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First-Principles Chemical Bonding Study of Manganese Carbodiimide, MnNCN, As Compared to Manganese Oxide, MnO Ryky Nelson,† Philipp M. Konze,† and Richard Dronskowski*,†,‡ †

Institute of Inorganic Chemistry, RWTH Aachen University, Landoltweg 1, 52056 Aachen, Germany Jülich-Aachen Research Alliance (JARA-FIT and JARA-HPC), RWTH Aachen University, 52056 Aachen, Germany

‡

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S Supporting Information *

ABSTRACT: We have performed an in-depth study of the chemical bonding in manganese oxide (MnO) and carbodiimide (MnNCN) from correlated spin-polarized density functional calculations. The chemical-bonding data were produced using the LOBSTER package, which has recently been enabled to process PAW-based output from Quantum ESPRESSO. Our results show that the ground states of MnO and MnNCN are similar, namely, antiferromagnetic structures whose axes are the MnO cubic [111] and the MnNCN hexagonal [001] axes, in agreement with experimental results. The results also evidence MnNCN being more covalent than MnO, in harmony with chemical intuition and spectroscopic data. In addition, the crystal orbital Hamilton population (COHP) analysis evidences that adopting the ground-state magnetic structures by MnO and MnNCN makes the cation−anion bonds optimized and annihilates obvious instability issues, that is, the existence of antibonding states in the vicinity of the Fermi level. We also detail the interactions involved in the systems using the recently introduced density-of-energy analysis and by partitioning the total and band-structure energies. While it is trivial that the total energy points toward the true magnetic ground state taken, the COHP integral of the metal−nonmetal bond is also capable of correctly delivering that particular information.

1. INTRODUCTION

Chemically, there is yet another way of modifying matter, namely, by changing hybridization through a chemically different divalent nonmetal atom. For example, the nitrogenbased pseudo-oxide of MnO reads manganese carbodiimide, MnNCN, in which the divalent carbodiimide anion NCN2− replaces the divalent oxide anion O2−. While the thermochemically stable binary transition-metal oxides are truly ancient phases dating back to alchemical times, the transition-metal carbodiimides are 21st century compounds and were quantumchemically predicted ahead of their synthesis only a decade ago, possibly for reasons of being metastable, that is, unstable (but inert) against the elements.5 Within a few years, the correlated carbodiimides of the 3d metals were then synthesized, namely, green MnNCN,6 black CuNCN,7 orange CoNCN, and lightbrown NiNCN8 as well as crimson FeNCN.9 All of them are semiconductors, just like their oxide cousins, and exhibit antiferromagnetic ground states; for MnNCN, the ground state was determined via spin-polarized neutron diffraction.10 The phase dubbed CuNCN is a many-body candidate giving rise to spin-liquid behavior11 and a resonating-valence bond (RVB) ground state.12 FeNCN has recently been shown to be an excellent battery material.13

Transition-metal oxides, in particular, the simple 1:1 binary phases with partially filled 3d shells such as MnO, FeO, CoO, NiO, and CuO, have been among the most fascinating objects of solid-state physics for many decades,1 and there is a plethora of experimental and theoretical studies dealing with such compounds. In a nutshell, these “correlated” oxides are characterized by an energetic competition between intra-atomic electron−electron repulsion on the metal, often dubbed “correlation” using physics jargon, and chemical bonding between the metal and the oxygen atom, dubbed “hybridization” or orbital mixing. Hence, many-body effects come to surface and lead to complicated phenomena such as metal− insulator transition, colossal magnetoresistance, charge ordering, orbital ordering, superconductivity, and a lot more.2 Indeed, some of the phases are so fundamental that they have shaped theoretical and experimental solid-state science. Nickel oxide, NiO, is the archetypal Mott−Hubbard insulator where band theory breaks down and must eventually be cured by various many-body approaches.3 Manganese oxide, MnO, is the first phase where magnetic-structure determination by means of neutron diffraction was successfully carried out.4 To physically modify such phases, there is also a strong tradition to substitute (usually coined as “to dope”) one transition metal by another one (that is, M1−xM′xO) such as to modify both electron count and the amount of correlation. © 2017 American Chemical Society

Received: August 17, 2017 Revised: September 21, 2017 Published: September 21, 2017 7778

DOI: 10.1021/acs.jpca.7b08218 J. Phys. Chem. A 2017, 121, 7778−7786

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The Journal of Physical Chemistry A

chemical bonding in MnO and MnNCN. Because U is not unique for a given element but depends on the details of the DFT+U implementation, it was determined using a linear response approach,35 which arrived at U = 4.07 and 4.33 eV for MnO and MnNCN, respectively. Hence, MnNCN is also slightly more correlated to begin with. For chemical-bonding analysis, we used the LOBSTER package,36,37 a tool to reconstruct electronic structures through projection of PAW-based wave functions onto atomic-like basis sets. LOBSTER has recently been improved to enable processing of PAW data from Quantum ESPRESSO, in addition to VASP and ABINIT. For the projection, we employed the local basis function as given by Bunge.38 Specifically, the orbitals included are 3s, 3p, 4s, and 3d for Mn, as well as 2s and 2p for O, N, and C. All crystallographic data for DFT calculations were taken from previous experiments.10,39 MnO crystallizes in space group Fm3̅m (225) with a = 4.440 Å (Mn on 4a, O on 4b), whereas MnNCN adopts R3m ̅ (166) with a = 3.358 Å and c = 14.247 Å (Mn on 3a, C on 3b, and N on 6c with z = 0.5855(3)). The unit cells of MnNCN and MnO are shown in Figure 1a,c, respectively.

This contribution aims at an in-depth study of the chemical bonding in manganese carbodiimide from (correlated) firstprinciples DFT calculations and a comparison with the corresponding oxide using exactly the same methodology. To the best of our knowledge, no such study has ever been attempted for a transition-metal carbodiimide. For the corresponding oxides, plenty of electronic-structure studies have already been performed,14−17 and their chemistry and physics have been examined.18 Details of the electronic structure of these materials have also been reviewed.19 MnO is a particular case where the nonmetallic behavior is seamlessly explained from the electron−electron repulsion parameter U, the bandwidth W, and the separation between manganese dand oxygen p-levels dubbed Δ as given by the Zaanen− Sawatzky−Allen framework.20 Previous studies21−23 have demonstrated, through partial DOS and population analyses, that MnO is classified as an intermediate Mott−Hubbard/ charge-transfer insulator with strong mixing of Mn 3d and O 2p at the Fermi level. Unfortunately, the nature of states in MnO at the Fermi level was never revealed using chemical-bonding analysis in any previous studies. This is one of the motivations of this study. The other questions that we would like to address through this work are the following: Into what category does MnNCN fit and how similar is the chemical bonding in MnO and MnNCN? Before doing the calculations, a moment of reflection reveals that the chemical bonding in transition-metal oxides cannot be purely ionic as the difference between Pauling electronegativities for Mn−Cu (1.5−1.9) and oxygen (3.5) is not exceedingly large and should allow for some covalency. This must also be the case for the bond between Mn (1.5) and N (3.0), where one expects even more covalency, as confirmed experimentally by means of X-ray emission spectroscopy (XES), X-ray absorption near edge structure (XANES), and resonant inelastic X-ray scattering (RIXS).24 A further investigation through analysis of the UV/vis absorption spectrum of MnNCN with the angular overlap model (AOM) and the Effective Hamiltonian Crystal Field (EHCF) method demonstrates clearly that the Mn−N bond is more covalent than the Mn−O bond.25 This paper is organized as follows: In section 2, methods and parameters used in the calculations are described in detail. Next, we present and discuss our results in section 3. Finally, we summarize and conclude our report in section 4.

2. METHODS AND COMPUTATIONAL DETAILS The density functional theory (DFT) results of this study employed the generalized-gradient approximation for solids (PBEsol),26 as implemented in Quantum ESPRESSO.27 All calculations were performed using the projector augmented wave (PAW) method.28,29 In order to achieve convergence, careful setups were made regarding the k-point mesh and the cutoff energy; the former was selected based on the Monkhorst−Pack scheme,30 and the latter was set to 1429 eV for MnO and 1225 eV for MnNCN. An energy convergence criterion of 13.6 × 10−6 eV was employed for the electronic loop. Following previous studies of MnO31 and MnNCN,24 which revealed the significance of the electron−electron Coulomb interaction (U) in Mn 3d orbitals, we also performed simulations with DFT+U32−34 using the simplified version developed by Cococcioni and de Gironcoli.35 A comparison of the calculations with and without U should allow one to show how electron−electron Coulomb interactions affect the

Figure 1. Crystal and magnetic structures of MnNCN and MnO starting with the crystallographic unit cells of (a) MnNCN and (c) MnO in which red, blue, green, and gray spheres indicate Mn, O, N, and C atoms, respectively. (b) AFM structure of MnNCN. (d) AFM1 and (e) AFM2 structures of MnO. Red and yellow spheres indicate Mn spin-up (majority) and -down (minority) atoms, respectively.

Finally, to understand how the chemical bonding is affected by magnetic interactions, we considered ferromagnetic (FM) and antiferromagnetic (AFM) structures. For MnO, two AFM structures were looked at, namely, AFM1 (Figure 1d) − antiferromagnetic between the (001) planes, and AFM2 (Figure 1e) − antiferromagnetic between the (111) planes; the latter is known from experiment. As for MnNCN, because LOBSTER can only handle collinear magnetic cases, we used a 7779


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Table 1. Total Energy (ΔE in eV per crystallographic unit cell), Band Gap (in eV), Mn Magnetic Moment (μMn in μB), Integrated COHP (ICOHP in eV per bond), and Band Energy (in eV) of MnO for the FM, AFM1, and AFM2 Structures GGA MnO ΔEa band gapb μMn ICOHPMn−Oc Eband

FM

AFM1

GGA+U AFM2

0.49 0.0

0.24 0.0

0.00 0.9

4.00 −1.28 −1002.28

4.06 −1.26 −1003.63

4.04 −1.30 −998.52

FM

AFM1

0.30 0.0 (↑), 3.8 (↓) 4.27 −1.20 −1036.63

AFM2

0.09 1.6

0.00 2.6

4.26 −1.19 −1040.15

4.22 −1.22 −1037.37

expt 4.1 ref 47 3.9 ± 0.4 ref 45 4.58 ref 44

ΔE is measured relative to the AFM2 total energy. bArrows indicate a certain spin direction (up or down) to which each given value belongs. Negative values of COHP and ICOHP are indicative of stabilization, just like other energies.

a c

Table 2. Same as Table 1 but for MnNCN GGA

GGA+U

MnNCN

FM

AFM

FM

AFM

ΔEa band gapb μMn (μB) ICOHPMn−Nc Eband

0.30 1.8 (↑), 2.0 (↓) 4.15 −1.33 −1072.67

0.00 0.8 4.07 −1.36 −1073.09

0.11 2.2 (↑), 3.2 (↓) 4.28 −1.25 −1108.15

0.00 2.2 4.25 −1.26 −1107.83

expt 3.4 ± 0.2 ref 24 4.72 ref 10

ΔE is measured relative to the AFM2 total energy. bArrows indicate a certain spin direction (up or down) to which each given value belongs. Negative values of COHP and ICOHP are indicative of stabilization, just like other energies.

a c

simplified version of the AFM structure measured in ref 10 − ferromagnetic within the (001) planes and antiferromagnetic between the (001) planes (Figure 1b). Puzzlingly, both experimentally found antiferromagnetic structures of MnO and MnNCN are ultimately linked to each other because the AFM2 structure of MnO indicates a switch of magnetic orientation along the cubic [111] axis. This very axis is identical to the hexagonal [001] axis along which the magnetic orientation flips in MnNCN. That being said, the AFM2 scenario of MnO does correspond to the AFM scenario of MnNCN.

gaps (MnO: 2.6 eV; MnNCN: 2.2 eV) are still underestimated as regards to the experimental ones,24,45 a well-known and notorious DFT issue. Although our somewhat approximate (see below) DFT band gaps differ from the experimental ones, they still show the same trend, namely, the MnO band gap being 15% wider than the one of MnNCN, in harmony with MnNCN being more covalent than MnO, which is the more ionic phase.46 Later on, we will see more signatures of MnNCN’s stronger covalency as compared to MnO. That being said, however, we note that the GGA+U self-consistent magnetic moments for the lowest-energy structures arrive at very similar numbers, namely, 4.22 (MnO) versus 4.25 μB (MnNCN). Finally, the band energies Eband given in the two tables, which are nothing but the sum of the Kohn−Sham one-electron eigenvalues, do not mirror the correct course of the total energies. This is a surprising result that needs further analysis, given later below. Figure 2 shows the total and partial density-of-states (DOS) of MnO and MnNCN as reconstructed from LOBSTER for different magnetic structures, using both GGA and GGA+U. Compared to the spin-polarized GGA scenario, introducing U pushes the Mn 3d band deeper into the valence region, as said before, for both MnO and MnNCN; for the former, there is still a significant Mn d-character at the Fermi level. While the GGA describes MnO and MnNCN as Mott insulators in their ground states (i.e., AFM2 and AFM, respectively, based on the total energy result), GGA+U yields them as intermediate charge-transfer/Mott insulators with a strong mix of Mn 3d and anion orbitals, easily seen from the top (FM) scenarios depicted in Figure 2a,d. This result on the nature of MnO is in agreement with previous studies.21−23 Upon changing from the ferromagnetic to the antiferromagnetic scenarios such as AFM1 and AFM2 for MnO as well as AFM for MnNCN (see Figure 2b−e), the differences are relatively subtle, mirroring the tiny energy lowerings (fractions of electron volts, Tables 1 and 2). Because antiferromagnetic

3. RESULTS AND DISCUSSION Tables 1 and 2 present Quantum ESPRESSO and LOBSTER data of MnO and MnNCN, respectively. The relative total energies (ΔE) reveal AFM2 and AFM as the ground states of MnO and MnNCN, respectively; this is in agreement with previous studies.6,10,40 In addition, the energetic course does not depend on U. A total energy difference calculation following the approach used in ref 41 yields nearest-neighbor (J1) and second-nearest-neighbor (superexchange, J2) magnetic exchange constants of −3.3 and −3.0 meV for MnO in the GGA+U case, very close to previous theoretical41 and experimental42 studies. For MnNCN, the two magnetic configurations studied do not allow determination of J1 and J2,43 and because additional magnetic calculations are outside of the scope of this paper, we leave them for future studies. Regarding the magnetic moment, the tables evidence that GGA +U results in a larger magnetic moment in all studied cases and that the magnetic moments are also closer to the experimental studies when including U.10,44 The reason for the larger magnetic moments lies in the electron−electron Coulomb interaction, U, which increases the splitting of the Mn d-band into the almost occupied (lower Hubbard) and unoccupied (upper Hubbard) bands. This larger splitting is also reflected in the increase of band gaps and the position of the Mn d-band in the valence region. However, despite the increase, the two band 7780


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practically no interaction with Mn but forming two double bonds with N, as indicated by the NCN2− Lewis formula. Next, we present charge-density plots of MnO and MnNCN, intended to provide a first but rough bonding analysis in the system of interest through a real-space charge-distribution picture. Figure 3a shows the charge density in the basal (001) plane of MnO for the AFM2 structure as obtained with GGA +U. We divide the plot into four sets to describe the charge density in the different spin channels and the upper (−5 to 0 eV) and lower valence bands (VBs) (−10 to −5 eV), respectively. There are two major observations: First, the figure reveals almost spherical charge densities around the atoms with high localizations, in particular, in the lower VBs, a clear sign of a nondirectional ionic bonding character. Second, it seems as if the spin-up and -down configurations basically form two symmetric sublattices. Otherwise, it seems practically impossible at this stage to correctly identify bonding, nonbonding, or antibonding effects because the phase of the interacting orbitals (constructive, zero, or destructive interference) is not contained in the charge-density plots. For comparison, the charge density of MnNCN in the (110) plane is shown in Figure 3b as evaluated for the AFM structure calculated with GGA+U. As before, the figure is divided into the same four subsets. The figure shows that, unlike in MnO, there must be more covalency in MnNCN’s bonding at both lower and upper VBs, simply because the atomic sphericity is smaller, with nonzero charge densities between Mn and N. Most obvious, however, is the NCN2− complex anion in the lower VB spin-down frame with its perfectly covalent double bonds between N and C. It also seems as if all of the MnNCN charge densities at the lower VB were more delocalized compared to their MnO counterparts. Again, this finding indicates that MnNCN is more covalent than MnO, in agreement with a previous charge-density analysis by Boyko and co-workers.24 To go beyond qualitative reasoning, the orbitals’ phases must be included; therefore, let us perform a crystal orbital Hamilton population (COHP) analysis,48 as will be discussed in the next paragraphs. In the following, we explain the nature of bonding in MnO and MnNCN and how it is affected by different magnetic structures and the electron−electron Coulomb U. Figure 4a−c displays the projected COHP (pCOHP) plots of the shortest Mn−O bond in MnO for various magnetic structures. Regardless of the magnetic structures and the size of U, the upper VB of the spin-up (majority) channel results from Mn− O antibonding interactions, whereas the lower one is from bonding interactions. The situation is clearly different in the spin-down (minority) channel because the whole VB is from bonding interactions. Furthermore, in the GGA model of the FM and AFM1 structures (Figure 4a,b), the Fermi level crosses strong Mn−O antibonding states, thereby indicating instability in these two structures. The use of GGA+U does not fully cure this instability because U basically shifts all of the Mn d-like states lower in energy but still leaves antibonding states as mixed into by O 2p levels in the vicinity of the Fermi level. Only by changing the magnetic structure to AFM2 (Figure 4c; for magnetic sketches, see Figure 4f) do these strong antibonding states close to the Fermi level get annihilated. This again demonstrates that AFM2 is the ground-state magnetic structure of MnO, supporting the total energy result. The strength of the Mn−O bond as quantified by the integrated COHP (ICOHP) values also confirms this stability of the AFM2 structure; as recorded in Table 1, the most

Figure 2. Total and partial DOS of MnO for (a) FM, (b) AFM1, and (c) AFM2 structures and of MnNCN for (d) FM and (e) AFM structures. The total DOS is indicated by the full black line, whereas the contributions from Mn 3d, O 2p (N 2p), and C 2p states are, respectively, indicated by lightest, darker, and darkest shadows. Red and blue indicate the DOS of the spin-up (majority) and -down (minority) electrons, respectively. The energy axis is shown relative to the Fermi level. In (f), a simplified depiction of the magnetic order is shown as a reference.

states have an identical number for majority/minority electrons, spin-up/-down frames are no longer necessary. For MnO, going from AFM1 to AFM2 (see magnetic sketches in Figure 2f) leads to a further lowering of the d-levels and an opening of a true band gap. The same can be seen for the AFM structure of MnNCN, the differences being even smaller. When comparing the occupied valence regions of the AFM2 structure of MnO (Figure 2c) and the AFM one of MnNCN (Figure 2e), both are heavily dominated by the 2p nonmetal levels (O for MnO and N for MnNCN), the quantitative difference being in the amount of dispersion, that is, the width of the 2p-centered part. For MnNCN, this energy region is broader by about 23% than the one of MnO, once again confirming the greater covalency of MnNCN. As a side remark, the role of the central C atom of the carbodiimide unit is mirrored from the fact that the carbon 2p states are lying in the center of this very region, with 7781


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Figure 3. Charge density of (a) MnO in the basal (001) plane for the ground-state magnetic structure (AFM2) and of (b) MnNCN in the (110) plane for the ground-state magnetic structure (AFM). Note the higher sphericity (more ionicity) in the MnO frames and the lower sphericity (more covalency) in the MnNCN frames.

GGA for the FM structure (Figure 4d, left), namely, the existence of the antibonding states in the vicinity of the Fermi level, albeit less pronounced than that in MnO. Figure 4d (right) makes it clear that the interaction U somehow cures this instability issue, although it also comes at a price, namely, a weakening of the Mn−N bond as indicated by its ICOHP value (ICOHPMn−N in Table 2). The effect of applying U on MnNCN is to shift the Mn d-like states lower in energy. The more profound effect, namely, the disappearance of the strong antibonding states from the vicinity of the Fermi level while maintaining the strength of the Mn−N bond, is realized when MnNCN adopts the AFM structure (Figure 4e). The change from the FM to AFM structures also strengthens the Mn−N bond, as mirrored by the increase of the ICOHPMn−N from −1.33 to −1.36 eV in GGA and from −1.25 to −1.26 eV in GGA+U. In contrast to the magnetic structure change, the inclusion of U does weaken the Mn−N bond (see Table 2), probably because U causes the Mn d-orbitals to be effectively more contracted and thus lessens the interactions between cations and anions. Nonetheless, for each exchange−correlation model, the Mn−N bond is optimized only for the correct magnetic scenario of MnNCN, as witnessed before for MnO. Likewise, the majority-spin contribution to the ICOHP is lower (−0.47 eV) than the minority-spin contribution (−0.74 eV), the aforementioned consequence of the exchange hole. The alert reader will have recognized that we have correlated the course of the total energy (decisive for the true ground state) with one particular bond (i.e., the strongest one) between the metal and the nearest nonmetal atoms, successfully so. While this is fine (and common practice in quantumchemical studies), a more comprehensive, possibly less arbitrary strategy would be to study the nature of all states in the VB. Hence, we also calculated the density-of-energy (DOE) function, which was recently introduced in the context of the “two-dimensional” Ge4Se3Te phase.50 Formally, the energydependent DOE is the sum of all interatomic (between the atoms) and all intra-atomic (on the atoms themselves)

negative ICOHPMn−O, that is, the strongest Mn−O bond, results from the AFM2 structure for both GGA and GGA+U calculations. Chemically speaking, it is the optimization of the Mn−O bond that lets MnO adopt the AFM2 magnetic structure. It is interesting to analyze how the ICOHP value of the Mn− O bond (−1.22 eV for AFM2 using GGA+U) can be further partitioned into those contributions that are due to the majority and minority spins. For a single Mn atom, the majority-spin contribution lowers the band-structure energy by an ICOHPα of −0.55 eV, considerably smaller than the minority-spin contribution (ICOHPβ = −0.72 eV). This can also be witnessed from Figure 4c (red curve) in which the upper antibonding contribution (between εF and −4 eV) cancels a significant part of the lower bonding contribution (−4 to −8 eV). In contrast, the minority-spin bonding contribution (in blue) is exclusively bonding below εF. Similar observations were made much earlier, almost 2 decades ago, in the spin-polarized bonding analyses of the 3d transition metals.49 Because of the exchange hole, the more contracted majority orbitals do not contribute as much to the total bonding as the more diffuse minority orbitals; hence, spin-polarized itinerant magnets have a wider lattice parameter than those in the non-spin-polarized scenario. The situation found for MnNCN is comparable but differs in a few but important details. Figure 4d,e depicts the pCOHP plots of the shortest Mn−N bond in MnNCN with the same description as before. Because the Mn−N bonds in MnNCN are the counterparts of the Mn−O bonds in MnO, the upper VB of the spin-majority channel of MnNCN also results from Mn−N antibonding interactions, whereas the lower one results from bonding interactions. Furthermore, the entire VB of the spin-minority channel again results from bonding interactions. A noticeable difference between the Mn−O and Mn−N pCOHPs is the interaction bandwidth, which is 23% wider in the latter (see also the DOS discussion above), indicating that the Mn−N bond is more covalent than the Mn−O bond. Furthermore, there is yet another sign of instability while using 7782


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being a DOS-similar function that is decorated, so to speak, by a stabilizing/destabilizing energy criterion that lets the DOE adopt both negative and positive values. To make stabilizing contributions go to the right side, we simply plot −DOE, similar to plotting −COHP or −pCOHP. Although we have witnessed that Eband does not correctly indicate the true ground state (see Tables 1 and 2), let us nonetheless look how it is composed. Also, we focus on the energetically lowest structures, AFM2 for MnO and AFM for MnNCN. Figure 5 displays the DOE of MnO (Figure 5a) and MnNCN (Figure 5b) ground states for an extended energy

Figure 5. Extended energy-range DOE plots of (a) MnO and (b) MnNCN in their stable AFM2 and AFM structures, respectively. The integrated DOEs are shown in red, while the energy axis is given relative to the Fermi level.

Figure 4. pCOHPs of the shortest Mn−O bond in MnO for (a) the FM, (b) AFM1, and (c) AFM2 structures and of the shortest Mn−N bond in MnNCN for the (d) FM and (e) AFM structures. pCOHPs of the spin majority and minority are indicated by the red and blue lines, respectively. The energy axis is shown relative to the Fermi level. In (f), a simplified depiction of the magnetic order is shown as a reference.

window going down into the semicore region to about −90 eV. On purpose, we start with these semicore contributions, which were labeled according to atom- and spin-resolved data (not shown here for brevity). Roughly between −40 and −80 eV, we find strongly negative (that is, highly stabilizing) energetic contributions to the DOE that consist of intra-atomic (on-site) manganese states, in particular, from Mn 3s (around −80 eV) and Mn 3p (around −50 eV) experiencing magnetic splitting by a few eV. Their atomic nature is obvious from the spiky behavior without any dispersion; therefore, it is justified to consider these energetic contributions (corresponding to the A = B entries in eq 1) as “crystal-field terms”48 being highly independent from covalent interactions to neighboring atoms. Around −20 eV, there are still atomic-like “on-site” contributions by O (MnO) and N and C atoms (MnNCN), almost as spiky as before. Upon moving into the valence-band region above −10 eV and up to the Fermi level, however, significant dispersion sets in and makes the DOE become much broader. This is the place to seek the interatomic (A ≠ B) stabilizing bonding interactions (witnessed before in the pCOHP plots in Figure 4), but they are outweighed by

interactions, and the DOE reveals all levels that stabilize and destabilize a system because it takes negative (energy-lowering) and positive (energy-increasing) values DOE(E) =

∑ ∑ ∑ ∑ Pμν(E)Hμν(E) A

μ μ∈A

B

ν ν∈B

(1)

Pμν and Hμν are the entries of the DOS and Hamiltonian matrices formed by the atomic orbitals μ and ν located on atoms A and B. If integrated up to the Fermi level, the DOE yields the band-structure energy, that is, the sum of all Kohn− Sham eigenvalues E band =

εF

∫−∞ DOE(E) dE

(2)

In contrast, the conventional DOS yields the number of electrons in the system. Hence, one may think of the DOE as 7783


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The Journal of Physical Chemistry A strongly destabilizing intra-atomic (A = B) interactions. That is to say that in this DOE representation of the band-structure energy, close to the Fermi level, the interatomic (A ≠ B) bonding is clearly overcompensated by the intra-atomic (A = B) terms; therefore, the DOE is entirely stabilizing in the semicore part and entirely destabilizing in the valence part, at least qualitatively. Integrated up to highest occupied bands (curves in the right frames), they result in band-structure energies (upper scale, compare with Tables 1 and 2) on the order of −500 eV per spin channel. Let us reiterate that the band energy, Eband, synonymous for the energy integral of the DOE, does not correctly indicate the correct magnetic state (see numerical data given in the Supporting Information), as found before. Fortunately enough, a much smaller but decisive part of the band-structure energy, namely, the integrated COHP values, does correctly identify the magnetic state, as seen before from Figure 4 and Tables 1 and 2. That being said, picking out the chemical bondings of the nearest metal−nonmetal interactions (Mn−O for MnO and Mn−N for MnNCN) is justified and by no means arbitrary; not only do these bonds represent the strongest interactions, but they are also optimized only for the correct ground state and “decide”, so to speak, about the magnetic structure eventually taken.

structures. Indeed, the size of the integrated COHP of the Mn−O and Mn−N bonds, although being on the order of just 1 eV and also being part of the magnetically indecisive bandstructure energy on the order of 1000 eV, correctly ranks the magnetic states as they are trivially given by the total energy on the order of 10 000 eV. Hence, the Mn−O and Mn−N interactions are the decisive ones both for structure and magnetism.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b08218. Projection of PAW-based electronic-structure data in LOBSTER and total-energy partitioning of MnO and MnNCN (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel. +49 241 8093642. ORCID

Philipp M. Konze: 0000-0002-7946-702X Richard Dronskowski: 0000-0002-1925-9624

4. CONCLUSIONS On the basis of explicitly correlated DFT+U electronicstructure calculations, we have analyzed the electronic structures of MnO and MnNCN and their chemical bonding as a function of correlation and magnetic structure. All data were generated using the LOBSTER suite, which has been enhanced to process PAW-based data from Quantum ESPRESSO. With respect to manganese oxide, our results support previous studies confirming the system as an intermediate charge-transfer/Mott insulator. Regarding the slightly more correlated carbodiimide phase, the theoretical data also confirm the system as an insulator of the same type. For both MnO and MnNCN, the relative orderings of the various magnetic states do not depend on the U correlation parameter. Despite being of the same charge-transfer/Mott insulator type, a direct comparison between MnO and MnNCN reveals both similarities and differences in their chemical bondings. For both chemical systems, the upper majority VB is comprised of antibonding Mn−O/Mn−N states that cancel, to some degree, the bonding states in the lower majority VB. In contrast, the entire minority VB is bonding, albeit smaller in magnitude. Nonetheless, as a consequence of the exchange hole, these coherently bonding interactions of the more diffuse minority orbitals contribute dominantly to the total bonding compared to the ones of the contracted majority orbitals, similar to what has been observed for the spin-polarized itinerant metals. In addition, our results concerning the charge distribution, the calculated band gaps, and also the dispersion of the VBs as observed in the local DOS and pCOHP plots furthermore clearly show that MnNCN is more covalent than MnO, in harmony with spectroscopic and electronegativity data. The chemical-bonding analysis also provides a reason why MnO and MnNCN adopt, respectively, the AFM2 and AFM structures. Only for these do the correct magnetic ground states fully annihilate formerly antibonding states from the vicinity of the Fermi level, thereby strengthening the strong Mn−O and Mn−N bonds and optimizing them for these magnetic

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Dr. Jan van Leusen for valuable discussions on magnetic materials. REFERENCES

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The Journal of Physical Chemistry A Phase and Analysis of Their Chemical Cause with the Density of Energy (DOE) Function. Angew. Chem., Int. Ed. 2017, 56, 10204− 10208.

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