Electron Localization Function and Compton Profiles of Cu2O - The

Mar 4, 2019 - Taking free-atom Compton profiles, the charge-transfer model is also applied. The first-principles calculations based on the GGA are per...
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Cite This: J. Phys. Chem. A XXXX, XXX, XXX−XXX

Electron Localization Function and Compton Profiles of Cu2O V. Maurya and K. B. Joshi*

J. Phys. Chem. A Downloaded from pubs.acs.org by WEBSTER UNIV on 03/05/19. For personal use only.

Department of Physics, M.L. Sukhadia University, Udaipur 313001, India ABSTRACT: The nature of bonding in the cubic cuprous oxide is studied by means of the theoretical tools, namely, the electron localization function and Compton profiles. The isotropic Compton profiles together with the anisotropies in the directional Compton profiles are presented. Taking free-atom Compton profiles, the charge-transfer model is also applied. The first-principles calculations based on the GGA are performed, and the self-interaction correction is incorporated, adopting the GGA+U approach. Both types of calculations are performed deploying the linearized augmented plane-wave (LAPW) method. The effect of selfinteraction correction on the electron localization function, Compton profiles, and anisotropies is discussed. The electron localization function reveals ionic behavior in the (110) plane and covalent nature in the Cu−O bond intersecting plane. The GGA+U exhibits more covalent nature. The two LAPW calculations of the Compton profiles show better agreement with the available experimental data than the free-atom profiles. Among all of the calculations undertaken, the GGA+U shows the best agreement with the experiment. The GGA+U calculation shows more anisotropic behavior in directional Compton profiles.

1. INTRODUCTION New strategies and tools have emerged to unravel the nature of bonding beyond the analyses based on charge density, gradient in charge densities, and charge population schemes.1,2 The electron localization function (ELF) is the modern theoretical tool to describe intraphase and interphase phenomenon and the characterization of bonding from the viewpoint of traditional Lewis theory.2,3 It enables us to understand the bond formation and track changes in chemical bonding on compound formation, particularly during the structural phase transition in crystals.4 Conceived originally by Mulliken5 and developed further by Bader, Luken, and Culberson6,7 it is possible to decompose the space into nonoverlapping regions. The ELF is viewed as the inverse probability of finding two electrons with the same spin at a given point in space. Consequently, spin pairing has larger ELF, and unpaired states have low ELF.8 The function is defined in terms of χσ(r), the ratio between the excess of kinetic energy density due to the Pauli exclusion principle, Dσ(r), and the kinetic energy density, D0σ(r)2,4,8 χσ (r) =

where tσ(r) =

3

© XXXX American Chemical Society

1 1 + χ (r)2

(4)

The ELF is normalized to the range [0,1]. The metals are very well described by the homogeneous electron gas. In such cases, eq 1 gives χ(r) = 1, suggesting, through eq 4, that ELF= 1/2. For the vacuum (no electron density) or the area between atomic orbitals, ELF = 0, whereas ELF = 1 corresponds to the highest degree of electron localization.9−11 Thus ELF enables a distinction between metallic, covalent bonding, the nonbonding, and core regions, which are not easily distinguishable from the charge density and their gradients. In contrast with the ELF, the Compton profile (CP) is determined in the momentum space. It is directly related to the electron momentum density (EMD) distribution.12,13 This can be calculated from the Fourier transform of the real-space ground-state wave function of crystalline solids. In the independent particle approximation

(1)

2 1 |∇ρσ (r)| 4 ρσ (r)

(3)

i

ELF = n(r) =

where Dσ0(r) = 5 (6π 2)2/3 ρσ5/3 (r) is the density of a homogeneous electron gas. Within the Kohn−Sham scheme, the increase in kinetic energy considering the Pauli principle is evaluated as2,4,8 Dσ (r) = tσ(r) −

∑ |∇ϕi|2

is the kinetic energy obtained from the Kohn−Sham orbitals, φi, and the second term is the von Weizsäcker kinetic energy density functional. The ELF at any point r is then

Dσ (r) Dσ0(r)

1 2

ψnk(p) =

+∞

∫−∞

ψnk(r) exp(−i p·r)dr

(5)

Received: December 17, 2018 Revised: February 18, 2019

(2) A

DOI: 10.1021/acs.jpca.8b12102 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A where Ψkn(r) is the wave function of an electron with wave vector k and nth band. The EMD, ρ(p), is obtained as ρ(p) =

∑ fnk |ψnk(p)|2

calculations. Because covalent bonds are directional, more anisotropy is generally found in crystals with covalent bonding. We apply the linearized augmented plane-wave (LAPW) method within the framework of density functional theory (DFT) to study ELF, CP, and the anisotropies in the DCPs of Cu2O.32−34 It is worth noting that only recently has it become possible to calculate the CP using the LAPW method.14,34 Interestingly, for Cu2O, the experimental profile is reported by Bandopadhyay et al., so we compare our calculations with the measurement.35 Calculations of the CP using the local density approximation (LDA) and Lee−Yang−Par exchange and correlation (XC) functionals are reported using the linear combination of Gaussian orbital (LCGO) and the linear combination of atomic orbital (LCAO) methods, respectively, by previous workers.36,37 The LCGO calculations are performed at the experimental lattice constant on a grid of few points to sample the irreducible Brillouin zone, which affects the accuracy of calculated CPs and anisotropies.36 Both previous calculations show opposite trends with the experiment. More often, the LDA, the generalized gradient approximation (GGA), and the Hartree−Fock method do not appropriately describe the localized electronic states such as the d states in transition-metal compounds. The electronic structure calculations give a diverging description of the band gap and discrepancies in the conduction band ordering.22,38−40 The electron correlation effects are important for such properties of transition-metal compounds. To handle the self-interaction correction with localized d states, an on-site Coulomb repulsion, the Hubbard U, term is added to the LDA or GGA, as prescribed by Liechtenstein et al. and Dudarev et al.41,42 In some cases, it gives a better description than the simple LDA or GGA calculations. Recently, many-body perturbation theory in the GW approximation has shown some improvements compared with other methods in semiconductors and insulators.40,43−45 The computationally demanding various GW schemes have been tested for transition-metal oxides, yet a universal scheme to describe ground-state electronic properties for transition-metal oxides is not yet available.40,46 We therefore use the GGA and the GGA +U approaches in current investigations. It is worth mentioning that the band gap is just one salient feature of electronic structure and does not contain entire microscopic information on the electronic structure. As illustrated before, the EMD and the nature of bonding are also equally significant. Therefore, in this work, we adopt both GGA47 and GGA +U40−42,48 approaches to compute the ELF, the CP, and the anisotropies in the DCPs of the Cu2O. Hitherto, to our knowledge, GGA+U calculations are not attempted to unleash DCP and the nature of bonding using the ELF. In the CP studies quantities are described in atomic units (au), where e = ℏ = m = 1 and c = 137.036, giving unit momentum of 1.9929 × 10−24 kg m s−1, unit energy of 27.212 eV, and unit length of 5.2917 × 10−10 m.

(6)

n,k

where, f kn is the occupation. The CP, J(pz), is then defined as J(pz ) =

∬ ρ(p) dpx dpy

(7)

Thus CP is a projection of EMD on the z axis. The directional Compton profile (DCP) is the 2D integration of the EMD ρ(p) over a plane perpendicular to the êhkl through pêhkl, where êhkl is the unit vector along the [hkl] direction. The DCP contains a distribution of the electron’s momenta along the [hkl] direction of the crystal J(p·ehkl ̂ = pz ) = Jhkl (pz ) =

∬ p(p)δ(p·ehkl̂ − pz) dp

(8)

The anisotropy can then be extracted by taking the difference ΔJ(pz ) = Jhkl (pz ) − Jh ′ k ′ l′(pz )

(9)

Through anisotropies the CP carries information of the occupied bands via f nk and the Fermi surface in metals and similar systems.14 It is interesting to note that when J(pz) is integrated over the entire momentum space it gives the total number of electrons of the unit under investigation. Within the validity of impulse approximation, it is possible to measure the CP using radionuclide in the laboratory and performing inelastic X-ray scattering measurements at the Synchrotron facilities.14−17 Furthermore, CP is a directional property, so one can calculate the DCP along a principal crystallographic direction. The DCP can also be measured if single crystals are used in the experiments.16 The DCPs allow us to find anisotropies in the EMD originating from the bands dispersion. These carry vital information, as all isotropic contributions are eliminated when differences in DCPs are taken. Thus one can unleash directional features of bonding by means of anisotropies. Cuprous oxide (Cu2O) is a semiconducting compound. It crystallizes in cubic, hexagonal (CdI2-type), and tetragonal (CdCl2-type) structures at different pressures.18−21 Under ambient conditions, Cu2O has the cubic structure.22,23 It is widely used in thin-film transistors, optoelectronic, and nanoelectronic, spintronic, and photovoltaic applications.24−26 Although there is progress, the nature of bonding in Cu2O is still not well understood.27 Initially, Cu2O was considered to be ionic, and partial occupancies in Cu-d,s,p shells were also suggested later.27−31 The X-ray diffraction and convergent beam electron diffraction measurement combined with multipole refinement confirmed charge depletion from Cu-3d states on hybridization.30 It is argued that weak Cu−Cu interaction and the covalent bonding in Cu2O are related to the stability of the cubic structure. An electron counting suggested a closedshell Cu+O2− configuration. A few studies suggested the coexistence of ionic−covalent bonding and metallic bonding through experimental and theoretical tools.28−31 So both Cu3d 4s,4p and the Cu−O charge-transfer models are suggested. In this endeavor, the central objective is to shed more light on the nature of bonding in cubic Cu2O. We examine the bonding characteristics by calculating the ELF along a few planes and computing anisotropies in the DCPs. Many charge-transfer models can be realized using the free-atom CPs, and anisotropies can be calculated using the first-principles

2. COMPUTATIONAL METHODS The cubic structure of Cu2O belongs to the space group Pn3m. We applied the first-principles LAPW method to compute the DCPs and ELF of Cu2O. Several XC functionals are available in the literature to construct the Kohn−Sham Hamiltonian for the crystal. We have used the Perdew−Burke−Ernzerhof (PBE) XC functional based on the GGA.47 The GGA+U method corrects the electronic states beyond GGA, treating 3d B

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radii used.27,45 In view of this, our results are also in accordance with previous findings. The LAPW method embodied in the Elk allows us to calculate the ELF directly using the tasks51−53.32 These enable us to evaluate the ELF in one, two, or three directions. After achieving the ground state, one can evaluate the kinetic energy defined in eq 3 from the Kohn−Sham orbitals generated by the charge and energy self-consistent calculations. The Dσ(r) defined in eq 2 is then evaluated from the charge density, its gradients, and the kinetic energy tσ(r). These are then combined to compute the ELF defined in eq 4.49,50 For cuprous oxide, the 3D-ELF is calculated at the grid size of 32 × 32 × 32 k points and plotted using VESTA.51 The calculation of a DCP along a specific crystallographic direction, due to valence electrons, is performed, taking the length of H+k, that is, hkmax = 20. The grid size 16 × 16 × 16 is taken for the k points. To get the CP of a polycrystalline material, the spherical average is required. It can be obtained by taking the average of the three (3D-Sphav) or six (6DSphav) DCPs using the following relations52−54

states of Cu specially. There are two specific ways; the around mean field and the self-interaction correction. The latter corrects for the electron self-interaction and hence is more important in systems like Cu2O. In the GGA+U potential, the U and J parameters are the average Coulomb and exchange interactions in the 3d shell and give the effective Ueff (= U − J).48 Essentially, the PBE+U calculations are undertaken, so, for the sake of discussion, the PBE and the PBE+U calculations will be referred to henceforth as GGA and GGA+U, respectively. Several combinations of U and J were considered, and the total energy calculations were performed. Figure 1

Sphav J3D (pz ) =

1 [10J100 (pz ) + 16J110 (pz ) + 9J111(pz )] 35 (10)

Sphav J6D (pz ) = 0.1088J100 (pz ) + 0.0708J110 (pz )

+ 0.0162J111(pz ) + 0.3526J210 (pz ) + 0.2877J211(pz ) + 0.1639J221(pz )

(11)

The contribution of core electrons is taken directly from the tables of Biggs et al. and Weiss et al.55,56 The free-atom profiles of only 3s, 3p, 3d, and 4s electrons are listed by Weiss et al.; therefore, the remaining free-atom profiles are taken from the tables of Biggs et al. All profiles are then normalized to the area of free-atom profiles, taking the number of electrons contributing to the profile into account.

Figure 1. Calculated band gap, lattice constant, and bulk modulus of Cu2O at various values of Ueff. The experimental values are marked by the horizontal lines.

shows a plot of the lattice constant, bulk modulus, and the band gap, taking various values of Ueff. The combination with Ueff = 8 eV gives the optimum values of the three crystal properties. The lattice constant and the bulk modulus show 0.02 and 3.5% deviation from the experiment.13,39 Because the self-interaction correction through Ueff is orbital-specific, a small variation in these crystal properties from previous calculations is likely due to the uncommon augmentation

3. RESULTS AND DISCUSSION Electron Localization Function. The calculated ELFs using GGA and GGA+U are displayed in Figure 2. These are calculated along the (110) planes shown in Figure 2a. The planes pass through the oxygen ions situated at the center of the tetragons. Moreover, the ELF along the second (110)

Figure 2. (a) Crystal structure and the (110) planes of Cu2O chosen to plot the ELF. The planes pass through the oxygen ions. The ELFs along (110) from (b) GGA (c) GGA+U. The ELF scale spans in the [0,1] range and increases from blue to red. C

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Figure 3. ELF in the (001) plane at z = 1 containing Cu ions from (a) GGA and (b) GGA+U. The ELF scale in the [0,1] span increases from blue to red.

Figure 4. ELF in the bond intersecting plane at z = 0.875 from (a) GGA and (b) GGA+U. The ELF scale in the [0,1] span increases from blue to red.

One can clearly see small ellipses with ELF ∼0.25 between the Cu locations along one of the diagonals. The ELF around Cu ions is nonspherical. The inclusion of self-interaction correction through GGA+U connects the elliptical ELF around Cu ions (Figure 3b) along another diagonal, and the value changes from 0.1 to 0.25. It has been argued and interpreted in terms of the covalent bond resulting from the hybridization of Cu-3d states with the Cu-4sp states.30,31,57,58 The symmetry allows for the hybridization of dz2 orbital with 4s accompanied by a charge transfer. To examine it, the ELF at the z = 0.875 plane is drawn in Figure 4. This is the plane midway across the Cu−O bond direction. A high value of ELF (∼0.8) points out charge localization to form the Cu−O bond. The ELF has anisotropic behavior. In this plane, shown in Figure 4b, the self-interaction correction also connects the elliptical ELF at the center with the ELF at the corners. However, the change in the value of ELF is smaller than that observed in the (110) and (001) planes. Thus self-interaction correction included via U signifies that the bonding between the Cu−Cu atoms and ELF point toward its metallic nature. Compton Profiles. Isotropic Compton Profiles. Both GGA and GGA+U schemes implemented in the LAPW method are deployed to compute the CPs. The LAPW method gives the DCPs. However, to perform the CP study of a poly crystalline material, we need the isotropic CP. This is deduced by taking the spherical average of the three or six DCPs, as previously described. The difference in the spherical average calculated from eqs 10 and 11 is plotted in Figure 5. The

plane is the mirror reflection of the ELF along the first (110) plane. The ELF deduced from GGA is drawn in Figure 2b, whereas that from the GGA+U is drawn in Figure 2c. In the color-coding scheme, a high degree of electron localization is shown in red, whereas regions of low ELF values are shown in blue. We note that at the Cu cation sites the value of ELF is low, whereas high ELF (0.85) is visible in the spherical regions around the sites of O anions. Such a large difference in the ELF is a signature of the enhanced ionic character of Cu−O bonding. It is known that in the case of a covalent bond a local maxima in ELF is found between the ions, whereas the ELF minimum isolates the closed shells. In Figure 2b,c, a narrow region of minimum ELF separating Cu and O valence shells suggests ionic bonding. In the interstitial region, the ELF has the absolute minima, giving no indication of any electron localization. Although this is in contrast with the excess charge pointed out by Zuo et al.,30 it is in line with the observations of Filippetti and Fiorentini.31 The effect of self-interaction correction on the ELF can be examined from Figure 2b,c. We notice negligible changes in the ELF around the oxygen ion. The elliptical shapes of ELF around Cu ions are opening (Figure 2c), and the value changes from 0.1 to 0.25, indicating a small shift toward a metallic bonding. Thus U correction tends to enhance Cu−Cu bonding. The interaction between the Cu atoms can be understood from the ELF plotted along the (001) plane at z = 1. The values obtained from both the GGA and GGA+U are shown in Figure 3. D

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Figure 5. Difference in isotropic valence Compton profiles derived from the spherical average of three and six directions.

Figure 6. Compton profiles of Cu2O from the charge-transfer model. The profiles from free-atom, semiempirical model, and the experiment (ref 35) are also shown. The difference profiles ΔJ(pz) = JTheory(pz) − JExpt.(pz) are shown in the lower panel. Experimental errors are also shown at a few points.

first, the Cu-4s1 electron from each copper atom is transferred to the O-2p4. Thus the valence configuration of the compound becomes (Cu-3d104s0) 2(O-2s2 2p 6). In the second, 0.2 electrons from the Cu-3d and 0.5 electrons from Cu-4s state are transferred to the O-2p4, leading to the (Cu-3d9.84s0.5)2(O2s22p5.4) configuration. We also include the CP calculated following the semiempirical approach proposed by Aguiar et al.53 This model smears the CP from semiempirical formula by the weight function W+ in the low momentum region and W− in the Fermi region. All of these CPs, including the free-atom and experimental CPs, are plotted in the upper panel of Figure 6. The lower panel exhibits scatter from the experiment in terms of the difference ΔJ = JTheory(pz) − JExpt.(pz) together with the experimental errors.

difference is too small to be observed by the experiment at 0.53 au resolution. Also, this is smaller than the experimental error reported for Cu2O. Here we see that the maximum difference is only 0.1% with respect to the 6D average in GGA as well as GGA+U calculations. The difference of this order was also noted in our previous work on BeTe.59 For further discussion, we use the valence profile deduced from the 3D-Sphav formula. To study the charge transfer, we considered a few configurations transferring electrons from the Cu-3d4s to O2s2p states. To construct the total profile, we added the contributions of 3s, 3p, 3d, and 4s electrons taken from the free-atom profiles tabulated by Weiss et al., whereas for remaining electrons, the tabulated data of Biggs et al. are used.55,56 We considered two charge-transfer models. In the E

DOI: 10.1021/acs.jpca.8b12102 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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Figure 7. Difference profiles ΔJ(pz) = JTheory(pz) − JExpt.(pz). The valence profiles are calculated taking a spherical average of three principal directions. The best charge-transfer configuration is also included. Experimental errors are also shown at a few points.

Figure 8. Anisotropies in the convoluted directional Compton profiles of Cu2O calculated from the GGA and GGA+U schemes. Extreme positions found and marked by arrows are ascribed to the specific positions related to the translations vectors of the crystal.

We can see an effective reduction in the difference ΔJ by the two approaches in this figure. Below 1.3 au, the values from both GGA and GGA+U approaches are larger than the experiment. The differences decrease gradually on moving toward the high momentum side, falling within the experimental error. It is also observed that current calculations are in better agreement with the experiment compared with the LCGO and LCAO calculations reported by previous workers.36,37 The trend shown by the LCAO calculation is different than the trend shown by the LCGO and current calculations. In the region beyond 3 au, the core electrons start to contribute in the CP. These electrons are less affected by the formation of crystals and can be well described by the freeatom profiles. This is why in this region the differences between theory and experiment are relatively negligible. Residual differences may be due to the nonuniform density of the powder sample, which affects the absorption correction

We note that the free-atom profile devoid of any charge transfer gives the largest difference. The semiempirical model also shows a large deviation from the experiment. The two charge-transfer models improve the agreement. Among all, the best agreement is shown by the (Cu-3d9.84s0.5)2(O-2s22p5.4) configuration. The ΔJ curves suggest the transfer of 0.7 electrons (0.2 from d and 0.5 from s) from each copper to the oxygen. It is known that free-atom profiles do not embody the crystalline effects. Therefore, the LAPW method is applied to compute the valence profiles with and without U in the GGA functional. To see the effect on the CPs, the difference (ΔJ) curves are plotted in Figure 7. The experimental errors are also marked on a difference curve. Here the difference curve corresponding to the (Cu-3d9.84s0.5)2(O-2s22p5.4) configuration showing the lowest difference with experiment in Figure 6 is also included. F

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The Journal of Physical Chemistry A due to semicore and core electrons. It also affects the correction of the multiple scattering in the sample, particularly in the high momentum side. Such an effect is less pronounced in samples of pellet form. On comparing the CPs from GGA and GGA+U, we notice that below 0.5 au, GGA values show less difference with the experiment. Thereafter, in the 0.5 ≤ pz ≤ 1.7 range, the GGA +U values come closer to the experiment. This is the region that is largely governed by the Cu-3d electrons. So the role of the self-interaction correction is to reduce the differences in this region and improve the agreement with experiment. Beyond 1.7 au, the differences are negligible, and both curves overlap each other. Among all, the difference curve corresponding to the GGA+U shows the best agreement with experiment in the entire range. Anisotropies. The DCPs calculated along the three principal directions, that is, [100], [110], and [111], are convoluted with a Gaussian of 0.53 au and normalized to 19.601 electrons in the 0−7 au range. Differences in the DCPs are calculated to get the anisotropies. Both the GGA and GGA+U schemes were applied to get the DCPs. Results are plotted in Figure 8. Two anisotropies with respect to [100] are 0.16 to 0.19 e/au at pz = 0. This is ∼1.7% of JVal Exp(0). The anisotropies are larger than the LCGO calculations.30 This is a manifestation of the dense grid considered in this work to evaluate the CPs, which is desired for the accurate evaluation. In terms of the bands occupancy, we may infer that there are more occupied states along the [100] direction up to 1 au than the occupied states contributing to the CP integral building the DCP along the [110] or [111] direction. As previously observed in the ELF, the hybridization of d electrons with the unoccupied higher s and p states gives rise to the nonspherical charge density around the copper ion. Symmetry allows the hybridization of dz2 and 4s, leaving the dz2 partly unoccupied.30,57,58 Therefore, anisotropies with respect to [100] are high in the lowmomentum region and are relatively reduced in the intermediate region. The anisotropies with respect to the [111] direction overlap each other beyond 2 au Thus the DCP along [111] has a major contribution in the two anisotropies from 1 to 2.5 au In the anisotropy curves, the positions of extrema are clearly visible. The arrows marked on the curves show the positions. These are directly related to the crystalline effects. In the two anisotropies with respect to the [111] direction, these are occurring at

2π 3 a

Figure 9. Anisotropies in the unconvoluted directional Compton profiles of Cu2O calculated from the GGA and GGA+U schemes. Extreme positions found and marked by arrows are ascribed to the specific positions related to the translation vectors of the crystal.

involving [111] show additional extreme positions at 0.67 related to the 1.34 au. Similarly, in the anisotropies involving the [110] direction, an additional extreme position appears at 0.55 au, which is related to 1.1 au. Interestingly, therefore, more structures are visible in the [111]−[110]. The unconvoluted anisotropies are ∼2.3% of JVal Exp(0). Although anisotropy measurements of the cuprous oxide are not available, the measured [100]−[110] anisotropy of La1.85Sr0.15CuO4 is reported by Sakurai et al. at room temperature.60 The measurement of this related compound is compared with the KKR-CPA calculations, and a reasonable agreement is found. The disagreement between theory and experiment has been partly ascribed to the underestimation of the dz2 character of electronic states of copper. Moreover, it is suggested that band structure calculations considering Coulomb repulsion may be useful. The GGA+U calculations take care of the dz2 character of electronic states. The calculated anisotropy shown in Figures 8 and 9 is ∼1.72 times lower than that in La1.85Sr0.15CuO4. Moreover, we have noted that the GGA+U calculations give more anisotropies in the Cu2O. In view of the results on La1.85Sr0.15CuO4, one may speculate that the scatter between theoretical and experimental anisotropy would increase if such calculations are carried out for this compound. Essentially, the presence of the La and Sr electronic states around Fermi energy and the crystal structure would also play a significant role in describing the electronic states using the GGA+U calculation and hence the CPs in La1.85Sr0.15CuO4. A fair comparison could only be done if the DCPs of Cu2O are available at very high statistics as well as high resolution, as done in the case of La2−xSrxCuO4. Then, it would give a more rigorous description of the effect of the Hubbard U parameter on the electronic states in such compounds. It is hoped that this work would invite such work on other similar compounds.

= 1.34 au. The [111]−[110] anisotropy

shows a minimum at

π 2 a

= 0.55au also. The only maximum 2π 2

appearing in the [110]−[100] anisotropy is at a = 1.1 au. Thus the anisotropies are well characterized by crystalline effects. Beyond 3 au, the anisotropies are negligibly small where only isotropic core electrons contribute. The anisotropies from GGA+U also exhibit positions of extremes at the same locations on the momentum axis. We see that the inclusion of self-interaction increases the magnitude of anisotropies derived from the GGA. This subtle rise is visible only up to 1.7 au. Similar conclusions can be drawn from the unconvoluted anisotropies plotted in Figure 9. On comparing Figures 8 and 9, we see that convolution at 0.53 au has smeared some fine structures. For instance, a maximum with a negative value of anisotropy appearing at 0.67 au in Figure 9 is missing in Figure 8. Likewise, a maximum with a positive value appearing at 0.55 au is also missing. In Figure 9, anisotropies

4. CONCLUSIONS The LAPW method is applied to compute the ELF, isotropic CPs, and the DCPs of cubic Cu2O. The GGA and GGA+U types of calculations are attempted. In the (110) plane and interstitial regions, the ELF exhibits ionic bonding between Cu and O. In the (001) plane consisting mainly of the Cu cations, G

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The Journal of Physical Chemistry A

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the ELF signifies weak metallic bonding with directional features. A high value of ELF (∼0.8) is found in the plane across the Cu−O bond, which is directional in nature. An enhanced Cu−Cu interaction is observed in the ELF obtained from the GGA+U. The self-interaction correction through U suggests enhanced metallic character of the Cu−Cu bond. The 3D and 6D spherical average formulas proposed to find the isotropic valence profiles differ by merely 0.1%, indicating essentially a negligible difference in the calculation procedures. The computed isotropic CP shows good agreement with the experiment. However, subtle differences are seen in the low momentum region. Among the charge-transfer configurations and semiempirical model, the (Cu-3d9.84s0.5)2(O-2s22p5.4) configuration scatters the least from the experiment. Among all calculations undertaken, the experiment gives the best agreement with the GGA+U. In the ELF, a covalent nature of bonding is also found, and consequently the anisotropies in the DCPs show the directional nature of bonding. The highly resolved DCP measurements on single-crystalline samples would be fruitful to see the differences arising from the selfinteraction and to examine the residual differences. It is hoped that our study will invite such work on other similar materials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

K. B. Joshi: 0000-0002-6415-8320 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We have benefited from the discussion with S. B. Dugdale, University of Bristol, U.K. The assistance is gratefully acknowledged. Partial financial support for the computational facility is provided under the RUSA grant of the MHRD, New Delhi. V.M. is grateful to the UGC, New Delhi for providing the SRF under its BSR scheme.

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DOI: 10.1021/acs.jpca.8b12102 J. Phys. Chem. A XXXX, XXX, XXX−XXX