Orbital centring

There may be special difficulties in reactions where the ordering of orbitals centred on the metal changes along the actual reaction path, because of configuration interaction and the non-crossing rule for states. 

Fig. 3.7 The Heitler-London configuration A(1) B(2) and A(2) B(1) (a) and (b) respectively, where 0A and represent the atomic 1s orbitals centred on atoms A and respectively, and 1 and 2 represent the coordinates of the two (indistinguishable) electrons, (c) The molecular orbital basis function in the singlet state where electrons 1 and 2 have opposite spin, (d) The up and down spin eigenfunctions corresponding to local exchange fields of opposite sign on A and B.

Pyridine is isoelectronic with benzene in which C — H is replaced by N. The two electrons which formed the w-bonds in — CH are now localized in a nonbonding hybrid orbital centred on the nitrogen atom. The substitution of N lowers the overall symmetry of the molecule from D to Cs, and the degenerate eu and e, orbitals are split into ax and ht type orbitals. The MOs of pyridine are given in Figure 2.20. 

Rather than find the most perfect MOs which satisfy eqn (10-2.4) (or eqns (10-2.5)), it is common practice to replace them by particular mathematical functions of a restricted nature. These functions will generally contain certain parameters which can then be optimized in accordance with eqn (10-2.4). Since these MOs are not completely flexible, we will have introduced a further approximation, the severity of which is determined by the degree of inflexibility in the form of our chosen functions. Typical of this kind of approximation is the one which expresses the space part of the MOs as various linear combinations of atomic orbitals centred on the same or different nuclei in the molecule. We write the space part of each of the approximate MOs as... 

In constructing a localized MO for the bond A—B it is necessary to specify an orbital centred on A (tpA) and an orbital centred on B (y ). In principle, provided symmetry about the bond axis is preserved (we are still considering only cr-bonded systems), our choice of tpA and pB is not restricted and we could use any well-defined mathematical function or combination of functions. Common sense, however, dictates that the most sensible functions to use for this purpose are the AOs of the free atoms A and B. There are three reasons why this is a sensible choice one mathematical, one chemical, and one practical. 

If the size of a donor or an acceptor molecule is large and the molecular orbital from which the electron leaves or at which it arrives is delocalized over the whole molecule, then, rigorously speaking, in eqns. (16) and (17) we must understand the distance R to be not the distance between the orbital centres, but the shortest distance from the donor to the acceptor. 

The fundamental assumption of the MO theory is that the 77-electrons can be considered separately from the e-electrons, which, with the atomic cores, may be considered to define the framework about which the 77-electrons move. The energy of the 77-electrons is then obtained in terms of Coulomb integrals ar related to the electronegativity (ar = a,. + r/3, at each atom r, being the energy needed to remove the 77-electron from the atom) and exchange integrals jSra between orbitals centred on atoms r and s. For carbon atoms Sr = 0 (usually). 

In order to eliminate the problems with the invariance, we proposed some time ago a topological approximation based on the so-called overlap determinant method [43]. This approximation is based on the transformation matrix T that describes the mutual phase relations of atomic orbitals centred on molecules R and P, and thus plays in this approach the same role as the so-called assigning tables in the overlap determinant method (Eq. 4)... 

Synthesis of the three observations led to Bohr s proposal of a planetary atom consisting of a heavy small stationary heavy nucleus and a number of orbiting electrons. Each electron, like a planet, had its own stable orbit centred at the atomic nucleus. The simplest atom, that of hydrogen, with atomic number 1 could therefore be described as a single electron orbiting a proton at a fixed, relatively large, distance. The mechanical requirement to stabilize the orbit is a balance between electrostatic and mechanical forces, expressed in simple electrostatic units, and particle momentum p = mv, as ... 

Mk is the number of basis orbitals centred on the fragment k. The total (MxN) matrix of the partitioned molecular orbital coefficients T, defined as... 

Following and elaborating on the approach of Karplus and Das (1961) Grant accepts a separation of op for carbon-13 into a term o ) for electrons in orbitals centred entirely on the carbon atom in question, and a term afor electrons in orbitals centred both on that atom as well as other atoms in the molecule (i.e., bonding ... 

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