## Introduction

Over the years, quantum chemistry (QC) methods have become essential tools for understanding and predicting a wide variety of molecular phenomena.[1] The growing popularity and success of QC in molecular sciences is mainly due to continuous advances in QC theories and software and the gradual increase in speed and power of the computational hardware.[1-3] The impact of QC on molecular science research is such that nowadays it is common practice in experimental papers to include QC calculations to complement and support their findings and conclusions.[4] Despite the growing popularity of QC, there are still molecular phenomena that cannot be analyzed with current QC programs due to limitations on either the theoretical methodologies or the availability of their computational implementations.

In these cases, molecular scientists are forced to either (a) write their own codes or scripts to process and manipulate the data obtained by using available QC codes, (b) modify the existing computational codes to implement their new theories, or in the worst case scenario (c) write from scratch their own computational codes to implement their new theories.

The two groups currently involved in the LOWDIN project faced all the aforementioned problems. As a result, each group decided separately to create their own computational codes from scratch to use them as platforms for the efficient implementation of new theoretical methods. Andrés Reyes' group developed the Any Particle Molecular Orbital code known as APMO[5, 6] and Roberto Flores-Moreno led the development of the electronic structure package named PARAKATA[7] (Butterfly in Purepecha, a native language of central-west Mexico).

At the time of the creation of APMO, GAMESS-NEO was the only freely available computational code capable of performing Nuclear Molecular Orbital (NMO) calculations of electrons and nuclei.[8] Despite all the computational possibilities offered by GAMESS-NEO, the code was only capable of handling two types of quantum species simultaneously (e.g., electrons and protons). APMO emerged as alternative to GAMESS-NEO, with the aim of extending electronic structure methods to any type and number of quantum particles such as nuclei, muons, positrons and pseudo-particles. The last existing version of APMO before it was merged into LOWDIN is capable of performing Hartree–Fock (HF) and MP2 calculations for systems containing any type and number of quantum species.[9-11]

On the other hand, PARAKATA implements Auxiliary Density Functional Theory (ADFT),[12, 13] Auxiliary Density Perturbation Theory (ADPT)[14, 15] and electron propagator theory (EPT) for electronic structure.[16-22] ADPT capabilities include calculation of electronic contribution to static and dynamic dipole polarizabilities,[14] hyper-polarizabilities,[23] electronic and nuclear Fukui function,[24, 25] and other related fields such us the Shannon's entropy response to molecular ionization.[26] As it is inherent to, extensive use is made of auxiliary functions. Similarly to the deMon2k program,[27] auxiliary functions are generated automatically.[28, 29] In contrast, PARAKATA uses Cartesian auxiliary functions.[30] In addition, auxiliary functions can also be used in Hartree–Fock and post-Hartree–Fock calculations, such as in electron propagator calculations. In the latter, it has been proven that only minor deviations are caused by using auxiliary functions and the gain in efficiency is considerable.[31]

With the aim of extending the electronic structure methods implemented in PARAKATA to study systems containing other types of quantum species, it was decided to merge PARAKATA and APMO in 2010. This merging process is about to culminate. The resulting code has been named LOWDIN as a tribute to Per Olov Löwdin.

This review article is organized as follows: the section on technical details presents the computational aspects of LOWDIN. The section on capabilities presents the features implemented in LOWDIN and summarizes some of the studies carried out with the code. The section on perspectives talks about the current research and points out theoretical and computational challenges of the APMO theory and the LOWDIN code and strategies for the future.