Second quantization formalism creation operators

However, before going into a detailed discussion of various relativistic Hamiltonians we will introduce an alternative form of the electronic Hamiltonian (3.4), which is useful for wavefunction-based correlation methods. It is obtained by switching to a particle-hole formalism and then introducing normal ordering. In the second-quantization formalism creation and annihilation operators refer to some specific set of (orthononnal) orbitals, and Slater determinants in Hilbert space translate into occupation-number veetors in Fock space. The annihilation operators in equation 3.4 by definition give zero when acting on the vacuum state... 

Most formulations of MCSCF theory are based on the second quantization formalism. We therefore review briefly in this section the basic definitions of the annihilation and creation operators, and the expansion of quantum mechanical operators in products of them. 

Second-quantization formalism was introduced into the theory of many-electron atoms by Judd [12]. This formalism enables one to give a simple and elegant description of both the rotation symmetry of a system and its permutational symmetry the tensorial properties of wave functions are translated to electron creation and annihilation operators, and the Pauli exclusion principle stems automatically from the anticommutation relations between these operators. 

This representation among others removes one more inconsistency in quantum chemistry one generally deals with the systems of constant composition i.e. of the fixed number of electrons. The expression eq. (1.178) allows one to express the matrix elements of an electronic Hamiltonian without the necessity to go in a subspace with number of electrons different from the considered number N which is implied by the second quantization formalism of the Fermi creation and annihilation operators and on the other hand allows to keep the general form independent explicitly neither on the above number of electrons nor on the total spin which are both condensed in the matrix form of the generators E specific for the Young pattern T for which they are calculated. 

Using the second quantization formalism simplifies everything greatly Antisym-metrization is achieved simply by putting all the operators creating electrons in the one-electron states of the A-th group to the left from those of the B-th group, provided B < A. The multipliers T can be considered as linear combinations of rows of Na creation operators a, .. 

Due to the fact that the SLG wave function belongs to the GF approximation (Section 1.7), it is subject to numerous selection rules characteristic of GF. Their explicit form can be easily obtained using the second quantization formalism. Since the operators of electron creation on the right and left HOs satisfy usual anticommutation relations for orthogonal basis and the number of particle operators have the usual form ... 

The basic elements of the second-quantization formalism are the annihilation and creation operators (Linderberg and (3hrn, 1973). The annihilation operator ap annihilates an electron in orbital creation operator ap (the conjugate of ap) creates an electron in orbital p. These operators satisfy 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... 

Besides of creation operators, the second quantized formalism also requires to the use of formal operators which remove (annihilate) electrons. 

Application of the second quantized formalism in quantum chemistry is merely an appHcation of simple algebraic rules followed by creation and annihilation operators. We have already been acquainted with one rule of this kind the anticommutator relation for creation operators of Eq. (2.11). The mutual commutator properties of creation and annihilation operators will be studied below. Again, the true annihilation operators will be considered as introduced above leaving open the question how a is related to a. ... 

Accordingly, the second quantized formalism can be generalized to the nonorthogonal case in two alternative manners one may keep either the adjoint relation of Eq. (13.1) or the simple anticommutation rule of Eq. (13.2). In the former case the commutation rules become more complicated, while in the latter case the annihilation operators will not be the adjoints of the corresponding creation operators. 

We shall mention that using the mixed second quantized formalism of Ref. [1], already mentioned, it is possible to present the chemical Hamiltonian (32)-(34) in a form in which each term of the Hamiltonian contains only creation and annihilation operators assigned to the corresponding atom or pair of atoms. To save place, we shall illustrate that only by considering the first term of Eq. (33)— all the other terms can be treated analogously. The first term in question is... 

Wick s theorem (3.109), which gives the prescription for handling products of creation and annihilation operators may, of course, be applied to the Hamiltonian operator, 5f, when it is expressed in the second-quantization formalism, eq. (3.91) ... 

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)... 

In formulating the second-quantized description of a system of noninteracting fermions, we shall, therefore, have to introduce distinct creation and annihilation operators for particle and antiparticle. Furthermore, since all the fermions that have been discovered thus far obey the Pauli Exclusion principle we shall have to make sure that the formalism describes a many particle system in terms of properly antisymmetrized amplitudes so that the particles obey Fermi-Dirac statistics. For definiteness, we shall in the present section consider only the negaton-positon system, and call the negaton the particle and the positon the antiparticle. 

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. 

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. 

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]

Surveying the history of the theory of optical lanthanide spectroscopy, we can discern several main features the usefulness of Lie groups, following their introduction by Racah (1949) the relevance of the method of second quantization, as demonstrated by the use of annihilation and creation operators for electrons and the inability of the Hartree-Fock method and its various elaborations to provide accurate values (say to within 1%) of such crucial quantities as the Slater integrals F (4f,4f) and the Sternheimer correction factors R , for a free ion. The success of the formal mathematics is in striking contrast to the failure of the machinery of computation. This turn of events has happened over a period of time when... 

Section 2.4 introduces creation and annihilation operators and the formalism of second quantization. Second quantization is an approach to dealing with many-electron systems, which incorporates the Pauli exclusion principle but avoids the explicit use of Slater determinants. This formalism is widely used in the literature of many-body theory. It is, however, not required for a comprehension of most of the rest of this book, and thus this section can be skipped without loss of continuity. 

We shall gradually construct the formalism of second quantization by showing how the properties of determinants can be transferred onto the algebraic properties of operators. We begin by associating a creation operator aj with each spin orbital We define aj by its action on an arbitrary Slater... 

All the properties of Slater determinants are contained in the anticommutation relations between two creation operators (Eq. (2.194)), between two annihilation operators (Eq. (2.208)), and between a creation and an annihilation operator (Eq. (2.217)). In order to define a Slater determinant in the formalism of second quantization, we introduce a vacuum state denoted by >. The vacuum state represents a state of the system that contains no electrons. It is normalized. 

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 formal similarities between the above treatment and the second quantized approach are obvious. The last result of Eq. (8.14) resembles very much to the second quantized representation of a one-electron operator, cf. Eq. (4.27), and the second quantized counterparts of all previous formulae can easily be identified. The correspondences that have been obtained so far are collected in Table 8.1. This shows that creation operators are analogs of ket functions, while annihilation operators correspond to fera-functions. The eigenprojector i>particle number operator Nj = aj does. The resolution of identity is analogous to the operator of the total number of particles. The... 

Besides the apparent similarities. Table 8.1 illustrates also the obvious formal differences between bras and kets and their second quantized counterparts. Namely, the corresponding symbols are mathematically very different. The bra and ket vectors are elements of a linear vector space over which quantum-mechanical operators are defined, while the creation and annihilation operators are defined over the abstract space of particle number represented wave functions serving as their carrier space. This carrier space leads to the concept of the vacuum state, which has no analog in the bra-ket formalism. Moreover, an essential difference is that the effect of second quantized operators depends on the occupancies of the one-electron levels in the wave function, since no annihilation is possible from an empty level and no electron can be created on an occupied spinorbital. At the same time, the occupancies of orbitals play no role in evaluating bra and ket expressions. Of course, both formalisms yield identical results after calculating the values of matrix elements. 

It is important to observe that the spin labels are not eliminated from the second quantized form of the Hamiltonian. They do not appear in the list of the integrals, however, which corresponds to the fact that the first quantized Hamiltonian is spin-independent and permits one to use the spin-free formalism. But it is essential to realize that creation and annihilation operators cannot be specified merely for spatial orbitals. 

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