Valence density model

A number of different atom-centered multipole models are available. We distinguish between valence-density models, in which the density functions represent all electrons in the valence shell, and deformation-density models, in which the aspherical functions describe the deviation from the IAM atomic density. In the former, the aspherical density is added to the unperturbed core density, as in the K-formalism, while in the latter, the aspherical density is superimposed on the isolated atom density, but the expansion and contraction of the valence density is not treated explicitly. 

The valence density model represents the one-electron density function for a molecule in a crystal as... 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

Figures, l-Alanine.Fits to noisy data Calculations A (experimental noise) and B (10% experimental noise). MaxEnt, deformation and error density profiles along the Cl-01 bond. Solid line Model valence density. Dashed line MaxEnt density A. Dot-dashed line MaxEnt density B. Dotted line valence-shells non-uniform prior.

There is almost quantitative agreement between the experimental model valence density (lower half of Fig. 11.2) and the result of an ab-initio local density functional calculation (upper part of Fig. 11.2). This agreement is also evident in... 

An X-ray atomic orbital (XAO) [77] method has also been adopted to refine electronic states directly. The method is applicable mainly to analyse the electron-density distribution in ionic solids of transition or rare earth metals, given that it is based on an atomic orbital assumption, neglecting molecular orbitals. The expansion coefficients of each atomic orbital are calculated with a perturbation theory and the coefficients of each orbital are refined to fit the observed structure factors keeping the orthonormal relationships among them. This model is somewhat similar to the valence orbital model (VOM), earlier introduced by Figgis et al. [78] to study transition metal complexes, within the Ligand field theory approach. The VOM could be applied in such complexes, within the assumption that the metal and the... 

A closer comparison of bond valence and electron density models is not possible because of the different underlying assumptions of the models. The forces in the bond valence model act between structureless point atoms, but the forces in the electron density model are exerted by electrons on nuclei and vice versa. This basic difference makes it difficult to compare the two models in greater detail. They are best seen as complementary, the electron density model providing important information about the nature of the bonding between the atoms, the bond valence model providing a simple tool for predicting structure and properties, particularly in cases where the structure is complex. 

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

The valence bond model constructs hybrid orbitals which contain various fractions of the character of the pure component orbitals. These hybrid orbitals are constructed such that they possess the correct spatial characteristics for the formation of bonds. The bonding is treated in terms of localised two-electron two-centre interactions between atoms. As applied to first-row transition metals, the valence bond approach considers that the 45, 4p and 3d orbitals are all available for bonding. To obtain an octahedral complex, two 3d, the 45 and the three 4p metal orbitals are mixed to give six spatially-equivalent directed cfisp3 hybrid orbitals, which are oriented with electron density along the principal Cartesian axes (Fig. 1-9). 

The formulation of spatially separated a and 7r interactions between a pair of atoms is grossly misleading. Critical point compressibility studies show [71] that N2 has essentially the same spherical shape as Xe. A total wave-mechanical model of a diatomic molecule, in which both nuclei and electrons are treated non-classically, is thought to be consistent with this observation. Clamped-nucleus calculations, to derive interatomic distance, should therefore be interpreted as a one-dimensional section through a spherical whole. Like electrons, wave-mechanical nuclei are not point particles. A wave equation defines a diatomic molecule as a spherical distribution of nuclear and electronic density, with a common quantum potential, and pivoted on a central hub, which contains a pith of valence electrons. This valence density is limited simultaneously by the exclusion principle and the golden ratio. 

The predicted course of reaction between a heteronuclear pair of atoms is shown in Figure 7.2. Promotion is once more modeled with isotropic compression of both types of atom. The more electropositive atom (at the lower quantum potential) reaches its valence state first and valence density starts to migrate from the parent core and transfers to an atom of the second kind, still below its valence state. The partially charged atom is more readily compressible to its promotion state, as shown by the dotted line. When this modified atom of the second kind reaches its valence state two-way delocalization occurs and an electron-pair bond is established as before. It is notable how the effective activation barrier is lowered with respect to both homonuclear (2Vq)i barriers to reaction. The effective reaction profile is the sum of the two promotion curves of atoms 1 and 2, with charge transfer. 

The models used in this review are due to Coppens and his co-workers [//]. First, the K formalism [ 11a) permits an estimation of the net charge of the atom and allows for the expansion contraction of the perturbated valence density for each atom, the density is described as,... 

Having secured a set of n values for phosphorus, the pseudoatom model was fitted to the four simulated data sets to test the effectiveness of the pseudoatoms model s formal deconvolution of multipolar valence density features from thermal vibrations smearing. Results are illustrated in Figure 4 as maps of the model static deformation densities ... 

FIGURE 6.37 The electron density for the if/g and ifil wave functions in the simple valence bond model for H2. (a) The electron density pg for if/g and Pu for calculated analytically as described in the text, (b) Three-dimensional isosurface of the electron density for the ipg wave function, as calculated numerically by Generalized Valence Bond Theory (GVB). 

Although it is possible to obtain good estimates of the electron correlation energy by either density functional or configuration interaction methods both of these methods (for different reasons) suffer from the same defect it is not possible to obtain a clear physical and chemical interpretation of the results of the calculation. The valence bond model, in principle and in practice, puts the physical interpretation as a top priority but pays a price in the complexity of its implementation. 

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