The joint probability distribution function of structure factors with rational indices. IV. The P1 case.

The method of the joint probability distribution functions of structure factors has been extended to reflections with rational indices. The most general case, space group P1, has been considered. The positional parameters are the primitive random variables of our probabilistic approach, while the reflection indices are kept fixed. Quite general joint probability distributions have been considered from which conditional distributions have been derived: these proved applicable to the accurate estimation of the real and imaginary parts of a structure factor, given prior information on other structure factors. The method is also discussed in relation to the Hilbert-transform techniques.

The joint probability distribution function of structure factors with rational indices. V. The estimates.

The probabilistic formulae [Giacovazzo, Siliqi & Fernández-Castaño (1999). Acta Cryst. A55, 512-524] relating standard and half-integral index reflections are modified for practical applications. The experimental tests prove the reliability of the probabilistic relationships. The approach is further developed to explore whether the moduli of the half-integral index reflections can be evaluated in the absence of phase information; i.e. by exploiting the moduli of the standard reflections only. The final formulae indicate that estimates can be obtained, even though the reliability factor is a constant.

The joint probability distribution function of structure factors with rational indices. VI. Cases with reduced dimensionality.

The probabilistic formulas relating standard and mixed type reflections (these last show integral and half-integral indices) are derived. It is shown that probabilistic estimates can be obtained by using particular sections of the three-dimensional reciprocal space. The concept of structure invariant is extended to define the wider class of structure quasi-invariant. Their statistical behaviour is briefly discussed with the help of some practical tests.

Self-assembly of NH-pyrazoles via intermolecular N-H.N hydrogen bonds.

The crystal structures of two NH-pyrazole derivatives forming intermolecular N-H.N hydrogen bonds are reported: 5-methyl-4-(3-methylpyrazol-5-yl)pyrazol-3-ol, C(8)H(10)N(4)O (P1), and 3-methyl-5-dihydro-1H-naphtho[1,2-d]pyrazole hemihydrochloride, C(12)H(12)N(2).-C(12)H(13)N(2)(+).Cl(-) (P2). 26 other structures are surveyed in order to obtain a deeper insight into the ways NH-pyrazoles self-assemble by means of intermolecular N-H.N hydrogen bonds in molecular crystals. A limited number of compounds form chains or dimers via homonuclear N(+)-H.N positive-charge-assisted hydrogen bonds, typical of proton sponges, which can be remarkably short [e.g. N.N 2.714 (3), N-H 1.09 (3), H.N 1.63 (3) Å, N-H.N 169 (3) degrees in (P2)]. Most pyrazoles, however, pack via neutral N-H.N bonds which are formally assisted by resonance (resonance-assisted hydrogen bond, RAHB) through the.N=C-C=C-NH. iminoenamine fragment, contained in the ring, giving rise to dimers, trimers, tetramers and infinite chains of pyrazole molecules. Surprisingly, the resonance does not appear to shorten the N-H.N bond with respect to the accepted mean value N.N 2.97 (10) Å for non-resonant N-H.N bonds. It is shown that this is due to the internal pi-delocalization of the pyrazole ring, which can be hardly increased by the hydrogen-bond interaction, except in symmetrically 3,5-substituted pyrazoles which display N.N distances as short as 2.82 Å, identical C-C and C-N distances in the two halves of the pyrazole molecule, and typical phenomena of N-H.N dynamical proton disorder, detectable by (15)N-CP/MAS solid-state NMR.