Supporting information
Crystallographic Information File (CIF) https://doi.org/10.1107/S1600536807032928/ci2407sup1.cif | |
Structure factor file (CIF format) https://doi.org/10.1107/S1600536807032928/ci2407Isup2.hkl |
CCDC reference: 657577
Key indicators
- Single-crystal X-ray study
- T = 293 K
- Mean (C-C) = 0.005 Å
- Disorder in main residue
- R factor = 0.038
- wR factor = 0.088
- Data-to-parameter ratio = 14.1
checkCIF/PLATON results
No syntax errors found
Alert level C PLAT042_ALERT_1_C Calc. and Rep. MoietyFormula Strings Differ .... ? PLAT241_ALERT_2_C Check High Ueq as Compared to Neighbors for Ag1 PLAT244_ALERT_4_C Low 'Solvent' Ueq as Compared to Neighbors for Cl1 PLAT301_ALERT_3_C Main Residue Disorder ......................... 4.00 Perc. PLAT764_ALERT_4_C Overcomplete CIF Bond List Detected (Rep/Expd) . 1.12 Ratio PLAT790_ALERT_4_C Centre of Gravity not Within Unit Cell: Resd. # 2 Cl O4
Alert level G FORMU01_ALERT_1_G There is a discrepancy between the atom counts in the _chemical_formula_sum and _chemical_formula_moiety. This is usually due to the moiety formula being in the wrong format. Atom count from _chemical_formula_sum: C19 H18 Ag1 Cl1 N6 O5 Atom count from _chemical_formula_moiety: PLAT199_ALERT_1_G Check the Reported _cell_measurement_temperature 293 K PLAT200_ALERT_1_G Check the Reported _diffrn_ambient_temperature . 293 K
0 ALERT level A = In general: serious problem 0 ALERT level B = Potentially serious problem 6 ALERT level C = Check and explain 3 ALERT level G = General alerts; check 4 ALERT type 1 CIF construction/syntax error, inconsistent or missing data 1 ALERT type 2 Indicator that the structure model may be wrong or deficient 1 ALERT type 3 Indicator that the structure quality may be low 3 ALERT type 4 Improvement, methodology, query or suggestion 0 ALERT type 5 Informative message, check
The ligand 1,3-bis[3-(2-pyridyl)-1H-pyrazole]propan-2-ol (L) was synthesized according to the method reported in the literature (Zhang et al., 2005). A solution of AgClO4 (22 mg, 0.1 mmol) in ethanol (10 ml) was added to a solution of L (35 mg, 0.1 mmol) in acetonitrile (20 ml) in a 50 ml beaker and the resulted solution was kept at room temperature in the dark. Single crystals of (I) suitable for X-ray analysis were obtained after 10 d (yield: 45%). Analysis calculated for (C19H18AgClN6O5): C 41.18, H 3.25, N 15.17%; found: C 41.36, H 3.64, N 14.91%.
H atoms were included in calculated positions and treated in the subsequent refinement as riding atoms, with C—H = 0.93 (aromatic) or 0.97 Å (methylene) and O—H = 0.82 Å, and with Uiso(H) = 1.2Ueq(C) and 1.5Ueq(O).
In recent years, 3-(2-pyridyl)pyrazole-based ligands have found a wide range of application in the area of coordination chemistry, because they can act as bridging or chelate ligands and exhibit a series of intriguing structures and potential applications as functional materials (Ruben et al., 2004; Steel et al., 2005). Nowadays, much attention has been focused on the synthetic approach and the structural control of coordination architectures (Bell et al., 2003; Paul et al., 2004). We report here the structure of a mononuclear silver complex, {1,3-bis[3-(2-pyridyl)pyrazole]propan-2-ol}silver(I) perchlorate.
In the title compound, the cation and anion both lie on crystallographic twofold rotation axes. In the cation, the twofold axis passes through atoms Ag1 and C1, and as a result the hydroxyl group is disordered. The AgI center is four-coordinated by four N donors from a 1,3-bis[3-(2-pyridyl)pyrazole]propan-2-ol ligand (Table 1). The coordination geometry around the AgI center can be described as a distorted tetrahedron (Fig. 1).
The AgI mononuclear units are linked to the perchlorate ions through O—H···O hydrogen bonds (Table 2) and weak C—H···O interactions (Desiraju et al., 1999; Barberà et al., 2002) leading to the formation of a one-dimensional chain (Fig. 2).
For general background, see: Bell et al. (2003); Paul et al. (2004); Ruben et al. (2004); Steel (2005); Zhang et al. (2005). For hydrogen-bonding, see: Barberà et al. (2002); Desiraju & Steiner (1999).
Data collection: SMART (Bruker, 1998); cell refinement: SAINT (Bruker, 1998); data reduction: SAINT; program(s) used to solve structure: SHELXS97 (Sheldrick, 1997); program(s) used to refine structure: SHELXL97 (Sheldrick, 1997); molecular graphics: SHELXTL (Bruker, 1998); software used to prepare material for publication: SHELXTL and PLATON (Spek, 2003).
[Ag(C19H18N6O)]ClO4 | F(000) = 1112 |
Mr = 553.71 | Dx = 1.784 Mg m−3 |
Monoclinic, C2/c | Mo Kα radiation, λ = 0.71073 Å |
Hall symbol: -C 2yc | Cell parameters from 946 reflections |
a = 8.687 (3) Å | θ = 2.6–22.8° |
b = 21.863 (9) Å | µ = 1.15 mm−1 |
c = 10.904 (4) Å | T = 293 K |
β = 95.482 (7)° | Block, colourless |
V = 2061.4 (14) Å3 | 0.20 × 0.20 × 0.18 mm |
Z = 4 |
Bruker SMART CCD area-detector diffractometer | 2135 independent reflections |
Radiation source: fine-focus sealed tube | 1497 reflections with I > 2σ(I) |
Graphite monochromator | Rint = 0.031 |
φ and ω scans | θmax = 26.5°, θmin = 2.5° |
Absorption correction: multi-scan (SADABS; Bruker, 1998) | h = −6→10 |
Tmin = 0.802, Tmax = 0.819 | k = −27→24 |
5960 measured reflections | l = −13→13 |
Refinement on F2 | Primary atom site location: structure-invariant direct methods |
Least-squares matrix: full | Secondary atom site location: difference Fourier map |
R[F2 > 2σ(F2)] = 0.038 | Hydrogen site location: inferred from neighbouring sites |
wR(F2) = 0.088 | H-atom parameters constrained |
S = 1.00 | w = 1/[σ2(Fo2) + (0.0393P)2 + 1.8827P] where P = (Fo2 + 2Fc2)/3 |
2135 reflections | (Δ/σ)max = 0.001 |
151 parameters | Δρmax = 0.53 e Å−3 |
0 restraints | Δρmin = −0.46 e Å−3 |
[Ag(C19H18N6O)]ClO4 | V = 2061.4 (14) Å3 |
Mr = 553.71 | Z = 4 |
Monoclinic, C2/c | Mo Kα radiation |
a = 8.687 (3) Å | µ = 1.15 mm−1 |
b = 21.863 (9) Å | T = 293 K |
c = 10.904 (4) Å | 0.20 × 0.20 × 0.18 mm |
β = 95.482 (7)° |
Bruker SMART CCD area-detector diffractometer | 2135 independent reflections |
Absorption correction: multi-scan (SADABS; Bruker, 1998) | 1497 reflections with I > 2σ(I) |
Tmin = 0.802, Tmax = 0.819 | Rint = 0.031 |
5960 measured reflections |
R[F2 > 2σ(F2)] = 0.038 | 0 restraints |
wR(F2) = 0.088 | H-atom parameters constrained |
S = 1.00 | Δρmax = 0.53 e Å−3 |
2135 reflections | Δρmin = −0.46 e Å−3 |
151 parameters |
Geometry. All e.s.d.'s (except the e.s.d. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell e.s.d.'s are taken into account individually in the estimation of e.s.d.'s in distances, angles and torsion angles; correlations between e.s.d.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell e.s.d.'s is used for estimating e.s.d.'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 > σ(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. |
x | y | z | Uiso*/Ueq | Occ. (<1) | |
Ag1 | 0.5000 | 0.510403 (19) | 0.7500 | 0.0676 (2) | |
Cl1 | 0.5000 | 0.76495 (5) | 1.2500 | 0.0524 (3) | |
N1 | 0.5693 (3) | 0.58693 (12) | 0.9020 (2) | 0.0499 (7) | |
N2 | 0.3199 (3) | 0.47008 (13) | 0.6097 (3) | 0.0523 (7) | |
N3 | 0.5527 (3) | 0.64717 (13) | 0.9227 (3) | 0.0550 (7) | |
C1 | 0.5000 | 0.7204 (3) | 0.7500 | 0.0764 (18) | |
H1B | 0.4098 | 0.7391 | 0.7082 | 0.092* | 0.50 |
C2 | 0.4355 (4) | 0.68233 (16) | 0.8498 (4) | 0.0610 (10) | |
H2A | 0.3847 | 0.7093 | 0.9040 | 0.073* | |
H2B | 0.3581 | 0.6545 | 0.8117 | 0.073* | |
C3 | 0.6549 (5) | 0.66635 (19) | 1.0135 (4) | 0.0652 (11) | |
H3A | 0.6642 | 0.7061 | 1.0438 | 0.078* | |
C4 | 0.7424 (5) | 0.61826 (18) | 1.0539 (3) | 0.0626 (10) | |
H4A | 0.8231 | 0.6179 | 1.1164 | 0.075* | |
C5 | 0.6859 (4) | 0.56931 (16) | 0.9822 (3) | 0.0494 (8) | |
C6 | 0.2745 (5) | 0.41184 (17) | 0.6109 (3) | 0.0629 (10) | |
H6A | 0.3154 | 0.3869 | 0.6750 | 0.075* | |
C7 | 0.1708 (5) | 0.38710 (19) | 0.5223 (4) | 0.0716 (11) | |
H7A | 0.1399 | 0.3465 | 0.5271 | 0.086* | |
C8 | 0.1142 (5) | 0.4230 (2) | 0.4277 (4) | 0.0732 (12) | |
H8A | 0.0438 | 0.4073 | 0.3662 | 0.088* | |
C9 | 0.1609 (4) | 0.48229 (19) | 0.4231 (4) | 0.0648 (10) | |
H9A | 0.1238 | 0.5071 | 0.3576 | 0.078* | |
C10 | 0.2638 (4) | 0.50556 (15) | 0.5161 (3) | 0.0474 (8) | |
O1 | 0.5842 (6) | 0.7655 (2) | 0.7817 (5) | 0.0669 (14) | 0.50 |
H1A | 0.6650 | 0.7632 | 0.7489 | 0.100* | 0.50 |
O2 | 0.5854 (3) | 0.80126 (13) | 1.1757 (3) | 0.0911 (10) | |
O3 | 0.3944 (4) | 0.72815 (17) | 1.1794 (4) | 0.1176 (13) |
U11 | U22 | U33 | U12 | U13 | U23 | |
Ag1 | 0.0799 (3) | 0.0611 (3) | 0.0565 (3) | 0.000 | −0.0203 (2) | 0.000 |
Cl1 | 0.0450 (7) | 0.0449 (7) | 0.0679 (8) | 0.000 | 0.0082 (6) | 0.000 |
N1 | 0.0484 (17) | 0.0463 (16) | 0.0551 (17) | 0.0007 (12) | 0.0061 (14) | −0.0066 (13) |
N2 | 0.0617 (19) | 0.0466 (17) | 0.0474 (17) | −0.0030 (13) | −0.0007 (14) | −0.0032 (13) |
N3 | 0.0512 (18) | 0.0479 (17) | 0.068 (2) | −0.0006 (13) | 0.0166 (15) | −0.0087 (14) |
C1 | 0.079 (4) | 0.051 (3) | 0.105 (5) | 0.000 | 0.040 (4) | 0.000 |
C2 | 0.056 (2) | 0.048 (2) | 0.082 (3) | 0.0067 (17) | 0.023 (2) | −0.0028 (19) |
C3 | 0.070 (3) | 0.061 (2) | 0.067 (3) | −0.015 (2) | 0.019 (2) | −0.027 (2) |
C4 | 0.064 (2) | 0.072 (3) | 0.052 (2) | −0.010 (2) | 0.0030 (18) | −0.0135 (19) |
C5 | 0.047 (2) | 0.059 (2) | 0.0423 (19) | −0.0041 (16) | 0.0093 (16) | −0.0089 (16) |
C6 | 0.078 (3) | 0.051 (2) | 0.059 (2) | −0.0030 (19) | 0.006 (2) | −0.0029 (18) |
C7 | 0.077 (3) | 0.061 (3) | 0.078 (3) | −0.013 (2) | 0.013 (2) | −0.020 (2) |
C8 | 0.068 (3) | 0.079 (3) | 0.069 (3) | −0.007 (2) | −0.007 (2) | −0.027 (2) |
C9 | 0.062 (2) | 0.079 (3) | 0.051 (2) | 0.006 (2) | −0.0058 (18) | −0.008 (2) |
C10 | 0.0418 (18) | 0.058 (2) | 0.0427 (18) | 0.0064 (15) | 0.0079 (14) | −0.0030 (16) |
O1 | 0.067 (3) | 0.052 (3) | 0.082 (4) | −0.020 (2) | 0.009 (3) | −0.020 (3) |
O2 | 0.086 (2) | 0.080 (2) | 0.113 (3) | 0.0011 (16) | 0.0363 (18) | 0.0338 (18) |
O3 | 0.077 (2) | 0.118 (3) | 0.156 (3) | −0.0239 (19) | 0.006 (2) | −0.072 (3) |
Ag1—N2 | 2.260 (3) | C2—H2A | 0.97 |
Ag1—N2i | 2.260 (3) | C2—H2B | 0.97 |
Ag1—N1 | 2.391 (3) | C3—C4 | 1.346 (5) |
Ag1—N1i | 2.391 (3) | C3—H3A | 0.93 |
Cl1—O3 | 1.395 (3) | C4—C5 | 1.387 (5) |
Cl1—O3ii | 1.395 (3) | C4—H4A | 0.93 |
Cl1—O2ii | 1.397 (3) | C5—C10i | 1.460 (5) |
Cl1—O2 | 1.397 (3) | C6—C7 | 1.368 (5) |
N1—C5 | 1.330 (4) | C6—H6A | 0.93 |
N1—N3 | 1.346 (4) | C7—C8 | 1.351 (6) |
N2—C6 | 1.333 (4) | C7—H7A | 0.93 |
N2—C10 | 1.337 (4) | C8—C9 | 1.361 (6) |
N3—C3 | 1.333 (5) | C8—H8A | 0.93 |
N3—C2 | 1.451 (5) | C9—C10 | 1.383 (5) |
C1—O1 | 1.257 (6) | C9—H9A | 0.93 |
C1—O1i | 1.257 (6) | C10—C5i | 1.460 (5) |
C1—C2i | 1.519 (5) | O1—O1i | 1.557 (10) |
C1—C2 | 1.519 (5) | O1—H1A | 0.82 |
C1—H1B | 0.96 | ||
N2—Ag1—N2i | 134.08 (15) | C1—C2—H2A | 108.9 |
N2—Ag1—N1 | 148.68 (10) | N3—C2—H2B | 108.9 |
N2i—Ag1—N1 | 72.03 (10) | C1—C2—H2B | 108.9 |
N2—Ag1—N1i | 72.03 (10) | H2A—C2—H2B | 107.7 |
N2i—Ag1—N1i | 148.68 (10) | N3—C3—C4 | 108.2 (3) |
N1—Ag1—N1i | 91.21 (14) | N3—C3—H3A | 125.9 |
O3—Cl1—O3ii | 109.6 (4) | C4—C3—H3A | 125.9 |
O3—Cl1—O2ii | 106.87 (19) | C3—C4—C5 | 105.1 (4) |
O3ii—Cl1—O2ii | 111.4 (2) | C3—C4—H4A | 127.5 |
O3—Cl1—O2 | 111.4 (2) | C5—C4—H4A | 127.5 |
O3ii—Cl1—O2 | 106.87 (19) | N1—C5—C4 | 110.7 (3) |
O2ii—Cl1—O2 | 110.8 (3) | N1—C5—C10i | 119.5 (3) |
C5—N1—N3 | 105.1 (3) | C4—C5—C10i | 129.8 (3) |
C5—N1—Ag1 | 112.2 (2) | N2—C6—C7 | 123.1 (4) |
N3—N1—Ag1 | 140.9 (2) | N2—C6—H6A | 118.4 |
C6—N2—C10 | 118.4 (3) | C7—C6—H6A | 118.4 |
C6—N2—Ag1 | 123.5 (2) | C8—C7—C6 | 118.4 (4) |
C10—N2—Ag1 | 118.0 (2) | C8—C7—H7A | 120.8 |
C3—N3—N1 | 111.0 (3) | C6—C7—H7A | 120.8 |
C3—N3—C2 | 128.6 (3) | C7—C8—C9 | 119.6 (4) |
N1—N3—C2 | 120.5 (3) | C7—C8—H8A | 120.2 |
O1—C1—O1i | 76.5 (6) | C9—C8—H8A | 120.2 |
O1—C1—C2i | 112.5 (3) | C8—C9—C10 | 119.8 (4) |
O1i—C1—C2i | 118.5 (3) | C8—C9—H9A | 120.1 |
O1—C1—C2 | 118.5 (3) | C10—C9—H9A | 120.1 |
O1i—C1—C2 | 112.5 (3) | N2—C10—C9 | 120.6 (3) |
C2i—C1—C2 | 113.6 (5) | N2—C10—C5i | 117.2 (3) |
O1—C1—H1B | 102.8 | C9—C10—C5i | 122.1 (3) |
C2i—C1—H1B | 103.5 | C1—O1—O1i | 51.7 (3) |
C2—C1—H1B | 103.5 | C1—O1—H1A | 109.3 |
N3—C2—C1 | 113.2 (3) | O1i—O1—H1A | 127.8 |
N3—C2—H2A | 108.9 | ||
N2—Ag1—N1—C5 | 157.0 (2) | C2—N3—C3—C4 | −179.3 (3) |
N2i—Ag1—N1—C5 | 6.1 (2) | N3—C3—C4—C5 | −0.2 (4) |
N1i—Ag1—N1—C5 | −147.0 (3) | N3—N1—C5—C4 | 0.3 (4) |
N2—Ag1—N1—N3 | −41.1 (4) | Ag1—N1—C5—C4 | 168.6 (2) |
N2i—Ag1—N1—N3 | 168.0 (3) | N3—N1—C5—C10i | −179.4 (3) |
N1i—Ag1—N1—N3 | 14.9 (3) | Ag1—N1—C5—C10i | −11.1 (4) |
N2i—Ag1—N2—C6 | 15.6 (3) | C3—C4—C5—N1 | −0.1 (4) |
N1—Ag1—N2—C6 | −124.4 (3) | C3—C4—C5—C10i | 179.6 (3) |
N1i—Ag1—N2—C6 | 175.0 (3) | C10—N2—C6—C7 | −1.5 (6) |
N2i—Ag1—N2—C10 | −159.8 (3) | Ag1—N2—C6—C7 | −176.8 (3) |
N1—Ag1—N2—C10 | 60.2 (3) | N2—C6—C7—C8 | 1.5 (6) |
N1i—Ag1—N2—C10 | −0.4 (2) | C6—C7—C8—C9 | −0.2 (6) |
C5—N1—N3—C3 | −0.5 (4) | C7—C8—C9—C10 | −1.0 (6) |
Ag1—N1—N3—C3 | −163.1 (3) | C6—N2—C10—C9 | 0.1 (5) |
C5—N1—N3—C2 | 179.3 (3) | Ag1—N2—C10—C9 | 175.8 (3) |
Ag1—N1—N3—C2 | 16.7 (5) | C6—N2—C10—C5i | 179.4 (3) |
C3—N3—C2—C1 | 77.9 (5) | Ag1—N2—C10—C5i | −5.0 (4) |
N1—N3—C2—C1 | −101.9 (4) | C8—C9—C10—N2 | 1.1 (5) |
O1—C1—C2—N3 | −69.5 (5) | C8—C9—C10—C5i | −178.1 (3) |
O1i—C1—C2—N3 | −155.9 (4) | C2i—C1—O1—O1i | 115.6 (4) |
C2i—C1—C2—N3 | 65.9 (2) | C2—C1—O1—O1i | −108.5 (4) |
N1—N3—C3—C4 | 0.5 (4) |
Symmetry codes: (i) −x+1, y, −z+3/2; (ii) −x+1, y, −z+5/2. |
D—H···A | D—H | H···A | D···A | D—H···A |
O1—H1A···O3iii | 0.82 | 2.21 | 3.017 (7) | 170 |
C8—H8A···O2iv | 0.93 | 2.60 | 3.315 (5) | 134 |
Symmetry codes: (iii) x+1/2, −y+3/2, z−1/2; (iv) −x+1/2, y−1/2, −z+3/2. |
Experimental details
Crystal data | |
Chemical formula | [Ag(C19H18N6O)]ClO4 |
Mr | 553.71 |
Crystal system, space group | Monoclinic, C2/c |
Temperature (K) | 293 |
a, b, c (Å) | 8.687 (3), 21.863 (9), 10.904 (4) |
β (°) | 95.482 (7) |
V (Å3) | 2061.4 (14) |
Z | 4 |
Radiation type | Mo Kα |
µ (mm−1) | 1.15 |
Crystal size (mm) | 0.20 × 0.20 × 0.18 |
Data collection | |
Diffractometer | Bruker SMART CCD area-detector |
Absorption correction | Multi-scan (SADABS; Bruker, 1998) |
Tmin, Tmax | 0.802, 0.819 |
No. of measured, independent and observed [I > 2σ(I)] reflections | 5960, 2135, 1497 |
Rint | 0.031 |
(sin θ/λ)max (Å−1) | 0.628 |
Refinement | |
R[F2 > 2σ(F2)], wR(F2), S | 0.038, 0.088, 1.00 |
No. of reflections | 2135 |
No. of parameters | 151 |
H-atom treatment | H-atom parameters constrained |
Δρmax, Δρmin (e Å−3) | 0.53, −0.46 |
Computer programs: SMART (Bruker, 1998), SAINT (Bruker, 1998), SAINT, SHELXS97 (Sheldrick, 1997), SHELXL97 (Sheldrick, 1997), SHELXTL (Bruker, 1998), SHELXTL and PLATON (Spek, 2003).
Ag1—N2 | 2.260 (3) | Ag1—N1 | 2.391 (3) |
N2—Ag1—N2i | 134.08 (15) | N2—Ag1—N1i | 72.03 (10) |
N2—Ag1—N1 | 148.68 (10) | N1—Ag1—N1i | 91.21 (14) |
Symmetry code: (i) −x+1, y, −z+3/2. |
D—H···A | D—H | H···A | D···A | D—H···A |
O1—H1A···O3ii | 0.82 | 2.21 | 3.017 (7) | 170 |
C8—H8A···O2iii | 0.93 | 2.60 | 3.315 (5) | 134 |
Symmetry codes: (ii) x+1/2, −y+3/2, z−1/2; (iii) −x+1/2, y−1/2, −z+3/2. |
In recent years, 3-(2-pyridyl)pyrazole-based ligands have found a wide range of application in the area of coordination chemistry, because they can act as bridging or chelate ligands and exhibit a series of intriguing structures and potential applications as functional materials (Ruben et al., 2004; Steel et al., 2005). Nowadays, much attention has been focused on the synthetic approach and the structural control of coordination architectures (Bell et al., 2003; Paul et al., 2004). We report here the structure of a mononuclear silver complex, {1,3-bis[3-(2-pyridyl)pyrazole]propan-2-ol}silver(I) perchlorate.
In the title compound, the cation and anion both lie on crystallographic twofold rotation axes. In the cation, the twofold axis passes through atoms Ag1 and C1, and as a result the hydroxyl group is disordered. The AgI center is four-coordinated by four N donors from a 1,3-bis[3-(2-pyridyl)pyrazole]propan-2-ol ligand (Table 1). The coordination geometry around the AgI center can be described as a distorted tetrahedron (Fig. 1).
The AgI mononuclear units are linked to the perchlorate ions through O—H···O hydrogen bonds (Table 2) and weak C—H···O interactions (Desiraju et al., 1999; Barberà et al., 2002) leading to the formation of a one-dimensional chain (Fig. 2).