International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. C, ch. 8.7, pp. 714725
Section 8.7.3. Charge densities^{a}732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 142603000, USA,^{b}Digital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 017542122, USA, and ^{c}Ecole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F92295 Châtenay Malabry CEDEX, France 
The charge density is related to the elastic Xray scattering amplitude F(S) by the expression or, for scattering by a periodic lattice, As F(h) is in general complex, the Fourier transform (8.7.3.1) requires calculation of the phases from a model for the charge distribution. In the centrosymmetric case, the freeatom model is in general adequate for calculation of the signs of F(h). However, for noncentrosymmetric structures in which phases are continuously variable, it is necessary to incorporate deviations from the freeatom density in the model to obtain estimates of the experimental phases.
Since the total density is dominated by the core distribution, differences between the total density ρ(r) and reference densities are important. The reference densities represent hypothetical states without chemical bonding or with only partial chemical bonding. Deviations from spherical, freeatom symmetry are obtained when the reference state is the promolecule, the superposition of freespace spherical atoms centred at the nuclear positions. This difference function is referred to as the deformation density (or standard deformation density) where is the promolecule density. In analogy to (8.7.3.1), the deformation density may be obtained from where and are in general complex.
Several other different density functions analogous to (8.7.3.3), summarized in Table 8.7.3.1, may be defined. Particularly useful for the analysis of effects of chemical bonding is the fragment deformation density, in which a chemical fragment is subtracted from the total density of a molecule. The fragment density is calculated theoretically and thermally smeared before subtraction from an experimental density. `Prepared' atoms rather than spherical atoms may be used as a reference state to emphasize the electrondensity shift due to covalent bond formation.

The electron density ρ(r) in the structurefactor expression can be approximated by a sum of nonnormalized density functions with scattering factor centred at Substitution in (8.7.3.4a) gives
When is the spherically averaged, freeatom density, (8.7.3.4b) represents the freeatom model. A distinction is often made between atomcentred models, in which all functions g(r) are centred at the nuclear positions, and models in which additional functions are centred at other locations, such as in bonds or lonepair regions.
A simple, atomcentred model with spherical functions g(r) is defined by This `kappa model' allows for charge transfer between atomic valence shells through the population parameter , and for a change in nuclear screening with electron population, through the parameter κ, which represents an expansion , or a contraction of the radial density distribution.
The atomcentred, spherical harmonic expansion of the electronic part of the charge distribution is defined by where p = ± when m is larger than 0, and is a radial function.
The real spherical harmonic functions and their Fourier transforms have been described in International Tables for Crystallography, Volume B, Chapter 1.2 (Coppens, 2001). They differ from the functions by the normalization condition, defined as . The real spherical harmonic functions are often referred to as multipoles, since each represents the component of the charge distribution ρ(r) that gives a nonzero contribution to the integral for the electrostatic multipole moment where the functions are the Cartesian representations of the real spherical harmonics (Coppens, 2001).
More general models include nonatom centred functions. If the wavefunction ψ in (8.7.2.4) is an antisymmetrized product of molecular orbitals ψ_{i}, expressed in terms of a linear combination of atomic orbitals (LCAO formalism), the integration (8.7.2.4) leads to with R_{μ} and R_{ν} defining the centres of and , respectively, , and the sum is over all molecular orbitals with occupancy . Expression (8.7.3.9) contains products of atomic orbitals, which may have significant values for orbitals centred on adjacent atoms. In the `chargecloud' model (Hellner, 1977), these products are approximated by bondcentred, Gaussianshaped density functions. Such functions can often be projected efficiently into the onecentre terms of the spherical harmonic multipole model, so that large correlations occur if both spherical harmonics and bondcentred functions are adjusted independently in a leastsquares refinement.
According to (8.7.2.4) and (8.7.3.9), the population of the twocentre terms is related to the onecentre occupancies. A molecularorbital based model, which implicitly incorporates such relations, has been used to describe local bonding between transitionmetal and ligand atoms (Becker & Coppens, 1985).
There are several physical constraints that an electrondensity model must satisfy. With the exception of the electroneutrality constraint, they depend strongly on the electron density close to the nucleus, which is poorly determined by the diffraction experiment.
Since a crystal is neutral, the total electron population must equal the sum of the nuclear charges of the constituent atoms. A constraint procedure for linear least squares that does not increase the size of the leastsquares matrix has been described by Hamilton (1964). If the starting point is a neutral crystal, the constraint equation becomes where , g being a general density function, and the are the shifts in the population parameters. For the multipole model, only the monopolar functions integrate to a nonzero value. For normalized monopole functions, this gives If the shifts without constraints are given by the vector y and the constrained shifts by y_{c}, the Hamilton constraint is expressed as where the superscript T indicates transposition, A is the leastsquares matrix of the products of derivatives, and Q is a row vector of the values of for elements representing density functions and zeros otherwise.
Expression (8.7.3.12) cannot be applied if the unconstrained refinement corresponds to a singular matrix. This would be the case if all population parameters, including those of the core functions, were to be refined together with the scale factor. In this case, a new set of independent parameters must be defined, as described in Chapter 8.1 on leastsquares refinements. Alternatively, one may set the scale factor to one and rescale the population parameters to neutrality after completion of the refinement. This will in general give a nonintegral electron population for the core functions. The proper interpretation of such a result is that a corelike function is an appropriate component of the density basis set representing the valence electrons.
The electron density at a nucleus i with nuclear charge must satisfy the electron–nuclear cusp condition given by where is the spherical component of the expansion of the density around nucleus i.
Only 1stype functions have nonzero electron density at the nucleus and contribute to (8.7.3.13). For the hydrogenlike atom or ion described by a single exponent radial function R(r) = N exp (−ζr), (8.7.3.13) gives , where Z is the nuclear charge, and is the Bohr unit. Thus, a modification of ζ for 1s functions, as implied by (8.7.3.6) and (8.7.3.7) if applied to H atoms, leads to a violation of the cusp constraint. In practice, the electron density at the nucleus is not determined by a limited resolution diffraction experiment; the single exponent function R(r) is fitted to the electron density away from, rather than at the nucleus.
Poisson's electrostatic equation gives a relation between the gradient of the electric field ∇^{2}Φ(r) and the electron density at r. As noted by Stewart (1977), this equation imposes a constraint on the radial functions R(r). For , the condition must be obeyed for to be finite at r = 0, which satisfies the requirement of the nondivergence of the electric field ∇V, its gradient ∇^{2}V, the gradient of the field gradient ∇^{3}V, etc.
According to the electrostatic Hellmann–Feynman theorem, which follows from the Born–Oppenheimer approximation and the condition that the forces on the nuclei must vanish when the nuclear configuration is in equilibrium, the nuclear repulsions are balanced by the electron–nucleus attractions (Levine, 1983). The balance of forces is often achieved by a very sharp polarization of the electron density very close to the nuclei (Hirshfeld & Rzotkiewicz, 1974), which may be represented in the Xray model by the introduction of dipolar functions with large values of ζ. The Hellmann–Feynman constraint offers the possibility for obtaining information on such functions even though they may contribute only marginally to the observed Xray scattering (Hirshfeld, 1984).
As the Hellmann–Feynman constraint applies to the static density, its application presumes a proper deconvolution of the thermal motion and the electron density in the scattering model.
8.7.3.4.1. Moments of a charge distribution^{1}
Use of the expectation value expression with the operator gives for the electrostatic moments of a charge distribution ρ(r) in which the are the three components of the vector r (α_{i} = 1, 2, 3), and the integral is over the complete volume of the distribution.
For l = 0, (8.7.3.16) represents the integral over the charge distribution, which is the total charge, a scalar function described as the monopole. The higher moments are, in ascending order of l, the dipole, a vector, the quadrupole, a secondrank tensor, and the octupole, a thirdrank tensor. Successively higher moments are named the hexadecapole (l = 4), the tricontadipole (l = 5), and the hexacontatetrapole (l = 6). An alternative, traceless, definition is often used for moments with . In the traceless definition, the quadrupole moment, , is given by where δ_{αβ} is the Kronecker delta function. The term , which is subtracted from the diagonal elements of the tensor, corresponds to the spherically averaged second moment of the distribution.
Expression (8.7.3.17) is a special case of the following general expression for the lthrank traceless tensor elements.
Though the traceless moments can be derived from the unabridged moments, the converse is not the case because the information on the spherically averaged moments is no longer present in the traceless moments. The general relations between the traceless moments and the unabridged moments follow from (8.7.3.18). For the quadrupole moments, we obtain with (8.7.3.17) and Expressions for the other elements are obtained by simple permutation of the indices.
For a site of point symmetry 1, the electrostatic moment of order l has (l + 1)(l + 2)/2 unique elements. In the traceless definition, not all elements are independent. Because the trace of the tensor has been set to zero, only 2l + 1 independent components remain. For the quadrupole there are 5 independent components of the form (8.7.3.19).
In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics (ITB, 2001) multiplied by , which defines the spherical electrostatic moments
The expressions for are listed in Volume B of International Tables for Crystallography (ITB, 2001); for the l = 2 moment, the have the well known form 3z^{2} − 1, xz, yz, (x^{2} − y^{2})/2, and xy, where x, y and z are the components of a unit vector from the origin to the point being described. The spherical electrostatic moments have (2l + 1) components, which equals the number of independent components in the traceless definition (8.7.3.18), as it should. The linear relationships are
In the multipole model [expression (8.7.3.7)], the charge density is a sum of atomcentred density functions, and the moments of a whole distribution can be written as a sum over the atomic moments plus a contribution due to the shift to a common origin. An atomic moment is obtained by integration over the charge distributions ρ_{total,i}(r) = ρ_{nuclear,i} − ρ_{e,i} of atom i, where the electronic part of the atomic charge distribution is defined by the multipole expansion where p = ± when m , and is a radial function.
We get for the jth moment of the valence density in which the minus sign arises because of the negative charge of the electrons.
We will use the symbol for the moment operators. We get where, as before, p = ±. The requirement that the integrand be totally symmetric means that only the dipolar terms in the multipole expansion contribute to the dipole moment. If we use the traceless definition of the higher moments, or the equivalent definition of the moments in terms of the spherical harmonic functions, only the quadrupolar terms of the multipole expansion will contribute to the quadrupole moment; more generally, in the traceless definition the lthorder multipoles are the sole contributors to the lth moments. In terms of the spherical moments, we get
Substitution with R_{l} = {(κ′ζ)^{n(l)+3}/[n(l) +2] !}r^{n(l)}exp(−ζr) and and subsequent integration over r gives where the definitions have been used (ITB, 2001). Since the functions are wavefunction normalized, we obtain Application to dipolar terms with n(l) = 2, L_{lm} = 1/π and M_{lm} = (3/4π)^{1/2} gives the x component of the atomic dipole moment as For the atomic quadrupole moments in the spherical definition, we obtain directly, using n(l) = 2, l = 2 in (8.7.3.29), and, for the other elements,
As the traceless quadrupole moments are linear combinations of the spherical quadrupole moments, the corresponding expressions follow directly from (8.7.3.31), (8.7.3.32) and (8.7.3.21). We obtain with n(2) = 2 and and analogously for the other offdiagonal elements.
The moments derived from the total density ρ(r) and from the deformation density Δρ(r) are not identical. To illustrate the relation for the diagonal elements of the secondmoment tensor, we rewrite the xx element as The promolecule is the sum over spherical atom densities, or If R_{i} = (X_{i}, Y_{i}, Z_{i}) is the position vector for atom i, each singleatom contribution can be rewritten as Since the last two integrals are proportional to the atomic dipole moment and its net charge, respectively, they will be zero for neutral spherical atoms. Substitution in (8.7.3.35) gives, with , and and, by substitution in (8.7.3.34), with in which and are the atomic dipole moment and the charge on atom i, respectively.
The last term in (8.7.3.38a) can be derived rapidly from analytical expressions for the atomic wavefunctions. Results for Hartree–Fock wavefunctions have been tabulated by Boyd (1977). Since the offdiagonal elements of the secondmoment tensor vanish for the spherical atom, the second term in (8.7.3.38a) disappears, and the offdiagonal elements are identical for the total and deformation densities.
The relation between the second moments μ_{αβ} and the traceless moments _{αβ} of the deformation density can be illustrated as follows. From (8.7.3.17), we may write Only the spherical density terms contribute to the integral on the right. Assuming for the moment that the spherical deformation is represented by the valenceshell distortion (i.e. neglect of the second monopole in the aspherical atom expansion), we have, with density functions ρ normalized to 1, for each atom and which, on substitution in (8.7.3.39), gives the required relation.
In general, the multipole moments depend on the choice of origin. This can be seen as follows. Substitution of in (8.7.3.16) corresponds to a shift of origin by R_{α}, or X, Y, Z in the original coordinate system. In three dimensions, we get, for the first moment, the charge q, and for the transformed first and second moments
For the traceless quadrupole moments, the corresponding equations are obtained by substitution of and r′ = r − R into (8.7.3.17), which gives Similar expressions for the higher moments are reported in the literature (Buckingham, 1970).
We note that the first nonvanishing moment is originindependent. Thus, the dipole moment of a neutral molecule, but not that of an ion, is independent of origin; the quadrupole moment of a molecule without charge and dipole moment is not dependent on the choice of origin and so on. The molecular electric moments are commonly reported with respect to the centre of mass.
The moments of a molecule or of a molecular fragment are obtained from the sum over the atomic moments, plus a contribution due to the shift to a common origin for all but the monopoles. If individual atomic coordinate systems are used, as is common if chemical constraints are applied in the leastsquares refinement, they must be rotated to have a common orientation. Expressions for coordinate system rotations have been given by Cromer, Larson & Stewart (1976) and by Su & Coppens (1994a).
The transformation to a common coordinate origin requires use of the originshift expressions (8.7.3.42)–(8.7.3.44), with, for an atom at , . The first three moments summed over the atoms i located at become and with α, β = x, y, z; and expressions equivalent to (8.7.3.44) for the traceless components _{αβ}.
Expression (8.7.3.16) for the outer moment of a distribution within a volume element may be written as with , and integration over the volume .
Replacement of ρ(r) by the Fourier summation over the structure factors gives where is the product of l coordinates according to (8.7.3.16), and μ^{l} represents the moment of the static distribution if the F(h) are the structure factors on an absolute scale after deconvolution of thermal motion. Otherwise, the moment of the thermally averaged density is obtained.
The integral is defined as the shape transform S of the volume For regularly shaped volumes, the integral can be evaluated analytically. A volume of complex shape may be subdivided into integrable subvolumes such as parallelepipeds. By choosing the subvolumes sufficiently small, a desired boundary surface can be closely approximated.
If the origin of each subvolume is located at , relative to a coordinate system origin at P, the total electronic moment relative to this origin is given by
Expressions for for and a subvolume parallelepipedal shape are given in Table 8.7.3.2. Since the spherical order Bessel functions that appear in the expressions generally decrease with increasing x, the moments are strongly dependent on the loworder reflections in a data set. An example is the shape transform for the dipole moment. Relative to an origin O, A shift of origin by leads to in agreement with (8.7.3.46).

The electrostatic potential Φ(r′) due to the electronic charge distribution is given by the Coulomb equation, where the constant k is dependent on the units selected, and will here be taken equal to 1. For an assembly of positive point nuclei and a continuous distribution of negative electronic charge, we obtain in which Z_{M} is the charge of nucleus M located at R_{M}.
The electric field E at a point in space is the gradient of the electrostatic potential at that point. As E is the negative gradient vector of the potential, the electric force is directed `downhill' and proportional to the slope of the potential function. The explicit expression for E is obtained by differentiation of the operator r − r′^{−1} in (8.7.3.50) towards x, y, z and subsequent addition of the vector components. For the negative slope of the potential in the x direction, one obtains which gives, after addition of the components,
The electric field gradient (EFG) is the tensor product of the gradient operator and the electric field vector E; It follows that in a Cartesian system the EFG tensor is a symmetric tensor with elements
The EFG tensor elements can be obtained by differentiation of the operator in (8.7.3.53) for E_{α} to each of the three directions β. In this way, the traceless result is obtained. We note that according to (8.7.3.57) the electric field gradient can equally well be interpreted as the tensor of the traceless quadrupole moments of the distribution −2ρ_{total}(r)/r − r′^{5} [see equation (8.7.3.17)].
Definition (8.7.3.56) and result (8.7.3.57) differ in that (8.7.3.56) does not correspond to a zerotrace tensor. The situation is analogous to the two definitions of the second moments, discussed above, and is illustrated as follows. The trace of the tensor defined by (8.7.3.56) is given by Poisson's equation relates the divergence of the gradient of the potential to the electron density at that point: Thus, the EFG as defined by (8.7.3.56) is not traceless, unless the electron density at r is zero.
The potential and its derivatives are sometimes referred to as inner moments of the charge distribution, since the operators in (8.7.3.50), (8.7.3.52) and (8.7.3.54) contain the negative power of the position vector. In the same terminology, the electrostatic moments discussed in §8.7.3.4.1 are described as the outer moments.
It is of interest to evaluate the electric field gradient at the atomic nuclei, which for several types of nuclei can be measured accurately by nuclear quadrupole resonance and Mössbauer spectroscopy. The contribution of the atomic valence shell centred on the nucleus can be obtained by substitution of the multipolar expansion (8.7.3.7) in (8.7.3.57). The quadrupolar (l = 2) terms in the expansion contribute to the integral. For the radial function with n(l) = 2, the following expressions are obtained: with in the case that n_{2} = 4 (Stevens, DeLucia & Coppens, 1980).
The contributions of neighbouring atoms can be subdivided into pointcharge, pointmultipole, and penetration terms, as discussed by Epstein & Swanton (1982) and Su & Coppens (1992, 1994b), where appropriate expressions are given. Such contributions are in particular important when short interatomic distances are involved. For transitionmetal atoms in coordination complexes, the contribution of neighbouring atoms is typically much smaller than the valence contribution.
Hirshfelder, Curtis & Bird (1954) and Buckingham (1959) have given an expression for the potential at a point outside a charge distribution: where summation over repeated indices is implied. The outer moments q, μ_{α}, Φ_{αβ} and Ω_{αβγ} in (8.7.3.61) must include the nuclear contributions, but, for a point outside the distribution, the spherical neutralatom densities and the nuclear contributions cancel, so that the potential outside the charge distribution can be calculated from the deformation density.
The summation in (8.7.3.61) is slowly converging if the charge distribution is represented by a single set of moments. When dealing with experimental charge densities, a multicentre expansion is available from the analysis, and (8.7.3.61) can be replaced by a summation over the distributed moments centred at the nuclear positions, in which case measures the distance from a centre of the expansion to the field point. The result is equivalent to more general expressions given by Su & Coppens (1992), which, for very large values of , reduce to the sum over atomic terms, each expressed as (8.7.3.61). The interaction between two charge distributions, A and B, is given by where ρ_{B} includes the nuclear charge distribution.
The electrostatic properties of a well defined group of atoms can be derived directly from the multipole population coefficients. This method allows the `lifting' of a molecule out of the crystal, and therefore the examination of the electrostatic quantities at the periphery of the molecule, the region of interest for intermolecular interactions. The difficulty related to the origin term, encountered in the reciprocalspace methods, is absent in the directspace analysis.
In order to express the functions as a sum over atomic contributions, we rewrite (8.7.3.51), (8.7.3.54) and (8.7.3.57) for the electrostatic properties at point P as a sum over atomic contributions. in which the exclusion of M = P only applies when the point P coincides with a nucleus, and therefore only occurs for the central contributions. Z_{M} and R_{M} are the nuclear charge and the position vector of atom M, respectively, while r_{P} and r_{M} are, respectively, the vectors from P and from the nucleus M to a point r, such that r_{P} = r − R_{P}, and r_{M} = r − R_{M} = r_{P} + R_{P} − R_{M} = r_{P} − R_{MP}. The subscript M in the second, electronic part of the expressions refers to density functions centred on atom M.
Expressions for the evaluation of (8.7.3.62)–(8.7.3.64) from the chargedensity parameters of the multipole expansion have been given by Su & Coppens (1992). They employ the Fourier convolution theorem, used by Epstein & Swanton (1982) to evaluate the electric field gradient at the atomic nuclei. A directspace method based on the Laplace expansion of 1/R_{p} − r was reported by Bentley (1981).
Expression (8.7.3.49) is an example of derivation of electrostatic properties by direct Fourier summations of the structure factors. The electrostatic potential and its derivatives may be obtained in an analogous manner.
In order to obtain the electrostatic properties of the total charge distribution, it is convenient to define the `total' structure factor including the nuclear contribution, where , the summation being over all atoms j with nuclear charge , located at . If Φ(h) is defined as the Fourier transform of the directspace potential, we have and which equals −4πρ_{total} according to the Poisson equation (8.7.3.59). One obtains with and, by inverse Fourier transformation of (8.7.3.65), (Bertaut, 1978; Stewart, 1979). Furthermore, the electric field due to the electrons is given by Thus, with (8.7.3.65), which implies Similarly, the h Fourier component of the electric field gradient tensor with trace is where i : k represents the tensor product of two vectors. This leads to the expression for the electric field gradient in direct space, (The elements of h : h are the products )
The components of E and the elements of the electric field gradient defined by (8.7.3.67a) and (8.7.3.68) are with respect to the reciprocallattice coordinate system. Proper transforms are required to get the values in other coordinate systems. Furthermore, to get the traceless tensor, the quantity −(4π/3)ρ_{e}(r) = −(4π/3V)F(h)exp(−2π ih · r) must be subtracted from each of the diagonal elements
The Coulombic selfelectronic energy of the crystal can be obtained from Since (Parseval's rule), the summation can be performed in reciprocal space, and, for the total Coulombic energy, where the integral has been replaced by a summation.
The summations are rapidly convergent, but suffer from having a singularity at h = 0 (Dahl & Avery, 1984; Becker & Coppens, 1990). The contribution from this term to the potential cannot be ignored if different structures are compared. The term at h = 0 gives a constant contribution to the potential, which, however, has no effect on the energy of a neutral system. For polar crystals, an additional term occurs in (8.7.3.69a,b), which is a function of the dipole moment D of the unit cell (Becker, 1990), To obtain the total energy of the static crystal, electron exchange and correlation as well as electron kinetic energy contributions must be added.
One can write the total energy of a system as where T is the kinetic energy, represents the exchange and electron correlation contributions, and , the Coulomb energy, discussed in the previous section, is given by where is the nuclear charge for an atom at position R_{i}, and R_{ij} = R_{j} − R_{i}.
Because of the theorem of Hohenberg & Kohn (1964), E is a unique functional of the electron density ρ, so that must be a functional of ρ. Approximate density functionals are discussed extensively in the literature (Dahl & Avery, 1984) and are at the centre of active research in the study of electronic structure of various materials. Given an approximate functional, one can estimate nonCoulombic contributions to the energy from the charge density ρ(r).
In the simplest example, the functionals are those applicable to an electronic gas with slow spatial variations (the `nearly free electron gas'). In this approximation, the kinetic energy T is given by with ; and the function . The exchangecorrelation energy is also a functional of ρ,with and .
Any attempt to minimize the energy with respect to ρ in this framework leads to very poor results. However, cohesive energies can be described quite well, assuming that the change in electron density due to cohesive forces is slowly varying in space.
An example is the system AB, with closedshell subsystems A and B. Let ρ_{A} and ρ_{B} be the densities for individual A and B subsystems. The interaction energy is written as This model is known as the Gordon–Kim (1972) model and leads to a qualitatively valid description of potential energy surfaces between closedshell subsystems. Unlike pure Coulombic models, this density functional model can lead to an equilibrium geometry. It has the advantage of depending only on the charge density ρ.
Frequently, the purpose of a charge density analysis is comparison with theory at various levels of sophistication. Though the charge density is a detailed function, the features of which can be compared at several points of interest in space, it is by no means the only level at which comparison can be made. The following sequence represents a progression of functions that are increasingly related to the experimental measurement. where the angle brackets refer to the thermally averaged functions.
The experimental information may be reduced in the opposite sequence:
The crucial step in each sequence is the thermal averaging (top sequence) or the deconvolution of thermal motion (bottom sequence), which in principle requires detailed knowledge of both the internal (molecular) and the external (lattice) modes of the crystal. Even within the generally accepted Born–Oppenheimer approximation, this is a formidable task, which can be simplified for molecular crystals by the assumption that the thermal smearing is dominated by the largeramplitude external modes. The procedure (8.7.3.65) requires an adequate thermalmotion model in the structurefactor formalism applied to the experimental structure amplitudes. The commonly used models may include anharmonicity, as described in Volume B, Chapter 1.2 (ITB, 2001), but assume that a density function centred on an atom can be assigned the thermal motion of that atom, which may be a poor approximation for the more diffuse functions.
The missing link in scheme (8.7.3.75) is the sequence . In order to describe the wavefunction analytically, a basis set is required. The number of coefficients in the wavefunction is minimized by the use of a minimal basis for the molecular orbitals, but calculations in general lead to poorquality electron densities.
If the additional approximation is made that the wavefunction is a single Slater determinant, the idempotency condition can be used in the derivation of the wavefunction from the electron density. A simplified twovalenceelectron twoorbital system has been treated in this manner (Massa, Goldberg, Frishberg, Boehme & La Placa, 1985), and further developments may be expected.
A special case occurs if the overlap between the orbitals on an atom and its neighbours is very small. In this case, a direct relation can be derived between the populations of a minimal basis set of valence orbitals and the multipole coefficients, as described in the following sections.
In general, the atomcentred density model functions describe both the valence and the twocentre overlap density. In the case of transition metals, the latter is often small, so that to a good approximation the atomic density can be expressed in terms of an atomic orbital basis set , as well as in terms of the multipolar expansion. Thus, in which are the density functions.
The orbital products can be expressed as linear combinations of spherical harmonic functions, with coefficients listed in Volume B, Chapter 1.2 (ITB, 2001), which leads to relations between the and . In matrix notation, where is a vector containing the coefficients of the 15 spherical harmonic functions with l = 0, 2, or 4 that are generated by the products of d orbitals. The matrix M is also a function of the ratio of orbital and densityfunction normalization coefficients, given in Volume B, Chapter 1.2 (ITB, 2001).
The dorbital occupancies can be derived from the experimental multipole populations by the inverse expression, (Holladay, Leung & Coppens, 1983).
The matrix M^{−1} is given in Table 8.7.3.3. Pointgroupspecific expressions can be derived by omission of symmetryforbidden terms. Matrices for point group 4/mmm (square planar) and for trigonal point groups (3, , 32, 3m, ) are listed in Tables 8.7.3.4 and 8.7.3.5, respectively. Point groups with and without vertical mirror planes are distinguished by the occurrence of both and functions in the latter case, and only in the former, with m being restricted to n, the order of the rotation axis. The functions can be eliminated by rotation of the coordinate system around a vertical axis through an angle given by .



In the Born–Oppenheimer approximation, the electrons rearrange instantaneously to the minimumenergy state for each nuclear configuration. This approximation is generally valid, except when very low lying excited electronic states exist. The thermally smeared electron density is then given bywhere R represents the 3N nuclear space coordinates and P(R) is the probability of the configuration R. Evaluation of (8.7.3.79) is possible if the vibrational spectrum is known, but requires a large number of quantummechanical calculations at points along the vibrational path. A further approximation is the convolution approximation, which assumes that the charge density near each nucleus can be convoluted with the vibrational motion of that nucleus, where ρ_{n} stands for the density of the nth pseudoatom. The convolution approximation thus requires decomposition of the density into atomic fragments. It is related to the thermalmotion formalisms commonly used, and requires that twocentre terms in the theoretical electron density be either projected into the atomcentred density functions, or assigned the thermal motion of a point between the two centres. In the LCAO approximation (8.7.3.9), the twocentre terms are represented by where χ_{μ} and χ_{ν} are basis functions centred at R_{μ} and R_{ν}, respectively. As the motion of a point between the two vibrating atoms depends on their relative phase, further assumptions must be made. The simplest is to assume a gradual variation of the thermal motion along the bond, which gives at a point r_{i} on the internuclear vector of length R_{μν} This expression may be used to assign thermal parameters to a bondcentred function.
Thermal averaging of the electron density is considerably simplified for modes in which adjacent atoms move in phase. In molecular crystals, such modes correspond to rigidbody vibrations and librations of the molecule as a whole. Their frequencies are low because of the weakness of intermolecular interactions. Rigidbody motions therefore tend to dominate thermal motion, in particular at temperatures for which kT (k = 0.7 cm^{−1}) is large compared with the spacing of the vibrational energy levels of the external modes (internal modes are typically not excited to any extent at or below room temperature).
For a translational displacement (u), the dynamic density is given by with ρ(r) defined by (8.7.3.81) (Stevens, Rees & Coppens, 1977). In the harmonic approximation, P(u) is a normalized threedimensional Gaussian probability function, the exponents of which may be obtained by rigidbody analysis of the experimental data. In general, for a translational displacement (u) and a librational oscillation (ω), If correlation between u and ω can be ignored (neglect of the screw tensor S), P(u, ω) = P(u)P(ω), and both types of modes can be treated independently. For the translations where F^{−1} is the inverse Fourier transform operator, and T_{tr}(h) is the translational temperature factor.
If R is an orthogonal rotation matrix corresponding to a rotation ω, we obtain for the librations in which f(h) has been averaged over the distribution of orientations of h with respect to the molecule;
Evaluation of (8.7.3.85) and (8.7.3.86) is most readily performed if the basis functions ψ have a Gaussiantype radial dependence, or are expressed as a linear combination of Gaussian radial functions.
For Gaussian products of s orbitals, the molecular scattering factor of the product ψ_{μ}ψ_{ν} = N_{μ}exp[−α_{μ}(r − r_{A})^{2}] × N_{ν}exp [−α_{ν} (r − r_{B})^{2}], where N_{μ} and N_{ν} are the normalization factors of the orbitals μ and ν centred on atoms A and B, is given by where the centre of density is defined by r_{c} = (α_{μ} r_{A} + α_{ν} r_{B})/(α_{μ} + α_{ν}).
For the translational modes, the temperaturefactor exponent is simply added to the Gaussian exponent in (8.7.3.88) to give For librations, we may write As , for a function centred at r, which shows that for ss orbital products the librational temperature factor can be factored out, or Expressions for P(ω) are described elsewhere (Pawley & Willis, 1970).
For general Cartesian Gaussian basis functions of the type the scattering factors are more complicated (Miller & Krauss, 1967; Stevens, Rees & Coppens, 1977), and the librational temperature factor can no longer be factored out. However, it may be shown that, to a first approximation, (8.7.3.90) can again be used. This `pseudotranslation' approximation corresponds to a neglect of the change in orientation (but not of position) of the twocentre density function and is adequate for moderate vibrational amplitudes.
Thermally smeared density functions are obtained from the averaged reciprocalspace function by performing the inverse Fourier transform with phase factors depending on the position coordinates of each orbital product where the orbital product χ_{μ}χ_{ν} is centred at . If the summation is truncated at the experimental limit of (sin θ)/λ, both thermal vibrations and truncation effects are properly introduced in the theoretical densities.
It is often important to obtain an estimate of the uncertainty in the deformation densities in Table 8.7.3.1. If it is assumed that the density of the static atoms or fragments that are subtracted out are precisely known, three sources of uncertainty affect the deformation densities: (1) the uncertainties in the experimental structure factors; (2) the uncertainties in the positional and thermal parameters of the density functions that influence ρ_{calc}; and (3) the uncertainty in the scale factor k.
If we assume that the uncertainties in the observed structure factors are not correlated with the uncertainties in the refined parameters, the variance of the electron density is given by where , u_{p} is a positional or thermal parameter, and the γ(u_{p}, k) are correlation coefficients between the scale factor and the other parameters (Rees, 1976, 1978; Stevens & Coppens, 1976).
Similarly, for the covariance between the deformation densities at points A and B, where it is implied that the second term includes the effect of the scale factor/parameter correlation.
Following Rees (1976), we may derive a simplified expression for the covariance valid for the space group . Since the structure factors are not correlated with each other, where the latter equality is specific for , and indicates summation over a hemisphere in reciprocal space. In general, the second term rapidly averages to zero as increases, while the first term may be replaced by its average with u = 2π r_{A} − r_{B}h_{max}, or cov(ρ_{obs, A}ρ_{obs, B}) (2/V^{2})C(u) × and σ^{2}(ρ_{obs}) , a relation derived earlier by Cruickshank (1949). A discussion of the applicability of this expression in other centrosymmetric space groups is given by Rees (1976).
The electrostatic moments are functions of the scale factor, the positional parameters x, y and z of the atoms, and their chargedensity parameters κ and . The standard uncertainties in the derived moments are therefore dependent on the variances and covariances of these parameters.
If M_{p} represents the m × m variance–covariance matrix of the parameters , and T is an n × m matrix defined by for the lth moment with n independent elements, the variances and covariances of the elements of m^{l} are obtained from If a moment of an assembly of pseudoatoms is evaluated, the elements of T include the effects of coordinate rotations required to transfer atomic moments into a common coordinate system.
A frequently occurring case of interest is the evaluation of the magnitude of a molecular dipole moment and its standard deviation. Defining where is the dipolemoment vector and G is the directspace metric tensor of the appropriate coordinate system. If Y is the 1 × 3 matrix of the derivatives , where M_{μ} is defined by (8.7.3.96a). The standard uncertainty in μ may be obtained from σ(μ) = σ(y)/2μ. Significant contributions often result from uncertainties in the positional and chargedensity parameters of the H atoms.
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