Second-quantized equations

C. L. anssen and H. F. Schaefer, Theor. Chim. Acta, 79,1 (1991). The Automated Solution of Second Quantization Equations with Applications to the Coupled-Cluster Approach. 

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). 

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. 

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

An elegant proof, in second quantization, that the 2-CSE imphes the Schrodinger equation, was given by Mazziotti and is as follows. 

Replacing into this equation the by its second quantization expression... 

While early work [16, 19] on the CSE assumed that Nakatsuji s theorem [37], proved in 1976 for the integrodifferential form of the CSE, remains valid for the second-quantized CSE, the author presented the first formal proof in 1998 [20]. Nakatsuji s theorem is the following if we assume that the density matrices are pure A-representable, then the CSE may be satisfied by and if and only if the preimage density matrix D satisfies the Schrodinger equation (SE). The above derivation clearly proves that the SE imphes the CSE. We only need to prove that the CSE implies the SE. The SE equation can be satisfied if and only if... 

By Eq. (6) the sum on the right-hand side of the above equation is equal to the energy E, and from Eq. (2) we realize that the sums on the left-hand side are just Hamiltonian operators in the second-quantized notation. Hence, when the 2-RDM corresponds to an A -particle wavefunction i//, Eq. (12) implies Eq. (13), and the proof of Nakatsuji s theorem is accomplished. Because the Hamiltonian is dehned in second quantization, the proof of Nakatsuji s theorem is also valid when the one-particle basis set is incomplete. Recall that the SE with a second-quantized Hamiltonian corresponds to a Hamiltonian eigenvalue equation with the given one-particle basis. Unlike the SE, the CSE only requires the 2- and 4-RDMs in the given one-particle basis rather than the full A -particle wavefunction. While Nakatsuji s theorem holds for the 2,4-CSE, it is not valid for the 1,3-CSE. This foreshadows the advantage of reconstructing from the 2-RDM instead of the 1-RDM, which we will discuss in the context of Rosina s theorem. 

Among the several 2-RDM-oriented methods that have been developed for the study of chemical systems, one of the most recent and promising techniques is based on the iterative solution of the second-order contracted Schrodinger equation (2-CSE) [1, 6, 15, 18, 36, 45-60, 62-65, 68, 70, 79-85, 103-111]. The 2-CSE was initially derived in 1976 in first quantization in the works of Cho [103], Cohen and Erishberg [104, 105], and Nakatsuji [106] and later on deduced in second quantization by Valdemoro [45] through the contraction of... 

The Hamiltonian matrix in Equation (15) is obtained from appropriate products of representations of second-quantized operators that act within the left block, right block, or partition orbital. For example, in the case of where... 

Upon doing so, the following set of equations is obtained (early references to the derivation of such equations include A. C. Wahl, J. Chem. Phys. 4T, 2600 (1964) and F. Grein and T. C. Chang, Chem. Phys. Lett. 12, 44 (1971) a more recent overview is presented in R. Shepard, p 63, in Adv. in Chem. Phys. LXIX, K. P. Lawley, Ed., Wiley-Interscience, New York (1987) the subject is also treated in the textbook Second Quantization Based Methods in Quantum Chemistry, P. Jprgensen and J. Simons, Academic Press, New York (1981))) ... 

In order to be able to write out all the terms of the direct Cl equations explicitly, the Hamiltonian operator is needed in a form where the integrals appear. This is done using the language of second quantization, which has been reviewed in the mathematical lectures. Since, in the MR-CI method, we will generally work with spin-adapted configurations a particularly useful form of the Hamiltonian is obtained in terms of the generators of the unitary group. The Hamiltonian in terms of these operators is written,... 

Equations (13.22) and (13.23) define the second-quantization form of an operator corresponding to a physical quantity, if its matrix elements are known in coordinate representations ((13.24) and (13.25)). Specifically, the operator of the total number of particles in a system will be... 

Expanding the wave function in a linear combination of pure spin functions could yield the correct secular equations and thus correct eigenvalues. However, such spin-only wave functions could not be considered complete since complete wave functions must describe both the spatial and spin motions of electrons and must be antisymmetric under exchange of any two electrons. It would be better to rewrite the VB model (18) in the second quantization form as given in Eq. (20), in which its eigenstates can be taken as a linear combination of Slater determinants or neutral VB structures. Then... 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

This general notation is deceptively simple. The bra is an excited determinant. There is an equation for each excited determinant, and each level of excitation leads to a different type of equation. Furthermore, the equations are all coupled, and they are non-linear in the amplitudes. However, they may be formulated in a quasilinear manner [27], and they have been solved for a wide range of CC schemes. The operator HN is the Hamiltonian written in second-quantized form minus the energy of the reference determinant, i.e. HN = H— < 0 /7 0 >. The subscript C restricts the operator product of HN and eT to connected terms. Once the CC equations have been solved, the CC correlation energy can be calculated from... 

The Hubbard picture is the most celebrated and simplest model of the Mott insulator. It is comprised of a tight-binding Hamiltonian, written in the second quantization formalism. Second quantization is the name given to the quantum field theory procedure by which one moves from dealing with a set of particles to a field. Quantum field theory is the study of the quantum mechanical interaction of elementary particles with fields. Quantum field theory is such a notoriously difficult subject that this textbook will not attempt to go beyond the level of merely quoting equations. The Hubbard Hamiltonian is ... 

Bucci and his coworkers (65,66) treated double resonance by means of a second quantization of the rf fields to avoid the use of the rotating frame. The total Hamiltonian 2 - is then given by equation (1) in which is the normal Hamiltonian of the isolated spin system in the absence of radiation but in the magnetic field, is the radiation Hamiltonian, and represents the interaction between the spins and the radiation field(s). 

In spite of the method s present utility and popularity, the quantum chemical community was slow to accept coupled cluster theory, perhaps because the earliest researchers in the field used elegant but unfamiliar mathematical tools such as Feynman-like diagrams and second quantization to derive working equations. Nearly 10 years after the essential contributions of Paldus and Cizek, Hurley presented a re-derivation of the coupled cluster doubles (CCD) equa-... 

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