Second quantization formalism operators

In second-quantization formalism, operators in coordinate space are replaced by operators defined in the space of occupation numbers... 

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... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

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. 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

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. 

The action of these operators on the expressions in Table 2 is governed by the set of commutation rules, specified in Eq. 3. Their raising or lowering character is made particularly transparent by the second-quantized formalism. As an example the + operator is seen to consist of a mi, a m. parts as these parts run through the operator sequences of the (t)3 multiplets they destroy a P spin on the l mr orbital and create an a spin on the same orbital, thus raising Ms by one. 

Wick s theorem (35) which gives us the prescription for treating a product of operators may of course be applied to the Hamiltonian, expressed in the second quantization formalism (29). This leads to the Hamiltonian in a form which is of primary importance in perturbation treatments. This form of the Hamiltonian which is called the normal product form is ... 

In the second quantization formalism the components of the T operator (104) may be expressed as follows... 

The term E - <4>0IHI 0) is the correlation energy of the closed shell ground state. As was shown in Section III the two-particle operator T2 can be expressed in the second quantization formalism as... 

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, .. 

Closely inspecting the operator terms entering the electronic Hamiltonian eq. (1.27) one can easily see that they are sums of equivalent contributions dependent on coordinates of one or two electrons only. Analogously in the second quantization formalism only the products of two and four Fermi operators appear in the Hamiltonian. Inserting the trial. Y-electron wave function of the (ground) state into the expression for the electronic energy yields its expectation value in terms of the expectation values of the one- and two-electron operators ... 

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... 

There have been a number of means proposed for circumventing superposition error. Mayer et al. advocated what they term a chemical Hamiltonian approach, which separates the physical part of this operator from that responsible for BSSE using a nonorthogonal second quantization formalism. However, the physical Hamiltonian is no longer variational and the wavefunction is constructed from orthonormalized molecular spin orbitals. Surjan et al. " further developed this approach and performed pilot applications on small complexes. 

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... 

In a given orbital basis, the Hartree-Fock description divides the orbital space into a set of occupied and virtual spin orbitals. From the Slater determinant any other determinant may be generated by replacing an occupied orbital by a virtual. Formally such operation is performed within the second quantization formalism by using the excitation operators... 

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. ... 

The expression for the matrix elements F y can be derived in the second quantized formalism in an elegant manner. This derivation relies on the physical picture behind the Hartree-Fock approximation. As known, in this model the electrons interact only in an averaged manner, so correlational effects are excluded. To derive such an averaged operator, we start again with the usual Hamiltonian ... 

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. 

If systems A and B are really isolated, their dynamical variables are independent and the corresponding fermion operators commute. Such a second-quantized formalism was used by Chang and Weinstein (1978). 

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. 

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... 

As said above, pd are the marginal distributions of the whole distribution D. To obtain them, the contraction mapping (CM) operation may be performed on D in order to reduce the number of variables from a fixed M number of particles to p, i.e., the order of contraction [26, 41]. In order to define this operation for the GC distribution which has no fixed number of particles, let us first sketch it for the MC and C distributions. For this goal, we introduce the p-RDMs in terms of the p-order replacement operators pe [42] in the second quantization formalism [43]... 

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) ... 

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... 

In Box 1.1, we summarize the fundamentals of the second-quantization formalism. In Section 1.4, we proceed to discuss the second-quantization representation of standard first-quantization operators such as the electronic Hamiltonian. 

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)... 

We have thus far only considered the relativistic quantum mechanical description of a single spin 0, mass m particle. We next turn to the problem of describing a system of n such noninteracting spin 0, mass m, particles. The most concise description of a system of such identical particles is in terms of an operator formalism known as second quantization. It is described in Chapter 8, The Mathematical Formalism of Quantum Statistics, and Hie reader is referred to that chapter for detailed exposition of the formalism. We here shall assume familiarity with it. 

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