Eigenvalue Moment Method

The preceeding Eigenvalue Moment Method (EMM) , can yield impressive results for singular, strongly coupled, multidimensional quantum systems, such as the quadratic Zeeman effect for superstrong magnetic fields. 

Eq. (8.69) is that all eigenvalues of P have magnitude less than unity. Eor moment methods, an equally important condition is realizability of the moment set. A sufficient condition for the realizability of Eq. (8.69) is that all components of P be nonnegative. Because... 

In our first exploration of the T1 and T2 conditions [5] we obtained results of the RDM method for the ground-state energy and dipole moment for a collection of small molecules and molecular ions, both closed-shell and open-shell systems. (We don t mean closed shell in a strict sense, and we only constrained the spin and spin multiplicity eigenvalues, not the elements of the RDM.) The choice of molecules and configurations largely followed Ref. [18]—a paper that, we think, reinvigorated the classical RDM approach. We showed that the addition of the T1 and T2 conditions (T2 without the off-diagonal block X)... 

For further confirmation of the mode-softening and a possible identification of the molecular nature of the over-damped mode, we used the rigid-body motion analysis of the thermal- parameters of the room temperature x-ray diffraction study. A thermal-motion analysis (TMA) program was used to calculate the components of the librational (L) and the translational (T) tensors with a least-square fit of the published thermal parameters ( ) of all nonhydrogen atoms of the molecule. The librational frequencies were calculated by the method of Cruickshank (7), using the appropriate eigenvalues of the L-tensor and the corresponding moments of inertia. 

One characteristic of all LR/EOM-CC methods is that a non-Hermitian eigenvalue problem finally has to be solved (which cannot be reduced to a standard eigenvalue problem as Eq. [43]) yielding left and right sets of solutions, and therefore also two sets of transition moments. 

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