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By I. David Brown

This booklet describes the bond valence version, an outline of acid-base bonding that is changing into more and more well known quite in fields similar to fabrics technology and mineralogy the place sturdy nation inorganic chemistry is necessary. contemporary advancements in crystal constitution decision have allowed the version to develop into extra quantitative. in contrast to different versions of inorganic chemical bonding, the bond valence version is easy, intuitive, and predictive, and will be used for analysing crystal buildings and the conceptual modelling of neighborhood in addition to prolonged constructions. this can be the 1st publication to discover the theoretical foundation of the version and to teach the way it should be utilized to man made and answer chemistry. It emphasizes the separate roles of the limitations of chemistry and of third-dimensional area through analysing the chemistry of solids. Many purposes of the version in physics, fabrics technological know-how, chemistry, mineralogy, soil technology, floor technological know-how, and molecular biology are reviewed. the ultimate bankruptcy describes how the bond valence version pertains to and represents a simplification of alternative types of inorganic chemical bonding.

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3. 3. A related basis set of angular functions has been proposed by Hirshfeld (1977). They are of the form cosn k , where k is the angle with a specified set of …n ‡ 1†…n ‡ 2†=2 polar axes. The Hirshfeld functions are identical to a sum of spherical harmonics with l ˆ n, n À 2, n À 4, F F F …0, 1† for n > 1, as shown elsewhere (Hirshfeld, 1977). The radial functions R…r† can be selected in different manners. Several choices may be made, such as …1:2:7:1† The angular functions  are based on the spherical harmonic functions Ylm defined by   !

2l ‡ 1 …l À jmj†3 1=2 m m Ylm …, '† ˆ …À1† Pl …cos † exp…im'†, 4 …l ‡ jmj†3 …1:2:7:2a† with Àl m l, where Pml …cos † are the associated Legendre polynomials (see Arfken, 1970). djmj Pl …x† , dxjmj i 1 dl h Pl …x† ˆ l l …x2 À 1†l : l32 dx The real spherical harmonic functions ylmp , 0 m ‡ or À are obtained as a linear combination of Ylm : ! …2l ‡ 1†…l À jmj†3 1=2 m ylm‡ …, † ˆ Pl …cos † cos m' 2…1 ‡ m0 †…l ‡ jmj†3 Pml …x† ˆ …1 À x2 †jmj=2 ˆ Nlm Pml …cos † cos m' ˆ …À1†m …Ylm ‡ Yl; Àm † l, p ˆ …1:2:7:2b† and ylmÀ …, † ˆ Nlm Pml …cos † sin m' ˆ …À1†m …Ylm À Yl; Àm †=2i: …1:2:7:2c† The normalization constants Nlm are defined by the conditions ‚ 2 ylmp d ˆ 1, …1:2:7:3a† R l …r† ˆ where the coefficient nl may be selected by examination of products of hydrogenic orbitals which give rise to a particular multipole (Hansen & Coppens, 1978).

8) show that the Fourier transform of a directspace spherical harmonic function is a reciprocal-space spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fourier-transform invariant. 2)]; the subscript p is either + or À. 6) is frequently truncated at the hexadecapolar …l ˆ 4† level. For atoms at positions of high site symmetry the first allowed functions may occur at higher l values. For trigonally bonded atoms in organic molecules the l ˆ 3 terms are often found to be the most significantly populated deformation functions.

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