Pure integrator asymptote

As explained in Figure 6.12 the pure integrator asymptote will pass through OdB at l.Orad/s (for A" = 1 in equation (6.67)) and the seeond-order element has an undamped natural frequeney of 2.0rad/s and a damping ratio of 0.5. 

The line-integral approximation [ ]), Eq. (213) with (211) or (216), to the exact u,(r [n]) seems to be especially accurate. First of all, its asymptotic behavior for pure-state systems is the same as the behavior of the exact r, in Eq. (134), therefore guaranteeing the proper asymptotic form of KS orbitals. Next, it satisfies exactly the Levy-Perdew [33] identity... 

The spatial integrations of Eq. (2.29), involving the coordinates of each atom, cover the volume V of the container. A reasonable definition of b Pi ) would require bound atoms to be within a molecular length of each other, and should not depend on the location of a center, such as the center of mass, of the molecule within the volume. Then will be independent of V asymptotically for large V. Similarly Xa° (. S") will be independent of V for large V. The factors of A, in Eq. (2.29) is purely for notational convenience, and disappears in formulae such as Eq. (3.4), p. 33. = 1 for an atomic case. 

Clearly, since the reactant wavepacket Xav is assumed in equation (72) to be both purely incoming and localized in the asymptotic region of channel av, its entire history Xaa(f) = e Xaa with t < 0 will also be confined to the asymptotic region of the reactant channel by virtue of the separation of the Hamiltonian H in equation (12). This history will therefore have zero overlap with Xca. and the correlation function Cci,, aa(f) = (Xci/lA aa(0) will be zero for all t < 0. Thus negative times will not contribute to the time integral in equation (72), which can therefore by replaced by equation (58). However, this still leaves the restriction that Xay must be composed solely of incoming waves. 

In order to remove this restriction, we can now turn the argument on its head and use the fact that the lower limit of the time integral in equation (58) is zero rather than minus infinity. In particular, since we have established that equation (58) is correct for a purely incoming reactant wavepacket, let us consider what happens to this equation when we add an arbitrary outgoing component Sx v to Xa.- that is also localized in the asymptotic region of channel av ... 

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