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Interacting Quantum Atoms Method for Crystalline Solids

Cite this: J. Phys. Chem. A 2021, 125, 40, 9011–9025
Publication Date (Web):October 1, 2021
https://doi.org/10.1021/acs.jpca.1c06574
Copyright © 2021 The Authors. Published by American Chemical Society
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Abstract

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An implementation of the Interacting Quantum Atoms method for crystals is presented. It provides a real space energy decomposition of the energy of crystals in which all energy components are physically meaningful. The new package ChemInt enables one to compute intra-atomic and inter-atomic energies, as well as electron population measures used for quantitative description of chemical bonds in crystals. The implementation is tested and applied to characteristic molecular and crystalline systems with different types of bonding.

Introduction

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Nowadays, many interesting materials, e.g., intermetallic compounds, belong to the families of chemical systems, where the usual valence approaches based on pure ionicity, covalency with 2-center 2-electron bonds, or “metallic bonding” are no longer valid. The interplay of all these features challenges our understanding with the emergence of new bonding scenarios beyond the well-accepted concepts. (1−6) Since chemical bonding analysis represents a way to understand inter-atomic interactions, it is also a way to understand material properties and their relations to specific electron counts or partial structures. For this sake, chemical bonding parameters like atomic charges and bond orders must be consistently related to their energetic counterparts.
This is achieved in the Interacting Quantum Atoms (IQA) method. (7−9) It represents an ideal framework to investigate covalent and ionic interactions on an equal footing. The sum of the individual atomic interaction energies forms a part of the recoverable total energy of the system. All routinely used quantum mechanical packages yield the total electronic energy. Usually this energy is given in terms of kinetic and potential energy contributions. Additionally, the potential energy is split into Coulomb and exchange-correlation parts. For deeper insight into the mutual interactions between the atoms, a suitable analysis of the total energy, respectively the wave function and its components, must be performed. A number of tools and methods are available for this task, but only the periodic energy decomposition analysis (pEDA) (10) has been used to decompose the total energy of extended systems. The decomposition is done in Hilbert space. Also the Crystal Orbital Hamilton Populations (COHP) (11) analysis does the decomposition in Hilbert space but decomposes the band structure energy instead. An alternative and appealing method to decompose the total energy components into contributions of and between position-space atomic regions is the approach of IQA. Unique features of IQA are that it is orbital invariant, methodologically independent of the type of basis set, applicable to correlated wave functions, and does not depend on an external reference state.
The IQA scheme is based on the idea that the one- and two-electron energy integrals computed over the total coordinate space can be evaluated separately over any set of nonoverlapping spatial domains. The connection to a sound chemical description is achieved by the utilization of domains defined by the Quantum Theory of Atoms in Molecules (QTAIM), i.e., by the atomic basins determined by the density gradient field. (12) With this, by successive integration of the one-electron components over each atomic basin, the total one-electron energy is decomposed into atomic contributions. The contributions describe the kinetic energy of the atomic domain as well as the attractive energy between the electrons and the nucleus enclosed within the atomic domain. The integration of the two-electron energy components over the atomic basins yields the intra-atomic and inter-atomic contributions that are used in the bonding analysis to judge the ionic and covalent bonding character. The whole procedure was adopted for molecular wave functions based on the Gauss-type of basis sets in the program Promolden. (13)
Until now, the IQA method has not been implemented for solids. Nevertheless, on the basis of careful calibration studies using a zeroth order approximation of the full IQA interaction energies, the so-called point-charge approximation of the covalent interaction energy and the ionic interaction energy (Madelung energy), was employed to understand chemical bonding and site occupation in ternary half-Heusler (MgAgAs-type) phases. (5) Finally, this scheme was employed to predict new phases with this kind of structure. (6) This may show the strength and scientific potential of the solid-state implementation of the exact methodology to be presented below.
The developed ChemInt software is coupled with the DGrid package. (14) This extends the capability to evaluate the molecular Gauss-type orbitals also to the utilization of Slater-type based molecular wave functions, as used in the ADF package, (15) as well as to access the numerical atomic orbitals expanding the wave functions for molecular and crystalline solid state systems computed with the FHI-aims code. (16) ChemInt is also capable to compute the delocalization indices for all those types of basis sets, including crystalline systems. (17) In particular, the bielectronic exchange energy integrals between position-space atomic regions for solid state wave functions are extremely challenging due to the huge number of participating orbitals, which made parallelization of ChemInt a necessary feature.
Theory and implementation of IQA for crystalline solids are outlined, reliability of the code is tested with an extensive set of molecules, and the method is applied to prototype solid-state compounds exhibiting different bonding scenarios.

Methods

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The total electronic energy of a molecule or solid with pairwise Coulomb interactions comprises the kinetic energy of the electrons T, electron–nucleus interactions Ene, electron–electron interactions Eee, and nucleus–nucleus interactions Enn
(1)
Given access to the first order density matrix ρ1(r; r′) and the pair density ρ2(r, r′), the energy can be written as
(2)
Suppose that the space is partitioned into a set of domains that exhaustively recover the total volume and are mutually exclusive. Then, the total energy can be decomposed as
(3)
The above energy terms can be written in compact form as the sum of domain self-energies and interdomain energies,
(4)
Domain self-energies are
(5)
(6)
where α ∈ A stands for all nuclei enclosed in domain A, and
(7)
Interdomain energies are
(8)
(9)
and
(10)
Notice that intradomain electron–electron terms are halved to represent total energies whereas interdomain terms represent nonequivalent interaction energies.
The approach to the energy decomposition given by eq 4 considers the crystal to be formed by interacting atoms, molecules, etc. One may, otherwise, take a reference unit formed by one or more domains, embedded in the crystal. We collect, in an additive manner, E = ∑GE(G), all energy contributions E(G) of reference unit G,
(11)
The interaction energy for reference unit G shall be organized by coordination spheres i,
(12)
where there are number of B neighbors in the coordination sphere i of domain A of the reference unit. The last term condenses the sum over the coordination spheres for each domain in the reference unit into a single sum over coordination shells j for all domains in the reference unit and, with interaction contributions within that shell. The multiplicity factor for the shell of those interactions (i.e., Li–Cl(1) or Cl–Li(1)) takes the form
(13)
where
(14)
and nA, nB are the numbers of symmetry equivalent atoms A and B in the reference unit, respectively.
The pair density can be viewed as the product of two quasi-independent densities minus a term accounting for the difference with the true pair distribution ρ2(r, r′) = ρ(r)ρ(r′) – ρxc(r, r′). (18) This leads in eq 8 to the Coulomb energy contribution
(15)
and the exchange-correlation energy contribution
(16)
With the exception of the exchange-correlation energy, all interdomain terms are classical. Thus, the interaction energy can be arranged as a sum of a classical electrostatic energy EclAB and a covalent energy ExcAB,
(17)
In the special case of the monodeterminantal wave function, the exchange-correlation density is directly given by the first order density matrix, ρxc = ρx = |ρ1(r; r′)|2.

Atomic Boundaries

Atomic domains defined by QTAIM theory are delimited by surfaces S that satisfy the zero-flux condition
(18)
where ρ(r) is the density at point r in the surface and n(r) is a normal vector to the surface. The manifold of all steepest ascent paths
(19)
terminating at given attractor (the common destination limt→∞r(t)) constitutes an atomic (QTAIM) basin. In the following, the QTAIM basins are used as the spatial domains over which the energetical decomposition is performed.
We note at this point that a representation of the surface compatible with atomic-centered grids is required to use the multi/bipolar expansions presented in the following sections. To alleviate the expense of the computation, we precomputed a trust sphere using the algorithm described by Rodríguez et al. (19)
Of fundamental relevance for solid state systems is achieving electroneutrality in the unit cell. In contrast to molecules, all atomic basins determined for a solid state system have finite volume. Thus, the atomic basins must recover the unit cell volume, a circumstance that seems to be problematic in case of atom-centered radial grids. We decided to use radial rays with symmetric spherical t-designs. (20)

Energy Integrals in the IQA Framework

The energy integrals for a system with perfect periodicity are performed over complex crystal orbitals ϕn,k for given band index n and reciprocal vector k
with R given by the unit cell vectors and the linear combination ψn,k(r – R) = ∑jcn,j(k) χj(r – aj) of atomic orbitals χj(r – aj), centered at aj = rj + R, and with coeficients cn,j(k) to be determined. In this paper, electronic structure calculations were done with FHI-aims, utilizing numerical atomic orbitals.
The band and translational symmetry labels are condensed here to a single label i ≡ (n, k). With this, the crystal orbitals ϕi determine the electron density as well as exchange pair density , where are overlap densities.
The density is replaced in eq 15, and the exchange pair density in eq 16,
(20)
with domains being now QTAIM atoms. Recall from eq 7 that B = A integrals have a prefactor.
Intra-atomic Coulomb integrals (A = B) are computed with the Laplace expansion of 1/|r – r′| around the position of the atomic nucleus (7)
(21)
where is the angular (r̂ = (θ,ϕ)) integral of the atomic density
(22)
weighted by real spherical harmonic Sl,m. The discriminant , with r< =  min(r, r′) and r> =  max(r, r′), ensures the convergence of the expansion.
The expansion is valid also for the exchange integral (7)
(23)
where . Atomic overlap densities ρi,jA are zero outside of the basin like the atomic density above.
The integral evaluation is more efficiently performed in the basis of overlap densities . The exchange term is diagonal in this basis with coefficients dij = 2(2 – δij) for monodeterminantal wave functions, (9) and no monadic diagonalization is required.
Two-center integrals (A ≠ B) are expanded around two poles, one at each atomic position, with a bipolar expansion (21−23)
(24)
where replaces as the discriminant. (7)
The exchange integral between two different basins is
(25)
Once the electronic structure problem is solved by whatever method, the self-energy of each atom (or fragment) in the unit cell is obtained. The interaction energy of each of them, e.g., A, with the rest of atoms of the lattice, appropriately classified by increasing distance, B, is obtained (). This infinite sum contains quickly convergent terms ExAB, which decay exponentially in insulators and polynomially in metals, and a potentially divergent series made up of EclAB electrostatic interactions. By separating Ecl into a short-range, penetration-like exponentially decaying term and a multipolar one that is summed via the Ewald construction, one gets rid of divergences. We have applied the Ewald construction with the QTAIM monopoles (net atomic charges) via the Environ program (24) and have summed directly the higher order multipolar corrections from the IQA integrations up to near neighbors. Even if the QTAIM monopoles are zero, those terms remain and constitute the well-known electrostatic interaction between uncharged atoms or molecules.

Approximations Based on Population Measures

Inter-atomic bielectronic integrals are a particularly expensive step. As seen in eqs 24 and 25, they involve a bipolar expansion. For classic electrostatic energy integrals of distant basins it is equivalent to use a multipolar expansion (25)
(26)
where
(27)
is a real multipole moment of the net charge density ρt(r) = ∑AZAδ(r – RA) – ρ(r) in basin A. is an angular factor (cf. eq 25 in ref (7)). Technically, this approximation is invalid if the circumsphere of the basin A (centered at the nuclear position) intersects the circumsphere of basin B (with center at nucleus B). As a remark, solid state systems normally have compact basins. This limits the extent of the spheres, and it is safe to say that classic interactions with neighbor atoms of the second and higher coordination spheres are well approximated with a multipolar expansion in these cases. Also, internuclear distances in the solid state systems are often large, exceeding 2 Å.
On the other hand, exchange integrals were proven numerically to lead to asymptotic convergent multipolar expansions, with low-order terms being a good approximation for second and further nearest neighbors. (26) The appropriate expression has the same structure as for classic interactions
(28)
but in terms of the sum of overlap density multipoles
(29)
Further approximation of bielectronic integrals is to consider only the first term in the multipolar expansion. For the classic electrostatic term it is the same as assuming spherical density distribution in the atomic domain. By Gauss law, a point charge located at the nucleus position (for A and B) produces a physically equivalent classic force. The classic electrostatic energy,
(30)
is proportional to the product of QTAIM net charges for both basins, where is the average electron population in basin A.
Inter-atomic exchange integrals also admit a similar expression (26)
(31)
that instead involves the delocalization index (DI) δAB between the atomic domains A and B
(32)
The delocalization index measures the number of electrons shared between domains (A and B) and is complemented by the localization index (LI)
(33)
Localization and delocalization indices must obey the following sum rule
(34)
The fluctuation of electron population σ2(A) in basin A is the result of electron pair sharing with atoms of the first coordination shell σ12(A), second coordination shell σ22(A), and so on.
Expressions 30 and 31 are the basis for our more generic scaled point-charge approximation (sPCA) for classic ionic and exchange (covalent) inter-atomic interactions. The scaling parameters scl and sx approximate the exact classic and exchange energy, respectively, with a scaled zeroth order multipolar approximation:
(35)

Computational Details

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Density functional theory calculations (PBE functional (27)) for crystalline solids were performed with FHI-aims, an all-electron full-potential electronic-structure code. The numeric atom-centered basis sets used were of standard “tight” type. The computed unit cells and Brillouin zone samplings are listed in Table 1. An external output option of FHI-aims (version 200112) is used to generate DGrid compatible wave function information. ChemInt interfaces DGrid to evaluate wave function properties at discrete points in space, determines the atomic QTAIM basins, and performs the IQA analysis with this space partitioning.
Table 1. Crystallographic Information and Brillouin Zone Sampling for the Calculated Materialsa
systemspace group typelattice parameters [Å]calculated cellk-point mesh
β-N2bP63/mmca = 4.050c1 × 1 × 15 × 5 × 3
  c = 6.604  
CO2Pa3̅a = 5.4942 (35)1 × 1 × 14 × 4 × 4
diamondFd3̅ma = 3.566606 (36)1 × 1 × 14 × 4 × 4
BN (zincblende)F4̅3ma = 3.61 (37)1 × 1 × 14 × 4 × 4
graphiteP63/mmca = 2.464 (38)2 × 2 × 14 × 4 × 3
  c = 6.711  
BN-bP63/mmca = 2.504323 (39)2 × 2 × 14 × 4 × 3
  c = 6.658852  
MgB2P6/mmma = 3.0846 (40)2 × 2 × 13 × 3 × 5
  c = 3.5199  
LiClFm3̅ma = 5.12952 (41)1 × 1 × 14 × 4 × 4
NaClFm3̅ma = 5.7915 (42)1 × 1 × 14 × 4 × 4
MgOFm3̅ma = 4.213 (43)1 × 1 × 14 × 4 × 4
AlFm3̅ma = 4.0494 (38)1 × 1 × 14 × 4 × 4
NaIm3̅ma = 4.235 (44)1 × 1 × 14 × 4 × 4
a

Lattice parameters are given for the crystallographic cell, and calculated cells are specified as multiples of the crystallographic cell.

b

The z coordinate for the Wyckoff site is calculated assuming a bond length of 1.108 Å and that the molecule center is located at the symmetry center of the space group P63/mmc, as in ref (45).

c

Lattice parameters taken from ref (46).

The set of molecules taken for the validation were computed with GAMESS, (28) ADF, (15) and FHI-aims, (16) employing HF, LDA, (29−31) PBE, (27) BLYP, (32,33) and B3LYP (34) functionals. The basis set taken within each program is cc-pvtz (GAMESS), TZ2P (ADF), and standard “tight” (FHI-aims). For each molecule, the geometry was optimized before integrating the energy components. The geometry was fixed for molecules used for comparison against the solid phase (N2, CO2, neopentane, and phenalene). Only the four central atoms of phenalene were fixed while the positions of the other atoms were optimized.

Results and Discussion

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IQA energy terms are examined for a number of molecular systems to evaluate the energy error. In subsequent exemplary application studies, some prototype compounds (diamond and zincblende structures, honeycomb networks, rocksalt structures, closest packings) covering covalent, ionic, and metallic situations were investigated.

Validation of the Implementation

Three approaches are explored to validate our implementation. First, similar results should be obtained with alternative implementations when comparable (molecular wave functions expanded with GTOs). In this document only the ethane system is shown (Table 2), but the equivalence was tested also with other simple systems reaching always similar accuracy. Second, properties like the volume, exchange-correlation energy, and total energy in the cell were checked against FHI-aims. The total energy requires an exhaustive computation of long- range interactions to be recovered. The electron count sum for the (crystallographic) unit cell provide a third way to check the validity of our implementation. Only the first is examined in this section and the other two in the next sections.
Table 2. Total Energy Components of Ethane Computed with ChemInt, Promolden, and GAMESSa
energyChemIntPromoldenGAMESS
total energy, E–79.731–79.732–79.730
kinetic energy, T79.24379.243–79.244
total potential energy, Ene–158.974–158.975–158.974
electron–electron energy, Eee67.55167.55267.551
Coulomb energy, ECoul80.52880.52980.528
exchange-correlation energy, Exc–12.977–12.977–12.977
a

The evaluated number of electrons per formula unit are 17.9992 (ChemInt) and 18.0001 (Promolden). Ex was re-scaled from the integration of the PBE functional. (47) Energy in Ha units.

For the first check, we take the same molecular wave function and perform an energy decomposition analysis with two independent IQA implementations: our code and the Promolden code. (13) We see that the integrated quantities have similar accuracy to those of Promolden and are consistent, beyond the chemical accuracy of approximately 1 mHa, with accurate full space integrations given by GAMESS (Table 2).
The precision of IQA decomposition is mainly determined by errors in the atomic boundary, the quality of the grid, and the truncation of the Laplace/bipolar expansion. Another possible source of error can be differences in the evaluation of the energy in ChemInt with respect to the SCF program. The energy difference between reconstructing the total energy from IQA components against the SCF energy is a measure of the error of our method. For the evaluated molecular systems, this energy error per atom is always below chemical accuracy (see Figure 1) with a modest choice of integration parameters. The boundary is determined with a precision of more than 5 × 10–4 Å normal to the surface using an ODE integrator that preserves a local relative error of the same magnitude. The grid has 600 radial points and 5780 points distributed on a symmetric spherical t-design grid. A maximum expansion order of l = 4 (l = 6) for regions near (far from) the nucleus is chosen. Increasing the precision of the surface determination does not yield a better integration. As well, the length of the expansion chosen here is enough to approximate bielectronic integrals. From our observations, the quality of the grid is the most determining factor once the rest of the parameters have been fixed like here.

Figure 1

Figure 1. Energy difference (per atom) between the reconstructed IQA energy and the SCF energy (energy error). The white band denotes the median. Boxes delimit the lower and upper quartiles Q3 and Q4, and fences delimit all energy errors from a sample that takes wave functions computed with GAMESS, ADF, or FHI-aims, employing HF, LDA, PBE, BLYP, and B3LYP functionals for every system. Ex was rescaled from the integration of the corresponding functional. (47) All errors fall below 1 mHa.

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Nevertheless, it is important to realize that periodic systems require a significantly larger number of orbitals than molecular systems. Computations with maximum resolution are not always feasible, and one must find a balance of precision and speed. Therefore, we are interested in the minimum required computation needed to achieve a reasonable accuracy level. For all the systems tested, we found that a surface represented with approximately 6000 points is enough to integrate the total volume in the crystallographic unit cell with a volume error ΔVcell = ∑AVA – Vcell below 0.200 Å3 if we combine it with a radial grid of at least 200 points (Table 3). The symmetrical t-design quadrature offers a more robust integration, devoid of occasional (large) volume integration errors present with Lebedev–Laikov grids. In particular, for half-Heusler LiMgN, the cell volume error would be 2.027 Å3 using a 5810-point Lebedev grid. Instead, it is 0.010 Å3 with a 5780-point t-design grid. The distribution of points offered by the new grids is more homogeneous, thus providing a better overall representation of basin surfaces. Besides this, volume errors take place at the boundary of the basins so the integration of the charge can, and this is the case from our observations, still be within chemically reasonable accuracy error (<0.01).
Table 3. Reconstructing the Crystallographic Unit Cell Volume from QTAIM Basin Volumes: Volume Error, ΔVcell, and Cell Charge, Qcell
compoundstructure typeΔVcell [Å3]Qcell [e]
LiClNaCl0.104–0.005
NaClNaCl0.1680.009
MgONaCl0.0390.002
CsClCsCl0.004–0.000
Cdiamond–0.0040.007
BNzincbende0.0200.000
Cgraphite0.0170.001
BNBN-b0.0090.001
Naα-W (bcc)–0.0280.001
AlCu (fcc)0.0100.000
MgB2AlB20.0140.002
LiMgNMgAgAs0.0100.006
N2β-N2–0.0070.008
CO2CO2–0.0730.000

Molecular Crystals

Molecular crystals are a suitable example to examine the type of strong bonds that one finds in molecules before shifting to the highly diverse bonds present in crystalline materials. For that matter, we analyze β-N2, and CO2 crystals. A priori, N–N and C–O bonds are only expected to differ slightly from a gas phase molecule. Indeed, covalent intramolecular energies in the solid phase bear a striking resemblance to bonds of equal internuclear distance between atoms in gas phase, as shown in Table 4. Intermolecular interactions in the solid barely debilitate the covalent stabilization provided by the N2 triple bonds. As well, the classic interaction EclNN between two nearest nitrogen atoms is almost unaltered. In this regard, the triple bond exhibited by a N2 molecule is transferable to a triple bond in β-N2.
Table 4. Comparison of IQA Two-Center Terms for Covalent Bonds in Molecular and Solid Phases β-N2 and CO2a
systemA–BRABδABEnnABEneABEneBAECoulABEclABExAB
N2 (mol.)N–N1.1083.04123.399–21.944–21.94420.7160.227–0.906
N2 (solid)N–N(1)1.1083.00023.399–21.952–21.95320.7260.220–0.903
CO2 (mol.)C–O1.1491.39622.114–25.443–13.41315.453–1.289–0.444
CO2 (solid)C–O(1)1.1491.32322.114–25.646–13.15215.269–1.415–0.426
CO2 (mol.)O–O2.2970.43314.743–16.981–16.98119.5630.343–0.054
CO2 (solid)O–O(2)2.2970.40714.743–17.081–17.08119.7930.373–0.052
a

Nearest-neighbor interactions are denoted with A–B(1), second neighbors with A–B(2), and so on. Energies in Ha; distances in Å.

On the other hand, while the C–O bond is similar in both environments, the strong polarization of this bond permits ionic interactions with neighbor molecules, with O–O being the most intense interaction. The C–O bond is mainly altered by its classic ionic component, which varies by 0.126 Ha in comparison to 0.018 Ha for the covalent energy ExCO. In both scenarios, a strong contribution from dipolar and quadrupolar components to the classic energy is found. The same effect is observed for intramolecular O–O interactions but is more attenuated.

Diamond and Zincblende Structure

Convergence of Bielectronic Integrals

A steady convergence with increasing multipolar order is observed for intra-atomic bielectronic integrals in diamond and zincblende BN (Figures 2 and 3). The Coulomb energy ECoulAA shows a dominating monopolar term (l = 0) followed by zero l = 1, 2 terms as demanded by symmetry. (48) The l = 3, 4 terms are nonzero due to nonspherical distribution of electrons in the basin. The l = 5 term vanishes again due to symmetry. These observations for diamond atoms are in close agreement with those of a carbon atom in methane or the quaternary atom in neopentane (cf. Supporting Information, Figure S2 and Figure S3).

Figure 2

Figure 2. Diamond: Convergence of bielectronic integrals inside the basin of the carbon atom, with increasing multipolar order l, and . Only their magnitude is plotted. Dashed lines indicate the trend of convergence for symmetry allowed terms. Disconnected dots are not allowed by symmetry and are nonzero due to numerical errors. Terms below 10–5 Ha are represented as dots at the bottom.

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Figure 3

Figure 3. BN (zincblende): Convergence of bielectronic intrabasin integrals with increasing multipolar order l. For point and lines explanations, see Figure 2.

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Convergence with l of nearest-neighbor inter-atomic electron–electron terms in diamond is slower than for intra-atomic terms (see Figures 2 and 4). The convergence of Coulomb and exchange components with the order of the multipolar expansion was examined to assess the possibility of truncating the series. Storage of higher-order terms contributes to a substantial increase of computation time, specially due to the increasing number of averaged overlap densities Rl,mi,j. Coulomb and exchange terms show the expected convergent behavior. However, terms of order l = 3, 4 are still important for the Coulomb integral. This forces us to take several orders in the expansion. This behavior could, as well, vary for other systems. Therefore, it is only a preliminary observation.

Figure 4

Figure 4. Diamond: Convergence of bielectronic interbasin integrals with increasing bipolar order L = lA + lB. Labels in the top right indicate an interaction of atom A with an atom B of the ith coordination sphere as A–B(i).

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The Coulomb integral ECoulAB between a boron atom and a nearest N atom is well approximated from the (lA, lB) = (0, 0) term (Figure 5). Higher order terms L = lA + lB > 0 clearly reflect the characteristics of the electron distribution in their respective basins. Terms with order L = 1, 2 result from the combinations {(0, 1), (1, 0), (1, 1), (2, 0), (0, 2)}. As we have seen above, since the symmetry in B and N sites is tetrahedral, l = 1, 2 terms are zero (Figure 3). The same argument justifies the absence of L = 5 terms. Thus, only with L = 0, 3, 4, and 6 are the corrresponding bicentric Coulomb terms nonzero. Only due to numerical accuracy are they nonzero. More distant interactions like the closest N–N and B–B exhibit the same pattern, but their contribution from higher L order terms is already small, especially for B–B interactions.

Figure 5

Figure 5. BN (zincblende): Convergence of bielectronic interbasin integrals with increasing bipolar order L = lA + lB. For point and lines explanations, see Figure 2.

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Self-Energies and Near Neighbor Interactions

Having a generic method to decompose atomic energies allows us to compare molecular and solid systems on the same footing. Diamond is a prototype of a covalent network with bonded quaternary carbon atoms. A carbon atom with the same coordination in a molecule is the quaternary carbon of neopentane.
Table 5 compares the population and intra-atomic energy components of this atom in the two environments. The average electron population of the quaternary C atom in neopentane is slightly lower than that in diamond carbon atoms. Both intra-atomic classic EclCC and exchange energies ExCC are larger in diamond due to the higher population of the atomic basins.
Table 5. IQA Monocentric Integrals Inside Basins with Td Point Symmetrya
systemA⟨NeA⟩λAσ2(A)TAEneAAECoulAAEclAAExAA
neopentaneC5.9293.7752.15437.781–89.69419.154–70.540–4.566
diamondC5.9993.8202.18037.917–90.12819.522–70.606–4.602
BNB2.8392.0750.76523.700–51.7537.923–43.830–3.008
 N9.1617.5171.64355.650–138.16236.638–101.524–7.019
a

Energies in Ha units.

Carbon atoms have a similar inter-atomic distance in diamond and neopentane, approximately 1.545 Å. To facilitate the discussion, the C – C distance in neopentane is fixed to be exactly the same as in diamond. This way, the nucleus–nucleus repulsion is identical. Full relaxation of the geometry does not change our conclusions. Inter-atomic interaction energies are presented in Table 6. As for intra-atomic terms ECoulCC, electron–electron Coulomb energies are larger in the solid phase. This is counterbalanced by EneCC to finally yield a smaller classic energy. In diamond, electron–nucleus interactions are symmetrical, EneAB = EneBA. The sum of all classic electrostatic terms nearly cancels to yield EclCC = 0.019 Ha for neopentane and EclCC = 0.014 Ha for diamond because the electroneutrality of the atom makes the L = 0 term vanish. The classical L > 0 energy term Ecl,L≠0AB has to be positive for neutral nonoverlapping charge distributions. (8)
Table 6. IQA Two-Center Terms for Relevant Interactions in Neopentane, Diamond, and Zincblende BNa
systemA–BmRABδABEnnABEneABEneBAECoulABEclABExAB
neopentaneC–C(1)41.5440.95712.335–12.058–12.09411.8360.019–0.288
diamondC–C(1)41.5440.91412.335–12.217–12.21712.1130.014–0.284
 C–C(2)122.5220.0427.554–7.551–7.5517.5490.001–0.006
BNB–N(1)41.5630.35711.848–15.502–6.6978.765–1.586–0.106
 N–N(2)62.5530.15210.158–13.290–13.29117.3900.967–0.030
 B–B(2)62.5530.0025.183–2.943–2.9431.6710.968–0.000
a

is the number of equivalent interactions A–B per reference unit (G = C1C2 or G = B1N1) (eq 13). Energies in Ha; distances in Å units.

Exchange energies ExCC are almost identical in neopentane and diamond. Conversely, the nearest-neighbor DI δCC in diamond is smaller due to higher delocalization of the wave function (more distant neighbors) in the solid.
Diamond presents a DI of δCC = 0.914 between nearest-neighbor atoms. The delocalization goes down to δCC = 0.044 for second nearest-neighbors. The variation of DIs for the third coordination shell obtained for equivalent interactions is too large to prevent a detailed numerical examination. Up to first and second nearest neighbors the sum rule (34) is shown in Table 7.
Table 7. Cumulative Electron Populations of Atomic Basins in Diamond and Boron Nitride (Zincblende) from Successive Inclusion of Delocalization Shells α with DI δαa
αcoordination sphereδα⟨NelA⟩cumα
0C–3.820
1C–C(1)0.9145.648
2C–C(2)0.0445.912
0B–2.075
1B–N(1)0.3572.789
2B–B(2)0.0022.801
0N–7.517
1N–B(1)0.3578.231
2N–N(2)0.1559.161
a

The sum rule of eq 34 approaches ⟨NelA⟩=6 as delocalization with more distant atoms is included. Boron has ⟨NelA⟩ = 2.839 electrons, and nitrogen has ⟨NelA⟩ = 9.161 electrons.

Boron nitride represents an example of a diamond-type structure with polar bonds. While the nearest-neighbor electrostatic interaction EclCC=0.014 Ha in diamond represents a destabilization, in B–N it is a stabilizing classic interaction between opposite charged atoms with an energy contribution of EclBN=–1.586 Ha (Table 6).
The nearest-neighbor exchange interaction ExBN is only one-third of the analogous nearest neighbors interaction ExCC in diamond. Note that this decrease can be estimated (31) from the strong decrease of the delocalization index from δCC = 0.957 to δBN = 0.357.
For BN, in overall, classic electrostatic energies between nearest neighbors are much larger than the exchange ones. This fact reveals the high importance of classic interactions in the short-range regime.
The closest B–N interactions involve 0.36 electron pair sharing (Table 7) that parallels the value of 0.37 obtained before. (49) Nitrogen atoms share 0.15 electron pairs with any other nearest nitrogen atom (to be compared with 0.35 electron pairs). The sum-rule for N is approached including interactions up to the second coordination shell (see Table 7). On the other hand, boron atoms have no significant electron sharing with other boron atoms. There are, however, 0.038 electrons still missing to recover the average number of electrons in the boron basin.
The localization indices are in agreement with results obtained before by some of us (Table 6).

Additive Interaction Energies

The additive interaction energy (cf. equation 11) for a 2-atom reference unit G = {C1, C2} ≡ C1C2 in diamond can be approximately obtained as follows. The covalent interaction energies between nearest neighbors (−0.284 × 4 = −1.136 Ha) and between second nearest neighbors (− 0.006 × 12 = −0.072 Ha) sum to a total covalent interaction energy of Ex(C1C2) = −1.208 Ha. Higher neighbors are omitted due to a very small value of remaining bond fluctuations . A kind of lower bound of the neglected covalent interaction can be estimated with the zeroth order approximation if we assign all remaining electrons to the third coordination shell, Ha. The stabilizing covalent interactions are counterbalanced by electrostatic interactions Ecl(C1C2) = 0.014 × 4 + 0.001 × 12 = 0.068 Ha (in diamond) from contributions between first and second nearest neighbors, respectively. The total interactions considered amount to Eint(C1C2) = −1.140 Ha.
For zincblende, again, a diatomic reference unit G = {B1, N1} ≡ B1N1 is chosen. The approximate covalent interaction energy up to second nearest neighbor terms sums nearest neighbor B–N(1) interactions (− 0.106 × 4 = −0.424 Ha), second nearest neighbors N–N(2) interactions (− 0.030 × 6 = −0.180 Ha), and second nearest neighbor B–B(2) interactions (0.000 × 6 = 0 Ha) giving Ex(B1N1) = −0.604 Ha. While the electron sum rule is already well satisfied for N, some 0.038 electrons are missing on B. If we use this number to approximate the maximum additional covalent energy, the third nearest neighbor B–N(3) interaction with a distance may be estimated to contribute maximally . It is interesting that a non-negligible covalent N–N(2) interaction has been found, which makes up 30% of the covalent bond energy. It can be considered as the residual bonding of the incompletely occupied N2.161– atoms, which is a necessary consequence of the polar bonding B–N(1) leading to the small DI δBN = 0.357.
The procedure applied for the covalent part is not feasible for the electrostatic part, because of the long-range of the electrostatic interactions and the well-known resulting nonconvergence of the simple coordination series energy in real space. The electrostatic part of the interaction per reference unit can only be approximated by the point charge approximation using the Madelung constant of zincblende fM,R = 1.63806. The validity of the PCA for this procedure can be estimated by determining the PCA scale factor scl for the computed first and second nearest neighbor Ecl(B1N1) energies using Q atomic net charges and the corresponding distances , and . It turns out that the PCA energy values are rather close to the exact IQA ones (, ), such that the PCA is well justified. The Madelung energy per unit cell G = B1N1 is calculated according to Ha, where R is the first-neighbor B–N distance. Although l = 1, 2 (dipoles, quadrupoles) are forbidden by symmetry, there are contributions from classic electrostatic terms with L > 0 due to the large atomic net charges present in the system. Owing to their absence in the Madelung formula, they must be accounted at least for nearest neighbors. We compute them with Ecl(G) – Ecl,L=0(G) = −0.026 Ha, where the last term is just the point-charge contribution (eq 30). The total electrostatic energy is then Ecl = −2.617 Ha. From these values, the total interaction energy amounts to Eint(B1N1) = −3.221 Ha. The fraction of covalent bond energy per G = B1N1 with respect to the total interaction energy yields a 19% contribution of covalent interaction. Based on this fraction, the interaction in BN (zincblende) is characterized by dominating 81% ionic interactions. Still, the covalency is non-negligible, and it represents the local type of bonding, while the electrostatic Coulomb interaction is a collective type of bonding as discussed previously. (50) While the covalent bonding A–B is a local property of A and B, the electrostatic interaction A–B has long-range consequences, alternating stabilizing and destabilizing terms. These two different types of bonding cannot be discussed on the same footing, i.e., in a local picture. For this reason, the energies per reference unit have been calculated for characterization.

Honeycomb Networks

Honeycomb networks can be found in graphite, hexagonal boron nitride, and the intermetallic superconductor MgB2. Table 8 shows nearest-neighbor interactions in the three hexagonal systems. In addition to the σ bonds discussed in the previous section, they might exhibit a certain degree of π double bond character. In graphite, it has been shown, (51) based on delocalization indices, that the shortest C–C bonds with a DI δCC = 1.20 have partial double bond character, whereas in MgB2 the number of shared electron pairs is the same as for a single bond δBB = 1.00. For comparison, similar values of δCC = 1.202 in graphite and δBB = 0.983 in MgB2 are obtained.
Table 8. IQA Two-Center Terms for Relevant Interactions in Hexagonal Graphite, BN, and MgB2a
A–BmRABδABEnnABEneABEneBAECoulABEclABExAB
C–C(1)ab31.4231.20213.391–13.098–13.10212.8460.037–0.372
C–C(2)ab62.4640.0547.731–7.692–7.6927.6550.002–0.007
C–C(3)ab32.8450.0376.696–6.679–6.6816.6650.001–0.004
C–C(4)c13.3560.0195.677–5.709–5.7095.7460.004–0.003
C–C(5)c6 + 33.6450.0065.227–5.245–5.2475.2660.001–0.001
B–N(1)ab31.4460.45212.810–16.751–7.0659.244–1.762–0.135
B–B(2)ab32.5040.0045.283–2.936–2.9351.6321.043–0.000
N–N(2)ab32.5040.21210.354–13.604–13.60517.8771.021–0.040
B–N(3)ab32.8920.0086.405–8.419–3.5644.685–0.893–0.001
B–N(4)c23.3290.0065.563–7.329–3.1104.101–0.776–0.001
N–N(5)c33.6300.0247.144–9.411–9.41012.4030.726–0.003
B–B(1)ab31.7810.9837.429–8.392–8.3929.5220.166–0.249
B–B(3)ab63.0850.0524.289–4.953–4.9535.7220.106–0.006
B–B(5)ab33.5620.0173.714–4.303–4.3034.9870.095–0.002
Mg–B(2)c122.5040.06112.681–14.830–10.96412.824–0.290–0.013
Mg–Mg(3)ab33.0850.00324.704–21.364–21.36418.4750.450–0.000
a

The reference units corresponding to are G = C1C2, G = B1N1, and G = MgB1B2. Energies in Ha; distances in Å units. Key: ab, intralayer interaction; c, interlayer interaction

Graphite nearest-neighbor C–C bonds display higher ExAB energies (covalent bond energies) than in the other systems with honeycomb networks. The nearest-neighbor B–N covalent bond is particularly weak. In the honeycomb layer, the Ex proportion of the B–N covalent bond compared to the C–C covalent bond, , parallels the proportion seen in diamond and zincblende BN, . Conversely, classic electrostatic interactions are different for each system studied here. BN is largely stabilized from collective classical interactions between ions. This finding provides conclusive support for interpreting B–N as a very polar bond. However, destabilizing second-neighbor ionic interactions overcome the stabilization achieved by nearest neighbors. In the long-distance limit, those classic ionic interactions would tend to partially cancel. (50) Graphite only has high order multipole contributions to the classic electrostatic energy. Thus, already for first neighbors, a low value of Ecl = 0.037 Ha is obtained. In MgB2, nearest B–B interactions display a considerable classic (interionic) destabilization that is compensated by Mg–B classic electrostatic stabilization.
As in zincblende BN (ExNN = −0.030 Ha), among second-neighbor interactions only N–N still has a notable exchange contribution ExNN = −0.040 Ha in the hexagonal phase.

Phase Stability: Cubic versus Hexagonal Structures

Diamond and graphite are the main allotropes of carbon. The experimental difference in enthalpies is known to be quite small, ΔH298 K = +1.895 kJ/mol, (52) requiring an exquisite degree of accuracy that challenges DFT functionals. Albeit the relative stability problem is already solved, (53) the internal forces leading to a greater stability of graphite are not completely known. The additional stability might result from the delocalized π bonding framework. However, it is not clear if the higher coordinated carbon atoms (via σ bonds) in diamond should be less stable. The origin of graphite stability could as well be attributed to a particular local property like a different hybridization. In the spirit of a recent publication that includes those contributions in an analytical model, (54) we perform a numerical exploration to elucidate why nature chooses graphite over diamond.
Similar to the case of diamond discussed in the Diamond and Zincblende Structure section, the case of graphite involves a combination of stabilizing covalent bond contributions – 0.004 × 3 = −1.170 Ha for ortho, meta, and para neighbors for the chosen C1C2 reference unit. The covalent part of the C–C interaction in the c direction amounts to = Ha to be added to yield final Ex(C1C2) = −1.182 Ha. The destabilizing electrostatic interactions amount to = = +0.126 Ha (intralayer) and Ha (interlayer) summing up to Ecl(C1C2) = +0.139 Ha. A total interaction energy of Eint(C1C2) = −1.043 Ha is obtained. Comparison of the total interaction energy to that of the diamond modification Eint(C1C2) = −1.140 Ha (Ex(C1C2) = −1.208 Ha, Ecl(C1C2) = +0.068 Ha) reveals that the covalent interactions up to third nearest neighbors are energetically smaller, and the destabilizing electrostatic ones are larger. This yields an interaction energy preference for the diamond structure of ΔEint(C1C2) = 0.097 Ha.
Thus, the interaction energies do not explain the obtained total energy difference ΔEFHI-aims(C1C2) = 0.009 Ha between both modifications giving a preference to the graphite modification. This finding suggests that, at least for the current calculation, the preference of the graphite modification is caused by intra-atomic electron reorganization terms. Indeed, inclusion of the intra-atomic energies using ΔE(C1C2) = E(C1C2, diamond) – E(C1C2, graphite) = ΔEself(C1C2) + ΔEint(C1C2) resolves this seeming discrepancy and yields the energetical preference of the hexagonal phases (Table 9). In the framework of Valence Bond Theory, one would argue, the sp3 hybridization of carbon in diamond requires more energy than the sp2 hybridization in graphite. Just as in the solid phase, the 4-connected central carbon atom in neopentane and the 3-connected one in phenalene display a consistent self-energy difference of ΔEself(2 × C1) = 0.068 Ha, thereby demonstrating that the hybridization of the carbon atom is related to the stability of graphite over diamond. This indicates that the relative stability of carbon allotropes is not ruled by nonlocal contributions (π delocalization) but by intra-atomic electron redistribution energies often coined as “hybridization energies”.
Table 9. Relative Phase Stability of Cubic and Hexagonal Phases for C and BN Compoundsa
phaseEx(G)Ecl(G)ΔEint(G)ΔEself(G)bΔE(G)ΔEFHI-aims(G)
C (cubic)–1.2080.06800.1460.0490.009
C (hex.)–1.1820.1390.097000
BN (cubic)–0.604–2.6170B: 00.0120.005
    N: 0.120  
BN (hex.)–0.539–2.6450.037B: 0.07100
    N: 0  
a

Values for a reference unit: G = C1C2 and G = B1N1. ΔE(G) values are referred to the most stable phase. Energies in Ha units.

b

Due to numerical difficulties integrating the total kinetic energy with mHa accuracy, atomic kinetic energies TA were scaled to recover the total kinetic energy from FHI-aims.

The covalent interaction part per diatomic unit G = B1N1 is calculated according to = = −0.528 Ha for ortho (3 B–N(1)), meta (3 B–B(2) + 3 N–N(2)), and para neighbors (3 B–N(3)) respectively. The B–N(4) and N–N(5) interactions in the c direction amount to Ha to yield a total covalent bond energy Ex(B1N1) = −0.539 Ha. Since the Madelung constant for this structure is not fixed by symmetry, it cannot be found tabulated in the literature. Therefore, the ionic electrostatic interaction energy Ha was computed with the Environ program using the QTAIM net charges. This value is slightly larger than Ha obtained for cubic phase. The low symmetry of hexagonal BN allows classic contributions not present in the cubic phase Ecl,L>0(B1N1) = 0.123 Ha that in sum counteract the point-charge Madelung energy. Thus, classic energy components account for Ecl(B1N1) = −2.645 Ha. The portion of covalent interactions of 17% is smaller than 19% obtained for the cubic variant. The total interaction energy obtained amounts to Eint(B1N1) = −3.184 Ha, which is of smaller size than Eint(B1N1) = −3.221 Ha obtained for cubic BN by 37 mHa. In contrast, the total energy difference per BN unit computed with FHI-aims is ΔEFHI-aims(B1N1) = 0.005 Ha energy preference to hexagonal BN. So, the total energies obtained from FHI-aims yield a preference for the hexagonal variants for both cases: carbon, and BN. In contrast, for both cases, the summed interaction energies yield a preference of the cubic structures. This seeming discrepancy is resolved again taking the intra-atomic energies into account as well. According to this finding it is the intra-atomic energies and not the interaction ones that are responsible for the preferred stability of the hexagonal phases (Table 9). For BN, an interesting competition between B and N species is found. Intra-atomic energies indicate a preference of B species for the zincblende environment, while N species prefer the hexagonal environment. In the end the higher stabilization of the N species in the hexagonal environment dominates, and from ΔE(B1N1) = E(B1N1, cubic) – E(B1N1, hex.), the preference for the hexagonal phase is obtained.

Rocksalt-Type Structures

Ionic crystals should present situations with extremely polar bonds. As a case in point we consider LiCl, NaCl, and MgO.
Table 10 shows short-range interactions in the rocksalt-type structure for various compounds. Even nearest-neighbor inter-atomic interactions have a small energy contribution from electron sharing. One can foresee from the delocalization indices that this is the case. Classical interactions take over the exchange component for those systems. As expected from their large QTAIM charges, the classic energy only converges when long-range interactions are also included.
Table 10. IQA Two-Center Terms for Relevant Interactions in LiCl, NaCl, and MgOa
A–BmRABδABEnnABEneABEneBAECoulABEclABExAB
Li–Cl(1)62.5650.04410.523–11.076–7.3747.762–0.165–0.007
Li–Li(2)63.6270.0001.313–0.920–0.9200.6450.118–0.000
Cl–Cl(2)63.6270.08442.164–44.393–44.39346.7410.118–0.012
Na–Cl(1)62.8960.06334.173–35.919–31.45233.061–0.138–0.010
Na–Na(2)64.0950.00015.635–14.391–14.39113.2460.099–0.000
Cl–Cl(2)64.0950.04437.344–39.274–39.27241.3010.100–0.005
Mg–O(1)62.1060.12524.116–29.218–20.67425.050–0.725–0.029
Mg–Mg(2)62.9790.00125.579–21.931–21.93118.8030.520–0.000
O–O(2)62.9790.09011.368–13.803–13.80316.7590.522–0.014
a

The net charges of QTAIM atoms are ±0.90e, ± 0.88e, and ±1.71e in LiCl, NaCl, and MgO, respectively. Energies in Ha; distances in Å units.

The additive energy for a reference unit G = LiCl is computed as follows. The covalent interaction energy from interactions with the first and second coordination sphere is Ex(LiCl) = −0.007 × 6 + 0.0 × 6–0.012 × 6 = −0.114 Ha where six Li–Cl(1), six Li–Li(2), and six Cl–Cl(2) interactions are considered. The lattice electrostatic energy assuming point charges is Ha based on a Madelung summation. The effects of nonsphericity of the charge distribution around ions has an effect that is appreciated only for nearest neighbor interactions, as seen from their scaling factors of scl(1)=0.98 and scl(2) = 1.00. To correct the classic energy for first-neighbors, Li–Cl(1), higher order multipolar contributions, taken from the IQA inter-atomic energy as Ha, are included. The total electrostatic energy is therefore Ecl(LiCl) = −0.2848 Ha. Thus, the covalent part contributes to the interaction energy Eint(LiCl) = −0.398 Ha.
Similarly, for a G = NaCl reference unit, covalent interactions contribute with Ex(NaCl) = −0.09 Ha. The classic energy, assuming point charges, is Ha, that is corrected by nonspherical terms Ha. The corrected classic energy Ecl(NaCl) = −0.230 Ha entails a 72% percent of the interaction energy Eint(NaCl) = −0.320 Ha.
MgO exhibits a stronger covalent stabilization than either LiCl or NaCl, Ex(MgO) = −0.258 Ha. However, this increase is paralleled by stronger classical interactions Ha generated by interacting point charges of ±1.71 e. The correction from nonspherical terms is also enhanced, Ha. In overall, the total classic energy is Ecl(MgO) = −1.215 Ha. Thus, the covalent part contributes less, , to the interaction energy Eint(MgO) = −1.473 Ha than the previous systems with rocksalt-type structure.
With respect to interaction energies, the sequence of increasing covalent character is MgO (17%) < NaCl (28%) < LiCl (29%).

Scaled Point-Charge Approximation (sPCA)

The kind of approximation discussed here has been used previously to predict the stability of phases with highly symmetric MgAgAs structure (6) and to discuss Fe–Fe bonding in FeGa3. (4)
The validity of those approximations is tested against formally exact integrals from IQA method. Table 11 compares the zero order (point charge) approximation for interactions where largest deviations are expected, that is, between nearest neighbors.
Table 11. Scaled Point-Charge Approximation (sPCA) (See Eq 35) of Inter-Atomic Bielectronic Integrals for Nearest-Neighbor Interactionsa
SystemRABQAQBδABEclABsclExABsx
β-N21.1080.0030.0032.9990.220b–0.9030.630
CO21.1492.280–1.1401.323–1.4151.182–0.4260.699
Graphite1.4220.000–0.0011.2020.037b–0.3720.832
BN (hex.)1.4462.214–2.2130.452–1.7620.982–0.1350.816
Diamond1.5450.0010.0010.9140.014b–0.2840.907
BN (cubic)1.5612.160–2.1600.357–1.5871.004–0.1060.881
MgB21.782–0.813–0.8130.9830.1660.845–0.2490.852
MgO2.1061.711–1.7120.129–0.7250.984–0.0290.884
LiCl2.5660.897–0.8990.044–0.1650.991–0.0070.819
Al2.8630.0000.0000.2730.0013b–0.0541.071
NaCl2.8960.875–0.8780.064–0.13760.980–0.0100.866
CsCl3.5400.826–0.8270.126–0.10190.996–0.0170.915
Na3.6670.0000.0000.1080.0008b–0.0130.836
a

Energies in Ha, distances in Å, and charges in e units.

b

For interactions between (nearly) noncharged atomic species, the scaling parameters scl become quite large, because the electrostatic interaction is then no longer dominated by a monopolar term. Nevertheless, the absolute error of this assumption is typically small, because the interactions are weak.

Molecular crystals like β-N2 or CO2 present rather short interactions that can not be modeled with the point-charge approximation. Rather like molecules, both solids show very strong covalent bonds as evidenced by their DIs and inter-atomic ExAB. This is in agreement with chemical wisdom. Covalent bonds break the spherical symmetry of the density leading to larger contributions from higher order multipoles to the classic energy. Even more, in N2 the atomic charge is zero but there is still a 0.220 Ha classic destabilization.
Graphite and diamond feature fractional double bonds and covalent single bonds, respectively, which display a corresponding increase of inter-atomic distances that leads to a corresponding decrease of EclCC.
An inflection point is seen for zincblende BN. Despite having also covalent single bonds like diamond, the classic energy closely follows the point-charge approximation scl ≈ 1, indicating that the electron density is close to the spherical one.
Ionic and metallic crystals typically display longer inter-atomic distances and conceptual bond orders <1. Therefore, their covalent and ionic interactions are expected to be accurately approximated with the simple scaled point-charge equations applying an intermediate scaling factor sx ≈ 0.75–0.85.

Conclusions

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The new software package ChemInt was developed to decompose the energy of crystalline solids according to the Interacting Quantum Atoms approach. The QTAIM basins are chosen as domains for the energy decomposition. Other possible spatial partitions (i.e., Becke fuzzy atoms, Hirshfeld atoms, ...) deserve future research and will assess the robustness of our conclusions. A number of key issues were addressed which limited the accuracy of the integration. The package ChemInt was used for the examination of some prototype crystalline solids. For the ionic solids studied, covalent interactions contribute 15–30% to the interaction energy. The scale parameters of our scaled point-charge approximation were obtained from the formally exact integration of covalent and classic inter-atomic interaction energies. While certain trends for these parameters are already visible, further investigations are needed to faithfully predict them for all kinds of bonding situations. The phase stability of graphite over diamond and hexagonal BN over zincblende BN was obtained in terms of chemically meaningful energy components. Future explorations along this line are a promising area of research. The obvious limitations of the application to very large systems can be significantly reduced in the future with the exploitation of crystal symmetries and treating only valence electrons. The new program may contribute toward a better understanding of interactions in crystalline materials, in particular in those which do not follow traditional valence rules. Extending the IQA method to planewave bases is perfectly possible, and it can be implemented on any electron structure package because it does not depend on the representation of the wave function in terms of atomic orbitals.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.1c06574.

  • Computational details of ChemInt calculations and supporting IQA results (PDF)

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Acknowledgments

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D.M.C. thanks J. L. Casals-Sainz for suggesting spherical t-design grids as an alternative to Lebedev–Laikov grids. A.M.P. and E.F. thank the Spanish MICINN, Project PGC2018-095953-B-I00, for funding.

References

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This article references 54 other publications.

  1. 1
    Hume-Rothery, W. Materials Science and Engineering; McGraw Hill: 1967; pp 3– 23.
    Google Scholar
    There is no corresponding record for this reference.
  2. 2
    Fredrickson, D. C.; Lee, S.; Hoffmann, R. The Nowotny Chimney Ladder Phases: Whence the 14 Electron Rule?. Inorg. Chem. 2004, 43, 6159– 6167,  DOI: 10.1021/ic049897h
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    2
    The Nowotny Chimney Ladder Phases: Whence the 14 Electron Rule?
    Fredrickson, Daniel C.; Lee, Stephen; Hoffmann, Roald
    Inorganic Chemistry (2004), 43 (20), 6159-6167CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    The late transition metal Nowotny chimney ladder phases (NCLs, TtEm; T, groups 7-9; E, groups 13 and 14) follow a 14 electron rule: the total no. of valence electrons per T atom is 14. We ext. a chem. explanation for this rule from extended Huckel calcns.; we focus on RuGa2, the parent NCL structure. A gap between filled and unfilled bands arises from the occupation of two Ga-Ga bonding/Ru-Ga nonbonding orbitals per RuGa2, independent of k-point. In addn., the five Ru d levels are filled. Together this makes for 7 filled bands at each k-point, or 14 electrons per Ru. We discuss the connections between this 14 electron rule and the 18 electron rule of organometallic complexes.
    >> More from SciFinder ®
    https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXlt1Sisrs%253D&md5=08e97f2117878bd85c54b3f77982ec0d
  3. 3
    Yannello, V. J.; Fredrickson, D. C. Generality of the 18- n Rule: Intermetallic Structural Chemistry Explained through Isolobal Analogies to Transition Metal Complexes. Inorg. Chem. 2015, 54, 11385– 11398,  DOI: 10.1021/acs.inorgchem.5b02016
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    3
    Generality of the 18-n Rule: Intermetallic Structural Chemistry Explained through Isolobal Analogies to Transition Metal Complexes
    Yannello, Vincent J.; Fredrickson, Daniel C.
    Inorganic Chemistry (2015), 54 (23), 11385-11398CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    Intermetallic phases exhibit a vast structural diversity in which electron count is known to be one controlling factor. However, chem. bonding theory has yet to establish how electron counts and structure are interrelated for the majority of these compds. Recently, a simple bonding picture for transition metal (T)-main group (E) intermetallics has begun to take shape based on isolobal analogies to mol. T complexes. This bonding picture is summarized in the 18-n rule: each T atom in a T-E intermetallic phase will need 18-n electrons to achieve a closed-shell 18-electron configuration, where n is the no. of electron pairs it shares with other T atoms in multicenter interactions isolobal to T-T bonds. In this Article, we illustrate the generality of this rule with a survey over a wide range of T-E phases. First, we illustrate how three structural progressions with changing electron counts can be accounted for, both geometrically and electronically, with the 18-n rule: (1) the transition between the fluorite and complex β-FeSi2 types for TSi2 phases; (2) the sequence from the marcasite type to the arsenopyrite type and back to the marcasite type for TSb2 compds.; and (3) the evolution from the AuCu3 type to the ZrAl3 and TiAl3 types for TAl3 phases. We then turn to a broader survey of the applicability of the 18-n rule through a study of the following 34 binary structure types: PtHg4, CaF2 (fluorite), Fe3C, CoGa3, Co2Al5, Ru2B3, β-FeSi2, NiAs, Ni2Al3, Rh4Si5, CrSi2, Ir3Ga5, Mo3Al8, MnP, TiSi2, Ru2Sn3, TiAl3, MoSi2, CoSn, ZrAl3, CsCl, FeSi, AuCu3, ZrSi2, Mn2Hg5, FeS2 (oP6, marcasite), CoAs3 (skutterudite), PdSn2, CoSb2, Ir3Ge7, CuAl2, Re3Ge7, CrP2, and Mg2Ni. Through these analyses, the 18-n rule is established as a framework for interpreting the stability of 341 intermetallic phases and anticipating their properties.
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  4. 4
    Wagner, F. R.; Cardoso-Gil, R.; Boucher, B.; Wagner-Reetz, M.; Sichelschmidt, J.; Gille, P.; Baenitz, M.; Grin, Y. On Fe–Fe Dumbbells in the Ideal and Real Structures of FeGa3. Inorg. Chem. 2018, 57, 12908– 12919,  DOI: 10.1021/acs.inorgchem.8b02094
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    4
    On Fe-Fe Dumbbells in the Ideal and Real Structures of FeGa3
    Wagner, Frank R.; Cardoso-Gil, Raul; Boucher, Benoit; Wagner-Reetz, Maik; Sichelschmidt, Joerg; Gille, Peter; Baenitz, Michael; Grin, Yuri
    Inorganic Chemistry (2018), 57 (20), 12908-12919CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    The intermetallic phase FeGa3 belongs to the rare examples of substances with transition metals where semiconducting behavior is found. The necessary electron count of 17 ve/fu can be formally derived from eight Fe-Ga and one Fe-Fe two-center-two-electron bond. The situation is reminiscent of the well-known Fe2(CO)9 scenario, where a direct Fe-Fe two-center-two-electron bond was shown to not be present. Fe-Fe interaction in FeGa3 and its substitution variants represents the crucial point for explanation of electronic, thermal transport, and optical properties of this material. Chem. bonding anal. in position space of FeGa3 and Fe2(CO)9 on the basis of the topol. of the electron localizability indicator distribution, QTAIM atoms, two- and three-center delocalization indexes, domain natural orbitals, IQA anal., and an evaluation of the Fe-Fe dissocn. energy yields a complete picture of the partially compensated Fe-Fe bond, which is nevertheless strong enough to be of decisive importance. Structural reinvestigation of differently synthesized single crystals leads to the compn. Fe1+xGa3 (0 ≤ x ≤ 0.018), where the addnl. Fe atoms are predicted from DFT/PBE calcns. to yield a magnetic moment of about 2 μB/Fe2 atom and metallic in-gap states. Accompanying magnetization and ESR measurements are consistent with this picture.
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  5. 5
    Bende, D.; Grin, Y.; Wagner, F. R. Covalence and Ionicity in MgAgAs-type Compounds. Chem. - Eur. J. 2014, 20, 9702– 9708,  DOI: 10.1002/chem.201400299
    [Crossref], [PubMed], [CAS], Google Scholar
    5
    Covalence and Ionicity in MgAgAs-Type Compounds
    Bende, David; Grin, Yuri; Wagner, Frank R.
    Chemistry - A European Journal (2014), 20 (31), 9702-9708CODEN: CEUJED; ISSN:0947-6539. (Wiley-VCH Verlag GmbH & Co. KGaA)
    MgAgAs-type "half-Heusler" compds. are known to realize two out of three possible at. arrangements of this structure type. The no. of transition metal components typically dets. which of the alternatives is favored. On the basis of DFT calcns. for all three variants of 20 eight- and eighteen-valence-electron compds., the exptl. obsd. structural variant was found to be detd. by basically two different bonding patterns. They are quantified by employing two complementary position-space bonding measures. The Madelung energy calcd. with the QTAIM effective charges reflects contributions of the ionic interactions to the total energy. The sum of nearest-neighbor delocalization indexes characterizes the covalent interactions through electron sharing. With the aid of these quantities, the energetic sequence of the three at. arrangements for each compd. is rationalized. The resulting systematic is used to predict a scenario in which an nontypical at. arrangement becomes most favorable.
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  6. 6
    Bende, D.; Wagner, F. R.; Sichevych, O.; Grin, Y. Chemical Bonding Analysis as a Guide for the Preparation of New Compounds: The Case of VIrGe and HfPtGe. Angew. Chem., Int. Ed. 2017, 56, 1313– 1318,  DOI: 10.1002/anie.201610029
    [Crossref], [CAS], Google Scholar
    6
    Chemical Bonding Analysis as a Guide for the Preparation of New Compounds: The Case of VIrGe and HfPtGe
    Bende, David; Wagner, Frank R.; Sichevych, Olga; Grin, Yuri
    Angewandte Chemie, International Edition (2017), 56 (5), 1313-1318CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The chem. bonding of transition metal compds. with a MgAgAs-type of crystal structure is analyzed with quantum chem. position-space techniques. The obsd. trends in QTAIM Madelung energy and nearest neighbor electron sharing explain the occurrence of recently synthesized MgAgAs-type compds., TiPtGe and TaIrGe, at the boundary to the TiNiSi-type crystal structure. These bonding indicators are used to identify favorable element combinations for new MgAgAs-type compds. The new phases-the high-temp. VIrGe and the low-temp. HfPtGe-showing this type of crystal structure are prepd. and characterized by powder x-ray diffraction and DTA.
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  7. 7
    Martín Pendás, Á.; Blanco, M. A.; Francisco, E. Two-electron Integrations in the Quantum Theory of Atoms in Molecules. J. Chem. Phys. 2004, 120, 4581– 4592,  DOI: 10.1063/1.1645788
    [Crossref], [PubMed], Google Scholar
    There is no corresponding record for this reference.
  8. 8
    Blanco, M. A.; Martín Pendás, Á.; Francisco, E. Interacting Quantum Atoms: A Correlated Energy Decomposition Scheme Based on the Quantum Theory of Atoms in Molecules. J. Chem. Theory Comput. 2005, 1, 1096– 1109,  DOI: 10.1021/ct0501093
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    8
    Interacting Quantum Atoms: A Correlated Energy Decomposition Scheme Based on the Quantum Theory of Atoms in Molecules
    Blanco, M. A.; Pendas, A. Martin; Francisco, E.
    Journal of Chemical Theory and Computation (2005), 1 (6), 1096-1109CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We make use of the Quantum Theory of Atoms in Mols. (QTAM) to partition the total energy of a many-electron system into intra- and interat. terms, by explicitly computing both the one- and two-electron contributions. While the general scheme is formally equiv. to that by Bader et al., we focus on the sepn. and computation of the at. self-energies and all the interaction terms. The partition is ultimately performed within the d. matrixes, in analogy with McWeeny's theory of electronic separability, and then carried onto the energy. It is intimately linked with the atomistic picture of the chem. bond, not only allowing the sepn. of different two-body contributions (point-charge-like, multipolar, total Coulomb, exchange, correlation, ...) to the interaction between a pair of atoms but also including an effective many-body contribution to the binding (self-energy, formally one-body) due to the deformation of the atoms within the many-electron system as compared to the free atoms. Many qual. ideas about the chem. bond can be quantified using this scheme.
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  9. 9
    Martín Pendás, Á.; Francisco, E.; Blanco, M. A. Two-electron Integrations in the Quantum Theory of Atoms in Molecules with Correlated Wave Functions. J. Comput. Chem. 2005, 26, 344– 351,  DOI: 10.1002/jcc.20173
    [Crossref], [PubMed], Google Scholar
    There is no corresponding record for this reference.
  10. 10
    Raupach, M.; Tonner, R. A Periodic Energy Decomposition Analysis Method for the Investigation of Chemical Bonding in Extended Systems. J. Chem. Phys. 2015, 142, 194105,  DOI: 10.1063/1.4919943
    [Crossref], [PubMed], [CAS], Google Scholar
    10
    A periodic energy decomposition analysis method for the investigation of chemical bonding in extended systems
    Raupach, Marc; Tonner, Ralf
    Journal of Chemical Physics (2015), 142 (19), 194105/1-194105/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The development and first applications of a new periodic energy decompn. anal. (pEDA) scheme for extended systems based on the Kohn-Sham approach to d. functional theory are described. The pEDA decomps. the bonding energy between two fragments (e.g., the adsorption energy of a mol. on a surface) into several well-defined terms: prepn., electrostatic, Pauli repulsion, and orbital relaxation energies. This is complemented by consideration of dispersion interactions via a pairwise scheme. One major extension toward a previous implementation [Philipsen and Baerends, J. Phys. Chem. B 110, 12470 (2006)] lies in the sep. discussion of electrostatic and Pauli and the addn. of a dispersion term. The pEDA presented here for an implementation based on AOs can handle restricted and unrestricted fragments for 0D to 3D systems considering periodic boundary conditions with and without the detn. of fragment occupations. For the latter case, reciprocal space sampling is enabled. The new method gives comparable results to established schemes for mol. systems and shows good convergence with respect to the basis set (TZ2P), the integration accuracy, and k-space sampling. Four typical bonding scenarios for surface-adsorbate complexes were chosen to highlight the performance of the method representing insulating (CO on MgO(001)), metallic (H2 on M(001), M = Pd, Cu), and semiconducting (CO and C2H2 on Si(001)) substrates. These examples cover diverse substrates as well as bonding scenarios ranging from weakly interacting to covalent (shared electron and donor acceptor) bonding. The results presented lend confidence that the pEDA will be a powerful tool for the anal. of surface-adsorbate bonding in the future, enabling the transfer of concepts like ionic and covalent bonding, donor-acceptor interaction, steric repulsion, and others to extended systems. (c) 2015 American Institute of Physics.
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  11. 11
    Dronskowski, R. Computational Chemistry of Solid State Materials; John Wiley & Sons, Ltd.: 2005.
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  12. 12
    Bader, R. F. W. Atoms in Molecules: A Quantum Theory (International Series of Monographs on Chemistry); Clarendon Press: 1990.
    Google Scholar
    There is no corresponding record for this reference.
  13. 13
    Martín Pendás, Á.; Francisco, E. Promolden: A QTAIM/IQA code. Available from the authors upon request by writing to [email protected]
    Google Scholar
    There is no corresponding record for this reference.
  14. 14
    Kohout, M. DGrid, ver. 5.2; Dresden: 2021.
    Google Scholar
    There is no corresponding record for this reference.
  15. 15
    te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T. Chemistry with ADF. J. Comput. Chem. 2001, 22, 931– 967,  DOI: 10.1002/jcc.1056
    [Crossref], [CAS], Google Scholar
    15
    Chemistry with ADF
    Te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; Van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T.
    Journal of Computational Chemistry (2001), 22 (9), 931-967CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    A review with 241 refs. We present the theor. and tech. foundations of the Amsterdam D. Functional (ADF) program with a survey of the characteristics of the code (numerical integration, d. fitting for the Coulomb potential, and STO basis functions). Recent developments enhance the efficiency of ADF (e.g., parallelization, near order-N scaling, QM/MM) and its functionality (e.g., NMR chem. shifts, COSMO solvent effects, ZORA relativistic method, excitation energies, frequency-dependent (hyper)polarizabilities, at. VDD charges). In the Applications section we discuss the phys. model of the electronic structure and the chem. bond, i.e., the Kohn-Sham MO (MO) theory, and illustrate the power of the Kohn-Sham MO model in conjunction with the ADF-typical fragment approach to quant. understand and predict chem. phenomena. We review the "Activation-strain TS interaction" (ATS) model of chem. reactivity as a conceptual framework for understanding how activation barriers of various types of (competing) reaction mechanisms arise and how they may be controlled, for example, in org. chem. or homogeneous catalysis. Finally, we include a brief discussion of exemplary applications in the field of biochem. (structure and bonding of DNA) and of time-dependent d. functional theory (TDDFT) to indicate how this development further reinforces the ADF tools for the anal. of chem. phenomena.
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  16. 16
    Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M. Ab Initio Molecular Simulations with Numeric Atom-Centered Orbitals. Comput. Phys. Commun. 2009, 180, 2175– 2196,  DOI: 10.1016/j.cpc.2009.06.022
    [Crossref], [CAS], Google Scholar
    16
    Ab initio molecular simulations with numeric atom-centered orbitals
    Blum, Volker; Gehrke, Ralf; Hanke, Felix; Havu, Paula; Havu, Ville; Ren, Xinguo; Reuter, Karsten; Scheffler, Matthias
    Computer Physics Communications (2009), 180 (11), 2175-2196CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
    We describe a complete set of algorithms for ab initio mol. simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of mol. and materials properties from quantum-mech. first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute "ab initio mol. simulations" (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on d.-functional theory (DFT) in the local and semilocal (generalized gradient) approxns., but an extension to hybrid functionals, Hartree-Fock theory, and MP2/GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron/full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio mol. dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calcn. to range from qual. tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs.
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  17. 17
    Baranov, A. I.; Kohout, M. Electron Localization and Delocalization Indices for Solids. J. Comput. Chem. 2011, 32, 2064– 2076,  DOI: 10.1002/jcc.21784
    [Crossref], [PubMed], [CAS], Google Scholar
    17
    Electron localization and delocalization indices for solids
    Baranov, Alexey I.; Kohout, Miroslav
    Journal of Computational Chemistry (2011), 32 (10), 2064-2076CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    The electron localization and delocalization indexes obtained by the integration of exchange-correlation part of pair d. over chem. meaningful regions of space, e.g., QTAIM atoms are valuable tools for the bonding anal. in mol. systems. However, among periodic systems only few simplest models were analyzed with this approach until now. This contribution reports implementation and evaluation of the localization and delocalization indexes on the basis of solid state DFT calcns. A comparison with the results of simple anal. model of Ponec was made. In addn., a small set of compds. with ionic (NaCl), covalent (diamond, graphite), and metallic (Na, Cu) bonding interactions was characterized using this method. Typical features of different types of bonding were discussed using the delocalization indexes. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011.
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  18. 18
    Chen, H.; Friesecke, G. Pair Densities in Density Functional Theory. Multiscale Model. Simul. 2015, 13, 1259– 1289,  DOI: 10.1137/15M1014024
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  19. 19
    Rodríguez, J. I.; Köster, A. M.; Ayers, P. W.; Santos-Valle, A.; Vela, A.; Merino, G. An Efficient Grid-Based Scheme to Compute QTAIM Atomic Properties without Explicit Calculation of Zero-Flux Surfaces. J. Comput. Chem. 2009, 30, 1082– 1092,  DOI: 10.1002/jcc.21134
    [Crossref], [PubMed], [CAS], Google Scholar
    19
    An efficient grid-based scheme to compute QTAIM atomic properties without explicit calculation of zero-flux surfaces
    Rodriguez, Juan I.; Koster, Andreas M.; Ayers, Paul W.; Santos-Valle, Ana; Vela, Alberto; Merino, Gabriel
    Journal of Computational Chemistry (2009), 30 (7), 1082-1092CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    We introduce a method to compute at. properties according to the "quantum theory of atoms in mols.". An integration grid in real space is partitioned into subsets, ωi. The subset, ωi, is composed of all grid points contained in the at. basin, Ωi, so that integration over Ωi is reduced to simple quadrature over the points in ωi. The partition is constructed from deMon2k's at. center grids by following the steepest ascent path of the d. starting from each point in the grid. We also introduce a technique that exploits the cellular nature of the grid to make the algorithm faster. The performance of the method is tested by computing properties of atoms and nonnuclear attractors (energies, charges, dipole, and quadrupole moments) for a set of representative mols.
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  20. 20
    Womersley, R. S. Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan; Springer International Publishing: Cham, Switzerland, 2018; pp 1243– 1285.
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  21. 21
    Blanco, M. A. Métodos Cuánticos Locales para la Simulación de Materiales Iónicos. Fundamentos, algoritmos y aplicaciones. Ph.D. thesis, University of Oviedo: 1997.
    Google Scholar
    There is no corresponding record for this reference.
  22. 22
    Kay, K. G.; Todd, H. D.; Silverstone, H. J. Bipolar Expansion for r12nYlm(θ12,ϕ12). J. Chem. Phys. 1969, 51, 2363– 2367,  DOI: 10.1063/1.1672353
    [Crossref], [CAS], Google Scholar
    22
    Bipolar expansion for r12nYtm(Θ12,Φ12)
    Kay, Kenneth G.; Todd, H. David; Silverstone, Harris J.
    Journal of Chemical Physics (1969), 51 (6), 2363-7CODEN: JCPSA6; ISSN:0021-9606.
    Explicit formulas for the radial functions V(n)l1l2l3l(r1,r2,R) in the bipolar expansion for r12nYlm(θ12, Φ12), r12nYlm(θ12, Φ12) = Σ(2λ + 1)1/2(2l3 + 1)1/2cλ(lm; l1m1)cl3(λ, m - m1; l2m2) × Yl1m1- (θ1, Φ1)Yl2m2(θ2, Φ2)Yl3m-m1-m2(θR, ΦR)V(n)l1l2l3l(r1, r2, R), where r12 = r1 - r2 - R, are derived by the use of the theory of generalized functions and Fourier transforms. When n ≤ -4 and n - l is odd, there are delta-function terms. In this approach the delta-function terms and the 4-region form of the expansion are obtained from a single, unified formula valid in all regions. Recurrence formulas for the V(n)l1l2l3l are given.
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  23. 23
    Buehler, R. J.; Hirschfelder, J. O. Bipolar Expansion of Coulombic Potentials. Phys. Rev. 1951, 83, 628– 633,  DOI: 10.1103/PhysRev.83.628
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  24. 24
    Luaña, Víctor. Environ: local environment determination of arbitrary positions in a crystal; Oviedo: 1992.
    Google Scholar
    There is no corresponding record for this reference.
  25. 25
    Kosov, D. S.; Popelier, P. L. A. Atomic Partitioning of Molecular Electrostatic Potentials. J. Phys. Chem. A 2000, 104, 7339– 7345,  DOI: 10.1021/jp0003407
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    25
    Atomic Partitioning of Molecular Electrostatic Potentials
    Kosov, D. S.; Popelier, P. L. A.
    Journal of Physical Chemistry A (2000), 104 (31), 7339-7345CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    The theory of atoms in mols. (AIM) defines bounded at. fragments in real space that generate transferable at. properties. As part of a program that investigates the topol. partitioning of electromagnetic properties based on the electron d., we have calcd. the exact at. electrostatic potential (AEP) of an AIM atom in a mol. Second we expand this at. electrostatic potential in terms of AIM electrostatic multipole moments based on spherical tensors. We prove that the convergence of this expansion is faster than previously assumed, even for complicated at. shapes.
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  26. 26
    Francisco, E.; Menéndez Crespo, D.; Costales, A.; Martín Pendás, Á. A Multipolar Approach to the Interatomic Covalent Interaction Energy. J. Comput. Chem. 2017, 38, 816– 829,  DOI: 10.1002/jcc.24758
    [Crossref], [PubMed], [CAS], Google Scholar
    26
    A multipolar approach to the interatomic covalent interaction energy
    Francisco, Evelio; Menendez Crespo, Daniel; Costales, Aurora; Martin Pendas, Angel
    Journal of Computational Chemistry (2017), 38 (11), 816-829CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
    Interat. exchange-correlation energies correspond to the covalent energetic contributions to an interat. interaction in real space theories of the chem. bond, but their widespread use is severely limited due to their computationally intensive character. In the same way as the multipolar (mp) expansion is customary used in biomol. modeling to approx. the classical Coulomb interaction between two charge densities ρA(r) and ρB(r), we examine in this work the mp approach to approx. the interat. exchange-correlation (xc) energies of the Interacting Quantum Atoms method. We show that the full xc mp series is quickly divergent for directly bonded atoms (1-2 pairs) albeit it works reasonably well most times for 1-n (n > 2) interactions. As with conventional perturbation theory, we show numerically that the xc series is asymptotically convergent and that, a truncated xc mp approxn. retaining terms up to l1 + l2 = 2 usually gives relatively accurate results, sometimes even for directly bonded atoms. Our findings are supported by extensive numerical analyses on a variety of systems that range from several std. hydrogen bonded dimers to typically covalent or arom. mols. The exact algebraic relationship between the monopole-monopole xc mp term and the inter-at. bond order, as measured by the delocalization index of the quantum theory of atoms in mols., is also established. © 2017 Wiley Periodicals, Inc.
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  27. 27
    Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865– 3868,  DOI: 10.1103/PhysRevLett.77.3865
    [Crossref], [PubMed], [CAS], Google Scholar
    27
    Generalized gradient approximation made simple
    Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
    Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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  28. 28
    Barca, G. M. J.; Bertoni, C.; Carrington, L.; Datta, D.; De Silva, N.; Deustua, J. E.; Fedorov, D. G.; Gour, J. R.; Gunina, A. O.; Guidez, E. Recent Developments in the General Atomic and Molecular Electronic Structure System. J. Chem. Phys. 2020, 152, 154102,  DOI: 10.1063/5.0005188
    [Crossref], [PubMed], [CAS], Google Scholar
    28
    Recent developments in the general atomic and molecular electronic structure system
    Barca, Giuseppe M. J.; Bertoni, Colleen; Carrington, Laura; Datta, Dipayan; De Silva, Nuwan; Deustua, J. Emiliano; Fedorov, Dmitri G.; Gour, Jeffrey R.; Gunina, Anastasia O.; Guidez, Emilie; Harville, Taylor; Irle, Stephan; Ivanic, Joe; Kowalski, Karol; Leang, Sarom S.; Li, Hui; Li, Wei; Lutz, Jesse J.; Magoulas, Ilias; Mato, Joani; Mironov, Vladimir; Nakata, Hiroya; Pham, Buu Q.; Piecuch, Piotr; Poole, David; Pruitt, Spencer R.; Rendell, Alistair P.; Roskop, Luke B.; Ruedenberg, Klaus; Sattasathuchana, Tosaporn; Schmidt, Michael W.; Shen, Jun; Slipchenko, Lyudmila; Sosonkina, Masha; Sundriyal, Vaibhav; Tiwari, Ananta; Galvez Vallejo, Jorge L.; Westheimer, Bryce; Wloch, Marta; Xu, Peng; Zahariev, Federico; Gordon, Mark S.
    Journal of Chemical Physics (2020), 152 (15), 154102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A discussion of many of the recently implemented features of GAMESS (General Atomic and Mol. Electronic Structure System) and LibCChem (the C + + CPU/GPU library assocd. with GAMESS) is presented. These features include fragmentation methods such as the fragment MO, effective fragment potential and effective fragment MO methods, hybrid MPI/OpenMP approaches to Hartree-Fock, and resoln. of the identity second order perturbation theory. Many new coupled cluster theory methods have been implemented in GAMESS, as have multiple levels of d. functional/tight binding theory. The role of accelerators, esp. graphical processing units, is discussed in the context of the new features of LibCChem, as it is the assocd. problem of power consumption as the power of computers increases dramatically. The process by which a complex program suite such as GAMESS is maintained and developed is considered. Future developments are briefly summarized. (c) 2020 American Institute of Physics.
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  29. 29
    Dirac, P. A. M. Note on Exchange Phenomena in the Thomas Atom. Math. Proc. Cambridge Philos. Soc. 1930, 26, 376– 385,  DOI: 10.1017/S0305004100016108
    [Crossref], [CAS], Google Scholar
    29
    Exchange phenomena in the Thomas atom
    Dirac, P. A. M.
    Proceedings of the Cambridge Philosophical Society (1930), 26 (), 376-85CODEN: PCPSA4; ISSN:0068-6735.
    Calcn. of the electron distribution in the state of lowest energy of an atom, for which a certain region of phase space is occupied with the max. d. of electrons and the remainder is empty. The calcn. gives a theoretical justification of the Thomas atom.
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  30. 30
    Bloch, F. Bemerkung zur Elektronentheorie des Ferromagnetismus und der elektrischen Leitfähigkeit. Eur. Phys. J. A 1929, 57, 545– 555,  DOI: 10.1007/BF01340281
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  31. 31
    Vosko, S. H.; Wilk, L.; Nusair, M. Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: a Critical Analysis. Can. J. Phys. 1980, 58, 1200– 1211,  DOI: 10.1139/p80-159
    [Crossref], [CAS], Google Scholar
    31
    Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis
    Vosko, S. H.; Wilk, L.; Nusair, M.
    Canadian Journal of Physics (1980), 58 (8), 1200-11CODEN: CJPHAD; ISSN:0008-4204.
    Various approx. forms for the correlation energy per particle of the spin-polarized homogeneous electron gas that have frequently been used in applications of the local spin d. approxn. to the exchange-correlation energy functional are assessed. By accurately recalcg. the RPA correlation energy as a function of electron d. and spin polarization, the inadequacies of the usual approxn. for interpolating between the para- and ferro-magnetic states are demonstrated and an accurate new interpolation formula is presented. A Pade approximant technique was used to accurately interpolate the recent Monte Carlo results. These results can be combined with the RPA spin-dependence so as to produce a correlation energy for a spin-polarized homogeneous electron gas with an estd. max. error of 1 mRy and thus should reliably det. the magnitude of non-local corrections to the local spin d. approxn. in real systems.
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  32. 32
    Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785– 789,  DOI: 10.1103/PhysRevB.37.785
    [Crossref], [PubMed], [CAS], Google Scholar
    32
    Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
    Lee, Chengteh; Yang, Weitao; Parr, Robert G.
    Physical Review B: Condensed Matter and Materials Physics (1988), 37 (2), 785-9CODEN: PRBMDO; ISSN:0163-1829.
    A correlation-energy formula due to R. Colle and D. Salvetti (1975), in which the correlation energy d. is expressed in terms of the electron d. and a Laplacian of the 2nd-order Hartree-Fock d. matrix, is restated as a formula involving the d. and local kinetic-energy d. On insertion of gradient expansions for the local kinetic-energy d., d.-functional formulas for the correlation energy and correlation potential are then obtained. Through numerical calcns. on a no. of atoms, pos. ions, and mols., of both open- and closed-shell type, it is demonstrated that these formulas, like the original Colle-Salvetti formulas, give correlation energies within a few percent.
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  33. 33
    Becke, A. D. Correlation Energy of an Inhomogeneous Electron Gas: A Coordinate-Space Model. J. Chem. Phys. 1988, 88, 1053– 1062,  DOI: 10.1063/1.454274
    [Crossref], [CAS], Google Scholar
    33
    Correlation energy of an inhomogeneous electron gas: a coordinate-space model
    Becke, A. D.
    Journal of Chemical Physics (1988), 88 (2), 1053-62CODEN: JCPSA6; ISSN:0021-9606.
    A coordinate-space model for dynamical correlations in an inhomogeneous electron gas is developed. The model treats opposite-spin and same-spin pairs sep., and it also accounts properly for correlation contributions to the kinetic energy. It gives identically zero correlation energy for 1-electron systems. Applications to the uniform electron gas and to the atoms H through Ar are made.
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  34. 34
    Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields. J. Phys. Chem. 1994, 98, 11623– 11627,  DOI: 10.1021/j100096a001
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    34
    Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields
    Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J.
    Journal of Physical Chemistry (1994), 98 (45), 11623-7CODEN: JPCHAX; ISSN:0022-3654.
    The unpolarized absorption and CD spectra of the fundamental vibrational transitions of the chiral mol. 4-methyl-2-oxetanone are calcd. ab initio. Harmonic force fields are obtained using d. functional theory (DFT), MP2 and SCF methodologies, and a [5s4p2d/3s2p] (TZ2P) basis set. DFT calcns. use the LSDA, BLYP, and Becke3LYP (B3LYP) d. functionals. Mid-IR spectra predicted using LSDA, BLYP, and B3LYP force fields are of significantly different quality, the B3LYP force field yielding spectra in clearly superior, and overall excellent, agreement with expt. The MP2 force field yields spectra in slightly worse agreement with expt. than the B3LYP force field. The SCF force field yields spectra in poor agreement with expt. The basis set dependence of B3LYP force fields is also explored: the 6-31G* and TZ2P basis sets give very similar results while the 3-21G basis set yields spectra in substantially worse agreement with expt.
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  35. 35
    Downs, R. T.; Somayazulu, M. S. Carbon Dioxide at 1.0 GPa. Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 1998, 54, 897– 898,  DOI: 10.1107/S0108270198001140
    [Crossref], [CAS], Google Scholar
    35
    Carbon dioxide at 1.0 GPa
    Downs, Robert T.; Somayazulu, M. S.
    Acta Crystallographica, Section C: Crystal Structure Communications (1998), C54 (7), 897-898CODEN: ACSCEE; ISSN:0108-2701. (Munksgaard International Publishers Ltd.)
    An x-ray diffraction study of single-crystal CO2 was undertaken at 1.00(5) GPa pressure. The crystal exhibits Pa‾3 symmetry with a cell edge of 5.4942(2) Å and a C-O bond length of 1.168(1) Å (cor. for thermal motion effects). An earlier claim of a new dry ice II phase at this pressure is unfounded.
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  36. 36
    Bindzus, N.; Straasø, T.; Wahlberg, N.; Becker, J.; Bjerg, L.; Lock, N.; Dippel, A.-C.; Iversen, B. B. Experimental Determination of Core Electron Deformation in Diamond. Acta Crystallogr., Sect. A: Found. Adv. 2014, 70, 39– 48,  DOI: 10.1107/S2053273313026600
    [Crossref], [CAS], Google Scholar
    36
    Experimental determination of core electron deformation in diamond
    Bindzus, Niels; Straaso, Tine; Wahlberg, Nanna; Becker, Jacob; Bjerg, Lasse; Lock, Nina; Dippel, Ann-Christin; Iversen, Bo B.
    Acta Crystallographica, Section A: Foundations and Advances (2014), 70 (1), 39-48CODEN: ACSAD7; ISSN:2053-2733. (International Union of Crystallography)
    Synchrotron powder X-ray diffraction data are used to det. the core electron deformation of diamond. Core shell contraction inherently linked to covalent bond formation is obsd. in close correspondence with theor. predictions. Accordingly, a precise and phys. sound reconstruction of the electron d. in diamond necessitates the use of an extended multipolar model, which abandons the assumption of an inert core. The present investigation is facilitated by negligible model bias in the extn. of structure factors, which is accomplished by simultaneous multipolar and Rietveld refinement accurately detg. an at. displacement parameter (ADP) of 0.00181(1) Å2. The deconvolution of thermal motion is a crit. step in exptl. core electron polarization studies, and for diamond it is imperative to exploit the monoat. crystal structure by implementing Wilson plots in detn. of the ADP. This empowers the electron-d. anal. to precisely administer both the deconvolution of thermal motion and the employment of the extended multipolar model on an exptl. basis.
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  37. 37
    Dobrzhinetskaya, L. F.; Wirth, R.; Yang, J.; Green, H. W.; Hutcheon, I. D.; Weber, P. K.; Grew, E. S. Qingsongite, Natural Cubic Boron Nitride: The First Boron Mineral from the Earth’s Mantle. Am. Mineral. 2014, 99, 764– 772,  DOI: 10.2138/am.2014.4714
    [Crossref], [CAS], Google Scholar
    37
    Qingsongite, natural cubic boron nitride: the first boron mineral from the Earth's mantle
    Dobrzhinetskaya, Larissa F.; Wirth, Richard; Yang, Jingsui; Green, Harry W.; Hutcheon, Ian D.; Weber, Peter K.; Grew, Edward S.
    American Mineralogist (2014), 99 (4), 764-772CODEN: AMMIAY; ISSN:0003-004X. (Mineralogical Society of America)
    Qingsongite (IMA 2013-30) is the natural analog of cubic boron nitride (c-BN), which is widely used as an abrasive under the name Borazon. The mineral is named for Qingsong Fang (1939-2010), who found the first diamond in the Luobusa chromitite. Qingsongite occurs in a rock fragment less than 1 mm across extd. from chromitite in deposit 31, Luobusa ophiolite, Yarlung Zangbu suture, southern Tibet at 29°13.86N and 92°11.41E. Five electron microprobe analyses gave B 48.54 ± 0.65 wt% (range 47.90-49.2 wt%); N 51.46 ± 0.65 wt% (range 52.10-50.8 wt%), corresponding to B1.113N0.887 and B1.087N0.913, for max. and min. B contents, resp. (based on 2 atoms per formula unit); no other elements that could substitute for B or N were detected. Crystallog. data on qingsongite obtained using fast Fourier transforms gave cubic symmetry, a = 3.61 ± 0.045 Å. The d. calcd. for the mean compn. B1.100N0.900 is 3.46 g/cm3, i.e., qingsongite is nearly identical to synthetic c-BN. The synthetic analog has the sphalerite structure, space group F‾43m. Mohs hardness of the synthetic analog is between 9 and 10; its cleavage is {011}. Qingsongite forms isolated anhedral single crystals up to 1 μm in size in the marginal zone of the fragment; this zone consists of ∼45 modal% coesite, ∼15% kyanite, and ∼40% amorphous material. Qingsongite is enclosed in kyanite, coesite, or in osbornite; other assocd. phases include native Fe; TiO2 II, a high-pressure polymorph of rutile with the αPbO2 structure; boron carbide of unknown stoichiometry; and amorphous carbon. Coesite forms prisms several tens of micrometers long, but is polycryst., and thus interpreted to be pseudomorphic after stishovite. Assocd. minerals constrain the estd. pressure to 10-15 GPa assuming temp. was about 1300 °C. Our proposed scenario for formation of qingsongite begins with a pelitic rock fragment that was subducted to mid-mantle depths where crustal B originally present in mica or clay combined with mantle N (δ15N = -10.4 ± 3‰ in osbornite) and subsequently exhumed by entrainment in chromitite. The presence of qingsongite has implications for understanding the recycling of crustal material back to the Earth's mantle since boron, an essential constituent of qingsongite, is potentially an ideal tracer of material from Earth's surface.
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  38. 38
    Didier, C.; Pang, W. K.; Guo, Z.; Schmid, S.; Peterson, V. K. Phase Evolution and Intermittent Disorder in Electrochemically Lithiated Graphite Determined Using in Operando Neutron Diffraction. Chem. Mater. 2020, 32, 2518– 2531,  DOI: 10.1021/acs.chemmater.9b05145
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    38
    Phase Evolution and Intermittent Disorder in Electrochemically Lithiated Graphite Determined Using in Operando Neutron Diffraction
    Didier, Christophe; Pang, Wei Kong; Guo, Zaiping; Schmid, Siegbert; Peterson, Vanessa K.
    Chemistry of Materials (2020), 32 (6), 2518-2531CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
    Since their commercialization in 1991, Li-ion batteries (LIBs) have revolutionized the authors' way of life, with LIB pioneers being awarded the 2019 Nobel Prize in Chem. Despite the widespread use of LIBs, many LIB applications are not realized due to performance limitations, detd. largely by the ability of electrode materials to reversibly host Li ions. Overcoming such limitations requires knowledge of the fundamental mechanism for reversible ion intercalation in electrode materials. The still-debated structure of the most common com. electrode material, graphite, during electrochem. lithiation is revisited using in operando neutron powder diffraction of a com. 18650 Li-ion battery. The authors ext. new structural information and present a comprehensive overview of the phase evolution for lithiated graphite. Charge-discharge asymmetry and structural disorder in the lithiation process are obsd., particularly surrounding phase transitions, and the phase evolution is kinetically influenced. Notably, the authors observe pronounced asymmetry over the compn. range 0.5 > x > 0.2, in which the stage 2L phase forms on discharge (delithiation) but not charge (lithiation), likely as a result of the slow formation of the stage 2L phase and the closeness of the stage 2L and stage 2 phase potentials. The authors reconcile the authors' measurements of this transition with a stage 2L stacking disorder model contg. an intergrown stage 2 and 2L phase. The authors resolve debate surrounding the intercalation mechanism in the stage 3L and stage 4L phase region, observing stage-specific reflections that support a 1st-order phase transition over the 0.2 > x > 0.04 range, in agreement with minor changes in the slope of the stacking axis length, despite relatively unchanging 00l reflection broadening. The authors' data support the previously proposed /ABA/ACA/ stacking for the stage 3L phase and an /ABAB/BABA/ stacking sequence of the stage 4L phase alongside exptl. derived at. parameters. Finally, at low Li content 0 < x < 0.04, an apparently homogeneous modification of the structure during both charge and discharge were found. Understanding the phase evolution and mechanism of structural response of graphite to lithiation under battery working conditions through in operando measurements may provide the information needed for the development of alternative higher performance electrode materials.
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  39. 39
    Li, M.-R.; Deng, Z.; Lapidus, S. H.; Stephens, P. W.; Segre, C. U.; Croft, M.; Paria Sena, R.; Hadermann, J.; Walker, D.; Greenblatt, M. Ba3(Cr0.97(1)Te0.03(1))2TeO9: in Search of Jahn-Teller Distorted Cr(II) Oxide. Inorg. Chem. 2016, 55, 10135– 10142,  DOI: 10.1021/acs.inorgchem.6b01047
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    39
    Ba3(Cr0.97(1)Te0.03(1))2TeO9: in Search of Jahn-Teller Distorted Cr(II) Oxide
    Li, Man-Rong; Deng, Zheng; Lapidus, Saul H.; Stephens, Peter W.; Segre, Carlo U.; Croft, Mark; Paria Sena, Robert; Hadermann, Joke; Walker, David; Greenblatt, Martha
    Inorganic Chemistry (2016), 55 (20), 10135-10142CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    A novel 6H-type hexagonal perovskite Ba3(Cr0.97(1)Te0.03(1))2TeO9 was prepd. at high pressure (6 GPa) and temp. (1773 K). Both TEM and synchrotron powder x-ray diffraction data demonstrate that Ba3(Cr0.97(1)Te0.03(1))2TeO9 crystallizes in P63/mmc with face-shared (Cr0.97(1)Te0.03(1))O6 octahedral pairs interconnected with TeO6 octahedra via corner-sharing. Structure anal. shows a mixed Cr2+/Cr3+ valence state with ∼10% Cr2+. The existence of Cr2+ in Ba3(Cr2+0.10(1)Cr3+0.87(1)Te6+0.03)2TeO9 is further evidenced by x-ray absorption near-edge spectroscopy. Magnetic properties measurements show a paramagnetic response down to 4 K and a small glassy-state curvature at low temp. The octahedral Cr2+O6 component is stabilized in an oxide material for the first time; the expected Jahn-Teller distortion of high-spin (d4) Cr2+ is not found, which is attributed to the small proportion of Cr2+ (∼10%) and the face-sharing arrangement of CrO6 octahedral pairs, which structurally disfavor axial distortion.
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  40. 40
    Tsirelson, V.; Stash, A.; Kohout, M.; Rosner, H.; Mori, H.; Sato, S.; Lee, S.; Yamamoto, A.; Tajima, S.; Grin, Y. Features of the Electron Density in Magnesium Diboride: Reconstruction from X-ray Diffraction Data and Comparison with TB-LMTO and FPLO Calculations. Acta Crystallogr., Sect. B: Struct. Sci. 2003, 59, 575– 583,  DOI: 10.1107/S0108768103012072
    [Crossref], [PubMed], [CAS], Google Scholar
    40
    Features of the electron density in magnesium diboride: reconstruction from x-ray diffraction data and comparison with TB-LMTO and FPLO calculations
    Tsirelson, V.; Stash, A.; Kohout, M.; Rosner, H.; Mori, H.; Sato, S.; Lee, S.; Yamamoto, A.; Tajima, S.; Grin, Yu.
    Acta Crystallographica, Section B: Structural Science (2003), B59 (5), 575-583CODEN: ASBSDK; ISSN:0108-7681. (Blackwell Publishing Ltd.)
    Features of the electron d. in MgB2 reconstructed from room-temp. single-crystal x-ray diffraction intensities using a multipole model are considered. Crystallog. data and at. coordinates are given. Topol. anal. of the total electron d. was applied to characterize the at. interactions in Mg diboride. The shared-type B-B interaction in the B-atom layer reveals that both σ and π components of the bonding are strong. A closed-shell-type weak B-B π interaction along the c axis of the unit cell also was found. The Mg-B closed-shell interaction exhibits a bond path that is significantly curved towards the vertical Mg-atom chain ([110] direction). The latter two facts reflect two sorts of bonding interactions along the [001] direction. Integration of the electron d. over the zero-flux at. basins reveals a charge transfer of ∼1.4(1) electrons from the Mg atoms to the B-atom network. The calcd. elec.-field gradients at nuclear positions are in good agreement with exptl. NMR values. The anharmonic displacement of the B atoms is also discussed. Calcns. of the electron d. by tight-binding linear muffin-tin orbital (TB-LMTO) and full-potential nonorthogonal local orbital (FPLO) methods confirm the results of the reconstruction from x-ray diffraction; for example, a charge transfer of 1.5 and 1.6 electrons, resp., was found.
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  41. 41
    Ievinǎ, A.; Straumanis, M.; Karlsons, K. Präzisionsbestimmung von Gitterkonstanten hygroskopischer Verbindungen (LiCl, NaBr). Z. Phys. Chem. 1938, 40B, 146– 150,  DOI: 10.1515/zpch-1938-4009
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  42. 42
    Walker, D.; Verma, P. K.; Cranswick, L. M.; Jones, R. L.; Clark, S. M.; Buhre, S. Halite-sylvite Thermoelasticity. Am. Mineral. 2004, 89, 204– 210,  DOI: 10.2138/am-2004-0124
    [Crossref], [CAS], Google Scholar
    42
    Halite-sylvite thermoelasticity
    Walker, David; Verma, Pramod K.; Cranswick, Lachlan M. D.; Jones, Raymond L.; Clark, Simon M.; Buhre, Stephan
    American Mineralogist (2004), 89 (1), 204-210CODEN: AMMIAY; ISSN:0003-004X. (Mineralogical Society of America)
    Unit-cell vols. of four single-phase intermediate halite-sylvite solid solns. have been measured to pressures and temps. of ∼28 kbar and ∼700 °C. Equation-of-state fitting of the data yields thermal expansion and compressibility as a function of compn. across the chloride series. The variation of the product α0·K0 is linear (ideal) in compn. between the accepted values for halite and sylvite. Taken sep., the individual values of α0 and K0 are not linear in compn. α0 Shows a max. near the consolute compn. (XNaCl = 0.64) that exceeds the value for either end-member. There is a corresponding min. in K0. The fact that the α0·K0 product is variable (and incidentally so well behaved as to be linear across the compn. series) reinforces the significance of the complementary maxima and min. in α0 and K0 (significantly, near the consolute compn.). These extrema in α0 and K0 provide an example of intermediate properties that do not follow simply from values for the end-members. Cell vols. across this series show small, well-behaved pos. excesses, consistent with K-Na substitution causing defects through lattice mismatches. Barrett and Wallace (1954) showed max. defect concns. in the consolute region. Defect-riddled, weakened structures in the consolute region are more easily compressed or more easily thermally expanded, providing an explanation for our obsd. α0 and K0 variations. These compliant, loosened lattices should resist diffusive transfer less than non-defective crystals and, hence, might be expected to show higher diffusivities. Tracer diffusion rates are predicted to peak across the consolute region as exchange diffusion rates drop to zero.
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  43. 43
    Ewais, E. M.; El-Amir, A. A.; Besisa, D. H.; Esmat, M.; El-Anadouli, B. E. Synthesis of Nanocrystalline MgO/MgAl2O4 Spinel Powders from Industrial Wastes. J. Alloys Compd. 2017, 691, 822– 833,  DOI: 10.1016/j.jallcom.2016.08.279
    [Crossref], [CAS], Google Scholar
    43
    Synthesis of nanocrystalline MgO/MgAl2O4 spinel powders from industrial wastes
    Ewais, Emad M. M.; El-Amir, Ahmed A. M.; Besisa, Dina H. A.; Esmat, Mohamed; El-Anadouli, Bahgat E. H.
    Journal of Alloys and Compounds (2017), 691 (), 822-833CODEN: JALCEU; ISSN:0925-8388. (Elsevier B.V.)
    This article reports a simple and cost-effective method to prep. ultrafine nanocryst. MgO/MgAl2O4 spinel (M-MA) powders from industrial wastes arising from aluminum and magnesium scraps. M-MA precursor powders were calcined at different temps. (650, 750, 850, 950, 1300-1500 °C). The calcined powders were characterized by XRD, FT-IR, DTA, FESEM, and HR-TEM. In particular, ultrafine MgO/MgAl2O3 powder was formed at a temp. of 650 °C with crystallite size of 4.8 nm and 7 nm, resp., as detd. by XRD. Optical properties of the M-MA spinel powders revealed that the optical reflectance is highly dependent on the calcination temp. A simple and cost-effective method to obtain ultrafine MgO/MgAl2O4 nanocryst. powders with expected unique properties was established. These synthesized spinel powders is a highly promised feedstock for refractory, ceramic and environmental applications.
    >> More from SciFinder ®
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  44. 44
    Barrett, C. S. X-ray Study of the Alkali Metals at Low Temperatures. Acta Crystallogr. 1956, 9, 671– 677,  DOI: 10.1107/S0365110X56001790
    [Crossref], [CAS], Google Scholar
    44
    X-ray study of the alkali metals at low temperatures
    Barrett, C. S.
    Acta Crystallographica (1956), 9 (), 671-7CODEN: ACCRA9; ISSN:0365-110X.
    Using a spectrometer having provision for cold-working and x-raying specimens in a high vacuum at low temps., B. found that Na partially transforms on cooling (below 36°K.) or on deforming (below 51°K.) to a close-packed hexagonal structure with stacking faults, having a = 3.767, c = 6.154 A., c/a = 1.634 at 5°K., this coexisting with body-centered cubic Na of a = 4.225 A. The body-centered cubic form at 78°K. has a = 4.235 A. Severe cold-working at 5°K. transforms about half of the material to the hexagonal form; subsequent reversion to cubic starts on heating to 60-75°K. and is completed at 100-110°K., or at lower temps. if there has been no cold-working. Reversion can be aided by cold-working at 45-100°K. High-purity, severely deformed Na recrystallizes at 98°K. Patterns of Li that has been cooled can be interpreted similarly; they indicate a phase of close-packed hexagonal structure with parameters a = 3.111, c = 5.093 A., c/a = 1.637 (which differ from the earlier, tentative ones (C.A. 46, 2386f)). This phase coexists with the body-centered cubic phase of a = 3.491 A., at 78°K. Confirming the earlier work (loc. cit.) hexagonal Li is converted to face-centered cubic by cold-working. K, Rb, and Cs retain their body-centered cubic structure after cooling and cold-working at 5°K., with a = 5.225, 5.585, and 6.045 A., resp., at 5°K. and with a = 5.247, 5.605, and 6.067 A. at 78°K.
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  45. 45
    Streib, W. E.; Jordan, T. H.; Lipscomb, W. N. Single-crystal X-Ray Diffraction Study of β Nitrogen. J. Chem. Phys. 1962, 37, 2962– 2965,  DOI: 10.1063/1.1733125
    [Crossref], [CAS], Google Scholar
    45
    Single-crystal x-ray diffraction study of β-nitrogen
    Streib, William E.; Jordan, Truman H.; Lipscomb, William N.
    Journal of Chemical Physics (1962), 37 (), 2962-5CODEN: JCPSA6; ISSN:0021-9606.
    Three-dimensional x-ray-diffraction data were collected from single crystals of N at 50°K. The observed general condition, l = 2n for hh2h-l reflections is in agreement with the previously assigned space group, P63/mmc, and Z = 2. The data agree equally well with two nearly phys. indistinguishable models, in each of which the mol. centers form a h.c.p. lattice. In one, the mol. is precessing about the z axis passing through its center, at an angle of 54.5 ±2.5° between z and the N.sbd.N bond, while in the other the N atoms are statistically distributed among the 24-fold positions with the mol. axis again at an angle of 54.5° relative to z.
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  46. 46
    Schuch, A. F.; Mills, R. L. Crystal Structures of the Three Modifications of Nitrogen 14 and Nitrogen 15 at High Pressure. J. Chem. Phys. 1970, 52, 6000– 6008,  DOI: 10.1063/1.1672899
    [Crossref], [CAS], Google Scholar
    46
    Crystal structures of the three modifications of nitrogen-14 and nitrogen-15 at high pressure
    Schuch, Adam F.; Mills, Robert L.
    Journal of Chemical Physics (1970), 52 (12), 6000-8CODEN: JCPSA6; ISSN:0021-9606.
    X-ray diffraction photographs were used to det. that high-pressure γ N2 is tetragonal with Z = 2 in special position f of space group P42/mnm. At an av. pressure and temp. of 4015 atm and 20.5°K, resp., the unit cell dimensions are a = 3.957 and c = 5.109 Å, giving a molar vol. in good agreement with that from p-V-T measurements. The β N2 solid modification, which is contiguous with the melting curve, remains hexagonal up to at least 4125 atm and 49°K where the unit cell consts. are a = 3.861 and c = 6.265 Å. Over the pressure range investigated, the c/a ratio is very close to the ideal value for closest packing of hard spheres. The at. positions in hexagonal (β) N2, which are known to be highly disordered at low pressure, show no evidence of ordering at the highest-pressures studied. The third allotrope, α N2, is cubic at 3785 atm and 19.6°K with Z = 4 in a unit cell 5.433 Å on a side. Both the Pa3 and P213 space groups, which have been reported at zero pressure, appear to be spatially possible at limiting high pressures for α N2. However, from diffraction measurements on the cubic solid at all pressures, it was impossible to prove the existence of the P213 structure. The measurements on pure 30N2 show that this isotope also exists in the same solid modifications as 28N2 with cell dimensions that are similar for both isotopes. However, the transition to γ N2 at 20°K for the mass-15 isotope occurs 400 atm lower than that for the mass-14 isotope.
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  47. 47
    Francisco, E.; Casals-Sainz, J. L.; Rocha-Rinza, T.; Martín Pendás, Á. Partitioning the DFT Exchange-Correlation Energy in Line with the Interacting Quantum Atoms Approach. Theor. Chem. Acc. 2016, 135, 170,  DOI: 10.1007/s00214-016-1921-x
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  48. 48
    Gelessus, A.; Thiel, W.; Weber, W. Multipoles and Symmetry. J. Chem. Educ. 1995, 72, 505– 508,  DOI: 10.1021/ed072p505
    [ACS Full Text ACS Full Text], [CAS], Google Scholar
    48
    Multipoles and symmetry
    Gelessus, Achim; Theil, Walter; Weber, Wolfgang
    Journal of Chemical Education (1995), 72 (6), 505-8CODEN: JCEDA8; ISSN:0021-9584. (Division of Chemical Education of the American Chemical Society)
    Group theor. derivations are outlined and explicit results listed on the connection between multipoles and symmetry.
    >> More from SciFinder ®
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  49. 49
    Bende, D. Chemical Bonding Models and Their Implications for Bonding-Property Relations in MgAgAs-Type and Related Compounds.; Ph.D. Dissertation; p 97.
    Google Scholar
    There is no corresponding record for this reference.
  50. 50
    Martín Pendás, Á.; Casals-Sainz, J. L.; Francisco, E. On Electrostatics, Covalency, and Chemical Dashes: Physical Interactions versus Chemical Bonds. Chem. - Eur. J. 2019, 25, 309– 314,  DOI: 10.1002/chem.201804160
    [Crossref], [PubMed], Google Scholar
    There is no corresponding record for this reference.
  51. 51
    Wagner, F. R.; Baranov, A. I.; Grin, Y.; Kohout, M. A Position-Space View on Chemical Bonding in Metal Diborides with AlB2 Type of Crystal Structure. Z. Anorg. Allg. Chem. 2013, 639, 2025– 2035,  DOI: 10.1002/zaac.201200523
    [Crossref], [CAS], Google Scholar
    51
    A Position-Space View on Chemical Bonding in Metal Diborides with AlB2 Type of Crystal Structure
    Wagner, Frank R.; Baranov, Alexey I.; Grin, Yuri; Kohout, Miroslav
    Zeitschrift fuer Anorganische und Allgemeine Chemie (2013), 639 (11), 2025-2035CODEN: ZAACAB; ISSN:0044-2313. (Wiley-VCH Verlag GmbH & Co. KGaA)
    On the basis of QTAIM and ELI-D partitioning of position space two- and three-center delocalization indexes were calcd. for fifteen MB2 phases with the crystal structure of AlB2 type. The bonding picture in main-group metal diborides is closest related to graphite with dominant covalent B-B bonding, albeit with lower effective bond order. For MgB2 an exceptionally large distant electron sharing was found. Transition-metal diborides display smaller effective bond orders B-B but higher effective bond orders TM-B and TM-TM than main-group metal diborides. The large chem. flexibility of this structure type is caused by counterbalancing effects of B-B bonding vs. M-B and M-M bonding. Different three-center fluctuation channels of bonds B-B are found for main-group and transition-metal diborides, namely B-B-B for the former and B-B-M for the latter. With the technique of ELI-D/QTAIM intersection the increasing importance of B2→4M bond charge fluctuations along each row of the periodic table can be recovered already at the topol. level of anal.
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  52. 52
    Barin, I. Thermochemical Data of Pure Substances; Wiley: 1995; Vol. I.
    [Crossref], Google Scholar
    There is no corresponding record for this reference.
  53. 53
    Grochala, W. Diamond: Electronic Ground State of Carbon at Temperatures Approaching 0 K. Angew. Chem., Int. Ed. 2014, 53, 3680– 3683,  DOI: 10.1002/anie.201400131
    [Crossref], [CAS], Google Scholar
    53
    Diamond: Electronic Ground State of Carbon at Temperatures Approaching 0 K
    Grochala, Wojciech
    Angewandte Chemie, International Edition (2014), 53 (14), 3680-3683CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
    The relative stability of graphite and diamond is revisited with hybrid d. functional theory calcns. The electronic energy of diamond is computed to be more neg. by 1.1 kJ mol-1 than that of graphite at T = 0 K and in the absence of external pressure. Graphite gains thermodn. stability over diamond at 298 K only because of the differences in the zero-point energy, sp. heat, and entropy terms for both polymorphs.
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  54. 54
    Popov, I. V.; Görne, A. L.; Tchougréeff, A. L.; Dronskowski, R. Relative Stability of Diamond and Graphite as Seen Through Bonds and Hybridizations. Phys. Chem. Chem. Phys. 2019, 21, 10961– 10969,  DOI: 10.1039/C8CP07592A
    [Crossref], [PubMed], [CAS], Google Scholar
    54
    Relative stability of diamond and graphite as seen through bonds and hybridizations
    Popov, Ilya V.; Gorne, Arno L.; Tchougreeff, Andrei L.; Dronskowski, Richard
    Physical Chemistry Chemical Physics (2019), 21 (21), 10961-10969CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    The relative stability of the 2 most important forms of elemental carbon, diamond and graphite, is readdressed from a newly developed perspective as derived from historically well-known roots. Unlike other theor. studies mostly relying on numerical methods, we consider an anal. model to gain fundamental insight into the reasons for the quasi-degeneracy of diamond and graphite despite their extremely different covalent bonding patterns. We derive the allotropes' relative energies and provide a qual. picture predicting a quasi-degenerate electronic ground state for graphite (graphene) and diamond at zero temp. Our approach also gives numerical ests. of the energy difference and interat. sepns. in good agreement with exptl. data and recent results of hybrid DFT modeling, although obtained with a much smaller numerical but highly transparent effort. An attempt to extend this treatment to the lowest energy allotropes of silicon proves to be successful as well.
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  • Abstract

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    Figure 1

    Figure 1. Energy difference (per atom) between the reconstructed IQA energy and the SCF energy (energy error). The white band denotes the median. Boxes delimit the lower and upper quartiles Q3 and Q4, and fences delimit all energy errors from a sample that takes wave functions computed with GAMESS, ADF, or FHI-aims, employing HF, LDA, PBE, BLYP, and B3LYP functionals for every system. Ex was rescaled from the integration of the corresponding functional. (47) All errors fall below 1 mHa.

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    Figure 2

    Figure 2. Diamond: Convergence of bielectronic integrals inside the basin of the carbon atom, with increasing multipolar order l, and . Only their magnitude is plotted. Dashed lines indicate the trend of convergence for symmetry allowed terms. Disconnected dots are not allowed by symmetry and are nonzero due to numerical errors. Terms below 10–5 Ha are represented as dots at the bottom.

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    Figure 3

    Figure 3. BN (zincblende): Convergence of bielectronic intrabasin integrals with increasing multipolar order l. For point and lines explanations, see Figure 2.

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    Figure 4

    Figure 4. Diamond: Convergence of bielectronic interbasin integrals with increasing bipolar order L = lA + lB. Labels in the top right indicate an interaction of atom A with an atom B of the ith coordination sphere as A–B(i).

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    Figure 5

    Figure 5. BN (zincblende): Convergence of bielectronic interbasin integrals with increasing bipolar order L = lA + lB. For point and lines explanations, see Figure 2.

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  • References

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    This article references 54 other publications.

    1. 1
      Hume-Rothery, W. Materials Science and Engineering; McGraw Hill: 1967; pp 3– 23.
      Google Scholar
      There is no corresponding record for this reference.
    2. 2
      Fredrickson, D. C.; Lee, S.; Hoffmann, R. The Nowotny Chimney Ladder Phases: Whence the 14 Electron Rule?. Inorg. Chem. 2004, 43, 6159– 6167,  DOI: 10.1021/ic049897h
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      2
      The Nowotny Chimney Ladder Phases: Whence the 14 Electron Rule?
      Fredrickson, Daniel C.; Lee, Stephen; Hoffmann, Roald
      Inorganic Chemistry (2004), 43 (20), 6159-6167CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      The late transition metal Nowotny chimney ladder phases (NCLs, TtEm; T, groups 7-9; E, groups 13 and 14) follow a 14 electron rule: the total no. of valence electrons per T atom is 14. We ext. a chem. explanation for this rule from extended Huckel calcns.; we focus on RuGa2, the parent NCL structure. A gap between filled and unfilled bands arises from the occupation of two Ga-Ga bonding/Ru-Ga nonbonding orbitals per RuGa2, independent of k-point. In addn., the five Ru d levels are filled. Together this makes for 7 filled bands at each k-point, or 14 electrons per Ru. We discuss the connections between this 14 electron rule and the 18 electron rule of organometallic complexes.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXlt1Sisrs%253D&md5=08e97f2117878bd85c54b3f77982ec0d
    3. 3
      Yannello, V. J.; Fredrickson, D. C. Generality of the 18- n Rule: Intermetallic Structural Chemistry Explained through Isolobal Analogies to Transition Metal Complexes. Inorg. Chem. 2015, 54, 11385– 11398,  DOI: 10.1021/acs.inorgchem.5b02016
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      3
      Generality of the 18-n Rule: Intermetallic Structural Chemistry Explained through Isolobal Analogies to Transition Metal Complexes
      Yannello, Vincent J.; Fredrickson, Daniel C.
      Inorganic Chemistry (2015), 54 (23), 11385-11398CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      Intermetallic phases exhibit a vast structural diversity in which electron count is known to be one controlling factor. However, chem. bonding theory has yet to establish how electron counts and structure are interrelated for the majority of these compds. Recently, a simple bonding picture for transition metal (T)-main group (E) intermetallics has begun to take shape based on isolobal analogies to mol. T complexes. This bonding picture is summarized in the 18-n rule: each T atom in a T-E intermetallic phase will need 18-n electrons to achieve a closed-shell 18-electron configuration, where n is the no. of electron pairs it shares with other T atoms in multicenter interactions isolobal to T-T bonds. In this Article, we illustrate the generality of this rule with a survey over a wide range of T-E phases. First, we illustrate how three structural progressions with changing electron counts can be accounted for, both geometrically and electronically, with the 18-n rule: (1) the transition between the fluorite and complex β-FeSi2 types for TSi2 phases; (2) the sequence from the marcasite type to the arsenopyrite type and back to the marcasite type for TSb2 compds.; and (3) the evolution from the AuCu3 type to the ZrAl3 and TiAl3 types for TAl3 phases. We then turn to a broader survey of the applicability of the 18-n rule through a study of the following 34 binary structure types: PtHg4, CaF2 (fluorite), Fe3C, CoGa3, Co2Al5, Ru2B3, β-FeSi2, NiAs, Ni2Al3, Rh4Si5, CrSi2, Ir3Ga5, Mo3Al8, MnP, TiSi2, Ru2Sn3, TiAl3, MoSi2, CoSn, ZrAl3, CsCl, FeSi, AuCu3, ZrSi2, Mn2Hg5, FeS2 (oP6, marcasite), CoAs3 (skutterudite), PdSn2, CoSb2, Ir3Ge7, CuAl2, Re3Ge7, CrP2, and Mg2Ni. Through these analyses, the 18-n rule is established as a framework for interpreting the stability of 341 intermetallic phases and anticipating their properties.
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    4. 4
      Wagner, F. R.; Cardoso-Gil, R.; Boucher, B.; Wagner-Reetz, M.; Sichelschmidt, J.; Gille, P.; Baenitz, M.; Grin, Y. On Fe–Fe Dumbbells in the Ideal and Real Structures of FeGa3. Inorg. Chem. 2018, 57, 12908– 12919,  DOI: 10.1021/acs.inorgchem.8b02094
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      4
      On Fe-Fe Dumbbells in the Ideal and Real Structures of FeGa3
      Wagner, Frank R.; Cardoso-Gil, Raul; Boucher, Benoit; Wagner-Reetz, Maik; Sichelschmidt, Joerg; Gille, Peter; Baenitz, Michael; Grin, Yuri
      Inorganic Chemistry (2018), 57 (20), 12908-12919CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      The intermetallic phase FeGa3 belongs to the rare examples of substances with transition metals where semiconducting behavior is found. The necessary electron count of 17 ve/fu can be formally derived from eight Fe-Ga and one Fe-Fe two-center-two-electron bond. The situation is reminiscent of the well-known Fe2(CO)9 scenario, where a direct Fe-Fe two-center-two-electron bond was shown to not be present. Fe-Fe interaction in FeGa3 and its substitution variants represents the crucial point for explanation of electronic, thermal transport, and optical properties of this material. Chem. bonding anal. in position space of FeGa3 and Fe2(CO)9 on the basis of the topol. of the electron localizability indicator distribution, QTAIM atoms, two- and three-center delocalization indexes, domain natural orbitals, IQA anal., and an evaluation of the Fe-Fe dissocn. energy yields a complete picture of the partially compensated Fe-Fe bond, which is nevertheless strong enough to be of decisive importance. Structural reinvestigation of differently synthesized single crystals leads to the compn. Fe1+xGa3 (0 ≤ x ≤ 0.018), where the addnl. Fe atoms are predicted from DFT/PBE calcns. to yield a magnetic moment of about 2 μB/Fe2 atom and metallic in-gap states. Accompanying magnetization and ESR measurements are consistent with this picture.
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    5. 5
      Bende, D.; Grin, Y.; Wagner, F. R. Covalence and Ionicity in MgAgAs-type Compounds. Chem. - Eur. J. 2014, 20, 9702– 9708,  DOI: 10.1002/chem.201400299
      [Crossref], [PubMed], [CAS], Google Scholar
      5
      Covalence and Ionicity in MgAgAs-Type Compounds
      Bende, David; Grin, Yuri; Wagner, Frank R.
      Chemistry - A European Journal (2014), 20 (31), 9702-9708CODEN: CEUJED; ISSN:0947-6539. (Wiley-VCH Verlag GmbH & Co. KGaA)
      MgAgAs-type "half-Heusler" compds. are known to realize two out of three possible at. arrangements of this structure type. The no. of transition metal components typically dets. which of the alternatives is favored. On the basis of DFT calcns. for all three variants of 20 eight- and eighteen-valence-electron compds., the exptl. obsd. structural variant was found to be detd. by basically two different bonding patterns. They are quantified by employing two complementary position-space bonding measures. The Madelung energy calcd. with the QTAIM effective charges reflects contributions of the ionic interactions to the total energy. The sum of nearest-neighbor delocalization indexes characterizes the covalent interactions through electron sharing. With the aid of these quantities, the energetic sequence of the three at. arrangements for each compd. is rationalized. The resulting systematic is used to predict a scenario in which an nontypical at. arrangement becomes most favorable.
      >> More from SciFinder ®
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    6. 6
      Bende, D.; Wagner, F. R.; Sichevych, O.; Grin, Y. Chemical Bonding Analysis as a Guide for the Preparation of New Compounds: The Case of VIrGe and HfPtGe. Angew. Chem., Int. Ed. 2017, 56, 1313– 1318,  DOI: 10.1002/anie.201610029
      [Crossref], [CAS], Google Scholar
      6
      Chemical Bonding Analysis as a Guide for the Preparation of New Compounds: The Case of VIrGe and HfPtGe
      Bende, David; Wagner, Frank R.; Sichevych, Olga; Grin, Yuri
      Angewandte Chemie, International Edition (2017), 56 (5), 1313-1318CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The chem. bonding of transition metal compds. with a MgAgAs-type of crystal structure is analyzed with quantum chem. position-space techniques. The obsd. trends in QTAIM Madelung energy and nearest neighbor electron sharing explain the occurrence of recently synthesized MgAgAs-type compds., TiPtGe and TaIrGe, at the boundary to the TiNiSi-type crystal structure. These bonding indicators are used to identify favorable element combinations for new MgAgAs-type compds. The new phases-the high-temp. VIrGe and the low-temp. HfPtGe-showing this type of crystal structure are prepd. and characterized by powder x-ray diffraction and DTA.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XitFGjurnO&md5=b12963b6469c882df9e614d1804625be
    7. 7
      Martín Pendás, Á.; Blanco, M. A.; Francisco, E. Two-electron Integrations in the Quantum Theory of Atoms in Molecules. J. Chem. Phys. 2004, 120, 4581– 4592,  DOI: 10.1063/1.1645788
      [Crossref], [PubMed], Google Scholar
      There is no corresponding record for this reference.
    8. 8
      Blanco, M. A.; Martín Pendás, Á.; Francisco, E. Interacting Quantum Atoms: A Correlated Energy Decomposition Scheme Based on the Quantum Theory of Atoms in Molecules. J. Chem. Theory Comput. 2005, 1, 1096– 1109,  DOI: 10.1021/ct0501093
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      8
      Interacting Quantum Atoms: A Correlated Energy Decomposition Scheme Based on the Quantum Theory of Atoms in Molecules
      Blanco, M. A.; Pendas, A. Martin; Francisco, E.
      Journal of Chemical Theory and Computation (2005), 1 (6), 1096-1109CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We make use of the Quantum Theory of Atoms in Mols. (QTAM) to partition the total energy of a many-electron system into intra- and interat. terms, by explicitly computing both the one- and two-electron contributions. While the general scheme is formally equiv. to that by Bader et al., we focus on the sepn. and computation of the at. self-energies and all the interaction terms. The partition is ultimately performed within the d. matrixes, in analogy with McWeeny's theory of electronic separability, and then carried onto the energy. It is intimately linked with the atomistic picture of the chem. bond, not only allowing the sepn. of different two-body contributions (point-charge-like, multipolar, total Coulomb, exchange, correlation, ...) to the interaction between a pair of atoms but also including an effective many-body contribution to the binding (self-energy, formally one-body) due to the deformation of the atoms within the many-electron system as compared to the free atoms. Many qual. ideas about the chem. bond can be quantified using this scheme.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXoslKnt7s%253D&md5=c2f580775113941b07db522473526339
    9. 9
      Martín Pendás, Á.; Francisco, E.; Blanco, M. A. Two-electron Integrations in the Quantum Theory of Atoms in Molecules with Correlated Wave Functions. J. Comput. Chem. 2005, 26, 344– 351,  DOI: 10.1002/jcc.20173
      [Crossref], [PubMed], Google Scholar
      There is no corresponding record for this reference.
    10. 10
      Raupach, M.; Tonner, R. A Periodic Energy Decomposition Analysis Method for the Investigation of Chemical Bonding in Extended Systems. J. Chem. Phys. 2015, 142, 194105,  DOI: 10.1063/1.4919943
      [Crossref], [PubMed], [CAS], Google Scholar
      10
      A periodic energy decomposition analysis method for the investigation of chemical bonding in extended systems
      Raupach, Marc; Tonner, Ralf
      Journal of Chemical Physics (2015), 142 (19), 194105/1-194105/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The development and first applications of a new periodic energy decompn. anal. (pEDA) scheme for extended systems based on the Kohn-Sham approach to d. functional theory are described. The pEDA decomps. the bonding energy between two fragments (e.g., the adsorption energy of a mol. on a surface) into several well-defined terms: prepn., electrostatic, Pauli repulsion, and orbital relaxation energies. This is complemented by consideration of dispersion interactions via a pairwise scheme. One major extension toward a previous implementation [Philipsen and Baerends, J. Phys. Chem. B 110, 12470 (2006)] lies in the sep. discussion of electrostatic and Pauli and the addn. of a dispersion term. The pEDA presented here for an implementation based on AOs can handle restricted and unrestricted fragments for 0D to 3D systems considering periodic boundary conditions with and without the detn. of fragment occupations. For the latter case, reciprocal space sampling is enabled. The new method gives comparable results to established schemes for mol. systems and shows good convergence with respect to the basis set (TZ2P), the integration accuracy, and k-space sampling. Four typical bonding scenarios for surface-adsorbate complexes were chosen to highlight the performance of the method representing insulating (CO on MgO(001)), metallic (H2 on M(001), M = Pd, Cu), and semiconducting (CO and C2H2 on Si(001)) substrates. These examples cover diverse substrates as well as bonding scenarios ranging from weakly interacting to covalent (shared electron and donor acceptor) bonding. The results presented lend confidence that the pEDA will be a powerful tool for the anal. of surface-adsorbate bonding in the future, enabling the transfer of concepts like ionic and covalent bonding, donor-acceptor interaction, steric repulsion, and others to extended systems. (c) 2015 American Institute of Physics.
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    11. 11
      Dronskowski, R. Computational Chemistry of Solid State Materials; John Wiley & Sons, Ltd.: 2005.
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    12. 12
      Bader, R. F. W. Atoms in Molecules: A Quantum Theory (International Series of Monographs on Chemistry); Clarendon Press: 1990.
      Google Scholar
      There is no corresponding record for this reference.
    13. 13
      Martín Pendás, Á.; Francisco, E. Promolden: A QTAIM/IQA code. Available from the authors upon request by writing to [email protected]
      Google Scholar
      There is no corresponding record for this reference.
    14. 14
      Kohout, M. DGrid, ver. 5.2; Dresden: 2021.
      Google Scholar
      There is no corresponding record for this reference.
    15. 15
      te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T. Chemistry with ADF. J. Comput. Chem. 2001, 22, 931– 967,  DOI: 10.1002/jcc.1056
      [Crossref], [CAS], Google Scholar
      15
      Chemistry with ADF
      Te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; Fonseca Guerra, C.; Van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler, T.
      Journal of Computational Chemistry (2001), 22 (9), 931-967CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      A review with 241 refs. We present the theor. and tech. foundations of the Amsterdam D. Functional (ADF) program with a survey of the characteristics of the code (numerical integration, d. fitting for the Coulomb potential, and STO basis functions). Recent developments enhance the efficiency of ADF (e.g., parallelization, near order-N scaling, QM/MM) and its functionality (e.g., NMR chem. shifts, COSMO solvent effects, ZORA relativistic method, excitation energies, frequency-dependent (hyper)polarizabilities, at. VDD charges). In the Applications section we discuss the phys. model of the electronic structure and the chem. bond, i.e., the Kohn-Sham MO (MO) theory, and illustrate the power of the Kohn-Sham MO model in conjunction with the ADF-typical fragment approach to quant. understand and predict chem. phenomena. We review the "Activation-strain TS interaction" (ATS) model of chem. reactivity as a conceptual framework for understanding how activation barriers of various types of (competing) reaction mechanisms arise and how they may be controlled, for example, in org. chem. or homogeneous catalysis. Finally, we include a brief discussion of exemplary applications in the field of biochem. (structure and bonding of DNA) and of time-dependent d. functional theory (TDDFT) to indicate how this development further reinforces the ADF tools for the anal. of chem. phenomena.
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    16. 16
      Blum, V.; Gehrke, R.; Hanke, F.; Havu, P.; Havu, V.; Ren, X.; Reuter, K.; Scheffler, M. Ab Initio Molecular Simulations with Numeric Atom-Centered Orbitals. Comput. Phys. Commun. 2009, 180, 2175– 2196,  DOI: 10.1016/j.cpc.2009.06.022
      [Crossref], [CAS], Google Scholar
      16
      Ab initio molecular simulations with numeric atom-centered orbitals
      Blum, Volker; Gehrke, Ralf; Hanke, Felix; Havu, Paula; Havu, Ville; Ren, Xinguo; Reuter, Karsten; Scheffler, Matthias
      Computer Physics Communications (2009), 180 (11), 2175-2196CODEN: CPHCBZ; ISSN:0010-4655. (Elsevier B.V.)
      We describe a complete set of algorithms for ab initio mol. simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of mol. and materials properties from quantum-mech. first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute "ab initio mol. simulations" (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on d.-functional theory (DFT) in the local and semilocal (generalized gradient) approxns., but an extension to hybrid functionals, Hartree-Fock theory, and MP2/GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron/full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio mol. dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calcn. to range from qual. tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs.
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    17. 17
      Baranov, A. I.; Kohout, M. Electron Localization and Delocalization Indices for Solids. J. Comput. Chem. 2011, 32, 2064– 2076,  DOI: 10.1002/jcc.21784
      [Crossref], [PubMed], [CAS], Google Scholar
      17
      Electron localization and delocalization indices for solids
      Baranov, Alexey I.; Kohout, Miroslav
      Journal of Computational Chemistry (2011), 32 (10), 2064-2076CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      The electron localization and delocalization indexes obtained by the integration of exchange-correlation part of pair d. over chem. meaningful regions of space, e.g., QTAIM atoms are valuable tools for the bonding anal. in mol. systems. However, among periodic systems only few simplest models were analyzed with this approach until now. This contribution reports implementation and evaluation of the localization and delocalization indexes on the basis of solid state DFT calcns. A comparison with the results of simple anal. model of Ponec was made. In addn., a small set of compds. with ionic (NaCl), covalent (diamond, graphite), and metallic (Na, Cu) bonding interactions was characterized using this method. Typical features of different types of bonding were discussed using the delocalization indexes. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011.
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    18. 18
      Chen, H.; Friesecke, G. Pair Densities in Density Functional Theory. Multiscale Model. Simul. 2015, 13, 1259– 1289,  DOI: 10.1137/15M1014024
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    19. 19
      Rodríguez, J. I.; Köster, A. M.; Ayers, P. W.; Santos-Valle, A.; Vela, A.; Merino, G. An Efficient Grid-Based Scheme to Compute QTAIM Atomic Properties without Explicit Calculation of Zero-Flux Surfaces. J. Comput. Chem. 2009, 30, 1082– 1092,  DOI: 10.1002/jcc.21134
      [Crossref], [PubMed], [CAS], Google Scholar
      19
      An efficient grid-based scheme to compute QTAIM atomic properties without explicit calculation of zero-flux surfaces
      Rodriguez, Juan I.; Koster, Andreas M.; Ayers, Paul W.; Santos-Valle, Ana; Vela, Alberto; Merino, Gabriel
      Journal of Computational Chemistry (2009), 30 (7), 1082-1092CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      We introduce a method to compute at. properties according to the "quantum theory of atoms in mols.". An integration grid in real space is partitioned into subsets, ωi. The subset, ωi, is composed of all grid points contained in the at. basin, Ωi, so that integration over Ωi is reduced to simple quadrature over the points in ωi. The partition is constructed from deMon2k's at. center grids by following the steepest ascent path of the d. starting from each point in the grid. We also introduce a technique that exploits the cellular nature of the grid to make the algorithm faster. The performance of the method is tested by computing properties of atoms and nonnuclear attractors (energies, charges, dipole, and quadrupole moments) for a set of representative mols.
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    20. 20
      Womersley, R. S. Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan; Springer International Publishing: Cham, Switzerland, 2018; pp 1243– 1285.
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    21. 21
      Blanco, M. A. Métodos Cuánticos Locales para la Simulación de Materiales Iónicos. Fundamentos, algoritmos y aplicaciones. Ph.D. thesis, University of Oviedo: 1997.
      Google Scholar
      There is no corresponding record for this reference.
    22. 22
      Kay, K. G.; Todd, H. D.; Silverstone, H. J. Bipolar Expansion for r12nYlm(θ12,ϕ12). J. Chem. Phys. 1969, 51, 2363– 2367,  DOI: 10.1063/1.1672353
      [Crossref], [CAS], Google Scholar
      22
      Bipolar expansion for r12nYtm(Θ12,Φ12)
      Kay, Kenneth G.; Todd, H. David; Silverstone, Harris J.
      Journal of Chemical Physics (1969), 51 (6), 2363-7CODEN: JCPSA6; ISSN:0021-9606.
      Explicit formulas for the radial functions V(n)l1l2l3l(r1,r2,R) in the bipolar expansion for r12nYlm(θ12, Φ12), r12nYlm(θ12, Φ12) = Σ(2λ + 1)1/2(2l3 + 1)1/2cλ(lm; l1m1)cl3(λ, m - m1; l2m2) × Yl1m1- (θ1, Φ1)Yl2m2(θ2, Φ2)Yl3m-m1-m2(θR, ΦR)V(n)l1l2l3l(r1, r2, R), where r12 = r1 - r2 - R, are derived by the use of the theory of generalized functions and Fourier transforms. When n ≤ -4 and n - l is odd, there are delta-function terms. In this approach the delta-function terms and the 4-region form of the expansion are obtained from a single, unified formula valid in all regions. Recurrence formulas for the V(n)l1l2l3l are given.
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    23. 23
      Buehler, R. J.; Hirschfelder, J. O. Bipolar Expansion of Coulombic Potentials. Phys. Rev. 1951, 83, 628– 633,  DOI: 10.1103/PhysRev.83.628
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    24. 24
      Luaña, Víctor. Environ: local environment determination of arbitrary positions in a crystal; Oviedo: 1992.
      Google Scholar
      There is no corresponding record for this reference.
    25. 25
      Kosov, D. S.; Popelier, P. L. A. Atomic Partitioning of Molecular Electrostatic Potentials. J. Phys. Chem. A 2000, 104, 7339– 7345,  DOI: 10.1021/jp0003407
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      25
      Atomic Partitioning of Molecular Electrostatic Potentials
      Kosov, D. S.; Popelier, P. L. A.
      Journal of Physical Chemistry A (2000), 104 (31), 7339-7345CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      The theory of atoms in mols. (AIM) defines bounded at. fragments in real space that generate transferable at. properties. As part of a program that investigates the topol. partitioning of electromagnetic properties based on the electron d., we have calcd. the exact at. electrostatic potential (AEP) of an AIM atom in a mol. Second we expand this at. electrostatic potential in terms of AIM electrostatic multipole moments based on spherical tensors. We prove that the convergence of this expansion is faster than previously assumed, even for complicated at. shapes.
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    26. 26
      Francisco, E.; Menéndez Crespo, D.; Costales, A.; Martín Pendás, Á. A Multipolar Approach to the Interatomic Covalent Interaction Energy. J. Comput. Chem. 2017, 38, 816– 829,  DOI: 10.1002/jcc.24758
      [Crossref], [PubMed], [CAS], Google Scholar
      26
      A multipolar approach to the interatomic covalent interaction energy
      Francisco, Evelio; Menendez Crespo, Daniel; Costales, Aurora; Martin Pendas, Angel
      Journal of Computational Chemistry (2017), 38 (11), 816-829CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)
      Interat. exchange-correlation energies correspond to the covalent energetic contributions to an interat. interaction in real space theories of the chem. bond, but their widespread use is severely limited due to their computationally intensive character. In the same way as the multipolar (mp) expansion is customary used in biomol. modeling to approx. the classical Coulomb interaction between two charge densities ρA(r) and ρB(r), we examine in this work the mp approach to approx. the interat. exchange-correlation (xc) energies of the Interacting Quantum Atoms method. We show that the full xc mp series is quickly divergent for directly bonded atoms (1-2 pairs) albeit it works reasonably well most times for 1-n (n > 2) interactions. As with conventional perturbation theory, we show numerically that the xc series is asymptotically convergent and that, a truncated xc mp approxn. retaining terms up to l1 + l2 = 2 usually gives relatively accurate results, sometimes even for directly bonded atoms. Our findings are supported by extensive numerical analyses on a variety of systems that range from several std. hydrogen bonded dimers to typically covalent or arom. mols. The exact algebraic relationship between the monopole-monopole xc mp term and the inter-at. bond order, as measured by the delocalization index of the quantum theory of atoms in mols., is also established. © 2017 Wiley Periodicals, Inc.
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    27. 27
      Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865– 3868,  DOI: 10.1103/PhysRevLett.77.3865
      [Crossref], [PubMed], [CAS], Google Scholar
      27
      Generalized gradient approximation made simple
      Perdew, John P.; Burke, Kieron; Ernzerhof, Matthias
      Physical Review Letters (1996), 77 (18), 3865-3868CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
      Generalized gradient approxns. (GGA's) for the exchange-correlation energy improve upon the local spin d. (LSD) description of atoms, mols., and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental consts. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential.
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    28. 28
      Barca, G. M. J.; Bertoni, C.; Carrington, L.; Datta, D.; De Silva, N.; Deustua, J. E.; Fedorov, D. G.; Gour, J. R.; Gunina, A. O.; Guidez, E. Recent Developments in the General Atomic and Molecular Electronic Structure System. J. Chem. Phys. 2020, 152, 154102,  DOI: 10.1063/5.0005188
      [Crossref], [PubMed], [CAS], Google Scholar
      28
      Recent developments in the general atomic and molecular electronic structure system
      Barca, Giuseppe M. J.; Bertoni, Colleen; Carrington, Laura; Datta, Dipayan; De Silva, Nuwan; Deustua, J. Emiliano; Fedorov, Dmitri G.; Gour, Jeffrey R.; Gunina, Anastasia O.; Guidez, Emilie; Harville, Taylor; Irle, Stephan; Ivanic, Joe; Kowalski, Karol; Leang, Sarom S.; Li, Hui; Li, Wei; Lutz, Jesse J.; Magoulas, Ilias; Mato, Joani; Mironov, Vladimir; Nakata, Hiroya; Pham, Buu Q.; Piecuch, Piotr; Poole, David; Pruitt, Spencer R.; Rendell, Alistair P.; Roskop, Luke B.; Ruedenberg, Klaus; Sattasathuchana, Tosaporn; Schmidt, Michael W.; Shen, Jun; Slipchenko, Lyudmila; Sosonkina, Masha; Sundriyal, Vaibhav; Tiwari, Ananta; Galvez Vallejo, Jorge L.; Westheimer, Bryce; Wloch, Marta; Xu, Peng; Zahariev, Federico; Gordon, Mark S.
      Journal of Chemical Physics (2020), 152 (15), 154102CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A discussion of many of the recently implemented features of GAMESS (General Atomic and Mol. Electronic Structure System) and LibCChem (the C + + CPU/GPU library assocd. with GAMESS) is presented. These features include fragmentation methods such as the fragment MO, effective fragment potential and effective fragment MO methods, hybrid MPI/OpenMP approaches to Hartree-Fock, and resoln. of the identity second order perturbation theory. Many new coupled cluster theory methods have been implemented in GAMESS, as have multiple levels of d. functional/tight binding theory. The role of accelerators, esp. graphical processing units, is discussed in the context of the new features of LibCChem, as it is the assocd. problem of power consumption as the power of computers increases dramatically. The process by which a complex program suite such as GAMESS is maintained and developed is considered. Future developments are briefly summarized. (c) 2020 American Institute of Physics.
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    29. 29
      Dirac, P. A. M. Note on Exchange Phenomena in the Thomas Atom. Math. Proc. Cambridge Philos. Soc. 1930, 26, 376– 385,  DOI: 10.1017/S0305004100016108
      [Crossref], [CAS], Google Scholar
      29
      Exchange phenomena in the Thomas atom
      Dirac, P. A. M.
      Proceedings of the Cambridge Philosophical Society (1930), 26 (), 376-85CODEN: PCPSA4; ISSN:0068-6735.
      Calcn. of the electron distribution in the state of lowest energy of an atom, for which a certain region of phase space is occupied with the max. d. of electrons and the remainder is empty. The calcn. gives a theoretical justification of the Thomas atom.
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    30. 30
      Bloch, F. Bemerkung zur Elektronentheorie des Ferromagnetismus und der elektrischen Leitfähigkeit. Eur. Phys. J. A 1929, 57, 545– 555,  DOI: 10.1007/BF01340281
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    31. 31
      Vosko, S. H.; Wilk, L.; Nusair, M. Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: a Critical Analysis. Can. J. Phys. 1980, 58, 1200– 1211,  DOI: 10.1139/p80-159
      [Crossref], [CAS], Google Scholar
      31
      Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis
      Vosko, S. H.; Wilk, L.; Nusair, M.
      Canadian Journal of Physics (1980), 58 (8), 1200-11CODEN: CJPHAD; ISSN:0008-4204.
      Various approx. forms for the correlation energy per particle of the spin-polarized homogeneous electron gas that have frequently been used in applications of the local spin d. approxn. to the exchange-correlation energy functional are assessed. By accurately recalcg. the RPA correlation energy as a function of electron d. and spin polarization, the inadequacies of the usual approxn. for interpolating between the para- and ferro-magnetic states are demonstrated and an accurate new interpolation formula is presented. A Pade approximant technique was used to accurately interpolate the recent Monte Carlo results. These results can be combined with the RPA spin-dependence so as to produce a correlation energy for a spin-polarized homogeneous electron gas with an estd. max. error of 1 mRy and thus should reliably det. the magnitude of non-local corrections to the local spin d. approxn. in real systems.
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    32. 32
      Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 37, 785– 789,  DOI: 10.1103/PhysRevB.37.785
      [Crossref], [PubMed], [CAS], Google Scholar
      32
      Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
      Lee, Chengteh; Yang, Weitao; Parr, Robert G.
      Physical Review B: Condensed Matter and Materials Physics (1988), 37 (2), 785-9CODEN: PRBMDO; ISSN:0163-1829.
      A correlation-energy formula due to R. Colle and D. Salvetti (1975), in which the correlation energy d. is expressed in terms of the electron d. and a Laplacian of the 2nd-order Hartree-Fock d. matrix, is restated as a formula involving the d. and local kinetic-energy d. On insertion of gradient expansions for the local kinetic-energy d., d.-functional formulas for the correlation energy and correlation potential are then obtained. Through numerical calcns. on a no. of atoms, pos. ions, and mols., of both open- and closed-shell type, it is demonstrated that these formulas, like the original Colle-Salvetti formulas, give correlation energies within a few percent.
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    33. 33
      Becke, A. D. Correlation Energy of an Inhomogeneous Electron Gas: A Coordinate-Space Model. J. Chem. Phys. 1988, 88, 1053– 1062,  DOI: 10.1063/1.454274
      [Crossref], [CAS], Google Scholar
      33
      Correlation energy of an inhomogeneous electron gas: a coordinate-space model
      Becke, A. D.
      Journal of Chemical Physics (1988), 88 (2), 1053-62CODEN: JCPSA6; ISSN:0021-9606.
      A coordinate-space model for dynamical correlations in an inhomogeneous electron gas is developed. The model treats opposite-spin and same-spin pairs sep., and it also accounts properly for correlation contributions to the kinetic energy. It gives identically zero correlation energy for 1-electron systems. Applications to the uniform electron gas and to the atoms H through Ar are made.
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    34. 34
      Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields. J. Phys. Chem. 1994, 98, 11623– 11627,  DOI: 10.1021/j100096a001
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      34
      Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields
      Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J.
      Journal of Physical Chemistry (1994), 98 (45), 11623-7CODEN: JPCHAX; ISSN:0022-3654.
      The unpolarized absorption and CD spectra of the fundamental vibrational transitions of the chiral mol. 4-methyl-2-oxetanone are calcd. ab initio. Harmonic force fields are obtained using d. functional theory (DFT), MP2 and SCF methodologies, and a [5s4p2d/3s2p] (TZ2P) basis set. DFT calcns. use the LSDA, BLYP, and Becke3LYP (B3LYP) d. functionals. Mid-IR spectra predicted using LSDA, BLYP, and B3LYP force fields are of significantly different quality, the B3LYP force field yielding spectra in clearly superior, and overall excellent, agreement with expt. The MP2 force field yields spectra in slightly worse agreement with expt. than the B3LYP force field. The SCF force field yields spectra in poor agreement with expt. The basis set dependence of B3LYP force fields is also explored: the 6-31G* and TZ2P basis sets give very similar results while the 3-21G basis set yields spectra in substantially worse agreement with expt.
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    35. 35
      Downs, R. T.; Somayazulu, M. S. Carbon Dioxide at 1.0 GPa. Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 1998, 54, 897– 898,  DOI: 10.1107/S0108270198001140
      [Crossref], [CAS], Google Scholar
      35
      Carbon dioxide at 1.0 GPa
      Downs, Robert T.; Somayazulu, M. S.
      Acta Crystallographica, Section C: Crystal Structure Communications (1998), C54 (7), 897-898CODEN: ACSCEE; ISSN:0108-2701. (Munksgaard International Publishers Ltd.)
      An x-ray diffraction study of single-crystal CO2 was undertaken at 1.00(5) GPa pressure. The crystal exhibits Pa‾3 symmetry with a cell edge of 5.4942(2) Å and a C-O bond length of 1.168(1) Å (cor. for thermal motion effects). An earlier claim of a new dry ice II phase at this pressure is unfounded.
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    36. 36
      Bindzus, N.; Straasø, T.; Wahlberg, N.; Becker, J.; Bjerg, L.; Lock, N.; Dippel, A.-C.; Iversen, B. B. Experimental Determination of Core Electron Deformation in Diamond. Acta Crystallogr., Sect. A: Found. Adv. 2014, 70, 39– 48,  DOI: 10.1107/S2053273313026600
      [Crossref], [CAS], Google Scholar
      36
      Experimental determination of core electron deformation in diamond
      Bindzus, Niels; Straaso, Tine; Wahlberg, Nanna; Becker, Jacob; Bjerg, Lasse; Lock, Nina; Dippel, Ann-Christin; Iversen, Bo B.
      Acta Crystallographica, Section A: Foundations and Advances (2014), 70 (1), 39-48CODEN: ACSAD7; ISSN:2053-2733. (International Union of Crystallography)
      Synchrotron powder X-ray diffraction data are used to det. the core electron deformation of diamond. Core shell contraction inherently linked to covalent bond formation is obsd. in close correspondence with theor. predictions. Accordingly, a precise and phys. sound reconstruction of the electron d. in diamond necessitates the use of an extended multipolar model, which abandons the assumption of an inert core. The present investigation is facilitated by negligible model bias in the extn. of structure factors, which is accomplished by simultaneous multipolar and Rietveld refinement accurately detg. an at. displacement parameter (ADP) of 0.00181(1) Å2. The deconvolution of thermal motion is a crit. step in exptl. core electron polarization studies, and for diamond it is imperative to exploit the monoat. crystal structure by implementing Wilson plots in detn. of the ADP. This empowers the electron-d. anal. to precisely administer both the deconvolution of thermal motion and the employment of the extended multipolar model on an exptl. basis.
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    37. 37
      Dobrzhinetskaya, L. F.; Wirth, R.; Yang, J.; Green, H. W.; Hutcheon, I. D.; Weber, P. K.; Grew, E. S. Qingsongite, Natural Cubic Boron Nitride: The First Boron Mineral from the Earth’s Mantle. Am. Mineral. 2014, 99, 764– 772,  DOI: 10.2138/am.2014.4714
      [Crossref], [CAS], Google Scholar
      37
      Qingsongite, natural cubic boron nitride: the first boron mineral from the Earth's mantle
      Dobrzhinetskaya, Larissa F.; Wirth, Richard; Yang, Jingsui; Green, Harry W.; Hutcheon, Ian D.; Weber, Peter K.; Grew, Edward S.
      American Mineralogist (2014), 99 (4), 764-772CODEN: AMMIAY; ISSN:0003-004X. (Mineralogical Society of America)
      Qingsongite (IMA 2013-30) is the natural analog of cubic boron nitride (c-BN), which is widely used as an abrasive under the name Borazon. The mineral is named for Qingsong Fang (1939-2010), who found the first diamond in the Luobusa chromitite. Qingsongite occurs in a rock fragment less than 1 mm across extd. from chromitite in deposit 31, Luobusa ophiolite, Yarlung Zangbu suture, southern Tibet at 29°13.86N and 92°11.41E. Five electron microprobe analyses gave B 48.54 ± 0.65 wt% (range 47.90-49.2 wt%); N 51.46 ± 0.65 wt% (range 52.10-50.8 wt%), corresponding to B1.113N0.887 and B1.087N0.913, for max. and min. B contents, resp. (based on 2 atoms per formula unit); no other elements that could substitute for B or N were detected. Crystallog. data on qingsongite obtained using fast Fourier transforms gave cubic symmetry, a = 3.61 ± 0.045 Å. The d. calcd. for the mean compn. B1.100N0.900 is 3.46 g/cm3, i.e., qingsongite is nearly identical to synthetic c-BN. The synthetic analog has the sphalerite structure, space group F‾43m. Mohs hardness of the synthetic analog is between 9 and 10; its cleavage is {011}. Qingsongite forms isolated anhedral single crystals up to 1 μm in size in the marginal zone of the fragment; this zone consists of ∼45 modal% coesite, ∼15% kyanite, and ∼40% amorphous material. Qingsongite is enclosed in kyanite, coesite, or in osbornite; other assocd. phases include native Fe; TiO2 II, a high-pressure polymorph of rutile with the αPbO2 structure; boron carbide of unknown stoichiometry; and amorphous carbon. Coesite forms prisms several tens of micrometers long, but is polycryst., and thus interpreted to be pseudomorphic after stishovite. Assocd. minerals constrain the estd. pressure to 10-15 GPa assuming temp. was about 1300 °C. Our proposed scenario for formation of qingsongite begins with a pelitic rock fragment that was subducted to mid-mantle depths where crustal B originally present in mica or clay combined with mantle N (δ15N = -10.4 ± 3‰ in osbornite) and subsequently exhumed by entrainment in chromitite. The presence of qingsongite has implications for understanding the recycling of crustal material back to the Earth's mantle since boron, an essential constituent of qingsongite, is potentially an ideal tracer of material from Earth's surface.
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    38. 38
      Didier, C.; Pang, W. K.; Guo, Z.; Schmid, S.; Peterson, V. K. Phase Evolution and Intermittent Disorder in Electrochemically Lithiated Graphite Determined Using in Operando Neutron Diffraction. Chem. Mater. 2020, 32, 2518– 2531,  DOI: 10.1021/acs.chemmater.9b05145
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      38
      Phase Evolution and Intermittent Disorder in Electrochemically Lithiated Graphite Determined Using in Operando Neutron Diffraction
      Didier, Christophe; Pang, Wei Kong; Guo, Zaiping; Schmid, Siegbert; Peterson, Vanessa K.
      Chemistry of Materials (2020), 32 (6), 2518-2531CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
      Since their commercialization in 1991, Li-ion batteries (LIBs) have revolutionized the authors' way of life, with LIB pioneers being awarded the 2019 Nobel Prize in Chem. Despite the widespread use of LIBs, many LIB applications are not realized due to performance limitations, detd. largely by the ability of electrode materials to reversibly host Li ions. Overcoming such limitations requires knowledge of the fundamental mechanism for reversible ion intercalation in electrode materials. The still-debated structure of the most common com. electrode material, graphite, during electrochem. lithiation is revisited using in operando neutron powder diffraction of a com. 18650 Li-ion battery. The authors ext. new structural information and present a comprehensive overview of the phase evolution for lithiated graphite. Charge-discharge asymmetry and structural disorder in the lithiation process are obsd., particularly surrounding phase transitions, and the phase evolution is kinetically influenced. Notably, the authors observe pronounced asymmetry over the compn. range 0.5 > x > 0.2, in which the stage 2L phase forms on discharge (delithiation) but not charge (lithiation), likely as a result of the slow formation of the stage 2L phase and the closeness of the stage 2L and stage 2 phase potentials. The authors reconcile the authors' measurements of this transition with a stage 2L stacking disorder model contg. an intergrown stage 2 and 2L phase. The authors resolve debate surrounding the intercalation mechanism in the stage 3L and stage 4L phase region, observing stage-specific reflections that support a 1st-order phase transition over the 0.2 > x > 0.04 range, in agreement with minor changes in the slope of the stacking axis length, despite relatively unchanging 00l reflection broadening. The authors' data support the previously proposed /ABA/ACA/ stacking for the stage 3L phase and an /ABAB/BABA/ stacking sequence of the stage 4L phase alongside exptl. derived at. parameters. Finally, at low Li content 0 < x < 0.04, an apparently homogeneous modification of the structure during both charge and discharge were found. Understanding the phase evolution and mechanism of structural response of graphite to lithiation under battery working conditions through in operando measurements may provide the information needed for the development of alternative higher performance electrode materials.
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    39. 39
      Li, M.-R.; Deng, Z.; Lapidus, S. H.; Stephens, P. W.; Segre, C. U.; Croft, M.; Paria Sena, R.; Hadermann, J.; Walker, D.; Greenblatt, M. Ba3(Cr0.97(1)Te0.03(1))2TeO9: in Search of Jahn-Teller Distorted Cr(II) Oxide. Inorg. Chem. 2016, 55, 10135– 10142,  DOI: 10.1021/acs.inorgchem.6b01047
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      39
      Ba3(Cr0.97(1)Te0.03(1))2TeO9: in Search of Jahn-Teller Distorted Cr(II) Oxide
      Li, Man-Rong; Deng, Zheng; Lapidus, Saul H.; Stephens, Peter W.; Segre, Carlo U.; Croft, Mark; Paria Sena, Robert; Hadermann, Joke; Walker, David; Greenblatt, Martha
      Inorganic Chemistry (2016), 55 (20), 10135-10142CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      A novel 6H-type hexagonal perovskite Ba3(Cr0.97(1)Te0.03(1))2TeO9 was prepd. at high pressure (6 GPa) and temp. (1773 K). Both TEM and synchrotron powder x-ray diffraction data demonstrate that Ba3(Cr0.97(1)Te0.03(1))2TeO9 crystallizes in P63/mmc with face-shared (Cr0.97(1)Te0.03(1))O6 octahedral pairs interconnected with TeO6 octahedra via corner-sharing. Structure anal. shows a mixed Cr2+/Cr3+ valence state with ∼10% Cr2+. The existence of Cr2+ in Ba3(Cr2+0.10(1)Cr3+0.87(1)Te6+0.03)2TeO9 is further evidenced by x-ray absorption near-edge spectroscopy. Magnetic properties measurements show a paramagnetic response down to 4 K and a small glassy-state curvature at low temp. The octahedral Cr2+O6 component is stabilized in an oxide material for the first time; the expected Jahn-Teller distortion of high-spin (d4) Cr2+ is not found, which is attributed to the small proportion of Cr2+ (∼10%) and the face-sharing arrangement of CrO6 octahedral pairs, which structurally disfavor axial distortion.
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    40. 40
      Tsirelson, V.; Stash, A.; Kohout, M.; Rosner, H.; Mori, H.; Sato, S.; Lee, S.; Yamamoto, A.; Tajima, S.; Grin, Y. Features of the Electron Density in Magnesium Diboride: Reconstruction from X-ray Diffraction Data and Comparison with TB-LMTO and FPLO Calculations. Acta Crystallogr., Sect. B: Struct. Sci. 2003, 59, 575– 583,  DOI: 10.1107/S0108768103012072
      [Crossref], [PubMed], [CAS], Google Scholar
      40
      Features of the electron density in magnesium diboride: reconstruction from x-ray diffraction data and comparison with TB-LMTO and FPLO calculations
      Tsirelson, V.; Stash, A.; Kohout, M.; Rosner, H.; Mori, H.; Sato, S.; Lee, S.; Yamamoto, A.; Tajima, S.; Grin, Yu.
      Acta Crystallographica, Section B: Structural Science (2003), B59 (5), 575-583CODEN: ASBSDK; ISSN:0108-7681. (Blackwell Publishing Ltd.)
      Features of the electron d. in MgB2 reconstructed from room-temp. single-crystal x-ray diffraction intensities using a multipole model are considered. Crystallog. data and at. coordinates are given. Topol. anal. of the total electron d. was applied to characterize the at. interactions in Mg diboride. The shared-type B-B interaction in the B-atom layer reveals that both σ and π components of the bonding are strong. A closed-shell-type weak B-B π interaction along the c axis of the unit cell also was found. The Mg-B closed-shell interaction exhibits a bond path that is significantly curved towards the vertical Mg-atom chain ([110] direction). The latter two facts reflect two sorts of bonding interactions along the [001] direction. Integration of the electron d. over the zero-flux at. basins reveals a charge transfer of ∼1.4(1) electrons from the Mg atoms to the B-atom network. The calcd. elec.-field gradients at nuclear positions are in good agreement with exptl. NMR values. The anharmonic displacement of the B atoms is also discussed. Calcns. of the electron d. by tight-binding linear muffin-tin orbital (TB-LMTO) and full-potential nonorthogonal local orbital (FPLO) methods confirm the results of the reconstruction from x-ray diffraction; for example, a charge transfer of 1.5 and 1.6 electrons, resp., was found.
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    41. 41
      Ievinǎ, A.; Straumanis, M.; Karlsons, K. Präzisionsbestimmung von Gitterkonstanten hygroskopischer Verbindungen (LiCl, NaBr). Z. Phys. Chem. 1938, 40B, 146– 150,  DOI: 10.1515/zpch-1938-4009
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    42. 42
      Walker, D.; Verma, P. K.; Cranswick, L. M.; Jones, R. L.; Clark, S. M.; Buhre, S. Halite-sylvite Thermoelasticity. Am. Mineral. 2004, 89, 204– 210,  DOI: 10.2138/am-2004-0124
      [Crossref], [CAS], Google Scholar
      42
      Halite-sylvite thermoelasticity
      Walker, David; Verma, Pramod K.; Cranswick, Lachlan M. D.; Jones, Raymond L.; Clark, Simon M.; Buhre, Stephan
      American Mineralogist (2004), 89 (1), 204-210CODEN: AMMIAY; ISSN:0003-004X. (Mineralogical Society of America)
      Unit-cell vols. of four single-phase intermediate halite-sylvite solid solns. have been measured to pressures and temps. of ∼28 kbar and ∼700 °C. Equation-of-state fitting of the data yields thermal expansion and compressibility as a function of compn. across the chloride series. The variation of the product α0·K0 is linear (ideal) in compn. between the accepted values for halite and sylvite. Taken sep., the individual values of α0 and K0 are not linear in compn. α0 Shows a max. near the consolute compn. (XNaCl = 0.64) that exceeds the value for either end-member. There is a corresponding min. in K0. The fact that the α0·K0 product is variable (and incidentally so well behaved as to be linear across the compn. series) reinforces the significance of the complementary maxima and min. in α0 and K0 (significantly, near the consolute compn.). These extrema in α0 and K0 provide an example of intermediate properties that do not follow simply from values for the end-members. Cell vols. across this series show small, well-behaved pos. excesses, consistent with K-Na substitution causing defects through lattice mismatches. Barrett and Wallace (1954) showed max. defect concns. in the consolute region. Defect-riddled, weakened structures in the consolute region are more easily compressed or more easily thermally expanded, providing an explanation for our obsd. α0 and K0 variations. These compliant, loosened lattices should resist diffusive transfer less than non-defective crystals and, hence, might be expected to show higher diffusivities. Tracer diffusion rates are predicted to peak across the consolute region as exchange diffusion rates drop to zero.
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    43. 43
      Ewais, E. M.; El-Amir, A. A.; Besisa, D. H.; Esmat, M.; El-Anadouli, B. E. Synthesis of Nanocrystalline MgO/MgAl2O4 Spinel Powders from Industrial Wastes. J. Alloys Compd. 2017, 691, 822– 833,  DOI: 10.1016/j.jallcom.2016.08.279
      [Crossref], [CAS], Google Scholar
      43
      Synthesis of nanocrystalline MgO/MgAl2O4 spinel powders from industrial wastes
      Ewais, Emad M. M.; El-Amir, Ahmed A. M.; Besisa, Dina H. A.; Esmat, Mohamed; El-Anadouli, Bahgat E. H.
      Journal of Alloys and Compounds (2017), 691 (), 822-833CODEN: JALCEU; ISSN:0925-8388. (Elsevier B.V.)
      This article reports a simple and cost-effective method to prep. ultrafine nanocryst. MgO/MgAl2O4 spinel (M-MA) powders from industrial wastes arising from aluminum and magnesium scraps. M-MA precursor powders were calcined at different temps. (650, 750, 850, 950, 1300-1500 °C). The calcined powders were characterized by XRD, FT-IR, DTA, FESEM, and HR-TEM. In particular, ultrafine MgO/MgAl2O3 powder was formed at a temp. of 650 °C with crystallite size of 4.8 nm and 7 nm, resp., as detd. by XRD. Optical properties of the M-MA spinel powders revealed that the optical reflectance is highly dependent on the calcination temp. A simple and cost-effective method to obtain ultrafine MgO/MgAl2O4 nanocryst. powders with expected unique properties was established. These synthesized spinel powders is a highly promised feedstock for refractory, ceramic and environmental applications.
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    44. 44
      Barrett, C. S. X-ray Study of the Alkali Metals at Low Temperatures. Acta Crystallogr. 1956, 9, 671– 677,  DOI: 10.1107/S0365110X56001790
      [Crossref], [CAS], Google Scholar
      44
      X-ray study of the alkali metals at low temperatures
      Barrett, C. S.
      Acta Crystallographica (1956), 9 (), 671-7CODEN: ACCRA9; ISSN:0365-110X.
      Using a spectrometer having provision for cold-working and x-raying specimens in a high vacuum at low temps., B. found that Na partially transforms on cooling (below 36°K.) or on deforming (below 51°K.) to a close-packed hexagonal structure with stacking faults, having a = 3.767, c = 6.154 A., c/a = 1.634 at 5°K., this coexisting with body-centered cubic Na of a = 4.225 A. The body-centered cubic form at 78°K. has a = 4.235 A. Severe cold-working at 5°K. transforms about half of the material to the hexagonal form; subsequent reversion to cubic starts on heating to 60-75°K. and is completed at 100-110°K., or at lower temps. if there has been no cold-working. Reversion can be aided by cold-working at 45-100°K. High-purity, severely deformed Na recrystallizes at 98°K. Patterns of Li that has been cooled can be interpreted similarly; they indicate a phase of close-packed hexagonal structure with parameters a = 3.111, c = 5.093 A., c/a = 1.637 (which differ from the earlier, tentative ones (C.A. 46, 2386f)). This phase coexists with the body-centered cubic phase of a = 3.491 A., at 78°K. Confirming the earlier work (loc. cit.) hexagonal Li is converted to face-centered cubic by cold-working. K, Rb, and Cs retain their body-centered cubic structure after cooling and cold-working at 5°K., with a = 5.225, 5.585, and 6.045 A., resp., at 5°K. and with a = 5.247, 5.605, and 6.067 A. at 78°K.
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    45. 45
      Streib, W. E.; Jordan, T. H.; Lipscomb, W. N. Single-crystal X-Ray Diffraction Study of β Nitrogen. J. Chem. Phys. 1962, 37, 2962– 2965,  DOI: 10.1063/1.1733125
      [Crossref], [CAS], Google Scholar
      45
      Single-crystal x-ray diffraction study of β-nitrogen
      Streib, William E.; Jordan, Truman H.; Lipscomb, William N.
      Journal of Chemical Physics (1962), 37 (), 2962-5CODEN: JCPSA6; ISSN:0021-9606.
      Three-dimensional x-ray-diffraction data were collected from single crystals of N at 50°K. The observed general condition, l = 2n for hh2h-l reflections is in agreement with the previously assigned space group, P63/mmc, and Z = 2. The data agree equally well with two nearly phys. indistinguishable models, in each of which the mol. centers form a h.c.p. lattice. In one, the mol. is precessing about the z axis passing through its center, at an angle of 54.5 ±2.5° between z and the N.sbd.N bond, while in the other the N atoms are statistically distributed among the 24-fold positions with the mol. axis again at an angle of 54.5° relative to z.
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    46. 46
      Schuch, A. F.; Mills, R. L. Crystal Structures of the Three Modifications of Nitrogen 14 and Nitrogen 15 at High Pressure. J. Chem. Phys. 1970, 52, 6000– 6008,  DOI: 10.1063/1.1672899
      [Crossref], [CAS], Google Scholar
      46
      Crystal structures of the three modifications of nitrogen-14 and nitrogen-15 at high pressure
      Schuch, Adam F.; Mills, Robert L.
      Journal of Chemical Physics (1970), 52 (12), 6000-8CODEN: JCPSA6; ISSN:0021-9606.
      X-ray diffraction photographs were used to det. that high-pressure γ N2 is tetragonal with Z = 2 in special position f of space group P42/mnm. At an av. pressure and temp. of 4015 atm and 20.5°K, resp., the unit cell dimensions are a = 3.957 and c = 5.109 Å, giving a molar vol. in good agreement with that from p-V-T measurements. The β N2 solid modification, which is contiguous with the melting curve, remains hexagonal up to at least 4125 atm and 49°K where the unit cell consts. are a = 3.861 and c = 6.265 Å. Over the pressure range investigated, the c/a ratio is very close to the ideal value for closest packing of hard spheres. The at. positions in hexagonal (β) N2, which are known to be highly disordered at low pressure, show no evidence of ordering at the highest-pressures studied. The third allotrope, α N2, is cubic at 3785 atm and 19.6°K with Z = 4 in a unit cell 5.433 Å on a side. Both the Pa3 and P213 space groups, which have been reported at zero pressure, appear to be spatially possible at limiting high pressures for α N2. However, from diffraction measurements on the cubic solid at all pressures, it was impossible to prove the existence of the P213 structure. The measurements on pure 30N2 show that this isotope also exists in the same solid modifications as 28N2 with cell dimensions that are similar for both isotopes. However, the transition to γ N2 at 20°K for the mass-15 isotope occurs 400 atm lower than that for the mass-14 isotope.
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    47. 47
      Francisco, E.; Casals-Sainz, J. L.; Rocha-Rinza, T.; Martín Pendás, Á. Partitioning the DFT Exchange-Correlation Energy in Line with the Interacting Quantum Atoms Approach. Theor. Chem. Acc. 2016, 135, 170,  DOI: 10.1007/s00214-016-1921-x
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    48. 48
      Gelessus, A.; Thiel, W.; Weber, W. Multipoles and Symmetry. J. Chem. Educ. 1995, 72, 505– 508,  DOI: 10.1021/ed072p505
      [ACS Full Text ACS Full Text], [CAS], Google Scholar
      48
      Multipoles and symmetry
      Gelessus, Achim; Theil, Walter; Weber, Wolfgang
      Journal of Chemical Education (1995), 72 (6), 505-8CODEN: JCEDA8; ISSN:0021-9584. (Division of Chemical Education of the American Chemical Society)
      Group theor. derivations are outlined and explicit results listed on the connection between multipoles and symmetry.
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    49. 49
      Bende, D. Chemical Bonding Models and Their Implications for Bonding-Property Relations in MgAgAs-Type and Related Compounds.; Ph.D. Dissertation; p 97.
      Google Scholar
      There is no corresponding record for this reference.
    50. 50
      Martín Pendás, Á.; Casals-Sainz, J. L.; Francisco, E. On Electrostatics, Covalency, and Chemical Dashes: Physical Interactions versus Chemical Bonds. Chem. - Eur. J. 2019, 25, 309– 314,  DOI: 10.1002/chem.201804160
      [Crossref], [PubMed], Google Scholar
      There is no corresponding record for this reference.
    51. 51
      Wagner, F. R.; Baranov, A. I.; Grin, Y.; Kohout, M. A Position-Space View on Chemical Bonding in Metal Diborides with AlB2 Type of Crystal Structure. Z. Anorg. Allg. Chem. 2013, 639, 2025– 2035,  DOI: 10.1002/zaac.201200523
      [Crossref], [CAS], Google Scholar
      51
      A Position-Space View on Chemical Bonding in Metal Diborides with AlB2 Type of Crystal Structure
      Wagner, Frank R.; Baranov, Alexey I.; Grin, Yuri; Kohout, Miroslav
      Zeitschrift fuer Anorganische und Allgemeine Chemie (2013), 639 (11), 2025-2035CODEN: ZAACAB; ISSN:0044-2313. (Wiley-VCH Verlag GmbH & Co. KGaA)
      On the basis of QTAIM and ELI-D partitioning of position space two- and three-center delocalization indexes were calcd. for fifteen MB2 phases with the crystal structure of AlB2 type. The bonding picture in main-group metal diborides is closest related to graphite with dominant covalent B-B bonding, albeit with lower effective bond order. For MgB2 an exceptionally large distant electron sharing was found. Transition-metal diborides display smaller effective bond orders B-B but higher effective bond orders TM-B and TM-TM than main-group metal diborides. The large chem. flexibility of this structure type is caused by counterbalancing effects of B-B bonding vs. M-B and M-M bonding. Different three-center fluctuation channels of bonds B-B are found for main-group and transition-metal diborides, namely B-B-B for the former and B-B-M for the latter. With the technique of ELI-D/QTAIM intersection the increasing importance of B2→4M bond charge fluctuations along each row of the periodic table can be recovered already at the topol. level of anal.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXjslKrsL0%253D&md5=83e2ac6f3e515d80935ad7ce5d85a8f9
    52. 52
      Barin, I. Thermochemical Data of Pure Substances; Wiley: 1995; Vol. I.
      [Crossref], Google Scholar
      There is no corresponding record for this reference.
    53. 53
      Grochala, W. Diamond: Electronic Ground State of Carbon at Temperatures Approaching 0 K. Angew. Chem., Int. Ed. 2014, 53, 3680– 3683,  DOI: 10.1002/anie.201400131
      [Crossref], [CAS], Google Scholar
      53
      Diamond: Electronic Ground State of Carbon at Temperatures Approaching 0 K
      Grochala, Wojciech
      Angewandte Chemie, International Edition (2014), 53 (14), 3680-3683CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)
      The relative stability of graphite and diamond is revisited with hybrid d. functional theory calcns. The electronic energy of diamond is computed to be more neg. by 1.1 kJ mol-1 than that of graphite at T = 0 K and in the absence of external pressure. Graphite gains thermodn. stability over diamond at 298 K only because of the differences in the zero-point energy, sp. heat, and entropy terms for both polymorphs.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2cXjtVGhu7w%253D&md5=5bab5280ef80da1881bd66667089c408
    54. 54
      Popov, I. V.; Görne, A. L.; Tchougréeff, A. L.; Dronskowski, R. Relative Stability of Diamond and Graphite as Seen Through Bonds and Hybridizations. Phys. Chem. Chem. Phys. 2019, 21, 10961– 10969,  DOI: 10.1039/C8CP07592A
      [Crossref], [PubMed], [CAS], Google Scholar
      54
      Relative stability of diamond and graphite as seen through bonds and hybridizations
      Popov, Ilya V.; Gorne, Arno L.; Tchougreeff, Andrei L.; Dronskowski, Richard
      Physical Chemistry Chemical Physics (2019), 21 (21), 10961-10969CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      The relative stability of the 2 most important forms of elemental carbon, diamond and graphite, is readdressed from a newly developed perspective as derived from historically well-known roots. Unlike other theor. studies mostly relying on numerical methods, we consider an anal. model to gain fundamental insight into the reasons for the quasi-degeneracy of diamond and graphite despite their extremely different covalent bonding patterns. We derive the allotropes' relative energies and provide a qual. picture predicting a quasi-degenerate electronic ground state for graphite (graphene) and diamond at zero temp. Our approach also gives numerical ests. of the energy difference and interat. sepns. in good agreement with exptl. data and recent results of hybrid DFT modeling, although obtained with a much smaller numerical but highly transparent effort. An attempt to extend this treatment to the lowest energy allotropes of silicon proves to be successful as well.
      >> More from SciFinder ®
      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXpvV2gsr8%253D&md5=b4455f0c481b1902ba27b55db857071b

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