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arXiv:1007.3112

Maximally entangled states in the Hydrogen molecule: The role of spin and correlation

Mohsen Babamoradi, Mehdi Heidari Saani, Mohammad Ali Vesaghi

Published Jul 19, 2010• 0 citations
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Abstract

By going beyond Hubbard Hamiltonian we reflected correlation effects accurately in the wavefunctions of H2. Using ab initio e-e interaction parameters resulted maximally entangled ground and third excited states. We assigned this maximally entangled character to the nonmagnetic (S=0) property of these states and also the minimally entangled character of the first excited states to its magnetic property. By switching on a magnetic field an entangled state with Sz = 0 can be extracted from a minimally entangled degenerate magnetic state. We suggest that presence of a moderate correlated system and a non-magnetic (Sz = 0) electronic state can be two criteria for finding maximally entangled electronic states in a realistic molecular system.

DOI
10.48550/arxiv.1007.3112

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By going beyond Hubbard Hamiltonian we reflected correlation effects accurately in the wavefunctions of H2. Using ab initio e-e interaction parameters resulted maximally entangled ground and third excited states. We assigned this maximally entangled character to the nonmagnetic (S=0) property of these states and also the minimally entangled character of the first excited states to its magnetic property. By switching on a magnetic field an entangled state with Sz = 0 can be extracted from a minimally entangled degenerate magnetic state. We suggest that presence of a moderate correlated system and a non-magnetic (Sz = 0) electronic state can be two criteria for finding maximally entangled electronic states in a realistic molecular system.
Many electron calculations on a simplest realistic two electron system i.e. H2molecule was applied and as the consequence correlation effects was reflected accurately in the wavefunctions of H2. Zanardi’s entanglement measurement, demonstrated that the maximum of entanglement for the ground state happens when U =J and this resolved the controversial conclusion of U = 0 for maximum entanglement. It was shown that the ground and third excited states are maximally entangled. These maximally entangled states and also the minimally entangled states are correlated to their spin’s property. The wavefunctions of the not magnetic (S = 0) ground and excited states explicitly depend on correlation parameters whereas the first excited states which is magnetic (S2 = 2 and Sz≠0) is not entangled. The second excited state is not magnetic but its wavefunction does not depend on correlation parameters therefore it is a moderately entangled state. In any case, by switching on a magnetic field an entangled state with Sz = 0 can be extracted from a not entangled degenerate magnetic state. We suggest that in a realistic molecular scale system, there is two criteria for finding maximally entangled electronic states, first the system should be in moderately correlated regime and second the system should have a non-magnetic (Sz = 0) electronic state.
Electron–electron correlation in quantum chemistry calculations can be analysed in terms of entanglement between electrons. Two exactly solvable models: two fixed spin-1/2 particles and two-electron two-site Hubbard model are used to define and discuss the entanglement as a function of the system parameters. Ab initio configuration interaction calculation for entanglement is presented for the H2 molecule. Qualitatively, entanglement and electron–electron correlation have similar behaviour. Thus, entanglement might be used as an alternative measure of electron correlation in quantum chemistry calculations.
An attempt is made to derive the parameters of the Hubbard Hamiltonian for the hydrogen molecule from state-of-the-art quantum chemistry calculations of the ground and excited states of the neutral and singly ionized molecules (eight states altogether). Calculations are carried out for interatomic distances in the range 0.6−−4.3A˚0.6--4.3\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}0.6−−4.3A˚. The results clearly indicate that the on-site repulsion UUU is not enough to fit the ab initio energies and that it is necessary to incorporate the intersite repulsion (usually referred to as JJJ). If JJJ is ignored, the fitted on-site repulsion approaches the atomic value very slowly as the interatomic distance is increased, a clearly unphysical behavior. For finite JJJ and at the equilibrium interatomic distance, the energies of the eight states mentioned above are reasonably fitted with an atomic orbital energy ${ϵ}_{0}\ensuremath{\approx}\ensuremath{-}24.6\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, Coulomb interaction parameters $U\ensuremath{\approx}11.6\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ and $J\ensuremath{\approx}8.9\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, and a hopping integral $t\ensuremath{\approx}7.9\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$.
ABSTRACT: In the present work, we have computed the entanglement between the electronic and nuclear motions in two molecular model systems: the one-dimensional hydrogen molecular ion (H2+) and the Shin-Metiu model, considering the molecules as a bipartite systems: electron and nuclear motion. For that purpose, we have computed the Born-Oppenheimer and non-Born-Oppenheimer (Born-Huang) wave function in terms of the Fourier Grid Hamiltonian basis that expands both the electronic and nuclear wave functions. Also, according to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, there is a much shorter but equivalent expansion in terms of the Schmidt bases for the electronic and nuclear sub-spaces. In these models of distinguishable coupled particles we have shown that the entanglements contents do not increase monotonically with the excitation energy. In the hydrogen molecular ion and in the ShinMetiu model, the entanglements contents for each Born-Oppenheimer electro-nuclear state is quantified through the von-Neumann and linear entropies and we have shown that entanglement serves as a witness of distinguishability of nuclear states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.
The electronic and magnetic correlations in diatomic molecules are studied rigorously on the basis of a Hubbard Hamiltonian. The problem is treated in three phases separately, firstly for electrons with nondegenerate orbitals, then for electrons with degenerate orbitals, and finally for a mixture of both. In this last case special emphasis is put on the interaction between the electrons in the nondegenerate orbital levels with those in the degenerate orbital levels. In this context some qualitative features of how electronic correlations influence the cohesive forces are sketched. We highlight the extent to which various factors such as the total number of electrons in the system, the Pauli principle, Hund's rule, the level splitting, and the relative size of the transfer integrals are influential in determining the ground-state configurations. Some simple physical guidelines are provided for accurately guessing the various possible ground-state configurations.
Electron correlation effects are essential for an accurate ab initio description of molecules. A quantitative a priori knowledge of the single- or multireference nature of electronic structures as well as of the dominant contributions to the correlation energy can facilitate the decision regarding the optimum quantum chemical method of choice. We propose concepts from quantum information theory as orbital entanglement measures that allow us to evaluate the single- and multireference character of any molecular structure in a given orbital basis set. By studying these measures we can detect possible artifacts of small active spaces.