Cubic harmonic

In fields like computational chemistry and solid-state and condensed matter physics the so-called atomic orbitals, or spin-orbitals, as they appear in textbooks[1][2][3] on quantum physics, are often partially replaced by cubic harmonics for a number of reasons.

hydrogen-like atomic orbitals with principal quantum number

are the spherical harmonics, which are solutions of the angular momentum operator.

In many cases, especially in chemistry and solid-state and condensed-matter physics, the system under investigation doesn't have rotational symmetry.

Biological and biochemical systems, like amino acids and enzymes often belong to low molecular symmetry point groups.

(Cubic harmonics representations are often listed and referenced in point group tables.)

In a Cartesian coordinate system the atomic orbitals are often expressed as with the cubic harmonics,[6][7][8]

LCAO and MO calculations in computational chemistry or tight binding calculations in solid-state physics use cubic harmonics as an atomic orbital basis.

The indices lc are denoting some kind of Cartesian representation.

For states of higher angular momentum quantum number

There is more freedom to choose a representation that fits the point group symmetry of the problem.

The cubic representations that are listed in the table are a result of the transformations, which are 45° 2D rotations and a 90° rotation to the real axis if necessary, like A substantial number of the spherical harmonics are listed in the Table of spherical harmonics.

The cubic harmonics often fit the symmetry of the potential or surrounding of an atom.

The representations of the cubic harmonics often have a high symmetry and multiplicity so operations like integrations can be reduced to a limited, or irreducible, part of the domain of the function that has to be evaluated.

The three p-orbitals are atomic orbitals with an angular momentum quantum number ℓ = 1.

The cubic harmonic expression of the p-orbitals with The five d-orbitals are atomic orbitals with an angular momentum quantum number ℓ = 2.

In many cases different linear combinations of spherical harmonics are chosen to construct a cubic f-orbital basis set.