Mauro Del Ben

Second-Order Møller–Plesset Perturbation Theory in the Condensed Phase: An Efficient and Massively Parallel Gaussian and Plane Waves Approach

A novel algorithm, based on a hybrid Gaussian and plane waves (GPW) approach, is developed for the canonical second-order Møller-Plesset perturbation energy (MP2) of finite and extended systems. The key aspect of the method is that the electron repulsion integrals (ia|λσ) are computed by direct integration between the products of Gaussian basis functions λσ and the electrostatic potential arising from a given occupied-virtual pair density ia. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. In particular, for condensed phase systems, this scheme is highly efficient. Furthermore, our implementation has low memory requirements and displays excellent parallel scalability up to 100 000 processes. In this way, canonical MP2 calculations for condensed phase systems containing hundreds of atoms or more than 5000 basis functions can be performed within minutes, while systems up to 1000 atoms and 10 000 basis functions remain feasible. Solid LiH has been employed as a benchmark to study basis set and system size convergence. Lattice constants and cohesive energies of various molecular crystals have been studied with MP2 and double-hybrid functionals.

Electron Correlation in the Condensed Phase from a Resolution of Identity Approach Based on the Gaussian and Plane Waves Scheme.

The second-order Møller-Plesset perturbation energy (MP2) and the Random Phase Approximation (RPA) correlation energy are increasingly popular post-Kohn-Sham correlation methods. Here, a novel algorithm based on a hybrid Gaussian and Plane Waves (GPW) approach with the resolution-of-identity (RI) approximation is developed for MP2, scaled opposite-spin MP2 (SOS-MP2), and direct-RPA (dRPA) correlation energies of finite and extended system. The key feature of the method is that the three center electron repulsion integrals (μν|P) necessary for the RI approximation are computed by direct integration between the products of Gaussian basis functions μν and the electrostatic potential arising from the RI fitting densities P. The electrostatic potential is obtained in a plane waves basis set after solving the Poisson equation in Fourier space. This scheme is highly efficient for condensed phase systems and offers a particularly easy way for parallel implementation. The RI approximation allows to speed up the MP2 energy calculations by a factor 10 to 15 compared to the canonical implementation but still requires O(N(5)) operations. On the other hand, the combination of RI with a Laplace approach in SOS-MP2 and an imaginary frequency integration in dRPA reduces the computational effort to O(N(4)) in both cases. In addition to that, our implementations have low memory requirements and display excellent parallel scalability up to tens of thousands of processes. Furthermore, exploiting graphics processing units (GPU), a further speedup by a factor ∼2 is observed compared to the standard only CPU implementations. In this way, RI-MP2, RI-SOS-MP2, and RI-dRPA calculations for condensed phase systems containing hundreds of atoms and thousands of basis functions can be performed within minutes employing a few hundred hybrid nodes. In order to validate the presented methods, various molecular crystals have been employed as benchmark systems to assess the performance, while solid LiH has been used to study the convergence with respect to the basis set and system size in the case of RI-MP2 and RI-dRPA.

Probing the structural and dynamical properties of liquid water with models including non-local electron correlation.

Water is a ubiquitous liquid that displays a wide range of anomalous properties and has a delicate structure that challenges experiment and simulation alike. The various intermolecular interactions that play an important role, such as repulsion, polarization, hydrogen bonding, and van der Waals interactions, are often difficult to reproduce faithfully in atomistic models. Here, electronic structure theories including all these interactions at equal footing, which requires the inclusion of non-local electron correlation, are used to describe structure and dynamics of bulk liquid water. Isobaric-isothermal (NpT) ensemble simulations based on the Random Phase Approximation (RPA) yield excellent density (0.994 g/ml) and fair radial distribution functions, while various other density functional approximations produce scattered results (0.8-1.2 g/ml). Molecular dynamics simulation in the microcanonical (NVE) ensemble based on Møller-Plesset perturbation theory (MP2) yields dynamical properties in the condensed phase, namely, the infrared spectrum and diffusion constant. At the MP2 and RPA levels of theory, ice is correctly predicted to float on water, resolving one of the anomalies as resulting from a delicate balance between van der Waals and hydrogen bonding interactions. For several properties, obtaining quantitative agreement with experiment requires correction for nuclear quantum effects (NQEs), highlighting their importance, for structure, dynamics, and electronic properties. A computed NQE shift of 0.6 eV for the band gap and absorption spectrum illustrates the latter. Giving access to both structure and dynamics of condensed phase systems, non-local electron correlation will increasingly be used to study systems where weak interactions are of paramount importance.

Forces and stress in second order Møller-Plesset perturbation theory for condensed phase systems within the resolution-of-identity Gaussian and plane waves approach.

The forces acting on the atoms as well as the stress tensor are crucial ingredients for calculating the structural and dynamical properties of systems in the condensed phase. Here, these derivatives of the total energy are evaluated for the second-order Møller-Plesset perturbation energy (MP2) in the framework of the resolution of identity Gaussian and plane waves method, in a way that is fully consistent with how the total energy is computed. This consistency is non-trivial, given the different ways employed to compute Coulomb, exchange, and canonical four center integrals, and allows, for example, for energy conserving dynamics in various ensembles. Based on this formalism, a massively parallel algorithm has been developed for finite and extended system. The designed parallel algorithm displays, with respect to the system size, cubic, quartic, and quintic requirements, respectively, for the memory, communication, and computation. All these requirements are reduced with an increasing number of processes, and the measured performance shows excellent parallel scalability and efficiency up to thousands of nodes. Additionally, the computationally more demanding quintic scaling steps can be accelerated by employing graphics processing units (GPUs) showing, for large systems, a gain of almost a factor two compared to the standard central processing unit-only case. In this way, the evaluation of the derivatives of the RI-MP2 energy can be performed within a few minutes for systems containing hundreds of atoms and thousands of basis functions. With good time to solution, the implementation thus opens the possibility to perform molecular dynamics (MD) simulations in various ensembles (microcanonical ensemble and isobaric-isothermal ensemble) at the MP2 level of theory. Geometry optimization, full cell relaxation, and energy conserving MD simulations have been performed for a variety of molecular crystals including NH3, CO2, formic acid, and benzene.

GW in the Gaussian and Plane Waves Scheme with Application to Linear Acenes

We present an implementation of G0W0 and eigenvalue-self-consistent GW (evGW) in the Gaussian and plane waves scheme for molecules. We calculate the correlation self-energy for imaginary frequencies employing the resolution of the identity. The correlation self-energy for real frequencies is then evaluated by analytic continuation. This technique allows an efficient parallel implementation and application to systems with several hundreds of atoms. Various benchmark calculations are presented. In particular, the convergence with respect to the most important numerical parameters is assessed for the benzene molecule. Comparisons with respect to other G0W0 implementations are reported for a set of molecules, while the performance of the method has been measured for water clusters containing up to 480 atoms in a cc-TZVP basis. Additionally, G0W0 has been applied for studying the influence of the ligands on the gap of small CdSe nanoparticles. evGW has been employed to calculate the HOMO-LUMO gaps of linear acenes, linear chains formed of connected benzene rings. Distinct differences between the closed and the open-shell (broken-symmetry) evGW HOMO-LUMO gaps for long acenes are found. In future experiments, a comparison of measured HOMO-LUMO gaps and our calculated evGW values may be helpful to determine the electronic ground state of long acenes.