In this context, LOWDIN has been fully coded in the FORTRAN 2003 standard, with some C/C++ bindings to external libraries. Although FORTRAN 2003 is not a full Object Oriented Programming (OOP) language, most OOP capabilities can be easily emulated, such as class definitions, some polymorphism, and inheritance.
The CORE program
CORE is the main program of LOWDIN. It implements a set of tools to load the input file, generate the molecular system, and run all the requested tasks. This program also includes the INTEGRALS and SCF programs.
Figure 2 presents an example of an input file. The minimum required blocks to run a calculation are GEOMETRY, TASKS, and CONTROL.
The GEOMETRY block provides the information needed to build the molecular system. The first column declares the type of the quantum species. As shown in Figure 2, e-(H) and e-(O) define the electrons of a Hydrogen and an Oxygen atom, respectively; u- defines a negative muon, O_16, H_1, and H_2 define 16O, 1H, and 2H nuclei, respectively.
The second column declares the basis sets. When the “dirac” basis is chosen, the particle is treated as a classical point charge. The third, fourth, and fifth columns declare the x,y,z coordinates of the particle basis set center.
The sixth column provides additional information via keywords addParticles and multiplicity. These keywords are used to change the default values. addParticles is used to modify the number of particles of a quantum species. As shown in the provided example, one electron is removed from the system. multiplicity defines the multiplicity for open shell calculations. In the example, an electronic multiplicity of 2 was chosen.
LOWDIN contains a large library of electronic basis sets. Some of the nuclear basis sets developed by Nakai[33, 34] and Hammes-Schiffer groups are also available. Home-made basis sets for nuclei, positrons, and muons have been generated by employing the even-tempered basis set scheme.
Any type of quantum particles or basis sets can be loaded even if they are not present in the main library. Users can upload new basis sets and quantum species through the addition or modification of some plain text files matching the information provided in the input file.
The TASKS block defines the type of calculation to be performed by LOWDIN. All capabilities are listed in section capabilities. Finally, CONTROL block contains all parameters needed to control the behavior of the program, such as thresholds, maximum number of SCF cycles, etc.
Once the input file is loaded in LOWDIN, the CORE program manages the execution of the requested tasks. The communication between the CORE program and the other subprograms is carried out through text files.
The INTEGRALS program
This program evaluates the one- and two-particle integrals. One-particle integrals such as overlap, kinetic, and nuclear attraction energy have been implemented for Gaussian basis functions of any angular momentum following Obara and Saika and Head-Gordon and Pople recursive schemes.
Two-particle interaction integrals of the type are calculated either with proprietary routines or with LIBINT library. Integrals of the type are calculated with the LIBINT library. Integrals can be stored on either memory or disk. Integrals stored on disk are collected in stacks containing a maximum of 30.000 of them. Several stacks are calculated simultaneously to exploit the computational power of the machine. Different schemes have been implemented to exploit the permutational symmetry of the integrals.
The SCF program
The SCF program has been designed to minimize the energy of a molecular system composed of multiple quantum species. In a multispecies calculation, LOWDIN creates Fock-like operators for every quantum species. Multiple-species SCF cycles can be performed using any of these three iteration schemes:
- Full convergence for each quantum species until global convergence is reached, as shown in Figure 3
- Full convergence of electrons, if any, followed by one iteration for each nonelectronic species
- One iteration for each quantum species until global convergence is achieved.
Convergence acceleration methods such as DIIS, level shifting, and optimal damping have been implemented and are fully operational for any type of quantum species.
LOWDIN has been designed to be capable of handling different representations of the wavefunctions of fermionic and bosonic species. The current version of code supports Hartree-products, Slater determinants, and symmetric permutations of spin-orbitals. The SCF program can perform Hartree, Hartree–Fock, and DFT[43, 44] calculations for any type of quantum species.
The DFT formalism implemented LOWDIN is an extension of the ADFT developed for electronic structure. At present, the electronic Slater–Dirac exchange[46, 47] and Vosko et al. correlation functionals. Electronic BLYP, B3LYP, PBE, and PBE0 functionals are currently being implemented. Nakai's Colle-Salvetti-type nuclear-electron correlation functional[44, 49] is also available. New interparticle correlation functionals for different combinations of quantum species are being developed.
LOWDIN contains a set of modules to perform post-Hartree–Fock calculations for systems comprised of any type and number of quantum species. Post-Hartree–Fock methods include second-order Møller–Plesset perturbation theory,[10, 42] second-order propagator theory for the calculation of any particle binding energies,[50, 51] and configuration interaction singles and doubles. We are currently implementing coupled cluster and third-order propagator methods for any type of quantum species.
Full-CI (FCI) calculations for only one type of quantum species can be performed in LOWDIN. These calculations employ the FCI routine developed by Knowles et al. for electrons.[53, 52] The FCI method can be used, for instance, to include electronic correlation using as reference the electronic wavefunction obtained from a APMO-HF calculation. In a similar fashion, nuclear quantum effects (NQEs) on electronic polarizabilities and Fukui functions can be obtained by ADPT calculations, based on the electronic density of an APMO–ADFT calculation. These methodologies will be extended to any type of quantum species in the near future.
APMO calculations for systems containing particles experiencing non-Coulombic interactions can be carried out in LOWDIN. To this aim, the interparticle interaction potentials are introduced as linear combinations of Gaussian functions. The resulting integrals are solved using the LIBINT library. With this methodology the quantum nature of pseudo-particles, such as dressed nuclei and quantum, can be studied. For instance, Figure 4 shows the density of two helium atoms inside a fullerene, where the helium–helium and helium–fullerene interactions are described as a two-particles potential and a central potential, respectively. The study of the quantum behavior of confined species in fullerenes is in progress in our group. In addition, LOWDIN can perform APMO calculations on systems constrained to one or two dimensions. These capabilities will have direct application to study of quantum wires and wells.
Figure 4. Left: Density contour (0.01 a.u.) of two Helium atoms confined in a fullerene. Right: HeHe and HeC60 potentials employed in the calculation.
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Study of NQEs
Earlier applications of APMO and LOWDIN concentrated on the investigation of hydrogen isotope effects on covalent and noncovalent interactions. Similar studies have been carried out by other research groups employing NMO methodologies.[56, 33, 57-66]
In a first series of studies we examined, the H/D/T isotope substitution effects on the geometry and electronic structure of hydrogen molecule, water, hydrogen halide dimers,[5, 67] and [XO3SOHOSO3X]− (X = H,K) complexes at the APMO/HF level. These studies revealed that nuclear delocalization impacts the geometry and stability of hydrogen bonds, affecting macroscopic properties. Our APMO/HF results qualitatively agree with the experimental isotopic substitution trends.
With the APMO/MP2, we have studied NQEs on the geometries and energies of small protonated rare gas clusters (RgnX+, Rg = He, Ne, Ar, X = H, D, T, and n = 13).[10, 69]. We proposed an APMO/MP2 energy decomposition analysis scheme that revealed how NQEs impact the electrostatic, polarization, charge transfer, and dispersion interactions contributing to the stabilization of the complexes.
A similar study was conducted for assessing the H/D secondary isotope effect on the binding energies of water–alkaline cation complexes, Alk+ (X2O)n (X = H, D; Alk = Li, Na, K; n = 1–4). Our results revealed that deuteration reduces the magnitude of the electrostatic and polarization interactions between the cation and the water molecule via an inductive effect. This important result is the first in a series of studies aiming for understanding hydrogen isotope effects on hydration, an important topic for biomolecular sciences.
An APMO/MP2 study conducted for the [CNHNC]− complex revealed the importance of including nuclear-electron correlation in APMO calculations. We discovered that while a NMO/HF calculation produced a minimal energy structure presenting an asymmetric hydrogen bond, the APMO/MP2 calculation produced a symmetric H-bond structure in agreement with available experimental data. This study confirmed the importance of including NQEs for an enhanced description of strong-low-barrier H-bonded systems.
We have proposed an extended propagator theory for electrons and other types of quantum particles. In a first application of the method, we analyzed NQEs and isotope effects on electron ionization energies of molecular systems. An energy decomposition analysis of the total ionization energies revealed that the calculated electron ionization energies correspond to nearly vertical processes. Our results reveal that including NQEs improves the estimation of ionization energies for systems experiencing Jahn–Teller effects in the photo-electron spectra.
The APMO propagator theory has also been applied to study proton detachment processes. We calculated the proton binding energies (PBEs) and proton affinities (PAs) for some inorganic and organic molecules, finding an excellent agreement between our predictions and the experimental values. We also combined our methodology with a systematic exploration of the energy surface of protonated water clusters, (H2O)nH+ (n = 1−7), to estimate the proton hydration free energy. Our prediction of −270.2 kcal/mol is in excellent agreement with other results reported in the literature. This work suggests that the second-order proton propagator method has the potential to become a powerful tool for predicting proton acid/base properties (Fig. 5). Currently, we are working on the implementation of higher-order approaches and the combination of the proton propagator with an implicit solvation model aiming for the accurate prediction of pKas.
Figure 5. The proton propagator, implemented in LOWDIN program, is a promising tool for predicting proton acid/base properties. Here, an example of how the proton binding energies (PBE) describe differences in the acidity of the protons of maleic acid.
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