Fock expansion truncated states

We can treat FD quantum-optical states as those of a real single-mode electromagnetic field, which fulfill the condition of truncated Fock expansion. These states can directly be generated by the truncation schemes (the quantum scissors) proposed by Pegg et al. [44] and then generalized by other authors [45-47]. Alternatively, one can analyze states obtained by a direct truncation of operators rather then of their Fock expansion. Such an operator truncation scheme, proposed by Leonski et al. [48-50], will be discussed in detail in the next chapter [51]. 

Kuang et al. [17] defined the normalized FD coherent states by truncating the Fock expansion of the conventional ID coherent states or equivalently by the action of the operator exp( v a ) (with proper normalization) on vacuum state. The Kuang et al. approach is similar to the Vaccaro-Pegg treatment [57] of the... 

Analogously to the generalized, CS in a FD Hilbert space, analyzed in Section IV. A, other states of the electromagnetic field can be defined by the action of the FD displacement or squeeze operators. In particular, FD displaced phase states and coherent phase states were discussed by Gangopadhyay [28]. Generalized displaced number states and Schrodinger cats were analyzed in Ref. 21 and generalized squeezed vacuum was studied in Ref. 34. A different approach to construction of FD states can be based on truncation of the Fock expansion of the well-known ID harmonic oscillator states. The same construction, as for the... 

One can propose another definition of a FD squeezed vacuum, such as by truncation of the Fock expansion of the conventional squeezed vacuum at the state. v). Thus, we define the truncated squeezed vacuum as follows [34]... 

In this paper, the systematic truncation of a distributed universal even-tempered basis set capable of supporting high precision in both mauix Haitree-Fock and second order many-body perturbation theory calculations is explored using the ground state of the boron fluoride molecule as a prototype. The truncation procedure adopted is based on the magnitude of the orbital expansion coefficients associated with a given basis function in each of the occupied orbitals. 

The truncation procedure explored in the present smdy is described in detail in section 2. An analysis of the orbital expansion coefficients for the ground state of the BF molecule is presented in section 3, where the truncated basis sets employed in the present study are defined. The results of both matrix Hartree-Fock calculations and second-order many-body perturbation theory studies are given in section 4 together with a discussion of the properties of the truncated basis sets. The final section, section 5, contains a discussion of the results and conclusions are given. 

If the set was complete, this would be an exact expansion, and any complete set could be used. Unfortunately, one is always restricted, for practical computational reasons, to a finite set of K basis functions. As such, it is important to choose a basis that will provide, as far as is possible, a reasonably accurate expansion for the exact molecular orbitals particularly, for those molecular orbitals which are occupied in To> and determine the ground state energy Eq, A later section of this chapter discusses the questions involved in the choice of a basis set and describes some of the art of choosing a basis set. For our purposes here, we need only assume that is a set of known functions. As the basis set becomes more and more complete, the expansion (3.133) leads to more and more accurate representations of the exact molecular orbitals, i.e., the molecular orbitals converge to those of Eq. (3.132), the true eigenfunctions of the Fock operator. For any finite basis set we will obtain molecular orbitals from the truncated expansion (3.133), which are exact only in the space spanned by the basis functions... 

A common truncation scheme is to include the Hartree—Fock determinant and only singly and doubly excited states of the Hartree—Fock determinant in the Cl expansion. This is known as a singly and doubly excited Cl (SDCI) or Cl with singles and doubles (CISD) ... 

Consider the truncated Cl-doubles (CID) expansion, which contains the Hartree-Fock deirami-nant together with all double excitations out of this determinant, and assume that, because of local symmetry, all single excitations vanish. The wave functions for the individual systems and may again be written in the form (11.2.1). The product of these two states therefore contains the term However, since this term contains only quadruple excitations, it is not included... 

A stationary point of the electronic energy occurs when W(ic) = 0. Applying Newton s method for the solution of nonlinear equations, we determine the Hartree—Fock state by truncating the GBT expansion (10.9.3) at first order and setting the resulting vector equal to zero. In the nth iteration, we solve the nonsymmetric linear Newton equations... 

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