Editing Spherical harmonics (section)

===Real form===
A real basis of spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> can be defined in terms of their complex analogues <math>Y_{\ell}^m: S^2 \to \Complex</math> by setting
<math display="block">\begin{align}
Y_{\ell m} &= \begin{cases}
\dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^m\, Y_\ell^{-m}\right) & \text{if}\ m < 0\\
Y_\ell^0 & \text{if}\ m=0\\
\dfrac{1}{\sqrt{2}} \left(Y_\ell^{-m} + (-1)^m\, Y_\ell^{m}\right) & \text{if}\ m > 0.
\end{cases}\\
&= \begin{cases}
\dfrac{i}{\sqrt{2}} \left(Y_\ell^{-|m|} - (-1)^{m}\, Y_\ell^{|m|}\right) & \text{if}\ m < 0\\
Y_\ell^0 & \text{if}\ m=0\\
\dfrac{1}{\sqrt{2}} \left(Y_\ell^{-|m|} + (-1)^{m}\, Y_\ell^{|m|}\right) & \text{if}\ m>0.
\end{cases}\\
&= \begin{cases}
\sqrt{2} \, (-1)^m \, \Im [{Y_\ell^{|m|}}] & \text{if}\ m<0\\
Y_\ell^0 & \text{if}\ m=0\\
\sqrt{2} \, (-1)^m \, \Re [{Y_\ell^m}] & \text{if}\ m>0.
\end{cases}
\end{align}
</math>
The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics <math>Y_{\ell}^m : S^2 \to \Complex</math> in terms of the real spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> are
<math display="block">
Y_{\ell}^{m} = \begin{cases}
\dfrac{1}{\sqrt{2}} \left(Y_{\ell |m|} - i Y_{\ell,-|m|}\right) & \text{if}\ m<0 \\[4pt]
Y_{\ell 0} &\text{if}\ m=0 \\[4pt]
\dfrac{(-1)^m}{ \sqrt{2}} \left(Y_{\ell |m|} + i Y_{\ell,-|m|}\right) & \text{if}\ m>0.
\end{cases}
</math>

The real spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> are sometimes known as ''tesseral spherical harmonics''.<ref>{{harvnb|Whittaker|Watson|1927|p=392}}.</ref> These functions have the same orthonormality properties as the complex ones <math>Y_{\ell}^m : S^2 \to \Complex</math> above. The real spherical harmonics <math>Y_{\ell m}</math> with {{math|''m'' > 0}} are said to be of cosine type, and those with {{math|''m'' < 0}} of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as
<math display="block">
Y_{\ell m} = \begin{cases}
 \left(-1\right)^m\sqrt{2} \sqrt{\dfrac{2\ell+1}{4\pi}\dfrac{(\ell-|m|)!}{(\ell+|m|)!}} \;
 P_\ell^{|m|}(\cos \theta) \ \sin( |m|\varphi )
 &\text{if } m<0
\\[4pt]
 \sqrt{\dfrac{ 2\ell+1}{4\pi}} \ P_\ell^m(\cos \theta)
 & \text{if } m=0
\\[4pt]
 \left(-1\right)^m\sqrt{2} \sqrt{\dfrac{2\ell+1}{4\pi}\dfrac{(\ell-m)!}{(\ell+m)!}} \;
 P_\ell^m(\cos \theta) \ \cos( m\varphi )
 & \text{if } m>0 \,.
\end{cases}
</math>

The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation.

See [[Table of spherical harmonics#Real spherical harmonics|here]] for a list of real spherical harmonics up to and including <math>\ell = 4</math>, which can be seen to be consistent with the output of the equations above.

==== Use in quantum chemistry ====
As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, the real functions span the same space as the complex ones would.

For example, as can be seen from the [[Table of spherical harmonics#Spherical harmonics|table of spherical harmonics]], the usual {{math|''p''}} functions (<math>\ell = 1</math>) are complex and mix axis directions, but the [[Table of spherical harmonics#Real spherical harmonics|real versions]] are essentially just {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}.