product approach may provide an alternative potentially efficient ab initio formalism suitable for computational studies of adiabatic potential energy surfaces more generally. Contribution to the Mark S. Gordon 65th Birthday Festschrift Issue.     

Symmetry and equivalence restrictions in electronic structure calculations

A simple method for obtaining MCSCF orbitals and CI natural orbitals adapted to degenerate point groups, with full symmetry and equivalence restrictions, is described. Among several advantages accruing from this method are the ability to perform atomic SCF calculations on states for which the SCF energy expression cannot be written in terms of Coulomb and exchange integrals over real orbitals, and the generation of symmetry-adapted atomic natural orbitals for use in a recently proposed method for basis set contraction.     

Three-dimensional numerical integration for electronic structure calculations

Two three-dimensional numerical schemes are presented for molecular integrands such as matrix alements of one-electron operators occuring in the Fock operator and expectation values of one-electron operators describing molecular properties. The schemes are based on a judicious partitioning of space so that product-Gauss integration rules can be used in each region. Convergence with the number of integration points is such that very high accuracy (8–10 digits) may be obtained with obtained with a modest number of points. The use of point group symmetry to reduce the required number of points is discussed. Examples are given for overlap, nuclear potential, and electric field gradient integrals.     

Extrapolating to the one-electron basis-set limit in electronic structure calculations

A simple, yet reliable, scheme based on treating uniformly singlet-pair and triplet-pair interactions is suggested to extrapolate atomic and molecular electron correlation energies calculated at two basis-set levels of ab initio theory to the infinite one-electron basis-set limit. The novel dual-level method is first tested on extrapolating the full correlation in single-reference coupled-cluster singles and doubles energies for the closed-shell systems CH2((1)A1), H2O, HF, N2, CO, Ne, and F2 with correlation-consistent basis sets of the type cc-pVXZ (X=D,T,Q,5,6) reported by Klopper [Mol. Phys. 6, 481 (2001)] against his own benchmark calculations with large uncontracted basis sets obtained from explicit correlated singles and doubles coupled-cluster theory. Comparisons are also reported for the same data set but using both single-reference Moller-Plesset and coupled-cluster doubles methods. The results show a similar, often better, accordance with the target results than Klopper's extrapolations where singlet-pair and triplet-pair energies are extrapolated separately using the popular X(-3) and X(-5) dual-level laws, respectively. Applications to the extrapolation of the dynamical correlation in multireference configuration interaction calculations carried out anew for He, H2, HeH+, He2 ++, H3+(1 (1)A'), H3+(1 (3)A'), BH, CH, NH, OH, FH, B2, C2, N2, O2, F2, BO, CO, NO, BN, CN, SH, H2O, and NH3 with standard augmented correlation-consistent basis sets of the type aug-cc-pVXZ (X=D,T,Q,5,6) are also reported. Despite lacking accurate theoretical or experimental data for comparison in the case of most diatomic systems, the new method also shows in this case a good performance when judged from the results obtained with the traditional schemes which extrapolate using the two largest affordable basis sets. For the Hartree-Fock and complete-active space self-consistent field energies, a simple pragmatic extrapolation rule is examined whose results are shown to compare well with the ones obtained from the best reported schemes.     

Sparse matrix multiplications for linear scaling electronic structure calculations in an atom-centered basis set using multiatom blocks

A sparse matrix multiplication scheme with multiatom blocks is reported, a tool that can be very useful for developing linear-scaling methods with atom-centered basis functions. Compared to conventional element-by-element sparse matrix multiplication schemes, efficiency is gained by the use of the highly optimized basic linear algebra subroutines (BLAS). However, some sparsity is lost in the multiatom blocking scheme because these matrix blocks will in general contain negligible elements. As a result, an optimal block size that minimizes the CPU time by balancing these two effects is recovered. In calculations on linear alkanes, polyglycines, estane polymers, and water clusters the optimal block size is found to be between 40 and 100 basis functions, where about 55-75% of the machine peak performance was achieved on an IBM RS6000 workstation. In these calculations, the blocked sparse matrix multiplications can be 10 times faster than a standard element-by-element sparse matrix package.