Orbitals localised

A wave function for the entire crystal based on localised orbitals... 

The most widely used qualitative model for the explanation of the shapes of molecules is the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie and Nyholm (25). The orbital correlation diagrams of Walsh (26) are also used for simple systems for which the qualitative form of the MOs may be deduced from symmetry considerations. Attempts have been made to prove that these two approaches are equivalent (27). But this is impossible since Walsh s Rules refer explicitly to (and only have meaning within) the MO model while the VSEPR method does not refer to (is not confined by) any explicitly-stated model of molecular electronic structure. Thus, any proof that the two approaches are equivalent can only prove, at best, that the two are equivalent at the MO level i.e. that Walsh s Rules are contained in the VSEPR model. Of course, the transformation to localised orbitals of an MO determinant provides a convenient picture of VSEPR rules but the VSEPR method itself depends not on the independent-particle model but on the possibility of separating the total electronic structure of a molecule into more or less autonomous electron pairs which interact as separate entities (28). The localised MO description is merely the simplest such separation the general case is our Eq. (6)... 

Schrodinger equation. Or alternatively, we may say, equally accurately, that the two electrons occupy two localised orbitals and %b, one at each end of the segment. These are just two different ways of interpreting the same total wave function. 

The two descriptions are useful in rather different contexts. If we are interested in the relative positions of the two electrons, then the interpretation in terms of localised orbitals gives a clearer description of the qualitative features of the overall probability distribution. On the other hand, if we are interested in the removal of an electron, the first description is more appropriate, for the remaining electron must occupy an orbital which is a solution of the original Schrodinger equation. Thus the electron must be removed from y)t or y>2. 

These both have the same energy so that, in the absence of other determining factors, the electrons go one into each with the same spin (or an equivalent state). This means that they are kept apart by the antisymmetry principle and so the energy Is lowered by the reduction of Coulomb repulsion. In this rather exceptional case, therefore, the orbitals are not all doubly occupied and we cannot carry out any simple transformation into localised orbitals. 

Heat capacity measurements on superconducting cuprates have been widely undertaken and recently Loram etal [25] reported condensation energies for a cuprate superconductor as a function of doping. In our theory the heat capacity per localised orbital (Cv IN0 is given by... 

In order to see if it is possible to neutralise this effect of the a-system we performed a second calculation which used localised orbitals for the a-system as well as for the Tt-system. In this calculation one perfect-pairing structure was used for the C-C bonds of the a-system. All orbitals were localised on the C-H fragments. Doubly occupied orbitals were used for the C-H bonds, and strictly localised singly occupied orbitals for the C-C bonds. This calculation again yields a rectangular geometry with a much lower resonance energy. The bond... 

Fig. 23 An illustration of the use of the NBO (or any localised orbital) procedure for analysing TB and TS interactions, using as an example, butane-1,4-diyl. The model includes two chromophore tt NBOs, tti and tt4, and the tr2 and 03 NBOs of the central C—C bridge bond. Firstly, the full Fock matrix, FN, in the basis of the NBOs is constructed, and the off-diagonal matrix elements are then deleted, to form a blank Fock matrix (top part of the figure). In the bottom part of the hgure, the Fock matrix is built up, starting with the blank matrix, and adding, in succession, the TS interaction between 7T and 7r4, the TB interaction with

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... 

Building on the use of Wannier functions to calculate properties, a very recent paper uses both electron Wannier functions and phonon Wannier functions (that is localised orbitals and vibrations) to calculate electron-phonon interaction, a property thought to be important in high Tc superconductors. 

While all classical VB-style approaches, even those using variational HAOs, can be formulated only in terms of localised atom-centred basis sets, MO theory can use basis sets of any type, including plane wave basis sets. There are no restrictions on the type of basis that can be used in modern VB (GVB or SC) calculations, and it would be interesting to see what is the size of the plane wave basis that would be required in order to reproduce the usually well-localised orbitals observed when working within atom-centred basis sets. 

The usually well-localised nature of the orbitals appearing in VB wavefunction makes spatial symmetry more difficult to use than in the MO case. In MO theory, symmetry can be introduced and utilised at the orbital level Each delocalised MO can be constructed as a symmetry-adapted linear combination (SALQ of basis functions, which is straightforward to implement in program code and can be exploited to achieve substantial computational savings. As a rule, the individual localised orbitals from VB wavefunctions are not S5mimetry-adapted, but transform into one another under the symmetry operations of the molecular point group. The use of symmetry of this type normally requires prior knowledge of the orbital shapes and positions and is very difficult to handle without human intervention. 

In order to produce qualitative information about bonding patterns and lone pairs, MO theory has to resort to orbital localisation procedures or more complicated interpretation techniques such as, for example, Bader s Atoms in Molecules (AIM) approach.However, orbital localisation procedures can be applied only to wavefunctions which remain invariant with respect to non-singular linear transformations of the orbitals, i.e. single-configuration and complete-active-space (CAS) MO wavefunctions and even for these, the use of orthogonal localised orbitals is not sufficient to reproduce all of the chemically-important information included in modern VB wavefunctions. Evidence of the superiority of the interpretational facilities offered by VB wavefunctions is provided by the continuing efforts to transform CAS wavefunctions to VB-style... 

However, as shown in ref. 92, if the orbitals in benzene are ordered as in eqn (4.2) and the active-space spin-coupling pattern is expressed in the Serber spin basis, it becomes very similar to that at the TS for the [1,5]-H shift in (Z)-l,3-pentadiene. An additional symmetry-constrained SC calculation on benzene produced an antipair solution which is just about 1 mhartree above the well-known unconstrained solution with localised orbitals. The orbitals from this antipair solution are shown in Fig. 9 (the orbitals from the standard solution are visually indistinguishable from i/ i). 

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