MADNESS: A Multiresolution, Adaptive Numerical Environment for Scientific Simulation

MADNESS (multiresolution adaptive numerical environment for scientific simulation) is a high-level software environment for solving integral and differential equations in many dimensions that uses adaptive and fast harmonic analysis methods with guaranteed precision based on multiresolution analysis and separated representations. Underpinning the numerical capabilities is a powerful petascale parallel programming environment that aims to increase both programmer productivity and code scalability. This paper describes the features and capabilities of MADNESS and briefly discusses some current applications in chemistry and several areas of physics

An efficient numerical quadrature for the calculation of the potential energy of wavefunctions expressed in the Daubechies wavelet basis

An efficient numerical quadrature is proposed for the approximate calculation of the potential energy in the context of pseudo potential electronic structure calculations with Daubechies wavelet and scaling function basis sets. Our quadrature is also applicable in the case of adaptive spatial resolution. Our theoretical error estimates are confirmed by numerical test calculations of the ground state energy and wave function of the harmonic oscillator in one dimension with and without adaptive resolution. As a byproduct we derive a filter, which, upon application on the scaling function coefficients of a smooth function, renders the approximate grid values of this function. This also allows for a fast calculation of the charge density from the wave function.Comment: 35 pages, 9 figures. Submitted to: Journal of Computational Physic

Wavelet-Based Linear-Response Time-Dependent Density-Functional Theory

Linear-response time-dependent (TD) density-functional theory (DFT) has been implemented in the pseudopotential wavelet-based electronic structure program BigDFT and results are compared against those obtained with the all-electron Gaussian-type orbital program deMon2k for the calculation of electronic absorption spectra of N2 using the TD local density approximation (LDA). The two programs give comparable excitation energies and absorption spectra once suitably extensive basis sets are used. Convergence of LDA density orbitals and orbital energies to the basis-set limit is significantly faster for BigDFT than for deMon2k. However the number of virtual orbitals used in TD-DFT calculations is a parameter in BigDFT, while all virtual orbitals are included in TD-DFT calculations in deMon2k. As a reality check, we report the x-ray crystal structure and the measured and calculated absorption spectrum (excitation energies and oscillator strengths) of the small organic molecule N-cyclohexyl-2-(4-methoxyphenyl)imidazo[1,2-a]pyridin-3-amine

Kinetic energy-free Hartree–Fock equations: an integral formulation

We have implemented a self-consistent feld solver for Hartree–Fock calculations, by making use of Multiwavelets and Multiresolution Analysis. We show how such a solver is inherently a preconditioned steepest descent method and therefore a good starting point for rapid convergence. A distinctive feature of our implementation is the absence of any reference to the kinetic energy operator. This is desirable when Multiwavelets are employed, because diferential operators such as the Laplacian in the kinetic energy are challenging to represent correctly. The theoretical framework is described in detail and the implemented algorithm is both presented in the paper and made available as a Python notebook. Two simple examples are presented, highlighting the main features of our implementation: arbitrary predefned precision, rapid and robust convergence, absence of the kinetic energy operator

Uncovering Structure-Property Relationships in Push-Pull Chromophores: A Promising Route to Large Hyperpolarizability and Two-Photon Absorption

In this investigation, we report the first hyperpolarizabilities and two-photon absorption cross sections of a large series of 12 push–pull cationic chromophores. All of these dyes show a dipolar acceptor+–π–donor structure, where the nature of the donor and acceptor units and π-bridge was synthetically tuned to allow insightful comparisons among the molecules. The hyperpolarizability was obtained through a solvatochromic method, by exploiting the rare negative solvatochromism exhibited by the investigated compounds. The two-photon absorption cross sections were determined through two-photon excited fluorescence measurements by means of a tunable nanosecond laser system for sample excitation. The nonlinear optical properties were discussed relatively to the photoinduced intramolecular charge transfer occurring in these donor–acceptor systems, investigated by femtosecond transient absorption experiments. We found a strong increase in hyperpolarizability upon increasing the molecular conjugation. Unexpectedly, the hyperpolarizability is almost unaffected by an increase in donor/acceptor strength and intramolecular charge transfer degree. Differently, the two-photon absorption cross sections of these dyes are enhanced by an increase in both molecular conjugation and intramolecular charge transfer efficiency. Several recent literature works have reported at the same time scattered information about the hyperpolarizability and two-photon absorption of small organic molecules. Our investigation is, to the best of our knowledge, the first attempt to uncover detailed structure–property relationships for these two nonlinear optical properties. Our results represent a promising route to achieve large hyperpolarizability and two-photon absorption in push–pull dyes and may drive the design of new efficient nonlinear optical materials

Approximating a Wavefunction as an Unconstrained Sum of Slater Determinants

The wavefunction for the multiparticle Schr\"odinger equation is a function of many variables and satisfies an antisymmetry condition, so it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. We present a method without any such constraints, by which we hope to obtain much more efficient expansions, and insight into the inherent structure of the wavefunction. We use an integral formulation of the problem, a Green's function iteration, and a fitting procedure based on the computational paradigm of separated representations. The core procedure is the construction and solution of a matrix-integral system derived from antisymmetric inner products involving the potential operators. We show how to construct and solve this system with computational complexity competitive with current methods.Comment: 30 page