Fock-space equation

This is the Bethe-Salpeter equation (BSE) in the effective-potential form. We can regard this equation as the projection of the Fock-space equation (34) onto the restricted space. 

Flow, control of, 265 Flow function on network, 258 Flow, optimal, method for, 261 Fock amplitude for one-particle system, 511 Fock space, 454 amplitudes, 570 description of photons, 569 representation of operators in, 455 Schrodinger equation in, 459 vectors in, 454 Focus, 326 weak, 328... 

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 ... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

Equation (4.1) is sometimes referred to as a state vector in Fock space and its use requires that the Hamiltonian be expressed in terms of operators that can act on such vectors. 

The relativistic coupled cluster method starts from the four-component solutions of the Drrac-Fock or Dirac-Fock-Breit equations, and correlates them by the coupled-cluster approach. The Fock-space coupled-cluster method yields atomic transition energies in good agreement (usually better than 0.1 eV) with known experimental values. This is demonstrated here by the electron affinities of group-13 atoms. Properties of superheavy atoms which are not known experimentally can be predicted. Here we show that the rare gas eka-radon (element 118) will have a positive electron affinity. One-, two-, and four-components methods are described and applied to several states of CdH and its ions. Methods for calculating properties other than energy are discussed, and the electric field gradients of Cl, Br, and I, required to extract nuclear quadrupoles from experimental data, are calculated. 

The orbitals in Ileft, span a complete Fock space of dimension 4P 1 since every orbital is associated with a Hilbert space of dimension 4 corresponding to the states —), 11 )> 14 )> I 44 ) as n Equation (2). Similarly, the orbitals in IRIGHT 44 span a complete Fock space of dimension 4k p. The idea of the DMRG algorithm is to construct a smaller optimized many-body basis /, with a specified reduced dimension M, to span the Fock space of the left block, and a... 

These O, are called Linear Combination of Atomic Orbitals Molecular Orbitals (LCAO MOs) and if they are introduced into the Hartree-Fock equations (eqns (10-2.5)), a simple set of equations (the Hartree-Fock-Roothaan equations) is obtained which can be used to determine the optimum coefficients Cti. For those systems where the space part of each MO is doubly occupied, i.e. there are two electrons in each 0, with spin a and spin respectively so that the complete MOs including spin are different, the total wavefunction is... 

Instead of supposing there to be a single Kohn-Sham potential, one can think of it as a vector in Fock space. For each sheet ft = N of the latter, there is a component vKS(r,N) and a corresponding set of Kohn-Sham equations. Density functional theory and Kohn-Sham theory hold separately on each sheet. Ensemble-average properties are then composed of weighted contributions from each sheet, computable sheet by sheet via the techniques of DFT and the KS equations. Nevertheless, though completely valid, this procedure would yield for the reactivity indices f(r), s(r), and S the results already obtained directly from Eqs. (28). We are left without proper definitions of chemical-reactivity indices for systems with discrete spectra at T = 0 [43]. 

Bloch Equation-Based Fock Space Theories... 

Another way is to start from the precursor of the Bloch equation, eq.(6.1.15), generalized to Fock space/69,76,93/ and use the Q and P projections for the n-valence sector. Writing the Fock space Bloch equation as... 

Bloch Equation Baaed Fock Space Theory ... 

Mukherjee/69/, use of the sufficiency conditions (7.3.9) amounts in effect to assuming that ft is a valence-universal wave-operator. In fact Haque has explicitly demonstrated/123/ that the use of a valence-universal ft in the Fock-space Bloch equation leads automatically to eqn (7.3.9) with the ad-hoc sufficiency requirement. We give the sketch of a general proof here, since it shows that the extra information content of a Fock-space ft, as opposed to a Hilbert space, can be used to advantage for ensuring the connectivity of the cluster amplitudes of S/93/. For a valence-universal ft, the Fock-space Bloch equation (6.1.15) leads to... 

We show below, following Mukherjee/91-94/, how a connected H can he obtained using a Valence-universal O which uses a normalization convention different from intermediate normalization. We start from the Fock-space Bloch Equation ... 

Kutzelnigg et. al795/ have discussed in detail the Fock—space classification of operators for quasi— complete model space. They also introduced a new type od IMS, called the isolated incomplete model space(IIMS). In IIMS, products of q—open operators are all q-open, never closed. As a result O. = 1., just as in a CMS. THe resulting CC equations for IIMS have thus exacly the same structure as in the CMS. 

One can set up the equations for the k=1=1 valence- problem in a hierarchical manner. The valence sectors are defined in terms of particle-hole rank. Thus the Fock space Bloch equation would be... 

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. 

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