A theorem on differential inequalities is proved. By means of this theorem the following result is established. Let psi (r 1 ,r 2 ,r 12 ) denote the ground-state wavefunction of a two-electron atom; then psi (r 1 ,r 2 r 12 )<or= psi (r 1 ,r 1 ,r' 12 ) for r' 12 <or=r 12 . Furthermore, a different proof is given, for the hydrogenic molecular ion, of a recent result of Lieb and Simon (1978) on the monotonicity of the electronic part of the Born-Oppenheimer energy with respect to the internuclear distance.
A theorem on differential inequalities is proved. By means of this theorem the following result is established. Let psi (r 1 ,r 2 ,r 12 ) denote the ground-state wavefunction of a two-electron atom; then psi (r 1 ,r 2 r 12 )<or= psi (r 1 ,r 1 ,r' 12 ) for r' 12 <or=r 12 . Furthermore, a different proof is given, for the hydrogenic molecular ion, of a recent result of Lieb and Simon (1978) on the monotonicity of the electronic part of the Born-Oppenheimer energy with respect to the internuclear distance.
We present a detailed study of the ground state behaviour of two-electron diatomic molecules. The ground state stability diagram for diatomic molecules in the Born–Oppenheimer approximation is obtained and the behaviour of the ground state near the stability line is studied. Two different cases are analysed: the homonuclear two-centre two-electron molecule with the internuclear distance as a free parameter and the diatomic two-electron molecule (in this case, the internuclear distance is determined by equilibrium conditions). Analytical and numerical results for these systems are presented.
We establish a new iterative method for computing rigorous lower bounds for the ground-state energies of all atoms and ions. The accuracy of the resulting bounds depends on two ingredients: the quality of the initial energy estimate and the quality of the used upper bound for the maximum number of bound electrons, Nc(Z), as a function of the nuclear charge Z>0. In particular, this new method allows us to establish the explicit lower bound E(N,Z)≥−0.8710N1/3Z2 for the ground-state energy E(N,Z) of the atom/ion with N∈N electrons and nuclear charge Z∈(0,N], thereby considerably improving the best known constant A=(3/2)1/3≈1.1447 in the estimate E(N,Z)≥−AN1/3Z2 for 0<Z≤N and coming much closer to the best possible constant given by the Thomas-Fermi asymptotics limN→∞∣E(N,N)∣N−7/3=−ETF(1)≈0.7687 derived in Adv. Math., 23(1) by Lieb and Simon. To demonstrate the potential and the limits of our method, we also compute explicit lower bounds for the ground-state energies E(N,N) of all atoms with 2≤N≤86 and we compare them with the bounds obtained in J. Chem. Phys., 62(8) by Hertel, Lieb, Thirring and the bounds obtained by Seco in his dissertation. Finally, we also provide a uniqueness result for the expectation values of the kinetic, potential and interaction energies on the ground-state space.
The quantum mechanics of two-electron systems is reviewed, starting with the ground state of the helium atom and heliumlike ions with central charge Z. The case of Z≥2 is rather straightforward. In contrast, for negative hydrogen ion with Z=1, the stability of H− cannot be achieved using a product of individual electron wavefunctions and requires explicit account of the anticorrelation among the two electrons. The wavefunction proposed by Chandrasekhar is revisited, where the permutation symmetry is first broken and then restored by a counterterm. More difficult problems can be studied using the same strategy such as the stability of hydrogenlike ions for any value of the proton-to-electron mass ratio M/m, the energy of the lowest spin-triplet state of helium and heliumlike ions, and the stability of the doubly excited hydrogen ion with unnatural parity. The positronium molecule, which was predicted years ago and discovered recently, can also be shown to be stable against spontaneous dissociation. Emphasis is placed on symmetry breaking, which can either spoil or improve the stability.
If two potentials V 1 (x) and V 2 (x) are ordered V 1 <V 2 , then the comparison theorem tells the author that the corresponding Schrodinger eigenvalues are ordered E 1 <E 2 . He presents some simple conditions which allow him to predict such spectral ordering for the ground state, even when the graphs of the potentials cross over. As illustrations, the truncated quartic oscillator and the Yukawa potential are studied. By allowing Coulomb 'tangents' to cross over the Yukawa in various ways, he is able to improve earlier energy upper bounds which he had obtained by representing the Yukawa potential as a smooth transformation of the Coulomb potential, and also to augment these results with lower bounds.
We derive eigenstate comparisons from ordered Schrödinger operators, extending energy bounds to overlaps, spatial distributions and nodes. For three form-ordered operators with discrete comparison ground states, suppose the upper ground-state energy lies in the lower operator's ground-state gap. Three comparison spectral values then bound all three squared ground-state overlaps and guarantee a simple discrete intermediate ground state. Related gap estimates control low-energy subspaces and individual excited-state overlaps under additional gap conditions. In one dimension and radially, a single crossing of the potential difference relative to the energy shift orders ground-state likelihood ratios and monotone observables; a nondecreasing difference suffices. The results cover asymmetric wells and shortened right Dirichlet walls, with coordinatewise comparisons for submodular reference potentials and coordinatewise nondecreasing perturbations. A two-sided Sturm argument orders corresponding internal nodes for nondecreasing potential differences, even between orthogonal one-dimensional excited states. For Yukawa ground states, Hulthén comparisons bound overlaps, radial moments and the density at the nucleus; the latter two-sided bounds hold for screening 0<δ<2/6. Optimizing scaled Hulthén majorants gives a lower bound of approximately 1.13611 for n2δc(n,0), where δc(n,0) is the critical screening for the nth s-wave bound state. This bound is optimal within that two-parameter comparison family.
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.