Quantum chemistry second quantization formalism

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. 

The eigenfunctions of the zeroth order Hamiltonian define the projection of the DCB equation onto the subspace of electronic solutions. This is a first and necessary step to apply QED theory in quantum chemistry. The resulting second quantized formalism is compatible with the non-relativistic spin-orbital formalism if the connection (unbarred spinors <-> alpha-spinorbitals) and (barred spinors beta spinorbitals) is made. This correspondence allows transfer to the relativistic domain of non-relativistic algorithms after the differences between the two formalism are accounted for. 

Up to this point we have tailored the second-quantization formalism in close connection to the independent-particle picture introduced before. However, the formalism can be generalized in an even more abstract fashion. For this we introduce so-called occupation number vectors, which are state vectors in Fock space. Fock space is a mathematical concept that allows us to treat variable particle numbers (although this is hardly exploited in quantum chemistry see for an exception the Fock-space coupled-cluster approach mentioned in section 8.9). Accordingly, it represents loosely speaking all Hilbert spaces for different but fixed particle numbers and can therefore be formally written as a direct sum of N-electron Hilbert spaces. 

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. ... 

As it is well known proper many body methods including Feynman diagrammatic techniques, developed in elementary particle physics, were transferred to solid-state physics many years ago. The introduction to quantum chemistry followed later, but only on the electronic level. So the question then appears Is it possible to formulate the full quantum chemical electron-vibrational Hamiltonian in a second quantization formalism The answer is negative. In fact the author did spend many years attempting to construct ideal representations by means of appropriate quasiparticle transformations (cf. equivalent FrOhlich type unitary transformations), but all variants, being either adiabatic- or nonadiabatic representations, did indeed fail. The reason lies actually on a deeper level than one would initially imagine. 

The basic theory of second quantization is found in most advanced textbooks on quantum mechanics but inclusion of relativity is not often considered. A good introduction to this topic is given by Strange [10] in his recent textbook on relativistic quantum mechanics. We will basically follow his arguments but make the additional assumption that a finite basis of Im Kramers paired 4-spinors is used to expand the Dirac equation. This brings the formalism closer to quantum chemistry where use of an (infinite) basis of plane waves, as is done in traditional introductions to the subject, is impractical. 

Chapter 2 introduces the basic techniques, ideas, and notations of quantum chemistry. A preview of Hartree-Fock theory and configuration interaction is used to motivate the study of Slater determinants and the evaluation of matrix elements between such determinants. A simple model system (minimal basis H2) is introduced to illustrate the development. This model and its many-body generalization N independent H2 molecules) reappear in all subsequent chapters to illuminate the formalism. Although not essential for the comprehension of the rest of the book, we also present here a self-contained discussion of second quantization. 

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... 

The matrix element in Eq. (1.4) is called a bracket. Accordingly, 1 is called a hra-function or fera-vector, while D2> is the fe t-vector. A large body of quantum chemistry can also be developed by using the abstract form of bra- and fc t-vectors, and in fact, there is some similarity between this formalism and that of second quantization. This point will be discussed in Sect. 8 in some detail. 

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). 

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