The Fock space

Let ( / (x) be a basis of M orthonormal spin orbitals, where the coordinates x represent collectively the spatial coordinates r and the spin coordinate a of the electron. A Slater determinant is an antisymmetrized product of one or more spin orbitals. For example, a normalized Slater determinant for N electrons may be written as 

We now introduce an abstract linear vector space - the Fock space - where each determinant is represented by an occupation-number (ON) vector k). 

the occupation number is 1 if (f)p is present in the determinant and 0 if it is absent. For an orthonormal set of spin orbitals, we define the inner product between two ON vectors jk) and m) as 

This definition is consistent with the overlap between two Slater determinants containing the same number of electrons. However, the extension of (1.1.3) to have a well-defined but zero overlap 

In a given spin-orbital basis, there is a one-to-one mapping between the Slater determinants with spin orbitals in canonical order and the ON vectors in the Fock space. Much of the terminology for Slater determinants is therefore used for ON vectors as well. Still, the ON vectors are not Slater determinants - unlike the Slater determinants, the ON vectors have no spatial structure but are just basis vectors in an abstract vector space. This Fock space can be manipulated as an ordinary inner-product vector space. For example, for two general vectors or states in the Fock space 

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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 ... 

We have carried out tins discussion in occupation number representation or coordinate representation each with a definite number N of particles. Similar results follow for the Fock space representation and the properties of grand ensembles. Averages over grand ensembles are also independent of time when the probabilities > are independent of time, whether the observable commutes with H or not. 

If not otherwise stated the four-component Dirac method was used. The Hartree-Fock (HF) calculations are numerical and contain Breit and QED corrections (self-energy and vacuum polarization). For Au and Rg, the Fock-space coupled cluster (CC) results are taken from Kaldor and co-workers [4, 90], which contains the Breit term in the low-frequency limit. For Cu and Ag, Douglas-Kroll scalar relativistic CCSD(T) results are used from Sadlej and co-workers [6]. Experimental values are from Refs. [91, 92]. 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

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. 

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. 

Let us dehne the Fock space associated with the lattice L by... 

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 present calculation uses the Fock-space scheme... 

The Fock-space Hamiltonian H is equivalent to the configuration-space Hamiltonian H insofar as both have the same matrix elements between n-electron Slater determinants. The main difference is that H has eigenstates of arbitrary particle number n it is, in a way, the direct sum of aU // . Another difference, of course, is that H is defined independently of a basis and hence does not depend on the dimension of the latter. One can also define a basis-independent Fock-space Hamiltonian H, in terms of field operators [11], but this is not very convenient for our purposes. 

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... 

Let us consider the 5s, 5p, 5d orbitals of lead and Is orbital of oxygen as the outercore and the ai, a2, os, tti, tt2 orbitals of PbO (consisting mainly of 6s, 6p orbitals of Pb and 2s, 2p orbitals of O) as valence. Although in the Cl calculations we take into account only the correlation between valence electrons, the accuracy attained in the Cl calculation of Ay is much better than in the RCC-SD calculation. The main problem with the RCC calculation was that the Fock-space RCC-SD version used there was not optimal in accounting for nondynamic correlations (see [136] for details of RCC-SD and Cl calculations of the Pb atom). Nevertheless, the potential of the RCC approach for electronic structure calculations is very high, especially in the framework of the intermediate Hamiltonian formulation [102, 131]. 

Let us confine our attention to the one-particle subspace of the Fock space. As the number operator N is conserved by virtue of Eq. (28), if we start from the one-particle subspace of the Fock space, we shall remain in this subspace during all the evolution. The transition amplitude Uki(t",t ) between the one-particle states ) d]k 0) and 11/) = n,+10) is given by the following scalar product in the holomorphic representation... 

Both are equivalent in terms of one-particle subspace of the Fock space which we will discuss in further detail later. 

The reader will recognize the analogy with the grand ensemble in statistical mechanics and with the Fock space in field theory. 

In the usual formalism of quantum mechanics, the first quantization formalism, observables are represented by operators and the wave functions are normal functions. In the method of second quantization, the wave functions are also expressed in terms of operators. The formalism starts with the introduction of an abstract vector space, the Fock space. The basis vectors of the Fock space are occupation number vectors, with each vector defined by a set of occupation numbers (0 or 1 for fermions). An occupation number vector represents a Slater determinant with each occupation number giving the occupation of given spin orbital. Creation and annihilation operators that respectively adds and removes electrons are then introduced. Representations of usual operators are expressed in terms of the very same operators. 

The Fock space F(m) is a sum of subspaces F(m,N) where each subspace F(m,N) contains all occupation number vectors that can be obtained by distributing N electrons into m spin orbitals. The subspace consisting of occupation number vectors with zero electrons contains a single vector, the true vacuum state,... 

This definition is consistent with the definition of overlap between two Slater-determinants having the same number of electrons. The overlap between Slater determinants having a different number of electrons is not defined. The extension to have a well-defined, but zero, overlap between two occupation number vectors with different numbers of electrons is a special feature of the Fock-space formulation that allows a unified description of systems with a different number of electrons. As a special case of Eq. (1.3), the vacuum state is defined to be normalized... 

The phase factor r(n) is introduced in order to endow the antisymmetry of many-electron wave functions in the Fock space, as we soon will see. The definition that ai operating on an occupation number vector gives zero if spin... 

The occupation number vectors are thus the common eigenvectors for the hermitian and commuting set of operators (aj a agag,. -.a am) and there is a one to one correspondence between an occupation vector and a set of eigenvalues for (aj av a ag,- -.a am). This is consistent with the definition of the occupation number vectors as being an orthonormal basis for the Fock space. 

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