Approximate energies of electronic states

Let us consider (within the RHF scheme) the simplest closed-shell qrstem with both electrons occupying the same orbital (p. The Slater determinant, called (G from the ground state) is built from two spinorbitals d = P oi and f 2 = PiP-We also have the virtual orbital (p2, corresponding to orbital energy S2, and we may form two other spinorbitals from it. We are now interested in the energies of all the possible excited states which can be formed from this pair of orbitals. These states will be represented as Slater determinants, built from (pi and (p2 orbitals with the appropriate electron occupancy. We will also assume that excitations do not deform the q orbitals (which is, of course, only partially true). Now all possible states may be listed by occupation of the ej and S2 orbital levels, see Table 8.2. 

Electronic Motion in the Mean Field Atonts and Molecules 

Now let us calculate the mean values of the Hamiltonian using the states mentioned above. Here we will use the Slater-Condon rules (p. 986), which soon produce in the MO representation  

The orbital energies of a molecule (calculated for the state with the doubly occupied (f i orbital) are  

the energies of the electronic states can be pressed in terms of orbital 

---

Chemical reactions of molecules at metal surfaces represent a fascinating test of the validity of the Born-Oppenheimer approximation in chemical reactivity. Metals are characterized by a continuum of electronic states with many possible low energy excitations. If metallic electrons are transferred between electronic states as a result of the interactions they make with molecular adsorbates undergoing reaction at the surface, the Born-Oppenheimer approximation is breaking down. 

Here, we pointed to the problem of theoretical representation, in particular, in two aspects of theory (i) the existence of highly mobile atoms at the surface such as hydrogen, which are usually not considered in the atomistic models and (ii) the importance of bandgaps and relative energy levels of electronic states, which is often distorted in local density approximations. In both respects, a quick fix to the problem is not very likely. However, as both theory and experiment continue to be developed and applied in common research projects, it can be expected that the actual understanding of the processes involved in reaction on model catalysts will substantially improve over the next 10 years. After all, the ability to trace reactions and to account for the position and charge state of each reactant is already a realization of what seemed 20 years ago a fiction rather than fact. 

An approximation stating that the motion of nuclei in ordinary molecular vibrations is slow relative to the motions of electrons. Thus, the nuclei can be held in fixed positions when doing calculations of electronic states. Such an assumption is useful in determining potential energy surfaces and is central in studying the quantum mechanical properties of molecules. See also Adiabatic Photoreaction Diabatic Photoreaction... 

Within the approximation that the valence electronic states can be described adequately as combinations of the above valence CSFs, the three JE, JE, and CSFs must be combined to form the three lowest energy valence electronic states of E symmetry. For the homonuclear case, the E CSF does not couple with the other two because it is of ungerade symmetry, while the other CSFs JE and1E have gerade symmetry and do combine. 

Abstract. We compute the velocity correlation function of electronic states close to the Fermi energy, in approximants of quasicrystals. As we show the long time value of this correlation function is small. This means a small Fermi velocity, in agreement with previous band structure studies. Furthermore the correlation function is negative on a large time interval which means a phenomenon of backscattering. As shown in previous studies the backscattering can explain unusual conduction properties, observed in these alloys, such as for example the increase of conductivity with disorder. 

Hartree—Fock Theory. In the lowest electronic state of most stable molecules the n orbitals of lowest energy are all doubly occupied, thus forming a closed shell. If, at least as a first approximation, such an electronic state is described by a single configuration, the wave function for this state can be written as... 

These are important results for practical (and approximate ) electrochemical calculations. As far as the occupancy of electronic states above the Fermi level is concerned, it becomes negligible so quickly (recall that the practical electrochemical scale is a few electron volts), and the occupancy is so high at the Fermi level that electrochemists usually use the rule that the only metal electrons they should count are those at the Fermi level they neglect electrons having energies below or above that of the Fermi level. 

---