Abstract
A systematic investigation of three different electron–electron entanglement measures, namely the von Neumann, the linear and the occupation number entropy at full configuration interaction level has been performed for the four helium-like systems hydride, helium, Li+ and Be2+ using a large number of different basis sets. The convergence behavior of the resulting energies and entropies revealed that the latter do in general not show the expected strictly monotonic increase upon increase of the one–electron basis. Overall, the three different entanglement measures show good agreement among each other, the largest deviations being observed for small basis sets. The data clearly demonstrates that it is important to consider the nature of the chemical system when investigating entanglement phenomena in the framework of Gaussian type basis sets: while in case of hydride the use of augmentation functions is crucial, the application of core functions greatly improves the accuracy in case of cationic systems such as Li+ and Be2+. In addition, numerical derivatives of the entanglement measures with respect to the nucleic charge have been determined, which proved to be a very sensitive probe of the convergence leading to qualitatively wrong results (i.e., the wrong sign) if too small basis sets are used.
Highlights
While the foundations of quantum theory were laid out already at the beginning of the 20th century (Planck, 1901; Einstein, 1905), their influence on chemical science become only apparent after the influential work of Schrödinger in the 1920’s (Schrödinger, 1926a,b,c,d), correctly predicting the non-relativistic energy eigenvalues for hydrogen-like systems from first principles
Application of the corresponding electronic Hamiltonian on the Slater determinant leads to the formulation of the HartreeFock (HF) method (Hartree, 1928a,b; Fock, 1930), in which a numerical solution of Schrödinger’s equation is obtained via a variational principle: by optimizing the coefficients used in the linear combination of atomic orbitals in an iterative way, the lowest energy and the best approximation to the wave function is obtained
Following Collins’ conjecture (Collins, 1993) the correlation energy of a system is proportional to the respective entropy of entanglement, typically expressed via the von Neumann entropy or related measures
Summary
While the foundations of quantum theory were laid out already at the beginning of the 20th century (Planck, 1901; Einstein, 1905), their influence on chemical science become only apparent after the influential work of Schrödinger in the 1920’s (Schrödinger, 1926a,b,c,d), correctly predicting the non-relativistic energy eigenvalues for hydrogen-like systems from first principles. Schrödinger’s formulation of the quantization as an eigenvalue problem of a wave equation essentially marked the starting point of modern electronic structure theory aimed at the quantum mechanical description of many-electron systems (Szabo and Ostlund, 1996; Levine, 1999; Helgaker et al, 2000; Cook, 2005). Application of the corresponding electronic Hamiltonian on the Slater determinant leads to the formulation of the HartreeFock (HF) method (Hartree, 1928a,b; Fock, 1930), in which a numerical solution of Schrödinger’s equation is obtained via a variational principle: by optimizing the coefficients used in the linear combination of atomic orbitals in an iterative way, the lowest energy and the best approximation to the wave function is obtained
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