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I need the bond length in Å for $\ce{O_2^{--}}$ from this geometry database page (type "O2--" for the input).

I get a bunch of possible values. I don't know much about those yet but after researching all those terms, I've understood that it's highly case-dependent.

I've read a few studies stating that the bond length error from 6-31G* basis sets is deficient with "GGA, hybrid-GGA and meta-GGA functionals", but I truly don't know which method on the left I should take for that, any help? Which basis set and method listed on that page, gives the smallest error for the bond length?

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3 Answers 3

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Starting from the point that it's impossible to obtain a "correct" bond length from a DFT calculation, due to the unknown exchange-correlation functional. By the way, in general terms DFTs functionals can be sorted in the so-called Jacob's ladder:

  1. Hartree-Fock
  2. LDA
  3. GGA
  4. meta-GGA
  5. hybrid-GGA
  6. Generalized random phase

The upper you go, the more accurate the calculation(with the same basis set) yet the more time-consuming.

Another parameter to take into consideration is the basis set: usually the bigger it is, the more accurately is possible to correctly fit the Slater exponential function of the cartesian orbital.

So to answer the question: the bond length of $\ce{O_2^{2-}}$ depends on which calculation level you want to take into account. Still, as the site you linked says

Maximum atom distance is 1.6765Å

To have another piece of data, you can try to use a different type of calculation changing the theory to Coupled Cluster (CC): the NIST is currently using these results as standard. From the site you linked

O-O distance at CCSD(T)=FULL/def2-TZVPP is 1.631 Å

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  • $\begingroup$ What do you mean by "usually the bigger it is, the more accurately is possible to correctly fit the Slater exponential function of the cartesian orbital"? $\endgroup$ Jan 31, 2023 at 9:14
  • $\begingroup$ Hartree-Fock is not a DFT functional. It has nothing to do with DFT in the first place, as it is a wavefunction based method. $\endgroup$
    – WIP
    Nov 16, 2023 at 13:26
  • $\begingroup$ That's why it is ranked in the 0th place... DFT is still based on HF, some functional have the exchange parameter right from HF. But yes, you are right @WIP: HF is not a functional $\endgroup$ Nov 17, 2023 at 21:19
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If your desire is to get the best bond length from that website, then I would begin by looking for the largest basis set (in this case it's aug-cc-pVQZ) and the method that is expected to recover the most correlation energy (in this case it's CCSD(T)=FULL), so the CCSD(T)=FULL/aug-cc-pVQZ bond length of 1.540 Å is a good starting point.

However there's caveats, and the most immediate ones that come to my mind are:

  • Since double-augmentation caused the CCSD(T)=FULL/aug-cc-pVTZ bond length to drop by about 0.16 Å (that's huge!) with CCSD(T)=FULL/daug-cc-pVTZ, I would want to calculate CCSD(T)=FULL/daug-cc-pVQZ to see if the corresponding aug-cc-pVQZ bond length of 1.540 Å drops too.

  • These calculations are likely done with a frozen core (only correlating 14 electrons instead of 18). Since it doesn't look like they've done any all-electron calculations (for which we'd see basis sets like cc-pCVXZ with the C being the important letter there), the best we can do is to use the calculation CCSD(T)=FULL that they said was done with an effective core potential and Def2TZVPP: 1.631 Å (this is the number in Andrea's answer). However, this is again not with a QZ basis set, so it might be wise to look at how much the all-electron calculation changes the value of the frozen-core calculation with the same basis set (but for frozen-core calculations they don't explicitly mention Def2TZVP, only TZVP, for which the bond length dropped by about 0.02 Å when correlating the core electrons).

  • Finally I'll point out that the "composite" methods mentioned at the very top of the page are meant to be better than single-energy calculations done with just one method and one basis set, but unfortunately the glossary doesn't say what "CBS-Q" means. I would hope that it means CCSD(T)=FULL with a basis set extrapolation done with the aug-cc-pVTZ and aug-cc-pVQZ results, which would mean 1.528 Å is likely the best bond length, with the caveat that this might go down by ~0.16 Å if doubly-augmenting the basis set, and it could go down by another ~0.02 Å if correlating all electrons (this is a very crude estimate of the all-electron correction). They also reported a bond length of 1.621 Å with the G4 composite method, which is not likely to be more accurate than CCSD(T)=FULL/CBS(aug-cc-pV(T,Q-> ∞)Z) but at least we know what it is.

Summary:

Method Corrections Bond Length (A)
CCSD(T)=FULL/aug-cc-pVQZ None 1.540
CCSD(T)=FULL/aug-cc-pVQZ CCSD(T)=FULL/daug-cc-pVTZ - CCSD(T)=FULL/aug-cc-pVTZ ≈1.380
CCSD(T)=FULL/aug-cc-pVQZ CCSD(T)=FULL/daug-cc-pVTZ - CCSD(T)=FULL/aug-cc-pVTZ & all-electron ≈1.360
CBS-Q None 1.528
CBS-Q CCSD(T)=FULL/daug-cc-pVTZ - CCSD(T)=FULL/aug-cc-pVTZ ≈1.37
CBS-Q CCSD(T)=FULL/daug-cc-pVTZ - CCSD(T)=FULL/aug-cc-pVTZ & all-electron ≈1.35
G4 None & all-electron ≈1.360
CCSD(T)=FULL/Def2TZVPP None 1.631 (Andrea's answer)

So the NR-CPN (non-relativistic, with clamped, point-sized nuclei) bond length is probably somewhere between 1.3 an 1.7 Å.

The real bond length could be obtained by doing a CBS extrapolation using FCI-level calculations with aug-cc-pCV8Z and aug-cc-pCV9Z, then adding corrections for relativity, QED, Born-Oppenheimer breakdown, finite nuclear size effects, etc.) and perhaps calculating a full potential energy curve and solving the rovibrational Schroedinger equation to find where the ground vibrational wavefucntion has its maximum (since a full non-Born-Oppenheimer calculation is unlikely to be possible for more than about 6 electrons and even less likely for an 14-18 electron system unless there's an algorithmic or hardware breakthrough).

You can also get "experimental" geometries from that same site, but they don't seem to have one for this molecule.

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Perhaps the question should be reformulated into under what circumstances O2 dianion is stable. There are no atoms that can bind two electrons, and given the electron affinity of O being 1.47 eV, and that of O2 being 0.9 eV, I would strongly expect that the O2 dianion is not stable in the gas phase. Indeed, the CCSD(T)/aug-cc-pV5Z energy for the dianion is significantly higher than the mono-anion. Yes, you can calculate the dianion species with the above methods, but even large basis sets like aug-cc-pV5Z do not allow the extra electron to escape. If more and more diffuse basis functions are added, the extra electron would drift to infinity, and you would be left with the monoanion. All of the above calculated results therefore are just artifacts of the chosen methods and basis set being unable to describe the real system. A O2 dianion can be stable in e.g. a matrix of cations, but if the stated question relates to the gas phase, then the answer is that there is no answer.

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