Simen Reine

The Dalton quantum chemistry program system

Dalton is a powerful general‐purpose program system for the study of molecular electronic structure at the Hartree–Fock, Kohn–Sham, multiconfigurational self‐consistent‐field, Møller–Plesset, configuration‐interaction, and coupled‐cluster levels of theory. Apart from the total energy, a wide variety of molecular properties may be calculated using these electronic‐structure models. Molecular gradients and Hessians are available for geometry optimizations, molecular dynamics, and vibrational studies, whereas magnetic resonance and optical activity can be studied in a gauge‐origin‐invariant manner. Frequency‐dependent molecular properties can be calculated using linear, quadratic, and cubic response theory. A large number of singlet and triplet perturbation operators are available for the study of one‐, two‐, and three‐photon processes. Environmental effects may be included using various dielectric‐medium and quantum‐mechanics/molecular‐mechanics models. Large molecules may be studied using linear‐scaling and massively parallel algorithms. Dalton is distributed at no cost from http://www.daltonprogram.org for a number of UNIX platforms.

Linear-scaling implementation of molecular response theory in self-consistent field electronic-structure theory

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field theories for the calculation of frequency-dependent molecular response properties and excitation energies is presented, based on a nonredundant exponential parametrization of the one-electron density matrix in the atomic-orbital basis, avoiding the use of canonical orbitals. The response equations are solved iteratively, by an atomic-orbital subspace method equivalent to that of molecular-orbital theory. Important features of the subspace method are the use of paired trial vectors (to preserve the algebraic structure of the response equations), a nondiagonal preconditioner (for rapid convergence), and the generation of good initial guesses (for robust solution). As a result, the performance of the iterative method is the same as in canonical molecular-orbital theory, with five to ten iterations needed for convergence. As in traditional direct Hartree-Fock and Kohn-Sham theories, the calculations are dominated by the construction of the effective Fock/Kohn-Sham matrix, once in each iteration. Linear complexity is achieved by using sparse-matrix algebra, as illustrated in calculations of excitation energies and frequency-dependent polarizabilities of polyalanine peptides containing up to 1400 atoms.

Linear scaling implementation of molecular electronic self-consistent field theory.

A linear-scaling implementation of Hartree-Fock and Kohn-Sham self-consistent field (SCF) theories is presented and illustrated with applications to molecules consisting of more than 1000 atoms. The diagonalization bottleneck of traditional SCF methods is avoided by carrying out a minimization of the Roothaan-Hall (RH) energy function and solving the Newton equations using the preconditioned conjugate-gradient (PCG) method. For rapid PCG convergence, the Lowdin orthogonal atomic orbital basis is used. The resulting linear-scaling trust-region Roothaan-Hall (LS-TRRH) method works by the introduction of a level-shift parameter in the RH Newton equations. A great advantage of the LS-TRRH method is that the optimal level shift can be determined at no extra cost, ensuring fast and robust convergence of both the SCF iterations and the level-shifted Newton equations. For density averaging, the authors use the trust-region density-subspace minimization (TRDSM) method, which, unlike the traditional direct inversion in the iterative subspace (DIIS) scheme, is firmly based on the principle of energy minimization. When combined with a linear-scaling evaluation of the Fock/Kohn-Sham matrix (including a boxed fitting of the electron density), LS-TRRH and TRDSM methods constitute the linear-scaling trust-region SCF (LS-TRSCF) method. The LS-TRSCF method compares favorably with the traditional SCF/DIIS scheme, converging smoothly and reliably in cases where the latter method fails. In one case where the LS-TRSCF method converges smoothly to a minimum, the SCF/DIIS method converges to a saddle point.

Variational and robust density fitting of four-center two-electron integrals in local metrics

Density fitting is an important method for speeding up quantum-chemical calculations. Linear-scaling developments in Hartree-Fock and density-functional theories have highlighted the need for linear-scaling density-fitting schemes. In this paper, we present a robust variational density-fitting scheme that allows for solving the fitting equations in local metrics instead of the traditional Coulomb metric, as required for linear scaling. Results of fitting four-center two-electron integrals in the overlap and the attenuated Gaussian damped Coulomb metric are presented, and we conclude that density fitting can be performed in local metrics at little loss of chemical accuracy. We further propose to use this theory in linear-scaling density-fitting developments.

Molecular gradient for second-order Møller-Plesset perturbation theory using the divide-expand-consolidate (DEC) scheme

We demonstrate that the divide-expand-consolidate (DEC) scheme--which has previously been used to determine the second-order Møller-Plesset (MP2) correlation energy--can be applied to evaluate the MP2 molecular gradient in a linear-scaling and embarrassingly parallel manner using a set of local Hartree-Fock orbitals. All manipulations of four-index quantities (describing electron correlation effects) are carried out using small local orbital fragment spaces, whereas two-index quantities are treated for the full molecular system. The sizes of the orbital fragment spaces are determined in a black-box manner to ensure that the error in the DEC-MP2 correlation energy compared to a standard MP2 calculation is proportional to a single input threshold denoted the fragment optimization threshold (FOT). The FOT also implicitly controls the error in the DEC-MP2 molecular gradient as substantiated by a theoretical analysis and numerical results. The development of the DEC-MP2 molecular gradient is the initial step towards calculating higher order energy derivatives for large molecular systems using the DEC framework, both at the MP2 level of theory and for more accurate coupled-cluster methods.

Attractive electron-electron interactions within robust local fitting approximations

An analysis of Dunlaps robust fitting approach reveals that the resulting two‐electron integral matrix is not manifestly positive semidefinite when local fitting domains or non‐Coulomb fitting metrics are used. We present a highly local approximate method for evaluating four‐center two‐electron integrals based on the resolution‐of‐the‐identity (RI) approximation and apply it to the construction of the Coulomb and exchange contributions to the Fock matrix. In this pair‐atomic resolution‐of‐the‐identity (PARI) approach, atomic‐orbital (AO) products are expanded in auxiliary functions centered on the two atoms associated with each product. Numerical tests indicate that in 1% or less of all Hartree–Fock and Kohn–Sham calculations, the indefinite integral matrix causes nonconvergence in the self‐consistent‐field iterations. In these cases, the two‐electron contribution to the total energy becomes negative, meaning that the electronic interaction is effectively attractive, and the total energy is dramatically lower than that obtained with exact integrals. In the vast majority of our test cases, however, the indefiniteness does not interfere with convergence. The total energy accuracy is comparable to that of the standard Coulomb‐metric RI method. The speed‐up compared with conventional algorithms is similar to the RI method for Coulomb contributions; exchange contributions are accelerated by a factor of up to eight with a triple‐zeta quality basis set. A positive semidefinite integral matrix is recovered within PARI by introducing local auxiliary basis functions spanning the full AO product space, as may be achieved by using Cholesky‐decomposition techniques. Local completion, however, slows down the algorithm to a level comparable with or below conventional calculations.

Multi-electron integrals

This review presents techniques for the computation of multi‐electron integrals over Cartesian and solid‐harmonic Gaussian‐type orbitals as used in standard electronic‐structure investigations. The review goes through the basics for one‐ and two‐electron integrals, discuss details of various two‐electron integral evaluation schemes, approximative methods, techniques to compute multi‐electron integrals for explicitly correlated methods, and property integrals.

The divide–expand–consolidate MP2 scheme goes massively parallel

For large molecular systems conventional implementations of second order Møller–Plesset (MP2) theory encounter a scaling wall, both memory- and time-wise. We describe how this scaling wall can be removed. We present a massively parallel algorithm for calculating MP2 energies and densities using the divide–expand–consolidate scheme where a calculation on a large system is divided into many small fragment calculations employing local orbital spaces. The resulting algorithm is linear-scaling with system size, exhibits near perfect parallel scalability, removes memory bottlenecks and does not involve any I/O. The algorithm employs three levels of parallelisation combined via a dynamic job distribution scheme. Results on two molecular systems containing 528 and 1056 atoms (4278 and 8556 basis functions) using 47,120 and 94,240 cores are presented. The results demonstrate the scalability of the algorithm both with respect to the number of cores and with respect to system size. The presented algorithm is thus highly suited for large super computer architectures and allows MP2 calculations on large molecular systems to be carried out within a few hours – for example, the correlated calculation on the molecular system containing 1056 atoms took 2.37 hours using 94240 cores.

Efficient linear-scaling second-order Møller-Plesset perturbation theory: The divide–expand–consolidate RI-MP2 model

The Resolution of the Identity second-order Møller-Plesset perturbation theory (RI-MP2) method is implemented within the linear-scaling Divide-Expand-Consolidate (DEC) framework. In a DEC calculation, the full molecular correlated calculation is replaced by a set of independent fragment calculations each using a subset of the total orbital space. The number of independent fragment calculations scales linearly with the system size, rendering the method linear-scaling and massively parallel. The DEC-RI-MP2 method can be viewed as an approximation to the DEC-MP2 method where the RI approximation is utilized in each fragment calculation. The individual fragment calculations scale with the fifth power of the fragment size for both methods. However, the DEC-RI-MP2 method has a reduced prefactor compared to DEC-MP2 and is well-suited for implementation on massively parallel supercomputers, as demonstrated by test calculations on a set of medium-sized molecules. The DEC error control ensures that the standard RI-MP2 energy can be obtained to the predefined precision. The errors associated with the RI and DEC approximations are compared, and it is shown that the DEC-RI-MP2 method can be applied to systems far beyond the ones that can be treated with a conventional RI-MP2 implementation.

An efficient density-functional-theory force evaluation for large molecular systems

An efficient, linear-scaling implementation of Kohn-Sham density-functional theory for the calculation of molecular forces for systems containing hundreds of atoms is presented. The density-fitted Coulomb force contribution is calculated in linear time by combining atomic integral screening with the continuous fast multipole method. For higher efficiency and greater simplicity, the near-field Coulomb force contribution is calculated by expanding the solid-harmonic Gaussian basis functions in Hermite rather than Cartesian Gaussians. The efficiency and linear complexity of the molecular-force evaluation is demonstrated by sample calculations and applied to the geometry optimization of a few selected large systems.