Half-electron method,

The half-electron method can also apply to triplet states. For this calciilatioit. IlyperCheni populates selected molecular orbitals with pairs of half electron s. The final energy is computed by assigning the proper spins. 

Although LHF is often a better theoretical treatment of open-shell systems than the RHF (half-electron) methods, it takes longer to compute. Separate matrices for electrons of each spin roughly double the length of the calculation. ... 

Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination. 

In the approach of Dewar and co-workers (34), termed the half-electron method , a physical model is considered in which an unpaired electron is replaced by two hypothetical half-electrons of opposite spin. For radicals containing one unpaired electron, the eigenvalue problem of this method is, in our opinion, identical with the method of Longuet-Higgins and Pople (27) ... 

For reasons given later, we shall most frequently use in applications the method of Longuet-Higgins and Pople. Recently, the half-electron method was extended to the lowest-energy open-shell states of any given symmetry and multiplicity (57). 

The description of configuration interaction given for rr-electron methods is also valid for all-valence-electron methods. Recently, two papers were published in which the half-electron method was combined with a modified CNDO method (69) and the MINDO/2 method was combined with the Roothaan method (70). Appropriate semiempirical parameters and applications of all-valence-electron methods are most probably the same as those reviewed for closed-shell systems (71). 

Calculated as differences between total SCF energies of the neutral hydrocarbon and the corresponding positive ion radicals were calculated by the half-electron method, (34). Method of Longuet-Higgins and Pople eq. (90) was employed, (61). 

Calculated by the half-electron method of Dewar et ai.262 These /-values correspond to the adiabatic ionization potential. 

H o.V G v and P v are the one-electron matrix, two-electron matrix and the density mtrix, respectively. Note that in this derivation the number of electrons (njO in a MO may take on a non-integer value (0physical system non-integer electrons have been used previously in, for example, the so-called half-electron method (Dewar, Hashmall et al., 1968). 

It is interesting that although this fact is known for more than a decade (55-58) its obvious connections with the half-electron method of Dewar (47) have never been pointed out before. At about the same time that Dewar published his first paper on the half-electron method (59) we did show (60) that Eq. (12) corresponds to treat the interelectronic repulsion as the interaction between two half-electron-pairs, which is not zero for one electron (two half-electrons). For closed-shell systems Eq. (11) and (12) give the same value for , but for = 1 the value of IF according to (12) is not the correct value... 

Figure 14 Different descriptions of the metal-metal bond orders in some dinuclear mmnlexes with bridging hydride ligands according to the eiectron counting method. The half-electron method does not o r ti take into account the 3-center-2-electron nature of the M-H-m interaction and thus resuits in a greater M-M predicted theoretically. In...

Figure 15 Examples of compounds that exhibit M- -x- -M antibonding interactions resulting from a 20-electron configuration that places a pair of electrons in a M-M antibonding orbital. In contrast, the formula employing the half-electron method defines each metal center as possessing an 18-electron configuration and so the true nature of the M-M interaction is not evident.

In addition to the half-arrow p,-LX description of the bonding for evaluating M-M bond orders, an alternative method merely apportions the electron associated with each hydride ligand equally to both metals (the half-electron method) [24-27]. Specifically, the M-M bond order is obtained by application of the formula w = (18n - A )/2, where m is the number of 2-center-2-electron M-M bonds, n is the number of M atoms, and N is the total electron count assuming that each bridging hydride ligand contributes one electron. For example, the half-electron method predicts an Re = Re double bond for [(CO)4Re]2(p--H)2 [25]. 

Mo—Mo Mo-Mo bond order [(ji-LX] halfarrow method Mo-Mo bond order half-electron method... 

Table 4 Comparison of calculated bond orders for a series of dinuclear molybdenum compounds with those predicted by two alternative electron-counting methods. In each case, the bond order predicted by the [p.-LX] half-arrow method corresponds exactly to that derived by consideration of the orbital occupations, whereas the half-electron method fails for compounds with bridging hydride and methyl ligands. Data ttiken from reference 6...

Consideration of both the Mayer bond order analysis and the molecular orbital analysis indicates that the Mo-Mo bond order predicted by the p,-LX half-arrow method for [CpMo(p,-02CH)]2(p,-PH2)(p,-H) is much more in accord with the theoretical calculations than is the bond order of two predicted by the half-electron method. A variety of other studies support the use of the p,-LX half-arrow electron-counting method for predicting M-M bond orders in dinuclear compounds [52-59]. In particular, Hoffmann and Albright have used MO theory to analyze the... 

CO)sM]2 (p,-H) (M = Cr, Mo, W) [11] represents a classic series of complexes with single 3-center-2-electron M-H-M interactions for which the half-electron method predicts an M-M single bond, whereas the p,-LX half-arrow method... 

JI.-LX "Half-Arrow" Method "Half-Electron" Method... 

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