Symmetry-adapted operators

IV. Projeetor Operators Symmetry Adapted Einear Combinations of Atomie Orbitals... 

IV. Projector Operators Symmetry Adapted Linear Combinations of Atomic Orbitals... 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

One more quantum number, that relating to the inversion (i) symmetry operator ean be used in atomie eases beeause the total potential energy V is unehanged when all of the eleetrons have their position veetors subjeeted to inversion (i r = -r). This quantum number is straightforward to determine. Beeause eaeh L, S, Ml, Ms, H state diseussed above eonsist of a few (or, in the ease of eonfiguration interaetion several) symmetry adapted eombinations of Slater determinant funetions, the effeet of the inversion operator on sueh a wavefunetion P ean be determined by ... 

One Must be Able to Evaluate the Matrix Elements Among Properly Symmetry Adapted N-Electron Configuration Eunctions for Any Operator, the Electronic Hamiltonian in Particular. The Slater-Condon Rules Provide this Capability... 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

To illustrate, again consider the H2O molecule in the coordinate system described above. The 3N = 9 mass weighted Cartesian displacement coordinates (Xl, Yl, Zl, Xq, Yq, Zq, Xr, Yr, Zr) can be symmetry adapted by applying the following four projection operators ... 

Symmetry tools are used to eombine these M objeets into M new objeets eaeh of whieh belongs to a speeifie symmetry of the point group. Beeause the hamiltonian (eleetronie in the m.o. ease and vibration/rotation in the latter ease) eommutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "bloek diagonal". That is, objeets of different symmetry will not interaet only interaetions among those of the same symmetry need be eonsidered. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

The importance of the characters of the symmetry operations lies in the fact that they do not depend on the specific basis used to form them. That is, they are invariant to a unitary or orthorgonal transformation of the objects used to define the matrices. As a result, they contain information about the symmetry operation itself and about the space spanned by the set of objects. The significance of this observation for our symmetry adaptation process will become clear later. 

To generate the above Ai and E symmetry-adapted orbitals, we make use of so-ealled symmetry projeetion operators Pe and Pa These operators are given in terms of... 

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. 

We now return to the symmetry analysis of orbital produets. Sueh knowledge is important beeause one is routinely faeed with eonstrueting symmetry-adapted N-eleetron eonfigurations that eonsist of produets of N individual orbitals. A point-group symmetry operator S, when aeting on sueh a produet of orbitals, gives the produet of S aeting on eaeh of the individual orbitals... 

The subset 0kv 0k2> 0k3,. . . formed from the complete set by means of the projection operator 0k is called /l-adapted or symmetry-adapted in the case when A is a symmetry operator. From Eqs. III.81 and III.86 it follows that the projection operators 0k commute with H and, using this property, the quantum-mechanical turn-over rule/ and Eq. III.91, we obtain... 

How do CMOs and LMOs differ The CMOs are symmetry-adapted eigenfunctions of the Fock (or Kohn-Sham) operator F, necessarily reflecting all the molecular point-group symmetries of F itself,26 whereas the LMOs often lack... 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

Because of round-off errors, symmetry contamination is often present even when the initial vector is properly symmetrized. To circumvent this problem, an effective scheme to reinforce the symmetry at every Lanczos recursion step has been proposed independently by Chen and Guo100 and by Wang and Carrington.195 Specifically, the Lanczos recursion is executed with symmetry-adapted vectors, but the matrix-vector multiplication is performed at every Lanczos step with the unsymmetrized vector. In other words, the symmetrized vectors are combined just before the operation Hq, and the resultant vector is symmetrized using the projection operators ... 

Finally, it must be mentioned that localized orbitals are not always simply related to symmetry. There are cases where the localized orbitals form neither a set of symmetry adapted orbitals, belonging to irreducible representations, nor a set of equivalent orbitals, permuting under symmetry operations, but a set of orbitals with little or no apparent relationship to the molecular symmetry group. This can occur, for example, when the symmetry is such that sev-... 

The operators S , S(II), S(in),... are the symmetry-adapted operators (Iachello and Oss, 1991). The construction of the symmetry-adapted operators of any molecule will become clear in the following sections where the cases of benzene (D6h) and of octahedral molecules (Oh) will be discussed. 

The construction of the symmetry-adapted operators and of the Hamiltonian operator of polyatomic molecules will be illustrated using the example of the benzene molecule. In order to do the construction, draw a figure corresponding to the geometric structure of the molecule (Figure 6.1). Number the degrees of freedom we wish to describe. 

The symmetry-adapted operators of benzene with symmetry D6h are those corresponding to these three couplings, that is,... 

The same method of construction of symmetry-adapted operators can be used for any isotopic substitution in which a H atom is replaced by D, for example, the molecule 2,4,6-C6H3D3 of Figure 6.4. This molecule has Dih symmetry (i.e., the symmetry is lowered). The symmetry-adapted operators can be obtained by inspection of Figure 6.4. The nearest-neighbor interactions are all identical (and are of the H-D type). The same is true for the third neighbor interactions. Thus the symmetry-adapted operators S(i) and S(III) are the same as... 

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