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.$$\int f(x)\text{d}x$$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:$$0\le \varphi \le 2\pi$$$$0\le \theta \le \pi$$ Polar coordinates can be converted from Cartesian $(r,\varphi ,\theta )\to (x,y,z)$:$$x=r\cos \varphi \cdot \sin \theta$$$$y=r\sin \varphi \cdot \sin \theta$$$$z=r\cos \theta$$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.$$\underset{z}{\underbrace{\int \underset{y}{\underbrace{\int \underset{x}{\underbrace{\int f(x,y,z)\text{d}x}}\text{d}y}}\text{d}z}}$$In polar coordinates, the strips of infinitesimal widths are converted to wedges and can be converted from Cartesian.$$\text{d}x\cdot \text{d}y\cdot \text{d}z=r^2\sin \theta \cdot \text{d}r\text{d}\theta \text{d}\varphi$$$$\iiint f(x,y,z)\text{d}x\text{d}y\text{d}z=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{r}^2\cdot f(r\cos \varphi \sin \theta ,r\sin \varphi \cos \theta ,r\cos \theta )\sin \theta \text{d}r\text{d}\theta \text{d}\varphi$$To solve spherical polar integral, begin at the inner most integral and proceed outwards.

Example

$\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-r}r\text{d}r\text{d}\theta \text{d}\varphi$

integrate the innermost integral, x.

$\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}(-r+1)e^{-r}{|}_0^{\mathrm{\infty }}\text{d}\theta \text{d}\varphi$

$\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}(-\mathrm{\infty }+1)e^{-\mathrm{\infty }}-(-0+1)e^{-0}\text{d}\theta \text{d}\varphi$

$\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}0-(-1)\text{d}\theta \text{d}\varphi$

$\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\text{d}\theta \text{d}\varphi$

Integrate the next integral, θ.

$\underset{0}{\overset{2\pi }{\int }}\theta {|}_0^{\pi }\text{d}\varphi$

$\underset{0}{\overset{2\pi }{\int }}(\pi -0)\text{d}\varphi$

Finally, integrate ϕ.

$\pi \varphi {|}_0^{2\pi }$

$\pi (2\pi -0)$

${2\pi }^2$Template:BookCat