Energy of Molecular Orbitals

The complexity of molecular systems precludes exact solution for the properties of their orbitals, including their energy levels, except in the very simplest cases. We can, however, approximate the energies of molecular orbitals by the variational method that finds their least upper bounds in the ground state as Eq. (6-16)... 

The molecular orbital theory of polyatomic molecules follows the same principles as those outlined for diatomic molecules, but the molecular orbitals spread over all the atoms in the molecule. An electron pair in a bonding orbital helps to bind together the whole molecule, not just an individual pair of atoms. The energies of molecular orbitals in polyatomic molecules can be studied experimentally by using ultraviolet and visible spectroscopy (see Major Technique 2, following this chapter). 

Most of what we know about the structure of atoms and molecules has been obtained by studying the interaction of electromagnetic radiation with matter. Line spectra reveal the existence of shells of different energy where electrons are held in atoms. From the study of molecules by means of infrared spectroscopy we obtain information about vibrational and rotational states of molecules. The types of bonds present, the geometry of the molecule, and even bond lengths may be determined in specific cases. The spectroscopic technique known as photoelectron spectroscopy (PES) has been of enormous importance in determining how electrons are bound in molecules. This technique provides direct information on the energies of molecular orbitals in molecules. 

Frost, A., and Musulin, B. (1953). /. Chem. Phys. 21, 572. The original description showing energies of molecular orbitals by means of inscribed polygons. 

Multiplying both terms in equation 1.131 by i//, extracting E, and integrating over the Cartesian coordinates, we obtain the energy of molecular orbitals (MOs) ... 

Corrected values (the energies of molecular orbitals wero decreased by a constant value). 

O Albert Haim, "The Relative Energies of Molecular Orbitals for Second-Row Homonuclear Diatomic Molecules The Effect of s-p Mixing," /. Chem. Educ., Vol. 68, 1991,737-738. 

Thus pure MM (MM by itself) can t calculate UV spectra, the shapes and energies of molecular orbitals, and electron distribution and derivative properties of this, like atomic charges, dipole moments, and more arcane molecular features like bond paths (associated with atoms-in-molecules theory, AIM [1]). 

We can get a good estimate of the absolute energies of molecular orbitals from photoelectron spectroscopy and electron transmission spectroscopy (see Chapter 2). Such experiments suggest energies for the HOMO and LUMO of butadiene to be -9.03 and +0.62 eV. respectively, whilst for ethene they are -10.51 and +1.78 eV, respectively. 

Photoelectron spectroscopy (PES) measures the energies (Table 1.3) of filled orbitals, and overcomes the problem that UV spectroscopy does not give good absolute values for the energies of molecular orbitals. 

Figure 9.7. Shapes and relative energies of molecular orbitals, (a) Bonding and antibonding orbitals, (b) Relative energies of orbitals and possible transitions...

The actual energies of molecular orbitals for diatomic molecules are intermediate between the extremes of this diagram, approximately in the region set off by the vertical lines. Toward the right within this region, closer to the separated atoms, the energy sequence is the normal one of O2 and F2 further to the left, the order of molecular orbitals is that of B2, C2 and N2, with a-g(2p) above TT (2p). 

Reactivity derives from energies of molecular orbitals ch3... 

A. HAIM, The relative energies of molecular orbitals for second-row homonuclear diatomic molecules. J. Chem. Educ., 68, Til (1991). 

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