Quantization, second

It is convenient to use the language of second quantization when treating a general many-body problem. Most quantum chemists are familiar with this formalism, and here we will only provide some brief reminders of the basic features. More thorough 

The simplest state is the state with no particles, the vacuum state, which we denote vac). This state is assumed to be normalized  

We can generate the other states in the Fock space by use of creation operators. The creation operator increases the number of particles in state i by 1  

The general state in Fock space may be constructed by applying a succession of creation 

Application of an annihilation operator to the vacuum state yields 0, and by Hermitian conjugation we must have 

When we work with a basis set composed of Slater determinants we are usually confronted with a large number of matrix elements involving one- and two-electron operators. The Slater-Condon rules (Appendix M) are doing the job to express these matrix elements by the one-electron and two-electron integrals. However, we may introduce an even easier tool called second quantization, which is equivalent to the Slater-Condon rules. 

In the second quantization formalism we introduce a reference state for the system under study, which is a Slater determinant (usually the Hartree-Fock wave function) composed of N orthonormal spinorbitals, where N is the number of electrons. This function will be denoted in short by o or in a more detailed way by F (ni, , oo)- The latter notation means that we have to do with a normal- 

Let us make quite a strange move, and consider operators that change the number of electrons in the system. To this end, let us define the creation operator of the electron going to occupy spinorbital k and the annihilation operator of an electron leaving spinorbital k  

Richard Feynman, in one of his books, says jokingly that he could not understand the very sense of the operators. If we annihilate or create an electron, then what about the system s electroneutrality Happily enough, these operators will always act in creator-annihilator pairs. 

The symbol 1 means that the spinorbital k is present in the Slater determinant, while Ofe means that this spinorbital is empty, i.e. is not present in the Slater determinant. The factors (1 — and ensure an important property of these operators, namely that 

In contrast to earlier approaches, we do not use the electrostatic interaction potential between electrons located at different particles and a perturbation theory with respect to free electrodynamic modes. We rather equate the electric interaction potential to zero and attribute the interaction exclusively to the transverse electric and magnetic modes. In 

The Hamiltonian describing this electron-photon interaction is conveniently given in terms of second quantization 

The first term in Eq. (8.1) represents the energy of the electrons. The operators 4 and c, are the creation and aimihilation operators for Fermions. By applying the product 4 Cj to an arbitrary many electron orbital, we obtain the occupation numter /j = 0,1 of state i , the application of the commutated product Ci4 renders 1 —/ . 

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Hamiltonian in the second-quantization fomi, only one //appears in this fmal so-called equation of motion (EOM) f//, <7/]+ = AJr 7 p(i e. in the second-quantized fomi, // and //are one and the same). 

One can, for example, express T in temis of a superposition of configrirations = Y.jCj whose amplitudes Cj have been detemiined from an MCSCF, Cl or MPn calculation and express Q in temis of second-quantization operators Offt that cause single-, double-, etc, level excitations (for the IP (EA)... 

J0rgensen P and Simons J 1981 Second Quantization-Based Methods in Quantum Chemistry (New York Aoademio) J0rgensen P and Simons J (eds) 1986 Geometrical Derivatives of Energy Surfaces and Molecular Properties (Boston, MA Reidel)... 

J0rgensen P and Simons J 1981 Second Quantization Based Methods in Quantum Chemistry (New York Academic) oh 4... 

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. 

The Hamilton operator (eq. (C.l)) in second quantization is given as (note that the summation now is over basis functions)... 

There are several advantage of second quantization over first quantization. 

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. 

Second quantization and configuration space description of spin 0 particles Schweber, S. S., An Introduction to Relativistic Quantum Field Theory, Harper and Row, New York, 1961. 

Second Quantized Description of a System of Noninteracting Spin Particles.—All the spin particles discovered thus far in nature have the property that particles and antiparticles are distinct from one another. In fact there operates in nature conservation laws (besides charge conservation) which prevent such a particle from turning into its antiparticle. These laws operate independently for light particles (leptons) and heavy particles (baryons). For the light fermions, i.e., the leptons neutrinos, muons, and electrons, the conservation law is that of leptons, requiring that the number of leptons minus the number of antileptons is conserved in any process. For the baryons (nucleons, A, E, and S hyperons) the conservation law is the... 

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. 

These ideas can be applied to electrochemical reactions, treating the electrode as one of the reacting partners. There is, however, an important difference electrodes are electronic conductors and do not posses discrete electronic levels but electronic bands. In particular, metal electrodes, to which we restrict our subsequent treatment, have a wide band of states near the Fermi level. Thus, a model Hamiltonian for electron transfer must contains terms for an electronic level on the reactant, a band of states on the metal, and interaction terms. It can be conveniently written in second quantized form, as was first proposed by one of the authors [Schmickler, 1986] ... 

In the second quantization representation, the Hamiltonian Hl describing the motion of the reactive -oscillator in the left potential well has the form... 

The Russian school of ETR (Levich, 1966 Dogonadze, 1971 Vorotyntsev et al, 1970) treats the medium polarization by a second-quantized Hamiltonian, written as... 

For anything bigger than the hydrogen atom, however, solving directly in terms of the coordinates and momenta becomes extremely difficult. Far more common is to express the wave function in terms of basis functions, introducing the idea of second quantization [45], A simple way to think of second quantization is that it describes the quantum mechanics, from the beginning, in terms of a set of basis functions. 

Here the operator af creates (and the operator a, removes) an electron at site i the nn denotes near-neighbors only, and /i,y = J drr/),/l(j)j denotes a Coulomb integral if i = j and a resonance integral otherwise. The second quantization form of this equation clearly requires a basis set. It is a model for the behavior of benzene - not a terribly accurate one, but one that helps us understand many things about its spectroscopy, its stability, its binding patterns, and other physical and chemical properties. 

Jprgensen P, Simmons J (1981) Second quantization-based methods in quantum chemistry. Academic, New York... 

In order to obtain the particle description required for quantum statistics, it may therefore be necessary to quantize the quantum-mechanical wave field a second time. This procedure, known as second quantization, starts from the wave field once quantized ... 

Second quantization transforms the Schrodinger particle density into a particle density operator,... 

The fact that every state may be occupied by several particles shows that the second quantization particles are bosons. However, in terms of different commutation relations an equivalent scheme may be obtained for fermions. To achieve this objective the wave functions are written in decomposed form as before ... 

The density matrix is defined in second quantization as an operator by an equation... 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

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

In (4.28) and (4.30), we have achieved our aim of expressing the Hamiltonian in the appropriate second quantized form for acting on the state vectors in Fock space. 

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