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Hydrogen wave functions and their lifetimes

Introduction

The hydrogen atom is one of the most interesting quantum systems for which the Schrödinger equation can be solved exactly. On this page the solutions for the time independent Schröndinger equation (stationary states) are listed. These solutions are labeled by three quantum numbers: n, l and m. These solutions are produced by a program I wrote in the Maxima language. This program can be very useful if you want to find the wavefunction for a high energy stationary state (large value of n), as doing this by hand becomes a large amount of work. Together with the wavefunctions I have also listed the transition rates: the chance the particle spontaneously transitions to a lower energy state in one second. The different transition rates in which an electron can decay together define the lifetime of the state \tau=\frac{{1}}{{{A}_{{1}}+{A}_{{2}}+\ldots+{A}_{{n}}}} , which is defined as the time after which there's only a \frac{{1}}{{e}} chance of the electron still being in the starting state.

List of wavefunctions

(n, l, m) Wavefunction Decays to With transition rates (1/s) Lifetime of state (s)
1, 0, 0
{{e^ {- {{r}\over{a}} }}\over{\sqrt{\pi}\,a^{{{3}\over{2}}}}}
\infty
2, 0, 0
-{{\left(r-2\,a\right)\,e^ {- {{r}\over{2\,a}} }}\over{2^{{{5}\over{2}}}\,\sqrt{\pi}\,a^{{{5}\over{2}}}}}
\infty
2, 1, -1
{{r\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{2\,a}}}}\over{8\,\sqrt{\pi}\,a^{{{5}\over{2}}}}}
1,0,0
6.268 \times 10^{+8}
1.595 \times 10^{-9}
2, 1, 0
{{r\,e^ {- {{r}\over{2\,a}} }\,\cos {\it θ}}\over{2^{{{5}\over{2}}}\,\sqrt{\pi}\,a^{{{5}\over{2}}}}}
1,0,0
6.268 \times 10^{+8}
1.595 \times 10^{-9}
2, 1, 1
{{r\,e^ {- {{r}\over{2\,a}} }\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{8\,\sqrt{\pi}\,a^{{{5}\over{2}}}}}
1,0,0
6.268 \times 10^{+8}
1.595 \times 10^{-9}
3, 0, 0
{{\left(2\,r^2-18\,a\,r+27\,a^2\right)\,e^ {- {{r}\over{3\,a}} }}\over{3^{{{9}\over{2}}}\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,1,1
2,1,0
2,1,-1
2.105 \times 10^{+6}
2.105 \times 10^{+6}
2.105 \times 10^{+6}
1.583 \times 10^{-7}
3, 1, -1
-{{\left(r^2-6\,a\,r\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{3\,a}}}}\over{81\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,0,0
1,0,0
2.246 \times 10^{+7}
1.673 \times 10^{+8}
5.268 \times 10^{-9}
3, 1, 0
-{{\left(2\,r^2-12\,a\,r\right)\,e^ {- {{r}\over{3\,a}} }\,\cos {\it θ}}\over{81\,\sqrt{2}\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,0,0
1,0,0
2.246 \times 10^{+7}
1.673 \times 10^{+8}
5.268 \times 10^{-9}
3, 1, 1
-{{\left(r^2-6\,a\,r\right)\,e^ {- {{r}\over{3\,a}} }\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{81\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,0,0
1,0,0
2.246 \times 10^{+7}
1.673 \times 10^{+8}
5.268 \times 10^{-9}
3, 2, -2
{{r^2\,\sin ^2{\it θ}\,e^{-2\,i\,{\it φ}-{{r}\over{3\,a}}}}\over{162\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,1,-1
6.468 \times 10^{+7}
1.545 \times 10^{-8}
3, 2, -1
{{r^2\,\cos {\it θ}\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{3\,a}}}}\over{81\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,1,0
2,1,-1
3.234 \times 10^{+7}
3.234 \times 10^{+7}
1.545 \times 10^{-8}
3, 2, 0
-{{e^ {- {{r}\over{3\,a}} }\,\left(3\,r^2\,\sin ^2{\it θ}-2\,r^2\right)}\over{\sqrt{2}\,3^{{{9}\over{2}}}\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,1,1
2,1,0
2,1,-1
failed
failed
3, 2, 1
{{r^2\,e^ {- {{r}\over{3\,a}} }\,\cos {\it θ}\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{81\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,1,1
2,1,0
3.234 \times 10^{+7}
3.234 \times 10^{+7}
1.545 \times 10^{-8}
3, 2, 2
{{r^2\,e^ {- {{r}\over{3\,a}} }\,\sin ^2{\it θ}\,e^{2\,i\,{\it φ}}}\over{162\,\sqrt{\pi}\,a^{{{7}\over{2}}}}}
2,1,1
6.468 \times 10^{+7}
1.545 \times 10^{-8}
4, 0, 0
-{{\left(r^3-24\,a\,r^2+144\,a^2\,r-192\,a^3\right)\,e^ {- {{r}\over{4\,a}} }}\over{1536\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,1,1
3,1,0
3,1,-1
2,1,1
2,1,0
2,1,-1
6.121 \times 10^{+5}
6.121 \times 10^{+5}
6.121 \times 10^{+5}
8.598 \times 10^{+5}
8.598 \times 10^{+5}
8.598 \times 10^{+5}
2.264 \times 10^{-7}
4, 1, -1
{{\left(r^3-20\,a\,r^2+80\,a^2\,r\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{4\,a}}}}\over{2^{{{19}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,0
3,2,-1
3,2,-2
3,0,0
2,0,0
1,0,0
3.477 \times 10^{+4}
1.043 \times 10^{+5}
2.086 \times 10^{+5}
3.066 \times 10^{+6}
9.673 \times 10^{+6}
6.822 \times 10^{+7}
1.229 \times 10^{-8}
4, 1, 0
{{\left(r^3-20\,a\,r^2+80\,a^2\,r\right)\,e^ {- {{r}\over{4\,a}} }\,\cos {\it θ}}\over{512\,\sqrt{5}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,1
3,2,0
3,2,-1
3,0,0
2,0,0
1,0,0
1.043 \times 10^{+5}
1.39 \times 10^{+5}
1.043 \times 10^{+5}
3.066 \times 10^{+6}
9.673 \times 10^{+6}
6.822 \times 10^{+7}
1.229 \times 10^{-8}
4, 1, 1
{{\left(r^3-20\,a\,r^2+80\,a^2\,r\right)\,\sin {\it θ}\,e^{i\,{\it φ}-{{r}\over{4\,a}}}}\over{2^{{{19}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,2
3,2,1
3,2,0
3,0,0
2,0,0
1,0,0
2.086 \times 10^{+5}
1.043 \times 10^{+5}
3.477 \times 10^{+4}
3.066 \times 10^{+6}
9.673 \times 10^{+6}
6.822 \times 10^{+7}
1.229 \times 10^{-8}
4, 2, -2
-{{\left(r^3-12\,a\,r^2\right)\,\sin ^2{\it θ}\,e^{-2\,i\,{\it φ}-{{r}\over{4\,a}}}}\over{2^{{{21}\over{2}}}\,\sqrt{3}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,1,-1
2,1,-1
7.041 \times 10^{+6}
2.063 \times 10^{+7}
3.612 \times 10^{-8}
4, 2, -1
-{{\left(r^3-12\,a\,r^2\right)\,\cos {\it θ}\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{4\,a}}}}\over{2^{{{19}\over{2}}}\,\sqrt{3}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,1,0
3,1,-1
2,1,0
2,1,-1
3.52 \times 10^{+6}
3.52 \times 10^{+6}
1.031 \times 10^{+7}
1.031 \times 10^{+7}
3.612 \times 10^{-8}
4, 2, 0
{{e^ {- {{r}\over{4\,a}} }\,\left(\left(3\,r^3-36\,a\,r^2\right)\,\sin ^2{\it θ}-2\,r^3+24\,a\,r^2\right)}\over{3072\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,1,1
3,1,0
3,1,-1
2,1,1
2,1,0
2,1,-1
1.173 \times 10^{+6}
4.694 \times 10^{+6}
1.173 \times 10^{+6}
3.439 \times 10^{+6}
1.375 \times 10^{+7}
3.439 \times 10^{+6}
3.612 \times 10^{-8}
4, 2, 1
-{{\left(r^3-12\,a\,r^2\right)\,e^ {- {{r}\over{4\,a}} }\,\cos {\it θ}\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{2^{{{19}\over{2}}}\,\sqrt{3}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,1,1
3,1,0
2,1,1
2,1,0
3.52 \times 10^{+6}
3.52 \times 10^{+6}
1.031 \times 10^{+7}
1.031 \times 10^{+7}
3.612 \times 10^{-8}
4, 2, 2
{{\left(\left(r^3-12\,a\,r^2\right)\,\cos ^2{\it θ}-r^3+12\,a\,r^2\right)\,e^{2\,i\,{\it φ}-{{r}\over{4\,a}}}}\over{2^{{{21}\over{2}}}\,\sqrt{3}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
Is 1 zero or nonzero?p; 3,1,1
2,1,1
7.041 \times 10^{+6}
2.063 \times 10^{+7}
3.612 \times 10^{-8}
4, 3, -3
{{r^3\,\sin ^3{\it θ}\,e^{-3\,i\,{\it φ}-{{r}\over{4\,a}}}}\over{6144\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,-2
1.379 \times 10^{+7}
7.248 \times 10^{-8}
4, 3, -2
{{r^3\,\cos {\it θ}\,\sin ^2{\it θ}\,e^{-2\,i\,{\it φ}-{{r}\over{4\,a}}}}\over{2^{{{21}\over{2}}}\,\sqrt{3}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,-1
3,2,-2
9.196 \times 10^{+6}
4.598 \times 10^{+6}
7.248 \times 10^{-8}
4, 3, -1
{{\left(5\,r^3\,\cos ^2{\it θ}-r^3\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{4\,a}}}}\over{2048\,\sqrt{3}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,0
3,2,-1
3,2,-2
failed
failed
4, 3, 0
{{e^ {- {{r}\over{4\,a}} }\,\left(5\,r^3\,\cos ^3{\it θ}-3\,r^3\,\cos {\it θ}\right)}\over{3072\,\sqrt{5}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,1
3,2,0
3,2,-1
failed
failed
4, 3, 1
{{e^ {- {{r}\over{4\,a}} }\,\left(5\,r^3\,\cos ^2{\it θ}-r^3\right)\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{2048\,\sqrt{3}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,2
3,2,1
3,2,0
9.196 \times 10^{+5}
7.357 \times 10^{+6}
5.518 \times 10^{+6}
7.248 \times 10^{-8}
4, 3, 2
{{r^3\,\cos {\it θ}\,\sin ^2{\it θ}\,e^{2\,i\,{\it φ}-{{r}\over{4\,a}}}}\over{2^{{{21}\over{2}}}\,\sqrt{3}\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,2
3,2,1
4.598 \times 10^{+6}
9.196 \times 10^{+6}
7.248 \times 10^{-8}
4, 3, 3
{{r^3\,e^ {- {{r}\over{4\,a}} }\,\sin ^3{\it θ}\,e^{3\,i\,{\it φ}}}\over{6144\,\sqrt{\pi}\,a^{{{9}\over{2}}}}}
3,2,2
1.379 \times 10^{+7}
7.248 \times 10^{-8}
5, 0, 0
{{\left(2\,r^4-100\,a\,r^3+1500\,a^2\,r^2-7500\,a^3\,r+9375\,a^4\right)\,e^ {- {{r}\over{5\,a}} }}\over{3\,5^{{{13}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,1,1
4,1,0
4,1,-1
3,1,1
3,1,0
3,1,-1
2,1,1
2,1,0
2,1,-1
2.151 \times 10^{+5}
2.151 \times 10^{+5}
2.151 \times 10^{+5}
3.017 \times 10^{+5}
3.017 \times 10^{+5}
3.017 \times 10^{+5}
4.297 \times 10^{+5}
4.297 \times 10^{+5}
4.297 \times 10^{+5}
3.521 \times 10^{-7}
5, 1, -1
-{{\left(2\,r^4-90\,a\,r^3+1125\,a^2\,r^2-3750\,a^3\,r\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{5\,a}}}}\over{3\,5^{{{13}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,0
4,2,-1
4,2,-2
4,0,0
3,2,0
3,2,-1
3,2,-2
3,0,0
2,0,0
1,0,0
1.885 \times 10^{+4}
5.656 \times 10^{+4}
1.131 \times 10^{+5}
7.375 \times 10^{+5}
1.496 \times 10^{+4}
4.488 \times 10^{+4}
8.977 \times 10^{+4}
1.638 \times 10^{+6}
4.951 \times 10^{+6}
3.439 \times 10^{+7}
2.377 \times 10^{-8}
5, 1, 0
-{{\left(4\,r^4-180\,a\,r^3+2250\,a^2\,r^2-7500\,a^3\,r\right)\,e^ {- {{r}\over{5\,a}} }\,\cos {\it θ}}\over{3\,\sqrt{2}\,5^{{{13}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,1
4,2,0
4,2,-1
4,0,0
3,2,1
3,2,0
3,2,-1
3,0,0
2,0,0
1,0,0
5.656 \times 10^{+4}
7.542 \times 10^{+4}
5.656 \times 10^{+4}
7.375 \times 10^{+5}
4.488 \times 10^{+4}
5.984 \times 10^{+4}
4.488 \times 10^{+4}
1.638 \times 10^{+6}
4.951 \times 10^{+6}
3.439 \times 10^{+7}
2.377 \times 10^{-8}
5, 1, 1
-{{\left(2\,r^4-90\,a\,r^3+1125\,a^2\,r^2-3750\,a^3\,r\right)\,e^ {- {{r}\over{5\,a}} }\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{3\,5^{{{13}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,2
4,2,1
4,2,0
4,0,0
3,2,2
3,2,1
3,2,0
3,0,0
2,0,0
1,0,0
1.131 \times 10^{+5}
5.656 \times 10^{+4}
1.885 \times 10^{+4}
7.375 \times 10^{+5}
8.977 \times 10^{+4}
4.488 \times 10^{+4}
1.496 \times 10^{+4}
1.638 \times 10^{+6}
4.951 \times 10^{+6}
3.439 \times 10^{+7}
2.377 \times 10^{-8}
5, 2, -2
{{\left(2\,r^4-70\,a\,r^3+525\,a^2\,r^2\right)\,\sin ^2{\it θ}\,e^{-2\,i\,{\it φ}-{{r}\over{5\,a}}}}\over{31250\,\sqrt{3}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,-1
4,3,-2
4,3,-3
4,1,-1
3,1,-1
2,1,-1
2.404 \times 10^{+3}
1.202 \times 10^{+4}
3.607 \times 10^{+4}
1.486 \times 10^{+6}
3.393 \times 10^{+6}
9.43 \times 10^{+6}
6.963 \times 10^{-8}
5, 2, -1
{{\left(2\,r^4-70\,a\,r^3+525\,a^2\,r^2\right)\,\cos {\it θ}\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{5\,a}}}}\over{15625\,\sqrt{3}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,0
4,3,-1
4,3,-2
4,1,0
4,1,-1
3,1,0
3,1,-1
2,1,0
2,1,-1
7.214 \times 10^{+3}
1.923 \times 10^{+4}
2.404 \times 10^{+4}
7.432 \times 10^{+5}
7.432 \times 10^{+5}
1.696 \times 10^{+6}
1.696 \times 10^{+6}
4.715 \times 10^{+6}
4.715 \times 10^{+6}
6.963 \times 10^{-8}
5, 2, 0
-{{e^ {- {{r}\over{5\,a}} }\,\left(\left(6\,r^4-210\,a\,r^3+1575\,a^2\,r^2\right)\,\sin ^2{\it θ}-4\,r^4+140\,a\,r^3-1050\,a^2\,r^2\right)}\over{46875\,\sqrt{2}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,1
4,3,0
4,3,-1
4,1,1
4,1,0
4,1,-1
3,1,1
3,1,0
3,1,-1
2,1,1
2,1,0
2,1,-1
1.442 \times 10^{+4}
2.164 \times 10^{+4}
1.442 \times 10^{+4}
2.477 \times 10^{+5}
9.91 \times 10^{+5}
2.477 \times 10^{+5}
5.655 \times 10^{+5}
2.262 \times 10^{+6}
5.655 \times 10^{+5}
1.571 \times 10^{+6}
6.287 \times 10^{+6}
1.571 \times 10^{+6}
6.963 \times 10^{-8}
5, 2, 1
{{\left(2\,r^4-70\,a\,r^3+525\,a^2\,r^2\right)\,e^ {- {{r}\over{5\,a}} }\,\cos {\it θ}\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{15625\,\sqrt{3}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,2
4,3,1
4,3,0
4,1,1
4,1,0
3,1,1
3,1,0
2,1,1
2,1,0
2.404 \times 10^{+4}
1.923 \times 10^{+4}
7.214 \times 10^{+3}
7.432 \times 10^{+5}
7.432 \times 10^{+5}
1.696 \times 10^{+6}
1.696 \times 10^{+6}
4.715 \times 10^{+6}
4.715 \times 10^{+6}
6.963 \times 10^{-8}
5, 2, 2
-{{e^ {- {{r}\over{5\,a}} }\,\left(\left(2\,r^4-70\,a\,r^3+525\,a^2\,r^2\right)\,\cos ^2{\it θ}-2\,r^4+70\,a\,r^3-525\,a^2\,r^2\right)\,e^{2\,i\,{\it φ}}}\over{31250\,\sqrt{3}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,3
4,3,2
4,3,1
4,1,1
3,1,1
2,1,1
3.607 \times 10^{+4}
1.202 \times 10^{+4}
2.404 \times 10^{+3}
1.486 \times 10^{+6}
3.393 \times 10^{+6}
9.43 \times 10^{+6}
6.963 \times 10^{-8}
5, 3, -3
-{{\left(r^4-20\,a\,r^3\right)\,\sin ^3{\it θ}\,e^{-3\,i\,{\it φ}-{{r}\over{5\,a}}}}\over{46875\,2^{{{3}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,-2
3,2,-2
2.585 \times 10^{+6}
4.544 \times 10^{+6}
1.402 \times 10^{-7}
5, 3, -2
-{{\left(r^4-20\,a\,r^3\right)\,\cos {\it θ}\,\sin ^2{\it θ}\,e^{-2\,i\,{\it φ}-{{r}\over{5\,a}}}}\over{31250\,\sqrt{3}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,-1
4,2,-2
3,2,-1
3,2,-2
1.723 \times 10^{+6}
8.619 \times 10^{+5}
3.029 \times 10^{+6}
1.514 \times 10^{+6}
1.402 \times 10^{-7}
5, 3, -1
-{{\left(\left(5\,r^4-100\,a\,r^3\right)\,\cos ^2{\it θ}-r^4+20\,a\,r^3\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{5\,a}}}}\over{2^{{{3}\over{2}}}\,\sqrt{3}\,5^{{{13}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,0
4,2,-1
4,2,-2
3,2,0
3,2,-1
3,2,-2
1.034 \times 10^{+6}
1.379 \times 10^{+6}
1.723 \times 10^{+5}
1.817 \times 10^{+6}
2.423 \times 10^{+6}
3.029 \times 10^{+5}
1.402 \times 10^{-7}
5, 3, 0
-{{e^ {- {{r}\over{5\,a}} }\,\left(\left(5\,r^4-100\,a\,r^3\right)\,\cos ^3{\it θ}+\left(60\,a\,r^3-3\,r^4\right)\,\cos {\it θ}\right)}\over{3\,\sqrt{2}\,5^{{{13}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,1
4,2,0
4,2,-1
3,2,1
3,2,0
3,2,-1
5.171 \times 10^{+5}
1.551 \times 10^{+6}
5.171 \times 10^{+5}
9.089 \times 10^{+5}
2.726 \times 10^{+6}
9.089 \times 10^{+5}
1.402 \times 10^{-7}
5, 3, 1
-{{e^ {- {{r}\over{5\,a}} }\,\left(\left(5\,r^4-100\,a\,r^3\right)\,\cos ^2{\it θ}-r^4+20\,a\,r^3\right)\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{2^{{{3}\over{2}}}\,\sqrt{3}\,5^{{{13}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,2
4,2,1
4,2,0
3,2,2
3,2,1
3,2,0
1.723 \times 10^{+5}
1.379 \times 10^{+6}
1.034 \times 10^{+6}
3.029 \times 10^{+5}
2.423 \times 10^{+6}
1.817 \times 10^{+6}
1.402 \times 10^{-7}
5, 3, 2
-{{\left(r^4-20\,a\,r^3\right)\,e^ {- {{r}\over{5\,a}} }\,\cos {\it θ}\,\sin ^2{\it θ}\,e^{2\,i\,{\it φ}}}\over{31250\,\sqrt{3}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,2
4,2,1
3,2,2
3,2,1
8.619 \times 10^{+5}
1.723 \times 10^{+6}
1.514 \times 10^{+6}
3.029 \times 10^{+6}
1.402 \times 10^{-7}
5, 3, 3
-{{\left(r^4-20\,a\,r^3\right)\,e^ {- {{r}\over{5\,a}} }\,\sin ^3{\it θ}\,e^{3\,i\,{\it φ}}}\over{46875\,2^{{{3}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,2,2
3,2,2
2.585 \times 10^{+6}
4.544 \times 10^{+6}
1.402 \times 10^{-7}
5, 4, -4
{{r^4\,\sin ^4{\it θ}\,e^{-4\,i\,{\it φ}-{{r}\over{5\,a}}}}\over{375000\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,-3
4.256 \times 10^{+6}
2.349 \times 10^{-7}
5, 4, -3
{{r^4\,\cos {\it θ}\,\sin ^3{\it θ}\,e^{-3\,i\,{\it φ}-{{r}\over{5\,a}}}}\over{46875\,2^{{{3}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,-2
4,3,-3
3.192 \times 10^{+6}
1.064 \times 10^{+6}
2.349 \times 10^{-7}
5, 4, -2
-{{\left(7\,r^4\,\cos ^4{\it θ}-8\,r^4\,\cos ^2{\it θ}+r^4\right)\,e^{-2\,i\,{\it φ}-{{r}\over{5\,a}}}}\over{187500\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,-1
4,3,-2
4,3,-3
2.28 \times 10^{+6}
1.824 \times 10^{+6}
1.52 \times 10^{+5}
2.349 \times 10^{-7}
5, 4, -1
{{\left(7\,r^4\,\cos ^3{\it θ}-3\,r^4\,\cos {\it θ}\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{5\,a}}}}\over{46875\,2^{{{3}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,0
4,3,-1
4,3,-2
failed
failed
5, 4, 0
{{e^ {- {{r}\over{5\,a}} }\,\left(35\,r^4\,\sin ^4{\it θ}-40\,r^4\,\sin ^2{\it θ}+8\,r^4\right)}\over{3\,2^{{{5}\over{2}}}\,5^{{{13}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,1
4,3,0
4,3,-1
failed
failed
5, 4, 1
{{e^ {- {{r}\over{5\,a}} }\,\left(7\,r^4\,\cos ^3{\it θ}-3\,r^4\,\cos {\it θ}\right)\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{46875\,2^{{{3}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,2
4,3,1
4,3,0
4.56 \times 10^{+5}
2.28 \times 10^{+6}
1.52 \times 10^{+6}
2.349 \times 10^{-7}
5, 4, 2
-{{e^ {- {{r}\over{5\,a}} }\,\left(7\,r^4\,\cos ^4{\it θ}-8\,r^4\,\cos ^2{\it θ}+r^4\right)\,e^{2\,i\,{\it φ}}}\over{187500\,\sqrt{7}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,3
4,3,2
4,3,1
1.52 \times 10^{+5}
1.824 \times 10^{+6}
2.28 \times 10^{+6}
2.349 \times 10^{-7}
5, 4, 3
{{r^4\,e^ {- {{r}\over{5\,a}} }\,\cos {\it θ}\,\sin ^3{\it θ}\,e^{3\,i\,{\it φ}}}\over{46875\,2^{{{3}\over{2}}}\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,3
4,3,2
1.064 \times 10^{+6}
3.192 \times 10^{+6}
2.349 \times 10^{-7}
5, 4, 4
{{r^4\,e^ {- {{r}\over{5\,a}} }\,\sin ^4{\it θ}\,e^{4\,i\,{\it φ}}}\over{375000\,\sqrt{\pi}\,a^{{{11}\over{2}}}}}
4,3,3
4.256 \times 10^{+6}
2.349 \times 10^{-7}
6, 0, 0
-{{\left(r^5-90\,a\,r^4+2700\,a^2\,r^3-32400\,a^3\,r^2+145800\,a^4\,r-174960\,a^5\right)\,e^ {- {{r}\over{6\,a}} }}\over{5\,2^{{{11}\over{2}}}\,3^{{{17}\over{2}}}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,1,1
5,1,0
5,1,-1
4,1,1
4,1,0
4,1,-1
3,1,1
3,1,0
3,1,-1
2,1,1
2,1,0
2,1,-1
8.943 \times 10^{+4}
8.943 \times 10^{+4}
8.943 \times 10^{+4}
1.194 \times 10^{+5}
1.194 \times 10^{+5}
1.194 \times 10^{+5}
1.691 \times 10^{+5}
1.691 \times 10^{+5}
1.691 \times 10^{+5}
2.451 \times 10^{+5}
2.451 \times 10^{+5}
2.451 \times 10^{+5}
5.348 \times 10^{-7}
6, 1, -1
{{\left(r^5-84\,a\,r^4+2268\,a^2\,r^3-22680\,a^3\,r^2+68040\,a^4\,r\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{6\,a}}}}\over{419904\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,2,0
5,2,-1
5,2,-2
5,0,0
4,2,0
4,2,-1
4,2,-2
4,0,0
3,2,0
3,2,-1
3,2,-2
3,0,0
2,0,0
1,0,0
9.598 \times 10^{+3}
2.879 \times 10^{+4}
5.759 \times 10^{+4}
2.43 \times 10^{+5}
9.421 \times 10^{+3}
2.826 \times 10^{+4}
5.653 \times 10^{+4}
4.458 \times 10^{+5}
7.828 \times 10^{+3}
2.348 \times 10^{+4}
4.697 \times 10^{+4}
9.556 \times 10^{+5}
2.859 \times 10^{+6}
1.973 \times 10^{+7}
4.079 \times 10^{-8}
6, 1, 0
{{\left(r^5-84\,a\,r^4+2268\,a^2\,r^3-22680\,a^3\,r^2+68040\,a^4\,r\right)\,e^ {- {{r}\over{6\,a}} }\,\cos {\it θ}}\over{6561\,2^{{{11}\over{2}}}\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,2,1
5,2,0
5,2,-1
5,0,0
4,2,1
4,2,0
4,2,-1
4,0,0
3,2,1
3,2,0
3,2,-1
3,0,0
2,0,0
1,0,0
2.879 \times 10^{+4}
3.839 \times 10^{+4}
2.879 \times 10^{+4}
2.43 \times 10^{+5}
2.826 \times 10^{+4}
3.768 \times 10^{+4}
2.826 \times 10^{+4}
4.458 \times 10^{+5}
2.348 \times 10^{+4}
3.131 \times 10^{+4}
2.348 \times 10^{+4}
9.556 \times 10^{+5}
2.859 \times 10^{+6}
1.973 \times 10^{+7}
4.079 \times 10^{-8}
6, 1, 1
{{\left(r^5-84\,a\,r^4+2268\,a^2\,r^3-22680\,a^3\,r^2+68040\,a^4\,r\right)\,e^ {- {{r}\over{6\,a}} }\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{419904\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,2,2
5,2,1
5,2,0
5,0,0
4,2,2
4,2,1
4,2,0
4,0,0
3,2,2
3,2,1
3,2,0
3,0,0
2,0,0
1,0,0
5.759 \times 10^{+4}
2.879 \times 10^{+4}
9.598 \times 10^{+3}
2.43 \times 10^{+5}
5.653 \times 10^{+4}
2.826 \times 10^{+4}
9.421 \times 10^{+3}
4.458 \times 10^{+5}
4.697 \times 10^{+4}
2.348 \times 10^{+4}
7.828 \times 10^{+3}
9.556 \times 10^{+5}
2.859 \times 10^{+6}
1.973 \times 10^{+7}
4.079 \times 10^{-8}
6, 2, -2
-{{\left(r^5-72\,a\,r^4+1512\,a^2\,r^3-9072\,a^3\,r^2\right)\,\sin ^2{\it θ}\,e^{-2\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{6561\,2^{{{15}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,-1
5,3,-2
5,3,-3
5,1,-1
4,3,-1
4,3,-2
4,3,-3
4,1,-1
3,1,-1
2,1,-1
1.861 \times 10^{+3}
9.309 \times 10^{+3}
2.792 \times 10^{+4}
4.497 \times 10^{+5}
1.021 \times 10^{+3}
5.109 \times 10^{+3}
1.532 \times 10^{+4}
8.626 \times 10^{+5}
1.878 \times 10^{+6}
5.147 \times 10^{+6}
1.19 \times 10^{-7}
6, 2, -1
-{{\left(r^5-72\,a\,r^4+1512\,a^2\,r^3-9072\,a^3\,r^2\right)\,\cos {\it θ}\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{6\,a}}}}\over{6561\,2^{{{13}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,0
5,3,-1
5,3,-2
5,1,0
5,1,-1
4,3,0
4,3,-1
4,3,-2
4,1,0
4,1,-1
3,1,0
3,1,-1
2,1,0
2,1,-1
5.585 \times 10^{+3}
1.489 \times 10^{+4}
1.861 \times 10^{+4}
2.248 \times 10^{+5}
2.248 \times 10^{+5}
3.065 \times 10^{+3}
8.175 \times 10^{+3}
1.021 \times 10^{+4}
4.313 \times 10^{+5}
4.313 \times 10^{+5}
9.394 \times 10^{+5}
9.394 \times 10^{+5}
2.573 \times 10^{+6}
2.573 \times 10^{+6}
1.19 \times 10^{-7}
6, 2, 0
{{e^ {- {{r}\over{6\,a}} }\,\left(\left(3\,r^5-216\,a\,r^4+4536\,a^2\,r^3-27216\,a^3\,r^2\right)\,\sin ^2{\it θ}-2\,r^5+144\,a\,r^4-3024\,a^2\,r^3+18144\,a^3\,r^2\right)}\over{128\,3^{{{17}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,1
5,3,0
5,3,-1
5,1,1
5,1,0
5,1,-1
4,3,1
4,3,0
4,3,-1
4,1,1
4,1,0
4,1,-1
3,1,1
3,1,0
3,1,-1
2,1,1
2,1,0
2,1,-1
1.117 \times 10^{+4}
1.675 \times 10^{+4}
1.117 \times 10^{+4}
7.495 \times 10^{+4}
2.998 \times 10^{+5}
7.495 \times 10^{+4}
6.131 \times 10^{+3}
9.197 \times 10^{+3}
6.131 \times 10^{+3}
1.437 \times 10^{+5}
5.751 \times 10^{+5}
1.437 \times 10^{+5}
3.131 \times 10^{+5}
1.252 \times 10^{+6}
3.131 \times 10^{+5}
8.579 \times 10^{+5}
3.431 \times 10^{+6}
8.579 \times 10^{+5}
1.19 \times 10^{-7}
6, 2, 1
-{{\left(r^5-72\,a\,r^4+1512\,a^2\,r^3-9072\,a^3\,r^2\right)\,e^ {- {{r}\over{6\,a}} }\,\cos {\it θ}\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{6561\,2^{{{13}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,2
5,3,1
5,3,0
5,1,1
5,1,0
4,3,2
4,3,1
4,3,0
4,1,1
4,1,0
3,1,1
3,1,0
2,1,1
2,1,0
1.861 \times 10^{+4}
1.489 \times 10^{+4}
5.585 \times 10^{+3}
2.248 \times 10^{+5}
2.248 \times 10^{+5}
1.021 \times 10^{+4}
8.175 \times 10^{+3}
3.065 \times 10^{+3}
4.313 \times 10^{+5}
4.313 \times 10^{+5}
9.394 \times 10^{+5}
9.394 \times 10^{+5}
2.573 \times 10^{+6}
2.573 \times 10^{+6}
1.19 \times 10^{-7}
6, 2, 2
{{e^ {- {{r}\over{6\,a}} }\,\left(\left(r^5-72\,a\,r^4+1512\,a^2\,r^3-9072\,a^3\,r^2\right)\,\cos ^2{\it θ}-r^5+72\,a\,r^4-1512\,a^2\,r^3+9072\,a^3\,r^2\right)\,e^{2\,i\,{\it φ}}}\over{6561\,2^{{{15}\over{2}}}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,3
5,3,2
5,3,1
5,1,1
4,3,3
4,3,2
4,3,1
4,1,1
3,1,1
2,1,1
2.792 \times 10^{+4}
9.309 \times 10^{+3}
1.861 \times 10^{+3}
4.497 \times 10^{+5}
1.532 \times 10^{+4}
5.109 \times 10^{+3}
1.021 \times 10^{+3}
8.626 \times 10^{+5}
1.878 \times 10^{+6}
5.147 \times 10^{+6}
1.19 \times 10^{-7}
6, 3, -3
{{\left(r^5-54\,a\,r^4+648\,a^2\,r^3\right)\,\sin ^3{\it θ}\,e^{-3\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{5038848\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,-2
5,4,-3
5,4,-4
5,2,-2
4,2,-2
3,2,-2
316.0
2.212 \times 10^{+3}
8.85 \times 10^{+3}
7.236 \times 10^{+5}
1.287 \times 10^{+6}
2.147 \times 10^{+6}
2.398 \times 10^{-7}
6, 3, -2
{{\left(r^5-54\,a\,r^4+648\,a^2\,r^3\right)\,\cos {\it θ}\,\sin ^2{\it θ}\,e^{-2\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{2^{{{15}\over{2}}}\,3^{{{17}\over{2}}}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,-1
5,4,-2
5,4,-3
5,2,-1
5,2,-2
4,2,-1
4,2,-2
3,2,-1
3,2,-2
948.2
3.793 \times 10^{+3}
6.637 \times 10^{+3}
4.824 \times 10^{+5}
2.412 \times 10^{+5}
8.584 \times 10^{+5}
4.292 \times 10^{+5}
1.431 \times 10^{+6}
7.157 \times 10^{+5}
2.398 \times 10^{-7}
6, 3, -1
{{\left(\left(5\,r^5-270\,a\,r^4+3240\,a^2\,r^3\right)\,\cos ^2{\it θ}-r^5+54\,a\,r^4-648\,a^2\,r^3\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{6\,a}}}}\over{256\,3^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,0
5,4,-1
5,4,-2
5,2,0
5,2,-1
5,2,-2
4,2,0
4,2,-1
4,2,-2
3,2,0
3,2,-1
3,2,-2
1.896 \times 10^{+3}
4.741 \times 10^{+3}
4.741 \times 10^{+3}
2.894 \times 10^{+5}
3.859 \times 10^{+5}
4.824 \times 10^{+4}
5.15 \times 10^{+5}
6.867 \times 10^{+5}
8.584 \times 10^{+4}
8.588 \times 10^{+5}
1.145 \times 10^{+6}
1.431 \times 10^{+5}
2.398 \times 10^{-7}
6, 3, 0
{{e^ {- {{r}\over{6\,a}} }\,\left(\left(5\,r^5-270\,a\,r^4+3240\,a^2\,r^3\right)\,\cos ^3{\it θ}+\left(-3\,r^5+162\,a\,r^4-1944\,a^2\,r^3\right)\,\cos {\it θ}\right)}\over{2519424\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,1
5,4,0
5,4,-1
5,2,1
5,2,0
5,2,-1
4,2,1
4,2,0
4,2,-1
3,2,1
3,2,0
3,2,-1
3.16 \times 10^{+3}
5.057 \times 10^{+3}
3.16 \times 10^{+3}
1.447 \times 10^{+5}
4.341 \times 10^{+5}
1.447 \times 10^{+5}
2.575 \times 10^{+5}
7.726 \times 10^{+5}
2.575 \times 10^{+5}
4.294 \times 10^{+5}
1.288 \times 10^{+6}
4.294 \times 10^{+5}
2.398 \times 10^{-7}
6, 3, 1
{{e^ {- {{r}\over{6\,a}} }\,\left(\left(5\,r^5-270\,a\,r^4+3240\,a^2\,r^3\right)\,\cos ^2{\it θ}-r^5+54\,a\,r^4-648\,a^2\,r^3\right)\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{256\,3^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,2
5,4,1
5,4,0
5,2,2
5,2,1
5,2,0
4,2,2
4,2,1
4,2,0
3,2,2
3,2,1
3,2,0
4.741 \times 10^{+3}
4.741 \times 10^{+3}
1.896 \times 10^{+3}
4.824 \times 10^{+4}
3.859 \times 10^{+5}
2.894 \times 10^{+5}
8.584 \times 10^{+4}
6.867 \times 10^{+5}
5.15 \times 10^{+5}
1.431 \times 10^{+5}
1.145 \times 10^{+6}
8.588 \times 10^{+5}
2.398 \times 10^{-7}
6, 3, 2
{{\left(r^5-54\,a\,r^4+648\,a^2\,r^3\right)\,\cos {\it θ}\,\sin ^2{\it θ}\,e^{2\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{2^{{{15}\over{2}}}\,3^{{{17}\over{2}}}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,3
5,4,2
5,4,1
5,2,2
5,2,1
4,2,2
4,2,1
3,2,2
3,2,1
6.637 \times 10^{+3}
3.793 \times 10^{+3}
948.2
2.412 \times 10^{+5}
4.824 \times 10^{+5}
4.292 \times 10^{+5}
8.584 \times 10^{+5}
7.157 \times 10^{+5}
1.431 \times 10^{+6}
2.398 \times 10^{-7}
6, 3, 3
-{{e^ {- {{r}\over{6\,a}} }\,\left(\left(r^5-54\,a\,r^4+648\,a^2\,r^3\right)\,\cos ^2{\it θ}-r^5+54\,a\,r^4-648\,a^2\,r^3\right)\,\sin {\it θ}\,e^{3\,i\,{\it φ}}}\over{5038848\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,4
5,4,3
5,4,2
5,2,2
4,2,2
3,2,2
8.85 \times 10^{+3}
2.212 \times 10^{+3}
316.0
7.236 \times 10^{+5}
1.287 \times 10^{+6}
2.147 \times 10^{+6}
2.398 \times 10^{-7}
6, 4, -4
-{{\left(r^5-30\,a\,r^4\right)\,\sin ^4{\it θ}\,e^{-4\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{6561\,2^{{{19}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,-3
4,3,-3
1.106 \times 10^{+6}
1.373 \times 10^{+6}
4.032 \times 10^{-7}
6, 4, -3
-{{\left(r^5-30\,a\,r^4\right)\,\cos {\it θ}\,\sin ^3{\it θ}\,e^{-3\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{1679616\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,-2
5,3,-3
4,3,-2
4,3,-3
8.297 \times 10^{+5}
2.765 \times 10^{+5}
1.03 \times 10^{+6}
3.433 \times 10^{+5}
4.032 \times 10^{-7}
6, 4, -2
{{\left(\left(7\,r^5-210\,a\,r^4\right)\,\cos ^4{\it θ}+\left(240\,a\,r^4-8\,r^5\right)\,\cos ^2{\it θ}+r^5-30\,a\,r^4\right)\,e^{-2\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{6561\,2^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,-1
5,3,-2
5,3,-3
4,3,-1
4,3,-2
4,3,-3
5.926 \times 10^{+5}
4.741 \times 10^{+5}
3.951 \times 10^{+4}
7.358 \times 10^{+5}
5.886 \times 10^{+5}
4.905 \times 10^{+4}
4.032 \times 10^{-7}
6, 4, -1
-{{\left(\left(7\,r^5-210\,a\,r^4\right)\,\cos ^3{\it θ}+\left(90\,a\,r^4-3\,r^5\right)\,\cos {\it θ}\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{6\,a}}}}\over{1679616\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,0
5,3,-1
5,3,-2
4,3,0
4,3,-1
4,3,-2
3.951 \times 10^{+5}
5.926 \times 10^{+5}
1.185 \times 10^{+5}
4.905 \times 10^{+5}
7.358 \times 10^{+5}
1.471 \times 10^{+5}
4.032 \times 10^{-7}
6, 4, 0
-{{e^ {- {{r}\over{6\,a}} }\,\left(\left(35\,r^5-1050\,a\,r^4\right)\,\sin ^4{\it θ}+\left(1200\,a\,r^4-40\,r^5\right)\,\sin ^2{\it θ}+8\,r^5-240\,a\,r^4\right)}\over{16796160\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,1
5,3,0
5,3,-1
4,3,1
4,3,0
4,3,-1
2.37 \times 10^{+5}
6.321 \times 10^{+5}
2.37 \times 10^{+5}
2.943 \times 10^{+5}
7.848 \times 10^{+5}
2.943 \times 10^{+5}
4.032 \times 10^{-7}
6, 4, 1
-{{e^ {- {{r}\over{6\,a}} }\,\left(\left(7\,r^5-210\,a\,r^4\right)\,\cos ^3{\it θ}+\left(90\,a\,r^4-3\,r^5\right)\,\cos {\it θ}\right)\,\sin {\it θ}\,e^{i\,{\it φ}}}\over{1679616\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,2
5,3,1
5,3,0
4,3,2
4,3,1
4,3,0
1.185 \times 10^{+5}
5.926 \times 10^{+5}
3.951 \times 10^{+5}
1.471 \times 10^{+5}
7.358 \times 10^{+5}
4.905 \times 10^{+5}
4.032 \times 10^{-7}
6, 4, 2
{{e^ {- {{r}\over{6\,a}} }\,\left(\left(7\,r^5-210\,a\,r^4\right)\,\cos ^4{\it θ}+\left(240\,a\,r^4-8\,r^5\right)\,\cos ^2{\it θ}+r^5-30\,a\,r^4\right)\,e^{2\,i\,{\it φ}}}\over{6561\,2^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,3
5,3,2
5,3,1
4,3,3
4,3,2
4,3,1
3.951 \times 10^{+4}
4.741 \times 10^{+5}
5.926 \times 10^{+5}
4.905 \times 10^{+4}
5.886 \times 10^{+5}
7.358 \times 10^{+5}
4.032 \times 10^{-7}
6, 4, 3
-{{\left(r^5-30\,a\,r^4\right)\,e^ {- {{r}\over{6\,a}} }\,\cos {\it θ}\,\sin ^3{\it θ}\,e^{3\,i\,{\it φ}}}\over{1679616\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,3
5,3,2
4,3,3
4,3,2
2.765 \times 10^{+5}
8.297 \times 10^{+5}
3.433 \times 10^{+5}
1.03 \times 10^{+6}
4.032 \times 10^{-7}
6, 4, 4
-{{\left(r^5-30\,a\,r^4\right)\,e^ {- {{r}\over{6\,a}} }\,\sin ^4{\it θ}\,e^{4\,i\,{\it φ}}}\over{6561\,2^{{{19}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,3,3
4,3,3
1.106 \times 10^{+6}
1.373 \times 10^{+6}
4.032 \times 10^{-7}
6, 5, -5
{{r^5\,\sin ^5{\it θ}\,e^{-5\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{33592320\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,-4
1.645 \times 10^{+6}
6.076 \times 10^{-7}
6, 5, -4
{{r^5\,\cos {\it θ}\,\sin ^4{\it θ}\,e^{-4\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{6561\,2^{{{19}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,-3
5,4,-4
1.316 \times 10^{+6}
3.291 \times 10^{+5}
6.076 \times 10^{-7}
6, 5, -3
-{{\left(9\,r^5\,\sin ^5{\it θ}-8\,r^5\,\sin ^3{\it θ}\right)\,e^{-3\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{20155392\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,-2
5,4,-3
5,4,-4
failed
failed
6, 5, -2
-{{\left(3\,r^5\,\cos {\it θ}\,\sin ^4{\it θ}-2\,r^5\,\cos {\it θ}\,\sin ^2{\it θ}\right)\,e^{-2\,i\,{\it φ}-{{r}\over{6\,a}}}}\over{2^{{{17}\over{2}}}\,3^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,-1
5,4,-2
5,4,-3
failed
failed
6, 5, -1
{{\left(21\,r^5\,\cos ^4{\it θ}-14\,r^5\,\cos ^2{\it θ}+r^5\right)\,\sin {\it θ}\,e^{-i\,{\it φ}-{{r}\over{6\,a}}}}\over{2^{{{19}\over{2}}}\,3^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,0
5,4,-1
5,4,-2
failed
failed
6, 5, 0
{{e^ {- {{r}\over{6\,a}} }\,\left(63\,r^5\,\cos {\it θ}\,\sin ^4{\it θ}-56\,r^5\,\cos {\it θ}\,\sin ^2{\it θ}+8\,r^5\,\cos {\it θ}\right)}\over{50388480\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,1
5,4,0
5,4,-1
failed
failed
6, 5, 1
{{\left(21\,r^5\,\cos ^4{\it θ}-14\,r^5\,\cos ^2{\it θ}+r^5\right)\,\sin {\it θ}\,e^{i\,{\it φ}-{{r}\over{6\,a}}}}\over{2^{{{19}\over{2}}}\,3^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{7}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,2
5,4,1
5,4,0
2.194 \times 10^{+5}
8.777 \times 10^{+5}
5.485 \times 10^{+5}
6.076 \times 10^{-7}
6, 5, 2
-{{e^ {- {{r}\over{6\,a}} }\,\left(3\,r^5\,\cos {\it θ}\,\sin ^4{\it θ}-2\,r^5\,\cos {\it θ}\,\sin ^2{\it θ}\right)\,e^{2\,i\,{\it φ}}}\over{2^{{{17}\over{2}}}\,3^{{{17}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,3
5,4,2
5,4,1
1.097 \times 10^{+5}
7.68 \times 10^{+5}
7.68 \times 10^{+5}
6.076 \times 10^{-7}
6, 5, 3
-{{e^ {- {{r}\over{6\,a}} }\,\left(9\,r^5\,\sin ^5{\it θ}-8\,r^5\,\sin ^3{\it θ}\right)\,e^{3\,i\,{\it φ}}}\over{20155392\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,4
5,4,3
5,4,2
3.657 \times 10^{+4}
5.851 \times 10^{+5}
1.024 \times 10^{+6}
6.076 \times 10^{-7}
6, 5, 4
{{r^5\,e^ {- {{r}\over{6\,a}} }\,\cos {\it θ}\,\sin ^4{\it θ}\,e^{4\,i\,{\it φ}}}\over{6561\,2^{{{19}\over{2}}}\,\sqrt{5}\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,4
5,4,3
3.291 \times 10^{+5}
1.316 \times 10^{+6}
6.076 \times 10^{-7}
6, 5, 5
{{r^5\,e^ {- {{r}\over{6\,a}} }\,\sin ^5{\it θ}\,e^{5\,i\,{\it φ}}}\over{33592320\,\sqrt{\pi}\,a^{{{13}\over{2}}}}}
5,4,4
1.645 \times 10^{+6}
6.076 \times 10^{-7}