Second-quantization formalism

The notation concerns are easily overcome by the following simple construct bearing the name of second quantization formalism.21 Let us consider the space of wave functions of all possible numbers of electrons and complement it by a wave function of no electrons and call the latter the vacuum state vac). This is obviously the direct sum of subspaces each corresponding to a specific number of electrons. It is called the Fock space. The Slater determinants eq. (1.137) entering the expansion eq. (1.138) of the exact wave function are uniquely characterized by subsets of spin-orbitals K = k,, k2. fc/v which are occupied (filled) in each given Slater determinant. The states in the list are the vectors in the carrier space of spin-orbitals (linear combinations of the functions of the (pk (x) = f ma (r, s) basis. We can think about the linear combinations of all Slater determinants, may be of different numbers of electrons, as elements of the Fock space spanned by the basis states including the vacuum one. 

21For a detailed description of second quantization in the context of Quantum Chemistry see [39-41] 

All the described features of how the creation and annihilation operators act on the Slater determinants constructed from the fixed basis of spin-orbitals are condensed in the set of the anticommutation rules  

The above construct is known as second quantization formalism. 

Much more important than the possibility of expressing the Slater determinants in terms of creation operators is the possibility of expressing all the operators acting upon the electron states in terms of the Fermi operators. We are not going to present the formal construct here (it is well described in books [39 11]), rather we are going to explain the situation. 

The antisymmetry of electronic fermion wave functions can lead to a cumbersome notation, especially when describing many-electron systems. In the method of second-quantization, only the occupied single particle levels are specified together with their occupation numbers. This is the only information which is compatible with the indistinguishability of electrons. 

The wave function of a many-electron system can be written as a Slater determinant (see App. B). However, a more convenient notation is provided by using the occupation number representation, whereby the A-electron de-terminantal function takes the form (March et al. 1967, Raimes 1972) 

Equation (4.1) is sometimes referred to as a state vector in Fock space and its use requires that the Hamiltonian be expressed in terms of operators that can act on such vectors. 

---

Where H contains all terms in (16) except (19). In eq. (16) all quantities were defined through the Cartesian coordinates. For further purposes it will be natural to work in normal coordinates Br. The normal coordinate in second quantized formalism is given as... 

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... 

A full account of the theory of relativistic molecular structure based on standard QED in the Furry picture will be found in a number of publications such as [7, Chapter 22], [8, Chapter 3]. These accounts use a relativistic second quantized formalism. For present purposes, it is sufficient to present the structure of BERTHA in terms of the unquantized effective Dirac-Coulomb-Breit (DCB) A-electron Hamiltonian ... 

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... 

The discussion in the previous sections dealt with the second quantization formalism for orthonormal spinorbitals. In this section, we will generalize the formalism to treat cases where the set of spin orbitals is non-orthogonal. Consider a set of n spin orbitals with the general metric... 

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. 

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

Second Quantization Photodissociation Hamiltonian, if we consider a system containing many molecules and fragments, it is convenient to use second quantization formalism. We have introduced above the matrix element for photodissociation (see eqs. 50 and 53-57). Based on it, one can write the total... 

Thus, Hj q describes directly the transition from the quasidiscrete state to fragments. According to second quantization formalism, it acts in an occupation-number space. 

One can introduce the tunneling Hamiltonian in a second quantization formalism (39) which can be written in the form... 

The other principal feature is the use of second-quantization formalism ... 

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. 

The latter of these two approximations cannot be justified. Through using second quantization formalism [82] Mayer showed that halving is a consequence of the LCAO approximation [83]. However, it does not mean that the second approximation is valid and the MP charges are the true atomic charges within the LCAO approximation. 

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 ... 

The whole problem of calculating fc/ (at least up to the third order) is now reduced to the calculation of individual terms (55)—(58). The second quantization formalism has the advantage that these terms can be calculated easily by making use of the diagrammatic technique which will be demonstrated in Section IV.B. 

We use the second quantization formalism to express the second order contribution to the correlation energy of the closed-shell ground state. By substituting the expression (67) for W in Eq. (72) we obtain... 

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

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, .. 

The second quantization formalism also greatly simplifies the treatment of the Hamiltonian and allows its analysis pertinent to the GF approximation for the wave function.23 Indeed, the total electron Hamiltonian H can be rewritten using the second quantization formalism according to the division of the orbital basis set into carrier subspace basis sets as introduced above ... 

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 ... 

---