Dalton Project: A Python platform for molecular- and electronic-structure simulations of complex systems.

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Intermediate data such as integrals are exposed to the platform and made accessible to the user in the form of NumPy arrays, and the resulting data are extracted, analyzed, and visualized. Complex computational protocols that may, for instance, arise due to a need for environment fragmentation and configuration-space sampling of biochemical systems are readily assisted by the platform. The platform is designed to host additional software libraries and will serve as a hub for future modular software development efforts in the distributed Dalton community.

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.

Explicitly correlated second-order Møller-Plesset perturbation theory in a Divide-Expand-Consolidate (DEC) context.

We present the DEC-RIMP2-F12 method where we have augmented the Divide Expand-Consolidate resolution-of-the-identity second-order Møller-Plesset perturbation theory method (DEC-RIMP2) [P. Baudin et al., J. Chem. Phys. 144, 054102 (2016)] with an explicitly correlated (F12) correction. The new method is linear-scaling, massively parallel, and it corrects for the basis set incompleteness error in an efficient manner. In addition, we observe that the F12 contribution decreases the domain error of the DEC-RIMP2 correlation energy by roughly an order of magnitude. An important feature of the DEC scheme is the inherent error control defined by a single parameter, and this feature is also retained for the DEC-RIMP2-F12 method. In this paper we present the working equations for the DEC-RIMP2-F12 method and proof of concept numerical results for a set of test molecules.

Geometry of the magic number H(+)(H2O)21 water cluster by proxy.

Abundance mass spectra, obtained upon carefully electrospraying solutions of tert-butanol (TB) in water into a mass spectrometer, display a systematic series of peaks due to mixed H(+)(TB)m(H2O)n clusters. Clusters with m + n = 21 have higher abundance (magic number peaks) than their neighbours when m ≤ 9, while for m > 9 they have lower abundance. This indicates that the mixed TB-H2O clusters retain a core hydrogen bonded network analogous to that in the famous all-water H(+)(H2O)21 cluster up to the limit m = 9. The limiting value corresponds to the number of dangling O-H groups pointing out from the surface of the degenerated pentagonal dodecahedral, considered to be the lowest energy form of H(+)(H2O)21; the experimental findings therefore support this geometry. The experimental findings are supported by ab initio quantum chemical calculations to provide a consistent framework for the interpretation of these kinds of experiments.

Wavelet formulation of the polarizable continuum model. II. Use of piecewise bilinear boundary elements.

The simplicity of dielectric continuum models has made them a standard tool in almost any Quantum Chemistry (QC) package. Despite being intuitive from a physical point of view, the actual electrostatic problem at the cavity boundary is challenging: the underlying boundary integral equations depend on singular, long-range operators. The parametrization of the cavity boundary should be molecular-shaped, smooth and differentiable. Even the most advanced implementations, based on the integral equation formulation (IEF) of the polarizable continuum model (PCM), generally lead to working equations which do not guarantee convergence to the exact solution and/or might become numerically unstable in the limit of large refinement of the molecular cavity (small tesserae). This is because they generally make use of a surface parametrization with cusps (interlocking spheres) and employ collocation methods for the discretization (point charges). Wavelets on a smooth cavity are an attractive alternative to consider: for the operators involved, they lead to highly sparse matrices and precise error control. Moreover, by making use of a bilinear basis for the representation of operators and functions on the cavity boundary, all equations can be differentiated to enable the computation of geometrical derivatives. In this contribution, we present our implementation of the IEFPCM with bilinear wavelets on a smooth cavity boundary. The implementation has been carried out in our module PCMSolver and interfaced with LSDalton, demonstrating the accuracy of the method both for the electrostatic solvation energy and for linear response properties. In addition, the implementation in a module makes our framework readily available to any QC software with minimal effort.

Ring planarity problem of 2-oxazoline revisited using microwave spectroscopy and quantum chemical calculations.

In a previous infrared, Raman, and microwave spectroscopic work,1 it was claimed that 2-oxazoline has a planar ring equilibrium conformation, and the ring-puckering potential function V(z) = 22.2(z(4) + 1.31z(2)) cm(-1), where z is a dimensionless reduced coordinate, was derived. This function poorly reproduces the rotational constants of the lowest and most important puckering states. The microwave spectrum has been reinvestigated and largely extended to include more than 4600 transitions of the ground state and six excited states of the ring-puckering vibration allowing accurate centrifugal distortion constants to be obtained for the first time. A new potential function V(z) = 38.8(z(4) - 0.65z(2)) cm(-1) has been determined. This function yields much better agreement between calculated and observed rotational constants, especially for the lowest puckering states, than the previous function and predicts a nonplanar ring equilibrium conformation. The barrier to ring planarity is determined to be 49(8) J/mol. The ground-state energy level is 35 cm(-1) above the barrier maximum. Theory predicts that three of the five Watson centrifugal distortion constants, ΔJK, ΔK, and δK, should vary with the puckering state, whereas ΔJ and δJ should be unaffected. It was found that ΔJK and ΔK indeed behave in the expected manner, while deviations were seen for the three other centrifugal distortion constants. The ab initio methods HF, MP2, CCSD, CCSD(T), and CCSD(T)-F12 with large basis sets as well as several DFT methods were used in an attempt to reproduce the low experimental barrier to the planar ring. Only the MP2 method yielded a satisfactory prediction of the barrier. The CCSD and the CCSD(T) calculations predict a planar ring, whereas the energy differences between a planar and a nonplanar ring obtained in the CCSD(T)-F12 computations are so small that a definite conclusion cannot be drawn.

Charge-constrained auxiliary-density-matrix methods for the Hartree-Fock exchange contribution.

Three new variants of the auxiliary-density-matrix method (ADMM) of Guidon, Hutter, and VandeVondele [J. Chem. Theory Comput. 6, 2348 (2010)] are presented with the common feature that they have a simplified constraint compared with the full orthonormality requirement of the earlier ADMM1 method. All ADMM variants are tested for accuracy and performance in all-electron B3LYP calculations with several commonly used basis sets. The effect of the choice of the exchange functional for the ADMM exchange-correction term is also investigated.

Insights into the dynamics of evaporation and proton migration in protonated water clusters from large-scale Born-Oppenheimer direct dynamics.

Large-scale on-the-fly Born-Oppenheimer molecular dynamics simulations using recent advances in linear scaling electronic structure theory and trajectory integration techniques have been performed for protonated water clusters around the magic number (H(2)O)(n)H(+) , for n = 20 and 21. Besides demonstrating the feasibility and efficiency of the computational approach, the calculations reveal interesting dynamical details. Elimination of water molecules is found to be fast for both cluster sizes but rather insensitive to the initial geometry. The water molecules released acquire velocities compatible with thermal energies. The proton solvation shell changes between the well-known Eigen and Zundel motifs and is characterized by specific low-frequency vibrational modes, which have been quantified. The proton transfer mechanism largely resembles that of bulk water but one interesting variation was observed.

Attractive electron-electron interactions within robust local fitting approximations.

An analysis of Dunlap's 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.

Analytical GIAO and hybrid-basis integral derivatives: application to geometry optimization of molecules in strong magnetic fields.

Analytical integral evaluation is a central task of modern quantum chemistry. Here we present a general method for evaluating differentiated integrals over standard Gaussian and mixed Gaussian/plane-wave hybrid orbitals. The main idea is to have a representation of basis sets that is flexible enough to enable differentiated integrals to be reinterpreted as standard integrals over modified basis functions. As an illustration of the method, we report a very simple implementation of Hartree-Fock level geometrical derivatives in finite magnetic fields for gauge-origin independent atomic orbitals, within the London program. As a quantum-chemical application, we optimize the structure of helium clusters and some well-known covalently bound molecules (water, ammonia and benzene) subject to strong magnetic fields.

MP2 energy and density for large molecular systems with internal error control using the Divide-Expand-Consolidate scheme.

Divide-Expand-Consolidate (DEC) is a local correlation method where the inherent locality of the electron correlation problem is used to express the correlated calculation on a large molecular system in terms of small independent fragment calculations employing small subsets of local HF orbitals. A crucial feature of the DEC scheme is that the sizes of the local orbital spaces are determined in a black box manner during the calculation. In this way it is ensured that the correlation energy has been determined to a predefined precision compared to a conventional calculation. In the present work we apply the DEC scheme to calculate the correlation energy as well as the electron density matrix for the insulin molecule using second order Møller-Plesset (MP2) theory. This is the first DEC calculation on a molecular system which is too large to be treated using a conventional MP2 implementation. The fragmentation errors for the insulin DEC calculation are carefully analyzed using internal consistency checks. It is demonstrated that size-intensive properties are determined to the same precision for small and large molecules. For example, the percentage of correlation energy recovered and the error per electron in the correlated density matrix depend only on the predefined precision and not on the molecular size.

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.

Harmonic Infrared and Raman Spectra in Molecular Environments Using the Polarizable Embedding Model.

We present a fully analytic approach to calculate infrared (IR) and Raman spectra of molecules embedded in complex molecular environments modeled using the fragment-based polarizable embedding (PE) model. We provide the theory for the calculation of analytic second-order geometric derivatives of molecular energies and first-order geometric derivatives of electric dipole moments and dipole-dipole polarizabilities within the PE model. The derivatives are implemented using a general open-ended response theory framework, thus allowing for an extension to higher-order derivatives. The embedding-potential parameters used to describe the environment in the PE model are derived through first-principles calculations, thus allowing a wide variety of systems to be modeled, including solvents, proteins, and other large and complex molecular environments. Here, we present proof-of-principle calculations of IR and Raman spectra of acetone in different solvents. This work is an important step toward calculating accurate vibrational spectra of molecules embedded in realistic environments.

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.

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.

On Resolution-of-the-Identity Electron Repulsion Integral Approximations and Variational Stability.

The definiteness of the Mulliken and Dirac electron repulsion integral (ERI) matrices is examined for different classes of resolution-of-the-identity (RI) ERI approximations with particular focus on local fitting techniques. For global RI, robust local RI, and nonrobust local RI we discuss the definiteness of the approximated ERI matrices as well as the resulting bounds of Hartree, exchange, and total energies. Lower bounds of Hartree and exchange energy contributions are crucial as their absence may lead to variational instabilities, causing severe convergence problems or even convergence to a spurious state in self-consistent-field optimizations. While the global RI approximation guarantees lower bounds of Hartree and exchange energies, local RI approximations are generally unbounded. The robust local RI approximation guarantees a lower bound of the exchange energy but not of the Hartree energy. The nonrobust local RI approximation guarantees a lower bound of the Hartree energy but not of the exchange energy. These issues are demonstrated by sample calculations on carbon dioxide and benzene using the pair atomic RI approximation.

Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method.

We compare the performance of three approximate methods for speeding up evaluation of the exchange contribution in Hartree-Fock and hybrid Kohn-Sham calculations: the chain-of-spheres algorithm (COSX; Neese , F. Chem. Phys. 2008 , 356 , 98 - 109 ), the pair-atomic resolution-of-identity method (PARI-K; Merlot , P. J. Comput. Chem. 2013 , 34 , 1486 - 1496 ), and the auxiliary density matrix method (ADMM; Guidon , M. J. Chem. Theory Comput. 2010 , 6 , 2348 - 2364 ). Both the efficiency relative to that of a conventional linear-scaling algorithm and the accuracy of total, atomization, and orbital energies are compared for a subset containing 25 of the 200 molecules in the Rx200 set using double-, triple-, and quadruple-ζ basis sets. The accuracy of relative energies is further compared for small alkane conformers (ACONF test set) and Diels-Alder reactions (DARC test set). Overall, we find that the COSX method provides good accuracy for orbital energies as well as total and relative energies, and the method delivers a satisfactory speedup. The PARI-K and in particular ADMM algorithms require further development and optimization to fully exploit their indisputable potential.

A ground-state-directed optimization scheme for the Kohn-Sham energy.

Kohn-Sham density-functional calculations are used in many branches of science to obtain information about the electronic structure of molecular systems and materials. Unfortunately, the traditional method for optimizing the Kohn-Sham energy suffers from fundamental problems that may lead to divergence or, even worse, convergence to an energy saddle point rather than to the ground-state minimum--in particular, for the larger and more complicated electronic systems that are often studied by Kohn-Sham theory nowadays. We here present a novel method for Kohn-Sham energy minimization that does not suffer from the flaws of the conventional approach, combining reliability and efficiency with linear complexity. In particular, the proposed method converges by design to a minimum, avoiding the sometimes spurious solutions of the traditional method and bypassing the need to examine the structure of the provided solution.

A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians.

Utilizing the fact that solid-harmonic combinations of Cartesian and Hermite Gaussian atomic orbitals are identical, a new scheme for the evaluation of molecular integrals over solid-harmonic atomic orbitals is presented, where the integration is carried out over Hermite rather than Cartesian atomic orbitals. Since Hermite Gaussians are defined as derivatives of spherical Gaussians, the corresponding molecular integrals become the derivatives of integrals over spherical Gaussians, whose transformation to the solid-harmonic basis is performed in the same manner as for integrals over Cartesian Gaussians, using the same expansion coefficients. The presented solid-harmonic Hermite scheme simplifies the evaluation of derivative molecular integrals, since differentiation by nuclear coordinates merely increments the Hermite quantum numbers, thereby providing a unified scheme for undifferentiated and differentiated four-center molecular integrals. For two- and three-center two-electron integrals, the solid-harmonic Hermite scheme is particularly efficient, significantly reducing the cost relative to the Cartesian scheme.

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.