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
CCDC reference: 2014342
Program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008).
Crystal data top
ClNa | V = −153.95 (3) Å3 |
Mr = 58.44 | Z = 4 |
a = 5.6035 (5) Å | F(000) = 112 |
b = 5.6035 (5) Å | Dx = 2.206 Mg m−3 |
c = 5.6035 (5) Å | Mo Kα radiation, λ = 0.71073 Å |
α = 90° | µ = 1.81 mm−1 |
β = 90° | T = 293 K |
γ = 90° | |
Data collection top
Radiation source: fine-focus sealed tube | Rint = 0.0000 |
β-filter monochromator | θmax = 64.9°, θmin = 7.3° |
1714 measured reflections | h = −4→8 |
1714 independent reflections | k = −14→14 |
1714 reflections with I > 2σ(I) | l = −14→14 |
Refinement top
Refinement on F2 | Primary atom site location: structure-invariant direct methods |
Least-squares matrix: full | Secondary 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 = 0.72 | Δρmax = 0.44 e Å−3 |
1714 reflections | Δρmin = −0.31 e Å−3 |
4 parameters | Extinction correction: SHELXL, Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4 |
0 restraints | Extinction 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 | x | y | z | Uiso*/Ueq | |
Na | 0.0000 | 0.0000 | 0.0000 | 0.00920 (2) | |
Cl | 0.5000 | 0.5000 | 0.5000 | 0.00757 (1) | |
Atomic displacement parameters (Å2) top | U11 | U22 | U33 | U12 | U13 | U23 |
Na | 0.00920 (2) | 0.00920 (2) | 0.00920 (2) | 0.000 | 0.000 | 0.000 |
Cl | 0.00757 (1) | 0.00757 (1) | 0.00757 (1) | 0.000 | 0.000 | 0.000 |
Geometric parameters (Å, º) top
Na—Cli | 2.8017 (3) | Na—Nai | 3.9623 (4) |
Na—Clii | 2.8017 (3) | Na—Naiii | 3.9623 (4) |
Na—Cliii | 2.8017 (3) | Na—Naix | 3.9623 (4) |
Na—Cliv | 2.8017 (3) | Cl—Nax | 2.8017 (3) |
Na—Clv | 2.8017 (3) | Cl—Navii | 2.8017 (3) |
Na—Clvi | 2.8017 (3) | Cl—Naxi | 2.8017 (3) |
Na—Nav | 3.9623 (4) | Cl—Naviii | 2.8017 (3) |
Na—Navii | 3.9623 (4) | Cl—Naxii | 2.8017 (3) |
Na—Naviii | 3.9623 (4) | Cl—Naix | 2.8017 (3) |
| | | |
Cli—Na—Clii | 180.0 | Clvi—Na—Nai | 45.0 |
Cli—Na—Cliii | 90.0 | Nav—Na—Nai | 60.0 |
Clii—Na—Cliii | 90.0 | Navii—Na—Nai | 120.0 |
Cli—Na—Cliv | 90.0 | Naviii—Na—Nai | 120.0 |
Clii—Na—Cliv | 90.0 | Cli—Na—Naiii | 135.0 |
Cliii—Na—Cliv | 180.0 | Clii—Na—Naiii | 45.0 |
Cli—Na—Clv | 90.0 | Cliii—Na—Naiii | 90.0 |
Clii—Na—Clv | 90.0 | Cliv—Na—Naiii | 90.0 |
Cliii—Na—Clv | 90.0 | Clv—Na—Naiii | 135.0 |
Cliv—Na—Clv | 90.0 | Clvi—Na—Naiii | 45.0 |
Cli—Na—Clvi | 90.0 | Nav—Na—Naiii | 60.0 |
Clii—Na—Clvi | 90.0 | Navii—Na—Naiii | 120.0 |
Cliii—Na—Clvi | 90.0 | Naviii—Na—Naiii | 180.0 |
Cliv—Na—Clvi | 90.0 | Nai—Na—Naiii | 60.0 |
Clv—Na—Clvi | 180.0 | Cli—Na—Naix | 90.0 |
Cli—Na—Nav | 135.0 | Clii—Na—Naix | 90.0 |
Clii—Na—Nav | 45.0 | Cliii—Na—Naix | 45.0 |
Cliii—Na—Nav | 135.0 | Cliv—Na—Naix | 135.0 |
Cliv—Na—Nav | 45.0 | Clv—Na—Naix | 45.0 |
Clv—Na—Nav | 90.0 | Clvi—Na—Naix | 135.0 |
Clvi—Na—Nav | 90.0 | Nav—Na—Naix | 120.0 |
Cli—Na—Navii | 45.0 | Navii—Na—Naix | 60.0 |
Clii—Na—Navii | 135.0 | Naviii—Na—Naix | 60.0 |
Cliii—Na—Navii | 45.0 | Nai—Na—Naix | 180.0 |
Cliv—Na—Navii | 135.0 | Naiii—Na—Naix | 120.0 |
Clv—Na—Navii | 90.0 | Nax—Cl—Navii | 90.0 |
Clvi—Na—Navii | 90.0 | Nax—Cl—Naxi | 90.0 |
Nav—Na—Navii | 180.0 | Navii—Cl—Naxi | 180.0 |
Cli—Na—Naviii | 45.0 | Nax—Cl—Naviii | 90.0 |
Clii—Na—Naviii | 135.0 | Navii—Cl—Naviii | 90.0 |
Cliii—Na—Naviii | 90.0 | Naxi—Cl—Naviii | 90.0 |
Cliv—Na—Naviii | 90.0 | Nax—Cl—Naxii | 90.0 |
Clv—Na—Naviii | 45.0 | Navii—Cl—Naxii | 90.0 |
Clvi—Na—Naviii | 135.0 | Naxi—Cl—Naxii | 90.0 |
Nav—Na—Naviii | 120.0 | Naviii—Cl—Naxii | 180.0 |
Navii—Na—Naviii | 60.0 | Nax—Cl—Naix | 180.0 |
Cli—Na—Nai | 90.0 | Navii—Cl—Naix | 90.0 |
Clii—Na—Nai | 90.0 | Naxi—Cl—Naix | 90.0 |
Cliii—Na—Nai | 135.0 | Naviii—Cl—Naix | 90.0 |
Cliv—Na—Nai | 45.0 | Naxii—Cl—Naix | 90.0 |
Clv—Na—Nai | 135.0 | | |
Symmetry codes: (i) x−1/2, y−1/2, z; (ii) x−1/2, y−1/2, z−1; (iii) x−1/2, y, z−1/2; (iv) x−1/2, y−1, z−1/2; (v) x, y−1/2, z−1/2; (vi) x−1, y−1/2, z−1/2; (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. |