Double commutators

Now the expectation (mean) value of any physical observable (A(t)) = Yv Ap(t) can be calculated using Eq. (22) for the auto-correlation case (/ = /). For instance, A can be one of the relaxation observables for a spin system. Thus, the relaxation rate can be written as a linear combination of irreducible spectral densities and the coefficients of expansion are obtained by evaluating the double commutators for a specific spin-lattice interaction X in the auto-correlation case. In working out Gm x) [e.g., Eq. (21)], one can use successive transformations from the PAS to the (X, Y, Z) frame, and the closure property of the rotation group to rewrite D2mG(Qp ) so as to include the effects of local segmental, molecular, and/or collective motions for molecules in LC. The calculated irreducible spectral densities contain, therefore, all the frequency and orientational information pertaining to the studied molecular system. 

The double commutator [[g, Tr /) (/], Tlp q may form new operators different from Q, and some of these new operators may not even be physical observables. When the double commutator conserves the operator Q, one speaks of the auto-correlation mechanism. Otherwise, one speaks of the cross-relaxation process. In other words, cross-relaxation is independent of the nature of the relaxation mechanism, but involves the interconversion between different operators. To facilitate such a possibility, it is desirable to write the density operator in terms of a complete set of orthogonal basis... 

In the expression for E, we apply the cumulant decomposition twice for the double commutator [[W, A2](i 2), 2](i 2)- Once again, only the fully contracted term contributes to the energy. The only way fully contracted terms arise is from double contractions in [W,A2] to produce a two-particle operator, which then doubly contracts with the final A2 commutator, to contribute to the energy. Since double contractions are involved in each step, no cumulant decomposition is involved for this term. There is no contribution from the three-particle... 

The similar double commutators but with the full Hamiltonian (instead of the residual interaction) correspond to m3 and mi sum rules, respectively, and so represent the spring and inertia parameters [24] in the basis of collective generators Qk and Pk- This allows to establish the connection of the SRPA with the sum rule approach [22,23] and local RPA ]24]. 

In order to find Jfps we first have to compute the double commutator... 

The Jacobi identity gives a relation between double commutators... 

Expressions are often encountered containing several commutators. The following double commutator is easily evaluated using Eq. (3.10)... 

It is seen that the particle rank is reduced by two as a result of the double commutator. 

This can be written according to Lie derivative, and if A is a generator for a global gauge transformation, then this double commutator vanishes. We are then left with... 

The /-selection rules are more difficult to obtain. The trick is to take matrix elements of the double commutator identity Eq. (38e). Apparently this trick is due to Dirac, Pauli, and Guttinger (see, e.g., Slater, 1960, Appendix 31). Using Eq. (31), the left side of Eq. (38e) can be expressed as... 

We note that, even if we start here from the same truncated basis B = B, B2,. . . , Bm as in the EOM method, the results are not necessarily the same, since (2.16) is a single-commutator secular equation whereas (1.50) is a double-commutator secular equation. It should be observed, however, that the column vectors d obtained by solving (2.15) are optimal in the sense of the variation principle, whereas this is not necessarily true for the vectors obtained by solving (1.49). In the following analysis, we will discuss the connection between these two approaches in somewhat greater detail. Since the variation principle (2.10) would provide an optimal approximation, the essential question is whether the theoretical and computational resources available today would permit the proper evaluation of the single-commutator matrix elements defined by (2.13) for a real many-particle system this remains to be seen. 

T = d>)( for the operator for the reference state. Starting from a given basis B in the operator space, we will further assume that we have constructed an approximate eigenoperator D = Bd by properly solving the double-commutator equations (1.49) and (1.50). The approximate unnormalized wave function for the final state is then given by the relation ... 

We note that the excitation operator Dt defined by (2.41) is expressed in terms of the linearly independent operators in the sets BT and TBr T, which form a new basis we note that the set BT as before contains exactly p linearly independent elements. It is evident that the coefficients in this basis are no longer necessarily optimal, and we may hence again start out with the double-commutator relations (1.49) and (1.50), etc.,... 

The matrix elements are essentially the same as those occurring in (1.53), except that the double-commutator character is now lost. This property is very essential as a simplification in evaluating the m2 matrix elements in the matrices occurring in (1.49) and (1.50), but it is of less importance in evaluating the few elements occurring in (2.44) and (2.47). The efforts involved in solving this computational problem is the price one has to pay to reach a more refined approximation. Solving the secular problem in the conventional way, one obtains two new normalized solutions... 

It is interesting to observe that, even if the coefficients d in the original approximate eigenelement D = Bd are determined by solving the equations (1.49) and (1.50) involving double commutators, the refined solutions (2.40) and (2.49) lead to situations where... 

Let us now study the connection between the Liouvillian formalism developed in terms of the Hilbert-Schmidt space and the double-commutator propagator theories or the EOM method. The special theories are based on linear equations of the type (1.49), or... 

This means that the approximate eigenbasis P diagonalizes both the double-commutator Liouvillian matrix and the metric matrix for an arbitrary reference function < >. 

These results indicate that the double-commutator Liouvillian matrix has rather different properties compared to the ordinary matrix (2.64), and that the eigenvalues in the form of energy differences have a different form. It should also be observed that the solutions to the linear equation system (2.83) may have a rather different character.13... 

In concluding this subsection, we note that the main purpose of the double-commutator approach is to provide a method, wherein one can evaluate the matrix element Lrs and A in (2.83) with the theory and computational tools available and then determine the approximate eigenvalues v without reference to the Hamiltonian formalism, i.e., without using the formula v = Ef - E, which contains the difference between two large numbers. In this connection, it may be of value to observe the validity of the relation (2.50) in the reverse order, i.e., / = /2 = /1, which implies that it is possible to evaluate the approximate eigenvalues / in the ordinary formalism based on (2.8) in terms of the quantity /, defined by (1.53) in the double-commutator formalism. It should also be observed that, if the original basis in the wave function space is built up from... 

There is little question that the double-commutator expressions in (3.46) and (3.47) greatly simplify the algebraic and computational aspects of the calculation of the m2 matrix elements occurring in the equation system (3.42) see Ref. 9. The theoretical results as to excitations, ionizations, etc., of certain many-particle systems are in such good agreement with experimental experience that one can probably only expect that part of this agreement will be lost, if one tries to refine the theory. 

Actual numerical calculations introduce double commutator EOM equations for excitation energies and for ionization potentials and electron affinities that, respectively, are... 

So far the reasons for introducing the double commutator form of the equations in (6) and (7) and then in (19) and (21) have not been mentioned, even though this form has required a more lengthy derivation. The... 

There is a price that is incurred by the use of the double commutator. Deexcitation operators of the form... 

We proved earlier that the excitation energy equation (19) yields the exact excitation energies and the exact de-excitation energies (just the negatives of the excitation energies) when any EOM complete operator basis is used. This is, likewise, a consequence of the symmetry of the symmetric double commutator and the commutator,... 

For convenience, we have introduced the double commutator, deiined as... 

The matrix A — B defines the second-order variation of the energy function and is often referred to as the Hessian matrix. The double-commutator form of the Hessian matrix allows these second-order terms to be expressed as a quadratic form. 

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