Quantitative electron delocalization in solids from maximally localized Wannier functions A. Otero-de-la-Roza,1, ∗ Angel´ Mart´ınPend´as,1 and Erin R. Johnson2 1Departamento de Qu´ımica F´ısica y Anal´ıtica, Facultad de Qu´ımica, Universidad de Oviedo, 33006 Oviedo, Spain 2Department of Chemistry, Dalhousie University, 6274 Coburg Road, Halifax, Nova Scotia, Canada B3H 4R2 (Dated: September 26, 2018) Electron delocalization is the quantum-mechanical principle behind chemical concepts such as aromaticity, resonance, and bonding. A common way to measure electron delocalization in the solid state is through the visualization of maximally localized Wannier functions, a method similar to using localized orbitals in molecular quantum chemistry. Although informative, this method can only provide qualitative information, and is essentially limited by the arbitrariness in the choice of orbital rotation. Quantitative orbital-independent interatomic delocalization indices can be calculated by integration inside atomic regions of probability densities obtained from the system's wavefunction. In particular, Bader's delocalization indices are very informative, but typically expensive to calculate. In this article, we present a fast method to obtain the localization and delocalization indices in a periodic solid under the plane-wave/pseudopotential approximation. The efficiency of the proposed method hinges on the use of grid-based atomic integration techniques and maximally localized Wannier functions. The former enables the rapid calculation of all atomic overlap integrals required in the construction of the delocalization indices. The latter allows discarding the overlaps between maximally localized Wannier functions whose centers are far enough apart. Using the new method, all localization and delocalization indices in solids with dozens of atoms can be calculated in hours on a desktop computer. Illustrative examples are presented and studied: some simple and molecular solids, polymeric nitrogen, intermolecular delocalization in ten phases of ice, and the self-ionization of ammonia under pressure. This work is an important step towards the quantitative description of chemical bonding in solids under pressure. INTRODUCTION the delocalized electron can be observed on A or on B, but not on both. Therefore, interatomic electron delocal- The interpretation of quantum chemical results is ization is directly calculated from the system's wavefunc- fraught with difficulty because traditional chemical con- tion as the (negative) covariance between the population cepts are not readily obtainable from the outcome of a distributions of two given atoms. In addition to being quantum mechanical calculation. Chemical models de- related to the traditional chemical concepts of electron veloped before the advent of quantum mechanics (aro- sharing and covalent bonding, electron delocalization is maticity, bond order, etc.) are not uniquely defined also behind physical phenomena such as magnetic ex- in terms of the system's wavefunction, yet they are in- change and electrical conductivity. valuable to understand and predict chemical phenom- A number of local properties, the most popular of ena. Over the years, many methods have been devel- which is the electron localization function[6, 7] (ELF), oped to extract chemical information from computed have been developed to interpret chemical bonding. How- wavefunctions. A common technique in the solid state ever, these quantities are only tangentially related to spa- is the interpretation of the band structure, and of the tial delocalization, and their interpretation is often not projections of Bloch states onto localized atomic-like straightforward. Other approaches are based on examin- orbitals. The equivalent of Bloch states in molecular ing the shape of localized orbitals, calculated using some quantum chemistry|orbitals|have long been used to ultimately arbitrary localization procedure.[8{11] In con- predict chemical reactivity. An alternative to orbital- trast to these methods, the calculation of the population based descriptors of chemical bonding is Bader's Quan- covariances in the QTAIM basins, known as localization tum Theory of Atoms in Molecules[1{5] (QTAIM), in and delocalization indices (DIs),[12] gives a direct quanti- which atomic regions (basins) are defined based on phys- tative measure of interatomic electron delocalization. DIs ical observables such as the electron density. have found ample use in the literature, with applications An important chemical concept is interatomic electron such as the calculation of bond orders and the position of delocalization. In quantum mechanical terms, delocal- electron pairs[13{18], NMR coupling constants[19], and ization between two atoms occurs when there is a statis- aromaticity[20], to name a few. Some of us recently used tical correlation between their electron populations. An DIs to detect delocalization error from density functional electron delocalized between A and B contributes to the approximations in halogen-bonded systems.[21] average electron population of both. It also imposes a The downside of DIs is that their computation is signif- negative correlation between the two populations, since icantly more complex than the aforementioned alterna- 2 tives. At the simplest level (Hartree-Fock or Kohn-Sham 0 for all B. The properties of variance and covariance density-functional theory), DIs require the calculation of ensure that: the overlaps between all pairs of occupied orbitals, in- 1 X tegrated over each atomic basin. While manageable in N = λ + δ (4) A A 2 AB gas-phase molecules, the sheer number of atomic over- B6=A lap integrals in infinite periodic solids makes the calcu- for every atom A in the system. Thus, the average num- lation of DIs much more challenging. To our knowledge, ber of electrons in atom A can be effectively partitioned the only practical approach to the calculation of DIs in into \localized" (first term) and \shared" (second term). condensed systems has been proposed by Baranov and The average values in the preceding equations are cal- Kohout[22] in the context of the augmented-plane-wave culated by integrating the relevant one- and two-particle (APW) method, and it requires a relatively large amount probability densities. Namely, the electron density (ρ(r)) of computational resources (specifically, memory). and the pair density (π(r ; r )). The latter is written in In this work, we present an alternative method 1 2 terms of an exchange-correlation density: to calculate DIs in periodic solids using the plane- wave/pseudopotential approach. The efficiency of the π(r1; r2) = ρ(r1)ρ(r2) − ρxc(r1; r2) (5) atomic basin integrations is greatly enhanced by using grid-specific integration methods, particularly Henkel- that measures the deviation of the pair density from the man et al.'s[23{25] and the Yu-Trinkle method.[26, 27] In independent-electron distribution. In a Hartree-Fock cal- addition, the use of maximally-localized Wannier func- culation, the exchange-correlation density contains only tions (MLWF) allows bypassing the calculation of all the exchange contribution: atomic overlaps involving functions whose centers are suf- ficiently far apart. The combination of these two ideas ρxc(r1; r2) = γ(r1; r2)γ(r2; r1) (6) results in a method that efficiently calculates all the DIs where γ(r ; r ) is the one-electron density matrix: in a periodic solid, and can be applied to systems with 1 2 dozens of atoms in the unit cell using a desktop computer. occ X ∗ The application of our new method to the calculation of γ(r1; r2) = i (r1) i(r2) (7) DIs in solids under pressure will allow the quantitative i description of bonding in such cases, where traditional In a Kohn-Sham density-functional theory (DFT) cal- chemical assumptions often do not apply. A few illus- culation, only the electron density is available, but the trative examples in this regard are provided in the last exchange-correlation density is approximated by replac- section. ing i with the Kohn-Sham orbitals. Since the exchange part of the exchange-correlation density contains most of its features (and the correct normalization) and Kohn- THEORY Sham and Hartree-Fock orbitals are similar, these as- sumptions yield delocalization indices that are still chem- Localization and delocalization indices ically meaningful,[28{30] at least in cases where strong correlation effects can be ignored. All equations and re- The average electron population, NA, of an atom A sults in the rest of the article use Kohn-Sham DFT. is given by integrating the electron density ρ(r) over its The localization and delocalization indices can be basin: shown to be[12]: Z α β NA = hn^Ai = ρ(r)dr; (1) λA = FAA + FAA (8) A α β δAB = 2(FAB + FAB) (9) In this work, we use the QTAIM atomic basins, which are σ defined as the region enclosed by zero-flux surfaces of the where the FAB are integrals of the exchange-correlation electron density.[4] Localization and delocalization are re- density over the basins of A and B: lated to the variance and covariance of the atomic pop- Z Z σ ulations. In particular, we define the localization (λA) FAB = ρxc(r1; r2)dr1dr2 (10) A B and delocalization (δAB) indices[12] as: X Aσ Bσ = Sji Sij (11) 2 2 ij λA = NA − Var(nA) = hn^Ai − (hn^Ai − hn^Ai ) (2) 2 δAB = −2 Cov(nA; nB) = −2(hn^An^Bi − hn^Ai ) (3) Aσ and the atomic overlap matrices Sij are defined as: Z where NA ≥ λA ≥ 0 and δAB ≥ 0. Inside a region where Aσ σ∗ σ Sij = i (r) j (r)dr (12) electrons are perfectly localized, λA = NA and δAB = A 3 Aσ ∗ ∗ ∗ The Sij are complex Hermitian matrices whose value the a , b , and c reciprocal directions (respectively) is depends on the particular set of orbitals used to describe equivalent to assuming periodic boundary conditions for σ the system. In contrast, FAB is a real symmetric matrix all one-electron states over a n1 × n2 × n3 supercell in that does not depend on the orbital rotation and, in con- real space (in the following, the supercell).