Closed Electron Shells

The pronounced tendency of radicals to engage in one-electron transfer reactions is well documented [3]. This reaction channel is favored because it provides the simplest way for radicals to lose their radical nature, i.e. to b ome species with an even number of electrons (closed-shell molecules). The direction of the electron flow between the radical X and the molecule Y depends on the oxidizing or reducing power of X and on the ability of Y to either donate or accept an electron the final result of the interaction between X and Y is then the either one-electron-reduced or -oxidized former radical (X" or X ) or the open-shell molecle (Y or Y"" ), (cf. Eq. 1) ... 

Assuming Eq. (49), this result shows that for canonical NSOs the operator F has an essentially degenerate eigenvalue spectmm that is, all the NO eigenvalues are the same (pi) and are equal to minus the vertical IP [84], Unfortunately, apart from the special case of the HE energy that may be viewed as the simplest 1 -RDM functional, none of the currently known functionals (including the exact functional for the total energy in two-electron closed-shell systems) have effective potentials that satisfy the formal relation (49). 

The formation of a complex Species by the association of two or more chemical entities (having, in general, electronic closed shells) is one of the most fundamental molecular processes in biology, in chemistry and in physics. Such a super-molecule represents the next higher level of physical complexity after the nucleus, the atom and the molecule. Its formation involves bonding interactions which are much weaker than the usual well defined covalent bonds. 

The prototypically zero oxidation state complexes of the group are the binary hexacarbonyls M(CO)6. Early studies of the electrochemistry of these 18-electron closed-shell systems in nonaqueous electrolytes has perhaps been seminal in understanding the electron-transfer reactions of more substituted systems and of metal carbonyls in general. 

A common alternative to writing a Hartree-Fock wavefunction as an explicit Slater determinant is to express it using a permutation operator P which permutes (switches) electrons around in MOs. Examine the Slater determinant for a two-electron closed-shell molecule, then try to rewrite the wavefunction using P. 

The Slater determinant for a two-electron closed-shell molecule is... 

The spin orbitals can be separated into a spatial part, if/ (orbital or molecular orbital) and a spin eigenfunction, a (or for the opposite spin). In restricted Hartee-Fock theory (RHF), the spatial part is independent on the spin state, in contrast to unrestricted Hartree-Fock (UHF) where it is spin dependent. Consequently, the RHF spin orbitals can be written as %i= whereas in UHF the corresponding relation is Xi= even number of electrons (closed shell system), but can easily be extended to treat also UHF1. 

In such a small cluster, the surface coverage is 70%, much higher than the percentage measured on flat surfaces covered by arylthiolates (33%).206 The formation of the Au102(SR)44 nanoparticle may be explained with the electronic rules of cluster stability. Considering that each gold atom contributes one valence electron, and that each of the 44 SR thiyl radicals formally takes one electron, a 58 electron closed shell is obtained. A deep analysis and discussion on this landmark achievement have been recently published.207,208... 

We saw that numerous ionization techniques exist that yield radical cations or radical anions, protonated or deprotonated molecules, and various adducts. These ions yield fragments with an even number of electrons (closed shell) or with an odd number of electrons (open shell). Even though the radical cations derived from electron ionization sources retain a privileged status in common mass spectrometry, the other ionization methods become increasingly common. Electron ionization is not possible for many categories of molecules. Therefore, we will not limit the discussion to radical cations. 

Although the molecular size is certainly an important factor that determines D, D also depends on the intermolecular interaction between the molecule and the solvent. A clear example has been reported for chemical reaction of aromatic molecules [15-19]. It was found that the D values of organic radicals are much smaller than those of electronically closed shell molecules with similar sizes and shapes. This change was attributed to the enhanced intermolecular interaction between the radicals and solvent molecules. It was further reported that D of cytochrome c in its native form is much larger than that in the unfolded state [25-27], This difference was attributed to the larger intermolecular interaction between the protein and water due to the unfolded conformation of the a-helices. 

For an N-electron closed-shell state with a dominant reference function, the exact wave-function 4 can be written as a superposition of various n-fold excited determinants on 4.4 may thought to be generated from by anwave operator... 

Trinuclear dusters are nearly always triangular, often heteronuclear, and very numerous. The M3(CO)i2 clusters of Fe, Ru, and Os have been much studied and can be used to illustrate some typical cluster chemistry. The Os3(CO)i2 cluster with the structure (16-XIII), where CO groups are denoted simply by lines, are electronically precise, namely, they have exactly the right number of electrons to provide each metal atom with an 18-electron, closed shell configuration. In such systems there is a total of 48 electrons, and each M—M bond is of order 1. As... 

Catalysis is obtained because the rate of the chemical step is often considerably more favorable at the odd-electron level than at the even-electron (closed-shell) level. Indeed, most of the time inorganic radicals react about 10 times faster than their isostructural diamagnetic analogs. For organic radicals, the rate enhancement is even larger [9]. 

We further neglect the commutator [fC, ri2 ] > which is justified in the so-called standard approximation A [11]. (For a two-electron closed-shell singlet state a formulation in which K does not arise is always exactly possible [10]). This allows us to rewrite (53) as... 

Comparing Equation (5) with (12), and (6) with (13), it is clear that while the SIC functional due to Perdew and Zunger [21] gives the HF energy for a two-electron closed-shell system, for an open-shell system this is no the case. We see that the difference found in the open-shell system, between hf and E, is related to the exchange contribution. Assuming that the SIC functional removes the self-interaction term in the KS equations, it follows that... 

Comparing the HF and KS-SIC-OEP-KLI equations, we see that for a two-electron closed-shell system the equations are the same this is not true for the open-shell system. 

The matrix element Prs is summed over all fdled MOs (from j/ to itn for the ground electronic state of a 2n-electron closed-shell molecule) an example of the calculation ofP was given in section 5.2.3.6e. The elements of the overlap matrix S are simply the overlap integrals ... 

If an approximate, n-electron, closed-shell wavefunction is written... 

In the case of a N-electron closed-shell, the self-consistent field to be felt by one electron arises from N-1 partners. The formal removal of — from (4) to give /f<(3) effects this adjustment. For the energy, this adjustment is introduced by the relation ... 

The PE spectra of the bis-cyclopentadienyl metal complexes, MCp2, illustrate how, in addition to information on metal electrons, information on bonding orbitals can be obtained. The PE spectra of the group 8 metallocenes, with 18-electron closed-shell configurations, are shown in Figure 11. ... 

Within the framework of molecular orbital theory, the simplest wavefunction satisfying this antisymmetric property is the determinant constructed from all the occupied MOs known as the Slater determinant. As an example, wc will consider a 2/7 electron closed-shell molecule that has doubly occupied levels, ... 

Let us consider (within the RHF scheme) the simplest two-electron closed-shell system with both electrons occupying the same orbital (pi. The Slater determinant, called ij/Q (G standing for the ground state) is built from two spinorbitals virtual orbital ipi, corresponding to orbital energy S2, and we may form two other spinorbitals from it. We are now interested in flie eneigies of all the possible excited states that can be formed from this pair of orbitals. These states will be represented as Slater determinants, built from and (p2 orbitals with the appropriate electron occupancy. We will also assume that excitations do not deform the cp orbitals (which is, of course, only partially true). Now all possible states may be listed by occupation of the ei and E2 orbital levels, as shown in Table 8.2. 

The Hartree-Fock method fn- a system with N electrons (closed-shell case) leads to a wave functitm... 

We have shown above that the Hartree-Fock function does not include any electron correlation. We must admit, however, that we have come to this conclusion on the basis of the two-electron, closed-shell case. This is a special situation, since both electrons have different spin coordinates (cr = and or = —j). Is it really true that the Hartiee-Fock function does not include any correlation of electronic motion ... 

Two atoms or molecules may react in many different ways (reaction channels). Even if in some conditions they do not react (e.g., the noble gases), the reason for this is that their kinetic energy is too low with respect to the corresponding reaction harrier, and the opening of their electronic closed shells is prohibitively expensive in the... 

Thus, the wave funetion is described by two electronic closed-shell configurations at infinite distance between the atoms. The situation is actually identical to what was obtained in the transition-state region for the cyclobutane reaction. The reason is also the same the two configurations (cr) and (fT ) become degenerate at dissociation and will mix with equal weights. It is clear that a wave function that describes the full potential curve for the dissociation of a single bond should have the form ... 

---