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Quantum theory of atoms in molecules and the orbital-free density functional theory (DFT) are combined in this work to study the spatial distribution of electrostatic and quantum electronic forces acting in stable crystals. The electron distribution is determined by electrostatic electron mutual repulsion corrected for exchange and correlation, their attraction to nuclei and by electron kinetic energy. The latter defines the spread of permissible variations in the electron momentum resulting from the de Broglie relationship and uncertainty principle, as far as the limitations of Pauli principle and the presence of atomic nuclei and other electrons allow. All forces are expressed via kinetic and DFT potentials and then defined in terms of the experimental electron density and its derivatives; hence, this approach may be considered as orbital-free quantum crystallography. The net force acting on an electron in a crystal at equilibrium is zero everywhere, presenting a balance of the kinetic Fkin(r) and potential forces F(r). The critical points of both potentials are analyzed and they are recognized as the points at which forces Fkin(r) and F(r) individually are zero (the Lagrange points). The positions of these points in a crystal are described according to Wyckoff notations, while their types depend on the considered scalar field. It was found that F(r) force pushes electrons to the atomic nuclei, while the kinetic force Fkin(r) draws electrons from nuclei. This favors formation of electron concentration bridges between some of the nearest atoms. However, in a crystal at equilibrium, only kinetic potential vkin(r) and corresponding force exhibit the electronic shells and atomic-like zero-flux basins around the nuclear attractors. The force-field approach and quantum topological theory of atoms in molecules are compared and their distinctions are clarified.

Supporting information

cif

Crystallographic Information File (CIF) https://doi.org/10.1107/S2052520620009178/px5027sup1.cif
Contains datablock I

hkl

Structure factor file (CIF format) https://doi.org/10.1107/S2052520620009178/px5027Isup2.hkl
Contains datablock I

CCDC reference: 2014342

Computing details top

Program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008).

(I) top
Crystal data top
ClNaV = 153.95 (3) Å3
Mr = 58.44Z = 4
a = 5.6035 (5) ÅF(000) = 112
b = 5.6035 (5) ÅDx = 2.206 Mg m3
c = 5.6035 (5) ÅMo Kα radiation, λ = 0.71073 Å
α = 90°µ = 1.81 mm1
β = 90°T = 293 K
γ = 90°
Data collection top
Radiation source: fine-focus sealed tubeRint = 0.0000
β-filter monochromatorθmax = 64.9°, θmin = 7.3°
1714 measured reflectionsh = 48
1714 independent reflectionsk = 1414
1714 reflections with I > 2σ(I)l = 1414
Refinement top
Refinement on F2Primary atom site location: structure-invariant direct methods
Least-squares matrix: fullSecondary atom site location: difference Fourier map
R[F2 > 2σ(F2)] = 0.015 w = 1/[σ2(Fo2) + (0.0467P)2]
where P = (Fo2 + 2Fc2)/3
wR(F2) = 0.035(Δ/σ)max < 0.001
S = 0.72Δρmax = 0.44 e Å3
1714 reflectionsΔρmin = 0.31 e Å3
4 parametersExtinction correction: SHELXL, Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4
0 restraintsExtinction coefficient: 0.210 (5)
Special details top

Geometry. All esds (except the esd in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell esds are taken into account individually in the estimation of esds in distances, angles and torsion angles; correlations between esds in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell esds is used for estimating esds involving l.s. planes.

Refinement. Refinement of F2 against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F2, conventional R-factors R are based on F, with F set to zero for negative F2. The threshold expression of F2 > 2sigma(F2) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F2 are statistically about twice as large as those based on F, and R- factors based on ALL data will be even larger.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/Ueq
Na0.00000.00000.00000.00920 (2)
Cl0.50000.50000.50000.00757 (1)
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Na0.00920 (2)0.00920 (2)0.00920 (2)0.0000.0000.000
Cl0.00757 (1)0.00757 (1)0.00757 (1)0.0000.0000.000
Geometric parameters (Å, º) top
Na—Cli2.8017 (3)Na—Nai3.9623 (4)
Na—Clii2.8017 (3)Na—Naiii3.9623 (4)
Na—Cliii2.8017 (3)Na—Naix3.9623 (4)
Na—Cliv2.8017 (3)Cl—Nax2.8017 (3)
Na—Clv2.8017 (3)Cl—Navii2.8017 (3)
Na—Clvi2.8017 (3)Cl—Naxi2.8017 (3)
Na—Nav3.9623 (4)Cl—Naviii2.8017 (3)
Na—Navii3.9623 (4)Cl—Naxii2.8017 (3)
Na—Naviii3.9623 (4)Cl—Naix2.8017 (3)
Cli—Na—Clii180.0Clvi—Na—Nai45.0
Cli—Na—Cliii90.0Nav—Na—Nai60.0
Clii—Na—Cliii90.0Navii—Na—Nai120.0
Cli—Na—Cliv90.0Naviii—Na—Nai120.0
Clii—Na—Cliv90.0Cli—Na—Naiii135.0
Cliii—Na—Cliv180.0Clii—Na—Naiii45.0
Cli—Na—Clv90.0Cliii—Na—Naiii90.0
Clii—Na—Clv90.0Cliv—Na—Naiii90.0
Cliii—Na—Clv90.0Clv—Na—Naiii135.0
Cliv—Na—Clv90.0Clvi—Na—Naiii45.0
Cli—Na—Clvi90.0Nav—Na—Naiii60.0
Clii—Na—Clvi90.0Navii—Na—Naiii120.0
Cliii—Na—Clvi90.0Naviii—Na—Naiii180.0
Cliv—Na—Clvi90.0Nai—Na—Naiii60.0
Clv—Na—Clvi180.0Cli—Na—Naix90.0
Cli—Na—Nav135.0Clii—Na—Naix90.0
Clii—Na—Nav45.0Cliii—Na—Naix45.0
Cliii—Na—Nav135.0Cliv—Na—Naix135.0
Cliv—Na—Nav45.0Clv—Na—Naix45.0
Clv—Na—Nav90.0Clvi—Na—Naix135.0
Clvi—Na—Nav90.0Nav—Na—Naix120.0
Cli—Na—Navii45.0Navii—Na—Naix60.0
Clii—Na—Navii135.0Naviii—Na—Naix60.0
Cliii—Na—Navii45.0Nai—Na—Naix180.0
Cliv—Na—Navii135.0Naiii—Na—Naix120.0
Clv—Na—Navii90.0Nax—Cl—Navii90.0
Clvi—Na—Navii90.0Nax—Cl—Naxi90.0
Nav—Na—Navii180.0Navii—Cl—Naxi180.0
Cli—Na—Naviii45.0Nax—Cl—Naviii90.0
Clii—Na—Naviii135.0Navii—Cl—Naviii90.0
Cliii—Na—Naviii90.0Naxi—Cl—Naviii90.0
Cliv—Na—Naviii90.0Nax—Cl—Naxii90.0
Clv—Na—Naviii45.0Navii—Cl—Naxii90.0
Clvi—Na—Naviii135.0Naxi—Cl—Naxii90.0
Nav—Na—Naviii120.0Naviii—Cl—Naxii180.0
Navii—Na—Naviii60.0Nax—Cl—Naix180.0
Cli—Na—Nai90.0Navii—Cl—Naix90.0
Clii—Na—Nai90.0Naxi—Cl—Naix90.0
Cliii—Na—Nai135.0Naviii—Cl—Naix90.0
Cliv—Na—Nai45.0Naxii—Cl—Naix90.0
Clv—Na—Nai135.0
Symmetry codes: (i) x1/2, y1/2, z; (ii) x1/2, y1/2, z1; (iii) x1/2, y, z1/2; (iv) x1/2, y1, z1/2; (v) x, y1/2, z1/2; (vi) x1, y1/2, z1/2; (vii) x, y+1/2, z+1/2; (viii) x+1/2, y, z+1/2; (ix) x+1/2, y+1/2, z; (x) x+1/2, y+1/2, z+1; (xi) x+1, y+1/2, z+1/2; (xii) x+1/2, y+1, z+1/2.
 

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