Creation operators, in second quantization

Let us begin by considering the relationship between spin-orbital transformations in first quantization and the transformations of creation operators in second quantization. Assume that we have constructed two Fock spaces, one from the set of first-quantization spin orbitals p and another from the transformed set in (3.2.1). What is the relationship between the second-quantization creation operators of the two Fock spaces In the original spin-orbital basis, an N-electron Slater determinant in first quantization... 

In terms of the creation operator of second quantization each energy level has an eigenfunction... 

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]

The essential new operators in second quantization are the creation operator (the creator) [79,292,350]... 

The conditions that a 3-RDM be 3-positive follow from writing the operators in Eq. (8) as products of three second-quantized operators [16, 17]. The resulting basis functions lie in four vector spaces according to the number of creation operators in the product the four sets of operators defining the basis functions in Eq. (8) are... 

The Dirac-Coulomb-Breit Hamiltonian H qb 1 rewritten in second-quantized form [6, 16] in terms of normal-ordered products of spinor creation and annihilation operators r+s and r+s+ut, ... 

In second quantization, the creation-operators corresponding to Eq. (5.2d) are written... 

We now introduce creation and annihilation operators ajj and an which create/annihilate e-h pairs at a given combination of sites n = (n, n1), i.e., 41°) = 14 = nen h), where 0) is the ground state. Using these operators, a generic monoexcitation configuration interaction Hamiltonian can be formulated as follows in second quantization notation,... 

One of the most important concepts of quantum chemistry is the Slater determinant. Most quantum chemical treatments are made just over Slater determinants. Nevertheless, in many problems the formulation over Slater determinants is not very convenient and the derivation of final expressions is very complicated. The advantage of second quantization lies in the fact that this technique permits us to arrive at the same expressions in a considerably simpler way. In second quantization a Slater determinant is represented by a product of creation and annihilation operators. As will be shown below, the Hamiltonian can also be expressed by creation and annihilation operators and thus the eigenvalue problem is reduced to the manipulation of creation and annihilation operators. This manipulation can be done diagrammatically (according to certain rules which will be specified later) and from the diagrams formed one can write down the final mathematical expression. In the traditional way a Slater determinant I ) is specified by one-electron functions as follows ... 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

These relationships can also be expressed in second-quantized form [4-8] by introducing Fermion creation and annihilation operators, which obey the anticommutation relations... 

The above anticommutation relations for second-quantization operators have been derived using the symmetry properties of one-determinant wave functions with relation to the permutation of the coordinates of particles. Since the second-quantization operators are only defined in the space of antisymmetric wave functions, the reverse statement is true -in second-quantization formalism the permutative symmetry properties of wave functions automatically follow from the anticommutation relations for creation and annihilation operators. We shall write these relations together in the form... 

To make further progress, the zero-order Hamiltonian and the perturbation must be written in second quantized form. Recall that the annihilation operator, a and the creation operator, a], satisfy the following anticommutation relations... 

In second quantized form the zero-order Hamiltonian may be written in terms of creation and annihilation field operators in the form... 

In this second-quantized form of the CASSCF wave function, Aj represents the creation operator for the reference state of block A, which contains all orbitals in the active space, and i+ stands for the creation operator in the ith occupied spin orbital. For convenience, the active space (or block A) is denoted as (Ao, M) (No electrons in M spatial orbitals). The reference state of block A corresponds to the lowest energy state in the No-electron subspace. Other No-electron block states and block states with different numbers of electrons are considered as excited block states [97], which are also required for CAS-BCCC calculations. Except for block A, each of the other blocks is defined to contain only one spin orbital. 

In course of the evaluation of the vacuum amplitudes, only those terms survive eventually in this step which are completely contracted, i.e. those with no creation or annihilation operators at all in the final expression. In other words, each (non-zero) vacuum amplitude is written as a superposition of just one- and two-particle matrix elements of various kinds (due to the one- and two-particle character of all atomic and molecular interactions), and including summations over the core, core-valence, valence and/or virtual orbitals. In certain cases, it has been found useful to combine the steps (1) and (3) and to evaluate the vacuum amplitudes directly from the rhs of the full matrix elements in second quantization. 

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... 

Contents Introduction. - Concept of Creation and Annihilation Operators. -Particle Number Operators. - Second Quantized Representation of Quantum Mechanical Operators. - Evaluation of Matrix Elements. - Advantages of Second Quantization. - Illustrative Examples. - Density Matrices. -Connection to Bra and Ket Formalism. - Using Spatial Orbitals. - Some Model Hamiltonians in Second Quantized Form. - The Brillouin Theorem. -Many-Body Perturbation Theory. -Second Quantization for Nonorthogonal Orbitals. - Second Quantization and Hellmann-Feynman Theorem. - Inter-molecular Interactions. - Quasiparticle Transformations. Miscellaneous Topics Related to Second Quantization -Problem Solutions. - References -Index. 

The particle number representation is conceptually very important because, strictly speaking, the abstract wave functions given in this representation serve as the carrier space of the second quantized creation operators. In other words, the creation operators act on the particle-number represented wave functions. 

Since a and p are often referred to as spin up ( ) and spin down ( ) states, and S" are called spin raising and lowering operators, respectively. Looking at Eqs. (17.4) one gets the impression that there is some analogy between creation/ annihilation operators and the spin operators S /S . To work out the connection between spin operators and second quantization, one has to analyze some points in more detail. The spin operators obey the following commutation rules ... 

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

Thus, all pairs of creation and/or annihilation operators anticommute except for the conjugate pairs of operators such as ap and ap. From these relationships, all other properties of the creation and annihilation operators - often referred to as the elementary operators of second quantization - follow. We note that equation (103) holds only for orthonormal sets of spin orbitals. For nonorthonormal spin orbitals, the Kronecker delta in equation (103) must be replaced by the overlap integral between the two spin orbitals. 

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 ... 

In second quantization, all operators and states can be constructed from a set of elementary creation and annihilation operators. In this section we introduce these operators and explore their basic algebraic properties. 

The Fock space as introduced in Chapter I is defined in terms of a set of orthonormal spin orbitals. In many situations - for example, during the optimization of an electronic state or in the calculation of the response of an electronic state to an external perturbation - it becomes necessary to carry out transformations between different sets of orthonormal spin orbitals. In this chapter, we consider the unitary transformations of creation and annihilation operators and of Fock-space states that are generated by such transformations of the underlying spin-orbital basis. In particular, we shall see how, in second quantization, the unitary transformations can be conveniently carried out by the exponential of an anti-Hermitian operator, written as a linear combination of excitation operators. 

In second quantization, the Pauli antisymmetry principle is incorporated through the algebraic properties of the creation and annihilation operators as discussed in Chapter 1. We note that, in density-functional theory (which bypasses the construction of the wave function and concentrates on the electron density), the fulfilment of the A -representability condition on the density represents a less trivial problem. A density is said to be N-representable if it can be derived from an antisymmetric wave function for N particles [1]. 

There is another commonly used notation known as second quantization. In this language the wave function is written as a series of creation operators acting on the vacuum state. A creation operator aj working on the vacuum generates an (occupied) molecular orbital i. 

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