Normalized orbitals

The problem of T[p] is cleverly dealt with by mapping the interacting many-electron system on to a system of noninteracting electrons. For a determinantal wave function of a system of N noninteracting electrons, each electron occupying a normalized orbital >p, (r), the Hamiltonian is given by... 

Electron A negatively charged atomic particle, normally orbiting the nucleus of an atom. 

Excitation or Activation. The disturbance of ah electron so that it passes from its normal orbit to one of higher energy content as the result of the absorption of radiation. It differs from ionization in the degree to which the electron is separated, as expressed by the equations ... 

L Sflp dn). As an example, consider the parallel overlap of a 2p and a 3d orbital, that is, S(2p7r3dn). The normalized orbitals are given by... 

We will, first of all, briefly resume the method1-2 3 of calculation by considering the particularly simple case of the electronic transitions of the rr electrons of ethylene. The problem may be treated, to a first approximation at least, as a mono-electronic problem. The normal orbital N is, as is well known, described by the 2p, orbitals of the bound atoms, written yx and y>2, and-the overlap integral S by ... 

Other combinations of hybrid orbitals are possible. For example, boron, whose normal orbital configuration is ... 

Electron configurations get more complicated in atoms with higher atomic numbers. Which orbitals the outermost electrons fill first is sometimes less than straightforward. In some cases, for instance, spin considerations can override the normal orbital filling sequence. Fortunately, though, the electron configurations of most... 

For a general many-electron molecule, equation (11) may be re-written in terms of normalized orbital and overlap densities... 

Small angle strain The strain that results from the angles being much less than those resulting from normal orbital overlap. In three- and four-membered rings, the bonds that form the ring are called bent or banana bonds. 

Note that the third term on the right-hand side of Eq. (25) is negative (i.e. energy lowering) regardless of the sign of Hji — i.e. regardless of whether (pj combines in phase or out of phase with cj -. The case of tp > 0 describes the normal orbital interaction, and that of < 0 the counterintuitive orbital interaction. 

In addition to the quantitative agreement of the spectra the orders of magnitude are also in favour of Bohr s model of the atom. For the radius of the normal orbit of the hydrogen atom, considered as a circle, we have by (10), 22, for /a=wi... 

Normal orbit of last electron added. See footnote, p. 186. 

In the normal state, hydrogen (1 H) has an electron in an orbit with the principal quantum number 1. As long as the orbit is regarded as an exact Kepler ellipse the subsidiary quantum number is undetermined. We shall see, however, on taking into account the relativity theory in 33, that the total angular momentum is also to be fixed by a quantum condition, without thereby appreciably altering the energy. The normal orbit of the electron is thus a lj-orbit. 

Too little is known spectroscopically regarding the next elements (7 N, 8 0, 9 F). The chemical evidence affirms that N, 0, F have an affinity for three, two, and one electrons, and the spectrum of 0 shows that the normal orbit of the last electron is a 2a-orbit. The eight-shell required by Kossel s theory must be reached in the case of the inert gas neon (10 Ne) we can assume, therefore, that the eight electrons added since Li are bound in orbits with the principal quantum number 2. The question as to how they are distributed among the 2X- and 2a-orbits we leave unanswered.3... 

The conception of the fully occupied eight-shell is confirmed by the well-known spectrum of sodium (11 Na). The normal orbit of the series electron is a 31-orbit, the p-orbit of maximum energy being a 3a-orbit. Outside the core, then, no more orbits occur with =2. We conclude from this, that the series of electrons for which n—2 is completed by the number 8 reached in the case of neon. Using the terminology of the X-ray spectra we call this structure... 

Since in the case of magnesium (12 Mg) the normal orbit of the series electron is again a S orbit, we assume, in accordance with the double valency, that the magnesium atom in the normal state has two equivalent 31-electrons in addition to the K- and L-shclls. 

In aluminium (13 Al), a 3a-orbit appears as the normal orbit. We see, therefore, that a system of three Sj-orbits cannot be formed as the outside shell. In the case of Li and C+ we arrived at a similar conclusion, namely, the impossibility of the existence of three lj- or 21-orbits. 

With regard to the following elements (15 P, 16 S, 17 Cl) the only relevant evidence at present available is the affinities for one, two, and three electrons and the spectrum of S which indicates that the normal orbit of the last bound electron is a 3a-orbit. The final element of the period is the inert gas argon (18 A), in which, again, a closed shell of 8 electrons must exist. The detailed construction of this shell is best considered from the standpoint of the following element potassium (19 K), the core of which must have this structure. 

---