supplementary materials


Acta Cryst. (2007). E63, i171    [ doi:10.1107/S160053680703615X ]

Dipotassium hexafluoridozirconate(IV) hydrogen fluoride, K2ZrF6·HF

A. V. Gerasimenko, N. A. Didenko and V. Y. Kavun

Abstract top

The title compound is built from edge-sharing centrosymmetric dimeric [Zr2F12]4- complexes, K+ cations and HF molecules. The ZrIV coordination polyhedron is a distorted monocapped octahedron. The K+ ions are coordinated by F atoms and HF molecules with potassium coordination numbers of 7 and 8. In addition, the structure is stabilized by strong F-H...F hydrogen bonds.

Comment top

The title compound, (I), is isostructural to K2Rb2(ZrF6)2.2H2O, (II), if hydrogen atoms are neglected (Tkachev et al., 1993). The asymmetric unit contains one crystallographically independent Zr atom, six fluorine atoms, two potassium cations and one hydrogen fluoride molecule. The Zr atoms are coordinated by seven F atoms in a distorted monocapped octahedral geometry. Two Zr-centred polyhedra are linked by double F bridges to form a centrosymmetric dimeric [Zr2F12]4− complex (Fig. 1) with a Zr···Zri distance of 3.6810 (2) Å. Parallel to the ac plane, layers of K1 atoms and HF molecules alternate with layers of K2 atoms and [Zr2F12]4− complexes (Fig. 2).

The coordination numbers (CN) of potassium atoms were calculated by the method of intersecting spheres (Serezhkin et al., 1997) with use of the program package TOPOS (Blatov, 2004). For K1, the CN is 8 (K1–F, 2.6214 (6)–2.9848 (9) Å) and for K2, the CN is 7 (K2–F, 2.6365 (6)–2.8774 (9) Å).

The HF molecule is coordinated to both potassium atoms and is involved as a donor in the strong hydrogen bond (2.3865 (10) Å) with the F6 atom (Fig. 2). Unlike the structure of (II), where the water molecule forms two hydrogen bonds with an F6 atom (2.88 Å) and a bridging F3 atom (2.74 Å) (Tkachev et al., 1993), in (I) there is only one hydrogen bond, resulting in a lengthened Zr–F6 bond (2.0790 (6) Å) and nearly equal Zr–F3 and Zr–F3i bridging bonds (2.1696 (6) and 2.1833 (6) Å). In (II), the corresponding bonds are 2.046, 2.188, and 2.199 Å, respectively (Tkachev et al., 1993).

Related literature top

For related literature, see: Blatov (2004); Serezhkin et al. (1997); Tkachev et al. (1993).

Experimental top

ZrO2 (6.2 g, 0.05 mol) was reacted with KHF2 (7.9 g, 0.1 mol) in a solution of hydrofluoric acid (40%, 30 ml). Upon slow evaporation at room temperature, crystals of K2ZrF6 precipitated first, and after they were separated, crystals of (I) precipitated next.

Refinement top

The H1 atom was found in a difference Fourier map and refined with Uiso(H1) = 1.5Ueq(F7). The maximum peak and deepest hole are located 0.51 Å and 0.97 Å, respectively, from Zr and K2.

Computing details top

Data collection: SMART (Bruker, 1998a); cell refinement: SAINT (Bruker, 2003); data reduction: SAINT; program(s) used to solve structure: SHELXTL (Bruker, 1998b); program(s) used to refine structure: SHELXTL; molecular graphics: SHELXTL; software used to prepare material for publication: SHELXTL.

Figures top
[Figure 1] Fig. 1. A view of the centrosymmetric dimeric [Zr2F12]4− complex, with displacement ellipsoids drawn at the 50% probability level.
[Figure 2] Fig. 2. The structure of (I), viewed along the c axis.
Dipotassium hexafluoridozirconate(IV) hydrogen fluoride top
Crystal data top
K2ZrF6·HFF(000) = 1136
Mr = 303.43Dx = 3.093 Mg m3
Orthorhombic, PbcaMo Kα radiation, λ = 0.71073 Å
Hall symbol: -P 2ac 2abCell parameters from 1939 reflections
a = 13.3641 (5) Åθ = 3.6–45.0°
b = 11.4457 (4) ŵ = 3.03 mm1
c = 8.5211 (3) ÅT = 203 K
V = 1303.40 (8) Å3Prism, colourless
Z = 80.44 × 0.37 × 0.27 mm
Data collection top
Bruker SMART 1000 CCD area-detector
diffractometer
5275 independent reflections
Radiation source: fine-focus sealed tube4834 reflections with I > 2σ(I)
graphiteRint = 0.030
Detector resolution: 8.33 pixels mm-1θmax = 45.0°, θmin = 3.1°
ω scansh = 2623
Absorption correction: gaussian
(SADABS in XPREP; Bruker, 2003)
k = 2022
Tmin = 0.443, Tmax = 0.535l = 1616
23595 measured reflections
Refinement top
Refinement on F2Secondary atom site location: difference Fourier map
Least-squares matrix: fullHydrogen site location: difference Fourier map
R[F2 > 2σ(F2)] = 0.030Only H-atom coordinates refined
wR(F2) = 0.071 w = 1/[σ2(Fo2) + (0.0278P)2 + 0.8258P]
where P = (Fo2 + 2Fc2)/3
S = 1.21(Δ/σ)max = 0.003
5275 reflectionsΔρmax = 1.58 e Å3
95 parametersΔρmin = 1.30 e Å3
0 restraintsExtinction correction: SHELXTL, Fc*=kFc[1+0.001xFc2λ3/sin(2θ)]-1/4
Primary atom site location: structure-invariant direct methodsExtinction coefficient: 0.0049 (2)
Crystal data top
K2ZrF6·HFV = 1303.40 (8) Å3
Mr = 303.43Z = 8
Orthorhombic, PbcaMo Kα radiation
a = 13.3641 (5) ŵ = 3.03 mm1
b = 11.4457 (4) ÅT = 203 K
c = 8.5211 (3) Å0.44 × 0.37 × 0.27 mm
Data collection top
Bruker SMART 1000 CCD area-detector
diffractometer
5275 independent reflections
Absorption correction: gaussian
(SADABS in XPREP; Bruker, 2003)
4834 reflections with I > 2σ(I)
Tmin = 0.443, Tmax = 0.535Rint = 0.030
23595 measured reflectionsθmax = 45.0°
Refinement top
R[F2 > 2σ(F2)] = 0.030Only H-atom coordinates refined
wR(F2) = 0.071Δρmax = 1.58 e Å3
S = 1.21Δρmin = 1.30 e Å3
5275 reflectionsAbsolute structure: ?
95 parametersFlack parameter: ?
0 restraintsRogers parameter: ?
Special details top

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.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/Ueq
Zr0.118901 (5)0.040452 (6)0.094477 (8)0.00846 (1)
K10.170769 (16)0.740441 (15)0.84850 (2)0.01749 (3)
K20.106835 (15)0.06846 (2)0.61088 (3)0.01853 (3)
F10.08118 (5)0.02918 (5)0.30531 (7)0.01728 (10)
F20.20020 (4)0.10800 (5)0.07447 (7)0.01528 (9)
F30.04308 (4)0.06348 (6)0.08217 (8)0.02043 (11)
F40.25374 (5)0.09337 (6)0.16899 (10)0.02846 (15)
F50.15406 (7)0.12846 (7)0.09782 (8)0.02574 (14)
F60.08494 (5)0.19919 (5)0.20043 (8)0.02147 (12)
F70.03330 (6)0.18673 (8)0.40966 (9)0.02981 (17)
H10.0108 (18)0.198 (2)0.335 (3)0.045*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Zr0.00841 (2)0.00835 (2)0.00861 (3)0.00061 (2)0.00108 (2)0.00001 (2)
K10.02166 (7)0.01343 (5)0.01738 (7)0.00241 (5)0.00038 (5)0.00296 (5)
K20.01207 (6)0.02388 (7)0.01963 (7)0.00100 (5)0.00039 (5)0.00405 (6)
F10.0187 (2)0.0189 (2)0.0143 (2)0.00307 (17)0.00298 (17)0.00444 (16)
F20.01427 (19)0.01355 (17)0.0180 (2)0.00254 (15)0.00253 (16)0.00328 (15)
F30.01218 (19)0.0261 (2)0.0230 (2)0.00429 (18)0.00479 (17)0.01411 (19)
F40.0159 (2)0.0211 (2)0.0483 (4)0.00098 (19)0.0129 (3)0.0108 (3)
F50.0333 (3)0.0260 (3)0.0179 (3)0.0053 (3)0.0052 (2)0.0075 (2)
F60.0268 (3)0.01294 (18)0.0247 (3)0.00303 (18)0.0042 (2)0.00626 (18)
F70.0254 (3)0.0383 (4)0.0258 (3)0.0039 (3)0.0074 (2)0.0036 (3)
Geometric parameters (Å, °) top
Zr—F51.9799 (7)K2—F4xiv2.6735 (7)
Zr—F42.0043 (7)K2—F6x2.7822 (7)
Zr—F22.0241 (5)K2—F12.8541 (6)
Zr—F12.0290 (6)K2—F72.8774 (9)
Zr—F62.0790 (6)K2—F7v3.0867 (10)
Zr—F32.1696 (6)K2—F3xv3.1381 (7)
Zr—F3i2.1833 (6)K2—K2v3.7657 (4)
Zr—K2ii3.8740 (2)K2—Zrxiv3.8740 (2)
Zr—K1iii3.9373 (2)K2—K1xvi4.0997 (3)
Zr—K1iv4.0823 (2)F1—K2v2.6506 (7)
Zr—K2v4.1180 (2)F1—K1iii2.7232 (6)
Zr—K2vi4.1364 (3)F2—K1iv2.6214 (6)
K1—F2vii2.6214 (6)F2—K2ii2.6365 (6)
K1—F4viii2.6411 (7)F2—K1iii2.8115 (6)
K1—F5ix2.7079 (8)F3—Zri2.1833 (6)
K1—F1x2.7232 (6)F3—K1iv2.8805 (7)
K1—F2x2.8115 (6)F3—K2vi3.1381 (7)
K1—F7xi2.8284 (8)F4—K1xvii2.6411 (7)
K1—F3vii2.8805 (7)F4—K2ii2.6735 (7)
K1—F7xii2.9848 (9)F4—K1xviii3.3629 (9)
K1—F4ix3.3629 (9)F5—K2vi2.6517 (7)
K1—Zrx3.9373 (2)F5—K1xviii2.7079 (8)
K1—Zrvii4.0823 (2)F6—K2iii2.7822 (6)
K1—K2xiii4.0997 (3)F7—K1xix2.8284 (8)
K2—F2xiv2.6365 (6)F7—K1xii2.9848 (9)
K2—F1v2.6506 (7)F7—K2v3.0867 (10)
K2—F5xv2.6517 (7)
F5—Zr—F483.98 (4)F7xi—K1—Zrvii72.45 (2)
F5—Zr—F2103.30 (3)F3vii—K1—Zrvii30.543 (12)
F4—Zr—F278.33 (3)F7xii—K1—Zrvii92.234 (18)
F5—Zr—F1172.54 (3)F4ix—K1—Zrvii93.154 (13)
F4—Zr—F193.53 (3)Zrx—K1—Zrvii159.923 (6)
F2—Zr—F183.00 (2)F2vii—K1—K2xiii124.870 (15)
F5—Zr—F688.09 (3)F4viii—K1—K2xiii77.459 (17)
F4—Zr—F678.15 (3)F5ix—K1—K2xiii39.595 (15)
F2—Zr—F6152.55 (3)F1x—K1—K2xiii79.959 (14)
F1—Zr—F684.52 (3)F2x—K1—K2xiii39.604 (12)
F5—Zr—F379.37 (3)F7xi—K1—K2xiii136.58 (2)
F4—Zr—F3143.64 (3)F3vii—K1—K2xiii154.800 (16)
F2—Zr—F374.45 (2)F7xii—K1—K2xiii102.494 (16)
F1—Zr—F3106.43 (3)F4ix—K1—K2xiii86.747 (13)
F6—Zr—F3132.73 (3)Zrx—K1—K2xiii57.592 (4)
F5—Zr—F3i97.71 (3)Zrvii—K1—K2xiii141.310 (6)
F4—Zr—F3i150.48 (3)F2xiv—K2—F1v171.13 (2)
F2—Zr—F3i129.02 (2)F2xiv—K2—F5xv80.38 (2)
F1—Zr—F3i81.00 (3)F1v—K2—F5xv90.99 (2)
F6—Zr—F3i72.48 (3)F2xiv—K2—F4xiv57.253 (19)
F3—Zr—F3i64.52 (3)F1v—K2—F4xiv119.55 (2)
F5—Zr—K2ii88.23 (3)F5xv—K2—F4xiv80.80 (3)
F4—Zr—K2ii40.11 (2)F2xiv—K2—F6x88.34 (2)
F2—Zr—K2ii39.196 (16)F1v—K2—F6x89.34 (2)
F1—Zr—K2ii94.399 (18)F5xv—K2—F6x61.36 (2)
F6—Zr—K2ii118.17 (2)F4xiv—K2—F6x133.22 (2)
F3—Zr—K2ii106.885 (17)F2xiv—K2—F194.437 (19)
F3i—Zr—K2ii168.149 (19)F1v—K2—F193.756 (18)
F5—Zr—K1iii145.96 (2)F5xv—K2—F1169.73 (2)
F4—Zr—K1iii85.09 (2)F4xiv—K2—F188.93 (2)
F2—Zr—K1iii42.802 (17)F6x—K2—F1127.72 (2)
F1—Zr—K1iii40.202 (17)F2xiv—K2—F7119.06 (2)
F6—Zr—K1iii120.88 (2)F1v—K2—F767.88 (2)
F3—Zr—K1iii90.912 (19)F5xv—K2—F7126.22 (3)
F3i—Zr—K1iii107.430 (19)F4xiv—K2—F7152.86 (3)
K2ii—Zr—K1iii63.310 (5)F6x—K2—F769.22 (2)
F5—Zr—K1iv87.86 (2)F1—K2—F764.05 (2)
F4—Zr—K1iv105.32 (2)F2xiv—K2—F7v117.85 (2)
F2—Zr—K1iv32.804 (16)F1v—K2—F7v63.44 (2)
F1—Zr—K1iv99.579 (17)F5xv—K2—F7v111.96 (2)
F6—Zr—K1iv174.36 (2)F4xiv—K2—F7v64.96 (2)
F3—Zr—K1iv42.430 (17)F6x—K2—F7v152.45 (2)
F3i—Zr—K1iv104.196 (17)F1—K2—F7v62.582 (19)
K2ii—Zr—K1iv65.620 (5)F7—K2—F7v101.78 (2)
K1iii—Zr—K1iv64.248 (3)F2xiv—K2—F3xv116.502 (18)
F5—Zr—K2v146.34 (3)F1v—K2—F3xv55.698 (17)
F4—Zr—K2v123.85 (3)F5xv—K2—F3xv53.77 (2)
F2—Zr—K2v100.980 (17)F4xiv—K2—F3xv72.63 (2)
F1—Zr—K2v32.783 (18)F6x—K2—F3xv101.703 (19)
F6—Zr—K2v80.783 (19)F1—K2—F3xv122.639 (18)
F3—Zr—K2v85.118 (18)F7—K2—F3xv123.12 (2)
F3i—Zr—K2v48.660 (19)F7v—K2—F3xv60.360 (19)
K2ii—Zr—K2v125.009 (5)F2xiv—K2—K2v138.945 (17)
K1iii—Zr—K2v63.039 (5)F1v—K2—K2v49.139 (14)
K1iv—Zr—K2v100.533 (5)F5xv—K2—K2v139.29 (2)
F5—Zr—K2vi31.28 (2)F4xiv—K2—K2v109.46 (2)
F4—Zr—K2vi109.10 (3)F6x—K2—K2v117.109 (18)
F2—Zr—K2vi90.120 (17)F1—K2—K2v44.617 (13)
F1—Zr—K2vi154.536 (18)F7—K2—K2v53.362 (19)
F6—Zr—K2vi110.88 (2)F7v—K2—K2v48.418 (16)
F3—Zr—K2vi48.175 (19)F3xv—K2—K2v90.694 (14)
F3i—Zr—K2vi84.499 (17)F2xiv—K2—Zrxiv29.024 (12)
K2ii—Zr—K2vi95.611 (5)F1v—K2—Zrxiv148.434 (15)
K1iii—Zr—K2vi128.158 (5)F5xv—K2—Zrxiv83.797 (19)
K1iv—Zr—K2vi63.915 (5)F4xiv—K2—Zrxiv28.880 (14)
K2v—Zr—K2vi127.033 (4)F6x—K2—Zrxiv114.644 (16)
F2vii—K1—F4viii83.77 (2)F1—K2—Zrxiv87.405 (13)
F2vii—K1—F5ix93.40 (2)F7—K2—Zrxiv138.215 (17)
F4viii—K1—F5ix96.23 (2)F7v—K2—Zrxiv89.697 (17)
F2vii—K1—F1x138.84 (2)F3xv—K2—Zrxiv97.590 (12)
F4viii—K1—F1x136.82 (2)K2v—K2—Zrxiv124.431 (9)
F5ix—K1—F1x89.00 (2)F2xiv—K2—K1xvi42.828 (13)
F2vii—K1—F2x161.06 (2)F1v—K2—K1xvi129.466 (15)
F4viii—K1—F2x81.60 (2)F5xv—K2—K1xvi40.607 (18)
F5ix—K1—F2x76.358 (19)F4xiv—K2—K1xvi74.689 (18)
F1x—K1—F2x58.032 (17)F6x—K2—K1xvi58.790 (16)
F2vii—K1—F7xi72.91 (2)F1—K2—K1xvi136.521 (14)
F4viii—K1—F7xi145.75 (3)F7—K2—K1xvi122.70 (2)
F5ix—K1—F7xi109.61 (2)F7v—K2—K1xvi135.492 (17)
F1x—K1—F7xi67.66 (2)F3xv—K2—K1xvi90.930 (13)
F2x—K1—F7xi125.37 (2)K2v—K2—K1xvi175.841 (10)
F2vii—K1—F3vii54.737 (17)Zrxiv—K2—K1xvi59.097 (4)
F4viii—K1—F3vii77.52 (2)Zr—F1—K2v122.73 (3)
F5ix—K1—F3vii147.81 (2)Zr—F1—K1iii111.05 (2)
F1x—K1—F3vii117.33 (2)K2v—F1—K1iii103.28 (2)
F2x—K1—F3vii132.321 (18)Zr—F1—K2128.62 (3)
F7xi—K1—F3vii68.60 (2)K2v—F1—K286.244 (18)
F2vii—K1—F7xii116.68 (2)K1iii—F1—K299.88 (2)
F4viii—K1—F7xii66.87 (2)Zr—F2—K1iv122.47 (3)
F5ix—K1—F7xii142.08 (2)Zr—F2—K2ii111.78 (2)
F1x—K1—F7xii82.98 (2)K1iv—F2—K2ii110.29 (2)
F2x—K1—F7xii67.95 (2)Zr—F2—K1iii107.91 (2)
F7xi—K1—F7xii101.41 (3)K1iv—F2—K1iii103.432 (19)
F3vii—K1—F7xii64.47 (2)K2ii—F2—K1iii97.567 (19)
F2vii—K1—F4ix71.922 (18)Zr—F3—Zri115.48 (3)
F4viii—K1—F4ix135.74 (3)Zr—F3—K1iv107.03 (2)
F5ix—K1—F4ix50.694 (19)Zri—F3—K1iv132.19 (3)
F1x—K1—F4ix78.296 (19)Zr—F3—K2vi100.81 (2)
F2x—K1—F4ix111.259 (18)Zri—F3—K2vi99.85 (2)
F7xi—K1—F4ix59.63 (2)K1iv—F3—K2vi92.473 (18)
F3vii—K1—F4ix113.649 (19)Zr—F4—K1xvii138.12 (3)
F7xii—K1—F4ix157.36 (2)Zr—F4—K2ii111.01 (3)
F2vii—K1—Zrx166.072 (14)K1xvii—F4—K2ii109.88 (2)
F4viii—K1—Zrx109.69 (2)Zr—F4—K1xviii99.42 (3)
F5ix—K1—Zrx81.870 (17)K1xvii—F4—K1xviii89.73 (2)
F1x—K1—Zrx28.747 (13)K2ii—F4—K1xviii89.27 (2)
F2x—K1—Zrx29.286 (11)Zr—F5—K2vi125.91 (4)
F7xi—K1—Zrx96.241 (19)Zr—F5—K1xviii125.86 (4)
F3vii—K1—Zrx130.140 (14)K2vi—F5—K1xviii99.80 (2)
F7xii—K1—Zrx73.412 (18)Zr—F6—K2iii133.89 (3)
F4ix—K1—Zrx95.194 (13)K1xix—F7—K295.21 (2)
F2vii—K1—Zrvii24.729 (12)K1xix—F7—K1xii94.38 (3)
F4viii—K1—Zrvii75.902 (18)K2—F7—K1xii166.20 (3)
F5ix—K1—Zrvii117.271 (17)K1xix—F7—K2v92.30 (2)
F1x—K1—Zrvii137.870 (15)K2—F7—K2v78.22 (2)
F2x—K1—Zrvii154.647 (13)K1xii—F7—K2v91.52 (2)
Symmetry codes: (i) −x, −y, −z; (ii) −x+1/2, −y, z−1/2; (iii) x, −y+1/2, z−1/2; (iv) x, y−1, z−1; (v) −x, −y, −z+1; (vi) x, y, z−1; (vii) x, y+1, z+1; (viii) −x+1/2, −y+1, z+1/2; (ix) −x+1/2, y+1/2, z+1; (x) x, −y+1/2, z+1/2; (xi) −x, y+1/2, −z+3/2; (xii) −x, −y+1, −z+1; (xiii) −x+1/2, y+1/2, z; (xiv) −x+1/2, −y, z+1/2; (xv) x, y, z+1; (xvi) −x+1/2, y−1/2, z; (xvii) −x+1/2, −y+1, z−1/2; (xviii) −x+1/2, y−1/2, z−1; (xix) −x, y−1/2, −z+3/2.
Hydrogen-bond geometry (Å, °) top
D—H···AD—HH···AD···AD—H···A
F7—H1···F60.87 (2)1.52 (2)2.3865 (10)172 (2)
Table 1
Selected geometric parameters (Å)
top
Zr—F51.9799 (7)K1—F2v2.8115 (6)
Zr—F42.0043 (7)K1—F7vi2.8284 (8)
Zr—F22.0241 (5)K1—F3ii2.8805 (7)
Zr—F12.0290 (6)K1—F7vii2.9848 (9)
Zr—F62.0790 (6)K2—F2viii2.6365 (6)
Zr—F32.1696 (6)K2—F1ix2.6506 (7)
Zr—F3i2.1833 (6)K2—F5x2.6517 (7)
K1—F2ii2.6214 (6)K2—F4viii2.6735 (7)
K1—F4iii2.6411 (7)K2—F6v2.7822 (7)
K1—F5iv2.7079 (8)K2—F12.8541 (6)
K1—F1v2.7232 (6)K2—F72.8774 (9)
Symmetry codes: (i) −x, −y, −z; (ii) x, y+1, z+1; (iii) −x+1/2, −y+1, z+1/2; (iv) −x+1/2, y+1/2, z+1; (v) x, −y+1/2, z+1/2; (vi) −x, y+1/2, −z+3/2; (vii) −x, −y+1, −z+1; (viii) −x+1/2, −y, z+1/2; (ix) −x, −y, −z+1; (x) x, y, z+1.
Table 2
Hydrogen-bond geometry (Å, °)
top
D—H···AD—HH···AD···AD—H···A
F7—H1···F60.87 (2)1.52 (2)2.3865 (10)172 (2)
Acknowledgements top

The authors thank the Russian Foundation for Basic Research (project No. 05–03-33298) for financial support.

references
References top

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