In Fock space

The general vector in Fock space may have components in some or all of the Hilbert subspaces, which means that it is now possible to consider states in which there is a superposition of different populations. Thus, we may represent the Fock space vector at an arbitrary time t by a symbol and expand this state in terms of its components in each subspace ... 

Thus (Xjf t,Ny is the component of with respect to XW> it is the probability density in Fock space that the system have a population N. We recognize it as nothing other than the Schrodinger wave function for N particles—Section 8.10. 

Fock Space Representation of Operators.—Let F be some operator that neither creates nor destroys particles, and is a known function in configuration space for N particles. In symbols such an operator must by definition have the following matrix elements in Fock space ... 

We now define the projection operator in Fock space as an obvious generalization of Eq. (8-164) ... 

For any linear operator 22 defined in Fock space, we can similarly prove, by following an argument like that leading to Eq. (8-189), that the trace in Fock space of WB is the grand-ensemble-average of 22 ... 

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. 

In (4.28) and (4.30), we have achieved our aim of expressing the Hamiltonian in the appropriate second quantized form for acting on the state vectors in Fock space. 

The operators W, A, occurring above, should be taken in the second-quantization form, free of explicit dependence on particle number, and Tr means the trace in Fock space (see e.g. [10] for details). Problems of existence and functional differentiability of generalized functionals F [n] and r [n] are discussed in [28] the functional F [n] is denoted there as Fi,[n] or Ffrac[n] or FfraoM (depending on the scope of 3), similarly for F [n]. Note that DMs can be viewed as the coordinate representation of the density operators. 

In summary, Erdahl s treatment is more general and allows a more concise formulation because he works in Fock space, conserving only the parity of the number of particles however, he finds it necessary to restrict the coefficients to be real. We work at fixed particle number and have no reason for the restriction to real coefficients. If the Hamiltonian should be general Hermitian, in which case the RDM must likewise be assumed to be general Hermitian, then our approach leads to Hermitian semidefinite conditions. 

W. Kutzelnigg, Quanmm chemistry in Fock space, in Aspects of Many-Body Effects in Molecules and Extended Systems (D. Mukherjee, ed.), Volume 50 of Lecture Notes in Chemistry, Springer-Verlag, Berlin, 1989, p. 35. 

II. Many-Body Theory in Fock-Space Formulation... 

II. MANY-BODY THEORY IN FOCK-SPACE FORMULATION... 

In the method based on the unitary transformation, we start by writing the exact wavefunction th in terms of the reference function and a unitary transformation operator in Fock space ... 

W. Kutzelnigg, Quantum chemistry in Fock space. I. The universal wave and energy operator. J. Chem. Phys. 77, 3081 (1982). 

Many electron systems such as molecules and quantum dots show the complex phenomena of electron correlation caused by Coulomb interactions. These phenomena can be described to some extent by the Hubbard model [76]. This is a simple model that captures the main physics of the problem and admits an exact solution in some special cases [77]. To calculate the entanglement for electrons described by this model, we will use Zanardi s measure, which is given in Fock space as the von Neumann entropy [78]. 

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]. 

FULL CLUSTER EXPANSION THEORIES IN FOCK SPACE 7.1 Preliminaries for a Fock Space Approach... 

FULL CLUSTER EXPANSION THEORIES IN FOCK SPACE... 

Pal et al/73(a)/ formulated a variational CC theory for energy differences in Fock space, which generates a hermitian H They employed the ansatz,... 

C. M. L. Rittby and R. J. Bartlett, Theor. Chim. Acta, 80, 469 (1991). Multireference Coupled Cluster Theory in Fock Space. 

The formulation of (59,60) in Fock space is (for d>(0) a Slater determinant built up from the spin orbitals (pi)... 

Instead of simple correspondence between occupation numbers and Hilbert spaces described above, we now establish another correspondence that makes possible to transfer a universal computing with fixed permanent interaction [Ozhigov 2002 (a) Ozhigov 2002 (b)] to the language of fermionic computing in Fock space of occupation numbers. 

Now the door is open for the representation of unitary transformations in Hilbert space required for quantum computing by transformations in Fock space. Consider Hermitian operator H in one-dimensional Hilbert space Tilt has the form Ho + Hi where... 

The operator He is defined in Fock space. Insertion of the expansions (87) in Eq(93) and integration over r yields ... 

We have also presented a recently developed size-extensive and size-consistent SS-MRCC approach based on a general model space. For this, the intermediate normalization convention of the wave operator has to be abandoned in favor of some appropriate size-extensive normalization. Suitable operators, defined in Fock space—described as closed, open and quasi-open— have to be introduced to ensure that the effective operator furnishing the target energy on diagonalization is a closed operator. 

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