Valence density

Al. no. Elemenl Symbol Al. wi. Mass number of common isotopes Per- iodic group - Valency + Density at 20°C (g cm ) Boiling point (°C)... 

When atoms occupy highly symmetrical sites, a further limitation of the current multipolar expansions is the limited order of the spherical harmonics employed, that do not usually extend past the hexadecapolar level (/ = 4). Only two multipolar studies published to date used spherical harmonics to orders higher than 1 = 4 graphite [15] and crystalline beryllium [16]. In the latter work, the most significant contribution to the valence density was indeed shown to be given by a pole of order 1 = 6. 

L-A la MaxEnt valence density from noise-free data... 

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... 

We stress here that any low-temperature valence density for a small organic molecule will have a comparably high dynamic range, so that even valence-only MaxEnt studies will always be likely to need a NUP if truncation ripples are to be avoided. 

BUSTER chooses the minimal grid necessary to avoid aliasing effects, based on the prior prejudice used and on the fall-off of the structure factor amplitudes with resolution for the 23 K L-alanine valence density reconstruction the grid was (64 144 64). The cell parameters for the crystal are a = 5.928(1)A b = 12.260(2)A c = 5.794(1) A [45], so that the grid step was shorter than 0.095 A along each axis. 

Figure 6(b) shows the difference between the MaxEnt valence density and the reference density, in the COO- plane. The error peaks in the bonding and lone-pair regions, where the deformation features are systematically lower than the reference map (negative contours). The deviation from the reference is largest in the region around the Cl atom valence shell, and reaches -0.406 e A 3. 

The <5q(x)) ms map peaks around the two oxygen atoms, where the valence density is highest the values of (8ry(x rms remain below 0.112 eA 3. This confirms that the deviations observed in the calculation A are indeed significant with respect to the intrinsic spread brought by the noise in the data. 

Sections of the density from one of these fits, which we will refer to as calculation B, are shown in Figure 7 the MaxEnt deformation density in the COO- plane is shown in Figure 7(a) Figure 7(b) is the difference between the MaxEnt valence density and the reference density in the same plane. The lower noise content of the data is clearly visible, when the map is compared with the one for calculation A in particular, the lone pairs on the oxygen atoms are better defined. The rms deviation from the reference is as low as 0.023 e A 3. 

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.

The scattering factor of the valence-density component in Eq. (3.16) is obtained by the Fourier transform... 

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. 

To preserve the shell structure of the spherical component of the valence density, the radial function of the bonded atom may be described by the isolated atom radial dependence, modified by the k expansion-contraction parameter. 

Combining the angular and radial functions discussed above leads to a valence-density formalism in which the density of each of the atoms is described as (Hansen and Coppens 1978)... 

The valence-shell populations are scaled after completion of the refinement by use of Ptj = Py/k. If the procedure is successful, the sum of FfJ should be close to the total number of valence electrons. The difference between these two quantities is used as a test of the adequacy of the valence density functions. 

Since the molecular volumes of neither the TTF nor the TCNQ molecule are easily described by a regular volume of integration, the parallelepiped subunit method was used and integration of the valence density was performed using Eqs. 

The spherically averaged atomic core and valence densities are obtained as the sum over products of the radial orbital functions, or, including normalization,... 

It is clear that in this approximation F(hkl) equals zero for all reflections for which h + k + l = 4n + 2. This is the reason that the observation of the (222) reflection of diamond led Bragg to conclude that bonding effects are detectable by X-ray diffraction (see chapter 3). If the Si atoms are not spherical, and their density contains antisymmetric components, such as dipolar or octupolar valence density functions, will be the complex conjugate of /fj1 and Eq. (1.12) is no longer valid. We can write = fc + ifa and /f, = fc — ifa, where c stands for the symmetric and a for the antisymmetric component of the atomic rest density. This gives... 

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... 

That the vibrational displacements of the valence shell electrons may be smaller than those of the core electrons can be qualitatively understood by considering the vibrations of two identical, strongly bonded atoms. When the atoms vibrate in phase, they behave as a rigid body, so all shells will vibrate equally. But when they vibrate out of phase, the density near the center of the bond will be stationary, assuming the average static overlap density to be independent of the vibrations. This apparently invariant component of the valence density would contribute to a lowering of the outer-shell temperature... 

A number of theoretical calculations are available for comparison with the experimental results on Be metal. The increase of the valence density in the tetrahedral holes is well reproduced by both the early augmented plane wave (APW) calculation of Inoue and Yamashita (1973), and the all-electron HF-LCAO calculation of Dovesi et al. (1982), but the latter gives somewhat better agreement with the experimental results. 

Provocative experimental evidence, at variance with conventional theory, is provided by the estimates of molecular diameters for diatomic molecules. Bonding theory requires the concentration of valence densities between the nuclei to increase as a function of bond order, in agreement with observed bond lengths (1.097, 1.208, 0.741 A) and force constants (22.95, 11.77, 5.75 Ncm-1) of the species N=N, 0=0 and H-H respectively. Molecular diameters can be measured by a variety of techniques based on gas viscosity, heat conductivity, diffusion and van der Waals equation of state. The results are in excellent agreement at values of 3.75, 3.61 and 2.72 A, for N2, O2 and H2, respectively. Conventional bonding theory cannot account for these results. 

In point-charge simulation this electronic rearrangement is of no immediate consequence except for the assumption of a reduced interatomic distance, which is the parameter needed to calculate increased dissociation energies. However, in Heitler-London calculation it is necessary to compensate for the modified valence density, as was done for heteronuclear interactions. The closer approach between the nuclei, and the consequent increase in calculated dissociation energy, is assumed to result from screening of the nuclear repulsion by the excess valence density. Computationally this assumption is convenient and effective. 

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. 

A second (green) octahedron, shown connected, surrounds a neighbouring C atom. Both arrangements are the same as the valence-density distribution in the CH4 molecule as derived on the basis of orbital angular-momentum minimization [65] to be discussed in more detail in the next chapter. 

The dominant interaction in molecules is covalent. The geometrical arrangement of atomic nuclei in molecules is mainly known from diffraction experiments in the solid state, and the persistence of such structures in the liquid, solution and gas phases is inferred from magnetic resonance and other spectroscopic studies. Apart from exceptional, nominally structureless, molecules to be discussed later on, most of these molecules may be considered as clusters of touching spheres, which represent atomic cores, and overlapping valence densities. 

A perspective drawing of a cyclohexane molecule is shown in Figure 5.18 - overlapping first ionization spheres of the carbon atoms define the outer perimeter. It is obvious that concentration of the valence density on the con-... 

Figure 6.1, predicting the same spherical structure of CH2 for H20 in a field-free environment. In a polarizing environment the excess valence density is displaced along Z to create what is known as a 2pz lone pair, shown schematically in Figure 6.2. In ammonia the N(2s22pl2p2y) valence shell...

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