Massimo Nespolo's research themes

The "Charge Distribution" method

Theoretical background

Nespolo et al. (2001) (pdf - 363Kb)

 

1. Introduction

The Charge Distribution method ( CD or CHARDI ), introduced by Hoppe et al. (1989),  is the most recent development of the classical theory of the bond strength (Pauling, 1929) and differs from the Bond Valence (BV) approach (e.g. Brown, 1978) in exploiting the true bond distances in a self-consistent computation, instead of employing empirical curves.

The core of the method consists in the computation of ECoN - Effective Coordination Number (Hoppe, 1979) around an atom M (q≥ 0) which is coordinated by atoms X (q< 0) (q is the formal oxidation number ). This is a real (non-integer) number which takes into account not only the number of atoms around a given atoms, but also their weight in terms of relative distances. For regular or quasi-regular coordination polyhedra (all the M-X distances identical) ECoN reduces to the classical coordination number.

The fractional contribution given by each type of X to ECoN of M is weighted with the formal oxidation numbers of the M and X atoms, and the sums around M and around X give what we call "charges". How to read these "charges" is described below.

CHARDI in practice works on an ionic-analogue of the real crystal: all the atoms in the structure are formally replaced by point charges. The method however does not make reference to an ionic model of bonding, but is applies equally well to covalent bonds and to hydrogen bonds (see section 4 below). It requires, however, that the compound be a normal valence compound, (Parthé 1996) ¹: in other words, a compound containing only bonds of type M-X, not of type M-M or X-X (ECoN does not suffer this limitation).

¹Rigorously speaking, normal valence compounds defined according to Parthé (1996) do not include cations with lone-pairs. This condition is not influential to apply CHARDI, and thus we include them in a slighlt enlarged definition.

2. Notation

M(ij) = cation site; i = atomic species, j = crystallographic type

A(rs) = anion site; r = atomic species, s = crystallographic type

h (ij), h (rs): multiplicities of the Wyckoff positions for M(ij) and A(rs) respectively

sof(ij), sof(rs): site occupancy fraction for M(ij) and A(rs) respectively

q (i), q(r): formal oxidation numbers of M(i) and A(r) respectively (independent from jand s)

Q(ij), Q(rs): computed 'charges'  of M(ij) and A(rs) respectively

d (ij→rs)_L:L-th distance between M(ij) and A(rs) [distances are ordered increasingly, sis a dummy index for the distances]

ⁿd (ij→r): weighted mean distance M(ij) and A(r), after convergence at the n-th cycle of calculation

3. Calculation procedure.

The first step consists in calculating the weighted mean distance (w.m.d.) as:

where

i. e. the w.m.d. at the zero-th stage is the shortest M-A distance. The "strength" of the d (ij→rs) _L bond is measured by the bond weight ⁿ BW(ij→rs) _L:

 

and the Effective Coordination Number ⁿ ECoN is defined both in terms of M(ij) and A(r), ⁿ ECoN(ij→r), and of M(ij) alone, ⁿ ECoN(ij):

The contribution by the A(rs) anion to ⁿECoN(ij→r) is:

The fraction of the formal charge q (ij) that the cation M(ij) shares with the anion A(rs) is obtained by simply multiplying ΔⁿECoN by q(ij), and by introducing a scale factor ⁿF(ij → r ) for hetero-ligand polyhedra (polyhedra with different anions):

The computed "charge" of the anions is then obtained by summing up each fraction of charge, taking however into account the ratio of the multiplicities of the respective Wyckoff positions (to avoid counting more than once the contributions from the same cation):

The "charge" of the cation M(ij) is computed as the weighted sum of q(rs)/ⁿQ(rs) for the anions A(rs) bonded to M(ij), where the weight is the fraction of shared charge q(ij)

A structure which is correctly solved and perfectly valence-balanced has both q(rs)/ⁿQ(rs) and q(ij)/Q(ij) ideally equal to 1. Structural strains affect the ratio q(rs)/ⁿQ(rs), which deviates from 1: in this case we speak of over-under-bonding (OUB) effect. Some of the q(rs)/ⁿQ(rs) ratios deviate from 1, but all the q(rs)/ⁿQ(rs) ratios corresponding to A(rs)bonded to a given cation enter in the calculation of Q(ij), and q(ij)/Q(ij) should not be affected by the OUB effect. Reasons for q(ij)/Q(ij) significantly deviating from 1 can be:

1. the refined structure model is inadequate including overlooked (light) atoms and disorder
2. wrong assignment of oxidation numbers, including the case of sites with isomorphous substitutions
3. presence of polyions, which are not accounted for by the M(ij)A(rs) patterns assumed by the CD method
4. the coordination of one or more cations is so distorted that the polyhedral description is too approximate.

When instead q(ij)/Q(ij) is reasonably close to 1, ⁿQ(rs),ⁿECoN(ij→r) and, with reference to each bond, ⁿBW(ij→r)_L, are suitable parameters to investigate structural changes, e. g. as a function of composition, temperature and pressure. Therefore, differently from the BV method, in the CD approach the computed "charges" (Q) of the cations and of the anions convey different information. The analysis of the structural details on the basis of the strength of each bond (termed "bond valence" in the BV and "bond weight" in the CD) is meaningful only when the structure is correctly refined and the empirical method itself is applicable. The CD method has in (q/Q)_cations ratio precisely this kind of criterion (Nespolo et al., 2001).

4. Structures containing hydrogen bonds

From the viewpoint of the coordination geometry, the hydrogen bond A1-H ···A2 has some features that make it unique and not treatable by the CD method in its original form:

1. the H atom has only two bonds except in the rare cases of polyfurcated hydrogen bonds;
2. very short A1-H distances;
3. large relative gap in length between the donor and the acceptor bonds, d₂₁ = d(ij→rs)₂ - d(ij→rs)₁.

These features result in a high ratio d(ij→rs)₂/d(ij→rs)₁ and in practice the bond with the acceptor, d(ij→rs)₂, is overlooked in the calculation of the weighted bond distance. The hydrogen bond is in this respect unique because other cations may show the same order of gap d₂₁ but the longer bond lengths result in a much lower d(ij→rs)₂/d(ij→rs)₁ratio.

The only empirical parameter employed by the CD method is the so-called contraction parameter, which consists of the exponent 6. This contraction parameter was chosen because it resulted in ¹ECoN identical with the classical coordination number for simple and regular structures (Hoppe, 1979). However, the decreasing weight it induces with the increasing distances d(ij→rs)_Lis too rapid for the case of hydrogen bond. For M(ij) = H a different contraction parameter has been obtained but analyzing a set of 119 structured refined by single-crystal neutron diffraction data extracted from the Inorganic Crystal Structure Database The contraction parameter found in this way is 1.6.

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The FORTRAN program

CHARDI-IT (CHARge DIstribution - ITerative algorithm) is a Fortran program that computes the Charge Distribution in non-molecular structures. The previous version of the program, CHARDIS99 (Nespolo et al., 1999) employed a non-iterative algorithm and had some limitations: in particular, structures with highly distorted polyhedra, hetero-ligand polyhedra (different species of anions) and hydrogen bonds were not treated correctly. These limitations have now been overcome by introducing the iterative algorithm originally proposed in Hoppe (1979) for the calculation of MEFIR and ECoN.

The CHARDI-IT program accepts three type of input: free-format, the ICSD old format, and bond distances. The first two formats require in input the cell constants, atomic coordinates and symmetry matrices. The program now accepts in input also bond distances: however also distances longer (at least 0.5ﾅ) than those commonly considered in a coordination polyhedron MUST be provided, because also longer distances ones may significantly contribute to ECoN. Instructions and examples are included in the downloadable ZIP file.

To decompress the archive, the Winzip utility is required. It can be download by clicking on the Winzip icon below. The explanatory text is in pdf (Portable Document Format). To open it, Acrobat Reader is required, and can be download by clicking on the corresponding icon below

Comments, bug reports and feedback are always

Update

A small but significant update has been on May 8, 2006, which addresses two points:

1. The limit number of independent atoms centring the coordination polyhedra has been incremented from 30 to 100
2. A bug that caused incomplete application of the lattice translations when fractional coordinates more negative than -0.5 occur has been fixed

Further and more important updates will follow shortly. Stay tuned!

DOWNLOAD CHARDI-IT (zip - 325Kb)

download electronic reprint (pdf - 363Kb)

 

 

References

 

Brown, I. D. (1978) Bond valences - a simple structural model for inorganic chemistry. Chem. Soc. Rev. 7, 359-376.

Hoppe, R. (1979). Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR). Z. Kristallogr. , 150, 23-52.

Hoppe, R., Voigt, S., Glaum, H., Kissel, J., Müller, H. P., Bernet, K. (1989). A new route to charge distributions in ionic solids. J. Less-Comm. Met. , 156, 105-122.

Nespolo, M., Ferraris, G.,  Ohashi, H. (1999). Charge Distribution as a tool to investigate structural details: meaning and application to pyroxenes. Acta Crystallogr. B55, 902-916.

Nespolo, M., Ferraris, G., Ivaldi, G., Hoppe, R, (2001). Charge Distribution as a tool to investigate structural details. II. Extension to hydrogen bonds, distorted and hetero-ligand polyhedra. Acta Crystallogr. B57, 652-664.

Parthé E. (1996). Elements of Inorganic Structural Chemistry, second edition. Petit-Lancy: K. Sutter Parthé

Pauling, L. (1929). The principles determining the structure of complex ionic crystals. J. Am. Chem. Soc. , 51, 1010-1026.