Orbital Energies and Ionization Potentials

Errors in ionization potentials are typically a few tenths of an electron volt, a small percentage of the total. Stewart gives tables with 

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Koopmans, T. Physica,/, 104 (1933).The physical reason for the agreement between orbital energies and ionization potentials was clarified by Mulliken, R. S. J. Chim. Phys. 46, 497 (1949). 

TABLE 4. Orbital energies and ionization potentials of cyclopropane ... 

The part of the Fock potential not defined through the Brillouin condition is often chosen on physical ground [e.g., to have the resultant orbital energies represent ionization potentials and electron afiinities (via Koopmans theorem)] (MeWeeney and Sutcliffe, 1976). For a reference state containing a set of occupied spin-orbitals that we denote by a, / , y, S and a set of unoccupied spin-orbitals denoted ni, i, p, t/, the Fock potential in Eq. (2.89) is defined by the BT only between occupied and unoccupied orbitals. From Eq. (2.89) we get... 

Koopmans theorem provides the theoretical justification for interpreting Hartree-Fock orbital energies as ionization potentials and electron affinities. For the series of molecules we are using, the lowest virtual orbital always has a positive orbital energy, and thus Hartree-Fock theory predicts that none of these molecules will bind an electron to form a negative ion. Hartree-Fock almost always provides a very poor description of the electron affinity, and we will not consider the energies of virtual orbitals further. 

So, within the limitations of the single-detenninant, frozen-orbital model, the ionization potentials (IPs) and electron affinities (EAs) are given as the negative of the occupied and virtual spin-orbital energies, respectively. This statement is referred to as Koopmans theorem [47] it is used extensively in quantum chemical calculations as a means for estimating IPs and EAs and often yields results drat are qualitatively correct (i.e., 0.5 eV). 

Figure 9. Determination of the first electron affinity, and the first and higher ionization potentials of formyl radical from the SCF orbital energies and electronic repulsion integrals, and K,j (cf. eqs. (90), (92), and (93)). The experimental value (112), 9.88 eV, for the first ionization potential corresponds to the theoretical value I . All entries are given in eV. With A and I a lower index stands for MO the upper one indicates the state multiplicity after ionization.

The individual terms, e , in Eq. (2.27) are termed one-electron orbital energies and correspond to the ionization potential (-e ) of an electron in MO i r assuming that no reorganization of the core nuclei, or the other (2N — 1) electrons, takes place during ionization. The total energy of the system [Eq. (2.27)] is clearly not the sum of the one-electron energies because electron-electron interaction terms are included twice. 

Fig. 10 Electrochemical energy level model for orbital mediated tunneling. Ap and Ac are the gas-and crystalline-phase electron affinities, 1/2(SCE) is the electrochemical potential referenced to the saturated calomel electrode, and provides the solution-phase electron affinity. Ev, is the Fermi level of the substrate (Au here). The corresponding positions in the OMT spectrum are shown by Ar and A0 and correspond to the electron affinity and ionization potential of the adsorbate film modified by interaction with the supporting metal, At. The spectrum is that of nickel(II) tetraphenyl-porphyrin on Au (111). (Reprinted with permission from [26])...

Figure 1 shows the electron attachment energies (AE) and ionization potentials (IP) of silyl substituted 7t-systems and related compounds [4], AE can be correlated with the energy level of the LUMO (lowest unoccupied molecular orbital) and IP can be correlated with the energy level of the HOMO (highest occupied molecular orbital). For a-substituted 7t-systems, the introduction of a silyl group produces a decrease in the tc -(LUMO) level. This effect is attributed to the interaction between a low-lying silicon-based unoccupied orbital such as the empty d orbital of silicon and the it orbital (d -p interaction) as shown in Fig. 2. Recent investigations on these systems, however, indicate that d orbitals on silicon are not necessarily required for interpreting this effect a-effects of SiR3 can also be explained by the interaction between Si-R a orbitals and the 7r-system.

The second [35], with the highest occupied Kohn-Sham orbital energy and is identified to be the negative of the first ionization potential... 

Sander applied DFT (B3LYP) theory to carbenic philicity, computing the electron affinities (EA) and ionization potentials (IP) of the carbenes." " The EA tracks the carbene s electrophilicity (its ability to accept electron density), whereas the IP represents the carbene s nucleophilicity (its ability to donate electron density). This approach parallels the differential orbital energy treatment. Both EA and IP can be calculated for any carbene, so Sander was able to analyze the reactivity of super electrophilic carbenes such as difluorovinylidene (9)" which is sufficiently electrophilic to insert into the C—H bond of methane. It even reacts with the H—H bond of dihydrogen at temperamres as low as 40 K, Scheme 7.2) ... 

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