Hamiltonians transformed

It is now shown how the abrupt changes in the eigenvalue distribution around the central critical point relate to changes in the classical mechanics, bearing in mind that the analog of quantization in classical mechanics is a transformation of the Hamiltonian from a representation in the variables pR, p, R, 0) to one in angle-action variables (/, /e, Qr, 0) such that the transformed Hamiltonian depends only on the actions 1r, /e) [37]. Hamilton s equations diR/dt = (0///00 j), etc.) then show that the actions are constants of the motion, which are related to the quantum numbers by the Bohr correspondence principle [23]. In the present case,... 

S iSy + S jS2i) — SiSj + SjSi The transformed Hamiltonian then is ... 

The terms linear in A appear now on the diagonal positions of the transformed Hamiltonian D... 

A Reinvestigation of Ramsey s Theory of NMR Coupiing Let us define a transformed Hamiltonian as h. 

As seen from equation (50), the ESC Hamiltonian is energy dependent and Hermitian. For a fixed value of E, the ESC Hamiltonian can be diagonalized and the resulting solutions, in principle, form a complete orthonormal set. The eigenfunctions of are identical to the large component of the Dirac spinor. When Z — 0, equations (38) and (44) give us the similarity transformed Hamiltonian... 

Now, substituting for X in equation (40) we get the FW or equivalent Hermitian similarity transformed Hamiltonian which is an expansion in terms of... 

The aim of our similarity-transformed Hamiltonian is to improve the computation of the correlation energy of conventional configuration-interaction (Cl) calculations. In this framework, the conventional wave function is multiplied by the correlation function 14,15,16)... 

The implementation of the above described correlation function is described in detail in Ref. (77). Figure 4 illustrates the efficiency of the method for the He ground-state energy as a function of the basis sets. Already for basis sets with s-and p-fimctions, the similarity-transformed Hamiltonian shows a better accuracy than the conventional Cl method. 

The similarity-transformed Hamiltonian method has so far been applied only to two-electron systems. Using closure (i.e., RI) approximations, this technique will be generalized to many-electron systems (IS). 

Of the methods listed above, only the noniterative CC approaches based on the partitioning of the similarity-transformed Hamiltonian (24-28) and the (C)R-CC approaches of refs 9,13-18,20,21, which employ the MMCC formalism (P. 13, 14, 18, 19, 21, 45, 106, 107), retain the simplicity and the black-box character of the standard CCSD(T) or CCSD(TQf) methods. One of the two goals of the present work is the development of a new class of the MMCC-based black-box methods for multiple bond breaking. 

If we transform Hamiltonian (16) into normal coordinates we arrive at the following expressions [14]... 

Step 4. Compute the transformed Hamiltonian H via the BCH equation (Eq. (8)). This is done by iterating a subroutine that computes the... 

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