Helmholz approximation

This property ensures that expression (11) satisfies the micro-reversibility principle [27]. Note that this result is maintained if the TJJ are kept in (6) and (9). Expression (5) of T b deserves a few comments. First, T b appears independent of the zero of energy of H due to the second term of the numerator [27]. Secondly, according to the Wolfsberg—Helmholz approximation [28] ... 

Some individual compounds have been studied using LCAO-MO theory in the Wolfsberg-Helmholz approximation (5). Although this method is somewhat more realistic and allows one to account for other properties (such as "charge-transfer bands, EPR, and NMR experiments) nevertheless, compared to the crystal field model it is much more laborious, it is only vahd for the individual case, and the choice of parameters in often rather arbitrary. 

The Wolfberg Helmholz Approximation and the Extended Hiickel Method... 

The angular overlap model is a relatively crude method which appears to yield results at least as good as those afforded by the crystal field model. As with all simple empirical models, the AOM depends on many approximations and assumptions which cannot be expected to be even approximately correct. Thus, for example, the parameter a is assumed to depend only on the identity of the metal and the ligand, and on the internuclear distance it is independent of the stoichiometry or stereochemistry. The theoretical basis for assuming the proportionality of the AOM matrix elements to overlap integrals is closely related to the Wolfs-berg-Helmholz approximation for the off-diagonal matrix elements of the one-electron operator ... 

K = 1.75. In the weighted Wolfsberg-Helmholz approximation [10], the coefficient K of Eq. (2) is replaced with another coefficient AT, which is given by... 

For example, according to the Wolfsberg-Helmholz approximation, 3s is a stronger overbinder, while, according to the Pople approximation, employed... 

EHM0 calculations performed using optimized geometries from ab initio calculations and the indicated value of K in the Wolfsberg-Helmholz approximation of 3 ... 

Ionization Energies (VOIE s), as well as the corresponding overlap Integral according to the Wolfsberg-Helmholz approximation ... 

This sequence is also established by comparing AOM parameters with the cor-responc g overlap integrals that are derived by substituting H l hi Eq. (4) using the Wolfsberg-Helmholz approximation [17]... 

In contrast to the useful conceptual framework provided by the approximate approach just described, the results of more detailed molecular orbital calculations have on the whole been rather disappointing. Thus, although some semi-empirical SCF treatments were attempted, most of the earlier MO calculations for metallocene systems (18, 161, 162, 163, 164,165) suffered from such deficiencies as the neglect of the a-framework, or the use of various one-electron Hamiltonians, for example the various Wolfsberg-Helmholz techniques. Of late, Drago and his coworkers have carried out further Extended Htickel type computations for a wide range of both metallocene and bis-arene species (153, 154), and similar... 

Detailed calculations using the Wolfsberg-Helmholz molecular orbital approximation have been carried out for some complexes. 

In general, the semiempirical approach has been widely adopted in inorganic systems, and the most common approximation is either to assume that Hu is directly proportional to the overlap integral Sij9 or related to it by a Wolfsberg-Helmholz type of formula (43) ... 

The EH method (developed by Wolfsberg and Helmholz and by Hoffmann) is an extension of the Hiickel method in which the pi-electron approximation is not made, but all valence electrons are treated. The method is thus applicable to nonplanar, as well as planar, molecules. The valence-electron Hamiltonian is taken as the sum of one-electron Hamiltonians //va, = 2(/ eff(0, where Hcft(i) is not explicitly defined. The valence-electron wave function is the antisymmetrized product of spin-... 

The more difficult resonance integrals to approximate are the off-diagonal ones. Wolfsberg and Helmholz (1952) suggested the following convention... 

The only models that exist for double layer structure in ionic liquids suggest a Helmholz layer at the electrode/solution interface [103, 104], If the reduction potential is below the point of zero charge (pzc) then this would result in a layer of cations approximately 5 A thick across which most of the potential would be dropped, making it difficult to reduce an anionic metal complex. Hence, the double layer models must be incorrect. 

If Hu is much less negative than H22, it is possible to apply second-order perturbation theory to eq. (37). Since the eigenvalues then contain only even powers Sj, S12,. .. and since it is clear that the Wolfsberg-Helmholz method anyhow is a dubious approximation to the M.O. if Sjg is not rather small compared to S1, it may be interesting to find the approximate eigenvalues accurate until second power of the overlap... 

Here ( and j span the n atomic orbitals in the basis set of valence orbitals (core electrons are neglected) and the coefficients 7 j- are chosen to diagonalize the Hamiltonian matrix. The quantity S,y is the overlap integral between orbital i and orbital j. EHT approximates the off-diagonal matrix elements Hi - by the Wolfsberg-Helmholz formula/98/... 

Nowadays, the success of the methods proposed by Hoffmann 50> and by Pople and Segal 51> among the chemists tends to promote the use of pure atomic orbital bases for all-valence treatments. The first method is a straightforward application of the Wolfsberg-Helmholz treatment of complexes to organic compounds and is called the Extended Hiickel Theory (EHT), because its matrix elements are parametrized in the same way as the Hiickel method with overlap for n electrons. The other method, known under the abbreviation Complete Neglect of Differential Overlap (CNDO), includes electron repulsion terms by extending to a orbitals the successful approximation of zero-differential overlap postulated for n electrons. 

In the first LCAO calculation on MnO, . Wolfberg and Helmholz [13] suggested as early as 1952 to use empirical data such as ionization potentials for the diagonal elements Frr of (5) since Frr represents the energy of an electron in /, in the field of the nuclear skeleton and the remaining valence electrons. For the off-diagonal elements, they introduced what has become known as the Wolfberg and Helmholz (WH) approximation... 

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