Hermiticity and time reversal

In section 5.2 the concepts of hermiticity and time-reversal where introduced in the discussion of the first order response of the wave function. In this section we shall see that these concepts allows us to determine whether the linear response function is real or imaginary. The linear response function is given by (51), but using the first-order response equation (49) it may be simplified to (54). This may reduce the precision in the numerical evaluation, but is of no consequence for the following arguments. In the notation of section 5.2 the linear response function at the closed-shell HF level of theory is accordingly written... 

Consider next the term involving the third derivative tensor 4. Again we split the response vectors into Hermitian and anti-Hermitian contributions such that we obtain a sum of terms B BgBc where the vectors B., Bg and Be all have well-defined hermiticities and time reversal structures. In order to be able to use (235) we first analyze the vector structure obtained by contracting the third derivative tensor with two vectors ... 

The matrices and are defined in perfect analogy with (239). Once again we see the structure of the gradient vector e /, but with the reference Hamiltonian Hq replaced by the two-index transformed Hamiltonian. To determine the vector structure of with respect to hermiticity and time reversal symmetry it suffices to look at the corresponding two-index transformed Hamiltonian... 

R/NR ratio—in this case about 75% of the ratio. The symmetry of the Hamiltonian about both diagonals is also found in the nonrelativistic case by a suitable ordering of the determinants, so the gains from Hermiticity and time reversal on the matrix itself are the same in both cases. 

Hermiticity and Time reversal, States Represented by Real Valued Wavefunctions... 

Hecht and Barron (1993) discuss the time reversal and Hermiticity characteristics of optical activity operators. They formulate the Raman optical activity observables for the four different forms of ROA in terms of matrix elements of the absorptive and dispersive parts of these operators. Rupprecht (1989) applied a matrix formalism for Raman optical activity to intensity sum rules. 

From the above observations we can draw the following conclusions The gradient vector appearing on the right hand side of the first-order response equation (223) generally has positive hermiticity. From (244) and (245) it is then clear that the Hessian and the matrix 5 should be multiplied with Hermi-tian and anti-Hermitian trial vectors, respectively. In the static case (O — O all trial vectors should consequently be restricted to have the Hermitian structure. Furthermore, tried vectors should have the time reversal symmetry of the property gradient since both 4 conserves time reversal symmetry. Then,... 

Properties (ii)-(iv) of the classical Liouville equation are a bit troublesome. The Hermiticity of the classical Liouville operator (f ) implies that its eigenvalues are real. Thus, p(X, t) must exhibit oscillatory temporal behavior and appears not to decay to a unique stationary state in the hmit t oo. This raises the question of how do we describe the irreversible decay of a system to a imique equilibrium state. The time-reversal invariance of... 

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