Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Next revision		Previous revision
physics_chemistry:orbitals:y [2016/10/03 23:00] – created Maurits W. Haverkortphysics_chemistry:orbitals:y [2017/02/23 09:45] (current) – Maurits W. Haverkort
Line 1: Line 1:
 +{{indexmenu_n>1}}
 +====== Spherical Harmonic (Y) ======
 +~~NOTOC~~
  
 +;;#
 +{{ :physics_chemistry:orbitals:basisy.nb|Download a notebook that generates this page}}
 +;;#
 +
 +###
 +The spherical harmonics are defined as
 +$$
 +Y_l^{(m)} = \frac{\sqrt{2 l+1}}{2 \sqrt{\pi }} \sqrt{\frac{(l-m)!}{(l+m)!}} e^{i m \phi } P_l^{(m)}(\cos (\theta )),
 +$$
 +with $l$ the angular momentum and $m$ the z projection of the angular momentum, $-l \leq m\leq l$. $P_l^{(m)}$ are the associated Legendre polynomials. For positive $m$ these are defined in terms of the unassociated Legendre polynomials as:
 +$$
 +\begin{align}
 +P_l^{(m)}(x) & = (-1)^m(1-x^2)^{m/2} \frac{d^m}{dx^m} P_l(x)\\
 + &= \frac{(-1)^m}{2^l l!}(1-x^2)^{m/2} \frac{d^{l+m}}{dx^{l+m}}(x^2-1)^l,
 +\end{align}
 +$$
 +using in the second line that:
 +$$
 +P_l(x)=\frac{1}{2^l l!}\frac{d^l}{dx^l} (x^2-1)^l.
 +$$
 +For negative $m$ the associated Legendre polynomials are defined as:
 +$$
 +P_l^{(-m)}(x) = (-1)^m \frac{(l-m)!}{(l+m)!} P_l^{(m)}(x).
 +$$
 +We thus include the //Condon-Shortley phase $(-1)^m$// in our definitions of the associated Legendre polynomials.
 +###
 +
 +###
 +The following table shows the spherical harmonics up to $l=6$. We list both the explicit function in terms of the angular coordinates $\theta$ and $\phi$ as well as the function in terms of the directional cosines $x$, $y$ and $z$. Using a right hand coordinate system the relation is: $x=\cos(\phi)\sin(\theta)$, $y=\sin(\phi)\sin(\theta)$ and $z=\cos(\theta)$. The plots show the surface defined by the equation $r={Y_l^{(m)}}^* Y_l^{(m)}$. The color of the surface is according to the phase with red for positive, cyan for negative, a mixture of yellow and green for pure positive imaginary values and a mixture of blue and magenta for pure negative imaginary values. We show a 3D image as well as a projection along the $x$, $y$ and $z$ direction.
 +###
 +
 +
 +----
 +----
 +===== $l=0$ =====
 +==== $m_l=0$ ====
 +$$
 +Y_{0}^{(0)}=\frac{1}{2 \sqrt{\pi }}\\
 +\phantom{Y_{0}^{(0)}}=\frac{1}{2 \sqrt{\pi }}
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby01.png?nolink |}}
 +
 +----
 +----
 +===== $l=1$ =====
 +==== $m_l=-1$ ====
 +$$
 +Y_{1}^{(-1)}=\frac{1}{2} \sqrt{\frac{3}{2 \pi }} e^{-i \phi } \sin (\theta )\\
 +\phantom{Y_{1}^{(-1)}}=\frac{1}{2} \sqrt{\frac{3}{2 \pi }} (x-i y)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby11.png?nolink |}}
 +
 +==== $m_l=0$ ====
 +$$
 +Y_{1}^{(0)}=\frac{1}{2} \sqrt{\frac{3}{\pi }} \cos (\theta )\\
 +\phantom{Y_{1}^{(0)}}=\frac{1}{2} \sqrt{\frac{3}{\pi }} z
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby12.png?nolink |}}
 +
 +==== $m_l=1$ ====
 +$$
 +Y_{1}^{(1)}=-\frac{1}{2} \sqrt{\frac{3}{2 \pi }} e^{i \phi } \sin (\theta )\\
 +\phantom{Y_{1}^{(1)}}=-\frac{1}{2} \sqrt{\frac{3}{2 \pi }} (x+i y)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby13.png?nolink |}}
 +
 +----
 +----
 +===== $l=2$ =====
 +==== $m_l=-2$ ====
 +$$
 +Y_{2}^{(-2)}=\frac{1}{4} \sqrt{\frac{15}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta )\\
 +\phantom{Y_{2}^{(-2)}}=\frac{1}{4} \sqrt{\frac{15}{2 \pi }} (x-i y)^2
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby21.png?nolink |}}
 +
 +==== $m_l=-1$ ====
 +$$
 +Y_{2}^{(-1)}=\frac{1}{2} \sqrt{\frac{15}{2 \pi }} e^{-i \phi } \sin (\theta ) \cos (\theta )\\
 +\phantom{Y_{2}^{(-1)}}=\frac{1}{2} \sqrt{\frac{15}{2 \pi }} z (x-i y)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby22.png?nolink |}}
 +
 +==== $m_l=0$ ====
 +$$
 +Y_{2}^{(0)}=\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(3 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{2}^{(0)}}=-\frac{1}{4} \sqrt{\frac{5}{\pi }} \left(x^2+y^2-2 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby23.png?nolink |}}
 +
 +==== $m_l=1$ ====
 +$$
 +Y_{2}^{(1)}=-\frac{1}{2} \sqrt{\frac{15}{2 \pi }} e^{i \phi } \sin (\theta ) \cos (\theta )\\
 +\phantom{Y_{2}^{(1)}}=-\frac{1}{2} \sqrt{\frac{15}{2 \pi }} z (x+i y)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby24.png?nolink |}}
 +
 +==== $m_l=2$ ====
 +$$
 +Y_{2}^{(2)}=\frac{1}{4} \sqrt{\frac{15}{2 \pi }} e^{2 i \phi } \sin ^2(\theta )\\
 +\phantom{Y_{2}^{(2)}}=\frac{1}{4} \sqrt{\frac{15}{2 \pi }} (x+i y)^2
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby25.png?nolink |}}
 +
 +----
 +----
 +===== $l=3$ =====
 +==== $m_l=-3$ ====
 +$$
 +Y_{3}^{(-3)}=\frac{1}{8} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \sin ^3(\theta )\\
 +\phantom{Y_{3}^{(-3)}}=\frac{1}{8} \sqrt{\frac{35}{\pi }} (x-i y)^3
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby31.png?nolink |}}
 +
 +==== $m_l=-2$ ====
 +$$
 +Y_{3}^{(-2)}=\frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta ) \cos (\theta )\\
 +\phantom{Y_{3}^{(-2)}}=\frac{1}{4} \sqrt{\frac{105}{2 \pi }} z (x-i y)^2
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby32.png?nolink |}}
 +
 +==== $m_l=-1$ ====
 +$$
 +Y_{3}^{(-1)}=\frac{1}{8} \sqrt{\frac{21}{\pi }} e^{-i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{3}^{(-1)}}=-\frac{1}{8} \sqrt{\frac{21}{\pi }} (x-i y) \left(x^2+y^2-4 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby33.png?nolink |}}
 +
 +==== $m_l=0$ ====
 +$$
 +Y_{3}^{(0)}=\frac{1}{4} \sqrt{\frac{7}{\pi }} \left(5 \cos ^3(\theta )-3 \cos (\theta )\right)\\
 +\phantom{Y_{3}^{(0)}}=\frac{1}{4} \sqrt{\frac{7}{\pi }} z \left(2 z^2-3 \left(x^2+y^2\right)\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby34.png?nolink |}}
 +
 +==== $m_l=1$ ====
 +$$
 +Y_{3}^{(1)}=-\frac{1}{8} \sqrt{\frac{21}{\pi }} e^{i \phi } \sin (\theta ) \left(5 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{3}^{(1)}}=\frac{1}{8} \sqrt{\frac{21}{\pi }} (x+i y) \left(x^2+y^2-4 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby35.png?nolink |}}
 +
 +==== $m_l=2$ ====
 +$$
 +Y_{3}^{(2)}=\frac{1}{4} \sqrt{\frac{105}{2 \pi }} e^{2 i \phi } \sin ^2(\theta ) \cos (\theta )\\
 +\phantom{Y_{3}^{(2)}}=\frac{1}{4} \sqrt{\frac{105}{2 \pi }} z (x+i y)^2
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby36.png?nolink |}}
 +
 +==== $m_l=3$ ====
 +$$
 +Y_{3}^{(3)}=-\frac{1}{8} \sqrt{\frac{35}{\pi }} e^{3 i \phi } \sin ^3(\theta )\\
 +\phantom{Y_{3}^{(3)}}=-\frac{1}{8} \sqrt{\frac{35}{\pi }} (x+i y)^3
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby37.png?nolink |}}
 +
 +----
 +----
 +===== $l=4$ =====
 +==== $m_l=-4$ ====
 +$$
 +Y_{4}^{(-4)}=\frac{3}{16} \sqrt{\frac{35}{2 \pi }} e^{-4 i \phi } \sin ^4(\theta )\\
 +\phantom{Y_{4}^{(-4)}}=\frac{3}{16} \sqrt{\frac{35}{2 \pi }} (x-i y)^4
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby41.png?nolink |}}
 +
 +==== $m_l=-3$ ====
 +$$
 +Y_{4}^{(-3)}=\frac{3}{8} \sqrt{\frac{35}{\pi }} e^{-3 i \phi } \sin ^3(\theta ) \cos (\theta )\\
 +\phantom{Y_{4}^{(-3)}}=\frac{3}{8} \sqrt{\frac{35}{\pi }} z (x-i y)^3
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby42.png?nolink |}}
 +
 +==== $m_l=-2$ ====
 +$$
 +Y_{4}^{(-2)}=\frac{3}{8} \sqrt{\frac{5}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta ) \left(7 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{4}^{(-2)}}=-\frac{3}{8} \sqrt{\frac{5}{2 \pi }} (x-i y)^2 \left(x^2+y^2-6 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby43.png?nolink |}}
 +
 +==== $m_l=-1$ ====
 +$$
 +Y_{4}^{(-1)}=\frac{3}{8} \sqrt{\frac{5}{\pi }} e^{-i \phi } \sin (\theta ) \cos (\theta ) \left(7 \cos ^2(\theta )-3\right)\\
 +\phantom{Y_{4}^{(-1)}}=-\frac{3}{8} \sqrt{\frac{5}{\pi }} z (x-i y) \left(3 \left(x^2+y^2\right)-4 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby44.png?nolink |}}
 +
 +==== $m_l=0$ ====
 +$$
 +Y_{4}^{(0)}=\frac{3 \left(35 \cos ^4(\theta )-30 \cos ^2(\theta )+3\right)}{16 \sqrt{\pi }}\\
 +\phantom{Y_{4}^{(0)}}=\frac{-72 z^2 \left(x^2+y^2\right)+9 \left(x^2+y^2\right)^2+24 z^4}{16 \sqrt{\pi }}
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby45.png?nolink |}}
 +
 +==== $m_l=1$ ====
 +$$
 +Y_{4}^{(1)}=-\frac{3}{8} \sqrt{\frac{5}{\pi }} e^{i \phi } \sin (\theta ) \cos (\theta ) \left(7 \cos ^2(\theta )-3\right)\\
 +\phantom{Y_{4}^{(1)}}=\frac{3}{8} \sqrt{\frac{5}{\pi }} z (x+i y) \left(3 \left(x^2+y^2\right)-4 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby46.png?nolink |}}
 +
 +==== $m_l=2$ ====
 +$$
 +Y_{4}^{(2)}=\frac{3}{8} \sqrt{\frac{5}{2 \pi }} e^{2 i \phi } \sin ^2(\theta ) \left(7 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{4}^{(2)}}=-\frac{3}{8} \sqrt{\frac{5}{2 \pi }} (x+i y)^2 \left(x^2+y^2-6 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby47.png?nolink |}}
 +
 +==== $m_l=3$ ====
 +$$
 +Y_{4}^{(3)}=-\frac{3}{8} \sqrt{\frac{35}{\pi }} e^{3 i \phi } \sin ^3(\theta ) \cos (\theta )\\
 +\phantom{Y_{4}^{(3)}}=-\frac{3}{8} \sqrt{\frac{35}{\pi }} z (x+i y)^3
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby48.png?nolink |}}
 +
 +==== $m_l=4$ ====
 +$$
 +Y_{4}^{(4)}=\frac{3}{16} \sqrt{\frac{35}{2 \pi }} e^{4 i \phi } \sin ^4(\theta )\\
 +\phantom{Y_{4}^{(4)}}=\frac{3}{16} \sqrt{\frac{35}{2 \pi }} (x+i y)^4
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby49.png?nolink |}}
 +
 +----
 +----
 +===== $l=5$ =====
 +==== $m_l=-5$ ====
 +$$
 +Y_{5}^{(-5)}=\frac{3}{32} \sqrt{\frac{77}{\pi }} e^{-5 i \phi } \sin ^5(\theta )\\
 +\phantom{Y_{5}^{(-5)}}=\frac{3}{32} \sqrt{\frac{77}{\pi }} (x-i y)^5
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby51.png?nolink |}}
 +
 +==== $m_l=-4$ ====
 +$$
 +Y_{5}^{(-4)}=\frac{3}{16} \sqrt{\frac{385}{2 \pi }} e^{-4 i \phi } \sin ^4(\theta ) \cos (\theta )\\
 +\phantom{Y_{5}^{(-4)}}=\frac{3}{16} \sqrt{\frac{385}{2 \pi }} z (x-i y)^4
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby52.png?nolink |}}
 +
 +==== $m_l=-3$ ====
 +$$
 +Y_{5}^{(-3)}=\frac{1}{32} \sqrt{\frac{385}{\pi }} e^{-3 i \phi } \sin ^3(\theta ) \left(9 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{5}^{(-3)}}=-\frac{1}{32} \sqrt{\frac{385}{\pi }} (x-i y)^3 \left(x^2+y^2-8 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby53.png?nolink |}}
 +
 +==== $m_l=-2$ ====
 +$$
 +Y_{5}^{(-2)}=\frac{1}{8} \sqrt{\frac{1155}{2 \pi }} e^{-2 i \phi } \sin ^2(\theta ) \cos (\theta ) \left(3 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{5}^{(-2)}}=-\frac{1}{8} \sqrt{\frac{1155}{2 \pi }} z (x-i y)^2 \left(x^2+y^2-2 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby54.png?nolink |}}
 +
 +==== $m_l=-1$ ====
 +$$
 +Y_{5}^{(-1)}=\frac{1}{16} \sqrt{\frac{165}{2 \pi }} e^{-i \phi } \sin (\theta ) \left(21 \cos ^4(\theta )-14 \cos ^2(\theta )+1\right)\\
 +\phantom{Y_{5}^{(-1)}}=\frac{1}{16} \sqrt{\frac{165}{2 \pi }} (x-i y) \left(-12 z^2 \left(x^2+y^2\right)+\left(x^2+y^2\right)^2+8 z^4\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby55.png?nolink |}}
 +
 +==== $m_l=0$ ====
 +$$
 +Y_{5}^{(0)}=\frac{1}{16} \sqrt{\frac{11}{\pi }} \left(63 \cos ^5(\theta )-70 \cos ^3(\theta )+15 \cos (\theta )\right)\\
 +\phantom{Y_{5}^{(0)}}=\frac{1}{16} \sqrt{\frac{11}{\pi }} z \left(-40 z^2 \left(x^2+y^2\right)+15 \left(x^2+y^2\right)^2+8 z^4\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby56.png?nolink |}}
 +
 +==== $m_l=1$ ====
 +$$
 +Y_{5}^{(1)}=-\frac{1}{16} \sqrt{\frac{165}{2 \pi }} e^{i \phi } \sin (\theta ) \left(21 \cos ^4(\theta )-14 \cos ^2(\theta )+1\right)\\
 +\phantom{Y_{5}^{(1)}}=-\frac{1}{16} \sqrt{\frac{165}{2 \pi }} (x+i y) \left(-12 z^2 \left(x^2+y^2\right)+\left(x^2+y^2\right)^2+8 z^4\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby57.png?nolink |}}
 +
 +==== $m_l=2$ ====
 +$$
 +Y_{5}^{(2)}=\frac{1}{8} \sqrt{\frac{1155}{2 \pi }} e^{2 i \phi } \sin ^2(\theta ) \cos (\theta ) \left(3 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{5}^{(2)}}=-\frac{1}{8} \sqrt{\frac{1155}{2 \pi }} z (x+i y)^2 \left(x^2+y^2-2 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby58.png?nolink |}}
 +
 +==== $m_l=3$ ====
 +$$
 +Y_{5}^{(3)}=-\frac{1}{32} \sqrt{\frac{385}{\pi }} e^{3 i \phi } \sin ^3(\theta ) \left(9 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{5}^{(3)}}=\frac{1}{32} \sqrt{\frac{385}{\pi }} (x+i y)^3 \left(x^2+y^2-8 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby59.png?nolink |}}
 +
 +==== $m_l=4$ ====
 +$$
 +Y_{5}^{(4)}=\frac{3}{16} \sqrt{\frac{385}{2 \pi }} e^{4 i \phi } \sin ^4(\theta ) \cos (\theta )\\
 +\phantom{Y_{5}^{(4)}}=\frac{3}{16} \sqrt{\frac{385}{2 \pi }} z (x+i y)^4
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby510.png?nolink |}}
 +
 +==== $m_l=5$ ====
 +$$
 +Y_{5}^{(5)}=-\frac{3}{32} \sqrt{\frac{77}{\pi }} e^{5 i \phi } \sin ^5(\theta )\\
 +\phantom{Y_{5}^{(5)}}=-\frac{3}{32} \sqrt{\frac{77}{\pi }} (x+i y)^5
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby511.png?nolink |}}
 +
 +----
 +----
 +===== $l=6$ =====
 +==== $m_l=-6$ ====
 +$$
 +Y_{6}^{(-6)}=\frac{1}{64} \sqrt{\frac{3003}{\pi }} e^{-6 i \phi } \sin ^6(\theta )\\
 +\phantom{Y_{6}^{(-6)}}=\frac{1}{64} \sqrt{\frac{3003}{\pi }} (x-i y)^6
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby61.png?nolink |}}
 +
 +==== $m_l=-5$ ====
 +$$
 +Y_{6}^{(-5)}=\frac{3}{32} \sqrt{\frac{1001}{\pi }} e^{-5 i \phi } \sin ^5(\theta ) \cos (\theta )\\
 +\phantom{Y_{6}^{(-5)}}=\frac{3}{32} \sqrt{\frac{1001}{\pi }} z (x-i y)^5
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby62.png?nolink |}}
 +
 +==== $m_l=-4$ ====
 +$$
 +Y_{6}^{(-4)}=\frac{3}{32} \sqrt{\frac{91}{2 \pi }} e^{-4 i \phi } \sin ^4(\theta ) \left(11 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{6}^{(-4)}}=-\frac{3}{32} \sqrt{\frac{91}{2 \pi }} (x-i y)^4 \left(x^2+y^2-10 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby63.png?nolink |}}
 +
 +==== $m_l=-3$ ====
 +$$
 +Y_{6}^{(-3)}=\frac{1}{32} \sqrt{\frac{1365}{\pi }} e^{-3 i \phi } \sin ^3(\theta ) \cos (\theta ) \left(11 \cos ^2(\theta )-3\right)\\
 +\phantom{Y_{6}^{(-3)}}=-\frac{1}{32} \sqrt{\frac{1365}{\pi }} z (x-i y)^3 \left(3 \left(x^2+y^2\right)-8 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby64.png?nolink |}}
 +
 +==== $m_l=-2$ ====
 +$$
 +Y_{6}^{(-2)}=\frac{1}{64} \sqrt{\frac{1365}{\pi }} e^{-2 i \phi } \sin ^2(\theta ) \left(33 \cos ^4(\theta )-18 \cos ^2(\theta )+1\right)\\
 +\phantom{Y_{6}^{(-2)}}=\frac{1}{64} \sqrt{\frac{1365}{\pi }} (x-i y)^2 \left(-16 z^2 \left(x^2+y^2\right)+\left(x^2+y^2\right)^2+16 z^4\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby65.png?nolink |}}
 +
 +==== $m_l=-1$ ====
 +$$
 +Y_{6}^{(-1)}=\frac{1}{16} \sqrt{\frac{273}{2 \pi }} e^{-i \phi } \sin (\theta ) \cos (\theta ) \left(33 \cos ^4(\theta )-30 \cos ^2(\theta )+5\right)\\
 +\phantom{Y_{6}^{(-1)}}=\frac{1}{16} \sqrt{\frac{273}{2 \pi }} z (x-i y) \left(-20 z^2 \left(x^2+y^2\right)+5 \left(x^2+y^2\right)^2+8 z^4\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby66.png?nolink |}}
 +
 +==== $m_l=0$ ====
 +$$
 +Y_{6}^{(0)}=\frac{1}{32} \sqrt{\frac{13}{\pi }} \left(231 \cos ^6(\theta )-315 \cos ^4(\theta )+105 \cos ^2(\theta )-5\right)\\
 +\phantom{Y_{6}^{(0)}}=\frac{1}{32} \sqrt{\frac{13}{\pi }} \left(-120 z^4 \left(x^2+y^2\right)+90 z^2 \left(x^2+y^2\right)^2-5 \left(x^2+y^2\right)^3+16 z^6\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby67.png?nolink |}}
 +
 +==== $m_l=1$ ====
 +$$
 +Y_{6}^{(1)}=-\frac{1}{16} \sqrt{\frac{273}{2 \pi }} e^{i \phi } \sin (\theta ) \cos (\theta ) \left(33 \cos ^4(\theta )-30 \cos ^2(\theta )+5\right)\\
 +\phantom{Y_{6}^{(1)}}=-\frac{1}{16} \sqrt{\frac{273}{2 \pi }} z (x+i y) \left(-20 z^2 \left(x^2+y^2\right)+5 \left(x^2+y^2\right)^2+8 z^4\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby68.png?nolink |}}
 +
 +==== $m_l=2$ ====
 +$$
 +Y_{6}^{(2)}=\frac{1}{64} \sqrt{\frac{1365}{\pi }} e^{2 i \phi } \sin ^2(\theta ) \left(33 \cos ^4(\theta )-18 \cos ^2(\theta )+1\right)\\
 +\phantom{Y_{6}^{(2)}}=\frac{1}{64} \sqrt{\frac{1365}{\pi }} (x+i y)^2 \left(-16 z^2 \left(x^2+y^2\right)+\left(x^2+y^2\right)^2+16 z^4\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby69.png?nolink |}}
 +
 +==== $m_l=3$ ====
 +$$
 +Y_{6}^{(3)}=-\frac{1}{32} \sqrt{\frac{1365}{\pi }} e^{3 i \phi } \sin ^3(\theta ) \cos (\theta ) \left(11 \cos ^2(\theta )-3\right)\\
 +\phantom{Y_{6}^{(3)}}=\frac{1}{32} \sqrt{\frac{1365}{\pi }} z (x+i y)^3 \left(3 \left(x^2+y^2\right)-8 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby610.png?nolink |}}
 +
 +==== $m_l=4$ ====
 +$$
 +Y_{6}^{(4)}=\frac{3}{32} \sqrt{\frac{91}{2 \pi }} e^{4 i \phi } \sin ^4(\theta ) \left(11 \cos ^2(\theta )-1\right)\\
 +\phantom{Y_{6}^{(4)}}=-\frac{3}{32} \sqrt{\frac{91}{2 \pi }} (x+i y)^4 \left(x^2+y^2-10 z^2\right)
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby611.png?nolink |}}
 +
 +==== $m_l=5$ ====
 +$$
 +Y_{6}^{(5)}=-\frac{3}{32} \sqrt{\frac{1001}{\pi }} e^{5 i \phi } \sin ^5(\theta ) \cos (\theta )\\
 +\phantom{Y_{6}^{(5)}}=-\frac{3}{32} \sqrt{\frac{1001}{\pi }} z (x+i y)^5
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby612.png?nolink |}}
 +
 +==== $m_l=6$ ====
 +$$
 +Y_{6}^{(6)}=\frac{1}{64} \sqrt{\frac{3003}{\pi }} e^{6 i \phi } \sin ^6(\theta )\\
 +\phantom{Y_{6}^{(6)}}=\frac{1}{64} \sqrt{\frac{3003}{\pi }} (x+i y)^6
 +$$
 +
 +{{:physics_chemistry:orbitals:plorby613.png?nolink |}}
 +
 +
 +
 +
 +====== Different orbital basis sets used ======
 +{{indexmenu>.#1|msort}}

You are here: Quanty - a quantum many body script language » Physics, Chemistry and Mathematics » One-particle orbitals » Spherical Harmonic (Y)

Print/export