In fields likecomputational chemistry andsolid-state andcondensed matter physics, the so-calledatomic orbitals, orspin-orbitals, as they appear in textbooks^[1][2][3] on quantum physics, are often partially replaced bycubic harmonics for a number of reasons. These harmonics are usually namedtesseral harmonics in the field of condensed matter physics in which the namekubic harmonics rather refers to the irreducible representations in the cubic point-group.^[4]

The${\displaystyle 2l+1}$hydrogen-like atomic orbitals with principal quantum number${\displaystyle n}$ and angular momentum quantum number${\displaystyle l}$ are often expressed as

${\displaystyle {\psi }_{nlm}(\mathbf{r})=R_{nl}(r)Y_l^m(\theta ,\phi )}$

in which the${\displaystyle R_{nl}(r)}$ is the radial part of the wave function and${\displaystyle Y_l^m(\theta ,\phi )}$ is the angular dependent part. The${\displaystyle Y_l^m(\theta ,\phi )}$ are thespherical harmonics, which are solutions of theangular momentum operator. The spherical harmonics are representations of functions of thefull rotation group SO(3)^[5] with rotational symmetry. In many fields of physics and chemistry these spherical harmonics are replaced by cubic harmonics because the rotational symmetry of the atom and its environment are distorted or because cubic harmonics offer computational benefits.

In many cases, especially inchemistry andsolid-state andcondensed-matter physics, the system under investigation doesn't have rotational symmetry. Often it has some kind oflower symmetry, with a specialpoint group representation, or it hasno spatial symmetry at all. Biological andbiochemical systems, likeamino acids andenzymes often belong to lowmolecular symmetry point groups. Thesolid crystals of the elements often belong to thespace groups and point groups with high symmetry. (Cubic harmonics representations are often listed and referenced inpoint group tables.) The system has at least a fixed orientation in three-dimensionalEuclidean space. Therefore, the coordinate system that is used in such cases is most often aCartesian coordinate system instead of aspherical coordinate system. In a Cartesian coordinate system theatomic orbitals are often expressed as

${\displaystyle {\psi }_{nlc}(\mathbf{r})=R_{nl}(r)X_{lc}(\mathbf{r})}$

with thecubic harmonics,^[6][7][8]${\displaystyle X_{lc}(\mathbf{r})}$, as abasis set.LCAO andMO calculations incomputational chemistry ortight binding calculations in solid-state physics use cubic harmonics as an atomic orbital basis. The indiceslc are denoting some kind of Cartesian representation.

For therepresentations of the spherical harmonics a spherical coordinate system is chosen with aprincipal axis in thez-direction. For the cubic harmonics this axis is also the most convenient choice. For states of higher angular momentum quantum number${\displaystyle l}$ and a higher dimension of${\displaystyle l(l+1)}$ the number of possible rotations orbasis transformations inHilbert space grows and so does the number of possible orthogonal representations that can be constructed on the basis of the${\displaystyle l(l+1)}$-dimensional spherical harmonics basis set. There is more freedom to choose a representation that fits the point group symmetry of the problem. The cubic representations that are listed inthe table are a result of the transformations, which are 45° 2D rotations and a 90° rotation to the real axis if necessary, like

${\displaystyle X_{lc}(\mathbf{r})=Y_l^0}$

${\displaystyle X_{l{c}^{\prime }}(\mathbf{r})=\frac{1}{i^{n_{c^{\prime }}}\sqrt{2}}\left(Y_l^m-Y_l^{-m}\right)}$

${\displaystyle X_{l{c}^{″}}(\mathbf{r})=\frac{1}{i^{n_{c^{″}}}\sqrt{2}}\left(Y_l^m+Y_l^{-m}\right)}$

A substantial number of the spherical harmonics are listed in theTable of spherical harmonics.

First of all, the cubic harmonics arereal functions, while spherical harmonics arecomplex functions. The complex numbers are two-dimensional with a real part and an imaginary part. Complex numbers offer very handsome and effective tools to tackle mathematical problems analytically but they are not very effective when they are used for numerical calculations. Skipping the imaginary part saves half the calculational effort in summations, a factor of four in multiplications and often factors of eight or even more when it comes to computations involving matrices.

The cubic harmonics often fit the symmetry of the potential or surrounding of an atom. A common surrounding of atoms in solids andchemical complexes is an octahedral surrounding with anoctahedral cubic point group symmetry. 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. A problem with the 48-fold octahedral O_h symmetry can be calculated much faster if one limits a calculation, like an integration, to the irreducible part of thedomain of the function.

Thes-orbitals only have a radial part.

${\displaystyle {\psi }_{n00}(\mathbf{r})=R_{n0}(r)Y_0^0}$

${\displaystyle s=X_{00}=Y_0^0=\frac{1}{\sqrt{4\pi }}}$

	n=1	2	3	4	5	6	7
Rn0

Thethree p-orbitals areatomic orbitals with anangular momentum quantum numberℓ = 1. The cubic harmonic expression of the p-orbitals

${\displaystyle p_z=N_1^c\frac{z}{r}=Y_1^0}$

${\displaystyle p_x=N_1^c\frac{x}{r}=\frac{1}{\sqrt{2}}\left(Y_1^{-1}-Y_1^1\right)}$

${\displaystyle p_y=N_1^c\frac{y}{r}=\frac{i}{\sqrt{2}}\left(Y_1^{-1}+Y_1^1\right)}$

with

${\displaystyle N_1^c={\left(\frac{3}{4\pi }\right)}^{1/2}}$

pz	px	py

Thefive d-orbitals areatomic orbitals with anangular momentum quantum numberℓ = 2. Theangular part of the d-orbitals are often expressed like

${\displaystyle {\psi }_{n2c}(\mathbf{r})=R_{n2}(r)X_{2c}(\mathbf{r})}$

Theangular part of the d-orbitals are thecubic harmonics${\displaystyle X_{2c}(\mathbf{r})}$

${\displaystyle d_{z^2}=N_2^c\frac{3z^2-r^2}{2r^2\sqrt{3}}=Y_2^0}$

${\displaystyle d_{xz}=N_2^c\frac{xz}{r^2}=\frac{1}{\sqrt{2}}\left(Y_2^{-1}-Y_2^1\right)}$

${\displaystyle d_{yz}=N_2^c\frac{yz}{r^2}=\frac{i}{\sqrt{2}}\left(Y_2^{-1}+Y_2^1\right)}$

${\displaystyle d_{xy}=N_2^c\frac{xy}{r^2}=\frac{i}{\sqrt{2}}\left(Y_2^{-2}-Y_2^2\right)}$

${\displaystyle d_{x^2-y^2}=N_2^c\frac{x^2-y^2}{2r^2}=\frac{1}{\sqrt{2}}\left(Y_2^{-2}+Y_2^2\right)}$

with

${\displaystyle N_2^c={\left(\frac{15}{4\pi }\right)}^{1/2}}$

dz2	dxz	dyz	dxy	dx2-y2

Theseven f-orbitals areatomic orbitals with anangular momentum quantum numberℓ = 3. often expressed like

${\displaystyle {\psi }_{n3c}(\mathbf{r})=R_{n3}(r)X_{3c}(\mathbf{r})}$

Theangular part of the f-orbitals are thecubic harmonics${\displaystyle X_{3c}(\mathbf{r})}$. In many cases different linear combinations of spherical harmonics are chosen to construct a cubic f-orbital basis set.

${\displaystyle f_{z^3}=N_3^c\frac{z(2z^2-3x^2-3y^2)}{2r^3\sqrt{15}}=Y_3^0}$

${\displaystyle f_{x{z}^2}=N_3^c\frac{x(4z^2-x^2-y^2)}{2r^3\sqrt{10}}=\frac{1}{\sqrt{2}}\left(Y_3^{-1}-Y_3^1\right)}$

${\displaystyle f_{y{z}^2}=N_3^c\frac{y(4z^2-x^2-y^2)}{2r^3\sqrt{10}}=\frac{i}{\sqrt{2}}\left(Y_3^{-1}+Y_3^1\right)}$

${\displaystyle f_{xyz}=N_3^c\frac{xyz}{r^3}=\frac{i}{\sqrt{2}}\left(Y_3^{-2}-Y_3^2\right)}$

${\displaystyle f_{z(x^2-y^2)}=N_3^c\frac{z\left(x^2-y^2\right)}{2r^3}=\frac{1}{\sqrt{2}}\left(Y_3^{-2}+Y_3^2\right)}$

${\displaystyle f_{x(x^2-3y^2)}=N_3^c\frac{x\left(x^2-3y^2\right)}{2r^3\sqrt{6}}=\frac{1}{\sqrt{2}}\left(Y_3^{-3}-Y_3^3\right)}$

${\displaystyle f_{y(3x^2-y^2)}=N_3^c\frac{y\left(3x^2-y^2\right)}{2r^3\sqrt{6}}=\frac{i}{\sqrt{2}}\left(Y_3^{-3}+Y_3^3\right)}$

with

${\displaystyle N_3^c={\left(\frac{105}{4\pi }\right)}^{1/2}}$

fz3	fxz2	fyz2	fxyz	fz(x2-y2)	fx(x2-3y2)	fy(3x2-y2)

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