DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Multi-center decomposition of molecular densities: A mathematical perspective.

The aim of this article is to analyze from a mathematical perspective some existing schemes to partition a molecular density into several atomic contributions with a specific focus on Iterative Stockholder Atom (ISA) methods. We provide a unified mathematical framework to describe the latter family of methods and propose a new scheme, named L-ISA (for linear approximation of ISA), which generalizes the so-called additive variational Hirshfeld method. We prove several important mathematical properties of the ISA and L-ISA minimization problems and show that the so-called ISA algorithms can be viewed as alternating minimization schemes, which, in turn, enables us to obtain new convergence results for these numerical methods. Specific mathematical properties of the ISA decomposition for diatomic systems are also presented. Numerical results on diatomic systems illustrate the proven mathematical properties.

Photoionization and core resonances from range-separated density-functional theory: General formalism and example of the beryllium atom.

We explore the merits of linear-response range-separated time-dependent density-functional theory (TDDFT) for the calculation of photoionization spectra. We consider two variants of range-separated TDDFT, namely, the time-dependent range-separated hybrid (TDRSH) scheme, which uses a global range-separation parameter, and the time-dependent locally range-separated hybrid (TDLRSH), which uses a local range-separation parameter, and compare with standard time-dependent local-density approximation (TDLDA) and time-dependent Hartree-Fock (TDHF). We show how to calculate photoionization spectra with these methods using the Sternheimer approach formulated in a non-orthogonal B-spline basis set with appropriate frequency-dependent boundary conditions. We illustrate these methods on the photoionization spectrum of the Be atom, focusing, in particular, on the core resonances. Both the TDRSH and TDLRSH photoionization spectra are found to constitute a large improvement over the TDLDA photoionization spectrum and a more modest improvement over the TDHF photoionization spectrum.

Photoionization and core resonances from range-separated time-dependent density-functional theory for open-shell states: Example of the lithium atom.

We consider the calculations of photoionization spectra and core resonances of open-shell systems using range-separated time-dependent density-functional theory. Specifically, we use the time-dependent range-separated hybrid (TDRSH) scheme, combining a long-range Hartree-Fock exchange potential and kernel with a short-range potential and kernel from a local density-functional approximation, and the time-dependent locally range-separated hybrid (TDLRSH) scheme, which uses a local range-separation parameter. To efficiently perform the calculations, we formulate a spin-unrestricted linear-response Sternheimer approach in a non-orthogonal B-spline basis set using appropriate frequency-dependent boundary conditions. We illustrate this approach on the Li atom, which suggests that TDRSH and TDLRSH are adequate simple methods for estimating the single-electron photoionization spectra of open-shell systems.

A posteriori error estimation for the non-self-consistent Kohn-Sham equations.

We address the problem of rigorously bounding the errors in the numerical solution of the Kohn-Sham equations due to (i) the finiteness of the basis set, (ii) the convergence thresholds in iterative procedures, and (iii) the propagation of rounding errors in floating-point arithmetic. In this contribution, we compute fully-guaranteed bounds on the solution of the non-self-consistent equations in the pseudopotential approximation in a plane-wave basis set. We demonstrate our methodology by providing band structure diagrams of silicon annotated with error bars indicating the combined error.

Insights into the π - π interaction driven non-covalent functionalization of carbon nanotubes of various diameters by conjugated fluorene and carbazole copolymers.

We investigate the interaction of polyfluorene and fluorene/carbazole copolymers bearing various functional groups and side chains with small to large diameter-from 1.7 nm to 9 nm-carbon nanotubes (CNTs) in vacuo. We use variable-charge molecular dynamics simulations based on the reactive force field ReaxFF. We show that non-covalent functionalization of nanotubes, driven by π - π interactions, is effective for all the polymers studied, thanks to their conjugated backbone and regardless of the presence of specific functional groups. The geometry at equilibrium of these polymer/CNT hybrids is analyzed in detail at the scale of each fluorene or carbazole unit. The role of both the functional groups and the alkyl chain length is analyzed in detail. Adsorption of the polymers on the nanotube sidewalls is shown to be either complete-with the whole chain physisorbed-or partial-due to intrachain coiling or interchain repulsion-depending on the initial geometry, number of polymers, and nanotube diameter. Energetic arguments supplement the described geometric features. Both energetic and geometric adsorption features are derived here for the first time for large diameter carbon nanotubes (up to 9 nm) and fluorene/carbazole copolymers having up to 30 monomers and bearing different functional groups. The force field ReaxFF and its available parameterization used for the simulations are validated, thanks to a benchmark and review on higher-level quantum calculations-for simple π - π interacting compounds made up of polycyclic aromatic molecules adsorbed on a graphene sheet or bilayer graphene. Although it is shown that the influence of the nanotube chirality on the adsorption pattern and binding strength cannot be discussed with our method, we highlight that an available force field such as ReaxFF and its parameterization can be transferable to simulate new systems without specific re-parameterization, provided that this model is validated against reference methods or data. This methodology proves to be a valuable tool for optimal polymer design for nanotube functionalization at no re-parameterization cost and could be adapted to simulate and assist the design of other types of molecular systems.

A new discretization for the polarizable continuum model within the domain decomposition paradigm.

We present a new algorithm to solve the polarizable continuum model equation in a framework compatible with the strategy previously developed by us for the conductor-like screening model based on Schwarz's domain decomposition method (ddCOSMO). The new discretization is systematically improvable and is fully consistent with ddCOSMO so that it reproduces ddCOSMO results for large dielectric constants.

Quantum, classical, and hybrid QM/MM calculations in solution: general implementation of the ddCOSMO linear scaling strategy.

We present the general theory and implementation of the Conductor-like Screening Model according to the recently developed ddCOSMO paradigm. The various quantities needed to apply ddCOSMO at different levels of theory, including quantum mechanical descriptions, are discussed in detail, with a particular focus on how to compute the integrals needed to evaluate the ddCOSMO solvation energy and its derivatives. The overall computational cost of a ddCOSMO computation is then analyzed and decomposed in the various steps: the different relative weights of such contributions are then discussed for both ddCOSMO and the fastest available alternative discretization to the COSMO equations. Finally, the scaling of the cost of the various steps with respect to the size of the solute is analyzed and discussed, showing how ddCOSMO opens significantly new possibilities when cheap or hybrid molecular mechanics/quantum mechanics methods are used to describe the solute.

Domain decomposition for implicit solvation models.

This article is the first of a series of papers dealing with domain decomposition algorithms for implicit solvent models. We show that, in the framework of the COSMO model, with van der Waals molecular cavities and classical charge distributions, the electrostatic energy contribution to the solvation energy, usually computed by solving an integral equation on the whole surface of the molecular cavity, can be computed more efficiently by using an integral equation formulation of Schwarz's domain decomposition method for boundary value problems. In addition, the so-obtained potential energy surface is smooth, which is a critical property to perform geometry optimization and molecular dynamics simulations. The purpose of this first article is to detail the methodology, set up the theoretical foundations of the approach, and study the accuracies and convergence rates of the resulting algorithms. The full efficiency of the method and its applicability to large molecular systems of biological interest is demonstrated elsewhere.

A variational formulation of the polarizable continuum model.

Continuum solvation models are widely used to accurately estimate solvent effects on energy, structural and spectroscopic properties of complex molecular systems. The polarizable continuum model (PCM) is one of the most versatile among the continuum models because of the variety of properties that can be computed and the diversity of methods that can be used to describe the solute from molecular mechanics (MM) to sophisticated quantum mechanical (QM) post-self-consistent field methods or even hybrid QM/MM methods. In this contribution, we present a new formulation of PCM in terms of a free energy functional whose variational parameters include the continuum polarization (represented by the apparent surface charges), the solute's atomic coordinates and-possibly-its electronic density. The problem of finding the optimized geometry of the (polarized) solute, with the corresponding self-consistent reaction field, is recast as the minimization of this free energy functional, simultaneously with respect to all its variables. The numerous potential applications of this variational formulation of PCM are discussed, including simultaneous optimization of solute's geometry and polarization charges and extended Lagrangian dynamics. In particular, we describe in details the simultaneous optimization procedure and we include several numerical examples.

Scalable evaluation of polarization energy and associated forces in polarizable molecular dynamics: II. Toward massively parallel computations using smooth particle mesh Ewald.

In this article, we present a parallel implementation of point dipole-based polarizable force fields for molecular dynamics (MD) simulations with periodic boundary conditions (PBC). The smooth particle mesh Ewald technique is combined with two optimal iterative strategies, namely, a preconditioned conjugate gradient solver and a Jacobi solver in conjunction with the direct inversion in the iterative subspace for convergence acceleration, to solve the polarization equations. We show that both solvers exhibit very good parallel performances and overall very competitive timings in an energy and force computation needed to perform a MD step. Various tests on large systems are provided in the context of the polarizable AMOEBA force field as implemented in the newly developed Tinker-HP package, which is the first implementation of a polarizable model that makes large-scale experiments for massively parallel PBC point dipole models possible. We show that using a large number of cores offers a significant acceleration of the overall process involving the iterative methods within the context of SPME and a noticeable improvement of the memory management, giving access to very large systems (hundreds of thousands of atoms) as the algorithm naturally distributes the data on different cores. Coupled with advanced MD techniques, gains ranging from 2 to 3 orders of magnitude in time are now possible compared to nonoptimized, sequential implementations, giving new directions for polarizable molecular dynamics with periodic boundary conditions using massively parallel implementations.

Polarizable molecular dynamics in a polarizable continuum solvent.

We present, for the first time, scalable polarizable molecular dynamics (MD) simulations within a polarizable continuum solvent with molecular shape cavities and exact solution of the mutual polarization. The key ingredients are a very efficient algorithm for solving the equations associated with the polarizable continuum, in particular, the domain decomposition Conductor-like Screening Model (ddCOSMO), which involves a rigorous coupling of the continuum with the polarizable force field achieved through a robust variational formulation and an effective strategy to solve the coupled equations. The coupling of ddCOSMO with nonvariational force fields, including AMOEBA, is also addressed. The MD simulations are feasible, for real-life systems, on standard cluster nodes; a scalable parallel implementation allows for further acceleration in the context of a newly developed module in Tinker, named Tinker-HP. NVE simulations are stable, and long-term energy conservation can be achieved. This paper is focused on the methodological developments, the analysis of the algorithm, and the stability of the simulations; a proof-of-concept application is also presented to attest to the possibilities of this newly developed technique.

Spectral method for the time-dependent Gross-Pitaevskii equation with a harmonic trap.

We study the numerical resolution of the time-dependent Gross-Pitaevskii equation, a nonlinear Schrödinger equation used to simulate the dynamics of Bose-Einstein condensates. Considering condensates trapped in harmonic potentials, we present an efficient algorithm by making use of a spectral-Galerkin method, using a basis set of harmonic-oscillator functions, and the Gauss-Hermite quadrature. We apply this algorithm to the simulation of condensate breathing and scissor modes.

Quantum Calculations in Solution for Large to Very Large Molecules: A New Linear Scaling QM/Continuum Approach.

We present a new implementation of continuum solvation models for semiempirical Hamiltonians that allows the description of environmental effects on very large molecular systems. In this approach based on a domain decomposition strategy of the COSMO model (ddCOSMO), the solution to the COSMO equations is no longer the computational bottleneck but becomes a negligible part of the overall computation time. In this Letter, we analyze the computational impact of COSMO on the solution of the SCF equations for large to very large molecules, using semiempirical Hamiltonians, for both the new ddCOSMO implementation and the most recent, linear scaling one, based on the fast multipole method. A further analysis is on the simulation of the UV/visible spectrum of a light-harvesting pigment-protein complex. All of the results show how the new ddCOSMO algorithm paves the way to routine computations for large molecular systems in the condensed phase.

Scalable Evaluation of Polarization Energy and Associated Forces in Polarizable Molecular Dynamics: II.Towards Massively Parallel Computations using Smooth Particle Mesh Ewald.

In this paper, we present a scalable and efficient implementation of point dipole-based polarizable force fields for molecular dynamics (MD) simulations with periodic boundary conditions (PBC). The Smooth Particle-Mesh Ewald technique is combined with two optimal iterative strategies, namely, a preconditioned conjugate gradient solver and a Jacobi solver in conjunction with the Direct Inversion in the Iterative Subspace for convergence acceleration, to solve the polarization equations. We show that both solvers exhibit very good parallel performances and overall very competitive timings in an energy-force computation needed to perform a MD step. Various tests on large systems are provided in the context of the polarizable AMOEBA force field as implemented in the newly developed Tinker-HP package which is the first implementation for a polarizable model making large scale experiments for massively parallel PBC point dipole models possible. We show that using a large number of cores offers a significant acceleration of the overall process involving the iterative methods within the context of spme and a noticeable improvement of the memory management giving access to very large systems (hundreds of thousands of atoms) as the algorithm naturally distributes the data on different cores. Coupled with advanced MD techniques, gains ranging from 2 to 3 orders of magnitude in time are now possible compared to non-optimized, sequential implementations giving new directions for polarizable molecular dynamics in periodic boundary conditions using massively parallel implementations.

Fast Domain Decomposition Algorithm for Continuum Solvation Models: Energy and First Derivatives.

In this contribution, an efficient, parallel, linear scaling implementation of the conductor-like screening model (COSMO) is presented, following the domain decomposition (dd) algorithm recently proposed by three of us. The implementation is detailed and its linear scaling properties, both in computational cost and memory requirements, are demonstrated. Such behavior is also confirmed by several numerical examples on linear and globular large-sized systems, for which the calculation of the energy and of the forces is achieved with timings compatible with the use of polarizable continuum solvation for molecular dynamics simulations.

Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations.

We present projected gradient algorithms designed for optimizing various functionals defined on the set of N-representable one-electron reduced density matrices. We show that projected gradient algorithms are efficient in minimizing the Hartree-Fock or the Muller-Buijse-Baerends functional. On the other hand, they converge very slowly when applied to the recently proposed BBk (k=1,2,3) functionals [O. Gritsenko et al., J. Chem. Phys. 122, 204102 (2005)]. This is due to the fact that the BBk functionals are not proper functionals of the density matrix.

The effective local potential method: implementation for molecules and relation to approximate optimized effective potential techniques.

We have recently formulated a new approach, named the effective local potential (ELP) method, for calculating local exchange-correlation potentials for orbital-dependent functionals based on minimizing the variance of the difference between a given nonlocal potential and its desired local counterpart [V. N. Staroverov et al., J. Chem. Phys. 125, 081104 (2006)]. Here we show that under a mildly simplifying assumption of frozen molecular orbitals, the equation defining the ELP has a unique analytic solution which is identical with the expression arising in the localized Hartree-Fock (LHF) and common energy denominator approximations (CEDA) to the optimized effective potential. The ELP procedure differs from the CEDA and LHF in that it yields the target potential as an expansion in auxiliary basis functions. We report extensive calculations of atomic and molecular properties using the frozen-orbital ELP method and its iterative generalization to prove that ELP results agree with the corresponding LHF and CEDA values, as they should. Finally, we make the case for extending the iterative frozen-orbital ELP method to full orbital relaxation.

The electronic ground-state energy problem: a new reduced density matrix approach.

We present here a formulation of the electronic ground-state energy in terms of the second order reduced density matrix, using a duality argument. It is shown that the computation of the ground-state energy reduces to the search of the projection of some two-electron reduced Hamiltonian on the dual cone of N-representability conditions. Some numerical results validate the approach, both for equilibrium geometries and for the dissociation curve of N(2).

An efficient sampling algorithm for variational Monte Carlo.

We propose a new algorithm for sampling the N-body density mid R:Psi(R)mid R:(2)R(3N)mid R:Psimid R:(2) in the variational Monte Carlo framework. This algorithm is based upon a modified Ricci-Ciccotti discretization of the Langevin dynamics in the phase space (R,P) improved by a Metropolis-Hastings accept/reject step. We show through some representative numerical examples (lithium, fluorine, and copper atoms and phenol molecule) that this algorithm is superior to the standard sampling algorithm based on the biased random walk (importance sampling).