Accurate unrestrained DDM refinement of crystal structures from highly distorted and low-resolution powder diffraction data

The structure of benzene:ethane co-crystal at 90 K is refined with anisotropic displacement parameters without geometric restraints from high-resolution synchrotron X-ray powder diffraction (XRPD) data using the derivative difference method (DDM) with properly chosen weighting schemes. The average C—C bond precision achieved is 0.005 Å and the H-atom positions in ethane are refined independently. A new DDM weighting scheme is introduced that compensates for big distortions of experimental data. The results are compared with density functional theory (DFT) calculations reported by Maynard-Casely et al. [(2016). IUCrJ, 3, 192–199] where a rigid-body Rietveld refinement was also applied to the same dataset due to severe distortions of the powder pattern attributable to experimental peculiarities. For the crystal structure of 2-aminopyridinium fumarate–fumaric acid formerly refined applying 77 geometric restraints by Dong et al. [(2013). Acta Cryst. C69, 896–900], an unrestrained DDM refinement using the same XRPD pattern surprisingly gave two times narrower dispersion of interatomic distances.

Computing details

For both compounds, program(s) used to refine structure: Powder diffraction, Derivative Difference Method (DDM) L.A. Solovyov, J. Appl. Cryst. 37 (2004) 743.

(I) Benzene:ethane co-crystal

Crystal data

3(C6H6)·C2H6	V = 1234.37 (1) Å^3
Mr = 264.39	Z = 3
Trigonal, R3	Dx = 1.068 Mg m^−3
Hall symbol: -R 3	T = 90 K
a = 15.97910 (3) Å	white
c = 5.58227 (2) Å

Data collection

Synchrotron, Mythen Detector diffractometer	Data collection mode: transmission
Specimen mounting: Oxford Cryostems Cryostream	Scan method: Stationary detector

Refinement

Least-squares matrix: full	77 parameters
Rwp = 0.085	0 restraints
Rexp = 0.060	H atoms treated by a mixture of independent and constrained refinement
RBragg = 0.032	(Δ/σ)max = 0.001
R(F) = 0.055

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
C1	0.00000	0.00000	0.3637 (10)	0.0292 (15)
C2	0.11930 (17)	0.33686 (19)	0.5380 (5)	0.0242 (12)
C3	0.11847 (17)	0.25917 (17)	0.1682 (5)	0.0237 (12)
C4	0.07061 (17)	0.26381 (16)	0.3734 (5)	0.0246 (11)
H1	0.063 (2)	0.035 (3)	0.301 (6)	0.038*
H2	0.08670	0.33866	0.67865	0.038*
H3	0.00563	0.21536	0.40375	0.038*
H4	0.08597	0.20837	0.05534	0.038*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
C1	0.0266 (18)	0.0266 (18)	0.034 (3)	0.0133 (9)	0.0000	0.0000
C2	0.0255 (10)	0.0393 (13)	0.0179 (13)	0.0238 (10)	0.0024 (11)	0.0016 (14)
C3	0.0248 (10)	0.0261 (10)	0.0226 (15)	0.0147 (9)	−0.0115 (11)	−0.0093 (12)
C4	0.0190 (9)	0.0215 (10)	0.0294 (19)	0.0072 (7)	−0.0012 (11)	0.0055 (11)

Geometric parameters (Å, º)

C1—C1^i	1.522 (11)	C3—C4	1.399 (4)
C2—C3^ii	1.390 (3)	C1—H1	0.94 (3)
C2—C4	1.380 (3)
C3^ii—C2—C4	120.5 (3)	C1^i—C1—H1	112 (2)
C2^ii—C3—C4	119.7 (2)	H1—C1—H1^iii	107 (2)
C2—C4—C3	119.9 (2)

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

(II) Bis(2-aminopyridinium) fumarate–fumaric acid (1/1)

Crystal data

C4H4O4·C4H2O4·2(C5H7N2)	β = 91.7434 (6)°
Mr = 420.38	γ = 91.7142 (7)°
Triclinic, P1	V = 488.27 (1) Å^3
Hall symbol: -P 1	Z = 1
a = 3.81737 (6) Å	Dx = 1.430 Mg m^−3
b = 9.60821 (16) Å	T = 298 K
c = 13.3630 (2) Å	yellow
α = 94.2993 (6)°

Data collection

Rigaku SmartLab diffractometer	Scan method: continuous
Specimen mounting: borosilicate capillary	2θmin = 5.000°, 2θmax = 70.000°, 2θstep = 0.020°
Data collection mode: transmission

Refinement

Least-squares matrix: full	107 parameters
Rwp = 0.068	0 restraints
Rexp = 0.023	H-atom parameters constrained
RBragg = 0.028	(Δ/σ)max = 0.010
R(F) = 0.039

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Ų)

	x	y	z	Uiso*/Ueq
O1	0.2564 (13)	0.4782 (4)	0.2052 (3)	0.067 (2)
O2	0.1137 (13)	0.2893 (5)	0.1017 (4)	0.069 (3)
O3	0.0968 (16)	0.2387 (4)	0.4857 (3)	0.066 (3)
O4	0.2729 (13)	0.3078 (5)	0.3406 (3)	0.052 (2)
N1	0.4783 (17)	0.0553 (6)	0.2605 (5)	0.059 (3)*
N2	0.2651 (16)	−0.0337 (5)	0.4014 (3)	0.058 (3)*
C1	0.4089 (17)	−0.0581 (9)	0.3132 (5)	0.047 (3)*
C2	0.4937 (19)	−0.1933 (7)	0.2739 (5)	0.050 (3)*
C3	0.6441 (17)	−0.2043 (7)	0.1829 (6)	0.050 (3)*
C4	0.7017 (18)	−0.0906 (7)	0.1248 (5)	0.052 (3)*
C5	0.6171 (17)	0.0365 (7)	0.1674 (5)	0.063 (3)*
C6	0.156 (2)	0.4152 (8)	0.1188 (6)	0.038 (4)*
C7	0.0909 (18)	0.5168 (9)	0.0433 (4)	0.046 (3)*
C8	0.134 (3)	0.3321 (9)	0.4268 (7)	0.049 (4)*
C9	0.0483 (19)	0.4786 (7)	0.4553 (4)	0.043 (3)*
H1	0.26268	0.41766	0.25335	0.0760*
H21	0.45008	−0.27346	0.30966	0.0760*
H31	0.71321	−0.29361	0.15684	0.0760*
H41	0.79130	−0.10171	0.05918	0.0760*
H51	0.66053	0.11683	0.13167	0.0760*
H11	0.43005	0.14159	0.28629	0.0760*
H91	0.06142	0.54431	0.40571	0.0760*
H71	0.18505	0.60930	0.05596	0.0760*
H22	0.21358	0.05192	0.42781	0.0760*
H23	0.15097	−0.09758	0.43481	0.0760*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O1	0.110 (5)	0.050 (4)	0.039 (4)	0.002 (5)	−0.018 (4)	0.005 (4)
O2	0.095 (5)	0.027 (4)	0.080 (5)	−0.013 (5)	−0.014 (4)	−0.018 (4)
O3	0.131 (6)	0.016 (4)	0.057 (6)	0.017 (4)	0.034 (4)	0.012 (4)
O4	0.072 (5)	0.036 (4)	0.049 (5)	0.020 (3)	0.027 (4)	0.004 (4)

Geometric parameters (Å, º)

N1—C1	1.365 (11)	C7—C7^i	1.344 (11)
N1—C5	1.369 (9)	C8—C9	1.480 (11)
C1—N2	1.323 (8)	C9—C9^ii	1.301 (12)
C1—C2	1.414 (11)	C6—O1	1.305 (9)
C2—C3	1.358 (10)	C6—O2	1.218 (9)
C3—C4	1.403 (10)	C8—O3	1.245 (10)
C4—C5	1.360 (10)	C8—O4	1.291 (10)
C6—C7	1.477 (11)
C1—N1—C5	119.4 (6)	O1—C6—O2	125.3 (7)
N1—C1—N2	116.6 (7)	O1—C6—C7	111.1 (6)
N1—C1—C2	120.3 (6)	O2—C6—C7	123.6 (7)
N2—C1—C2	123.1 (7)	C6—C7—C7^i	122.6 (10)
C1—C2—C3	117.4 (6)	O3—C8—O4	121.6 (7)
C2—C3—C4	123.5 (6)	O3—C8—C9	121.6 (8)
C3—C4—C5	116.0 (6)	O4—C8—C9	116.6 (7)
N1—C5—C4	123.2 (7)	C8—C9—C9^ii	123.1 (9)
C5—N1—C1—C2	2.7 (10)	O4—C8—C9—C9^ii	−170.5 (8)
C1—N1—C5—C4	−2.2 (10)	C8—C9—C9^ii—C8^ii	180.0 (9)
C5—N1—C1—N2	−178.0 (6)	N1—C1—C2—C3	−0.1 (10)
O2—C6—C7—C7^i	14.9 (11)	N2—C1—C2—C3	−179.3 (6)
O1—C6—C7—C7^i	−163.9 (7)	C1—C2—C3—C4	−3.3 (10)
C6—C7—C7^i—C6^i	180.0 (7)	C2—C3—C4—C5	3.9 (10)
O3—C8—C9—C9^ii	4.6 (14)	C3—C4—C5—N1	−1.1 (10)

Symmetry codes: (i) −x, −y+1, −z; (ii) −x, −y+1, −z+1.

Hydrogen-bond geometry (Å, º)

D—H···A	D—H	H···A	D···A	D—H···A
O1—H1···O4	0.9000	1.6300	2.530 (6)	180.00
N1—H11···O4	0.9000	1.8300	2.724 (7)	172.00
N2—H22···O3	0.9000	1.9700	2.870 (6)	180.00
N2—H23···O3^iii	0.9000	2.0200	2.911 (7)	178.00

Symmetry code: (iii) −x, −y, −z+1.