Why isn't Cahn Ingold Prelog principle used to name the isomers of complex compound having square planar geometry & different ligands attached to it?

For a complex compound with coordination number 4, say having square planar geometry, and say all the ligands attached to the central metal atom are different, **then we know that we will get 3 geometrical isomers**. 


Now my doubt is like in organic chemistry if we had 4 different groups around a carbon atom, we still named those isomers as Z and E isomers (by Cahn Ingold Prelog principle) . Similarly why can't we name the isomers as Z and E for the case of complex compound with square planar arrangement having four different ligands attached to it? 


That is why can't we name those 3 geometrical isomers as Z and E isomers of the complex compound by Cahn Ingold Prelog principle?




### Part I


The CIP rules in fact *are* used to determine the priority. However, a different notation system has been developed for inorganic nomenclature of coordination compounds.


For the [square planar](https://en.wikipedia.org/wiki/Square_planar_molecular_geometry) geometry, *SP*-4-*x* descriptor (an instance of ‘[polyhedral symbol](https://en.wikipedia.org/wiki/Polyhedral_symbol)’) is used, where *x* is a CIP-based *priority number* of the ligand in *trans* (opposite) position relative to the ligand with the highest priority number (1).


In $\ce{[Mabcd]}$ complexes (ligand priority a > b > c > d, priority numbers 1, 2, 3, 4, respectively), for the 3 possible isomers, the descriptors are (forgive me the use of left/right superscript/subscript typography to depict square planar geometry, for the sake of simplicity):


* $\ce{^\style{font-weight:bold}{a}\_cM^b\_\style{font-weight:bold}{d}}$    *SP*-4-4
* $\ce{^\style{font-weight:bold}{a}\_dM^b\_\style{font-weight:bold}{c}}$    *SP*-4-3
* $\ce{^\style{font-weight:bold}{a}\_dM^c\_\style{font-weight:bold}{b}}$    *SP*-4-2


For $\ce{[Ma2b2]}$ (priority numbers 1, 1, 2, 2) and $\ce{[Ma2bc]}$ (priority numbers 1, 1, 2, 3) complexes, where only 2 isomers are possible, the *cis*/*trans* notation can be used as well. The *E*/*Z* notation from organic chemistry is not used at all for coordination compounds central atom configuration.


Examples:


* $\ce{^{\style{color:grey}{(1)}{ }I}\_{\style{color:grey}{(3)}{ }Cl}Pt^{Br{ }\style{color:grey}{(2)}}\_{F{ }\style{color:grey}{(4)}}}$  

(*SP*-4-4)-bromidochloridofluoridoiodidoplatinate(2−)  

*(hypothetical)*
* $\ce{^{\style{color:grey}{(1)}{ }Cl}\_{\style{color:grey}{(2)}{ }NH\_3}Pt^{NH\_3{ }\style{color:grey}{(2)}}\_{Cl{ }\style{color:grey}{(1)}}}$  

(*SP*-4-1)-diamminedichloroplatinum(II)  

*trans*-diamminedichloroplatinum(II)
* $\ce{^{\style{color:grey}{(1)}{ }Cl}\_{\style{color:grey}{(1)}{ }Cl}Pt^{NH\_3{ }\style{color:grey}{(2)}}\_{NH\_3{ }\style{color:grey}{(2)}}}$  

(*SP*-4-2)-diamminedichloroplatinum(II)  

*cis*-diamminedichloroplatinum(II)  

[cisplatin](https://en.wikipedia.org/wiki/Cisplatin)
* $\ce{^{\style{color:grey}{(1)}{ }Cl}\_{\style{color:grey}{(2)}{ }Ph\_{3}P}Ir^{PPh\_{3}{ }\style{color:grey}{(2)}}\_{CO{ }\style{color:grey}{(3)}}}$  

(*SP*-4-3)-carbonylchloridobis(triphenylphosphane)iridium(I)  

*trans*-carbonylchloridobis(triphenylphosphane)iridium(I)  

[Vaska's complex](https://en.wikipedia.org/wiki/Vaska%27s_complex) (a different IUPAC name descriptor claimed there is [disputed](https://en.wikipedia.org/wiki/Talk:Vaska%27s_complex#IUPAC_name))


(References: see Loong's comment below the question.)


### Part II


Now, your question is self-contradictory. You would like to use *E*/*Z* (two) descriptors for three different isomers.


Maybe you wanted to ask something like  

**Why are there 3 geometrical isomers of square planar [Mabcd] complexes, but only 2 isomers of tetrasubstituted ethene C2abcd?**  

(Note the latter statement is not true)


It's because for $\style{font-weight:bold}{\ce{[Mabcd]}}$, the four different ligands are at the vertices (corners) of a square. Which can be abstracted as a ‘closed loop’, i.e. a [cycle graph](https://en.wikipedia.org/wiki/Cycle_graph). The combinatorics can be abstracted to **free [circular permutations](http://mathworld.wolfram.com/CircularPermutation.html)**, $P'\_{n}={1 \over 2} (n-1)!$,  

for $n=4$: $P'\_{4}=3$.


For $\style{font-weight:bold}{\ce{C2abcd}}$, the situation is somewhat different. This is a combination of **positional isomers** $\ce{xyC=Czw}$, where the set $\{x,y\}$ is $\{\text{a}, \text{combination}\_2(\{\text{b},\text{c},\text{d}\})\}$, and $\{z,w\}$ is a complement. There are (concidentally?) 3 of them (${n \choose k} = {{n!} \over {k! (n-k)!}}, {3 \choose 2} = 3$):


* $\ce{abC=Ccd}$
* $\ce{acC=Cbd}$
* $\ce{adC=Cbc}$


(Here the different sortings of substituents at each carbon does not matter.)


However, the isomerism one is talking about when using the *E*/*Z* descriptors applies to **geometrical isomers** of a single positional isomer (take e.g. $\ce{abC=Ccd}$). Now, the sorting of substituents at each carbon *does* matter. What makes the isomers different is matching vs non-matching sort order at each carbon:


* $\ce{abC=Ccd}$    ≡    $\ce{baC=Cdc}$    (ascending-ascending, descending-descending – matching)
* $\ce{abC=Cdc}$    ≡    $\ce{baC=Ccd}$    (ascending-descending, descending-ascending – non‑matching)


which are 2 possibilities, and correspond to *E* and *Z* configurations, respectively (provided that the lexicographical order corresponds with the CIP priority order).


So, the **total number of C2abcd isomers is 6** (3 positional isomers, each has 2 geometrical isomers). One think of the alkene as an object similar to the square planar metal complex, where the C=C can be arranged in the center in two orientations, horizontal and vertical, hence 2× as many possibilities than ${\ce{[Mabcd]}}$.


(I hope exactly zero mathematicians and chemists were harmed during the reading.)


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