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In this question and answers, I found that Basis Set Exchange provides information about the orbitals (e.g., the orbital exponents and contraction coefficients). For example, the following example is orbital data of the oxygen:

enter image description here

However, I cannot understand the details about these values and meanings; specifically I have three questions as follows.

  1. I understood that first 6 lines show 6 orbital exponent values and 6 contraction coefficient values in 6-31; however, each line of next 3 and 1 parts has "one" orbital exponent value and "two" contraction coefficient values. Why does one orbital have two coefficients?

  2. I believe that the oxygen orbital has 4 kinds of orbitals (i.e., 2s, 2px, 2py, and 2pz), but I cannot understand which value corresponds to 2s and which value corresponds to 2px.

Postscript using a concrete example.

1. They are Pople-type basis set. Namely, s and p have the same exponents but different contraction coefficients. Hence, one set of exponents, two sets of coefficients. (Maybe format Gaussian somehow looks clearer) 2. Px/Py/Pz have the same coef and exp.

Thank you for this comment from Lancashire3000. I add the following example for describing my confusing. I believe that "31" of 2px in the 6-31G basis set can be written (but omit the spherical harmonics) as:

$\phi_{2px} = \sum_{i=1}^3 d_i \exp(-\zeta_i r^2) + d_4 \exp(-\zeta_4 r^2)$.

So, if px, py, pz have the same exp and coef, totally we have four exponents, $\zeta_1, \zeta_2, \zeta_3, \zeta_4$, and four coefficients, $d_1, d_2, d_3, d_4$. But the Basis Set Exchange file provides two contraction coefficients in each one; **that is, we seem to have $d_1, d'_1, d_2, d'_2, d_3, d'_3, d_4, d'_4$.

$d$ is for the s orbital and $d'$ is for the px, py, pz orbitals?

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    $\begingroup$ +1 but we have a policy of one question per post. I've commented out your third question, which you can ask separately. This is probably the first time ever that I've answered to separate questions in one post's answer, only because it was easy enough for me to do it this time. Please ask your third question separately (if you aren't able to figure out the answer based on what you've learned about the P-type orbitals in my answer). $\endgroup$ Dec 19, 2021 at 7:24
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    $\begingroup$ 1. They are Pople-type basis set. Namely, s and p have the same exponents but different contraction coefficients. Hence, one set of exponents, two sets of coefficients. (Maybe format Gaussian somehow looks clearer) 2. Px/Py/Pz have the same coef and exp. $\endgroup$ Dec 19, 2021 at 19:44

2 Answers 2

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"I understood that first 6 lines show 6 orbital exponent values and 6 contraction coefficient values in 6-31; however, each line of next 3 and 1 parts has "one" orbital exponent value and "two" contraction coefficient values. Why does one orbital have two coefficients?"

When it becomes confusing to figure out what's going on in a basis set, I find it useful to look at the same basis set in other formats.

The CFOUR / ACESII / MRCC format for that same basis set looks like this:

O:6-31G(d,p)
6-31G + polarization on all atoms

  3
    0    1    2
    3    2    1
   10    4    1

0.5484671660D+04 0.8252349460D+03 0.1880469580D+03 0.5296450000D+02 0.1689757040D+02 
0.1553961625D+02 0.5799635340D+01 0.3599933586D+01 0.1013761750D+01 0.2700058226D+00 

0.1831074430D-02  0.0000000000D+00 0.00000000 
0.1395017220D-01  0.0000000000D+00 0.00000000 
0.6844507810D-01  0.0000000000D+00 0.00000000 
0.2327143360D+00  0.0000000000D+00 0.00000000 
0.4701928980D+00  0.0000000000D+00 0.00000000 
0.0000000000D+00 -0.1107775495D+00 0.00000000 
0.3585208530D+00  0.0000000000D+00 0.00000000 
0.0000000000D+00 -0.1480262627D+00 0.00000000 
0.0000000000D+00  0.1130767015D+01 0.00000000 
0.0000000000D+00  0.0000000000D+00 0.1000000000D+01 

0.1553961625D+02 0.3599933586D+01 0.1013761750D+01 0.2700058226D+00 

0.7087426823D-01 0.00000000 
0.3397528391D+00 0.00000000 
0.7271585773D+00 0.00000000 
0.0000000000D+00 0.1000000000D+01 

0.8000000000D+00 

1.0000000 

You can see here that all four of the P-type functions have the same exponents as the L-type functions in your version, and both versions show two "contraction coefficients" for each of these exponents, but in my version one contraction coefficient for each of these exponents is always zero, meaning that there's only ever one non-zero contraction coefficient for each exponent here.

"I believe that the oxygen orbital has 4 kinds of orbitals (i.e., 2s, 2px, 2py, and 2pz), but I cannot understand which value corresponds to 2s and which value corresponds to 2px."

Notice that the first column says "S", "L", and "D". The "L" in your case represents "SP" which just means that there's an S-type function with the given parameters, and a P-type function with the same parameters. In my version of the basis set, I've explained how to find the S-type and P-type functions in my answer to your recent post here: The orbital exponent in Gaussian. Examine the following part:

    0    1    2
    3    2    1
   10    4    1

This means that:

  • For S-type functions (L=0), we have 10 exponents and 3 contractions.
  • For P-type functions (L=1), we have 4 exponents and 2 contractions, and
  • For D-type functions (L=2), we have 1 exponent and 1 contraction.

You are then given the S-type exponents followed by the S-type contraction coefficients then the P-type exponents (0.1553961625D+02 0.3599933586D+01 0.1013761750D+01 0.2700058226D+00) followed by their contraction coefficients, etc. You can then match these four P-type exponents and their corresponding contraction coefficients with the ones in your version of the basis set.

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    $\begingroup$ I believe the L is written SP in Gaussian's format. These old Pople basis sets used a shared set of exponents for s and p orbitals to make them a little more affordable to optimize. $\endgroup$
    – Tyberius
    Dec 19, 2021 at 16:04
  • $\begingroup$ @Tyberius good point! My version shows 0.1553961625D+02 in both the S and P sections, for example. You could probably write an answer that complements the missing information in mine! $\endgroup$ Dec 19, 2021 at 16:16
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    $\begingroup$ @Tyberius not for optimization, but for integral computation which is cheaper when the functions share exponents. This restriction, however, results in a poorer accuracy for the basis set. $\endgroup$ Dec 19, 2021 at 18:43
  • $\begingroup$ I still don't understand but 2s, 2px, 2py, and 2pz orbitals have the same exponent values? In the above example, four 2s orbitals have four exponent values: 0.1553961625D+02 0.3599933586D+01 0.1013761750D+01 0.2700058226D+00, and four 2px orbitals also have the same four exponent values: 0.1553961625D+02 0.3599933586D+01 0.1013761750D+01 0.2700058226D+00? $\endgroup$
    – neco
    Dec 19, 2021 at 18:58
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    $\begingroup$ @neco I'm glad to hear it, thank you for getting back to me! $\endgroup$ Jan 6 at 1:00
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L is the name Pople used for this type of shell, but I like how the Gaussian format labels these functions (SP) as it makes it more explicit what is being described. Your overall description of the (GAMESS?) basis format is correct, but this use of two contraction coefficients is almost exclusively seen in the older Pople (x-yzG) and STO-nG basis sets.

In the original STO-nG paper[1], they decided that rather than using a separate set of exponents for the 2s and 2p orbitals, they would use a single exponent for each of them. Even more so than today, calculation speed was highly valued at the time this basis set was made. By creating this L shell, integral (particularly two-electron integral) evaluations could done more quickly with more reuse of intermediates. As an example that they give in the paper, if you have a two electron integral involving four P functions, you can reuse the same exponential and error function intermediates to compute $3^4=81$ different integrals (same intermediates for $p_x$, $p_y$, $p_z$). If you have integrals involving 4 L functions, these intermediates can be reused for $4^4=256$ different integrals.

Overall they found that the time per integral for small molecules was roughly half when using these L functions compared to having separate S and P exponents. The downside to using the same exponents for S and P functions is that it makes the basis less flexible. Combine this with the relatively minor speedup by today's standards and you can see why this practice fell out of favor.

So this specification for L tells you the exponents used for 2s and 2p (x/y/z) in the first column and then the next two columns give the separate contraction coefficients for the 2s and 2p respectively. You likely won't see this three column format outside of the Pople and STO-nG basis sets because more modern basis sets have found different ways to speed up integral evaluation without combining primitive exponents.

References:

  1. Hehre, W. J.; Stewart, R. F.; Pople, J. A. Self‐Consistent Molecular‐Orbital Methods. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals. J. Chem. Phys. 1969, 51 (6), 2657–2664. DOI: 10.1063/1.1672392.
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  • $\begingroup$ Thank you for the detailed answer! I could understand about the exponents and coefficients in the basis set. $\endgroup$
    – neco
    Jan 6 at 1:00

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