Experimental determination of spin-dependent electron density by joint refinement of X-ray and polarized neutron diffraction data

New crystallographic tools were developed to access a more precise description of the spin-dependent electron density of magnetic crystals. The method combines experimental information coming from high-resolution X-ray diffraction (XRD) and polarized neutron diffraction (PND) in a unified model. A new algorithm that allows for a simultaneous refinement of the charge- and spin-density parameters against XRD and PND data is described. The resulting software MOLLYNX is based on the well known Hansen-Coppens multipolar model, and makes it possible to differentiate the electron spins. This algorithm is validated and demonstrated with a molecular crystal formed by a bimetallic chain, MnCu(pba)(H₂O)₃·2H₂O, for which XRD and PND data are available. The joint refinement provides a more detailed description of the spin density than the refinement from PND data alone.

Computing details

Program(s) used to refine structure: SHELXL97 (Sheldrick, 2008).

(I)

Crystal data

C7H16CuMnN2O11	Z = 4
Mr = 422.69	F(000) = 856
Orthorhombic, Pnma	Dx = 1.998 Mg m^−3
a = 12.7158 (3) Å	Mo Kα radiation, λ = 0.71073 Å
b = 21.3520 (5) Å	µ = 2.48 mm^−1
c = 5.1745 (2) Å	T = 10 K
V = 1404.92 (7) Å^3	0.27 × 0.16 × 0.06 mm

Data collection

Radiation source: fine-focus sealed tube	Rint = 0.000
Absorption correction: gaussian CrysAlisPro, Oxford Diffraction Ltd., Version 1.171.34.44 (release 25-10-2010 CrysAlis171 .NET) (compiled Oct 25 2010,18:11:34) Numerical absorption correction based on gaussian integration over a multifaceted crystal model	θmax = 53.3°, θmin = 3.2°
Tmin = 0.615, Tmax = 0.880	h = 0→28
8505 measured reflections	k = 0→47
8505 independent reflections	l = 0→11
6880 reflections with > 3σ(I)

Refinement

Refinement on F^2	0 restraints
Least-squares matrix: full	Primary atom site location: structure-invariant direct methods
R[F^2 > 2σ(F^2)] = 0.029	Secondary atom site location: difference Fourier map
wR(F^2) = 0.062	Hydrogen site location: inferred from neighbouring sites
S = 0.96	H atoms treated by a mixture of independent and constrained refinement
8505 reflections	w = 1/[σ^2(Fo^2) + (0.0177P)^2 + 1.3214P] where P = (Fo^2 + 2Fc^2)/3
140 parameters

Special details

Geometry. All s.u.'s (except the s.u. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell s.u.'s are taken into account individually in the estimation of s.u.'s in distances, angles and torsion angles; correlations between s.u.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell s.u.'s is used for estimating s.u.'s 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 > 2σ(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
Cu	0.050679 (8)	0.2500	0.150042 (19)	0.00415 (2)
Mn	0.0000	0.5000	0.0000	0.00501 (2)
O1	−0.06303 (3)	0.423844 (19)	0.22854 (9)	0.00660 (5)
O2	0.09575 (4)	0.42089 (2)	−0.12554 (9)	0.00716 (6)
O3	0.11523 (3)	0.317983 (19)	−0.06201 (9)	0.00643 (5)
O4	−0.11791 (4)	0.47868 (2)	−0.29552 (9)	0.00889 (6)
O5	0.17489 (6)	0.2500	0.46535 (14)	0.00904 (9)
O6	0.21400 (4)	0.41212 (3)	0.42287 (10)	0.01069 (7)
N	−0.03674 (4)	0.31734 (2)	0.28494 (10)	0.00587 (6)
C1	0.07282 (4)	0.37043 (2)	−0.01580 (10)	0.00557 (6)
C2	−0.01646 (4)	0.37163 (2)	0.18463 (10)	0.00536 (6)
C4	−0.10898 (6)	0.2500	0.62761 (15)	0.00669 (9)
C3	−0.12204 (5)	0.31020 (3)	0.47259 (12)	0.00734 (7)
H1	−0.1630 (17)	0.2500	0.761 (4)	0.015 (5)*
H2	−0.0382 (16)	0.2500	0.706 (4)	0.013 (4)*
H3	−0.1877 (12)	0.3103 (7)	0.388 (3)	0.015 (3)*
H4	−0.1224 (11)	0.3445 (7)	0.585 (3)	0.015 (3)*
H5	−0.1509 (15)	0.5102 (9)	−0.342 (4)	0.036 (5)*
H6	−0.1047 (13)	0.4602 (8)	−0.426 (3)	0.021 (4)*
H7	0.1673 (14)	0.2199 (8)	0.559 (3)	0.034 (5)*
H8	0.1986 (14)	0.4119 (8)	0.574 (4)	0.028 (4)*
H9	0.2733 (15)	0.4166 (8)	0.413 (4)	0.034 (5)*

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
Cu	0.00437 (3)	0.00378 (3)	0.00429 (3)	0.000	0.00067 (2)	0.000
Mn	0.00614 (3)	0.00385 (3)	0.00503 (4)	0.00021 (3)	0.00029 (3)	0.00065 (3)
O1	0.00760 (13)	0.00505 (11)	0.00715 (14)	0.00133 (10)	0.00143 (11)	0.00047 (10)
O2	0.00841 (13)	0.00557 (12)	0.00749 (14)	0.00003 (10)	0.00203 (11)	0.00171 (11)
O3	0.00692 (12)	0.00532 (11)	0.00705 (13)	0.00076 (10)	0.00147 (11)	0.00037 (10)
O4	0.01031 (15)	0.00899 (14)	0.00737 (14)	0.00099 (12)	−0.00087 (12)	−0.00066 (12)
O5	0.0095 (2)	0.0099 (2)	0.0077 (2)	0.000	−0.00093 (17)	0.000
O6	0.00880 (15)	0.01390 (17)	0.00935 (16)	0.00074 (13)	0.00066 (13)	−0.00134 (14)
N	0.00629 (14)	0.00500 (12)	0.00632 (14)	0.00004 (10)	0.00167 (11)	0.00052 (11)
C1	0.00607 (15)	0.00516 (13)	0.00547 (15)	0.00006 (12)	0.00049 (12)	0.00050 (12)
C2	0.00559 (15)	0.00484 (13)	0.00565 (15)	0.00029 (11)	0.00073 (12)	0.00027 (12)
C4	0.0080 (2)	0.0062 (2)	0.0058 (2)	0.000	0.00127 (19)	0.000
C3	0.00745 (16)	0.00606 (14)	0.00850 (18)	0.00082 (13)	0.00319 (14)	0.00084 (14)

Geometric parameters (Å, º)

Cu—N^i	1.9469 (5)	Mn—O4^ii	2.1894 (5)
Cu—N	1.9469 (5)	O1—C2	1.2827 (6)
Cu—O3^i	1.9962 (4)	O2—C1	1.2524 (7)
Cu—O3	1.9962 (4)	O3—C1	1.2657 (7)
Cu—O5	2.2709 (7)	N—C2	1.2960 (7)
Mn—O1	2.1645 (4)	N—C3	1.4637 (7)
Mn—O1^ii	2.1645 (4)	C1—C2	1.5378 (7)
Mn—O2^ii	2.1811 (4)	C4—C3^i	1.5243 (7)
Mn—O2	2.1811 (4)	C4—C3	1.5243 (7)
Mn—O4	2.1894 (5)
N^i—Cu—N	95.22 (3)	O1—Mn—O4^ii	91.613 (18)
N^i—Cu—O3^i	83.962 (18)	O1^ii—Mn—O4^ii	88.387 (18)
N—Cu—O3^i	165.67 (2)	O2^ii—Mn—O4^ii	90.758 (18)
N^i—Cu—O3	165.67 (2)	O2—Mn—O4^ii	89.242 (18)
N—Cu—O3	83.962 (18)	O4—Mn—O4^ii	180.0
O3^i—Cu—O3	93.30 (2)	C2—O1—Mn	112.66 (3)
N^i—Cu—O5	98.021 (19)	C1—O2—Mn	113.66 (4)
N—Cu—O5	98.021 (19)	C1—O3—Cu	111.36 (4)
O3^i—Cu—O5	96.252 (18)	C2—N—C3	120.42 (5)
O3—Cu—O5	96.252 (18)	C2—N—Cu	113.79 (4)
O1—Mn—O1^ii	180.000 (14)	C3—N—Cu	125.73 (3)
O1—Mn—O2^ii	102.260 (16)	O2—C1—O3	125.19 (5)
O1^ii—Mn—O2^ii	77.740 (16)	O2—C1—C2	117.60 (5)
O1—Mn—O2	77.740 (16)	O3—C1—C2	117.17 (5)
O1^ii—Mn—O2	102.260 (16)	O1—C2—N	127.92 (5)
O2^ii—Mn—O2	180.0	O1—C2—C1	118.35 (5)
O1—Mn—O4	88.387 (18)	N—C2—C1	113.71 (4)
O1^ii—Mn—O4	91.613 (18)	C3^i—C4—C3	114.98 (7)
O2^ii—Mn—O4	89.242 (18)	N—C3—C4	110.87 (5)
O2—Mn—O4	90.758 (18)

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