Hamiltonian transformed Fock space

Kutzelnigg and Koch/77/ also introduced a unitary Fock space cluster operator O = exp(o) with an antihermitian Fock space cr, as in eq. (7.2.3). In this case, the transformed Fock space hamiltonian L should be brought into a form such that... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

The main idea of TFD is the following (Santana, 2004) for a given Hamiltonian which is written in terms of annihilation and creation operators, one applies a doubling procedure which implies extending the Fock space, formally written as Ht = H H. The physical variables are described by the non-tilde operators. In a second step, a Bogolyubov transformation is applied which introduces a rotation of the tilde and non-tilde variables and transforms the non-thermal variables into temperature-dependent form. This formalism can be applied to quite a large class of systems whose Hamiltonian operators can be represented in terms of annihilation and creation operators. 

L. Meissner and R. J. Bartlett, J. Chem. Phys., 94, 6670 (1991). Transformation of the Hamiltonian in Excitation Energy Calculations Comparison Between Fock-Space Multireference Coupled-Cluster and Equation-of-Motion Coupled-Cluster Methods. 

Indeed, when Mr < Mt, the disconnected component of the left-hand side of Eq. (97), i.e. the expression P Ck,open Hn,oPen )i vanishes, since cluster amplitudes defining T, Eq. (41), satisfy equations (78) with n = 1,..., Mr-Equation (99) represents a generalization of the exact Eq. (88) to truncated EOMXCC schemes. Again, the only significant difference between the EOMXCC equations (98) and (99) and their EOMCC analogs (48) and (47), respectively, is the similarity transformed Hamiltonian used by both theories. As in the EOMCC theory, Eqs. (98) and (99) have the same general form (in particular, they rely on the same similarity transformed Hamiltonian) for all the sectors of Fock space. 

Some properties of the Fock space transformations W and effective Hamiltonians h and, thus, of the resulting h, appear to differ from those obtciined by Hilbert space transformations. For example, their canonical unitary W is not separable and yields an h and, thus, an h with disconnected diagrams on each degenerate subspace. However, the analogous U of Eq. (5.13) may be shown to be separable 71), and the resulting He on each complete subspace flo is fully linked , as proven by Brandow [8]. These differences are not explained. 

The Fock space multireference CC methods and the intermediate Hamiltonian techniques (see e.g. Refs. [24-29] and references therein), as well as closely related similarity transformed EOMCC [30-33] are methods particularly suited for calculation of excited/ionized states with a multireference character. Recently, a Brillouin-Wigner formulation of Fock space CC has also been derived [34]. 

The Fourier transformation method enables us to immediately write the momentum space equations as soon as the SCF theory used to describe the system under consideration allows us to build one or several effective Fock Hamiltonians for the orbitals to be determined. This includes a rather large variety of situations ... 

Contents Experimental Basis of Quantum Theory. -Vector Spaces and Linear Transformations. - Matrix Theory. -- Postulates of Quantum Mechanics and Initial Considerations. - One-Dimensional Model Problems. - Angular Momentum. - The Hydrogen Atom, Rigid, Rotor, and the H2 Molecule. - The Molecular Hamiltonian. - Approximation Methods for Stationary States. - General Considerations for Many-Electron Systems. - Calculational Techniques for Many-Electron Systems Using Single Configurations. - Beyond Hartree-Fock Theory. 

The solution of equation 6.11 with the hamiltonian 3.39 and under the conditions 6.12-6.14 is conceptually similar to the solution of equation 3.4 under the LCAO assumption 3.46, although (not unexpectedly) a number of computational complications arise [5]. Averaging over k-space and transformations between real and reciprocal space have to be painstakingly carried out. A major difference is that while for the molecular problem one solution of the Fock equations is sufficient, for the crystal problem the periodicized Fock equations are a function of k and therefore the Abs X Abs variational problem must be solved a number of times equal to the number of sampling points within the first Brillouin zone. At the time of writing, these computational difificulties limit the applicability of the crystal orbital method to rather small molecules and unit cells [6]. 

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