This article presents the use of free particle models to obtain quantum rules for planar boron clusters, with nuclearities in the range from seven to twenty. The information obtained from the models is being compared with electronic structure calculations based on the DFT method. Separate rules for in-plane and out-of-plane bonding are derived. In-plane bonding is precise on the cluster boundary and forms a network of alternating triangular 3c-2e bonds on the inside. The out-of-plane bonding is strongly delocalized and only depends on the global shape and size of the cluster.

A well-defined class of planar or quasi-planar elongated boron clusters, of type B q2 713n , serves as a basis to identify the valence bond picture of delocalized boron networks. The origin of the series is the B 2 7 cluster, which exhibits r-aromaticity. The cluster generating step is the repetitive expansion by three boron atoms in the direction of elongation. Specific electron counting rules are obtained for p-bonding, peripheral r-bonding and multicenter inner r-bonding. A valence bond structure is introduced which explains the remarkable regularity in the bonding pattern. The analysis supports 4c-2e bonds as an alternative to the common 3c-2e bonds. The results are validated by symmetry induction and ab initio calculations. K E Y W O R D S boron clusters, computational chemistry, induction method, multicenter bonding 1 | I N TR ODU C TI ONThe recent literature reports on a wide variety of planar and bowl shaped boron clusters. Proposed structures are usually based on theoretical calculations, but for some cases structures could be confirmed by photoelectron or infrared spectroscopy [1] on clusters produced by laser evaporation. [2][3][4] A further special feature of some clusters which are shaped like two concentric rings is the almost barrierless rotatory motion of the inner ring with respect to the outer ring. [5] This motion has been compared to a Wankel motor at the molecular scale. [6] In view of the rich variety of shapes and properties, which challenges accepted concepts of chemical bonding, boron is said to be the new carbon. As opposed to carbon, it is known to adopt multicenter bonds which have to be accomodated in a proper theoretical scheme. [7][8][9] To build a consistent valence bond picture that would apply to all these clusters, a gradual approach is required based on a well-defined set of structures. For this, we chose the particular family of the so-called elongated boron clusters. The aim is to obtain a set of rules that rationalize the electronic structure calculations on a series of structures extending from B 2 7 to B 22 28 . [10][11][12][13][14][15][16][17][18] Int J Quantum Chem. 2018;118:e25575.

Valence bonds within the perimeter of disk-like boron clusters with a concentric topology follow simple 4n and 8n electron counting rules.

The E ⊗ e Jahn-Teller Hamiltonian in the Bargmann-Fock representation gives rise to a system of two coupled first-order differential equations in the complex field, which may be rewritten in the Birkhoff standard form. General leapfrog recurrence relations are derived, from which the quantized solutions of these equations can be obtained. The results are compared to the analogous quantization scheme for the Rabi Hamiltonian.

The Bargmann-Fock representation of the Rabi Hamiltonian is expressed by a system of two coupled first-order differential equations in the complex field, which may be rewritten in a canonical form under the Birkhoff transformation. The transformation gives rise to leapfrog recurrence relations, from which the eigenvalues and eigenvectors could be obtained. The interesting feature of this approach is that it generates integer quantum numbers, which relate the solutions to the Juddian baselines. The relationship with Braak's integrability claim [PRL 107, 100401 (2011)] is discussed.