Second quantization approach

We introduce the creation/annihilation operators for electrons such that cf creates an electron in state i, i.e., 

Thus we see that the result is zero if the state is occupied, i.e., if n, = 1. The phase factor has to do with the requirement of antisymmetry of the wavefunction for Fermi-ions, i.e., the ordering of the states plays an important role. Likewise we have 

The Hamiltonian for the molecule-surface electron system is expressed as 

In second quantization language we have the following hamiltonian [247, 156, 248] for the problem 

In order to evaluate the matrix elements Hij, it is convenient to expand the interaction potential as 

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P. R. Surjan. Second Quantized Approach to Quantum Chemistry An Elementary Introduction, Springer-Verlag, New York, 1989. 

Surjan PR (1989) Second quantized approach to quantum chemistry. Springer-Verlag, Berlin... 

Moszynski R, Jeziorski B, Szalewicz K (1994) Many-body theory of exchange effects in intermolecular interactions. Second-quantization approach and comparison with full configuration interaction results. J Chem Phys 100 1312-1325... 

In this chapter it is convenient to use both the second quantization approach and the traditional wavefunction approach, depending on the particular discussion. In the opinion of the author, some aspects of the MCSCF method are best understood in terms of a traditional wavefunction-oriented approach while other aspects are best understood in terms of the algebraic manipulations of operators for which the method of second quantization is best suited. This notation will be briefly reviewed so that the... 

In order to develop computationally manageable expressions for the RMEs, second quantization form of the SOMF operator is usually employed in actual calculations. As described elsewhere [112], the SOMF operator in the second quantization approach written in the basis of the one-electron orbitals p, q used to construct the CSFs is given by ... 

Finally, a formalism for calculating spin-orbit coupling (SOC) effects in organic molecules based on Riuner spin eigenfunctions and the second quantization approach has been derived and implemented for the semiempirical MNDOC-CI model. It allows for a straightforward determination of SOC surfaces within the context of MR-CI calculations at the same level of theory as PESs. 

Approximate analytical methods, based on a second quantization approach, have been developed by HOPAGKER and LEVINE /93/, making use of curvilinear coordinates, to describe the non-adiabatic transitions leading to a population inversion of the products vibrational states. 

Surjan PR (1989) Second quantized approach to quantum chemistry. Springer, Berlin Tachibana A (1987) Int J Quantum Chem 21 181-190 Tachibana A, Parr RG (1992) hit J Quantum Chem 41 527-555... 

We may interpret the terms in eqn (2.19) as follows. The first term on the right-hand side represents the transfer of an electron from the spin-orbital Xj(r, ct) to the spin-orbital Xi(r, vice versa), with an energy scale The terms i = j in the sum represent the single-particle on-site energy, while the other terms represent the hybridization of the electrons between different orbitals. The second term on the right-hand side represents electron-electron interactions, the most important being the direct Coulomb interaction when i = j and k = I, as we discuss in Section 2.6. For readers not famihar with the second quantization approach, Appendix A describes a first quantization representation of the first term on the right-hand side of eqn (2.19). 

For readers not familiar with the second quantization approach. Appendix A describes a first quantization representation and solution of the eqn (3.1). 

Among possible approaches, the so-called second quantization plays an important role. The ultimate goal of the second quantized approach to the many-electron problem is to offer a formalism which is substantially simpler than the traditional one in many cases. As a matter of fact, most difficulties of the traditional or first quantized approach arises from the Pauli principle which requires the wave function W of Eq. (1.1) to be antisymmetric in the electronic variables. This is an additional requirement which does not result from the Schrodinger equation and requires a special formalism the using of Slater determinants for constructing appropriate solutions to Eq. (1.1). The Slater determinant is not a very pictorial mathematical entity, and the evaluation of matrix elements over determinantal wave functions makes the first quantized quantum chemistry somewhat complicated for beginners. In the second quantized... 

Physically measurable quantities in quantum mechanics are associated with expectation values or matrix elements of the corresponding operators. It is therefore of fundamental importance to elaborate efficient methods for the calculation of such matrix elements. One of the most attractive features of the second quantized approach is its simplicity in evaluating matrix elements. To appreciate this, one must get some practice in the formalism. 

Throughout the previous sections we become acquainted with the basic notations and rules of second quantization. It is time now to utilize the benefits of this formalism. In this section the usefulness of the second quantized approach will be illustrated from the practical point of view on a few selected simple examples. In fact, this formalism can only be appreciated by those who really use it in practice. It is hoped that the following development and examples will help the reader to appreciate the beauty and power of this formalism and to give him/her the possibility to apply it in his/her work. Before turning to the concrete examples, some general remarks are made in order to put the second quantized formalism to its proper place. Some of the following remarks have already been noted before, but it appears to be useful to collect here the most important points. 

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... 

This section is devoted to give an overview on the second quantized forms of various model Hamiltonians used extensively in the everyday practice of quantum chemistry and theoretical solid-state physics. In many scientific publications different quantum chemical models and approximations are introduced or defined by means of the second quantized approach. These models might be as simple as the Hiickel model, for example. Quite often no specific features of second quantization are utilized, but this formalism is used as a convenient language to define various model Hamiltonians. It seems to be useful therefore to review the most frequently applied model Hamiltonians. For further reading we refer to the brief monograph by Del Re et al. (1980). A simple description of the semiempirical schemes discussed below, not using second quantization, can be found in Naray et al. (1987). 

In the preceding chapters we have seen the advantages of the second quantized approach rather from the formalistic point of view. Here we take another standpoint and show an example where second quantization involves a somewhat different interpretation of the results. The example will be somewhat peculiar (Surjan et al. 1988) we shall study the significance of the Hellmann-Feynman theorem in evaluating the first derivatives of the energy, which is topical in the field of optimizing molecular geometries, exponents of basis orbitals, etc. 

Here E and E are the exact energies of the two individual molecules A and B when they are isolated, while E" is the exact energy of the supersystem (molecular complex, for example). Theoretically, these quantities can be obtained from the exact solution of the Schrodinger equation for the corresponding systems. (We remain within the nonrelativistic Born-Oppenheimer model.) This requires the definition of the Hamiltonians H", H and H" , and one feels challenged to handle these Hamiltonians in a common (e.g., perturbational) scheme. This point is not at all trivial especially if approximate model Hamiltonians are used. In what follows we shall consider this issue emphasizing the points where the second quantized approach can help to clarify the situation. 

Second quantized approach to quantum chemistry an elementary introduction with 11 figures / Peter R. Surjan. p. cm. 

Again to save space, some other related methods such as Green s function formalism or the diagrammatic perturbation theory, which are usually treated with second quantization on an equal footing, are not presented here. Merely the second quantized approach (particle number representation) will be elaborately discussed. However, a short review of some recent developments partly connected to the author s own work is included to illustrate the value and actuality of second quantization. 

The author is convinced that second quantization is the simplest approach to the many-electron problem, and it should be available not only for physicists but for every chemist engaged in quantum chemistry. If this does not come about from the present treatment, it is the fault of the presentation and not of the second quantized approach. 

The reaction volume hamiltonian can be used for large systems in which the reaction is a three center reaction. For such a system it will be advantageous to treat the SN — 9 perpendicular vibrations within the second quantization approach. By minimizing the gradient we define a reference volume in which all reaction following motions as e.g. umbrella and other concerted motion is taken care of through the reference position p). Small am-... 

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