Transformed Hamiltonians Theory

The one-particle Dirac equation for a particle with spin 1/2 in the external potential of the nucleus Vext can then be written as 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

A representation of the one-electron Dirac equation which is decoupled in the electronic and the charge-conjugated degrees of freedom is achieved by a unitary transformation of (3.1) 

The operator X maintains the exact relationship between the large and the small component 

An expansion in (E — V)/2me2 is the basis of the method of elimination of the small component, which in its classical version is only of limited use because the expansion 

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

We are now equipped with all of the basic concepts of the CC/EOMCC theory which are necessary to explain the noniterative MMCC approaches to ground and excited electronic states. In this section, we focus on the exact MMCC theory. The approximate MMCC schemes for excited electronic states, including the externally corrected MMCC approaches and the CR-EOMCCSD(T) theory, and their most recent analog based on the left eigenstates of the similarity-transformed Hamiltonian, are discussed in Section 3. 

In vibration-rotation theory, the /., / and contributions to the contact-transformed Hamiltonian are commonly evaluated directly from the relationships (7.59), (7.63), (7.65) and (7.66). This is because the particularly simple commutation relationships which exist between the normal coordinate operator Q, its conjugate... 

CCSD(T) method. The question then naturally arises as to how these methods can be extended to excited states. For the iterative methods, the extension is straightforward by analyzing the correspondence between terms in the CC equations and in H, one can define an H matrix for these methods, even though it is not exactly of the form of a similarity-transformed Hamiltonian. If one follows the linear-response approach, one arrives at the same matrix in the linear response theory, one starts from the CC equations, rather than the CC wave function, and no CC wave function is assumed. This matrix also arises in the equations for derivatives of CC amplitudes. In linear response theory, this matrix is sometimes called the Jacobian [19]. The upshot is that excited states for methods such as CCSDT-1, CCSDT-2, CCSDT-3, and CC3 can be obtained by solving eigenvalue equations in a manner similar to those for methods such as CCSD and CCSDT. 

Briefly, the aim of Lie transformations in Hamiltonian theory is to generate a symplectic (that is, canonical) change of variables depending on a small parameter as the general solution of a Hamiltonian system of differential equations. The method was first proposed by Deprit [75] (we follow the presentation in Ref. 76) and can be stated as follows. 

In addition to discussing specific renormalized CC methods, we have reviewed the MMCC formalism, which is the key concept behind all renormalized CC approaches. In this discussion, we have included the most recent biorthogonal formulation of the MMCC theory employing the left eigenstates of the similarity-transformed Hamiltonian which leads to the CR-CC(2,3) and other CR-CC(ma, / / ) approaches. We have discussed the similarities and differences between the original MMCC theory of Piecuch and Kowalski, introduced in Refs. [11,24,34], and the biorthogonal MMCC formalism of Piecuch and Wloch, introduced in Ref. [45] and further elaborated on in Ref. [46]. In particular, we have pointed out how the biorthogonal formulation of the MMCC theory enables one to eliminate the overlap denominators, which are... 

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