Symmetry adapted

Kutzelnigg W 1992 Does the polarization approximation converge for large-rto a primitive or a symmetry-adapted wavefunction Chem. Phys. Lett. 195 77... 

Bukowski R, Sadie] J, Jeziorski B, Jankowski P, Szalewicz K, Kucharski S A, Williams H L and Rice B M 1999 Intermolecular potential of carbon dioxide dimer from symmetry-adapted perturbation theory J. Chem. Phys. 110 3785... 

NakatsujI H and Nakal H 1990 Theoretical study on molecular and dissociative chemisorptions of an O2 molecule on an Ag surface dipped adcluster model combined with symmetry-adapted cluster-configuration interaction method Chem. Phys. Lett. 174 283-6... 

Miller W H 1983 Symmetry-adapted transition-state theory and a unified treatment of multiple transition states J. Phys. Chem. 87 21... 

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... 

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. 

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. 

Define the symmetry adapted "eore" and "valenee" orbitals of the following systems ... 

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 ... 

To identify the states which arise from a given atomic configuration and to construct properly symmetry-adapted determinental wave functions corresponding to these symmetries, one must employ L and Ml and S and Ms angular momentum tools. One first identifies those determinants with maximum Ms (this then defines the maximum S value that occurs) within that set of determinants, one then identifies the determinant(s) with maximum Ml (this identifies the highest L value). This determinant has S and L equal to its Ms and Ml values (this can be verified, for example for L, by acting on this determinant with f2 in the form... 

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 ... 

Remembering that the projection process does not lead to a normalized function, although it does generate a function of pure symmetry, one can finally write down the normalized symmetry-adapted singlet functions as ... 

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... 

Within a variational treatment, the relative eontributions of the spin-and spaee-symmetry adapted CSFs are determined by solving a seeular problem for the eigenvalues (Ei) and eigenveetors (C ) of the matrix representation H of the full many-eleetron Hamiltonian H within this CSF basis ... 

The spin- and spatial- symmetry adapted N-eleetron funetions referred to as CSFs ean be formed from one or more Slater determinants. For example, to deseribe the singlet CSF eorresponding to the elosed-shell orbital oeeupaney, a single Slater determinant... 

It is essential to realize that the energies (i H Oi> of the CSFs do n represent the energies of the true electronic states Ek the CSFs are simply spin- and spatial-symmetry adapted antisymmetric functions that form a basis in terms of which to expand the true electronic states. For R-values at which the CSF energies are separated widely, the true Ek are rather well approximated by individual (i H Oi> values such is the case near Rg. 

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 ... 

The hi, b 1 and a2 bloeks are formed in a similar manner. The eigenvalues of eaeh of these bloeks provide the squares of the harmonie vibrational frequeneies, the eigenveetors provide the normal mode displaeements as linear eombinations of the symmetry adapted... 

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

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. 

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

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... 

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... 

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