Using projection operators

We have already encountered the projection operator formalism in Appendix 9A, where an apphcation to the simplest system-bath problem—a single level interacting with a continuum, was demonstrated. This formalism is general can be applied in different ways and flavors. In general, a projection operator (or projector) P is defined with respect to a certain sub-space whose choice is dictated by the physical problem. By definition it should satisfy the relationship = P (operators that satisfy this relationship are called idempotent), but other than that can be chosen to suit our physical intuition or mathematical approach. For problems involving a system interacting with its equilibrium thermal environment a particularly convenient choice is the thermal projector. An operator that projects the total system-bath density operator on a product of the system s reduced density operator and the 

Since Trep = 1 P is indeed idempotent, P = P. The complementary projector Q is defined simply by Q = 1 — P. 

The projection operator P is chosen according to our stated need We want an equation of motion that will describe the time evolution of a system in contact with a thermally equilibrated bath. Pp ofEq. (10.87) is the density operator of just this system, and its dynamics is determined by the time evolution of the system s density operator O. Finding an equation of motion for this evolution is our next task. 

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There is also an interesting alternative approach by Aharonov et al. [18], who start by using projection operators, n = n) n to partition the Hamiltonian... 

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. 

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... 

In the derivation of normal modes of vibration we started with a set of displacements of individual atoms. By determining the reducible representation Ltot and decomposing it, we calculated the number of normal modes of each symmetry species. We could determine what these modes are by solving a secular equation. We could alternatively have used projection operators to determine the symmetry-adapted combinations. 

Using projection operators constructed from mode eigenvectors, Pio/v( ) (or the influence spectrum for another dynamical variable of interest) can be decomposed into subspectra arising from a a variety of molecular processes. The projection operator is given by... 

Rather than solve this equation by standard techniques and develop the connection between rate coefficient and density, p, as originally done by Smoluchowski, Northrup and Hynes used projection operator techniques to obtain the probability that a reactant pair survives at a time t after formation, P t) = /drp(r, t) as usual. They found that the survival probability satisfies an equation (which is derived in Appendix D)... 

Consider the four functions of Problem 5.2 which form a basis for a reducible representation T of Using projection operators find the orthonormal basis functions which reduce T. Assume (/, / ) = 5y. 

The first comment briefly describes an alternative (and very popular) derivation of the same result (3.14), using projection operators.5 0 Define the... 

Thus we need to form LCAOs of the indicated symmetry types, and this can be done by using projection operators for these representations of D6/l. However, it is advantageous to approach this task from a less direct point of view in order to arrive ultimately at an easier and more general approach to the type of problem it represents. 

There are essentially two types of ECP s in general use, one which follows Phillips and Kleinmans original suggestion and uses explicit core orbitals in the projection operators, and one which uses projection operators on the orbital angular momentum with... 

Determine the symmetry of the cr bonded MOs in square-pyramidal ML5. Use projection operators to find the LCAO Molecular Orbitals for ML5, assuming d2sp2 hybridization to predominate. 

This energy is six-fold degenerate since the states with [[ 1 0 0]] all have the same energy c ,(r) = 1. The eigenfunctions are linear combinations of the i/vO ) in eq. (14) which are of the correct symmetry. To find these IRs we need to know the subspaces spanned by the m basis, which consists of the six permutations of m [1 0 0], and then use projection operators. But actually we have already solved this problem in Section 6.4 in finding the... 

The sets of variables (coordinates) for the complete list of functions are in Table 17.10. Basis functions of the correct symmetry may be found by using projection operators. For the y th IR in eq. (51),... 

We can see that the non-uniqueness of the pseudopotential and of the open-shell hamiltonian have similar origins. Following Roothaan36 the total open-shell hamiltonian may be written in terms of the basic operator Pa by using projection operators to define the particular form of the operator for each sub-space ... 

We have already noticed that the are positive if is Hermitian can, in fact, be interpreted as the normalization factors of a hierarchy of states obtained using projection operators. The positiveness of b implies the positiveness of /) , and vice versa. The parameters a are positive if H is Hermitian with positive eigenvalues a can in fact be interpreted as the expectation values of on a hierarchy of states obtained using the memory function approaches. The positiveness of a implies that of and vice versa. 

Fluctuation relations for the shear viscosities and the twist viscosities were originally derived by Forster [28] using projection operator formalism and by Sarman and Evans analysing the linear response of the SLLOD equations [24]. They were very complicated, i. e. rational functions of TCFI s. The reason for this is that the conventional canonical ensemble was used. In this ensemble one... 

A.K. Bhatia, A. Temkin, Galculation of aotoionization of He and H using projection-operator formalism, Phys. Rev. A11 (1975) 2018. 

In the numerical solution of the SCF orbital equations kinetic balance restrictions are not required, as this condition will be satisfied exactly. However, in the numerical solution of MCSCF equations for purely correlating orbitals, difficulties may arise if the orbital energy , becomes too negative (Bierori etal. 1994 Indelicate 1995,1996 Kim et al. 1998). Here it is suggested that we use projection operators to eliminate the functions that correspond to the negative continuum. 

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),... 

On the other hand, for strictly non-empirical calculations, one may apply beforehand and get the ( y)-type equations (70) directly, without finding the irreducible pairs first. The action of Q (the k i,j part) in Eq. (70) is equivalent to that of and Eq. (70) has a solution for any ij) whether it is degenerate with any other 3S kl) of or not. Equations (70) and the corresponding ones to all orders , Eqs. (100), are the easiest to deal with since they avoid the necessity for finding the irreducible pair states first. They are the ones adopted in many-electron theory. The relation between irreducible pairs and the 3S ij)-type pairs has been given in detail in Reference 9b and the results of perturbation theory - using projection operators have been summarized there. They will not be repeated here. 

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