Editing Gaussian orbital (section)

===Cartesian coordinates===

In Cartesian coordinates, Gaussian-type orbitals can be written in terms of exponential factors in the <math>x</math>, <math>y</math>, and <math>z</math> directions as well as an exponential factor <math>\alpha</math> controlling the width of the orbital. The expression for a Cartesian Gaussian-type orbital, with the appropriate normalization coefficient is

:<math>\Phi(x,y,z;\alpha,i,j,k)=\left(\frac{2\alpha}{\pi}\right)^{3/4}\left[\frac{(8\alpha)^{i+j+k}i!j!k!}{(2i)!(2j)!(2k)!}\right]^{1/2}x^i y^j z^k e^{-\alpha(x^2+y^2+z^2)}</math>

In the above expression, <math>i</math>, <math>j</math>, and <math>k</math> must be integers. If <math>i+j+k=0</math>, then the orbital has spherical symmetry and is considered an s-type GTO. If <math>i+j+k=1</math>, the GTO possesses axial symmetry along one axis and is considered a p-type GTO. When <math>i+j+k=2</math>, there are six possible GTOs that may be constructed; this is one more than the five canonical d orbital functions for a given angular quantum number. To address this, a linear combination of two d-type GTOs can be used to reproduce a canonical d function. Similarly, there exist 10 f-type GTOs, but only 7 canonical f orbital functions; this pattern continues for higher angular quantum numbers.<ref>{{cite book |last1=Cramer |first1=Christopher J. |title=Essentials of computational chemistry : theories and models |date=2004 |publisher=Wiley |location=Chichester, West Sussex, England |isbn=9780470091821 |pages=167 |edition=2nd}}</ref>