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Many general chemistry textbooks introduced the concept "hybridization" to construct a symmetry-adapted VB-type wavefunction. In the textbooks, usually the minimal basis is used and without optimizing all resonance forms.

My question is, in the ab initio valence bond calculation with extended basis set and optimizing many resonance forms,

(i) do people still use hybridization to construct the wavefuntion?

(ii) if the answer to question (i) is yes, does the result differ from the textbook description? For instance, the lone pair electron in the $\ce{H2O}$ molecule? (is there any unitary equivalent result like orbital localization in MO-based method?)

I asked the question (ii), since NBO analysis tends to give non-equivalent composition for lone pairs in $\ce{H2O}$, one sp, the other p. I am wondering what is the answer in ab initio valence bond method.

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    $\begingroup$ Do I understand correctly that you are looking for a Valence Bond Description? If so, you might have already read Ab initio valence-bond calculations of $\ce{H2O}$? I personally don't quite understand the benefits of VB theory compared with other approaches as CASSCF, MBPT, CC, CI or DFT - but that might well be because I do not understand VB theory completely. $\endgroup$ – Martin - マーチン Apr 10 '14 at 6:12
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    $\begingroup$ @Martin Proponents of VB feel that it is intuitively superior to MO type calculations because the calculations are performed with configurations that resemble "Lewis structures". However, VB configurations are generally non-orthogonal, and so anywhere there is an overlap of configurations computation, it has to be done explicitly. Thus MO based methods beat the pants off of VB in terms of computation speed. $\endgroup$ – Eric Brown Jun 1 '14 at 6:57
  • $\begingroup$ @Martin (cont.) In the full limit of VB-CI and MO-CI, the theories are equivalent. Since both methods yield the same observables, MO theory wins out from a practical level. (Unless one desires a VB-like description of the wavefunction, which I think the poster desires, hence the NBO analysis) $\endgroup$ – Eric Brown Jun 1 '14 at 6:58
  • $\begingroup$ @Eric You are very right with your first statement. However the second statement is only half true. VB theory is naturally capable of describing strongly correlated electrons. In MO theory you only generate one configuration, so this case is impossible to consider. Therefore you need more advanced methods like MCSCF. So for any easy molecule, complete description, your statement holds, for more complex systems you should rather state: VB = MCSCF. $\endgroup$ – Martin - マーチン Jun 1 '14 at 12:30
  • $\begingroup$ @Martin I am assuming CI is full, multi reference CI, "full limit" $\endgroup$ – Eric Brown Jun 1 '14 at 19:13
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(i) do people still use hybridization to construct the wavefuntion?

Yes and no. No: Only atomic orbitals are included in the base set. Yes: if the wavefunction is optimized, linear combinations of atomic orbitals are allowed, and we all know that one S-orbital + some P-orbitals give hybridized orbitals.

For $\ce{H2O}$, the lowest 3 molecular orbitals are generally these (see below):the lowest 3 molecular orbitals of $\ce{H2O}$

At first, this result seems unlike the SP$^3$ bonds between hydrogen and carbon. There is a trick, however. It is called "localization of the orbitals", meaning that we generate linear combinations of molecular orbitals to generate new ones. Below, I'll give you an example in which we linear combinations of molecular orbitals $\varphi_2$ and $\varphi_3$ generate orbitals that look more like the SP$^3$ orbitals from text books.

Linear combinations of molecular orbitals $\varphi_2$ and $\varphi_3$ generate orbitals that look more like the SP$^3$ orbitals from text books

(ii) if the answer to question (i) is yes, does the result differ from the textbook description? For instance, the lone pair electron in the $\ce{H2O}$ molecule? (is there any unitary equivalent result like orbital localization in MO-based method?)

As you may have noticed above, the results differ slightly, but I can elaborate.

Lone pairs are given by $\varphi_4$ and $\varphi_5$ below. The higher 2 molecular orbitals of $\ce{H2O}$ and the LUMO and LUMO+1

For $\varphi_4$, note that p$_z$ is along the z-axis, which is in the plane of the oxygen and hydrogen atoms: it is the 2-fold rotation axis.

For $\varphi_5$, note that p$_y$ is along the y-axis, which is perpendicular to the plane of the oxygen and hydrogen atoms: it is out of the plane.

For $\varphi_6$ and $\varphi_7$, note that they are unoccupied orbitals (i.e. lumo and lumo+1)

Now, orbital localization can be applied, linear combinations of $\varphi_4$ and $\varphi_5$ will give 2 lone pair orbitals, that stick slightly out of plane. I haven't tried drawing those yet, as it is not so easy to do so in this representation in 2D, so I'l try to switch to another representation...

enter image description here

As you see, those localized molecular orbitals closely resemble the SP$^3$ lone pair orbitals on oxygen, as predicted by valence bond theory. The only difference is the shape & probability of the small counter-part of the lobe; I included phase, not probability density here.

Does this answer your question?

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  • $\begingroup$ I think this completely misses the point of the question. You are describing a molecular orbital approach and the interpretation with localised orbitals. So far so good, unfortunately the question is about ab initio valence bond theory and the use of hybridised orbitals within the construction of the wave function. (Also I think the localisation schemes you apply in your images do not add up. And especially in $\varphi_2\pm\varphi_3$ the angle does not make any sense. Also in none of the localisation algorithms will the oxygen lone pairs be equivalent.) $\endgroup$ – Martin - マーチン Nov 15 '18 at 12:59
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I can answer your first question definitively from my experience in quantum chemistry. Yes, hybrid orbital basis sets are still used, albeit minimally. Typically, these basis sets are used in QM/MM in hopes to scale down the QM computation. There are severe limitations in this approach. You tend to fix the geometry to hybrid orbitals, as you can only overlap certain orbitals. It becomes "ball and stick" chemistry. For this reason, it is not popular. Better to use a traditional QM technique on a smaller region where you will have reactivity.

I can attempt to answer your second question from a comprehensive standpoint as I have never done ab-initio calculations myself using hybrid orbitals. The results should be similar for geometries of simple molecules and break down for increasingly complex molecules. This is because non-VB electrons can undergo orbital overlap.

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  • $\begingroup$ It is news to me that there are uses of valence bond calculations in QM/MM approaches. They certainly would not scale down the calculation cost. All in all, I have troubles how this would answer any part of the question. $\endgroup$ – Martin - マーチン Mar 13 '17 at 5:32

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