Orbitals physical meaning

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

This result is remarkably simple as compared to the usual methods. For a spin-polarised potential V, Kraft, Oppeneer, Antonov and Eschrig (1995) used the elimination method and found the corrections as a sum of 9 terms, which is equivalent to our Eq.(ll). They notice that three terms of their sum have a known physical meaning (spin-orbit, Darwin and mass-velocity corrections), but the other terms have no special name . 

The physical meaning of our final equation is best seen on eqn 39. The term containing w is essentially the self-energy correction introduced by Mulliken in his analysis of electronegativities to account for the average repulsion of electrons occupying the same orbital. In order to get an idea of the orders of magnitude, let us apply eqn 39 to a model computation of FeCO, made to compare the ClPSl results of Berthier et al. [11] with those of a simple orbital scheme. Consider one of the two x systems of FeCO, treated under the assumption of full localization (and therefore strict cr — x separation)... 

The second term s may be called the operator for spin angular momentum of the photon. However, the separation of the angular momentum of the photon into an orbital and a spin part has restricted physical meaning. Firstly, the usual definition of spin as the angular momentum of a particle at rest is inapplicable to the photon since its rest mass is zero. More importantly, it will be seen that states with definite values of orbital and spin angular momenta do not satisfy the condition of transversality. 

Eigenfunctions that accompany these eigenvalues have a clear physical meaning that corresponds to electron attachment or detachment. These functions are known as Dyson orbitals, Feynman-Dyson amplitudes, or generalized overlap amplitudes. For ionization energies, they are given by... 

In further studies of chemistry and physics, you will learn that the wave functions that are solutions to the Schrodinger equation have no direct, physical meaning. They are mathematical ideas. However, the square of a wave function does have a physical meaning. It is a quantity that describes the probability that an electron is at a particular point within the atom at a particular time. The square of each wave function (orbital) can be used to plot three-dimensional probability distribution graphs for that orbital. These plots help chemists visualize the space in which electrons are most likely to be found around atoms. These plots are... 

Similar probability functions (including the factor 4nr2) for the 2s, 2p, 3s, 3p, and 3d orbitals are also shown in Fig. 2.4. Note that although the radial function for the 2s orbital is both positive (r < 2a0/Z) and negative (r > 2u0/Z), the probability function is everywhere positive (as or course it must be to have any physical meaning) as a result or the squaring operation. 

The simplest way to illustrate physical meaning of these quantities is to consider the perturbations of orthogonally twisted ethylene for which SAB = yAB = <5ab = yab = 0 holds via (1) return to planarity or (2) substitution at one end of the C=C bond. For (1), localized orbitals interact, yAB 0, but their energies are the same, 5AB = 0. Since delocalized orbitals become eventually HOMO and LUMO of planar ethylene, they do not have the same energy, Sab 0, but they do not interact, yab = 0. For (2), orthogonal-substituted ethylene, the situation is different. In the localized basis SAB 0, but the interaction is not present due to the symmetry yAn = 0. (A and 2 S belong to different irreducible representations.) For the delocalized description the energies of these orbitals are the same 5ab = 0 since the orbitals are equally distributed over both carbon atoms. But yab 0, since a and b are not canonical orbitals. 

Now let us inquire into the physical meaning of the two conditions Hu -H22 and H12 (= H21) = 0. If we consider the basis (< )1 and 2) of the secular equation (Eq. [2]) as the diabatic components of the adiabatic electronic eigenfunction (a diabatic function describes the energy of a particular spin-coupling or atomic orbital occupancies,14 while the adiabatic function represents the surface of the real state), the crossing condition (real or avoided) is fulfilled when the two diabatic components and ( >2 cross each other, and this happens when H11 = H22, that is, when the energy of the two diabatic potentials (H11 is... 

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