I'm new to basis sets. I read that for 3-21G, given the following basis set for carbon (from Basis Set Exchange):
C S
172.2560000 0.0617669
25.9109000 0.3587940
5.5333500 0.7007130
C SP
3.6649800 -0.3958970 0.2364600
0.7705450 1.2158400 0.8606190
C SP
0.1958570 1.0000000 1.0000000
You form three 'atomic orbitals' (is that the right term?) as:
let f(alpha) = (2*alpha/pi)^(3/4) * exp(-alpha*r^2)
O1 = 0.061 f(172) + 0.358 f(25) + 0.7 f(5.5)
O2 = -.39 f(3.66) + 1.21 f(.77) + 1 f(.195)
O3 = 0.23 f(3.66) + 0.86 f(.77) + 1 f(.195)
I truncated the numbers to make it easier to type...
Assuming that is correct, that makes sense to me. But when I look at the 6-31G basis set data for carbon:
C S
3047.5249000 0.0018347
457.3695100 0.0140373
103.9486900 0.0688426
29.2101550 0.2321844
9.2866630 0.4679413
3.1639270 0.3623120
C SP
7.8682724 -0.1193324 0.0689991
1.8812885 -0.1608542 0.3164240
0.5442493 1.1434564 0.7443083
C SP
0.1687144 1.0000000 1.0000000
C SP
0.0438000 1.0000000 1.0000000
From the naming nomenclature, I thought the only difference would be 6 primitive gaussians for the inner shell which I see, but . . . I was not expecting that third SP line. I don't know how to use it.
I want to write:
O1 = .0018 f(3047) ...
O2 = -.119 f(7.86) + ... + 1.0 f(.168)
O3 = 0.068 f(7.86) + ... + 1.0 f(.168)
Just like I did for the 3-21G case.
Maybe I could add:
O4 = -.119 f(7.86) + ... + 1.0 f(0.0438)
O5 = 0.068 f(7.86) + ... + 1.0 f(0.0438)
But I don't think this is right because I read online that there should be 9 orbitals formed (although they didn't explain how to get them).
Thanks for the help!
Update
Ok, so I looked up 6-311G instead of 6-31G. However, I still don't see how to form 9 orbitals from 6-31G:
C S
3047.5249000 0.0018347
457.3695100 0.0140373
103.9486900 0.0688426
29.2101550 0.2321844
9.2866630 0.4679413
3.1639270 0.3623120
C SP
7.8682724 -0.1193324 0.0689991
1.8812885 -0.1608542 0.3164240
0.5442493 1.1434564 0.7443083
C SP
0.1687144 1.0000000 1.0000000
I really need to see all 9 orbitals written out explicitly.
Update 2 Ok, with more research I finally found this
So it looks like I'm missing a function. I need to define:
f1(alpha) = (2*alpha/pi)^(3/4) exp(-alpha r^2)
f2(alpha, x) = (128 alpha^5/pi^3) x exp(-alpha r^2)
Then the 9 orbitals are:
O1 = .001 f1(3047) + .014 f1(457) + ... + .362 f1(3.16)
O2 = -.119 f1(7.68) + -.16 f1(1.88) + 1.14 f1(.54)
O3 = 1.0 f1(.168)
O4 = .068 f2(7.68, x) + .316 f2(1.88, x) + .744 f2(.54, x)
O5 = 1.0 f2(.168, x)
O6 = .068 f2(7.68, y) + .316 f2(1.88, y) + .744 f2(.54, y)
O7 = 1.0 f2(.168, y)
O8 = .068 f2(7.68, z) + .316 f2(1.88, z) + .744 f2(.54, z)
O9 = 1.0 f2(.168, z)
Can anyone confirm this? Or, if not confirm, specify what's wrong?
Thanks!