Linear scaling computation of the Fock matrix. V. Hierarchical Cubature for numerical integration of the exchange-correlation matrix

Quantum Solvers for Plane-Wave Hamiltonians: Abridging Virtual Spaces Through the Optimization of Pairwise Correlations

For many-body methods such as MCSCF and CASSCF, in which the number of one-electron orbitals is optimized and independent of the basis set used, there are no problems with using plane-wave basis sets. However, for methods currently used in quantum computing such as select configuration interaction (CI) and coupled cluster (CC) methods, it is necessary to have a virtual space that is able to capture a significant amount of electron-electron correlation in the system. The virtual orbitals in a pseudopotential plane-wave Hartree-Fock calculation, because of Coulomb repulsion, are often scattering states that interact very weakly with the filled orbitals. As a result, very little correlation energy is captured from them. The use of virtual spaces derived from the one-electron operators has also been tried, and while some correlations are captured, the amount is quite low. To overcome these limitations, we have been developing new classes of algorithms to define virtual spaces by optimizing orbitals from small pairwise CI Hamiltonians, which we term as correlation optimized virtual orbitals with the abbreviation COVOs. With these procedures, we have been able to derive virtual spaces, containing only a few orbitals, which are able to capture a significant amount of correlation. The focus in this manuscript is on using these derived basis sets to target full CI (FCI) quality results for H2 on near-term quantum computers. However, the initial results for this approach were promising. We were able to obtain good agreement with FCI/cc-pVTZ results for this system with just 4 virtual orbitals, using both FCI and quantum simulations. The quality of the results using COVOs suggests that it may be possible to use them in other many-body approaches, including coupled cluster and Møller-Plesset perturbation theories, and open up the door to many-body calculations for pseudopotential plane-wave basis set methods.

Intermediate electrostatic field for the elongation method

A simple way to improve the accuracy of the fragmentation methods is proposed. The formalism was applied to the elongation (ELG) method at restricted open-shell Hartree-Fock (ROHF) level of theory. The α-helix conformer of polyglycine was taken as a model system. The modified ELG method includes a simplified electrostatic field resulting from point-charge distribution of the system's environment. In this way the long-distance polarization is approximately taken into account. The field attenuates during the ELG process to eventually disappear when the final structure is reached. The point-charge distributions for each ELG step are obtained from charge sensitivity analysis (CSA) in force-field atoms resolution. The presence of the intermediate field improves the accuracy of ELG calculations. The errors in total energy and its kinetic and potential contributions are reduced by at least one-order of magnitude. In addition the SCF convergence of ROHF scheme is improved.

A variational linear-scaling framework to build practical, efficient next-generation orbital-based quantum force fields

We introduce a new hybrid molecular orbital/density-functional modified divide-and-conquer (mDC) approach that allows the linear-scaling calculation of very large quantum systems. The method provides a powerful framework from which linear-scaling force fields for molecular simulations can be developed. The method is variational in the energy, and has simple, analytic gradients and essentially no break-even point with respect to the corresponding full electronic structure calculation. Furthermore, the new approach allows intermolecular forces to be properly balanced such that non-bonded interactions can be treated, in some cases, to much higher accuracy than the full calculation. The approach is illustrated using the second-order self-consistent charge density-functional tight-binding model (DFTB2). Using this model as a base Hamiltonian, the new mDC approach is applied to a series of water systems, where results show that geometries and interaction energies between water molecules are greatly improved relative to full DFTB2. In order to achieve substantial improvement in the accuracy of intermolecular binding energies and hydrogen bonded cluster geometries, it was necessary to extend the DFTB2 model to higher-order atom-centered multipoles for the second-order self-consistent intermolecular electrostatic term. Using generalized, linear-scaling electrostatic methods, timings demonstrate that the method is able to calculate a water system of 3000 atoms in less than half of a second, and systems of up to one million atoms in only a few minutes using a conventional desktop workstation.

O(N) methods in electronic structure calculations

Linear-scaling methods, or O(N) methods, have computational and memory requirements which scale linearly with the number of atoms in the system, N, in contrast to standard approaches which scale with the cube of the number of atoms. These methods, which rely on the short-ranged nature of electronic structure, will allow accurate, ab initio simulations of systems of unprecedented size. The theory behind the locality of electronic structure is described and related to physical properties of systems to be modelled, along with a survey of recent developments in real-space methods which are important for efficient use of high-performance computers. The linear-scaling methods proposed to date can be divided into seven different areas, and the applicability, efficiency and advantages of the methods proposed in these areas are then discussed. The applications of linear-scaling methods, as well as the implementations available as computer programs, are considered. Finally, the prospects for and the challenges facing linear-scaling methods are discussed.

Linear Scaling Hierarchical Integration Scheme for the Exchange-Correlation Term in Molecular and Periodic Systems

An adaptive numerical integration scheme for efficient evaluation of the exchange-correlation term using localized basis functions and atom-centered grids is presented. The method treats molecules and systems with periodic boundary conditions on an equal footing. Its computational efficiency and O(N) scaling with the system size is achieved by a hierarchical spatial grouping of basis functions and grid points using an octree. This allows for an efficient screening of negligible contributions and an optimal use of hardware-optimized matrix-matrix multiplication subroutines, such as BLAS. The implementation of the method within the TURBOMOLE program package demonstrates consistent accuracy and efficiency across molecular and periodic systems.

Parallel algorithm for the computation of the Hartree-Fock exchange matrix: Gas phase and periodic parallel ONX

In this paper we present an efficient parallelization of the ONX algorithm for linear computation of the Hartree-Fock exchange matrix [J. Chem. Phys. 106, 9708 (1997)]. The method used is based on the equal time (ET) partitioning recently introduced [J. Chem. Phys. 118, 9128 (2003)] and [J. Chem. Phys. 121, 6608 (2004)]. ET exploits the slow variation of the density matrix between self-consistent-field iterations to achieve load balance. The method is presented and some benchmark calculations are discussed for gas phase and periodic systems with up to 128 processors. The current parallel ONX code is able to deliver up to 77% overall efficiency for a cluster of 50 water molecules on 128 processors (2.56 processors per heavy atom) and up to 87% for a box of 64 water molecules (two processors per heavy atom) with periodic boundary conditions.

Energy gradients with respect to atomic positions and cell parameters for the Kohn-Sham density-functional theory at the Γ point

The application of theoretical methods based on density-functional theory is known to provide atomic and cell parameters in very good agreement with experimental values. Recently, construction of the exact Hartree-Fock exchange gradients with respect to atomic positions and cell parameters within the Gamma-point approximation has been introduced. In this article, the formalism is extended to the evaluation of analytical Gamma-point density-functional atomic and cell gradients. The infinite Coulomb summation is solved with an effective periodic summation of multipole tensors. While the evaluation of Coulomb and exchange-correlation gradients with respect to atomic positions are similar to those in the gas phase limit, the gradients with respect to cell parameters needs to be treated with some care. The derivative of the periodic multipole interaction tensor needs to be carefully handled in both direct and reciprocal space and the exchange-correlation energy derivative leads to a surface term that has its origin in derivatives of the integration limits that depend on the cell. As an illustration, the analytical gradients have been used in conjunction with the QUICCA algorithm to optimize one-dimensional and three-dimensional periodic systems at the density-functional theory and hybrid Hartree-Fock/density-functional theory levels. We also report the full relaxation of forsterite supercells at the B3LYP level of theory.

DISTRIBUTED ADAPTIVE MULTIVARIATE FUNCTION VISUALIZATION

In this article, we introduce a new function visualization method and demonstrate that numerical integration and visualization of multi-dimensional functions are closely related. Adaptive numerical integration is utilized to reduce the number of function evaluations, and generate time series data. The integration region is partitioned into a uniform grid. A grid cell can be sampled many times, or is not sampled at all, depending on the function properties and the integration rule. Function properties are extracted during the process of function evaluation. An aging technique helps visualize functions by retaining the most recently sampled areas and making the older ones transparent. This also results in giving the non-smooth areas more attention than the smooth areas. The new function visualization method gives a view of the whole function while elaborating on important areas such as ridges and troughs, which are critical in many fields, including numerical integration. A Grid service, called Integration Service, is used to solve computationally intensive integration problems. Remote visualization based on the adaptive method helps monitor the progress of a computation, and can be utilized for computational steering. The data are filtered by the server and transferred to the client, which is responsible for visualization mapping and rendering.

Linear scaling computation of the Fock matrix. VIII. Periodic boundaries for exact exchange at the Gamma point

A translationally invariant formulation of the Hartree-Fock (HF) Gamma-point approximation is presented. This formulation is achieved through introduction of the minimum image convention (MIC) at the level of primitive two-electron integrals, and implemented in a periodic version of the ONX algorithm [E. Schwegler, M. Challacombe, and M. Head-Gordon, J. Chem. Phys. 106, 9708 (1997)] for linear scaling computation of the exchange matrix. Convergence of the HF-MIC Gamma-point model to the HF k-space limit is demonstrated for fully periodic magnesium oxide, ice, and diamond. Computation of the diamond lattice constant using the HF-MIC model together with the hybrid PBE0 density functional [C. Adamo, M. Cossi, and V. Barone, THEOCHEM 493, 145 (1999)] yields a0=3.569 A with the 6-21G* basis set and a 3x3x3 supercell. Linear scaling computation of the HF-MIC exchange matrix is demonstrated for diamond and ice in the condensed phase.

Linear scaling computation of the Fock matrix. VII. Periodic density functional theory at the Gamma point

Linear scaling quantum chemical methods for density functional theory are extended to the condensed phase at the Gamma point. For the two-electron Coulomb matrix, this is achieved with a tree-code algorithm for fast Coulomb summation [M. Challacombe and E. Schwegler, J. Chem. Phys. 106, 5526 (1997)], together with multipole representation of the crystal field [M. Challacombe, C. White, and M. Head-Gordon, J. Chem. Phys. 107, 10131 (1997)]. A periodic version of the hierarchical cubature algorithm [M. Challacombe, J. Chem. Phys. 113, 10037 (2000)], which builds a telescoping adaptive grid for numerical integration of the exchange-correlation matrix, is shown to be efficient when the problem is posed as integration over the unit cell. Commonalities between the Coulomb and exchange-correlation algorithms are discussed, with an emphasis on achieving linear scaling through the use of modern data structures. With these developments, convergence of the Gamma-point supercell approximation to the k-space integration limit is demonstrated for MgO and NaCl. Linear scaling construction of the Fockian and control of error is demonstrated for RBLYP6-21G* diamond up to 512 atoms.

Efficient computation of the exchange-correlation contribution in the density functional theory through multiresolution

A new algorithm is presented to improve the efficiency of the computation of exchange-correlation contributions, a major time-consuming step in a density functional theory (DFT) calculation. The new method, called multiresolution exchange correlation (mrXC), takes advantage of the variation in resolution among the Gaussian basis functions and shifts the calculation associated with low-resolution (smooth) basis function pairs to an even-spaced cubic grid. The cubic grid is much less dense in the vicinity of the nuclei than the atom-centered grid and the computation on the former is shown to be much more efficient than on the latter. MrXC does not alter the formalism of the current standard algorithm based on the atom-centered grid (ACG), but instead employs two fast and accurate transformations between the ACG and the cubic grid. Preliminary results with local density approximation have shown that mrXC yields three to five times improvement in efficiency with negligible error. The extension to DFT functionals with generalized gradient approximation is also briefly discussed.

Linear scaling computation of the Fock matrix. VII. Parallel computation of the Coulomb matrix

We present parallelization of a quantum-chemical tree-code for linear scaling computation of the Coulomb matrix. Equal time partition is used to load balance computation of the Coulomb matrix. Equal time partition is a measurement based algorithm for domain decomposition that exploits small variation of the density between self-consistent-field cycles to achieve load balance. Efficiency of the equal time partition is illustrated by several tests involving both finite and periodic systems. It is found that equal time partition is able to deliver 91%-98% efficiency with 128 processors in the most time consuming part of the Coulomb matrix calculation. The current parallel quantum chemical tree code is able to deliver 63%-81% overall efficiency on 128 processors with fine grained parallelism (less than two heavy atoms per processor).

Linear-scaling formation of Kohn-Sham Hamiltonian: Application to the calculation of excitation energies and polarizabilities of large molecular systems

We present calculations of excitation energies and polarizabilities in large molecular systems at the local-density and generalized-gradient approximation levels of density-functional theory (DFT). Our results are obtained using a linear-scaling DFT implementation in the program system DALTON for the formation of the Kohn-Sham Hamiltonian. For the Coulomb contribution, we introduce a modification of the fast multipole method to calculations over Gaussian charge distributions. It affords a simpler implementation than the original continuous fast multipole method by partitioning the electrostatic Coulomb interactions into "classical" and "nonclassical" terms which are explicitly evaluated by linear-scaling multipole techniques and a modified two-electron integral code, respectively. As an illustration of the code, we have studied the singlet and triplet excitation energies as well as the static and dynamic polarizabilities of polyethylenes, polyenes, polyynes, and graphite sheets with an emphasis on the trends observed with system size.

Efficient and reliable numerical integration of exchange-correlation energies and potentials

An adaptive numerical integrator for the exchange-correlation energy and potential is presented. It uses the diagonal elements of the exchange-correlation potential matrix as a grid generating function. The only input parameter is the requested grid tolerance. In combination with a defined cell function the adaptive grid generation scales almost linear with the number of basis functions in a system. With the adaptive numerical integrator the self-consistent field energy error, which is due to the numerical integration of the exchange-correlation energy, converges with increasing adaptive grid size to a reference value. The performance of the adaptive numerical integration is analyzed using molecules with first, second, and third row elements. Especially for transition metal systems the adaptive numerical integrator shows considerably improved performance and reliability.

A novel parallel algorithm for large-scale Fock matrix construction with small locally distributed memory architectures: RT parallel algorithm

We developed a novel parallel algorithm for large-scale Fock matrix calculation with small locally distributed memory architectures, and named it the "RT parallel algorithm." The RT parallel algorithm actively involves the concept of integral screening, which is indispensable for reduction of computing times with large-scale biological molecules. The primary characteristic of this algorithm is parallel efficiency, which is achieved by well-balanced reduction of both communicating and computing volume. Only the density matrix data necessary for Fock matrix calculations are communicated, and the data once communicated are reutilized for calculations as many times as possible. The RT parallel algorithm is a scalable method because required memory volume does not depend on the number of basis functions. This algorithm automatically includes a partial summing technique that is indispensable for maintaining computing accuracy, and can also include some conventional methods to reduce calculation times. In our analysis, the RT parallel algorithm had better performance than other methods for massively parallel processors. The RT parallel algorithm is most suitable for massively parallel and distributed Fock matrix calculations for large-scale biological molecules with more than thousands of basis functions.