Orbitals and Their Energies

We can describe the electronic structure of a many-electron atom in terms of orbitals like those of the hydrogen atom. Thus, we continue to designate orbitals as k, 2p and so forth. Further, these orbitals have the same general shapes as the corresponding hydrogen orbitals. 

To explain this fact, we must consider the forces between the electrons and how these forces are affected by the shapes of the orbitals. We wiU, however, forgo this analysis until Chapter 7. 

In a many-electron atom, can we predict unambiguously whether the 4s orbital is lower or higher in energy than the Sr/orbitals  

Not all of the orbitals in the r = 4 shell are shown in this figure. Which subshells are missing  

Orbitals in any subshell are degenerate (have same energy) 

---

We shall illustrate these rules first with H2 and then with other diatomic molecules. The same principles apply to polyatomic molecules, but their molecular orbitals are more complicated and their energies are harder to predict. Mathematical software for calculating the molecular orbitals and their energies is now widely available, and we shall show some of the results that it provides. 

Total Energies Total energies should be related to orbitals and their energies. They should be size-extensive as well. 

Vol. 38 E. Lindholm, L. Asbrink, Molecular Orbitals and their Energies, Studied by the Semiempirical HAM Method. X, 288 pages. 1985. 

In the Xa scattered wave approximation, the exchange potential for spin-up electrons may be different from that for spin-down electrons. In particular, when unpaired electrons are present, the exchange potentials, and hence the spin-up and spin-down orbitals and their energy levels, are different. Thus, MO calculations are performed using a spin-unrestricted formalism so that separate orbital energy levels are given for spin-up (a) and spin-down (p) electrons. 

Notes (a) Secular determinant corresponding to Eq. (16) (h) symmetry-factored determinant (c) symmetrized orbitals in terms of p atomic orbitals (d) molecular orbitals and their energies ) given by = a—Xj3. 

Here, 2 is an operator appropriate to the quantity of interest, ij/ is the wave function for the system being studied, and a is the result of an experimental measurement. Experimental quantities include the energy, the charge density, the momentum, the dipole moment, etc. The molecular orbitals and their energies are not experimental quantities and the wave function for a system need not be expressed in terms of molecular orbitals. In fact, the wave functions themselves would not be necessary if there were a direct way in which to calculate the electron density distribution for a molecule since all of the properties may be derived from the density. ... 

Since we now have a one-electron problem, the Kohn-Sham equations (2.4) can be solved in a self-consistent manner. We obtain a set of orbitals and their energies, much as in HF theory. The density function, p(r, can be found as the sum of the squares of the w/, for the occupied orbitals. From p(r) the expectation value of the energy can be found, as well as other one-electron properties. Just as in the HF method, the total electronic energy is equal to the sum of the energies of the occupied orbitals, minus a correction because the electron-electron interactions have been counted twice. 

The situation is different in the Hiickel model, for example, since no electron repulsion is included and therefore the equations are linear, the orbitals and their energies are independent of the number of electrons in the molecule or its state and therefore, within the confines of this model one may construct approximations to any state of the molecule. This is not to say that such functions are realistic, just that the model is consistent with their construction ... 

An empirical investigation do the orbitals and their energy eigenvalues converge smoothly to some limiting functions as the occupied orbitals do. 

Whatever procedure is used to calculate the molecular orbitals and their energies, a certain number of electron and energy indices can be defined from them, which are related to the essential chemical, physicochemical and biochemical properties of molecules. For a detailed discussion see ref. 2. Here we simply recall the main definitions ... 

When designing the electrolyte solvent and salt compounds one can consult the molecular orbital methods with some programs such as the Gaussian [63]. The molecular orbital method calculates the most stable conformation as well as the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) levels of the compound concerned. Figure 2.11 indicates the relationship between molecular orbitals and their energy levels in a molecule. When the oxidation takes place an electron is ranoved from the HOMO. When the reduction occurs an electron is inserted into the LUMO. Therefore, the lower HOMO level... 

Fig. 2.U The relationship between molecular orbitals and their energy levels. The arrow refers to an electron and its direction does the direction of electron spin...

These real-amplitude molecular orbitals and their energies are illustrated in Fig. 11.4. 

---