Group theory projection operators

The projection operator is one of the most useful concepts in the application of group theory to chemical problems [25, 26], It is an operator which takes the non-symmetry-adapted basis of a representation and projects it along new directions in such a way that it belongs to a specific irreducible representation of the group. The projection operator is represented by P in the following form ... 

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. 

Per-Olov Lowdin established both the Quantum Chemistry Group at Uppsala and the Quantum Theory Project at Gainesville through his personal and scientific stature and he was unquestionably the Director of both operations. Several prominent scientists were invited during the first years to be Associate Directors and to maintain the spirit when "the Force" was absent. Yngve Ohrn was invited to join the faculty at Florida as Associate Professor in 1966 and he chose to become a member of the Quantum Theory Project. He assumed the Associate Directorship of the Quantum Theory Project in 1967 and proved quickly to have the qualities that allowed him to deal effectively with the university administration at all levels as well as with issues within the rather diverse and extensive body of scientists and students of the Project. Fairness, compassion, and persistence are words that come to my mind when I characterize Yngve s leadership. 

The outline of the review is as follows in the next section (Sect. 2) we introduce the basic ideas of effective Hamiltonian theory based on the use of projection operators. The effective Hamiltonian (1-5) for the ligand field problem is constructed in several steps first by analogy with r-electron theory we use the group product function method of Lykos and Parr to define a set of n-electron wavefimctions which define a subspace of the full -particle Hilbert space in which we can give a detailed analysis of the Schrodinger equation for the full molecular Hamiltonian H (Sect. 3 and 4). This subspace consists of fully antisymmetrized product wavefimctions composed of a fixed ground state wavefunction, for the electrons in the molecule other than the electrons which are placed in states, constructed out of pure d-orbitals on the... 

Within a group theoretical analysis, this is normally accomplished with projection operators. For a discussion of this method, see the group theory texts listed in Footnote I of Chapter 3. 

In the SRS perturbation theory the symmetry forcing operator appears only in the energy expression. Thus, in this formalism VT = 1 and VQ = VA, where "A is the projection operator on the appropriate representation of the symmetric group 4. Moreover, the interpolation function "e(C) appearing in Eq. (6) is not calculated from Eq. (5) but is taken directly from the RS theory (10, 21). The SRS perturbation corrections to the interaction energy, "J srsi are given by,... 

The projection operator for a given irreducible representation, F, from standard group theory is given by ... 

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