Electron Field Operators

Here, using electron field operator, momentum density is expressed as... 

The electron field operator V>(x) for the direct space is given by... 

In going from the second to the third line in the above equation we have used the fact that the electron field operators are nilpotent and this also is the reason that the exponentiation in the fourth line works. The end result is true because all the operators a° a] commute. 

The Furry bound state interaction picture of quantum electrodynamics76 relies on an expansion of the second-quantized electron field operator in terms of single-particle solutions of the Dirac equation for a static external field. This external field may be thought of as some mean atomic or molecular potential, whose single-particle spectrum can be divided into positive- and negative-energy branches. This can always be done for the usual elements of the Periodic Table, although problems arise for super-heavy atomic nuclei. 

In order to discuss interacting electrons, it is often advantageous to introduce electron field operators or second quantization, which means that the expansion coefficients as t) of Eq. (3.2) become operators rather than scalars. These operators are non-Hermitian so that also their ad joints aj(t) are needed. These operators are postulated to satisfy the anticommutation relations (at equal times) ... 

Angular momentiun is an important example of quantum mechanical operators in terms of electron field operators. The components of the angular momentiun vector j = I + s are the generators of the 3-dimensional rotation group. In our units, the operator of orbital angular momentum is... 

Many molecular hamiltonians commute with the total spin angular momentum operator, a fact that leads to the consideration of transformation properties of electron field operators under rotations in spin space. Basis functions, natural for such studies, are... 

The electron propagator is obtained when the basis field operators are restricted to the electron field operators A, = a and becomes... 

The SCF spin orbitals and their corresponding electron field operators are labeled in order of increasing energies Cm, making the iV first orbitals occupied in the iV-electron ground state... 

It follows from the symmetry properties of the 3 symbols and from the anticommutation relations of the electron field operators that the triplet operators do not exist for L even, while the singlet operators exist only for such L. 

In this section, a few examples are given of the approximate determination of matrix elements of electron field operators. Atomic units are used throughout. 

As in general all the y-coefficients do not vanish one has to assume a more general reference state than the single determinant SCF state. This is the rather well-known problem of finding the consistent reference state for the Random Phase Approximation (RPA). It also means that the field operator basis can be enlarged and can for instance include the iV-electron occupation number operators (in this discussion, electron field operators and their adjoints are used referring to a basis of spin orbitals that are the natural spin orbitals of the reference state, as will be discussed below, i.e., the spin orbitals that diagonalize the one-matrix)... 

A key quantity is the vector potential A f,t) = A f), which satisfies a wave equation analogously to the wave equation for the electron field operators ip ). It is chosen divergence free V A r) = 0. Invoking zero scalar potential and this choice is usually referred to as the Coulomb gauge. 

Field-dependent basis orbitals as vj (r, G, B) — Uj (f) exp(- (BxG)-f) should be used, resulting in a similar transformation of the electron field operators. Such field-dependent orbitals would guarantee gauge origin independence even in a finite basis. In this expression, G is the position of orbital Uj with respect to an arbitrary origin and the electron coordinate r is given in the same coordinate system. Usually the atomic orbitals are centered at the nuclear positions. In such cases, Rg takes the place of G for the orbital Vj f, Rg, By. 

Such a treatment can, with advantage, be expressed in terms of the superoperators introduced in Eq. (4.19) and in terms of a basis of field operators. The basis of fermion-like operators Xj = a, aj[aja, ,a aja, a ap, - is chosen, such that the electron field operators correspond to the SCF spin orbitals. The field operator space supports a scalar product (XjlXj) = ([A , X,]+) = Tr /9[Xl,Xj]+, where p is the density operator defined in Eq. (4.33). The superoperator identity and the superoperator hamiltonian operate on this space of fermion-like field operators and, in particular, Xi HXj) = [x/, [H,Xj - J. ) = Tt p[xI[H,X ] U. ... 

Restricting f to only some simple fermion-like products of electron field operators will generate only certain types of diagrams that can then be summed to all orders. For instance, self-energy diagrams in third order of the ring and ladder types, which can easily be generalized in any order. It is notable that between consecutive interaction lines, there only occur one hole line and two particle lines or vice versa [see Fig. 9.1]. 

The treatment of orbital overlap in conjunction with the use of nonorthogonal basis sets deserves particular attention in treatments in terms of electron field operators. The definition of creation and annihilation operators and their anticommutation rules are basic for this development. Let s( ) be a set of atomic spin orbitals used to define the creation operators... 

According to the definition of the electron field operators in a basis (Eqs. (10.1) and (10.2)), the basis electron field operator transformations are... 

In order to derive the PPP-hamiltonian, it is assumed that the nuclear framework of the molecular system is invariant under the point group Cg, which contains the operations of identity and a reflection. This is not always true for all molecules to which the model is applied, and then the magnitude of the perturbation caused by the noninvariant part of the nuclear potential must be examined. The electron field operators t) are expressed as the sum of two components, each transforming according to an irreducible representation of C, ... 

If we introduce a spin orbital basis us( ), in which to expand the electron field operators... 

The electron field operators in analogy with the time-dependent case can now be expressed as... 

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