Orbital-free quantum crystallography: view on forces in crystals

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 F_kin(r) and potential forces F(r). The critical points of both potentials are analyzed and they are recognized as the points at which forces F_kin(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 F_kin(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 v_kin(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.

Computing details

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

(I)

Crystal data

ClNa	V = −153.95 (3) Å^3
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

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

Refinement on F^2	Primary atom site location: structure-invariant direct methods
Least-squares matrix: full	Secondary atom site location: difference Fourier map
R[F^2 > 2σ(F^2)] = 0.015	w = 1/[σ^2(Fo^2) + (0.0467P)^2] where P = (Fo^2 + 2Fc^2)/3
wR(F^2) = 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+0.001xFc^2λ^3/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 0.210 (5)

Special details

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 (Ų)

	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 (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
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)

Na—Cl^i	2.8017 (3)	Na—Na^i	3.9623 (4)
Na—Cl^ii	2.8017 (3)	Na—Na^iii	3.9623 (4)
Na—Cl^iii	2.8017 (3)	Na—Na^ix	3.9623 (4)
Na—Cl^iv	2.8017 (3)	Cl—Na^x	2.8017 (3)
Na—Cl^v	2.8017 (3)	Cl—Na^vii	2.8017 (3)
Na—Cl^vi	2.8017 (3)	Cl—Na^xi	2.8017 (3)
Na—Na^v	3.9623 (4)	Cl—Na^viii	2.8017 (3)
Na—Na^vii	3.9623 (4)	Cl—Na^xii	2.8017 (3)
Na—Na^viii	3.9623 (4)	Cl—Na^ix	2.8017 (3)
Cl^i—Na—Cl^ii	180.0	Cl^vi—Na—Na^i	45.0
Cl^i—Na—Cl^iii	90.0	Na^v—Na—Na^i	60.0
Cl^ii—Na—Cl^iii	90.0	Na^vii—Na—Na^i	120.0
Cl^i—Na—Cl^iv	90.0	Na^viii—Na—Na^i	120.0
Cl^ii—Na—Cl^iv	90.0	Cl^i—Na—Na^iii	135.0
Cl^iii—Na—Cl^iv	180.0	Cl^ii—Na—Na^iii	45.0
Cl^i—Na—Cl^v	90.0	Cl^iii—Na—Na^iii	90.0
Cl^ii—Na—Cl^v	90.0	Cl^iv—Na—Na^iii	90.0
Cl^iii—Na—Cl^v	90.0	Cl^v—Na—Na^iii	135.0
Cl^iv—Na—Cl^v	90.0	Cl^vi—Na—Na^iii	45.0
Cl^i—Na—Cl^vi	90.0	Na^v—Na—Na^iii	60.0
Cl^ii—Na—Cl^vi	90.0	Na^vii—Na—Na^iii	120.0
Cl^iii—Na—Cl^vi	90.0	Na^viii—Na—Na^iii	180.0
Cl^iv—Na—Cl^vi	90.0	Na^i—Na—Na^iii	60.0
Cl^v—Na—Cl^vi	180.0	Cl^i—Na—Na^ix	90.0
Cl^i—Na—Na^v	135.0	Cl^ii—Na—Na^ix	90.0
Cl^ii—Na—Na^v	45.0	Cl^iii—Na—Na^ix	45.0
Cl^iii—Na—Na^v	135.0	Cl^iv—Na—Na^ix	135.0
Cl^iv—Na—Na^v	45.0	Cl^v—Na—Na^ix	45.0
Cl^v—Na—Na^v	90.0	Cl^vi—Na—Na^ix	135.0
Cl^vi—Na—Na^v	90.0	Na^v—Na—Na^ix	120.0
Cl^i—Na—Na^vii	45.0	Na^vii—Na—Na^ix	60.0
Cl^ii—Na—Na^vii	135.0	Na^viii—Na—Na^ix	60.0
Cl^iii—Na—Na^vii	45.0	Na^i—Na—Na^ix	180.0
Cl^iv—Na—Na^vii	135.0	Na^iii—Na—Na^ix	120.0
Cl^v—Na—Na^vii	90.0	Na^x—Cl—Na^vii	90.0
Cl^vi—Na—Na^vii	90.0	Na^x—Cl—Na^xi	90.0
Na^v—Na—Na^vii	180.0	Na^vii—Cl—Na^xi	180.0
Cl^i—Na—Na^viii	45.0	Na^x—Cl—Na^viii	90.0
Cl^ii—Na—Na^viii	135.0	Na^vii—Cl—Na^viii	90.0
Cl^iii—Na—Na^viii	90.0	Na^xi—Cl—Na^viii	90.0
Cl^iv—Na—Na^viii	90.0	Na^x—Cl—Na^xii	90.0
Cl^v—Na—Na^viii	45.0	Na^vii—Cl—Na^xii	90.0
Cl^vi—Na—Na^viii	135.0	Na^xi—Cl—Na^xii	90.0
Na^v—Na—Na^viii	120.0	Na^viii—Cl—Na^xii	180.0
Na^vii—Na—Na^viii	60.0	Na^x—Cl—Na^ix	180.0
Cl^i—Na—Na^i	90.0	Na^vii—Cl—Na^ix	90.0
Cl^ii—Na—Na^i	90.0	Na^xi—Cl—Na^ix	90.0
Cl^iii—Na—Na^i	135.0	Na^viii—Cl—Na^ix	90.0
Cl^iv—Na—Na^i	45.0	Na^xii—Cl—Na^ix	90.0
Cl^v—Na—Na^i	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.