Quantum Chemistry/Integrals in polar coordinates

Integration in spherical polar coordinates is a triple integral in the polar coordinate system. In the Cartesian coordinate system, points in space are defined by their position along the x, y, and z axes. Integration divides the space under a curve into infinitesimal widths along the x-axis. The sum of these strips is the area under the curve.$$\int f(x)~{\textrm {d}}x$$In spherical polar coordinates, the frame of reference is changed so all points are defined by its position about the origin. The distance from the origin is defined by the radius r and the angle of displacement longitudinally, θ, and latitudinally ϕ. The angular components are limited in their value because of their periodic nature. Their ranges are defined as:$$0\leq \phi \leq 2\pi$$$$0\leq \theta \leq \pi$$ Polar coordinates can be converted from Cartesian ${\textstyle (r,\phi ,\theta )\rightarrow (x,y,z)}$:$$x=r\cos \phi \cdot \sin \theta$$$$y=r\sin \phi \cdot \sin \theta$$$$z=r\cos \theta$$For integration in higher dimensions, integration by parts is used. A triple integral is in three dimensions and therefore has three variables. The outer integral relates to the outer variable, and the inner integrals to the inner variable.$$\underbrace {\int \underbrace {\int \underbrace {\int f(x,y,z)~{\textrm {d}}x} _{x}~{\textrm {d}}y} _{y}~{\textrm {d}}z} _{z}$$In polar coordinates, the strips of infinitesimal widths are converted to wedges and can be converted from Cartesian.$${\textrm {d}}x\cdot {\textrm {d}}y\cdot {\textrm {d}}z=r^{2}\sin \theta \cdot {\textrm {d}}r~{\textrm {d}}\theta ~{\textrm {d}}\phi$$$$\iiint f(x,y,z){\textrm {d}}x~{\textrm {d}}y~{\textrm {d}}z=\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{\infty }r^{2}\cdot f(r\cos \phi \sin \theta ,r\sin \phi \cos \theta ,r\cos \theta )\sin \theta ~{\textrm {d}}r~{\textrm {d}}\theta ~{\textrm {d}}\phi$$To solve spherical polar integral, begin at the inner most integral and proceed outwards.

Example

$\int _{0}^{2\pi }\int _{0}^{\pi }\int _{0}^{\infty }e^{-r}r~{\textrm {d}}r~{\textrm {d}}\theta ~{\textrm {d}}\phi$

integrate the innermost integral, x.

$\int _{0}^{2\pi }\int _{0}^{\pi }{\bigl (}-r+1{\bigr )}e^{-r}|_{0}^{\infty }~{\textrm {d}}\theta ~{\textrm {d}}\phi$

$\int _{0}^{2\pi }\int _{0}^{\pi }{\bigl (}-\infty +1{\bigr )}e^{-\infty }-{\bigl (}-0+1{\bigr )}e^{-0}~{\textrm {d}}\theta ~{\textrm {d}}\phi$

$\int _{0}^{2\pi }\int _{0}^{\pi }0-{\bigr (}-1{\bigr )}~{\textrm {d}}\theta ~{\textrm {d}}\phi$

$\int _{0}^{2\pi }\int _{0}^{\pi }~{\textrm {d}}\theta ~{\textrm {d}}\phi$

Integrate the next integral, θ.

$\int _{0}^{2\pi }\theta |_{0}^{\pi }~{\textrm {d}}\phi$

$\int _{0}^{2\pi }(\pi -0)~{\textrm {d}}\phi$

Finally, integrate ϕ.

$\pi \phi |_{0}^{2\pi }$

$\pi ({2\pi }-0)$

${2\pi }^{2}$