l-Cysteine-I at 30 K

Moggach, S.A.; Clark, S.J.; Parsons, S.

doi:10.1107/S1600536805023688

organic compounds

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ISSN: 2056-9890

Volume 61| Part 8| August 2005| Pages o2739-o2742

https://doi.org/10.1107/S1600536805023688

L-Cysteine-I at 30 K

Stephen A. Moggach,^a Stewart J. Clark ^b and Simon Parsons ^a ^*

^aSchool of Chemistry, The University of Edinburgh, King's Buildings, West Mains Road, Edinburgh EH9 3JJ, Scotland, and ^bDepartment of Physics, The University of Durham, South Road, Durham DH1 3LE, England
^*Correspondence e-mail: s.parsons@ed.ac.uk

(Received 13 July 2005; accepted 25 July 2005; online 30 July 2005)

The crystal structure of the orthorhombic phase I of L-cysteine, C₃H₇NO₂S, has been determined at 30 K. The molecule adopts a gauche⁺ conformation and the structure consists of zwitterions connected into sinusoidal layers by N—H⋯O hydrogen bonds. Further N—H⋯O hydrogen bonds connect the structure into a three-dimensional array. Under ambient conditions, the thiol H atom is disordered in such a way as to form intermolecular S—H⋯S and S—H⋯O hydrogen bonds. At 30 K the structure is ordered with retention of the S—H⋯S contacts [S⋯S = 3.8489 (4) Å, S—H⋯S = 2.66 (3) Å and S—H⋯S = 150.8 (16)°].

Comment

The amino acid L-cysteine (Fig. 1) is known to crystallize in two polymorphic forms, viz. an orthorhombic phase (P2₁2₁2₁, Z′ = 1) and a monoclinic phase (P2₁, Z′ = 2). We refer to these as L-cysteine-I and L-cysteine-II, respectively. The crystal structure of L-cysteine-I was determined by Kerr & Ashmore (1973 ) by X-ray diffraction and then again by Kerr et al. (1975 ) by neutron diffraction. Both of these studies were at ambient temperature. L-Cysteine-II was characterized at ambient temperature by Harding & Long (1968 ) and later by Görbitz & Dalhus (1996 ) at 120 K; both of these determinations employed X-ray diffraction. Two new polymorphs (one orthorhombic and the other monoclinic) have recently been characterized by us at elevated pressure (Moggach et al., 2005 ).

Both L-cysteine-I and L-cysteine-II crystallize with the molecule as its zwitterionic tautomer (Fig. 1). In principle, the N1—C2—C1—S1 torsion angle (χ₁) can adopt values of ca 60° (the gauche+ conformer, g⁺), −60° (g⁻) and 180° (trans or t). In L-cysteine-I at 30 K, this parameter is 70.66 (9)°, which compares with a value of 65.3° as determined by X-ray diffraction at room temperature. This is consistent with the finding of Görbitz (1990 ) that in small molecules there is a strong preference for the g⁺ conformation.

Intermolecular interactions in both forms of L-cysteine are dominated by N—H⋯O hydrogen bonds. In L-cysteine-I, the shortest of these, N1—H7⋯O2, lies along c to form a C(5) chain (Bernstein et al., 1995 ). The second shortest hydrogen bond, N1—H5⋯O1, links molecules into C(5) chains, which run along a. The combination of these two C(5) chains yields a layer composed of R₄⁴(16) ring motifs (Fig. 2). The layer is parallel to the ac plane, though it is not planar, having a sinusoidal appearance when viewed in projection down c. The last of the N—H⋯O interactions, N1—H6⋯O2, acts to link the layers together along the b direction. Pairs of N1—H6⋯O2 contacts form R₃²(9) ring motifs (Fig. 3).

Although the crystal structures of both polymorphs of L-cysteine are dominated by N—H⋯O hydrogen bonding, the thiol group is also capable of forming hydrogen bonds. Hydrogen bonds where Csp³—SH groups act as donors are very weak, leading to red shifts of only ca 20 cm⁻¹ in vibrational spectra (Desiraju & Steiner, 1999 ). This weakness often results in disorder in the H-atom position, and thus geometric data for `well behaved' S—H⋯X interactions are rather sparse.

The structure of L-cysteine-II is unusual in thiol chemistry because it contains ordered thiol groups; intermolecular S—H⋯S and S—H⋯O hydrogen bonds are formed by the two molecules that make up the asymmetric unit. The H⋯S and S⋯S distances in L-cysteine-II are 2.78 (4) and 4.080 (1) Å, respectively (Görbitz & Dalhus, 1996). These are similar to other systems, e.g. hydrogen sulfide (2.68–2.74 and 3.985–4.027 Å; Cockcroft & Fitch, 1990 ) and hexakis(mercaptomethyl)benzene (ca 2.8 and 4.0 Å; Mallinson et al., 1997 ) quoted in a survey by Desiraju & Steiner (1999).

The thiol group is disordered in the crystal structure of L-cysteine-I at room temperature. Different components of the disorder lead to the formation of S—H⋯O and S—H⋯S hydrogen bonds, but the latter is marginally favoured. This result is consistent with the results of DFT calculations, which place the S—H⋯S structure 4.11 kJ mol⁻¹ lower in energy. This energy difference is small, and it suggests that the disorder may be frozen out at low enough temperatures.

This proves to be the case, and at 30 K the thiol H atom in L-cysteine-I is ordered (Fig. 4), forming an S—H⋯S hydrogen bond, with parameters given in Table 2. The geometrical parameters of this interaction are S⋯S = 3.8489 (4) Å, H⋯S = 2.66 (3) Å and S—H⋯S = 150.8 (16)°. This bond is shorter than that in L-cysteine-II and the other systems cited above. The S—H⋯S interactions form an infinite hydrogen-bonded chain which zigzags along c. These interactions support the R₃²(9) ring motifs in connecting the sinusoidal layers formed by R₄⁴(16) ring motifs (Fig. 3).

At 0.06 Å², the isotropic displacement parameter of the thiol H atom is high relative to those of the other atoms in the system (0.008 − 0.017 Å²). This suggests that the thiol H atom is still quite mobile at 30 K, and its behaviour at still lower temperatures would be of considerable interest.

Figure 1
The molecular structure of L-cysteine as observed in the crystal structure of orthorhombic L-cysteine at 30 K and ambient pressure. The displacement ellipsoids are drawn at the 50% probability level, and the H atoms as circles of arbitrary radius.

Figure 2
Hydrogen-bonded layers in L-cysteine-I via N1—H7⋯O2^iv and N1—H5⋯O1ⁱⁱ interactions. These build R₄⁴(16) rings. This view is along b. See Table 2

for symmetry codes.

Figure 3
The layers shown in Fig. 2

are connected by N1—H6⋯O2ⁱⁱⁱ hydrogen bonds. The hydrogen bonds illustrated in Fig. 2

are shown in orange; the hydrogen bonds that connect the layers are shown in black. This view is along c.

Figure 4
Difference map showing location of the thiol H atom. Contours are drawn at 0.4 (green), 0.6 (blue) and 0.8 eÅ⁻³ (red).

Experimental

Crystals of orthorhombic L-cysteine-I were obtained from Sigma (99%, catalogue number G, 1002) and used as received.

Crystal data

C₃H₇NO₂S
M_r = 121.16
Orthorhombic, P 2₁ 2₁ 2₁
a = 8.1435 (4) Å
b = 11.9365 (5) Å
c = 5.4158 (3) Å
V = 526.44 (4) Å³
Z = 4
D_x = 1.529 Mg m⁻³
Mo Kα radiation
Cell parameters from 4210 reflections
θ = 3.0–31.0°
μ = 0.50 mm⁻¹
T = 30 K
Block, colourless
0.40 × 0.20 × 0.17 mm

Data collection

Bruker–Nonius APEX CCD area-detector diffractometer
ω scans
Absorption correction: multi-scan(SADABS; Sheldrick, 2004 )T_min = 0.775, T_max = 0.920
4686 measured reflections
1516 independent reflections
1474 reflections with I > 2σ(I)
R_int = 0.021
θ_max = 30.8°
h = −11 → 9
k = −17 → 17
l = −6 → 7

Refinement

Refinement on F²
R[F² > 2σ(F²)] = 0.017
wR(F²) = 0.047
S = 1.03
1514 reflections
93 parameters
All H-atom parameters refined
w = 1/[σ²(F²) + ( 0.02P)² + 0.04P] where P = [max(F_o²,0) + 2F_c²]/3
(Δ/σ)_max = 0.001
Δρ_max = 0.27 e Å⁻³
Δρ_min = −0.18 e Å⁻³
Absolute structure: Flack (1983 ), 592 Friedel pairs
Flack parameter: −0.02 (5)

Table 1
Selected geometric parameters (Å, °)

S1—C1	1.8237 (10)
S1—H1	1.31 (3)
C1—C2	1.5223 (13)
C2—C3	1.5359 (13)
C2—N1	1.4843 (12)
C3—O1	1.2444 (12)
C3—O2	1.2623 (11)

S1—C1—C2	113.91 (6)
C1—C2—C3	111.11 (8)
C1—C2—N1	110.73 (7)
C3—C2—N1	110.96 (8)
C2—C3—O1	116.98 (8)
C2—C3—O2	116.87 (8)
O1—C3—O2	126.14 (10)

Table 2
Hydrogen-bond geometry (Å, °)

D—H⋯A	D—H	H⋯A	D⋯A	D—H⋯A
S1—H1⋯S1ⁱ	1.30 (3)	2.66 (3)	3.8489 (4)	151 (2)
N1—H5⋯O1ⁱⁱ	0.83 (2)	1.97 (2)	2.7694 (11)	162 (2)
N1—H6⋯O2ⁱⁱⁱ	0.87 (1)	2.12 (1)	2.9451 (11)	159 (2)
N1—H7⋯O2^iv	0.89 (2)	1.87 (2)	2.7546 (11)	170 (1)
C1—H2⋯O1^iv	0.96 (2)	2.56 (2)	3.2748 (13)	132 (1)
C2—H4⋯S1^v	0.93 (1)	2.85 (1)	3.7770 (9)	175 (1)
Symmetry codes: (i) $[-x+{\script{1\over 2}}, -y+2, +z-{\script{1\over 2}}]$ ; (ii) $[+x+{\script{1\over 2}}, -y+{\script{3\over 2}}, -z+1]$ ; (iii) $[-x+{\script{3\over 2}}, -y+2, +z+{\script{1\over 2}}]$ ; (iv) x, y, z+1; (v) $[-x+1, +y-{\script{1\over 2}}, -z+{\script{3\over 2}}]$ .

H atoms were located in a difference map. The aim of this structure determination was to determine the position of the H atom attached to S1, and therefore all H atoms were refined independently with isotropic displacement parameters. Two reflections were omitted, one as an outlier, the other because it was obscured by the beam stop.

The ab initio calculations were performed with the plane-wave pseudopotential implementation of density functional theory (DFT) using the CASTEP code (Segall et al., 2002 ). Plane-wave basis sets have many benefits compared with conventionally used quantum chemistry basis sets; in particular, there exists a simple parameter, the cutoff energy, to determine the completeness of the basis. This gives us confidence that the wavefunction can describe any properties without bias towards any other particular result (Clark et al., 1998 ). In our calculations, the many-body exchange and correlation interactions are described using the generalized gradient approximation (Perdew & Wang, 1992 ). Such calculations are capable of giving accurate and reliable structural and electronic information. Ultrasoft pseudopotentials (Vanderbilt, 1990 ) are used to describe the electron-ion interactions. A cut-off energy of 380 eV is used, which converged the total energy of the system to 1.0 meV atom⁻¹. The Monkhorst–Pack k-point sampling scheme (Monkhorst & Pack, 1976 ) was used to perform the integrations in k-space over the first Brillouin zone with the grids for each cell chosen to be dense enough to also converge the total energy to 1.0 meV atom⁻¹. For each structure considered, the geometry (atomic positions and unit-cell parameters) was optimized using a conjugate gradient algorithm. The tolerances used give energy differences between structures accurate to better than 1.0 meV.

Data collection: APEX (Bruker, 2004 ); cell refinement: SAINT (Bruker, 2004); data reduction: SAINT; program(s) used to solve structure: SIR92 (Altomare et al., 1994 ); program(s) used to refine structure: CRYSTALS (Betteridge et al., 2003 ); molecular graphics: CAMERON (Watkin et al., 1996 ), DIAMOND (Crystal Impact, 2004 ), MERCURY (Bruno et al., 2002 ; Taylor & Macrae, 2001 ), MCE Fourier Map Viewer (Hušák & Kratochvila, 2003 ) and SHELXTL (Sheldrick, 2001 ); software used to prepare material for publication: CRYSTALS, and PLATON (Spek, 2003 ) as incorporated into WinGX (Farrugia, 1999 ).

Supporting information

Crystal structure: contains datablocks global, I. DOI: https://doi.org/10.1107/S1600536805023688/ya6256sup1.cif

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S1600536805023688/ya6256Isup2.hkl

Computing details top

Data collection: APEX (Bruker, 2004); cell refinement: SAINT; data reduction: SAINT (Bruker, 2004); program(s) used to solve structure: SIR92 (Altomare et al., 1994); program(s) used to refine structure: CRYSTALS (Betteridge et al., 2003); molecular graphics: CAMERON (Watkin et al., 1996), DIAMOND (Crystal Impact, 2004), Mercury (Bruno et al., 2002; Taylor & Macrae, 2001), MCE Fourier Map Viewer (Husak & Kratochvila, 2003) and SHELXTL (Sheldrick, 2001); software used to prepare material for publication: CRYSTALS, and PLATON (Spek, 2003) as incorporated into WinGX (Farrugia, 1999).

L-cysteine top

Crystal data top

C₃H₇NO₂S	F(000) = 256
M_r = 121.16	D_x = 1.529 Mg m⁻³
Orthorhombic, P2₁2₁2₁	Mo Kα radiation, λ = 0.71073 Å
Hall symbol: P 2ac 2ab	Cell parameters from 4210 reflections
a = 8.1435 (4) Å	θ = 3.0–31.0°
b = 11.9365 (5) Å	µ = 0.50 mm⁻¹
c = 5.4158 (3) Å	T = 30 K
V = 526.44 (4) Å³	Block, colourless
Z = 4	0.40 × 0.20 × 0.17 mm

Data collection top

Bruker–Nonius APEX CCD area-detector diffractometer	1474 reflections with I > 2σ(I)
Graphite monochromator	R_int = 0.021
ω scans	θ_max = 30.8°, θ_min = 3.0°
Absorption correction: multi-scan (SADABS; Sheldrick, 2004)	h = −11→9
T_min = 0.775, T_max = 0.920	k = −17→17
4686 measured reflections	l = −6→7
1516 independent reflections

Refinement top

Refinement on F²	Secondary atom site location: difference Fourier map
Least-squares matrix: full	Hydrogen site location: difference Fourier map
R[F² > 2σ(F²)] = 0.017	All H-atom parameters refined
wR(F²) = 0.047	w = 1/[σ²(F²) + ( 0.02P)² + 0.04P] where P = [max(F_o²,0) + 2F_c²]/3
S = 1.03	(Δ/σ)_max = 0.001
1514 reflections	Δρ_max = 0.27 e Å⁻³
93 parameters	Δρ_min = −0.18 e Å⁻³
0 restraints	Absolute structure: Flack (1983), 592 Friedel pairs
Primary atom site location: structure-invariant direct methods	Absolute structure parameter: −0.02 (5)

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
S1	0.41463 (3)	1.022635 (19)	0.60963 (5)	0.0087
C1	0.43549 (12)	0.88319 (8)	0.74364 (18)	0.0083
C2	0.59208 (12)	0.82254 (7)	0.66819 (17)	0.0065
C3	0.61052 (11)	0.81952 (7)	0.38602 (19)	0.0068
N1	0.73776 (10)	0.87498 (7)	0.78553 (16)	0.0071
O1	0.51302 (9)	0.75892 (6)	0.26998 (15)	0.0111
O2	0.72196 (8)	0.87985 (6)	0.29348 (13)	0.0082
H1	0.345 (3)	0.989 (2)	0.406 (5)	0.062 (6)*
H2	0.4310 (17)	0.8914 (12)	0.920 (3)	0.017 (4)*
H3	0.3379 (17)	0.8389 (11)	0.696 (3)	0.008 (3)*
H4	0.5824 (17)	0.7491 (11)	0.725 (3)	0.005 (3)*
H5	0.8201 (19)	0.8409 (12)	0.738 (3)	0.015 (4)*
H6	0.7427 (16)	0.9454 (11)	0.748 (3)	0.008 (3)*
H7	0.7250 (16)	0.8690 (12)	0.949 (3)	0.008 (3)*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
S1	0.00811 (10)	0.00808 (10)	0.00992 (11)	0.00130 (8)	−0.00065 (8)	0.00039 (8)
C1	0.0078 (4)	0.0095 (4)	0.0076 (4)	0.0010 (3)	0.0019 (3)	0.0011 (3)
C2	0.0065 (4)	0.0071 (3)	0.0060 (4)	−0.0006 (3)	0.0004 (3)	−0.0001 (3)
C3	0.0077 (4)	0.0070 (3)	0.0057 (4)	0.0023 (3)	−0.0001 (3)	0.0004 (3)
N1	0.0076 (3)	0.0091 (3)	0.0045 (4)	0.0004 (3)	−0.0001 (3)	0.0000 (3)
O1	0.0113 (3)	0.0135 (3)	0.0086 (3)	−0.0042 (3)	−0.0010 (3)	−0.0014 (3)
O2	0.0093 (3)	0.0100 (3)	0.0051 (3)	−0.0018 (2)	0.0004 (3)	0.0002 (2)

Geometric parameters (Å, º) top

S1—C1	1.8237 (10)	C2—H4	0.933 (13)
S1—H1	1.31 (3)	C3—O1	1.2444 (12)
C1—C2	1.5223 (13)	C3—O2	1.2623 (11)
C1—H2	0.961 (16)	N1—H5	0.826 (16)
C1—H3	0.989 (14)	N1—H6	0.866 (14)
C2—C3	1.5359 (13)	N1—H7	0.894 (16)
C2—N1	1.4843 (12)

C1—S1—H1	95.3 (10)	C3—C2—H4	108.5 (9)
S1—C1—C2	113.91 (6)	N1—C2—H4	108.7 (8)
S1—C1—H2	107.4 (9)	C2—C3—O1	116.98 (8)
C2—C1—H2	110.3 (8)	C2—C3—O2	116.87 (8)
S1—C1—H3	108.0 (8)	O1—C3—O2	126.14 (10)
C2—C1—H3	110.4 (8)	C2—N1—H5	107.9 (11)
H2—C1—H3	106.5 (12)	C2—N1—H6	110.2 (9)
C1—C2—C3	111.11 (8)	H5—N1—H6	111.5 (13)
C1—C2—N1	110.73 (7)	C2—N1—H7	107.3 (9)
C3—C2—N1	110.96 (8)	H5—N1—H7	111.4 (14)
C1—C2—H4	106.7 (9)	H6—N1—H7	108.5 (13)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
S1—H1···S1ⁱ	1.30 (3)	2.66 (3)	3.8489 (4)	150.8 (16)
N1—H5···O1ⁱⁱ	0.825 (15)	1.972 (15)	2.7694 (11)	162.1 (16)
N1—H6···O2ⁱⁱⁱ	0.866 (13)	2.120 (13)	2.9451 (11)	159.1 (15)
N1—H7···O2^iv	0.894 (16)	1.870 (16)	2.7546 (11)	169.6 (13)
C1—H2···O1^iv	0.961 (16)	2.557 (15)	3.2748 (13)	131.6 (11)
C2—H4···S1^v	0.932 (14)	2.848 (13)	3.7770 (9)	174.6 (12)

Symmetry codes: (i) −x+1/2, −y+2, z−1/2; (ii) x+1/2, −y+3/2, −z+1; (iii) −x+3/2, −y+2, z+1/2; (iv) x, y, z+1; (v) −x+1, y−1/2, −z+3/2.

Acknowledgements

We thank the EPSRC for funding, and Dr A. Goeta (University of Durham, England) for his helpful experimental advice.

References

Altomare, A., Cascarano, G., Giacovazzo, G., Guagliardi, A., Burla, M. C., Polidori, G. & Camalli, M. (1994). J. Appl. Cryst. 27, 435. CrossRef Web of Science IUCr Journals Google Scholar
Bernstein, J., Davis, R. E., Shimoni, L. & Chang, N.-L. (1995). Angew. Chem. Int. Ed. Engl. 34, 1555–1573. CrossRef CAS Web of Science Google Scholar
Betteridge, P. W., Carruthers, J. R., Cooper, R. I., Prout, K. & Watkin, D. J. (2003). J. Appl. Cryst. 36, 1487. Web of Science CrossRef IUCr Journals Google Scholar
Bruker (2004). APEX and SAINT (Version V7.12A). Bruker AXS Inc., Madison, Wisconsin, USA. Google Scholar
Bruno, I. J., Cole, J. C., Edgington, P. R., Kessler, M., Macrae, C. F., McCabe, P., Pearson, J. & Taylor, R. (2002). Acta Cryst. B58, 389–397. Web of Science CSD CrossRef CAS IUCr Journals Google Scholar
Clark, S. J., Ackland, G. J. & Crain, J. (1998). Europhys. Lett. 44, 578–584. Web of Science CrossRef CAS Google Scholar
Cockcroft, J. K. & Fitch, A. N. (1990). Z. Kristallogr. 193, 1–19. CrossRef CAS Web of Science Google Scholar
Crystal Impact (2004). DIAMOND. Version 3.0. Crystal Impact, Postfach 1251, 53002 Bonn, Germany. https://www.crystalimpact.com/diamond. Google Scholar
Desiraju, G. R. & Steiner, T. (1999). The Weak Hydrogen Bond in Structural Chemistry and Biology. Oxford University Press. Google Scholar
Farrugia, L. J. (1999). J. Appl. Cryst. 32, 837–838. CrossRef CAS IUCr Journals Google Scholar
Flack, H. D. (1983). Acta Cryst. A39, 876–881. CrossRef CAS Web of Science IUCr Journals Google Scholar
Görbitz, C. H. (1990). Acta Chem. Scand. 44, 584–590. Google Scholar
Görbitz, C. H. & Dalhus, B. (1996). Acta Cryst. C52, 1756–1759. CSD CrossRef Web of Science IUCr Journals Google Scholar
Harding, M. M. & Long, H. A. (1968). Acta Cryst. B24, 1096–1102. CSD CrossRef CAS IUCr Journals Web of Science Google Scholar
Hušák, M. & Kratochvila, B. (2003). J. Appl. Cryst. 36, 1104. CrossRef IUCr Journals Google Scholar
Kerr, K. A. & Ashmore, J. P. (1973). Acta Cryst. B29, 2124–2127. CSD CrossRef CAS IUCr Journals Web of Science Google Scholar
Kerr, K. A., Ashmore, J. P. & Koetzle, T. F. (1975). Acta Cryst. B31, 2022–2026. CSD CrossRef CAS IUCr Journals Web of Science Google Scholar
Mallinson, P. R., MacNicol, D. D., McCormack, K. L., Yufit, D. S., Gall, J. H. & Henderson, R. K. (1997). Acta Cryst. C53, 90–92. CSD CrossRef CAS Web of Science IUCr Journals Google Scholar
Moggach, S. A., Allan, D. R., Clark, S. J., Gutmann, M. J., Parsons, S., Pulham, C. R. & Sawyer, L. (2005). In preparation. Google Scholar
Monkhorst, H. J. & Pack, J. D. (1976). Phys. Rev. B, 13, 5188–5192. CrossRef Web of Science Google Scholar
Perdew, J. P. & Wang, Y. (1992). Phys. Rev. B, 46, 12947–12954. CrossRef Web of Science Google Scholar
Segall, M. D., Lindan, P. J. D., Probert, M. J., Pickard, C. J., Hasnip, P. J., Clark, S. J. & Payne, M. C. (2002). J. Phys. Condens. Matter, 14, 2717–2744. Web of Science CrossRef CAS Google Scholar
Sheldrick, G. M. (2001). SHELXTL. Version 6.01. University of Göttingen, Germany, and Bruker AXS Inc., Madison, Wisconsin, USA. Google Scholar
Sheldrick, G. M. (2004). SADABS. University of Göttingen, Germany. Google Scholar
Spek, A. L. (2003). J. Appl. Cryst. 36, 7–13. Web of Science CrossRef CAS IUCr Journals Google Scholar
Taylor, R. & Macrae, C. F. (2001). Acta Cryst. B57, 815–827. Web of Science CrossRef CAS IUCr Journals Google Scholar
Vanderbilt, D. (1990). Phys. Rev. B, 41, 7892–7895. CrossRef Web of Science Google Scholar
Watkin, D. J., Prout, C. K. & Pearce, L. J. (1996). CAMERON. Chemical Crystallography Laboratory, Oxford, UK. Google Scholar

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