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We introduce a method for computing quantum mechanical forces through surface integrals over the stress tensor within the framework of density functional theory. This approach avoids the inaccuracies of traditional force calculations using the Hellmann-Feynman theorem when applied to multiresolution wavelet representations of orbitals. By integrating the quantum mechanical stress tensor over surfaces that enclose individual nuclei, we achieve highly accurate forces that exhibit superior consistency with the potential energy surface. Extensive benchmarks show that surface integrals over the stress tensor offer a robust and reliable alternative to the direct use of the Hellmann-Feynman theorem for force computations in DFT with discontinuous basis sets, particularly in cases where wavelet-based methods are employed. In addition, we integrate this approach with machine learning techniques, demonstrating that the forces obtained through surface integrals are sufficiently accurate to be used as training data for machine-learned potentials. This stands in contrast to forces calculated using the Hellmann-Feynman theorem, which do not offer this level of accuracy., Comment: 8 pages, 4 figures, 1 table

Wavelets and Multiwavelets have lately been adopted in Quantum Chemistry to overcome challenges presented by the two main families of basis sets: Gaussian atomic orbitals and plane waves. In addition to their numerical advantages (high precision, locality, fast algorithms for operator application, linear scaling with respect to system size, to mention a few) they provide a framework which narrows the gap between the theoretical formalism of the fundamental equations and the practical implementation in a working code. This realization led us to the development of the Python library called VAMPyR, (Very Accurate Multiresolution Python Routines). VAMPyR encodes the binding to a C++ library for Multiwavelet calculations (algebra, integral and differential operator application) and exposes the required functionality to write simple Python code to solve among others, the Hartree--Fock equations, the generalized Poisson Equation, the Dirac equation and the time-dependent Schr\"odinger equation up to any predefined precision. In this contribution we will outline the main features of Multiresolution Analysis using multiwavelets and we will describe the design of the code. A few illustrative examples will show the code capabilities and its interoperability with other software platforms.

The many-body wave function of an $N$-electron system within a relativistic framework can be described by the Dirac equation. Unfortunately, the Dirac operator $\hat{\mathfrak{D}}$ is unbounded and in case we would describe anions we will observe the variational collapse of wavefunction of $N+1$ electron. Thus, it is necessary to avoid it and an alternative approach is based on applying the square of the Dirac operator $\hat{\mathfrak{D}}^{2}$. This approach is especially suitable for a multiwavelet framework: its implementation in an integral equation form is readily available and the completeness of the multiwavelet basis guarantees that the matrix representation of $\hat{\mathfrak{D}}^{2}$ in the occupied space is identical to the square of the matrix representation of $\hat{\mathfrak{D}}$. This equivalence is not valid in a finite basis set. The use of the $\hat{\mathfrak{D}}^{2}$ operator within a multiwavelet framework is able to achieve higher precision compared to the $\hat{\mathfrak{D}}$, consistent with the quadratic nature of the error estimates in a variational framework., Comment: Submitted to Theoretical Chemistry Accounts

The importance of relativistic effects in quantum chemistry is widely recognized, not only for heavier elements but throughout the periodic table. At the same time, relativistic effect are strongest in the nuclear region, where the description of electrons through linear combination of atomic orbitals becomes more challenging. Furthermore, the choice of basis sets for heavier elements is limited compared to lighter elements where precise basis sets are available. Thanks to the framework of multiresolution analysis, multiwavelets provide an appealing alternative to overcome this challenge: they lead to robust error control and adaptive algorithms that automatically refine the basis set description until the desired precision is reached. This allows to achieve a proper description of the nuclear region. In this work we extended the Multiwavelet-based code MRChem to the scalar zero-order regular approximation (ZORA) framework. We validated our implementation comparing the total energies for a small set of elements and molecules. To confirm the validity of our implementation, we compared both against a radial numerical code for atoms and the plane-wave based code exciting.

New techniques in core-electron spectroscopy are necessary to resolve the structures of oxides of $f$-elements and other strongly correlated materials that are present only as powders and not as single crystals. Thus, accurate quantum chemical methods need to be developed to calculate core spectroscopic properties in such materials. In this contribution, we present an important development in this direction, extending our fully adaptive real-space multiwavelet basis framework to tackle the 4-component Dirac-Coulomb-Breit Hamiltonian. We show that Multiwavelets are able to reproduce one-dimensional grid-based approaches. They are however a fully three-dimensional approach which can later on be extended to molecules and materials. Our Multiwavelet implementation attained precise results irrespective of the chosen nuclear model, provided that the error threshold is tight enough and the chosen polynomial basis is sufficiently large. Furthermore, our results confirmed that in two-electron species, the magnetic and Gauge contributions from $s$-orbitals are identical in magnitude and can account for the experimental evidence from $K$ and $L$ edges.

We present a multiwavelet-based implementation of a quantum/classical polarizable continuum model. The solvent model uses a diffuse solute-solvent boundary and a position-dependent permittivity, lifting the sharp-boundary assumption underlying many existing continuum solvation models. We are able to include both surface and volume polarization effects in the quantum/classical coupling, with guaranteed precision, due to the adaptive refinement strategies of our multiwavelet implementation. The model can account for complex solvent environments and does not need a posteriori corrections for volume polarization effects. We validate our results against a sharp-boundary continuum model and find very good correlation of the polarization energies computed for the Minnesota solvation database.

MRChem is a code for molecular electronic structure calculations, based on a multiwavelet adaptive basis representation. We provide a description of our implementation strategy and several benchmark calculations. Systems comprising more than a thousand orbitals are investigated at the HF level of theory, with an emphasis on scaling properties. With our design, terms which formally scale quadratically with the system size, in effect have a better scaling because of the implicit screening introduced by the inherent adaptivity of the method: all operations are performed to the requested precision, which serves the dual purpose of minimizing the computational cost and controlling the final error precisely. Comparisons with traditional Gaussian-type orbitals based software, show that MRChem can be competitive with respect to performance.

We present the extension of the quantum/classical polarizable fluctuating charge model to the calculation of single residues of quadratic response functions, as required for the computational modeling of two-photon absorption cross-sections. By virtue of a variational formulation of the quantum/classical polarizable coupling, we are able to exploit an atomic orbital-based quasienergy formalism to derive the additional coupling terms in the response equations. Our formalism can be extended to the calculation of arbitrary order response functions and their residues. The approach has been applied to the challenging problem of one- and two-photon spectra of rhodamine 6G (R6G) in aqueous solution. Solvent effects on one- and two-photon spectra of R6G in aqueous solution have been analyzed by considering three different approaches, from a continuum (QM/PCM) to two QM/MM models (non-polarizable QM/TIP3P and polarizable QM/FQ). Both QM/TIP3P and QM/FQ simulated OPA and TPA spectra show that the inclusion of discrete water solvent molecules is essential to increase the agreement between theory and experiment. QM/FQ has been shown to give the best agreement with experiments.

PCMSolver is an open-source library for continuum electrostatic solvation. It can be combined with any quantum chemistry code and requires a minimal interface with the host program, greatly reducing programming effort. As input, PCMSolver needs only the molecular geometry to generate the cavity and the expectation value of the molecular electrostatic potential on the cavity surface. It then returns the solvent polarization back to the host program. The design is powerful and versatile: minimal loss of performance is expected, and a standard single point self-consistent field implementation requires no more than 2 days of work. We provide a brief theoretical overview, followed by two tutorials: one aimed at quantum chemistry program developers wanting to interface their code with PCMSolver, the other aimed at contributors to the library. We finally illustrate past and ongoing work, showing the library's features, combined with several quantum chemistry programs., Comment: 42 pages, 1 table, 15 figures

Using multiwavelets, we have obtained total energies and corresponding atomization energies for the GGA-PBE and hybrid-PBE0 density functionals for a test set of 211 molecules with an unprecedented and guaranteed $\mu$Hartree accuracy. These quasi-exact references allow us to quantify the accuracy of standard all-electron basis sets that are believed to be highly accurate for molecules, such as Gaussian-type orbitals (GTOs), all-electron numeric atom-centered orbitals (NAOs) and full-potential augmented plane wave (APW) methods. We show that NAOs are able to achieve the so-called chemical accuracy (1 kcal/mol) for the typical basis set sizes used in applications, for both total and atomization energies. For GTOs, a triple-zeta quality basis has mean errors of ~10kcal/mol in total energies, while chemical accuracy is almost reached for a quintuple-zeta basis. Due to systematic error cancellations, atomization energy errors are reduced by almost an order of magnitude, placing chemical accuracy within reach also for medium to large GTO bases, albeit with significant outliers. In order to check the accuracy of the computed densities, we have also investigated the dipole moments, where in general, only the largest NAO and GTO bases are able to yield errors below 0.01 Debye. The observed errors are similar across the different functionals considered here.