Ionization radii

Hence the Markovian distribution is common for all non-Markovian ones but only in the limit X[> —> oo. This distribution completely ignores the nonstationary annihilation and therefore does not depend on the exciton concentration and lifetime. The difference between /o(r) and fm(r) becomes more pronounced when xD is reduced (Fig. 3.99). Under diffusion control both of them have a well-pronounced maximum near the effective ionization radius. However, the nearcontact contribution of the nonstationary annihilation increases with shortening xD, on account of the main maximum. Finally (as xD > 0), the UT distribution tends to become exponential, as W/(r), while the Markovian one remains unchanged. 

On compression to the ionization radius of an atom the equivalent of one electron becomes decoupled from the atomic core and finds itself in an impenetrable hollow sphere at constant potential, conveniently defined as V = 0. This problem, which is closely related to the problem of an electron confined to a one-dimensional finite line segment, has been studied in great detail. The Hamiltonian... 

For an atom, compressed to its ionization radius, the decoupled (valence) electron hence is spread across the sphere of radius r0 at uniform density, and its wave function is the step function... 

The physical nature of an electron in the valence state is not obvious. Although decoupled from the atomic core it remains associated with the core, because of environmental confinement. It therefore corresponds to unit negative charge confined to a sphere, described by the ionization radius of the atom concerned, but excluded from the core region of the atom. 

On extending this idea to a quantitative study of the valence state, ionization radius, which is characteristic of each atom, is the important parameter. When using the Hartree-Fock-Slater method to calculate the ionization radii of non-hydrogen atoms the boundary condition is introduced by multiplying... 

The most general solution to the wave equation of a spherically confined particle is the Fourier transform of this Bessel function, i.e. the box function defined by ro- Such a wave function, which terminates at the ionization radius, has a uniform amplitude throughout the sphere, defined before (3.36)... 

An unexpected feature of Table 5.1 is the remarkable similarity between the energies calculated from the characteristic radius rc and those calculated from the ionization radius r0, for the same interactions, but with bond orders increased by unity. It means that the steric factor which is responsible for the increase in bond order i.e. screening of the internuclear repulsion) is also correctly described by an adjustment to r o to compensate for modified valence density. Calculating backwards from first-order D0 = 210 kjmol-1, an effective zero-order C-C bond length of 1.72 A is obtained. 

An electron in the valence state is confined to a sphere, defined by the ionization radius of the atom, and with electronic charge uniformly distributed. Such a charge density is correctly described by a wave function of constant amplitude within the sphere, and vanishing outside. The only parameter that differentiates between atoms of different type is the characteristic ionization radius, which is also a measure of the classical atomic property of electronegativity. 

An example of an electron in a phase-locked cavity has been encountered in the study of compressed atoms [76, 24]. Isotropic compression of an atom, simulated by imposing a finite boundary condition on the electronic wave function, i.e. linv xpe = 0, r0 < oo, raises electronic energy levels, until an electron is decoupled from the core at a characteristic atomic ionization radius. This electron then exists in a field-free cavity with a spherical Bessel wave function. In the ground state... 

The effective nuclear charge on a ligand atom was established as Ze = ka 3, where a is a characteristic radius which can now be identified with the ionization radius of the atom, and k — 0.8 was found empirically to produce effective nuclear charges relative to Z H) = 1. This formula follows from the formulation... 

Equivalent point charges to represent lone pairs were calculated as a/2, from the ionization radius of the central atom and their effective distance from the nucleus was related to the same parameter. These values are based on the assumption that lone-pair densities should relate to valence densities and that their distance from the nucleus should depend on atomic size... 

Overabundance by a factor 1000 can be accreted in less than 10 years if the effective accretion radius is 3 x 10 km, smaUer than the magnetosphere and much smaller than the ionization radius R, of the Stromgren sphere (being for A stars about 3 light-years or 3 x 10 km). Inside Rq the hydrogen atoms are nearly completely (at 10 K, 99 percent) ionized, and the concentration of free... 

A word of warning about terminology is necessary at this point. While E = 0 corresponds to the ionization threshold for the free hydrogen atom, when speaking about the confined version the situation is not so clear. The radius in Equation (2) has been called the ionization radius or critical cage radius [4,5]. Here we avoid such usage. An additional and important comment on this point is formulated towards the end of this subsection. 

Each atom has a characteristic ionization radius [15] and a characteristic value of valence-state quantum potential, identified with electronegativity [11]. It is only the artificial compression barrier that keeps the activated electron confined. In the real world an activated electron is free to interact with its environment and initiate chemical reaction. The activation is rarely caused by uniform compression and, more typically, is due to thermal, collisional, or catalytic activation. 

From the latter, an ionization radius (valence electron radius)... 

Starting from the numerically optimized valence density in a ground-state valence shell, the radius of an equivalent sphere, which accommodates this total density at a uniform level, is readily calculated by simple geometry. The correspondence with Hartree-Fock values of ionization radius is almost exact, with the added advantage of higher accuracy for the chemically important second period elements where the HF results are notoriously unreliable [14,18]. 

On looking for a relationship between ionization radius and the chemistry of homonuclear covalent interaction, the classification into single and multiple bonds is followed as a first approximation. An immediate observation, valid for most single bonds, is a constant value of the dimensionless distance... 

With the relationship between ionization radius and bond order in hand, the calculation of covalent interaction parameters becomes an almost trivial exercise. The common volume, e, between overlapping spheres of radius tq at characteristic separations d for given bond order, and considered proportional to dissociation energy, varies in a quantized fashion similar to d. This allows definition of a dimensionless dissociation energy D = Dro K, as explained in the paper on Covalent Interaction (see p. 93), AT is a dimensional constant. Noting the connection of e with spherical volume, one looks for a dependence of the type... 

Interacting elementary wave packets are expected to coalesce into larger wave packets. All extranuclear electrons on an atom therefore together constitute a single spherical standing wave with internal structure, commensurate with a logarithmic optimization pattern. In the activated valence state, the central core of the wave packet is compressed into a miniscule sphere, compared to the valence shell which dominates the extranuclear space up to the ionization radius. 

Keywords Atomic wave model Electron density Golden-spiral optimization Ionization radius Self-similarity... 

Keywords Bond order Dipole moment Force constant General covalence Ionization radius Golden ratio... 

For ease of reference, we tabulate atomic ionization radius (ro), bond order (b) and exponents (n) of lowest-order observed homonuclear interactions in Table 6. 

These results are in line with the small ionization radius of hydrogen, which shows that its entire charge sphere becomes embedded into a larger sphere on molecular formation. The effective point position of the proton relative to the wave structure of the larger atom decides the bond order. 

It is not the purpose of this chapter to produce and present a new force field. We rather want to provide a theoretical basis for MM and therefore also to be able to efficiently produce generic force-field parameters. As it stands, one parameter (ionization radius) is needed to initiate the derivation of all other parameters to model all bond orders of any covalent interaction. It is therefore reassuring to note that the uniform valence density within a characteristic atomic sphere has the same symmetry as the Is hydrogen electron. The first-order covalent interaction between any pair of atoms can therefore be modeled directly by the simple Heitler-London method for hydrogen to predict d, D and kr [44]. The results are in agreement with those of the simpler number-theory simulation [38], which is therefore preferred for general use. 

Apart from the relative abundance of isotopes, all other basic data usually shown on a periodic table of the elements are predicted by elementary number theory. Based on the periodic table shown as an Appendix, the chemically important parameters of ionization radius and electronegativity have been used, together with the golden section and golden spirals, to derive all essential parameters pertaining to covalent interaction and the optimization of molecular structure by MM. The detailed results are described in the preceding chapters in this volume. 

---