# H2

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In quantum chemistry, linear combinations of atomic orbitals (LCAO) have long been used to construct molecular orbitals. This Demonstration considers the Bohm trajectories of two initially separated excited metastable hydrogen atoms, separated by an internuclear distance . Keep in mind that in this special case, the standard LCAO method must be adjusted to fit, and it does not give exact solutions to the Schrödinger equation, but only first approximations.

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The most stable electronic state of an atom is called the ground state. An excited atom, which is called a Rydberg atom (with a very high principal quantum number ), in which the electron is not in the lowest-energy orbit, immediately emits energy in the form of light when the electron falls back into lower-energy orbit. Metastable states are excited atoms with a long lifetime before they return to the ground state by emitting photons.

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An example of a metastable atom with a lifetime of about 0.12 seconds is the () excited state of the hydrogen atom, which undergoes radiative decay to the ground state (), only by two-photon transitions [6].

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As in standard treatments of the hydrogen atom, the nuclei are assumed to have infinite mass and thus remain stationary in space, with the specified internuclear distance parameter in the direction. Thus the model consists of electrons that are electrostatically attracted to the stationary protons. This approximation neglects the kinetic energy of the nuclei, and the system is approximated as two separate one-electron molecular orbitals. In the Bohm picture, the electron acts like an actual particle, its velocity at any instant being fully determined by the gradient of the phase function, which in turn depends on the LCAO representation of the total molecular wavefunction. Dewdney and Malik [1], Holland [2] and Colijn and Vrscay [3] first examined the Bohmian trajectories associated with eigenstates for the hydrogen atom.

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Pauli's exclusion principle, which states that two identical electrons cannot occupy the same spin-orbital within a quantum system, plays no role here since the two occupied molecular orbitals of the hydrogen molecule have opposite spins. There are four possible Bohm trajectories for two electrons in depending on the initial position , , and the internuclear distance parameter . If , you see only one electron with two possible orbits. In the interference region of the two hydrogen atoms, the component of the velocity is not equal to zero.

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H

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In the graphic you see the wave density (if enabled); four possible orbits of two electrons, where one trajectory (blue) depends on the initial starting point (, , ), and the initial starting points of the four trajectories (black points, shown as small spheres) and the actual position (colored points, shown as small spheres).

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