Radial decay

Boys (1950) proposed an alternative to the use of STOs. All that is required for there to be an analytical solution of the general four-index integral formed from such functions is that the radial decay of the STOs be changed from e to e. That is, the AO-like functions are chosen to have the form of a Gaussian function. The general functional form of a normalized Gaussian-type orbital (GTO) in atom-centered Cartesian coordinates is... 

Although they are convenient from a computational standpoint, GTOs have specific features that diminish their utility as basis functions. One issue of key concern is the shape of the radial portion of the orbital. For s type functions, GTOs are smooth and differentiable at the nucleus (r = 0), but real hydrogenic AOs have a cusp (Figure 6.1). In addition, all hydrogenic AOs have a radial decay that is exponential in r while the decay of GTOs is exponential in... 

The monomer density profile of the layer of grafted chains follows the usual radial decay and strongly depends on the grafting density. A representative example is shown in Fig. 3. 

It is a Gaussian PDF with zero mean so it decays as the radial distance to the origin increases in any direction. If there exist two or more models that fit the measurement equally well, using this radially decaying prior distribution helps trimming down the set of the optimal parameters to the one with the smallest 2-norm. 

V = jl under good and theta-solvent conditions, respectively). Hence, the blob picture enables one to derive the power law for the radial decay in polymer density ... 

The radial decay in polymer density corresponds to a radial decrease in local stretching of the arms, dr/dn = p/[r Cp(r)] = At the same... 

According to a nonlinear Poisson-Boltzmann analysis, the initial radial decay of f from a cylindrical chain segment is much steeper than predicted by a Debye-Hiickel analysis see Figure 3 (28). The steep decay lessens at large distance, and eventually adopts an asymptotic functional form compatible with a Debye-Hiickel approximation. However, to superimpose across this distant region the predictions of the Poisson-Boltzmann analysis onto those formidated... 

The radial decay in polymer density corresponds to a radial decrease in local stretching of its arms, dr/dn -/[r c(r)]". At the same time, the local stretching of the branches controls the elastic tension t and, thereby, the size of the elastic blob, feiastic-feBT/tSfl(adn/dr) V /( l Within the blob approach, feiastic(r)-f(r). Hence, the radial increase in the size of the concentration blob i r) ensures also the decrease in local tension in the arms of the star, tlkeT-f lr. 

Figure 7.6 (a) The two Is orbitals for the H2 molecule sketched as noninteracting orbitals, to the right is a plot of the radial decay of the Is orbitals away from the nuclear centres, (b) The MOs for the H2 molecule. In the 1 (Tg+ SALC the AOs reinforce one another in the internuclear region, building up negative charge between the two positive nuclei. In the combination the two s-orbitals have opposite phase and cancel each other at the bond centre. 

The basis functions used in constructing MOs are the AOs based on the hydrogen atom solutions of the Schrddinger equation discussed in Appendix 9, with the proviso that accurate energies will require flexibility in the radial decay constants. Before moving on to molecules more complex than H2, it is worth looking at the shapes of the AOs relevant for the first row of the periodic table. We have already used the shapes of s-, p- and d-functions to discuss the symmetry of particular AOs (e.g. the d-orbitals of the central metal atom in transition metal complexes were covered in Section 5.8). These shapes are based on the... 

This demonstrates that the linear combination of AOs using the radial decay factors for atoms cannot give a complete picture of chemical bond formation. The radial profiles of the orbitals also have to be allowed to adapt to account for the changing environment the electron experiences on moving from the AOs to MOs. Once this is done, the potential energy is decreased and the kinetic energy increased due to the contraction of the orbitals around the nuclei. 

Equation (A10.9) shows that (T) = - U), which is the correct result for the Coulomb power law n = —1). That the virial theorem is obeyed is also confirmation that the exponent in the 5) function (Equation (A 10.3)) has the optimum form we are correct to use exp( -r) rather than some other radial decay, such as exp( r) with C 7 1, as will be checked in Problem AlO.l. The electron is distributed as a function of r, so the decay constant affects the averaging process and so is important in calculating the expectation values of the energies. 

We will find that the potential energy is important and stabilizing the bond. The fault lies in the use of the AOs to construct MOs. We have treated these as rigid entities with a radial decay set for the atomic state, but this is not appropriate to the molecular environment. 

The picture is not yet complete, and Problem A 10.1 indicates why. The s-orbitals used to construct the bonding and antibonding combinations have radial decays optimized for the atomic state. In the H2 ion there are two nuclei, and so there is no reason why this decay factor should be suited to describe the MO. To address this we will optimize the energy with respect to the radial decay of the basis functions. 

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