Hamiltonian second-order iterated

For w = 1 or 2 they have the general form of a radial eigenvalue problem arising from some Hamiltonian. In fact, the radial parts of the nonrelativistic hydrogenic Hamiltonian, Klein-Gordon, and second-order iterated Dirac Hamiltonians with 1/r potential can all be expressed in this form for w = 1 and suitable choices of the parameters , rj, x. Similarly, the three-dimensional isotropic harmonic oscillator radial equation has this form for w = 2. 

It is shown that eigenvalue prohlems associated with contracted Hamiltonians may be solved by iteration procedures and particularly that there exists a second-order iteration procedure which is a generalization of the convention variational principle. 

In this subsection we will combine the general ideas of the iterative perturbation algorithms by unitary transformations and the rotating wave transformation, to construct effective models. We first show that the preceding KAM iterative perturbation algorithms allow us to partition at a desired order operators in orthogonal Hilbert subspaces. Its relation with the standard adiabatic elimination is proved for the second order. We next apply this partitioning technique combined with RWT to construct effective dressed Hamiltonians from the Floquet Hamiltonian. This is illustrated in the next two Sections III.E and III.F for two-photon resonant processes in atoms and molecules. 

We develop the partitioning technique with the use of the iterative KAM perturbation algorithms. We derive an effective Hamiltonian of second order. The scheme we show can be easily extended to higher orders. 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

Although the DKS Hamiltonian is linear in Vgff = F c + Vh, the second-order DK Hamiltonian 9- (1 )- Hence, during the SCF iterations,... 

The no-virtual-pair Dirac-Coulomb-Breit Hamiltonian, correct to second order in the fine-structure constant a, provides the framework for four-component methods, the most accurate approximations in electronic structure calculations for heavy atomie and molecular systems, ineluding aetinides. Electron correlation is taken into aeeount by the powerful coupled eluster approaeh. The density of states in actinide systems necessitates simultaneous treatment of large manifolds, best achieved by Fock-space coupled eluster to avoid intruder states, which destroy the convergence of the CC iterations, while still treating a large number of states simultaneously, intermediate Hamiltonian sehemes are employed. 

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