Symmetry-adapted pair

Just as for the pentagon, any orthogonal pair of linear combination of the degenerate pair of 2, coordinates is also a symmetry-adapted pair of deformation... 

This is a nine electron system, which autoionizes into the following five available open channels Is - 2p D, Is — 2p S, Is — 2s2p P°, Is — 2s2p P° and Is - 2s S. As written symbolically, each of these electron rearrangements is represented by the corresponding asymptotic" symmetry-adapted pair function, nasirup )-... 

In this chapter we employ a special version of the second-order Moller-Plesset [13] perturbation theory (MP2/CA) which is especially useful in applications to closed-shell atoms. In this approach the first-order wave function consists of spinor-bitals and symmetry-adapted pair functions (SAFE). A detailed presentation of the MP2/CA method can be found in Ref. [14]. For sake of describing the nomenclature used, we repeat here the basic equations. 

It is convenient to transform the set of pair functions Ujj corresponding to spinorbitals defined by the same pair of shell quantum numbers (nili,rijlj), and different magnetic quantum numbers, into a set of symmetry-adapted pair functions (SAPF), m(T 1,2), which satisfy the equations [14]... 

Within the MP2/CA approach the correlation energy of an N-electron atom can be decomposed in a way which is very advantageous from the points of view of physical intuitions as well as computational methodology. At the first important step, the total second-order energy is decomposed into symmetry-adapted pair (SAP) energies E >(T ) defined by the individual SAPFs u(T 1,2)... 

Milet A, Moszynski R, Wormer P E S and van der Avoird A 1999 Hydrogen bonding in water olusters pair and many-body interaotions from symmetry-adapted perturbation theory J. Phys. Chem. A 103 6811-19... 

For example, the three NH bonding and three NH antibonding orbitals in NH3, when symmetry adapted within the C3V point group, cluster into ai and e mos as shown in the Figure below. The N-atom localized non-bonding lone pair orbital and the N-atom Is core orbital also belong to ai symmetry. 

An example will help illustrate these ideas. Consider the formaldehyde molecule H2CO in C2v symmetry. The configuration which dominates the ground-state waveflinction has doubly occupied O and C 1 s orbitals, two CH bonds, a CO a bond, a CO n bond, and two 0-centered lone pairs this configuration is described in terms of symmetry adapted orbitals as follows (Iai22ai23ai2lb2 ... 

These equations state that the three Ish orbitals can be combined to give one Ai orbital and, since E is degenerate, one pair of E orbitals, as established above. With knowledge of the ni, the symmetry-adapted orbitals can be formed by allowing the projectors... 

From the information on the right side of the C3v eharaeter table, translations of all four atoms in the z, x and y direetions transform as Ai(z) and E(x,y), respeetively, whereas rotations about the z(Rz), x(Rx), and y(Ry) axes transform as A2 and E. Henee, of the twelve motions, three translations have A and E symmetry and three rotations have A2 and E symmetry. This leaves six vibrations, of whieh two have A symmetry, none have A2 symmetry, and two (pairs) have E symmetry. We eould obtain symmetry-adapted vibrational and rotational bases by allowing symmetry projeetion operators of the irredueible representation symmetries to operate on various elementary eartesian (x,y,z) atomie displaeement veetors. Both Cotton and Wilson, Deeius and Cross show in detail how this is aeeomplished. 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

An orbital correlation diagram can be constructed by examining the symmetry of the reactant and product orbitals with respect to this plane. The orbitals are classified by symmetry with respect to this plane in Fig. 11.9. For the reactants ethylene and butadiene, the classifications are the same as for the consideration of electrocyclic reactions on p. 610. An additional feature must be taken into account in the case of cyclohexene. The cyclohexene orbitals tr, t72. < i> and are called symmetry-adapted orbitals. We might be inclined to think of the a and a orbitals as localized between specific pairs of carbon... 

Form symmetry-adapted combinations (SAC) of POs. This is accomplished by combining couples into in-phase and out-of-phase pairs across symmetry elements. Figure 5 shows examples. 

A T structure with the strongest ct-donor D trans to the empty site (I in Scheme 1) is preferred in the case of three pure cr-donor ligands. The presence of a ir-acceptor ligand also makes the T structure more stable. When one of the ligands is a tt-donor, X, a Y structure of type II (Scheme 1) is observed. This structure permits the formation of a w bond between the empty metal d orbital and the lone pair of X. No such tt bond is present in the T structure since all symmetry adapted d orbitals are filled. This partial M—X multiple bond stabilizes Y over T. 

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