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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 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.

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.

The phase transition occurring in a square 2-D spin lattice governed by an anisotropic Heisenberg Hamiltonian has been studied according to two recently proposed methods. The first one, the Dressed Cluster Method, provides excellent evaluations of the cohesive energy, the discontinuity of its derivative around the critical (isotropic) value of the anisotropy parameter confirms the first-order character of the phase transition. Nevertheless the method introduces two distinct reference functions (either N\'eel or XY) which may in principle force the discontinuity. The Real Space Renormalization Group with Effective Interactions does not reach the same numerical accuracy but it does not introduce a reference function and the phase transition appears qualitatively as due to the existence of two domains, with specific fixed points. The method confirms the dependence of the spin gap on the anisotropy parameter occurring in the Heisenberg-Ising domain.