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Journal of Modern Physics

Maximally Entangled States in the Hydrogen Molecule: The Role of Spin and Correlation

Mohammad Ali Vesaghi, Mohsen Babamoradi, Mehdi Heidari Saani

Published Jan 1, 2011• 1 citation
Publisher

Abstract

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.

DOI
10.4236/jmp.2011.27079

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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.
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.
We theoretically investigate the quantum correlations (in terms of concurrence of indistinguishable electrons) in a prototype molecular system (hydrogen molecule). With the assistance of the standard approximations of the linear combination of atomic orbitals and the configuration interaction methods we describe the electronic wavefunction of the ground state of the H 2 molecule. Moreover, we managed to find a rather simple analytic expression for the concurrence (the most used measure of quantum entanglement) of the two electrons when the molecule is in its lowest energy. We have found that concurrence does not really show any relation to the construction of the chemical bond.
We use the concept of quantum entanglement to give a physical meaning to the electron correlation energy in systems of interacting electrons. The electron correlation is not directly observable, being defined as the difference between the exact ground state energy of the many-electron Schr ̈odinger equation and the Hartree–Fock energy. Using the configuration interaction method for the hydrogen molecule, we calculate the correlation energy and compare it with the entanglement as a function of the nucleus– nucleus separation. In the same spirit, we analyze a dimer of ethylene, which represents the simplest organic conjugate system, changing the relative orientation and distance of the molecules to obtain the configuration corresponding to maximum entanglement.
We compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes.
Low dimensional spin systems are a class of interacting many body systems which often can suitably be described by physical models with possible experimental realizations. A series of inorganic, organometallic and organic systems representing different spin ½ models have been synthesized and their thermodynamics have been investigated. In most of the cases, antiferromagnetic coupling is favored and the system can be well described by Heisenberg model where the unpaired spins interact isotropically to the nearest neighbor with an intrachain coupling constant (J). Isothermal heat capacity plots as a function of field at low temperature captures the Quantum Critical Point (QCP) experimentally, owing to change in symmetry of the ground state. This change of phase is associated with the occurrence of first order quantum phase transition in NH4CuPO4·H2O which has been established by studying field dependent behavior of magnetization and entanglement. On one hand entanglement being a nonlocal property of the spins takes its maximum value when the applied magnetic field is zero. On the other hand magnetization describing the local properties of the individual spins takes its maximum value upon reaching the saturation magnetic field at the expense of entanglement. It signifies a physical picture of quantum information sharing and gives an to opportunity capture the critical magnetic field. We have investigated this scenario in NH4CuPO4·H2O and determined the critical field which shows an excellent match with the value obtained from the field dependent specific heat data.
The first study of the quantum dynamics for a highly excited ('Rydberg') H 2 molecule in a magnetic field is presented and compared with the well studied atomic case. Wavefunctions from an accurate quantal calculation have been used to obtain the corresponding Wigner functions by means of a transformation to a basis where the latter are analytical. Unlike the atom, the dynamics of the excited electron in the molecular case experiences two, interfering, energy regimes. This study consider the case near electronic scaled energies -0.5 and -2.3, which would correspond to predominantly regular motion in the atomic case. In the molecular case there is evidence for chaotic behaviour in this energy range. Quantum states for the molecule may divide into those where the underlying dynamics is a perturbed version of the hydrogen atom case, and others where different configuration-space and phase-space structures are apparent.