Spin in Second Quantization

In the formalism of second quantization as presented in the previous chapter, there is no reference to electron spin - the intrinsic angular momentum of the electron. In nonrelativistic theory, many important simplifications follow by taking spin explicitly into account. In the present chapter, we develop the theory of second quantization further so as to allow for an explicit description of electron spin. Although no fundamentally new concepts of the second-quantization formalism are introduced, the results obtained here are essential for an efficient description of molecular electronic systems in the nonrelativistic limit. 

---

The incorporation of spin in second quantization leads to operators with different spin synunetry properties as demonstrated in Section 2.2. Thus, spin-free interactions are represented by operatOTs that are totally symmetric in spin space and thus expressed in terms of orbital excitation operators that affect alpha and beta electrons equally, whereas pure spin interactions are represented by excitation operators that affect alpha and beta electrons differently. For the efficient and transparent manipulation of these operators, we shall apply the standard machinery of group theory. More specifically, we shall adopt the theory of tensor operators for angular momentum in quantum mechanics and develop a useful set of tools for the construction and classification of states and operators with definite spin symmetry properties. 

Most of this chapter utilizes the first-quantized formulation of the ROMs introduced above. However, some concepts related to separabihty and extensiv-ity are more easily discussed in second quantization, and the second-quantized formalism is therefore employed in Section IE. Introducing an orthonormal spin-orbital basis 1 ) = dj 0), the elements of the p-RDM are expressed directly in second quantization as... 

Here is the fragment wavefunction and iJ)q is the Q-state wave-function in second quantization representation. Further, as an example, we have limited consideration here to systems of integral total spin Fermi systems can be treated in a similar way. 

The Hamiltonian is assumed to be spin-independent. It can then be written, in second quantized form, in terms of the spin-averaged excitation operators (the generators of the unitary group )... 

We next turn to the question of whether the total spin operator commutes with Wj v In Ref. [8] it is shown that the components of the total spin operator can be expressed in second-quantized form by means of the relationships ... 

In (A.3) the velocity form of the dipole approximation is used. The factor of in p(E) cancels with the normalization for the plane wave, thus providing the correct continuum limit (L oo). If it is assumed that 0> is a closed-shell state, the two terms on the right-hand side of (A.2) yield identical results in (A.3). Therefore, we simplify the notation by combining the two terms and suppress the spin designations. The electronic momentum operator for our system, expressed in second quantized notation, is given by... 

In second quantization, the numerical vector-coupling coefficients (the Aff and ) appear as matrix elements of creation and annihilation operators X] and jc> The operator X creates an electron in an orthonormal spin orbital io), where /(j) = /) (j), and (T = a or p. Similarly, operator destroys an electron in the orthonormal spin orbital ia). In quantum chemistry problems in which the number of particles is conserved, the Xj and will always occur in pairs. The role of these operators is easily illustrated by showing their operation on a specific type of CSF, namely a Slater determinant. Thus, as an example, for the determinant... 

Now let US consider the process of adding an electron to one of the virtual spin orbitals Xr fo produce the (JV + l)-electron single determinant " V> = XrXiXi where again the remaining spin orbitals are identical to those in " o>- In second quantization, this would be accomplished by creating an electron in Xr... 

The challenge is to devise an algorithm for the spin-orbit case that incorporates as much as possible of the machinery that makes this nonrelativistic approach so efficient. We begin by writing the spin-orbit operator in second quantization. Looking at the one-electron contribution, we may write this as... 

In second quantization, the electronic Hamiltonian operator is expressed as a linear combination of strings of creation and annihilation operators. The following form is appropriate for a spin-free, nonrelativistic electronic system ... 

The density function can also be expressed in second quantization [10]. Introducing the notation q = tiqlqtni ms for spin-orbitals, expression (9.12) becomes... 

In second quantization, the singlet one- and two-electron spin tensor operators have the following representations ... 

Before proceeding to determine the form of the operators in second quantization, we recall that the matrix elements between Slater determinants depend on the spatial form of the spin orbitals. Since the ON vectors are independent of the spatial form of spin orbitals, we conclude that the second-quantization f )eratws - in contrast to their first-quantization counterparts - must depend on the spatial form of the spin orbitals. 

Combining the results of Sections 1.4.1 and 1.4.2, we may now construct the full second-quantization representation of the electronic Hamiltonian operator in the Bom-Oppenheimer approximation. Although not strictly needed for the development of the second-quantization theory in this chapter, we present the detailed form of this operator as an example of the construction of operators in second quantization. In the absence of external fields, the second-quantization nonrelativistic and spin-free molecular electronic Hamiltonian is given by... 

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... 

---