Spin-orbitals quantization representation

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 occupation number vectors are basis vectors in an abstract linear vector space and specify thus only the occupation of the spin orbitals. The occupation number vectors contain no reference to the basis set. The reference to the basis set is built into the operators in the second quantization formalism. Observables are described by expectation values of operators and must be independent of the representation given to the operators and states. The matrix elements of a first quantization operator between two Slater determinants must therefore equal its counterpart of the second quantization formulation. For a given basis set the operators in the Fock space can thus be determined by requiring that the matrix elements between two occupation number vectors of the second quantization operator, must equal the matrix elements between the corresponding two Slater determinants of the corresponding first quantization operators. Operators that are considered in first quantization like the kinetic energy and the coulomb repulsion conserve the number of electrons. In the Fock space these operators must be represented as linear combinations of multipla of the ajaj... 

Other two-electron operators are the mass-polarization and the spin-orbit coupling operator. A two-electron operator gives non-vanishing matrix elements between two Slater determinants if the determinants contain at least two electrons and if they differ in the occupation of at most two pairs of electrons. The second quantization representation of a two-electron operator must thus have the structure... 

In the second-quantization representation the atomic interaction operators are given by relations (13.22) and (13.23), which do not include the operators themselves in coordinate representations, but rather their one-electron and two-electron matrix elements. Therefore, in terms of irreducible tensors in orbital and spin spaces, we must expand the products of creation and annihilation operators that enter (13.22) and (13.23). In this approach, the tensorial properties of one-electron wave functions are translated to second-quantization operators. 

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

The operator of the energy of electrostatic interaction of electrons in (14.65) is represented as a sum of second-quantization operators, and the appropriate submatrix element of each term is proportional to the energy of electrostatic interaction of a pair of equivalent electrons with orbital Lu and spin S12 angular momenta. The values of these submatrix elements are different for different pairing states, since, as follows from (14.66), the two-electron submatrix elements concerned are explicitly dependent on L12, and, hence, implicitly - on S12 (sum L12 + S12 is even). It is in this way that, in the second-quantization representation for the lN configuration, the dependence of the energy of electrostatic interaction on the angles between the particles shows up. This dependence violates the central field approximation. 

In this chapter we have found the relationship between the various operators in the second-quantization representation and irreducible tensors of the orbital and spin spaces of a shell of equivalent electrons. In subsequent chapters we shall be looking at the techniques of finding the matrix elements of these operators. 

The resolvent in eq. (1.208) is called the one-electron Green s function and the notation for it reads G (z). The integration contour may be set in such a way that it encloses all the poles of the resolvent corresponding to the occupied MOs giving by this the required total projection operator. In the spin-orbital occupation number and the second quantization representations related to each other, one can write the operator projecting to the occupied (spin)-MO as an operator of the number of particles in it. Indeed, the expression... 

The results of response theory are most conveniently cast into a formalism based on second quantization. In a second quantized representation of (66), the spin-operators will transfer to triplet excitation operators which are weighted by the integrals over the orbital parts. The -component will have the form... 

In the method of second quantization, a primitive expansion term is represented by a ket (or a bra, depending on how the expansion term is used). These expansion kets may be regarded as members of an abstract space, but, for our purposes here, they will be treated simply as an alternate representation of a determinant formed from orthonormal spin orbitals. An example of the correspondence between a determinant and a ket may be written for a three-electron determinant in a spin-orbital basis of dimension 4 as... 

We may interpret the terms in eqn (2.19) as follows. The first term on the right-hand side represents the transfer of an electron from the spin-orbital Xj(r, ct) to the spin-orbital Xi(r, vice versa), with an energy scale The terms i = j in the sum represent the single-particle on-site energy, while the other terms represent the hybridization of the electrons between different orbitals. The second term on the right-hand side represents electron-electron interactions, the most important being the direct Coulomb interaction when i = j and k = I, as we discuss in Section 2.6. For readers not famihar with the second quantization approach, Appendix A describes a first quantization representation of the first term on the right-hand side of eqn (2.19). 

The Hartree-Fock model leads to an effective one-electron Hamiltonian, called the Fockian F. The second quantized representation of the Fockian has that same form as any other one-electron operator. In the basis of orthogonalized spin-orbitals one can write ... 

Let us determine the second quantized representation of the above spin operators. (For more details, see Jorgensen and Simons 1981). Consider first which is a sum of one-electron operators thus, in terms of spin orbitals we can write ... 

Let us for a moment consider the nonrelativistic case of the second-quantized representation of operators. Here, time reversal flips the spin function between a and p, and in a spin-restricted formalism the basis of Kramers pairs is [(ppa, 4>p(> > where cj) is the spatial part of the orbital. Normally the operators used in nonrelativistic calculations are real and spin free, and we can then use the representation... 

The phase factor Fp is equal to +1 if there are an even number of electrons in the spin orbitals Q < P (i.e. to the left of P in the ON vector) and equal to — 1 if there are an odd number of electrons in these spin orbitals. As we shall see, this factor is necessary to obtain a representation of wave fimctions and operators consistent with first quantization. The requirement (1.2.2) that op produces zero when it operates on a vector with = 1 is in agreement with the fact that a Slater determinant vanishes if a spin orbital appears twice. 

The recipe for constructing a second-quantization representation of a one-electron operator is therefore to use (1.4.2) with the integrals (1.4.12). For real spin orbitals, the integrals exhibit the following permutational symmetry... 

The second-quantization operators are projections of the exact operators onto a basis of spin orbitals. For an incomplete basis, the second-quantization representation of an operator product therefore depends on when the projection is made. For a complete basis, however, the representation is exact and independent of when the projection is made. 

In Box 1.2, we summarize some of the characteristics of operators in the first and second quantizations. The dependence on the spin-oibital basis is different in the two representations. In first quantization, the Slater determinants depend on the spin-orbital basis whereas the operators are independent of the spin orbitals. In the second-quantization formalism, the ON vectors are basis vectors in a linear vector space and contain no reference to the spin-orbital basis. Instead, the reference to the spin-orbital basis is made in the operators. We also note that, whereas the first-quantization operators depend explicitly on the number of electrons, no such dependence is found in the second-quantization operators. 

Having considered the representation of states and operators in second quantization, let us now turn our attention to expectation values. As in first quantization, the evaluation of expectation values is carried out by means of density matrices [4]. Consider a general one- and two-electron Hermitian operator in the spin-orbital basis... 

The density matrix in the spin-orbital representation was introduced in second quantization for the evaluation of one-electron expectation values in the following form... 

We are now in a position to write up the second-quantization representation of the nonrelativistic and spin-free molecular electronic Hamiltonian in the orbital basis ... 

The second-quantization representation of the effective spin-orbit operator now becomes... 

From the expression for the spin-orbit operator (2.2.47), we note that the second-quantization representation of a mixed (spin and space) operator depends on both the spin of the electron and the functional form of the orbitals (2.2.48). For comparison, the pure spin operators in Section 2.2.2 are independent of the functional form of the orbitals, whereas the spin-ftee operators in Section 2.2.1 depend on the orbitals but have the same amplitudes (integrals) for alpha and beta spins. Mixed spin operators are treated in Exercises 2.1 and 2.2. 

Comparing (2.3.34) and (2.2.40), we note that the three components of the spin-orbit operator are treated alike in the Cartesian form (2.2.40) but differently in the spin-tensor form (2.3.34). The spin-tensor representation (2.3.34), on the other hand, separates the spin-orbit operator into three terms, each of which produces a well-defined change in the spin projection. From the discussion in this section, we see that the singlet and triplet excitation operators (in Cartesian or spin-tensor form) allow for a compact representation of the second-quantization operators in the orbital basis. The coupling of more than two elementary operators to strings or linear combinations of strings that transform as irreducible spin tensor operators is described in Section 2.6.7. 

At this point, a comment on terminology is in order. In Chapter 1, we used the term ON vector for second-quantization vectors that correspond to Slater determinants in first quantization. The unusual term ON vector was employed in order to make a clear distinction between the first-and second-quantization representations of the same object and to emphasize the separate and independent structure of second quantization. We shall from now on use the conventional term Slater determinant or simply determinant for ON vectors, bearing in mind that determinants in second quantization are just vectors with elements representing spin-orbital occupations. 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

In the previous discussion of the symmetry of the Dirac equation (chapter 6), it was shown that the Dirac equation was symmetric under time reversal, and that the fermion functions occur in Kramers pairs where the two members are related by time reversal. We will have to deal with a variety of operators, and in most cases the methodologies will be developed in the absence of an external magnetic field, or with the magnetic field considered as a perturbation. Consequently, we can make the developments in terms of a basis of Kramers pairs, which are the natural representation of the wave function in a system that is symmetric under time reversal. The development here is primarily the development of a second-quantized formalism. We will use the term Kramers-restricted to cover techniques and methods based on spinors that in some well-defined way appear as Kramers pairs. The analogous nonrelativistic situation is the spin-restricted formalism, which requires that orbitals appear as pairs with the same spatial part but with a and spins respectively. Spin restriction thus appears as a special case of Kramers restriction, because of the time-reversal connection between a and spin functions. 

Our spin basis is - in contrast to the orbital basis - complete. Therefore, for pure spin operators we have none of the problems associated with the representation of product operators discussed in Section 1.5, and the usual first-quantization commutation relations hold also for the second-quantization spin operators. For example, we may easily verify that the commutator between the second-quantization raising and lowering curators is the same as in first quantization ... 

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