Configuration interaction second quantization

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... 

The operator of the energy of electrostatic interaction of electrons in (14.65) is represented as a sum of second-quantization operators, and the appropriate submatrix element of each term is proportional to the energy of electrostatic interaction of a pair of equivalent electrons with orbital Lu and spin S12 angular momenta. The values of these submatrix elements are different for different pairing states, since, as follows from (14.66), the two-electron submatrix elements concerned are explicitly dependent on L12, and, hence, implicitly - on S12 (sum L12 + S12 is even). It is in this way that, in the second-quantization representation for the lN configuration, the dependence of the energy of electrostatic interaction on the angles between the particles shows up. This dependence violates the central field approximation. 

In order to calculate the spin-angular parts of matrix elements of the two-particle operator (1) with an arbitrary number of open shells, it is necessary to consider all possible distributions of shells upon which the second quantization operators are acting. In [2] they are found to be grouped into 42 different distributions, subdivided into 4 different classes. This also explains why operator (1) is written as the sum of four complex terms. The first term represents the case when all second-quantization operators act upon the same shell (distribution 1 in [2]), the second describes the situation when these operators act upon the two different shells (distributions 2-10), third and fourth are in charge of the interactions upon three and four shells respectively (distributions 11-18 and 19-42). Such expression is particularly convenient to take into account correlation effects, because it describes all possible superpositions of configurations for the case of two-electron operator. 

We now introduce creation and annihilation operators ajj and an which create/annihilate e-h pairs at a given combination of sites n = (n, n1), i.e., 41°) = 14 = nen h), where 0) is the ground state. Using these operators, a generic monoexcitation configuration interaction Hamiltonian can be formulated as follows in second quantization notation,... 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

In order to have a more complete picture of the many-body problem for more general or complicated cases that DFT could help to treat, it is necessary to make a correspondence with the use of many-body perturbation theory. Under this wider classification of perturbation theory are included all the methods that treat electron correlation beyond the Hartree-Fock level, including configuration interaction, coupled cluster, etc. This perturbational approach has traditionally been known as second quantization, and its power for some applications can be seen when dealing with problems beyond the standard quantum mechanics. 

The incorporation of electron correlation effects in a relativistic framework is considered. Three post Hartree-Fock methods are outlined after an introduction that defines the second quantized Dirac-Coulomb-Breit Hamiltonian in the no-pair approximation. Aspects that are considered are the approximations possible within the 4-component framework and the relation of these to other relativistic methods. The possibility of employing Kramers restricted algorithms in the Configuration Interaction and the Coupled Cluster methods are discussed to provide a link to non-relativistic methods and implementations thereof. It is shown how molecular symmetry can be used to make computations more efficient. 

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. 

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