Atomic valence function

As soon as covalency comes into play, small interatomic distances with small coordination numbers are preferred. This is because covalent interactions do not fall off as but much faster. For example, the overlap integral Sij between two atomic orbitals goes to zero quite rapidly because the atomic valence functions themselves scale as as has been indicated in Section 2.2. Thus, if the cooalent contribution to the cohesive energy increases, smaller coordination numbers (4) are favored over larger coordination numbers (6) although both would be equally suitable for a purely Coulomb interaction. A more detailed discussion has been given by Pearson [201]. 

Three basis sets (minimal s-p, extended s-p and minimal s-p with d functions on the second row atoms) are used to calculate geometries and binding energies of 24 molecules containing second row atoms, d functions are found to be essential in the description of both properties for hypervalent molecules and to be important in the calculations of two-heavy-atom bond lengths even for molecules of normal valence. 

The starting point to obtain a PP and basis set for sulphur was an accurate double-zeta STO atomic calculation4. A 24 GTO and 16 GTO expansion for core s and p orbitals, respectively, was used. For the valence functions, the STO combination resulting from the atomic calculation was contracted and re-expanded to 3G. The radial PP representation was then calculated and fitted to six gaussians, serving both for s and p valence electrons, although in principle the two expansions should be different. Table 3 gives the numerical details of all these functions. 

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 dispose of the simplest problems first. Any orbital-basis theory of molecular electronic structure which purports to be interpretable as a theory of valence is committed to the use of atom-centred functions (or, at least, functions which go over into atomic orbitals for some values of their parameters).7) We therefore wish to stay as... 

Hiickel s application of this approach to the aromatic compounds gave new confidence to those physicists and chemists following up on the Hund-Mulliken analysis. It was regarded by many people as the simplest of the quantum mechanical valence-bond methods based on the Schrodinger equation. 66 Hiickel s was part of a series of applications of the method of linear combination of atom wave functions (atomic orbitals), a method that Felix Bloch had extended from H2+ to metals in 1928 and that Fowler s student, Lennard-Jones, had further developed for diatomic molecules in 1929. Now Hiickel extended the method to polyatomic molecules.67... 

In the early calculations of IR spectra of molecules, small basis sets (e.g., STO-3G, 3-21G, or 4-31G) were used because of limitations of computational power. At present typically a basis set consists of split valence functions (double zeta) with polarization functions placed on the heavy atoms (i.e., non-hydrogens) of the molecule (the so-called DZ+P or 6-31 G basis set). Such basis sets have been... 

Random structure methods have proved useful in solving structures from X-ray powder diffraction patterns. The unit cell can usually be found from these patterns, but the normal single-crystal techniques for solving the structure cannot be used. A variation on this technique, the reverse Monte Carlo method, includes in the cost function the difference between the observed powder diffraction pattern and the powder pattern calculated from the model (McGreevy 1997). It is, however, always necessary to include some chemical information if the correct structure is to be found. Various constraints can be added to the cost function, such as target coordination numbers or the deviation between the bond valence sum and atomic valence (Adams and Swenson 2000b Swenson and Adams 2001). 

Valence maps may alternatively be presented in a way which gives a direct impression of the atom s probability density function, a function which indicates the probability of finding the atoms at a particular point in space. This is calculated by inverting the valence function using eqn (11.3) ... 

Raising to an inverse power has the effect of converting minima into maxima, and the division by the atomic valence gives p, a value of 1.0 at an ideal location. If N is set equal to 16, the maxima become quite sharp and the resultant p, map contains peaks that, under suitable conditions, resemble the probability density function for the atom at room temperature as shown for... 

The concepts which we need for understanding the structural trends within covalently bonded solids are most easily introduced by first considering the much simpler system of diatomic molecules. They are well described within the molecular orbital (MO) framework that is based on the overlapping of atomic wave functions. This picture, therefore, makes direct contact with the properties of the individual free atoms which we discussed in the previous chapter, in particular the atomic energy levels and angular character of the valence orbitals. We will see that ubiquitous quantum mechanical concepts such as the covalent bond, overlap repulsion, hybrid orbitals, and the relative degree of covalency versus ionicity all arise naturally from solutions of the one-electron Schrodinger equation for diatomic molecules such as H2, N2, and LiH. 

Fig. 3.13 Left-hand panel The electronic structure of an sp-valent diatomic molecule as a function of the internuclear separation. Labels Et and Ep mark the positions of the free-atomic valence s and p levels respectively and = ( s + p). The quantity Rx is the distance at which the and upper tr9 levels cross. The region between the upper and lower n levels has been shaded to emphasize the increase in their separation with decreasing distance that is responsible for this crossing. Right-hand panel The self-consistent local density approximation electronic structure for C2 and Si2 whose equilibrium internuclear separations are marked by RCz and RSa respectively. (After Harris 1984.)...

The G and L values may be regarded as free parameters, but in practice they can be estimated from spectroscopic data. When the atomic valence orbitals include d and f functions, the number of unique integrals increases considerably, and the estimation of appropriate values from spectroscopy becomes considerably more complicated. 

A modified INDO model that is not entirely obsolete is the symmetric orthogonal-ized INDO (SINDOl) model of Jug and co-workers, first described in 1980 (Nanda and Jug 1980). The various conventions employed by SINDOl represent slightly different modifications to INDO theory than those adopted in the MINDO/3 model, but the more fundamental difference is the inclusion of d functions for atoms of the second row in the periodic table. Inclusion of such functions in the atomic valence basis set proves critical for handling hyper-valent molecules containing these atoms, and thus SINDO1 performs considerably better for phosphorus-containing compounds, for instance, than do otlier semiempirical models that lack d functions (Jug and Schulz 1988). 

The next approximation is the expression of each molecular orbital p as a linear combination of atomic orbitals (LCAO) (Equation A2.4), where atomic orbital functions and cf are coefficients that give the contribution of each atomic orbital to the molecular orbital. The

basis functions discussed in Section 1.2. Valence atomic orbitals are ordinarily chosen for the basis. 

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