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Progress of Theoretical and Experimental Physics

Application of the Dyson-type boson mapping for low-lying electron excited states in molecules

Sadao Ohkido, Masahide Takahashi

Published Jun 1, 2015• 0 citations
Publisher

Abstract

In many-electron systems, particularly when it is the case that the system consists of quasi-particles having almost no interaction with each other, the approaches to considering a quasi-particle to be a basic particle are important. We propose a bosonization method of the system in which a quasi-particle consists of electron–hole pairs. Low-lying electron excited states are approximately considered as the collective motion states of the quasi-particles that correspond to the quantized normal mode oscillations, and each of these quasi-particles consists of the electron–hole pairs. We consider the fundamental model Hamiltonian, and transform this into the quasi-particle representation Hamiltonian. Then, we transform these quasi-particles into bosons by using the Dyson-type boson mapping method. The obtained Dyson-type boson Hamiltonian enables a straightforward interpretation of the low-lying electron excited states and facilitates their calculation. To confirm the usefulness of this method, we apply this to the |$\pi$|-electrons system of the ethylene molecule.

DOI
10.1093/ptep/ptv068

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This Brief Report deals with the solution of the shell-model Hamiltonian in the boson representation. In the present approach, the fermion Hamiltonian is exactly mapped to the boson space using the Dyson boson mapping. The problem of mapping the basis states is resolved by employing the intrinsic boson basis. The method has been applied to a model Hamiltonian consisting of a monopole pairing and a deformation force. It is observed that only s and d bosons are dominant in the intrinsic wave function for the model case studied. This is synonymous to the interacting boson model approach.
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The nonunitary character of the Dyson boson mapping is explored in a model case with the monopole pairing interaction. It is shown that because of the very special form of this interaction, the calculations in the intrinsic state formalism with the particle-particle representation of the Hamiltonian reproduce the exact results of the original fermion space. In contrast, we show that with the particle-hole representation of the Hamiltonian the intrinsic state approach produces erroneous results. Of course, upon employing the physical boson states we recover the exact results in the particle-hole represenation as well.
We propose an approximate mapping of a molecular Hamiltonian to a Hamiltonian of qubits, which allows for high accuracy quantum chemistry calculations of vertical excitation energies of some molecules. The mapping is based on separating of a very small active space of only two orbitals and on modeling the electronic excitations in the remaining orbitals by a set of qubits or, equivalently, by a set of oscillators. This approach is inspired by the Random Phase Approximation (RPA), in which the excitations of electron gas are described by bosonic degrees of freedom. As a result, the Hamiltonian of the molecule is reduced to that of a system-bath model. The "system" part of the Hamiltonian describes the two molecular orbitals -- the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) -- which are populated by two electrons. Two qubits are sufficient to encode the Hamiltonian of such a system. The "bath" consists of oscillators or, equivalently, of two level systems with each of them corresponding to an electron excitation from a doubly occupied orbital below the Fermi level to an empty orbital above the Fermi level. We hope that this mapping can inspire new approaches and algorithms aimed at calculating excitation energies of molecules on near term quantum computers.