Hamiltonian, second-quantized groups

An approach to constructing CSFs and matrix elements of the Hamiltonian that initially appears quite different from the symmetric group approach can be developed by considering the second-quantized form of the Hamiltonian. If we have an orthonormal... 

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,... 

Successful model building is at the very heart of modern science. It has been most successful in physics but, with the advent of quantum mechanics, great inroads have been made in the modelling of various chemical properties and phenomena as well, even though it may be difficult, if not impossible, to provide a precise definition of certain qualitative chemical concepts, often very useful ones, such as electronegativity, aromaticity and the like. Nonetheless, all successful models are invariably based on the atomic hypothesis and quantum mechanics. The majority, be they of the ah initio or semiempirical type, is defined via an appropriate non-relativistic, Born-Oppenheimer electronic Hamiltonian on some finite-dimensional subspace of the pertinent Hilbert or Fock space. Consequently, they are most appropriately expressed in terms of the second quantization formalism, or even unitary group formalism (see, e.g. [33]). 

The Hamiltonian is assumed to be spin-independent. It can then be written, in second quantized form, in terms of the spin-averaged excitation operators (the generators of the unitary group )... 

In this section we shall marginally discuss two selected topics of quantum chemistry emphasizing the points where second quantization is useful. First in Sect. 17.1 the second quantized representation of spin operators will be introduced and they will be applied to construct a model Hamiltonian. Section 17.2 will explain the connection between second quantization and the unitary group approach. 

These time-reversal reductions affect the expressions for the second-quantized Kramers-restricted Hamiltonian. Both forms given in chapter 9 contain integrals with an odd number of bars, and in both, these terms vanish if the group has no quaternion irreps. If we want to use the Hamiltonian as expressed in terms of the excitation operators Ep and such as in a coupled-cluster calcula-... 

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