Modeling density

The analyst now has available the complete details of the chemical composition of a gasoline all components are identified and quantified. From these analyses, the sample s physical properties can be calculated by using linear or non-linear models density, vapor pressure, calorific value, octane numbers, carbon and hydrogen content. 

The phenomenon can be illustrated by considering a model density q(x), from which diffraction data can be computed at arbitrarily high resolution. The (normalised) exponential factor needed to reconstruct q(x) by MaxEnt modulation of a chosen prior-prejudice distribution mix) can be written as... 

Figure 1. Amplitudes of the Fourier coefficients of log(model density, from a multipolar tit to 23 K diffraction data protect [45]. Continuous line m(x) = uniform distribution. Dotted line m(x) = core and valence monopoles. The vertical bar marks the experimental resolution limit 0.463 A. 

To check this prediction, a number of MaxEnt charge density calculations have been performed with the computer program BUSTER [42] on a set of synthetic structure factors, obtained from a reference model density for a crystal of L-alanine at 23 K. The set of 1500 synthetic structure factors, complete up to a resolution of 0.555 A [45], was calculated from a multipolar expansion of the density, with the computer program VALRAY[ 46],... 

The value of the mis deviation from the reference density can be deceptively low, due to the fact that in the intermolecular regions the model density is virtually the same as the one made of spherical-valence shells, which was used as a NUP. The agreement between the MaxEnt map and the reference model is very close in those regions. 

Both the determination of the effective number of scatterers and the associated rescaling of variances are still in progress within BUSTER. The value of n at the moment is fixed by the user at input preparation time for charge density studies, variances are also kept fixed and set equal to the observational c2. An approximate optimal n can be determined empirically by means of several test runs on synthetic data, monitoring the rms deviation of the final density from the reference model density (see below). This is of course only feasible when using synthetic data, for which the perfect answer is known. We plan to overcome this limitation in the future by means of cross-validation methods. 

A test of the computational strategy outlined in the previous paragraph has been performed on a set of synthetic noisy structure factor amplitudes. The diffraction data were computed from the same model density for L-alanine at 23 K as the one used for the noise-free calculations described in Section 3.1. 

Gaussian noise has been added onto the structure factor amplitudes squared as computed from the L-alanine model density for each datum, the amount of noise added was proportional to the experimental esd for the corresponding intensity measurement ... 

The core and valence monopole populations used for the MaxEnt calculation were the ones of the reference density (electrons in the asymmetric unit iw = 12.44 and nvalence = 35.56). The phases and amplitudes for this spherical-atom structure, union of the core fragment and the NUP, are already very close to those of the full multipolar model density to estimate the initial phase error, we computed the phase statistics recently described in a multipolar charge density study on 0.5 A noise-free data [56],... 

For a number of 1907 acentric reflexions up to 0.463 A resolution, the mean and rms phase angle differences between the noise-free structure factors for the full multipolar model density and the structure factors for the spherical-atom structure (in parentheses we give the figures for 509 acentric reflexions up to 0.700A resolution only) were (Acp) = 1.012(2.152)°, rms(A( >) = 2.986(5.432)° while... 

Larkin, R.G. and Clark, J.E., Modeling density changes in hazardous disposal well plumes, in Underground Injection Science and Technology, Tsang, C.F. and Apps, J.A., Eds., Elsevier, New York, 2007. 

The density functional (DF) method has been successful and quite useful in correlating experimental results when model densities are used in the calculations. In fact, the equations characteristic of the DF method can be derived from a variational approach as Kohn and Sham showed some time ago. In this approach, when model densities are introduced, it is not always possible to relate such densities to corresponding wave functions this is the N-representability problem. Fortunately, for any normalized well behaved density there exists a Slater single determinant this type of density is then N-representable. The problem of approximately N-representable density functional density matrices has been recently discussed by Soirat et al. [118], In spite... 

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. 

Static deformation density maps can be compared directly with theoretical deformation densities. For tetrafluoroterephthalonitrile (l,4-dicyano-2,3,5,6-tetra-fluorobenzene) (Fig. 5.13), a comparison has been made between the results of a density-functional calculation (see chapter 9 for a discussion of the density-functional method), and a model density based on 98 K data with a resolution of (sin 0//)max = 1.15 A -1 (Hirshfeld 1992). The only significant discrepancy is in the region of the lone pairs of the fluorine and nitrogen atoms, where the model functions are clearly inadequate to represent the very sharp features of the density distribution. 

The topological analysis of the total density, developed by Bader and coworkers, leads to a scheme of natural partitioning into atomic basins which each obey the virial theorem. The sum of the energies of the individual atoms defined in this way equals the total energy of the system. While the Bader partitioning was initially developed for the analysis of theoretical densities, it is equally applicable to model densities based on the experimental data. The density obtained from the Fourier transform of the structure factors is generally not suitable for this purpose, because of experimental noise, truncation effects, and thermal smearing. 

For analysis of experimental results, the static model density must be used to eliminate noise, truncation effects, and thermal smearing. Some caution is called for, because the reciprocal space representation of the Laplacian is a function of F(H) H2, and thus has poor convergence properties.2 This difficulty is only partly circumvented by use of the model density, as high-resolution detail may be quite dependent on the nature of the model functions, as is evident in the experimental study of the quartz polymorph coesite discussed in chapter 11. 

The promolecule density shows (3, — 1) critical points along the bond paths, just like the molecule density. But, as the promolecule is hypothetical and violates the exclusion principle, it would be incorrect to infer that the atoms in the promolecule are chemically bonded. In a series of topological analyses, Stewart (1991) has compared the model densities and promolecule densities of urea,... 

With the topological analysis of the total charge density, the distinction between a covalent and a closed-shell ionic interaction can be based on the value of the Laplacian and its components at the bond critical point. Such an analysis will be most conclusive when done on a series of related compounds, analyzed with identical basis sets, as the topological values of the model density from experimental data have been found to be quite dependent on the choice of basis functions. 

Phenol, from irradiated benzene, 3 181-182 Phenolic macrocycle, 39 140-141 Phenomenological modeling density-functional theory, 38 465 oxidized (-t3) and reduced (-H) clusters, four-iron clusters, 38 462 64 Phenylacetic acid, irradiation of, 5 201 Phenylalanine... 

The perturbation A(T f + 2T ) describes the replacement of model densities and inter-nuclear distances by the values that are appropriate for the molecule under scrutiny. Similarly, appropriate reference atomic energies must be used in the atomic-like formula (4.15) to get A °. Ingeniously selected references require small corrections. Nature helps a lot in that matter by keeping the changes of p(r) as small as possible. The bond energy theory is rooted in Eq. (4.47). 

An alternative approach to improve upon Hartree-Fock models involves including an explicit term to account for the way in which electron motions affect each other. In practice, this account is based on an exacf solution for an idealized system, and is introduced using empirical parameters. As a class, the resulting models are referred to as density functional models. Density functional models have proven to be successful for determination of equilibrium geometries and conformations, and are (nearly) as successful as MP2 models for establishing the thermochemistry of reactions where bonds are broken or formed. Discussion is provided in Section II. 

None of the semi-empirical models perform as well as Hartree-Fock models (except STO-3G), local density models, density functional models or MP2 models. PM3 provides the best overall description, although on the basis of mean absolute errors alone, all three models perform to an acceptable standard. Given the large difference in cost of application, semi-empirical models clearly have a role to play in structure determination. 

Hybrid Density Functional Models. Density Functional Models... 

Local Density Models. Density Functional Models which are based on the assumption that the Electron Density is constant (or slowly varying) throughout all space. 

Variational. Methods for which the calculated energy represents an upper bound to the exact (experimental) energy. Hartree-Fock and Configuration Interaction Models are variational while Moller-Plesset Models, Density Functional Models and Semi-Empirical Models are not variational. 

Fig. 10. Hybridized model densities of states for UN. The full rectangles are the original unhybridized densities of states (see Fig. 48). The broken rectangles are the additional projected densities of states due to hybridization. In a vertical line are the contributions to the local atomic and angular momentum projected densities of states. The electron transfer, in terms of the fractional occupancy (F) of the unhybridized f-band, is shown...

A. Khan. A liquid water model density variation from supercooled to superheated states, prediction of H-bonds and temperature limits. /. Phys. Chem., 104, 11268-11274, 2000. 

Local density models = density functional models in which the electron density is assumed to vary only slowly throughout all space. 

Non-local density functional models or gradient corrected density functional models = density functional models which take explicitly into account the nonuniformity in electron distributions, thus improving on the local density model. 

Figure 5. The energy gap in the normal state determined by fitting to the specific heat data with a model density of states by Loram et al. is shown in green. The tunneling experiments well below l c give the gap shown in red in agreement with ARPES shown in blue.

Keywords computer modelling, density functional theory, dipole moment, dipole polarizability,... 

Keywords microscopic models density functional SCC-DFTB CM3 charges mezoscopic... 

Once the multipole analysis of the X-ray data is done, it provides an analytical description of the electron density that can be used to calculate electrostatic properties (static model density, topology of the density, dipole moments, electrostatic potential, net charges, d orbital populations, etc.). It also allows the calculation of accurate structure factors phases which enables the calculation of experimental dynamic deformation density maps [16] ... 

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