Orbitals transformation, equivalent

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 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. 

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... 

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 ... 

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... 

If we want the transformation described by exp(iA) to preserve orthonormality of the spin-orbitals or, equivalently, to preserve the anticommutation relations [see discussion following Eq. (1.5)]... 

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. 

Evidence for the structure of 7r-allylpalladium dichloride (VI) shows that the terminal carbons are equivalent and that the three carbons are in a plane facing the metal. The simplest description of the bonding between the TT-allyl group and the palladium atom is one which is somewhat analogous to the description given for the 7r-ethylene-platinum system (5). In complex (VI) the TT-allyl group has the symmetry and the carbon 2p orbitals transform under C2 symmetry as + irreducible representations. 

A special situation arises when the two orbitals are equivalent in the sense that they have the same energy and can be transformed into each other by a symmetry operation such as rotation. The triplet state then can be lower in energy than a singlet state in which both electrons reside in one of the orbitals. This is the case for O2, for which the ground state is a triplet and the lowest excited state is a singlet. 

A complication arises for functions of d or higher symmetry. There are five real d orbitals, which transform as xy, xz, yz, x —y, and z, that are called pure d functions. The orbital commonly referred to as is actually Iz —x —y. An alternative scheme for the sake of fast integral evaluation is to use the six Cartesian orbitals, which are xy, xz, yz, x, y, and These six orbitals are equivalent to the five pure d fimctions plus one additional spherically symmetric function x +y +z ). Calculations using the six d functions often yield a very slightly lower energy due to this additional function. Some ab initio programs give options to control which method is used, such as Sd, 6d, pure-J, or Cartesian. Pure- f is equivalent to 5d and Cartesian is equivalent to 6d. Similarly, If and 10/ are equivalent to pure-/ and Cartesian / functions, respectively. 

With regard to the different points of view outlined in (a), (b) and (c), it should be pointed out that these differences arise mainly from the use of localized (a, LMO), or canonical (CMO, b, and c) molecular orbitals. In principle LMOs and CMOs are equivalent and are related by a unitary transformation. This can be illustrated by the C=C bonding in acetylene. 

The MO and VB methods provide altema ive but equivalent descriptions of the bonding in a molecule. A set of molecular orbitals can always be transformed into a corresponding set of more localized orbitals, and vice versa. For example, according to the MO de-... 

Step 2. The set of CMOs orthogonal molecular orbitals (LMOs) Xj using, e.g., Ruedenberg s localization criterion205. This is achieved by multiplying up with an appropriate unitary transformation matrix L ... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

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)... 

The third example in Scheme 3.64 represents the cation-radical of 1,3,6,8-tetraazatricyclo [4.4.1.F ]dodecane. Zwier et al. (2002) prodnced evidence of instantaneous electron delocalization over the four equivalent nitrogen atoms. This extensive delocalization in a completely saturated system is a principal featnre of the third example and reveals the consequences of orbital interactions throngh space and bonds. The space—bond delocalization can serve as a driving force for the cation-radical rearrangements as it has recently been exemplified by transformation of the phenyl-honsane cation-radical into a mixtnre of phenylbicyclononenes (Gerken et al. 2005). 

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