Second-quantized operator strings

Fiow does Wick s theorem help us in evaluating matrix elements of second-quantized operators Recall that any matrix element of an operator may be written as a vacuum expectation value by simply writing its left- and right-hand determinants as operator strings acting on the vacuum state, I ). The... 

The construction of the coupled cluster amplitude equations is somewhat more complicated than the energy equation in that the latter requires only reference expectation values of the second-quantized operators. For the amplitude equations, we now require matrix elements between the reference, o, on the right and specific excited determinants on the left. We must therefore convert these into reference expectation value expressions by writing the excited determinants as excitation operator strings acting on Oq. For example, a doubly excited bra determinant may be written as... 

Often, rather than the simple strings of second quantized operators (e.g., al ala a ), spin- and/or space-symmetry-adapted linear combinations of these operators are utilized. Since the EOM matrix equations are... 

In the second-quantized operators (31) and (32), the summation over the particle indices i,j,... runs over all the electron states of the (complete one-electron) spectrum. If these operators act to the right upon the reference state, i.e. the many-electron vacuum of the particle-hole formalism, some of these (strings of) creation and annihilation operators create excitations while other gives simply zero, i.e. no contribution. For the pure vacuum, in particular, the behavior of the second-quantized operators can be read off quite easily because the creation operators appear left of the annihilation operators in expressions (31) and (32), respectively. 

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. 

In general, a given sequence of creation and annihilation operators is said to be normal ordered, if all the creation operators appear left of all annihilation operators. Such an ordering of the operator strings simplifies the manipulation of operator products as well as the evaluation of their matrix elements, as the action of these operators can be read off immediately. In the particle-hole formalism, its hereby obvious that we can annihilate only those particles or holes which exist initially in fact, an existing hole is nothing else than that there is no electron in this hole state. In this formalism, therefore, an operator in second quantization is normal ordered with regard to the reference state [Pg.190]

In second quantization. Slater determinants are expressed as products or strings of creation operators aj, working on the vacuum state... 

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 ... 

Second quantization treats operators and wave functions in a unified way - they are all expressed in terms of the elementary creation and annihilation operators. This property of the second-quantization formalism can, for example, be exploited to express modifications to the wave function as changes in the operators. To illustrate the unified description of states and operators afforded by second quantization, we note that any ON vector may be written compactly as a string of creation operators working on the vacuum state (1.2.4)... 

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