New parallel optimal-parameter fast multipole method (OPFMM)

Divide-and-Conquer Linear-Scaling Quantum Chemical Computations

Fragmentation and embedding schemes are of great importance when applying quantum-chemical calculations to more complex and attractive targets. The divide-and-conquer (DC)-based quantum-chemical model is a fragmentation scheme that can be connected to embedding schemes. This feature article explains several DC-based schemes developed by the authors over the last two decades, which was inspired by the pioneering study of DC self-consistent field (SCF) method by Yang and Lee (J. Chem. Phys. 1995, 103, 5674-5678). First, the theoretical aspects of the DC-based SCF, electron correlation, excited-state, and nuclear orbital methods are described, followed by the two-component relativistic theory, quantum-mechanical molecular dynamics simulation, and the introduction of three programs, including DC-based schemes. Illustrative applications confirmed the accuracy and feasibility of the DC-based schemes.

The grid-based fast multipole method – a massively parallel numerical scheme for calculating two-electron interaction energies

A grid-based fast multipole method has been developed for calculating two-electron interaction energies for non-overlapping charge densities.

Systematic fragmentation method and the effective fragment potential: an efficient method for capturing molecular energies

The systematic fragmentation method fragments a large molecular system into smaller pieces, in such a way as to greatly reduce the computational cost while retaining nearly the accuracy of the parent ab initio electronic structure method. In order to attain the desired (sub-kcal/mol) accuracy, one must properly account for the nonbonded interactions between the separated fragments. Since, for a large molecular species, there can be a great many fragments and therefore a great many nonbonded interactions, computations of the nonbonded interactions can be very time-consuming. The present work explores the efficacy of employing the effective fragment potential (EFP) method to obtain the nonbonded interactions since the EFP method has been shown previously to capture nonbonded interactions with an accuracy that is often comparable to that of second-order perturbation theory. It is demonstrated that for nonbonded interactions that are not high on the repulsive wall (generally >2.7 A), the EFP method appears to be a viable approach for evaluating the nonbonded interactions. The efficacy of the EFP method for this purpose is illustrated by comparing the method to ab initio methods for small water clusters, the ZOVGAS molecule, retinal, and the alpha-helix. Using SFM with EFP for nonbonded interactions yields an error of 0.2 kcal/mol for the retinal cis-trans isomerization and a mean error of 1.0 kcal/mol for the isomerization energies of five small (120-170 atoms) alpha-helices.

An error-controlled fast multipole method

We present a two-stage error estimation scheme for the fast multipole method (FMM). This scheme can be applied to any particle system. It incorporates homogeneous as well as inhomogeneous distributions. The FMM error as a consequence of the finite representation of the multipole expansions and the operator error is correlated with an absolute or relative user-requested energy threshold. Such a reliable error control is the basis for making reliable simulations in computational physics. Our FMM program on the basis of the two-stage error estimation scheme is available on request.

Second-order Møller-Plesset perturbation energy obtained from divide-and-conquer Hartree-Fock density matrix

The density matrix (DM) obtained from Yang's [Phys. Rev. Lett. 66, 1438 (1991)] divide-and-conquer (DC) Hartree-Fock (HF) calculation is applied to the explicit second-order Møller-Plesset perturbation (MP2) energy functional of the HF DM, which was firstly mentioned by Ayala and Scuseria [J. Chem. Phys. 110, 3660 (1999)] and was improved by Surján [Chem. Phys. Lett. 406, 318 (2005)] as DM-Laplace MP2. This procedure, termed DC-DM MP2, requires the HF DM of holes, for which we propose two evaluation schemes in DC manner. Numerical studies reveal that the DC-DM MP2 energy deviation from canonical MP2 is the same order of magnitude as DC-HF energy deviation from conventional HF whichever type of hole DM is adopted. It is also confirmed that the central processing unit time of DC-DM MP2 is less than that of DM-Laplace MP2 because the DC-HF DM is sparser than conventional DM.

Efficient implementation of the fast multipole method

A number of computational techniques are described that reduce the effort related to the continuous fast multipole method, used for the evaluation of Coulomb matrix elements as needed in Hartree-Fock and density functional theories. A new extent definition for Gaussian charge distributions is proposed, as well as a new way of dividing distributions into branches. Also, a new approach for estimating the error caused by truncation of multipole expansions is presented. It is found that the use of dynamically truncated multipole expansions gives a speedup of a factor of 10 in the work required for multipole interactions, compared to the case when all interactions are computed using a fixed multipole expansion order. Results of benchmark calculations on three-dimensional systems are reported, demonstrating the usefulness of our present implementation of the fast multipole method.

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).

Revisiting infinite lattice sums with the periodic fast multipole method

The evaluation of lattice sums as well as stress lattice sums encountered in the periodic fast multipole method is reinvestigated. Simple, accurate, and efficient recurrence expressions for such sums following the ideas of the renormalization method are derived. The first few nonzero lattice sum terms in a three-dimensional cubic lattice are computed and given in Tables. The practical considerations accompanying the computation of the sums such as convergence and accuracy are discussed.

Direct determination of multipole moments of Cartesian Gaussian functions in spherical polar coordinates

A new way of generating the multipole moments of Cartesian Gaussian functions in spherical polar coordinates has been established, bypassing the intermediary of Cartesian moment tensors. A new set of recurrence relations have also been derived for the resulting analytic integral values. The new method furnishes a conceptually simple and numerically efficient evaluation procedure for the multipole moments. The advantages over existing methods are documented. The results are relevant for the linear scaling quantum theories based on the fast multipole method.

Development of hardware accelerator for molecular dynamics simulations: a computation board that calculates nonbonded interactions in cooperation with fast multipole method

Evaluation of long-range Coulombic interactions still represents a bottleneck in the molecular dynamics (MD) simulations of biological macromolecules. Despite the advent of sophisticated fast algorithms, such as the fast multipole method (FMM), accurate simulations still demand a great amount of computation time due to the accuracy/speed trade-off inherently involved in these algorithms. Unless higher order multipole expansions, which are extremely expensive to evaluate, are employed, a large amount of the execution time is still spent in directly calculating particle-particle interactions within the nearby region of each particle. To reduce this execution time for pair interactions, we developed a computation unit (board), called MD-Engine II, that calculates nonbonded pairwise interactions using a specially designed hardware. Four custom arithmetic-processors and a processor for memory manipulation ("particle processor") are mounted on the computation board. The arithmetic processors are responsible for calculation of the pair interactions. The particle processor plays a central role in realizing efficient cooperation with the FMM. The results of a series of 50-ps MD simulations of a protein-water system (50,764 atoms) indicated that a more stringent setting of accuracy in FMM computation, compared with those previously reported, was required for accurate simulations over long time periods. Such a level of accuracy was efficiently achieved using the cooperative calculations of the FMM and MD-Engine II. On an Alpha 21264 PC, the FMM computation at a moderate but tolerable level of accuracy was accelerated by a factor of 16.0 using three boards. At a high level of accuracy, the cooperative calculation achieved a 22.7-fold acceleration over the corresponding conventional FMM calculation. In the cooperative calculations of the FMM and MD-Engine II, it was possible to achieve more accurate computation at a comparable execution time by incorporating larger nearby regions.