The invariom model and its application: refinement of D,L-serine at different temperatures and resolution

Three X-ray data sets of the same D,L-serine crystal were measured at temperatures of 298, 100 and 20 K. These data were then evaluated using invarioms and the Hansen & Coppens aspherical-atom model. Multipole populations for invarioms, which are pseudoatoms that remain approximately invariant in an intermolecular transfer, were theoretically predicted using different density functional theorem (DFT) basis sets. The invariom parameters were kept fixed and positional and thermal parameters were refined to compare the fitting against the multi-temperature data at different resolutions. The deconvolution of thermal motion and electron density with respect to data resolution was studied by application of the Hirshfeld test. Above a resolution of sinθ/λ ≈ 0.55 Å⁻¹, or d ≈  0.9 Å, this test was fulfilled. When the Hirshfeld test is fulfilled, a successful modeling of the aspherical electron density with invarioms is achieved, which was proven by Fourier methods. Molecular geometry improves, especially for H atoms, when using the invariom method compared to the independent-atom model, as a comparison with neutron data shows. Based on this example, the general applicability of the invariom concept to organic molecules is proven and the aspherical density modeling of a larger biomacromolecule is within reach.

S1. Comment

see manuscript

Computing details

For all compounds, program(s) used to refine structure: Koritsanszky et al., (2003); molecular graphics: Koritsanszky et al., (2003), PLATON (Spek, 2003); software used to prepare material for publication: Koritsanszky et al., (2003), enCIFer 1.1.

Figures

(298K) (±) 2-Amino-3-hydroxypropionic acid

Crystal data

C3H7NO3	F(000) = 224.0
Mr = 105.10	Dx = 1.535 Mg m^−3
Monoclinic, P21/a	Melting point: 235 K
Hall symbol: -P 2yab	Mo Kα radiation, λ = 0.71073 Å
a = 10.7355 (9) Å	Cell parameters from 13549 reflections
b = 9.1456 (7) Å	θ = 3.0–43.3°
c = 4.8304 (4) Å	µ = 0.14 mm^−1
β = 106.46 (1)°	T = 298 K
V = 454.83 (7) Å^3	Rectangular, colorless
Z = 4	0.5 × 0.45 × 0.35 mm

Data collection

Bruker APEX on a Huber Type 512 diffractometer	Rint = 0.023
Graphite monochromator	θmax = 44.4°
ω–scans	h = −9→4
25178 measured reflections	k = −17→17
3551 independent reflections	l = −20→20
2703 reflections with I > 2σ I

Refinement

Refinement on F	0 restraints
Least-squares matrix: full	All H-atom parameters refined
R[F^2 > 2σ(F^2)] = 0.022	w1 = 1/[s^2(Fo)]
wR(F^2) = 0.026	(Δ/σ)max < 0.001
S = 1.39	Δρmax = 0.20 e Å^−3
2703 reflections	Δρmin = −0.21 e Å^−3
93 parameters

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

	x	y	z	Uiso*/Ueq
O(1)	0.66393 (3)	0.40214 (4)	0.43070 (6)	0.03
O(2)	0.83195 (4)	0.50585 (4)	0.75250 (7)	0.033
O(3)	0.93151 (3)	0.16868 (4)	0.72969 (8)	0.034
N(1)	0.65241 (3)	0.16997 (4)	0.74433 (7)	0.023
C(1)	0.74963 (4)	0.40565 (4)	0.66748 (7)	0.023
C(2)	0.75489 (3)	0.27886 (4)	0.87766 (7)	0.022
C(3)	0.88895 (4)	0.20938 (5)	0.97056 (8)	0.027
H(2)	0.7320 (7)	0.3221 (8)	1.0656 (16)	0.035 (2)
H(11)	0.5640 (9)	0.2268 (10)	0.6878 (17)	0.048 (2)
H(12)	0.6497 (8)	0.0921 (11)	0.898 (2)	0.051 (2)
H(13)	0.6653 (9)	0.1174 (10)	0.566 (2)	0.049 (2)
H(31)	0.9576 (8)	0.2904 (10)	1.0945 (17)	0.046 (2)
H(32)	0.8885 (8)	0.1141 (10)	1.1162 (19)	0.053 (2)
H(4)	0.9032 (8)	0.0721 (11)	0.6743 (19)	0.043 (2)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O(1)	0.0380 (2)	0.02510 (10)	0.02340 (10)	−0.00120 (10)	0.00300 (10)	0.00480 (10)
O(2)	0.0428 (2)	0.02440 (10)	0.02840 (10)	−0.00950 (10)	0.00940 (10)	−0.00210 (10)
O(3)	0.02540 (10)	0.03010 (10)	0.0445 (2)	−0.00570 (10)	0.01710 (10)	−0.01010 (10)
N(1)	0.02190 (10)	0.02110 (10)	0.02500 (10)	0.00180 (10)	0.00820 (10)	0.00360 (10)
C(1)	0.0286 (2)	0.01860 (10)	0.01990 (10)	−0.00010 (10)	0.00880 (10)	−0.00070 (10)
C(2)	0.02430 (10)	0.02140 (10)	0.01800 (10)	0.00150 (10)	0.00810 (10)	0.00060 (10)
C(3)	0.02250 (10)	0.0294 (2)	0.02650 (10)	0.00160 (10)	0.00410 (10)	0.00130 (10)

Geometric parameters (Å, º)

O(1)—C(1)	1.2487 (5)	N(1)—H(13)	1.030 (10)
O(2)—C(1)	1.2585 (5)	C(1)—C(2)	1.5316 (5)
O(3)—C(3)	1.4150 (5)	C(2)—C(3)	1.5201 (5)
O(3)—H(4)	0.947 (10)	C(2)—H(2)	1.080 (7)
N(1)—C(2)	1.4884 (5)	C(3)—H(31)	1.096 (9)
N(1)—H(11)	1.048 (9)	C(3)—H(32)	1.121 (9)
N(1)—H(12)	1.034 (10)
C(3)—O(3)—H(4)	109.0 (5)	N(1)—C(2)—C(3)	111.71 (3)
C(2)—N(1)—H(11)	106.5 (5)	N(1)—C(2)—H(2)	107.0 (4)
C(2)—N(1)—H(12)	108.6 (5)	C(1)—C(2)—C(3)	111.48 (3)
C(2)—N(1)—H(13)	114.6 (5)	C(1)—C(2)—H(2)	107.8 (4)
H(11)—N(1)—H(12)	108.7 (6)	C(3)—C(2)—H(2)	109.1 (4)
H(11)—N(1)—H(13)	109.9 (6)	O(3)—C(3)—C(2)	111.50 (3)
H(12)—N(1)—H(13)	108.4 (6)	O(3)—C(3)—H(31)	107.6 (4)
O(1)—C(1)—O(2)	125.77 (4)	O(3)—C(3)—H(32)	111.8 (4)
O(1)—C(1)—C(2)	117.84 (3)	C(2)—C(3)—H(31)	108.2 (4)
O(2)—C(1)—C(2)	116.38 (3)	C(2)—C(3)—H(32)	109.8 (4)
N(1)—C(2)—C(1)	109.62 (3)	H(31)—C(3)—H(32)	107.7 (6)

(100K) (±) 2-Amino-3-hydroxypropionic acid

Crystal data

C3H7NO3	F(000) = 224.0
Mr = 105.10	Dx = 1.542 Mg m^−3
Monoclinic, P21/a	Melting point: 235 K
Hall symbol: -P 2yab	Mo Kα radiation, λ = 0.71073 Å
a = 10.7621 (10) Å	Cell parameters from 9829 reflections
b = 9.1771 (9) Å	θ = 16.2–40.2°
c = 4.7883 (5) Å	µ = 0.14 mm^−1
β = 106.76 (2)°	T = 100 K
V = 452.83 (9) Å^3	Rectangular, colorless
Z = 4	0.5 × 0.45 × 0.35 mm

Data collection

Bruker APEX on a Huber Type 512 diffractometer	Rint = 0.029
Graphite monochromator	θmax = 57.6°
ω–scans	h = −4→8
38455 measured reflections	k = −21→21
5146 independent reflections	l = −24→24
4109 reflections with I > 2σ I

Refinement

Refinement on F	0 restraints
Least-squares matrix: full	All H-atom parameters refined
R[F^2 > 2σ(F^2)] = 0.023	w1 = 1/[s^2(Fo)]
wR(F^2) = 0.024	(Δ/σ)max < 0.001
S = 1.24	Δρmax = 0.23 e Å^−3
4109 reflections	Δρmin = −0.22 e Å^−3
93 parameters

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

	x	y	z	Uiso*/Ueq
O(1)	0.65971 (2)	0.40588 (2)	0.42808 (5)	0.012
O(2)	0.82960 (2)	0.50982 (2)	0.75169 (5)	0.013
O(3)	0.92941 (2)	0.17379 (2)	0.72353 (5)	0.014
N(1)	0.65294 (2)	0.17254 (2)	0.74409 (5)	0.01
C(1)	0.74714 (2)	0.40934 (2)	0.66646 (5)	0.01
C(2)	0.75448 (2)	0.28247 (3)	0.87869 (5)	0.009
C(3)	0.88934 (2)	0.21473 (3)	0.97025 (6)	0.011
H(2)	0.7325 (7)	0.3234 (8)	1.0664 (16)	0.020 (2)
H(11)	0.5643 (9)	0.2289 (10)	0.6837 (17)	0.030 (2)
H(12)	0.6531 (9)	0.0944 (11)	0.904 (2)	0.037 (2)
H(13)	0.6664 (9)	0.1217 (10)	0.561 (2)	0.035 (2)
H(31)	0.9566 (8)	0.2970 (9)	1.0920 (16)	0.025 (2)
H(32)	0.8894 (8)	0.1195 (10)	1.1160 (16)	0.027 (2)
H(4)	0.9019 (8)	0.0771 (10)	0.6736 (17)	0.025 (2)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O(1)	0.01460 (6)	0.01086 (6)	0.00987 (7)	−0.00053 (5)	0.00144 (5)	0.00166 (5)
O(2)	0.01635 (7)	0.01052 (6)	0.01157 (8)	−0.00359 (5)	0.00396 (6)	−0.00086 (5)
O(3)	0.01123 (6)	0.01288 (6)	0.01697 (9)	−0.00212 (5)	0.00720 (6)	−0.00381 (6)
N(1)	0.00949 (5)	0.00947 (6)	0.01064 (8)	0.00049 (4)	0.00363 (5)	0.00128 (5)
C(1)	0.01157 (7)	0.00838 (6)	0.00847 (8)	−0.00019 (5)	0.00359 (5)	−0.00024 (5)
C(2)	0.01012 (6)	0.00950 (6)	0.00812 (8)	0.00061 (5)	0.00350 (5)	0.00035 (5)
C(3)	0.00958 (6)	0.01267 (7)	0.01108 (9)	0.00070 (6)	0.00229 (6)	0.00030 (6)

Geometric parameters (Å, º)

O(1)—C(1)	1.2529 (3)	N(1)—H(13)	1.039 (10)
O(2)—C(1)	1.2632 (3)	C(1)—C(2)	1.5323 (3)
O(3)—C(3)	1.4201 (3)	C(2)—C(3)	1.5226 (3)
O(3)—H(4)	0.944 (10)	C(2)—H(2)	1.062 (7)
N(1)—H(11)	1.050 (9)	C(3)—H(31)	1.091 (8)
N(1)—H(12)	1.048 (10)	C(3)—H(32)	1.118 (9)
C(3)—O(3)—H(4)	108.1 (5)	C(1)—C(2)—H(2)	108.1 (4)
H(11)—N(1)—H(12)	110.4 (6)	C(3)—C(2)—H(2)	109.1 (4)
H(11)—N(1)—H(13)	109.0 (6)	O(3)—C(3)—C(2)	111.19 (2)
H(12)—N(1)—H(13)	109.7 (7)	O(3)—C(3)—H(31)	108.3 (4)
O(1)—C(1)—O(2)	125.86 (2)	O(3)—C(3)—H(32)	111.5 (4)
O(1)—C(1)—C(2)	117.81 (2)	C(2)—C(3)—H(31)	107.5 (4)
O(2)—C(1)—C(2)	116.32 (2)	C(2)—C(3)—H(32)	109.1 (4)
C(1)—C(2)—C(3)	111.329 (19)	H(31)—C(3)—H(32)	109.1 (5)

(20K) (±) 2-Amino-3-hydroxypropionic acid

Crystal data

C3H7NO3	F(000) = 224.0
Mr = 105.10	Dx = 1.541 Mg m^−3
Monoclinic, P21/a	Melting point: 235 K
Hall symbol: -P 2yab	Mo Kα radiation, λ = 0.71073 Å
a = 10.7764 (10) Å	Cell parameters from 10633 reflections
b = 9.1947 (9) Å	θ = 16.4–40.3°
c = 4.7788 (5) Å	µ = 0.14 mm^−1
β = 106.87 (2)°	T = 20 K
V = 453.13 (9) Å^3	Rectangular, colorless
Z = 4	0.5 × 0.45 × 0.35 mm

Data collection

Bruker APEX on a Huber Type 512 diffractometer	Rint = 0.027
Graphite monochromator	θmax = 57.4°
ω–scans	h = −4→8
38628 measured reflections	k = −21→21
5136 independent reflections	l = −24→24
4292 reflections with I > 2σ I

Refinement

Refinement on F	0 restraints
Least-squares matrix: full	All H-atom parameters refined
R[F^2 > 2σ(F^2)] = 0.020	w1 = 1/[s^2(Fo)]
wR(F^2) = 0.022	(Δ/σ)max < 0.001
S = 1.30	Δρmax = 0.22 e Å^−3
4292 reflections	Δρmin = −0.24 e Å^−3
93 parameters

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

	x	y	z	Uiso*/Ueq
O(1)	0.65863 (2)	0.40703 (2)	0.42739 (4)	0.008
O(2)	0.82879 (2)	0.51099 (2)	0.75170 (4)	0.008
O(3)	0.92877 (2)	0.17538 (2)	0.72150 (4)	0.009
N(1)	0.65325 (2)	0.17344 (2)	0.74378 (4)	0.007
C(1)	0.74647 (2)	0.41044 (2)	0.66619 (5)	0.006
C(2)	0.75445 (2)	0.28355 (2)	0.87889 (5)	0.006
C(3)	0.88949 (2)	0.21634 (2)	0.96983 (5)	0.007
H(2)	0.7323 (7)	0.3266 (8)	1.0683 (15)	0.017 (2)
H(11)	0.5631 (9)	0.2275 (10)	0.6859 (16)	0.027 (2)
H(12)	0.6515 (8)	0.0966 (10)	0.8979 (19)	0.029 (2)
H(13)	0.6661 (8)	0.1228 (9)	0.5615 (18)	0.027 (2)
H(31)	0.9586 (7)	0.2992 (8)	1.0924 (15)	0.019 (2)
H(32)	0.8924 (7)	0.1206 (9)	1.1175 (16)	0.021 (2)
H(4)	0.9045 (7)	0.0799 (9)	0.6731 (16)	0.019 (2)

Atomic displacement parameters (Ų)

	U^11	U^22	U^33	U^12	U^13	U^23
O(1)	0.00876 (5)	0.00731 (5)	0.00610 (6)	−0.00043 (4)	0.00064 (4)	0.00097 (4)
O(2)	0.00950 (5)	0.00679 (5)	0.00720 (6)	−0.00213 (4)	0.00222 (4)	−0.00038 (4)
O(3)	0.00763 (5)	0.00849 (5)	0.00943 (6)	−0.00099 (4)	0.00436 (4)	−0.00187 (4)
N(1)	0.00635 (5)	0.00634 (5)	0.00684 (6)	−0.00015 (4)	0.00228 (4)	0.00060 (4)
C(1)	0.00683 (5)	0.00548 (5)	0.00525 (7)	−0.00021 (4)	0.00200 (4)	−0.00002 (4)
C(2)	0.00638 (5)	0.00603 (5)	0.00531 (7)	0.00023 (4)	0.00217 (4)	0.00030 (4)
C(3)	0.00619 (5)	0.00815 (6)	0.00659 (7)	0.00060 (4)	0.00152 (4)	0.00026 (5)

Geometric parameters (Å, º)

O(1)—C(1)	1.2539 (3)	N(1)—H(13)	1.032 (9)
O(2)—C(1)	1.2643 (3)	C(1)—C(2)	1.5333 (3)
O(3)—C(3)	1.4216 (3)	C(2)—C(3)	1.5236 (3)
O(3)—H(4)	0.926 (9)	C(2)—H(2)	1.076 (7)
N(1)—H(11)	1.054 (9)	C(3)—H(31)	1.107 (8)
N(1)—H(12)	1.025 (9)	C(3)—H(32)	1.123 (8)
C(3)—O(3)—H(4)	109.2 (4)	C(1)—C(2)—H(2)	107.1 (4)
H(11)—N(1)—H(12)	107.8 (6)	C(3)—C(2)—H(2)	109.6 (4)
H(11)—N(1)—H(13)	109.3 (6)	O(3)—C(3)—C(2)	111.129 (17)
H(12)—N(1)—H(13)	109.3 (6)	O(3)—C(3)—H(31)	107.9 (3)
O(1)—C(1)—O(2)	125.88 (2)	O(3)—C(3)—H(32)	110.9 (4)
O(1)—C(1)—C(2)	117.815 (18)	C(2)—C(3)—H(31)	108.2 (4)
O(2)—C(1)—C(2)	116.298 (18)	C(2)—C(3)—H(32)	110.3 (4)
C(1)—C(2)—C(3)	111.284 (16)	H(31)—C(3)—H(32)	108.4 (5)