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Between sigma-pi bond and 2 bent bonds, is it actually equiavalence? Or there is specific molecule that require to be bent bonds and cannot exist as pi bond and vice versa? I felt like we actually don't need pi bond model at all if we consider it all to be bent bond
If it not, then how can we consider which chemical result will be bent bond, and which will form pi bond?
In the famous words of George Box, "All models are wrong. Some are useful."
The bent bond and sigma/pi descriptions are two apparently distinct models for representing the underlying structure of electron density of molecules. Both are oversimplifications, but the key question (as with any model) is which is better for accounting for all of the known molecular structures and predicting the results of observations we have not yet made.
There are examples that support both. Bent-bonds have been cited famously as better predicting the conformation of propene, but the sigma/pi gives more intuitive results for reactions like the Diels-Alder. As in the case of the VB vs MO theory debate, both models can be elaborated with greater complexity to the point that they give indistinguishable results, but the sigma/pi approach has largely taken over because in its basic form it is more intuitive and gives accurate predictions in most cases.
The general concept of a bent bond remains useful for highly strained systems such as cyclopropanes, where the electron density of a single bond does not appear to align with the axis connecting the atoms.
To answer your final question,
If it not, then how can we consider which chemical result will be bent bond, and which will form pi bond?
it is rare that the sigma/pi model will fail you in modeling a molecule with a double bond, and since it is the prevailing model pedagogically, you would do best to use it preferentially and only invoke bent bonds for strained single-bond systems.
