Classical electron orbit

We can now understand why a hydrogen atom does not collapse, the way a classical electron orbit would. Suppose we took the electronic distribution about the nucleus, and cut all distances in half. 

Here we restrict ourselves to the case 5 W and to the states with m = n—1, i.e. the states with meiximal possible values of the magnetic quantum number m. Parameters of the classiced electron orbit are determined from equilibrium conditions analogous to the quantization of angular momentum ... 

In a number of classic papers Hohenberg, Kohn and Sham established a theoretical framework for justifying the replacement of die many-body wavefiinction by one-electron orbitals [15, 20, 21]. In particular, they proposed that die charge density plays a central role in describing the electronic stnicture of matter. A key aspect of their work was the local density approximation (LDA). Within this approximation, one can express the exchange energy as... 

In the classical picture of an electron orbiting round the nucleus it would not surprise us to discover that the electron and the nucleus could each spin on its own axis, just like the earth and the moon, and that each has an angular momentum associated with spinning. Unfortunately, although quantum mechanical treatment gives rise to two new angular momenta, one associated with the electron and one with the nucleus, this simple physical... 

Other approximate, more empirical methods are the extended Huckel 31> and hybrid-based Hiickel 32. 3> approaches. In these methods the electron repulsion is not taken into account explicitly. These are extensions of the early Huckel molecular orbitals 4> which have successfully been used in the n electron system of planar molecules. On account of the simplest feature of calculation, the Hiickel method has made possible the first quantum mechanical interpretation of the classical electronic theory of organic chemistry and has given a reasonable explanation for the chemical reactivity of sizable conjugated molecules. 

It is necessary to postulate a dynamic charge distribution as in the well-known, but unrealistic planetary model of the atom. A stable electronic orbit can only be maintained by a constantly accelerated electron, which according to the principles of electrodynamics constitutes a source of radiation. The stability of the atom can simply not be accounted for in terms of classical mechanics. A radically different description of electronic behaviour is required. As a matter of fact, a radically different system of mechanics is required to describe electronic motion correctly and this is where a theoretical understanding of chemistry must start. 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

Figure 3. Contour piot of the eiectronic density of a (tripiet) eigenstate strongly scarred by the antisymmetric stretch orbit (left), in 2D configuration space (spanned by the electrons distances ri and r-2 from the nucleus, in the collinear configurations considered here). This eigenstate belong to the N = 9 series. The solid lines depict the associated classical periodic orbit. Autoionization rates of antisymmetric stretch singlet states (right) of the Nth autoionizing series of the helium spectrum, in ID (squares), 2D (circles), and 3D (diamonds) configuration space.

EPR. The frozen solution EPR spectrum of 75 shown in Fig. 13 exhibits rhombic symmetry typical of a Ni(III) bis(dithiolene) complex (126), with g-tensor values of gx = 2.13, gy = 2.04, g, = 1.99. The nickel dimer can thus be viewed as a classical bis(dithiolene) moiety with a 3B3g ground state where the odd electron orbital composed primarily of a metal dyZ orbital and sulfur 2pz orbitals. The Ni(II) in the pz, as expected, is EPR silent (19, 122). 

Umt[p is the classical Coulomb energy and Exc[p] is the XC energy. It is the functional form of this XC functional, which is usually approximated in absence of an exact expression. The one-electron orbitals ipk(r) are obtained through self-... 

Resultant energy curves in H2 and H2. u, Burrau s curve for H2 . b, Curve for H2 for non-interacting electrons, c, Approximate curve for H2 with interacting electrons. The small circle in the crook of curve b, represents the equilibrium position and energy on Hutchisson s classical crossed-orbit model of H2. Units same as figure 1 (note different scales of ordinates for Ha and H2+). 

A second type of force between water molecules and the metal consists of the dispersion forces. Dispersion forces (or London forces) can be seen classically as follows A time-averaged picture of any atom shows spherical symmetry because the charge due to the electrons orbiting around the nucleus is smoothed out in time. An instantaneous picture of, say, a hydrogen atom, would, however, show a proton here and an electron there—two charges separated by a distance. Hence, every atom has an instantaneous dipole moment of course, the time average of all these dipole moments is zero. This instantaneous dipole will induce an instantaneous dipole in a contiguous atom, and an instantaneous dipole-dipole force will arise. When these... 

In a classical Bohr orbit, the electron makes a complete journey in 0.15 fs. In reactions, the chemical transformation involves the separation of nuclei at velocities much slower than that of the electron. For a velocity 105 cm/s and a distance change of 10 8 cm (1 A), the time scale is 100 fs. This is a key concept in the ability of femtochemistry to expose the elementary motions as they actually occur. The classical picture has been verified by quantum calculations. Furthermore, as the deBroglie wavelength is on the atomic scale, we can speak of the coherent motion of a single-molecule trajectory and not of an ensemble-averaged phenomenon. Unlike kinetics, studies of dynamics require such coherence, a concept we have been involved with for some time. 

EHMOcalculations on 111 and 100 metal planes have indicated that the surface electron orbitals are quite localized (refs. 14-16). This supports the premise that these surface sites can be considered as "surface complexes". With this assumption classical inorganic techniques can... 

A way out of this dilemma was suggested by Bohr15 in 1913 [9, 10]. He retained the classical picture of electrons orbiting the nucleus in accord with Newton s laws, but subject to the constraint that the angular momentum of an electron must be an integral multiple of h/2n ... 

There are two ways of describing the transfer of an electron from its site in the solvent into the orbitals of a substrate. One is the classical mechanism, which involves an overlap of the orbitals of the substrate and those of e, and the formation of a transition state followed by a rapid electron transfer. The other is a non-classical electron tunnelling... 

---