# How to form 6-31G basis set; specific instructions

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.

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!

• Could you possibly be looking at 6-311G? – Tyberius Dec 3 '17 at 4:21
• Tyberius is correct - in Pople notation, you're looking at a 6-311G basis set. That said, for a 6-31G, there will be 10 atomic orbitals (not 9, as you state). – Todd Minehardt Dec 3 '17 at 4:40
• @Tyberius : I just went back to Basis Set Exhange, and yes, it seems that I inadvertently downloaded the 6-311G set instead of the 6-31G. – ACasey Dec 3 '17 at 4:51
• @ToddMinehardt : you're right, I had 6-311G instead of 6-31G. But still, I only see how to form 3 atomic orbitals: 1) the 'S' data 2) the first set of coefficients of the first 'SP' + the second 'SP', 3) the second set of coefficients of the first 'SP' + the second 'SP'. I don't know if that is right... Could you please explain how you get 10 orbitals? – ACasey Dec 3 '17 at 4:56
• My mistake - it is 9, not 10. – Todd Minehardt Dec 4 '17 at 13:28

O1 = 0.061 f(172) + 0.358 f(25) + 0.7 f(5.5)

Depending on the integral scheme, one can reuse certain intermediates of some integrals for others and save some computational time. However, modern programs tend to work better with segmented basis sets (i.e. basis sets that do not allow for reuse, because the $\alpha$ exponents are all different). The results are also of higher quality. IMHO, the Pople basis sets are only of historical interest or for hobbyists with very limited resources.