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For example, the nitrogen atom ground state has three unpaired electrons of parallel spin, so that the total spin is 3/2 and the multiplicity is 4. The lower energy and increased stability of the atom arise because the high-spin state has unpaired electrons of parallel spin, which must reside in different spatial orbitals according to the Pauli ...
Each has two electrons of opposite spin in the π* level so that S = 0 and the multiplicity is 2S + 1 = 1 in consequence. In the first excited state, the two π* electrons are paired in the same orbital, so that there are no unpaired electrons. In the second excited state, however, the two π* electrons occupy different orbitals with opposite spin.
It is particularly important for systems at atomic length scales, such as individual atoms, protons, or electrons. A triplet state occurs in cases where the spins of two unpaired electrons, each having spin s = 1/2, align to give S = 1, in contrast to the more common case of two electrons aligning oppositely to give S = 0, a spin singlet.
That is, the spin of the excited electron is still paired with the ground state electron (a pair of electrons in the same energy level must have opposite spins, per the Pauli exclusion principle). In a triplet state the excited electron is no longer paired with the ground state electron; that is, they are parallel (same spin). Since excitation ...
Since metals can display multiple oxidation numbers, the exact definition of how many "valence electrons" an element should have in elemental form is somewhat arbitrary, but the following table lists the free electron densities given in Ashcroft and Mermin, which were calculated using the formula above based on reasonable assumptions about ...
A birefringent crystal is a material that has an optic axis with the property that the light has a different index of refraction for light polarized parallel to the axis than it has for light polarized perpendicular to the axis. Light polarized parallel to the axis are called "extraordinary rays" or "extraordinary photons", while light ...
The vertical axis of the band diagram represents the energy of an electron, which includes both kinetic and potential energy. The horizontal axis represents position, often not being drawn to scale. Note that the Heisenberg uncertainty principle prevents the band diagram from being drawn with a high positional resolution, since the band diagram ...
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, .