Hartree–Fock calculations with linearly scaling memory usage

Multinode Multi-GPU Two-Electron Integrals: Code Generation Using the Regent Language

The computation of two-electron repulsion integrals (ERIs) is often the most expensive step of integral-direct self-consistent field methods. Formally it scales as O(N⁴), where N is the number of Gaussian basis functions used to represent the molecular wave function. In practice, this scaling can be reduced to O(N²) or less by neglecting small integrals with screening methods. The contributions of the ERIs to the Fock matrix are of Coulomb (J) and exchange (K) type and require separate algorithms to compute matrix elements efficiently. We previously implemented highly efficient GPU-accelerated J-matrix and K-matrix algorithms in the electronic structure code TeraChem. Although these implementations supported the use of multiple GPUs on a node, they did not support the use of multiple nodes. This presents a key bottleneck to cutting-edge ab initio simulations of large systems, e.g., excited state dynamics of photoactive proteins. We present our implementation of multinode multi-GPU J- and K-matrix algorithms in TeraChem using the Regent programming language. Regent directly supports distributed computation in a task-based model and can generate code for a variety of architectures, including NVIDIA GPUs. We demonstrate multinode scaling up to 45 GPUs (3 nodes) and benchmark against hand-coded TeraChem integral code. We also outline our metaprogrammed Regent implementation, which enables flexible code generation for integrals of different angular momenta.

Linear-Scaling Implementation of Multilevel Hartree-Fock Theory

We introduce a new algorithm for the construction of the two-electron contributions to the Fock matrix in multilevel Hartree-Fock (MLHF) theory. In MLHF, the density of an active molecular region is optimized, while the density of an inactive region is fixed. The MLHF equations are solved in a reduced molecular orbital (MO) basis localized to the active region. The locality of the MOs can be exploited to reduce the computational cost of the Fock matrix: the cost related to the inactive density becomes linear scaling, while the iterative cost related to the active density is independent of the system size. We demonstrate the performance of this new algorithm on a variety of systems, including amino acid chains, water clusters, and solvated systems.

Low-Order Scaling G0W0 by Pair Atomic Density Fitting

We derive a low-scaling G0W0 algorithm for molecules using pair atomic density fitting (PADF) and an imaginary time representation of the Green's function and describe its implementation in the Slater type orbital (STO)-based Amsterdam density functional (ADF) electronic structure code. We demonstrate the scalability of our algorithm on a series of water clusters with up to 432 atoms and 7776 basis functions and observe asymptotic quadratic scaling with realistic threshold qualities controlling distance effects and basis sets of triple-ζ (TZ) plus double polarization quality. Also owing to a very small prefactor, a G0W0 calculation for the largest of these clusters takes only 240 CPU hours with these settings. We assess the accuracy of our algorithm for HOMO and LUMO energies in the GW100 database. With errors of 0.24 eV for HOMO energies on the quadruple-ζ level, our implementation is less accurate than canonical all-electron implementations using the larger def2-QZVP GTO-type basis set. Apart from basis set errors, this is related to the well-known shortcomings of the GW space-time method using analytical continuation techniques as well as to numerical issues of the PADF approach of accurately representing diffuse atomic orbital (AO) products. We speculate that these difficulties might be overcome by using optimized auxiliary fit sets with more diffuse functions of higher angular momenta. Despite these shortcomings, for subsets of medium and large molecules from the GW5000 database, the error of our approach using basis sets of TZ and augmented double-ζ (DZ) quality is decreasing with system size. On the augmented DZ level, we reproduce canonical, complete basis set limit extrapolated reference values with an accuracy of 80 meV on average for a set of 20 large organic molecules. We anticipate our algorithm, in its current form, to be very useful in the study of single-particle properties of large organic systems such as chromophores and acceptor molecules.

Polarization consistent basis sets using the projector augmented wave method: a renovation brought by PAW into Gaussian basis sets

A new family of polarization consistent basis set, combined with the projector augmented wave method, was introduced. The basis sets are compact and have good performance as compared to conventional all-electron basis sets in DFT calculations.

Building a More Predictive Protein Force Field: A Systematic and Reproducible Route to AMBER-FB15

The increasing availability of high-quality experimental data and first-principles calculations creates opportunities for developing more accurate empirical force fields for simulation of proteins. We developed the AMBER-FB15 protein force field by building a high-quality quantum chemical data set consisting of comprehensive potential energy scans and employing the ForceBalance software package for parameter optimization. The optimized potential surface allows for more significant thermodynamic fluctuations away from local minima. In validation studies where simulation results are compared to experimental measurements, AMBER-FB15 in combination with the updated TIP3P-FB water model predicts equilibrium properties with equivalent accuracy, and temperature dependent properties with significantly improved accuracy, in comparison with published models. We also discuss the effect of changing the protein force field and water model on the simulation results.

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.

Sparse Projected-Gradient Method As a Linear-Scaling Low-Memory Alternative to Diagonalization in Self-Consistent Field Electronic Structure Calculations

Large-scale electronic structure calculations usually involve huge nonlinear eigenvalue problems. A method for solving these problems without employing expensive eigenvalue decompositions of the Fock matrix is presented in this work. The sparsity of the input and output matrices is preserved at every iteration, and the memory required by the algorithm scales linearly with the number of atoms of the system. The algorithm is based on a projected gradient iteration applied to the constraint fulfillment problem. The computer time required by the algorithm also scales approximately linearly with the number of atoms (or non-null elements of the matrices), and the algorithm is faster than standard implementations of modern eigenvalue decomposition methods for sparse matrices containing more than 50 000 non-null elements. The new method reproduces the sequence of semiempirical SCF iterations obtained by standard eigenvalue decomposition algorithms to good precision.

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.

Difficulties in applying pure Kohn-Sham density functional theory electronic structure methods to protein molecules

Self-consistency-based Kohn-Sham density functional theory (KS-DFT) electronic structure calculations with Gaussian basis sets are reported for a set of 17 protein-like molecules with geometries obtained from the Protein Data Bank. It is found that in many cases such calculations do not converge due to vanishing HOMO-LUMO gaps. A sequence of polyproline I helix molecules is also studied and it is found that self-consistency calculations using pure functionals fail to converge for helices longer than six proline units. Since the computed gap is strongly correlated to the fraction of Hartree-Fock exchange, test calculations using both pure and hybrid density functionals are reported. The tested methods include the pure functionals BLYP, PBE and LDA, as well as Hartree-Fock and the hybrid functionals BHandHLYP, B3LYP and PBE0. The effect of including solvent molecules in the calculations is studied, and it is found that the inclusion of explicit solvent molecules around the protein fragment in many cases gives a larger gap, but that convergence problems due to vanishing gaps still occur in calculations with pure functionals. In order to achieve converged results, some modeling of the charge distribution of solvent water molecules outside the electronic structure calculation is needed. Representing solvent water molecules by a simple point charge distribution is found to give non-vanishing HOMO-LUMO gaps for the tested protein-like systems also for pure functionals.

A new parametrizable model of molecular electronic structure

A new electronic structure model is developed in which the ground state energy of a molecular system is given by a Hartree-Fock-like expression with parametrized one- and two-electron integrals over an extended (minimal + polarization) set of orthogonalized atom-centered basis functions, the variational equations being solved formally within the minimal basis but the effect of polarization functions being included in the spirit of second-order perturbation theory. It is designed to yield good dipole polarizabilities and improved intermolecular potentials with dispersion terms. The molecular integrals include up to three-center one-electron and two-center two-electron terms, all in simple analytical forms. A method to extract the effective one-electron Hamiltonian of nonlocal-exchange Kohn-Sham theory from the coupled-cluster one-electron density matrix is designed and used to get its matrix representation in a molecule-intrinsic minimal basis as an input to the parametrization procedure--making a direct link to the correlated wavefunction theory. The model has been trained for 15 elements (H, Li-F, Na-Cl, 720 parameters) on a set of 5581 molecules (including ions, transition states, and weakly bound complexes) whose first- and second-order properties were computed by the coupled-cluster theory as a reference, and a good agreement is seen. The model looks promising for the study of large molecular systems, it is believed to be an important step forward from the traditional semiempirical models towards higher accuracy at nearly as low a computational cost.

Assessment of density matrix methods for linear scaling electronic structure calculations

Purification and minimization methods for linear scaling computation of the one-particle density matrix for a fixed Hamiltonian matrix are compared. This is done by considering the work needed by each method to achieve a given accuracy in terms of the difference from the exact solution. Numerical tests employing orthogonal as well as non-orthogonal versions of the methods are performed using both element magnitude and cutoff radius based truncation approaches. It is investigated how the convergence speed for the different methods depends on the eigenvalue distribution in the Hamiltonian matrix. An expression for the number of iterations required for the minimization methods studied is derived, taking into account the dependence on both the band gap and the chemical potential. This expression is confirmed by numerical tests. The minimization methods are found to perform at their best when the chemical potential is located near the center of the eigenspectrum. The results indicate that purification is considerably more efficient than the minimization methods studied even when a good starting guess for the minimization is available. In test calculations without a starting guess, purification is more than an order of magnitude more efficient than minimization.