Localised hybrid orbitals

The valence bond model constructs hybrid orbitals which contain various fractions of the character of the pure component orbitals. These hybrid orbitals are constructed such that they possess the correct spatial characteristics for the formation of bonds. The bonding is treated in terms of localised two-electron two-centre interactions between atoms. As applied to first-row transition metals, the valence bond approach considers that the 45, 4p and 3d orbitals are all available for bonding. To obtain an octahedral complex, two 3d, the 45 and the three 4p metal orbitals are mixed to give six spatially-equivalent directed cfisp3 hybrid orbitals, which are oriented with electron density along the principal Cartesian axes (Fig. 1-9). 

Similarly, the bonding in tetrahedral complexes of first row transition-metal ions is considered in terms of four equivalent sp3 hybrid orbitals (which are constructed from the 4s and 4p orbitals of the metal) oriented towards the vertices of a tetrahedron (Fig. 1-10). For a further discussion of the application of the valence bond method to transition-metal complexes, the reader is referred to publications by Pauling.4) The essential feature is that the bonding consists of localised, two-centre two-electron bonds. 

Unfortunately QM/MM potentials are not devoid of problems. The most severe ones are probably the division of covalent bonds across the QM and MM regions and the lack of explicit polarisation of the MM approach. The first of these two difficulties has been looked at by several groups who have proposed different schemes to deal with the problem Warshel and Levitt [299] have used a single hybrid orbital on the MM atom in the QM/MM region a similar approach has been proposed subsequently by Rivail and co-workers [312, 355, 373] with their frozen orbital (or excluded orbital) in which the continuity between the two critical regions is assured by a strictly localised bond orbital (SLBO) obtained from model compounds. Another popular approach introduces link atoms [300, 310, 315] between QM and MM covalently bonded atoms to cap the valency of the QM atoms the link atoms, usually hydrogen, do not interact with the MM atoms. These are not, by any means, the only ways of dealing with this problem. However, so far it does not seem to have an obvious solution. 

The (assumed localised) orbitals are assumed to be constructed by pairwise addition of pairs of hybrid orbitals the e matrix can be transformed to a tri-diagonal form to reflect this model of the electronic structure. 

Of the five r/-orbitals, it is assumed that only 2.56 are capable of bonding, the remaining 2.44 being localised atomic d-orbitals, which are non-bonding, and capable of receiving electrons with parallel spins as long as is permitted by Hund s Rule. With chromium the sixth electron is divided as shown in Table 1.1. Now the r/ip-hybrid orbital should in theory accommodate 6.56 electrons... 

The tet-PbO elementary (unit) cell is presented in Fig. 2.19. Tet-PbO has a well-defined layered structure. The bonds between the layers are a result of Pb—Pb interactions [1]. The layered structure is parallel to the (001) plane. The layers are arranged in a sandwich Pb—0—Ph structure. Within the sandwich, each Pb ion is located at the top of a Pb04 pyramid and all the Ph—O distances are identical and equal to 2.32 A. To a first approximation, the Pb ions can be considered as cubic packed. Oxygen ions use sp hybrid orbitals to form electron pairs with the lead atoms [2]. The bonds can be considered as approximately covalent. This localised electron distrihution in PbO determines the latter s high electric resistance of 10 ohm cm [3],... 

The GHO basis can therefore provide a localised, directional set of orbitals (hybrids) which do not have the principal qualitative disadvantage of the usual hybrid sets they can be mutually orientated in any directions. What is more the directions taken up by the GHOs can be decided variationally and not by the unitary properties of a hybridisation matrix . This conclusion means that the use of a GHO basis provides both a localised bonding picture and simultaneously a theoretical validation of the VSEPR rules. Thus, it is not necessary, for example, to contrast the hybrid method and the VSEPR method for molecular geometries (30) they are complementary. 

Atomic orbitals of this mixed type are usually referred to as hybrids (or more specifically digonal dimensional model, they reinforce on one side of the nucleus and partly cancel on the other. Hence the transformation is from the delocalised s and p description to a description in terms of two equivalent orbitals, localised cn opposite sides of the nucleus. Again a similar transformation may be applied to a 4 X 4 determinant describing a system with two electrons in each of these orbitals. 

Hybrid functionals have also proved useful in localised orbital approaches. The B3LYP functional used in CRYSTAL06 can be written as... 

Hybrid functionals other than B3LYP have also proved successful in localised orbital methods. Yang and Dolg found that the hybrid functional B3PW including spin-orbit coupling reproduced the band gap of BiB30g well, while Prodan et u/." found that the HSE (Heyd, Scuseria and Emzerhof) screened coulomb hybrid potential described the oxides of uranium and plutonium well. The performance of hybrid functionals is discussed by Cora et Just as there is no universal... 

Boron Trichloride, BCI3. Here the molecule is planar, with the boron atom at the centre of an equilateral triangle of chlorine atoms (Fig. 46). The valence state must be described in terms of three similar hybrid AO s pointing towards the comers of the triangle. Such orbitals can be formed by mixing 2s and two 2p AO s, 2p and 2p say they lie in the plane of the latter and are precisely equivalent (Fig. 47). If the so-called trigonal hybrids are denoted by h, hg and hg, the appropriate boron valence state must be B(ls2 h h2 hg ). The hybrid AO s overlap chlorine 3p AO s, directed towards the boron atom, to form localised MO s similar to those in beryllium chloride. 

Although a double hydrogen bridge is certain, the precise nature of the bonds involved is still uncertain. Clearly they are abnormal and a recent suggestion is that they are formed by an overlap of the sp tetrahedral hybrids and hydrogen Is orbitals. Each electron pair is then less localised than usual, extending over three centres (Fig. 122). This accounts for the... 

Canonical form A hypothetical representation of a molecule, ion or radical, in which the electrons are portrayed as being present in discrete localised orbitals, i.e. a Lewis structure. The real molecule, ion or radical actually exists as the weighted average of all the possible canonical forms, which is called the resonance hybrid. 

Lewis structure A representation of a molecule, ion, or radical, that shows the electrons as being contained in discrete localised orbits. If there are two or more such structures for any given compound, they may also be called canonical forms, which when combined give the resonance hybrid. 

The existence of an anomaly in the spectrum of Ba+ had been known for many years. It was commented on by Saunders et al. [213], who remarked the perturbation. .. appears to be of a novel type but could advance no explanation for it. The realisation that this is due to a near-critical potential barrier effect came much later [212]. Hartree-Fock calculations were used to establish that the n/ wavefunctions of Ba+ become resonantly localised in the inner well of a double-well potential around n = 5. As a consequence, all the low n orbitals become distinctly bimodal or hybrid in character (as can be seen in fig. 5.12) with peaks in both the inner and the outer potential wells. This results in (i) an unusual course in the quantum defects for the nf series they are not constant, but depend strongly on the energy, despite the absence of any local perturbation (ii) a pronounced intensity anomaly in the 5d — nf excitation series, which was first noted by Roig and Tondello [214] and (iii) an anomaly in the course of the spin-orbit splittings for the first members of the series, both of which can be explained on the basis of the bimodal character of the... 

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