The post on applying VSEPR ("valence shell electron pair repulsion") theory to the geometry of ClF3 has proved perennially popular. So here is a follow-up on another little molecue, F3SN. As the name implies, it is often represented with an S≡N bond. Here I take a look at the conventional analysis.
This is as follows:
Now for a calculation[1]; ωB97XD/Def2-TZVP, where the wavefunction is analysed using ELF (electron localisation function), which is a useful way of locating the centroids of bonds and lone pairs (click on diagram below to see 3D model).
So we have achieved the same result as classical VSEPR, but using partial rather than full electron pairs to do so. We got the same result with ClF3 before. So perhaps this variation could be called "valence shell partial electron pair repulsions" or VSPEPR.
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I note Sason has now joined in. I just checked with him about N2(2+), which I did recollect was done a few years ago along with the original C2 work. He tells me that N22+ "has an even stronger 4th bond than C2 (simple electronegativity effect)". Yes, a characteristic of charge-shift bonds are NNAs, and in ELF they manifest by bifurcating the ELF basin into two, as with fluorine. In C2, this bifurcation seems to be extreme.
Yes, Henry, we are back to charge-shift bonding in this molecule. We were delayed because of the controvercy with the coalition, but we might get back to it after the paper I sent you is accepted.
We'll definitely be in touch.
Sason
Since we have been discussing multireference methods (for C2), I thought I might check on any such character for the molecule which is actually the topic of this post, NSF3, 96% of the wavefunction is in fact the closed shell one. In contrast, C2 is 73.2%, N22+ is 68.7% and B22- has two dominant configurations, each 47%.
Henry, Did you check the Laplacian for C2? Sason
Regarding the Laplacian, I should first explain that the QTAIM (and also the ELF basin centroids) reduce down to a point along the C-C axis (they are obliged to be axially symmetric). At the QTAIM LCP/BCP (3,-1) centroids, the ∇2ρ(r) values are -0.29 (twice) and -0.72 for the central point (CASSCF(8,8)/Def2-TZVPP calculation, doi: 10.14469/ch/191851 and 10.14469/hpc/222). These appear to be normal "covalent" values. The full Laplacian function takes the form of a torus surrounding the central bond and the following best shows the properties of this function in three-dimensional space (red = -0.6 for ∇2ρ(r), blue = +0.6).
You can see other examples of this function for ethane and benzene for comparison here.
For completeness I also include the ELF function, contoured at a value of 0.82, which is again axially symmetric.
I should add to my last comment that the value ∇2ρ(r) at the mid point of the CC bond probably reflects mostly the "normal" first σ-bond rather than the much weaker 4th bond. If the 4th bond does "bifurcate" (a long weak bond), then the value of ∇2ρ(r) closer to the atoms may be a better reflection of its properties than the mid point.
Here is another view of ∇2ρ(r), this time at an isosurface value of +0.1. The blue torus around the bond arises from the π-electrons, for which the ∇2ρ(r) is +0.1. This tends to suggest that the π-torus has charge-shifted character which is not observed along the bond-axis. There is of course another region of +ve ∇2ρ(r) which is fully enclosed inside the red (-ve) region. This also serves as a reminder that QTAIM values at the centroid (along the bond axis) may not properly reflect the full region of space.
I am a bit skeptic about the idea that this figure is related to charge shift nature of pi-electrons. You may get a similar figure from ethane's C-C bond. Please see the figure that I had produced for the profile of the ∇2ρ(r) of C2 in my comment above. The same type of figure can be made for any bond having a shared character, i.e. negative ∇2ρ(r) in the bond's LCP.
Might I suggest another approach? Computing the difference in properties between the closed shell CASSCF singlet and the open shell CASSCF triplet. This should reveal differences induced by the spin coupling of just these electrons (i.e. the putative 4th bond).
If e.g. the Laplacian were computed for both states and the two cubes subtracted, it might reveal something interesting?
I did try a CASSCF calculation with triplet spin state specified (and the resulting energy was ~15 kcal/mol higher than the singlet) but clearly a new trick is needed to write out the WFN file in appropriate open shell form? My WFN appeared to be for the closed shell natural orbitals.
I would also comment that using just CASSCF (which includes only static, and incomplete dynamic correlation effects) might also be influencing things. As you point out Cina, a CASPT2 approach (which does include both static and more complete dynamic correlation) might be needed.
I am not comfortable with electron density difference maps. Indeed, you cannot get a precise physical insight out of them. To make such a map, you should first violate PEP, that means introduction of an unphysical reference state. It can make a big trouble in some cases.
Density difference maps have been used in conceptual DFT, by eg Paul Ayers. One of his examples is the density difference map for FCCF and its cation; removal of an electron actually increases the density along the sigma nodal planes. You can see a picture of this here. I do not know of any difference maps for the Laplacian of the density.
I do agree that difference maps are pretty popular but popularity does not change physical principles. In such a delicate case I do prefer not to introduce any arbitrary reference state. About the difference maps of Laplacian you are right, I have not seen such a map either. About wfn of triplet C2, have you tried converting fchk to wfn? If you write fchk by density=current keyword, it should remain as it is. Then you can use fchk for AIM analysis directly.
Since the total density is in principle a directly observable property, I am not sure I understand the issue regarding reference states?
Re: use of .fchk files. This might be fine for QTAIM, but the Topmod09 ELF program does not accept them. But I am trying one or two other things.
You can convert fchk to WFN/X by means of AIMAll, then use the wfn/x for ELF computations. About difference maps; comparing the density of singlet and triplet species seems OK to me but many subtract an arbitrary density from a particular density. For example, subtract density of two fragments of a complex from that of the complex. In such cases the sum of fragment densities violates PEP. Though, as I confessed now, I see no violation of PEP in this case. I am sorry, but responding while I am in a party is a bit difficult. Some times I need to concentrate that is very difficult now!
Thanks for the tip. In fact this helps isolate the issue to the program being used to subtract two cubes (cubgen). The individual unsubtracted cubes display more or less OK. Thus the CASSCF(8,8)/Def2-TZVPP Laplacian for the triplet C2 is shown below. It contains an extra feature exo- to the bond, where the two unpaired electrons take up residence. This additional feature has a -ve Laplacian (red).
You can find the position of unpaired electrons by analyzing alpha-electron density. Please see (DOI:10.1039/C5CP04280A). Laplacian of alpha electron density should be intriguing! Sorry, I have no access to computational facilities now otherwise I would do it myself.
Sometimes, there is "behind the scenes" stuff going on. Cina asks about analyzing the α-electron density. The keywords (in Gaussian) for obtaining this would be
CASSCF(8,8)/Def2TZVPP output=wfn pop=NOAB density=current
which should write the appropriate WFN file containing natural orbitals for both α and β density suitable for further analysis. I have however hit a problem with this approach which is currently being resolved, currently with no precise timeline for solution. So I cannot provide the analysis as per above just yet.
Thank you for considering my comment. However, you don't need to obtain separated wfn for the alpha electron density. The information already is encoded in the wfn file that you made from fchk. AIMStudio can perform analysis on alpha, beta, and spin density on an open shell wfn. Laplacian of spin density has also unique features. It shows you the shape of orbitals that unpaired electrons are in. Please see the same paper that I addressed above on U2@C80.
The problem is a bit deeper than that, since the .fchk file I generated does not seem to carry the information either (and hence it cannot be propagated to the wfn). This seems to be associated with the CASSCF keyword, since its not a problem with DFT or HF. There was no problem for example with obtaining the spin density for the electride properties shown on this blog.
More analysis of the above point. A CASSCF function uses (spin) restricted or RO formalism, and does not write out separate density matrices for the alpha and beta spins. QTAIM and ELF require UHF (spin unrestricted) style density matrices for open shell systems to be present in the WFN (or FCHK) outputs. To quote an expert I consulted "code will have to be written".
Henry, Cina: What I picked up from the discussion is only one sure thing: the density for C2 is larger than in HCCH. But what kind of Laplacian elludes me. For me the laplacian is a probe of the balance of kinetic energy (density) vs. potential energy (density). If it can apply bond-wise, then it is helpful. I recall by the way Henry, a quintuple CrCr bond you looked at with a whoopingly large positive Laplacian. Sason
Re Quintuple bond. Yes, that was obtained using single-reference DFT (∇2ρ(r) +1.45). I think it seems agreed that Cr-Cr bonds really do have to be treated with multireference methods (probably more so than C2). But then the size of the active space becomes the question, and whether dynamic correlation also needs adding in.
I rely on a parameter for bonding analysis that measures sum of all overlap matrixes between a pair of atoms, the delocalization index within the context on AIM. However, presence of two NNAs between two C atoms makes the analysis indeed complicated. I tend to say that comparing ethyne, C2 has more electrons shared. Please see this paper for a non-technical explanation on DI and its relation to covalency: http://pubs.rsc.org/en/content/articlelanding/2015/cp/c5cp05777a About Laplacian, I prefer to look at Laplacian from the perspective that Paul Popelier looks at it. Laplacian is a magnifier that shows where electrons are concentrated, where thay are depleted. My coleagues and I at least in one particular case could show the shape of orbitals on the basis of Laplacian: http://pubs.rsc.org/is/content/articlehtml/2015/cp/c5cp04280a see these figures:
I expect that analyzing the spin density in C2-triplet shows the position of single electrons.
We briefly departed from C2 (and NSF3) to consider an old analysis of mine on a Cr-Cr quintuple bond. There I now tried out a CASSCF(14,14) calculation to try to include as much of the active space as possible. Emboldened by its success (i.e. the program did not crash, and successfully converged to a solution), I thought I might as well try out a CASSCF(12,12)/Def2-TZVPP calculation on C2 itself (doi: 10042/196111). I was surprised to find that the QTAIM simplified; the two NNAs in the previous density vanished, and now we have a regular result of just a single LCP (BCP) along the axis. So Cina, given the complexity you mention above is now simplified, can you tell us what the QTAIM now shows?
On a separate issue, to recover all the dynamic correlation in a system, one can try a complete CI. A CASSCF(12,12) is not such a complete CI of course, because it does not include the majority of the unoccupied virtual orbitals. It does however include all the occupied ones, including all the cores. So how much if any dynamic correlation does such a semi-complete CI actually recover (as distinct to the static correlation from the multi-reference character).
I am closely following your blog but unfortunately I am currently connected to the web via my mobile phone. If you can kindly check the value of delocalization index for CC bond then we can compare it with that of ethyne and see whether DI of C2 is larger or that of ethyne dominates. The DI value for ethyne is given above for CAS (6, 6) computation. I don't think that increasing the active space affects the DI value of the ethyne much. If it changes, it should decrease. Now, if the DI for C2 at CAS (12, 12) is larger than that of CC bond at CAS (6, 6), we can safely say C2 has a bond with higher order than three. Otherwise I would perform a larger CAS computation for ethyne.
I leave a comment regarding CrCr bond here. Like C2, I would definitely check the DI value for CrCr bond from the SUM file of the AIM computations. To the best of my knowledge DI is the best and the most straightforward index of bond order within the context of AIM theory.
Reading my own comment, I noticed I wrote it a bit unclear. By analyzing the spin density of triplet C2 one must find the shape of atomic orbitals in which single electrons are placed. Anyhow, could you get the wave function of triplet C2? If not, what that Laplacian map in these comments is referring to. Isn't it triplet C2?
The triplet diagram I put up was approximated by a closed-shell natural orbital WFN, with occupation numbers of ~1.0 and ~1.0. I argued that since one must square the coefficients to get the density, this open shell "singlet" would give a similar Laplacian to the equivalent open shell "triplet". As I indicated elsewhere, the program will need code written to write out a proper open shell WFN file for the triplet.
I wanted to follow up Sason's comment about Cr-Cr with an evaluation of whether a CASSCF calculation, along the lines we have been discussing for C2 and other species, might change the value of ∇2ρ(r) for the Cr-Cr system. Its a bit too long to include here, and so I have posted instead.
Can you do MRCI?
If so, then MRCI/6-31g* is almost as good as FCI with the same basis set.
Sason
For efficient MRCI calculation, the commercial programs MOLCAS or MOLPRO are probably best. The free programs FIREFLY, GAMESS and ORCA also have the capability. Of these, we really only have access to ORCA here on our high performance cluster and I have no experience of the MRCI modules.
Well, I connected my mobile phone to our computational resources, thanks to diverse number of apps that provide SSH connection, and performed AIM computation on C2 wave function, obtained at CAS(12,12). The AIM computations were fast and extremely accurate (the Lagrangian of atom that is -1/4 atomic integral of Laplacian is only 8 x 10^-9; the smallest Lgrangian I have ever seen).
However, delocalization index between C atoms is smaller than that of CC bond in ethyne. It is just 2.1 e (compared with 2.2 e for CC bond in ethyne at CAS(6,6)/def2-TZVP level). This result is not convincing for me yet. As the active space grows larger, the correlation between electrons increases thus alpha and beta electrons cannot delocalize as easily as they did at a non-correlated/weakly correlated method. That is why delocalization index decrease by increasing the electron correlation. Here, I think the only way to solve this mystry is to obtain wave function of ethyne at the same level as C2. So, we need CAS(12,12)/def2-TZVP wave function for HCCH. I could not go further than CAS(6,6) for C2 and HCCH on our systems since we have a limitation of ~ 5 TB for memory usage. I could not get a CAS(8,8) wave function for C2 with 5 TB of memory. How about you? Can you get the wave function of HCCH from CAS(12,12)?
Indeed, the total DI for the CASSCF(12,12) wavefunction is 2.102. I will add that for HCCH shortly (due to a power failure in our computer room, no calculations have been possible for about 24 hours).
I might also be able to manage a CASSCF(12,14) for C2 itself before our computers refuse, thus adding in a bit more of the virtual orbitals.
A CASSCF(12,14)/Def2-TZVPP calculation for C2 (probably the practical upper limit) gives DI = 2.1204 (doi: 10.14469/ch/191870). Only missing now is the effect of any further dynamic correlation, achievable from e.g. a MRCI (multi-reference configuration interaction).
Here are QTAIM results deriving from a CASSCF(12,12)/Def2-TZVPP calculation on C2 and HCCH, including full geometry optimisation at this level.
1. Data for C2 is published at doi: 10.14469/hpc/242 and 10.14469/ch/191866, DI(A,B) = 2.09967
2. Data for HCCH is published at 10.14469/hpc/243 and 10.14469/ch/191867, DI(A,B) = 2.18559
So Cina's values above are replicated, and we may take these values as convergent for the active space.
But I remind that the two electrons derived from half of the covalent H-C bonds are going to be a weak perturbation. Any 4th bond formed by spin coupling of this electron pair may have different (i.e. charge shifted) character from the underlying covalent bond, and to me the correlation between such an electron pair and the DI remains to be established. What for example is the DI for F2, another charge shifted bond?
I performed an additional analysis on C2 and HCCH molecules using IQA (interacting quantum atom) method of Pendas et. al. There was a minute chance that in C2/HCCH the electron sharing/exchange-correlation energy has a particular non-regular relationship. Let me explain what I am talking about. First, IQA analysis is if not the most accurate, for sure one of the most accurate bonding analyses. It does not invoke any arbitrary reference state and/or does not introduce any non-physical state/energy component. Thus, following Occam's razor, I tend to say this analysis is the most accurate energy-based bonding analysis. Within the context of IQA one can define a first order relationship between the delocalization index and exchange-correlation energy, DI/r = m Vxc; where DI is the delocalization index, r is inter-nuclear distance, Vxc is the exchange-correlation contribution to the binding energy and m is a universal constant see (DOI:10.1039/C5CP05777A and DOI: 10.1039/C5CP05222J). In certain cases DI and Vxc that are electronic and energetic measures of covalency within the context of QTAIM/IQA analyses do not follow the above mentioned relationship. Unfortunately, here this is not the case (if it was, it would be absolutely amazing). Here are exact energy-based bonding analysis of C2 and ethyne: C2: Vxc : -5.8199131018E-01 au Vne + Ven: -2.8927657586E+01 au Vee: 1.3219363532E+01 au Vnn: 1.5263919917E+01 au HCCH (CC bond): Vxc: -6.2956199228E-01 au Vne +Ven: -3.0351391878E+01 au Vee: 1.4145927949E+01 au Vnn: 1.5700902141E+01 au Vxc, Vne/en, Vee and Vnn denote exchange-correlation, nuclear-electron and electron nuclear attractions, electron-electron repulsion and nuclear-nuclear repulsion energies, respectively, defined within the QTAIM atomic basins On the basis of DI and Vxc analyses I would say that either C2 does not have a quadruple bond or though very unlikely, QTAIM is not a proper theory to identify the forth bond of this system. I have to read all literature of C2 to see which side can convince me :) My analysis is against quadruple C2, unfortunately.
A reminder that the enthalpies of HCCH, HCC(.) and CC are all known, and indicate that it is easier to remove the second HC than the first by about 17 kcal/mol. If the theoretical analysis cannot reproduce this by the means above, then there must be some missing component in the theoretical analysis (or there is some unsuspected error in the measured thermochemistry).
I went through a paper by Frenking and Hermann DOI:10.1002/anie.201301485 that discuesses several factors including bond length, bond dissociation energy (the ~17 kcal/mol stability of the supposedly forth bond of C2) and force constant. Actually, I find their discussions pretty convincing and in line with the conclusion on the basis of QTAIM/IQA analyses. In fact, I think C2 is way different from the classical picture that simple MO analysis provides for this molecule, i.e. a doubly bonded system; it is not even similar to a triply bonded ethyne! The story of C2 reminds me a story by Rumi, Perisan poet. He says four friends, one Persian, one Turk, one Greek and one Arab had some money and wanted to by grape but since they did not know the meaning of grape in other languages they were arguing for a long time on the same thing! In my humble opinion the forth bond indeed forms i.e. fourth electrons couple weakly. However, since the fourth elecrons are "almost free" (very weakly coupled) and hence pretty much delocalized around the nuclei of their own atoms (in AIM sense) they seriously affect the "triple bond" between two carbon atoms of ethyne. The immidiate effect of these free electrons is "localization" of bonded electrons. Thus, in C2 we have a forth bond that strongly weakens a triple bond; it is an ironic situation! Now, VB counts the number of electron pairs and finds four couples. In the meantime AIM measures DI but finds less DI than ethyne because the forth electron interferes with the other three and localizes them as a result of PEP. The localization of electrons elongates the bond and lowers the force constant too. Therefore, anyone who looks at force constant/bond length as a measure of bond order will be confused. Now, I think I understand why Prof Shaik says C2 is quadruply bonded and I understand why others cannot accept it. It is a linguistic problem. Please correct me if I am missing something.