Projection operator method

Lu D-H, Zhao M and Truhlar D G 1991 Projection operator method for geometry optimization with... 

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. 

Symmetry properties which have so far been successfully treated by the projection operator method, include translational symmetry in crystals, cyclic systems, spin, orbital and total angular momenta, and further applications are in progress. ... 

This equation does not provide a closed expression for hk and to close it we use projection operator methods. We introduce a projection operator V defined by... 

The hydrodynamic equations can be derived from the MPC Markov chain dynamics using projection operator methods analogous to those used to obtain... 

Apply the projection-operator method to obtain the molecular orbital expression shown in Fig. 9 and verify the energies. 

The projection-operator technique will be employed in several examples presented in the following chapter and Chapter 12. For. the quantitative interpretation of molecular spectra both electronic and vibrational, molecular symmetry plays an all-important role. The correct linear combinations of electronic wavefunctions, as well as vibrational coordinates, are formed with the aid of the projection-operator method. 

From the above analysis, as shown in the last column of Table 2, it should be obvious that two linear combinations of the internal coordinates Ar can be formed whose symmetry corresponds to the IRs k and B2. They are Arj + Ar2(A]) and Ar — Ar2(B2), as can be verified by inspection of the character table. Tbe choice of the correct linear combinations is in this case simple. However, more generally, it can be made by application of the projection-operator method described in Section 8.11. 

We shall use the projection operator method to derive the Pauli master equation. With the Liouville equation, we separate the Liouville operator into two parts ... 

Use the projection operator method to determine the SALCs required for the formation of M—X a bonds by the eight ligands in an Mo2Xs ion of symmetry D4/i. 

Because chemists seem to have become increasingly interested in employing vibration spectra quantitatively—or at least semiquantitatively—to obtain information on bond strengths, it seemed mandatory to augment the previous treatment of molecular vibrations with a description of the efficient F and G matrix method for conducting vibrational analyses. The fact that the convenient projection operator method for setting up symmetry coordinates has also been introduced makes inclusion of this material particularly feasible and desirable. 

The normal mode displacements are sketched in Figure 9.4. The notation u, v for the degenerate pair of H symmetry and , //, ( for the T2 triplet is standard. Actually, these projections had already been done in Section 6.4, but this example has been worked in full here to illustrate the projection operator method of finding normal modes. 

In applying the projection operator method for the calculation of symmetrized linear combinations of eigenfunctions, we shall need the effect of a space-group function operator (R ) on the FE eigenfunction mk(r), which is... 

Explicit orthogonality constraints can be removed by transformingthe hamiltonian so that it only acts on a specific subspace (e.g. the valence space) of the all-electron space. This can be done formally by the use of the projection operator method (see Huzinaga and Cantu18 or Kahn, Baybutt, and Truhlar20). If the core function is written as in equation (6) a projection operator may be defined for each valence electron p ... 

Beyond the systems and applications described in this chapter, projection-operator methods can, for example, be used to study the dynamics near glass transitions [77] and the propagation of wave functions in systems with non-resonant transitions. The latter application has recently been analyzed in connection with the decomposition of the spectral density [78] showing the wide range of applicability of the proposed schemes. 

To describe the dynamics of an isolated system (for example, how the isolated system relaxes to equilibrium), one often follows the dynamics of the population pm(t) of the system. For this purpose, the projection operator method [1,5-7] is employed. A main function of the projection operator D is to divide p t) into two parts... 

The projection operator method splits the density matrix into diagonal part and off-diagonal part. Discuss why this method is powerful for treating non-equilibrium statistical mechanics ... 

Because the postulated diabatic state is never computed explicitly, the NADP formalism completely avoids the conceptual difficulties associated with this state in the LCP and projection operator methods. Reduction of the vibrational computation to solution of Eq. (9.3) removes the difficult issue of vibrational completeness inherent in vibrational close-coupling theory and in the method of Schneider et al. 

One can also obtain the Fano formula explicitly from a diagrammatic many-body expansion. There exists a wide variety of alternative theoretical approaches, some of which (e.g. coordinate rotation, projection operator methods, etc) will not even be described in the present monograph. 

The microscopic basis of the phenomenological rate law can be investigated by making use of projection operator methods. The time evolution of the microscopic density fields, discussed in Section IV, is given by the Liouville equation... 

We make frequent use of these now standard projection operator methods. Although no details are given, the various results are easily derived by application of the operator identity [z - A] = [z — A] -t-[z — A] A... 

We use the same symbol as in Section IX.D, but this should not lead to confusion.) Now, however, since F(12)=F(12, t=0) is arbitrary rather than an equilibrium distribution function, it is no longer true that F(12) = 0. Thus using projection operator methods on the Laplace transform of (11.1), we find... 

The rate kernel expression in (3.13) can be obtained by using projection operator methods to derive an equation for the unreacted pair probability P t),... 

Normalization of SALCCS) derived from the projection operator method must account for the doubled contribution of relative to H, and H<, while maintaining the opposite wave function signs for relative to H, and H,. 

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