Charge density fitting

Gallant, R. T. and A. St-Amant. 1996. Linear scaling for the charge density fitting procedure of the linear combination of Gaussian-type orbitals density functional method. Chem. Phys. Lett. 256, 569. 

R. T. Gallant and A. St-Amant, Chem. Phys. Lett., 256 569 (1996). Linear Scaling for the Charge Density Fitting Procedure of the Linear Combination of Gaussian-Type Orbitals Density Functional Method. 

Notice that the density of a complete p, d, f, atomic subshell, or an incomplete sub shell in the central field approximation is rotationally invariant [58]. Thus only s-type charge density fitting functions are needed in any atomic central-field calculation. However if the central-field approximation is not invoked then very-high angular momenta are required to fit the density. From a practical point of view it might be better to set off center s-type fitting functions. 

Frequent approximations made in TB teclmiques in the name of achieving a fast method are the use of a minimal basis set, the lack of a self-consistent charge density, the fitting of matrix elements of the potential. 

Her workers to fit the exchange-correlation potential and the charge density (in the Coulomb potential) to a linear combination of Gaussian-typc functions. 

The density fitting functions may or may not be the same as those used in expanding the orbitals. The fitting constants a are chosen so that the Coulomb energy arising from the difference between the exact and fitted densities is minimized, subject to the constraint of charge conservation. The J integrals then become... 

The first step is to use tp) and tp) to create a fitted charge density that has the same value and slope as the true charge density on the hard core spheres. This is thus a continuous, differentiable fit. As input we need the value and slope of the charge density on the a spheres. This is directly obtainable form the one center expansion. 

Conventional implementations of MaxEnt method for charge density studies do not allow easy access to deformation maps a possible approach involves running a MaxEnt calculation on a set of data computed from a superposition of spherical atoms, and subtracting this map from qME [44], Recourse to a two-channel formalism, that redistributes positive- and negative-density scatterers, fitting a set of difference Fourier coefficients, has also been made [18], but there is no consensus on what the definition of entropy should be in a two-channel situation [18, 36,41] moreover, the shapes and number of positive and negative scatterers may need to differ in a way which is difficult to specify. 

Since the HF model already gives good charge densities, and reliably predicts many other diverse properties, it seems reasonable to expect that the charge densities produced from this model will be better than those from conventional least squares fitting. In other words, quantum knowledge is built into the model. 

The multipole model reduces the crystal electron density to a number of parameters, which can be fitted to experimental structure factors. For CU2O, structure factors for the (531) and higher-order reflections out to (14,4,2) were taken from X-ray measurements. Weak (ooe) (with o for odd and e for even) and very weak (eeo) reflections were also taken from X-ray work. Fig. 6 shows a three-dimensional plot of the difference between the static crystal charge density obtained from the multipole fitting to... 

Application to hexacyanobenzene indicates an improved fit to the 120 K experimental data (Druck and Kotuglu 1984). But interpretation of the results is not straightforward, because such a model does not deconvolute charge density and thermal motions effects, and is not well suited for comparison with theory and derivation of electrostatic properties. 

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

The implications of a charge density least-squares refinement can be visualized by calculation of the deformation density corresponding to the least-squares fitted model. 

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