# What would follow in the series sigma, pi and delta bonds?

I realise, that this question is a stretch, but I was wondering, how would a bonding orbital be called if it was formed from two $f_{x(x^2−3y^2)}$ or $f_{y(3x^2−y^2)}$ orbitals. Have there been any suggestions on this, was it anywhere proposed or discussed? I am not arguing about the existence of such a thing, but as a thought experiment, how would it be called? Phi, $\varphi$?

I am also throwing in a reference request here, since I am interested in any literature related to the topic.

Some background:

• A $\sigma$ orbital is an orbital, that has no nodal plane between the bonding partners. It is $C_\infty$ symmetric with respect to the bonding axis (in first approximation - of course $d_{xy}$ orbitals can align in a $\sigma$ fashion, too, but for the sake of the argument, let's not consider this.) The point group of this orbital can further be assumed as $D_\mathrm{\infty h}$. It shares therefore most features of an $s$ orbital, hence $s$igma, $\sigma$.
The most prominent example for this is probably the dihydrogen, $\ce{H2}$, molecule.
• A $\pi$ orbital has one nodal plane and the bonding axis of the involved atoms is part of this plane. It is therefore asymmetric, $C_1$, with respect to this plane. However, it is $C_\mathrm{s}$ symmetric with respect to the plane that is perpendicular, also sharing the bonding axis. The orbital has further the point group $C_\mathrm{2v}$. It has therefore most features of a $p$ orbital, hence $p$i, $\pi$.
The most prominent examples for $\pi$ bonds are probably ethylene, $\ce{C2H4}$, and acetylene, $\ce{C2H2}$.
• A $\delta$ orbital has two nodal planes between the bonding partners. These are perpendicular to each other. The bond orbital is asymmetric with respect to both of them. As a result, it is $C_\mathrm{s}$ symmetric with respect to the bisecting mirror planes. The point group of this orbital is $D_\mathrm{2h}$. Therefore it has shares features with a $d$ orbital, hence $d$elta, $\delta$.
One example is the $\ce{[Re2Cl8]^{2-}}$ ion, see Wikipedia's quadruple bond.
• I imagine two $f$ orbitals aligning with the bonding axis being simultaneously a $C_3$ symmetry axis, giving it an overall $D_\mathrm{3h}$ point group. Since there is no greek letter starting with an $f$, I think the logical thing would be to choose $\phi$ as it sounds a like. (I cannot be the only one thinking that.)

The following graphic should somewhat demonstrate what I mean:

• I think $\phi$ would be the correct name (unless you want digamma $\digamma$), and following the pattern, I would imagine three nodal planes. Other than that, I expect the $\phi$-bond exists only in theoretical models. – Ben Norris Mar 19 '15 at 10:39
• Wikipedia has a tiny section mentioning the possibility of $f$-orbital overlap, indeed forming a phi bond. – Nicolau Saker Neto Mar 19 '15 at 11:55
• @NicolauSakerNeto I just printed the paper and will have a look at that when I get home... it looks promising. – Martin - マーチン Mar 19 '15 at 12:10
• @BenNorris and others.. shouldn't we make one of these comments into an answer? – Geoff Hutchison May 20 '15 at 18:29
• @Geoff I have a couple of publications on my desk, that I found based on the tiny section. Yet I have not come around to write them up in an answer. But I hope to do that soon. (That browser tab has been open since I asked the question.) – Martin - マーチン May 21 '15 at 2:38

tl;dr The next in the series is called φ bond. There is even a tiny Wikipedia article about it.

Nicolau pointed me to the Wikipedia article, that had at the time a tiny section about the φ symmetry of the bond. Ben also kindly agreed with my naming proposition. I'd like to back up just a little bit an quote one sentence of this article:

The type of interaction between atomic orbitals can be further categorized by the molecular-orbital symmetry labels σ (sigma), π (pi), δ (delta), φ (phi), γ (gamma) etc. paralleling the symmetry of the atomic orbitals s, p, d, f and g. The number of nodal planes containing the internuclear axis between the atoms concerned is zero for σ MOs, one for π, two for δ, etc.

Basically this was what I was already thinking, but since the section is quite tiny overlooked it accordingly. But since this is a reference request, I'll go a little deeper.

The concept of the bond is actually quite old and I was able to track it down to at least 1984, where Bruce E. Bursten and Geoffrey A. Ozin "demonstrate[d] for the first time the existance of φ bonds between metal atoms."[1] A little further back, 1978, N. Rösch and A. Streitwieser already performed SCF-Xα calculations on thorocene and uranocene, where they came across f orbital contributions in bonding. Yet, they did not coin the phrase φ bond.[2] Following that, there have been more theoretical studies on the diuranium molecule, with increasing level of theory.[3] One of the more recent studies is by Laura Gagliardi and Björn O. Roos, which is available free of charge (see below).[4] Xin Wu and Xin Lu also studied the phenomenon in endohedral matallofullerenes, i.e. diuranium in a Buckminster fullerene.[5]

It is quite safe to assume, that there are more studies about this research topic or in a more general aspect the research on molecules with high bond orders. For this purpose I will include a couple of good reads.[6]