Delta Bond Or Pi Bond, Which Is More Strong? [closed]








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It is pretty evident that sigma bonds are more stronger than pi bonds. (due to the extent of overlapping)


But when there are 4 lobes of two d-orbitals involved in overlapping as that is the case in formation of 'delta bond', is it more stronger than pi bond ?


[it would of great help, if you can list down the sources backing your answers too]




In the case of transition elements, these bonds are involved in the transition from an isolated atom to a transition metal solid. A simple approach for understanding this transition by keeping an intuitive description is to use atomic orbitals, precisely a linear combination of atomic orbitals (LCAO) basis set of the [Tight-binding approximation](https://en.wikipedia.org/wiki/Tight_binding). A useful picture summarizing this approximation is this Hamiltonian : $\hat{H} = - t \sum\_{i,j,\sigma} c\_{j \sigma}^\dagger c\_{i \sigma} + h.c$. which means that an electron can hop from an atomic site $i$ to another atomic site $j$ with a transfer integral $t$ as the strength of this motion leadind a bandstructure.


If you do not understand this theory, you just need to keep in mind that the parameter $t$ can be related to the strength of the bond, therefore this approximation can give a quantitative strength of bonds relative to the overlap between orbitals. In the basic application of the theory, the Hamiltonian is made of 10 Slater-Koster (SK) hopping parameters between $s, p$ and $d$ orbitals : $ss\sigma$, $sp\sigma$, $ss\sigma$,$pp\sigma$, $pp\pi$, $pd\sigma$, $pd\pi$, $dd\sigma$, $dd\pi$, $dd\delta$. The strength of the SK parameters can give a quantitative approximation of the strength of the bond. I suppose that you are mainly interested in $dd\sigma$, $dd\pi$, $dd\delta$. These parameters can be obtained by a fit of a higher level of theory or experimental bandstructure if available. For a modeller, the values can be obtained easily, a reliable source is [this book](https://books.google.cm/books?id=av9MBQAAQBAJ&printsec=frontcover&hl=fr#v=onepage&q&f=false) where some pages appears online. Another more complicated source is the [NRL tight-bnding](http://esd.cos.gmu.edu/tb/index.html).


[![Tight-binding Slater-Koster parameters](https://i.stack.imgur.com/9a87l.jpg)](https://i.stack.imgur.com/9a87l.jpg)


The sign of the SK parameters is not interesting for your question, only the magnitude matters : we will consider SK parameters as positive. For example, for Chromium (Cr) :


1. $dd\sigma| \rightarrow 0.84 $ eV (81.46 kJ.mol$^{-1}$)
2. $dd\pi \rightarrow 0.50 $ eV (48.22 kJ.mol$^{-1}$)
3. $dd\delta \rightarrow 0.04 $ eV ( 3.51 kJ.mol$^{-1}$)


Obviously, the $\sigma-$bond is the strongest and the $\delta-$bond is the weakest bond formed but the overlap between two d-orbitals, about 10 times weaker than the $\pi-$bond. This result seems surprising maybe counterintuitive, but it is intuitive according to the picture above. The strength of a bond can be enhanced if the electron lies between the centers of the potential of the nucleus. An overlap has a coulomb cost reduces by the exchange interaction and more significantly by the potential of the nucleus if the overlap is not far from the nucleus : this is the case of a $\sigma-$bond. As shown in the figure above, in the case of $d$-orbital, the overlap $dd\pi$ is greater and has a $\sigma$ bond character but is simply farther from the nucleus : $dd\pi < dd\sigma$, however $dd\pi \gg dd\delta$ because of a planar overlap $dd\delta$.


The overlap $dd\delta$ is very similar to $pp\pi$, the only case where a $\delta-$bond can have more strength than a $\pi-$bond is when the $\pi-$bond between two $p$ orbitals.


[Image source](https://iramis.cea.fr/en/Phocea/Vie_des_labos/Ast/ast_visu.php?id_ast=309)


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