Chemistry - What does this lambda–sigma notation mean?

Solution 1:

The notation refers to the valency (number of valence electrons involved in bonds, i.e., either 3 or 5) and coordination number (number of substituents attached) in organophosphorus compounds. For example, it could refer to the tautomeric forms of phosphonate esters:¹

So the λ³σ³ refers to a P with 3 bonding valence electrons (the lone pair doesn't count) and 3 substituents attached while λ⁵σ⁴ refers to a P with all five of its valence electrons involved in bonds and 4 substituents attached.

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Reference

1. Kraszewski, A.; Stawinski, J. H-Phosphonates: Versatile synthetic precursors to biologically active phosphorus compounds. Pure Appl. Chem. 2009, 79 (12), 2217–2227. DOI: 10.1351/pac200779122217.

Solution 2:

To add to the currently accepted answer, this is what I found from IUPAC publications.

In Treatment of Variable Valence in Organic Nomenclature (Lambda Convention) (IUPAC Recommendations 1983):

These recommendations, designated by Lm (lambda), provide a general method for indicating nonstandard valence states of formally neutral skeletal atoms in parent hydrides.

The bonding number, $n$, of a skeletal atom in a parent hydride is the sum of the total number of valence bonds to adjacent skeletal atoms, if any, and the number of attached hydrogen atoms.

A nonstandard bonding number of a neutral skeletal atom in a parent hydride is indicated by the symbol λⁿ. [...] Derivatives formed by substitution of hydrogen atoms of the parent hydride are named by means of prefixes and/or suffixes in the established manner.

Regarding the sigma notation, it seems to have been included in a tentative draft of Nomenclature of Organic Chemistry, dating from 1973. However, it was omitted from the eventual 1979 publication for the following reason (taken from the same source above):

In the first publication of the provisional Section D rules, the symbol σ^m, where $m$ is the number of σ bonds terminating at the heteroatom, was suggested either as an alternative to or in conjunction with the λⁿ symbol. However, the number of hydrogen atoms attached to the heteroatom would have to be included in this symbolism for a parent hydride, which is not consistent with the basic principles of substitutive nomenclature. In organic nomenclature, implied (or expressed) valence requirements of a skeletal atom in a parent hydride are satisfied with hydrogen atoms after other structural requirements are met. Hence, this symbolism was not included in the revision of the Section D rules for the 1979 edition of the IUPAC Organic Rules.