Equivalent orbital transformation

The procedures outlined above for atoms also apply to small molecules, e.g. LiH, Lig, Ng, HF, Fj, etc. For these, the theory starts with H.F. SCF MO s the l /s refer to MO pairs. The inner shells, however, can be taken directly from free atoms after an equivalent orbital transformation has been performed on say the (l(r,) (lmedium effects on the innermost shells... 

The same equivalent orbital transformation which applied to works on the correlation part, Eqs. (77) and (153), of the wave function and energy as well and gives the generalized London-van der Waals terms at all r. These are the pairs in Eq. (152b) that have two localized orbitals and rj with spatial parts on different molecules. Denoting the sum of all such intermolecular pairs by E, we have... 

This wave function has many features in common with that of the previous model. While rp2s is spherically symmetric, yj2J> has a nodal plane through the centre of symmetry. A similar transformation can be applied and we can use two equivalent orbitals... 

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. 

In this case we can transform these into three equivalent orbitals which are s-p hybrids (called trigonal hybrids) pointing towards the vertices of an equilateral triangle, so that the angle between neighbouring directions is 120°. The actual transformation is... 

It is not immediately clear from the form in which these are written that they are equivalent functions, that is, differ only in their orientation, but it is easily confirmed that they do transform into each other if the axes are rotated through 120°. Again it can be shown that the determinant of -functions has the same value as (18). One other point about this set of equivalent orbitals is that there appears to be no preferential direction in which any one of the vertices of this triangle may be chosen. The choice is, in fact, arbitrary and any set of three equivalent directions perpendicular to the z direction would suffice. This only applies for an atomic wave function, of course. In molecules (such as planar XY3) there may be a preferred choice of axes on account of symmetry. This will be clear from some examples considered in the next section. 

The case of four electrons in the atomic orbitals 2s, 2px, 2py, and 2pz can be handled in a similar manner. Here we can transform the expression into four equivalent orbitals given by... 

This function is zero for all points equidistant front the two nuclei (that is, it has a nodal plane). The orbital y>j is large in the region between the nuclei (where lsA and 1% overlap and the electrostatic potential is low) and is generally referred to as a bonding orbital. Similarly, y>t, which keeps its electron away from the internuclear region, is antibonding. The two functions and yj2 are analogous to the symmetric and antisymmetric orbitals for the one-dimensional model. A similar transformation can be applied and two equivalent orbitals constructed. These are... 

The localised equivalent orbitals are connected with these by the transformations... 

The final molecule of this series is methane, the tetrahedral structure of which follows if a fourth unit positive charge is removed from the nucleus in the ammonia lone-pair direction. There are now four equivalent bonding orbitals, which may be represented approximately as linear combinations of carbon s-p hybrid and hydrogen Is functions. The transformation from molecular orbitals into equivalent orbitals or vice versa is exactly the same as for the neon atom. 

The equivalent orbitals corresponding to these are obtained by applying the reverse transformation and are... 

The main result that emerges from the discussions of particular eases is that it has proved possible to give a description of a molecule in terms of equivalent orbitals which are approximately localised, but which can be-transformed into delocalised molecular orbitals without any change in the value of the total wave function. The equivalent orbitals are closely associated with the interpretation of a chemical bond in the theory, for, in a saturated molecule, the equivalent orbitals are mainly localised about two atoms, or correspond to lone-pair electrons. Double and triple bonds in molecules such as ethylene and acetylene are represented as bent single bonds, although the rather less localised o-n description is equally valid. 

Another property of these equivalent orbitals is that they include, in themselves effects of delocalisation. Such effects are most important in conjugated molecules, although they arc present in all molecules to a greater or lesser extent. In a highly conjugated system such as benzene only limited amount of localisation can be achieved by transforming the orbitals. 

For large molecules, the equivalent-orbital analysis is the most convenient starting point for a molecular-orbital treatment. In a molecule such as a long-chain paraffin it is possible to write approximate equivalent orbitals corresponding to each bond and then to apply a transformation to obtain the delocalised molecular orbitals. Simple assumption about, the interaction of neighbouring bonds will then lead to estimates of the relative stability of the various energy levels.9... 

A semi-empirical calculation method for ionization potentials has been developed, using the fact that a Slater determinant is defined only up to a unitary transformation (see Sect. 4.4) the canonical molecular orbitals Hartree-Fock operator F for a closed-shell system, can be replaced by equivalent orbitals eo, almost completely localized. 

The symmetry projection of the wavefunction is equivalent to a particular orbital transformation among the occupied orbitals of the wavefunction. If the CSF expansion is full within these sets of symmetry-related orbitals, no new CSFs will be generated by this orbital transformation. This type of wavefunction could have been computed directly in terms of symmetry orbitals with no loss of generality. (In fact, the CSF expansion expressed in terms of symmetry orbitals will usually result in fewer expansion terms because the symmetry blocking of the individual CSFs allows those of the incorrect symmetries to be deleted from the expansion.) However, if the CSF expansion is not full within these orbital sets, it is possible that the symmetry transformation of the orbitals will generate new CSF expansion terms. The coefficients of these new CSF expansion terms are determined by the old expansion coefficients and the symmetry transformation coefficients. For example, consider the case of two H2 molecules, described in terms of localized orbitals, separated by a reflection plane. Assume that the localized description of the two H2 molecules is of the form... 

Eq. (221) shows that the wavefunction change induced by the orbital variations for a fixed set of expansion coefficients is equivalent to a transformation of the CSF expansion coefficient for a fixed set of orbitals. Eq. (222) shows that variations of the coefficients C and orbital transformation coefficients T do not need to be considered separately. Only the combined effect, expressed as variations of the coefficient matrix C, is required to allow an arbitrary two-electron wavefunction change. This occurs in this case because the wavefunction is expanded in the full Cl set of CSFs. This demonstrates that a redundancy exists between the orbital coefficient variations and the CSF expansion coefficient variations and that this redundancy may be eliminated by considering only the CSF coefficient variations for some fixed set of orbitals. Other solutions to this redundancy will now be considered. 

Quenching of the Orbital Angular Momentum As electrons will have orbital momentum around the given axis (say az-axis), if the orbital that it occupies can be transformed into an entirely equivalent orbital (which should be degenerate also with it) by a simple rotation around the axis. The electron in the d2 -... 

The method of Edmiston and Ruedenberg is based on the generalization of equivalent orbitals [3-5]. It is known that a one-determinant wavefonction for closed-shell system is invariant (to a phase factor of unity) to any unitary transformation of the tp, canonical molecular orbitals (CMOs). As both the Coulomb and the exchange interaction energy terms contain the same... 

Coulson and Lennard-Jones have shown that relative charge distributions are better represented by a unitary transformation of the H.F. MO determinant which leaves it unchanged but expresses it in terms of localized or "equivalent orbitals. In a symmetric case, like CH4, this transformation is unique,f but where all the bonds are not equivalent, as in CgH, a criterion is required. The transformation may be defined, for instance, by requiring the magnitude of the exchange terms to be a minimum. This would give the most localized and the most nearly classical bonds. ... 

This problem too has an orbital and a correlation part. The orbital part was solved by Lennard-Jones, > who showed that in an H.F. SCF MO determinant, , electrons are already localized with respect to one another (e.g. in CH4, HgO, Ne,.. . ) as mentioned in Section VII. This relative distribution of electrons with respect to one another is better described by a unitary "equivalent or "localized orbital transformation t which leaves (f>o unchanged ... 

In dealing with intermolecular forces the many-electron theory again starts with H.F, SCF MO s, this time on the composite system of interacting molecules. In He He, H.F. accounts for the gradual distortion of atomic orbitals Is, Is, IsJ, Isf into the equivalent orbitals ( i,j 2,%, 24) at shorter r. Ransil s H.F, could be transformed at various r to get these jj s. Nesbet has treated the Ng molecule all the way to dissociation by the Hartree-Fock method directly in terms of equivalent orbitals. ... 

We need in general to find then how the complete set of symmetry equivalent orbitals x/ transforms under the symmetry operations of the group. To do this we need the characters Z)p. These may be generated by operating on each of the orbitals of the set x/j with R. If Rx/ = djXi then c// may equal +1 (if R sends X/ to itself), "1 (if R sends Xi to minus itself), and 0 (if R sends X/ to some other member of the set). Dr is then given by... 

A second example, H2 O, is depicted in Figure 3. Ib-e. But, before we can proceed with the discussion, we describe another useful orbital transformation localization of symmetry orbitals. Figure 3.1b shows the two bonding molecular orbitals (MOs) of H2O taken from a Hartree-Fock (HF) calculation. The 3aj orbital has even symmetry (++), while the Ibj orbital has odd symmetry (-F-). If we take the two linear combinations yT/2(3aj) -y TT flbj) of these orbitals, we see that two equivalent orbitals are produced (shown on the right side of the row). These are designated as and a j. because they are bond orbitals localized between O and the left and right H atoms, respectively. It is evident by inspection that each of these localized MOs closely resembles the a bond MO of OH shown in Figure 3.1a. 

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