III-defined concepts in chemistry: rigid force constants vs. compliance constants as bond strength descriptors for the triple bond in diboryne

In a recent publication in this journal, the interpretation of the Braunschweig's diboryne as a true triple bond is questioned.


Introduction
Computational chemistry has reached a high degree of maturity and comprehension making it one of the most active research areas in modern chemical and physical research in general. Predictions concerning single molecules, molecular clusters or even the solid state in combination with detailed information from apparatus based experiments are currently providing the ingredients to an auspicious revolution in the borderland between theory and experiment. Many, but not all, computational chemistry applications deal with observable properties. Here, one can always try to nd an experiment, which allows either falsication or conrmation of the computer simulation. This is in sharp contrast to the second major application area of computational chemistry, the underpinning of chemical concepts, where a comparison with experiment is not always possible. Therefore, from time to time, even seemingly trivial questions frequently lead to highly active discussions in the scientic community. Especially the idea of bond orders and localized orbitals, are under debate. 1 In a recent publication, 2 the interpretation of the boronboron triple bond in Holger Braunschweig's diboryne NHC-BB-NHC (NHC ¼ N-heterocyclic carbene) 3 as a triple bond is questioned. The analysis by Köppe and Schnöckel is based, inter alia, on the calculation of rigid coupling force constants. Nevertheless, since it is known for a long time that the use of rigid force constants as bond strength descriptors is by no means straightforward, 4 we conducted the present study in order to evaluate the robustness of the numerical data.

Computations
To analyze the numerical stability of rigid coupling force constants, as computed by Schnöckel und Köppe in ref. 2, we calculated, in a rst step, the 3 Â 3 matrix for H 2 O at the CCSD(T)/aug-cc-pvqz level of theory. Scheme 1 shows the matrix of force constants for water in terms of two different coordinate systems (I and II). Both systems have in common the two stretching coordinates OH(1) and OH (2). They differ only with respect to the third coordinate, which is the angle H-O-H or the H/H "stretching" coordinate, respectively. The rigid force constant matrix (coordinate system I) predicts a OH bond strength of 8.34 mdynÅ À1 , while aer transformation into coordinate system II the OH force constant signicantly changes to 8.97 mdynÅ À1 , simulating a stronger bond. Even more importantly, the stretch/stretch coupling constant does not only dramatically change its absolute value (from 0.11 to 0.47 mdynÅ À1 ), again simulating a strong electronic coupling, but also the algebraic sign from À to +. Any interpretation as a bond strength descriptor, is thus invalid. On the other side, looking at the relaxed force constant matrices (compliance matrix 5 ) expressed in both coordinate systems I and II, a numerically stable sub-matrix can be identied. With 8.34 mdynÅ À1 , the value of the relaxed force constant (inverse of the compliance constant) depicts a lower threshold of all possible coordinate systems. The same is true for the coupling constants. Both, value and sign, are no longer dependent on the coordinate system. In fact, the real OH/OH coupling constant vanishes completely.
Coming back to the diboryne question, the seemingly high BB/BC coupling force constant computed by Schnöckel and Köppe of +0.16 mdynÅ À1 piques one's curiosity: is it "real" or is it an artifact of the coordinate system selected by the authors? This is important since, in the words of Schnöckel and Köppe "the interaction force constant means the interaction between two bonds, i.e. whether or not and to which extent there is a restoring force within the two bonds".
As part of our ongoing project to develop unique numerical descriptors for chemical concepts, we introduced the method of generalized compliance constants (GCC) some years ago, as an extension to the theory of compliance matrices, valid for arbitrary non-stationary or stationary points on the potential energy hypersurface. 6 The entries of the inverted Hessian matrix, the compliance constants, do not suffer from coordinate dependencies and can thus be assumed to be much more transferable between similar chemical environments. 7 In order to separate real coupling phenomena from deceptive ones suggested by numerical artefacts, wein a second steptherefore recomputed (1) the rigid force constants for Schnöckel's model system 1 of Braunschweig's diboryne, applying three different coordinate systems (see below) and (2) relaxed force constants applying our GCC formalism. In the following we denote the different coordinate systems according to Schnöckel's and Köppe's symmetry coordinates (S1); Peter Pulay's natural internal coordinates (S2); primitive internal z-matrix coordinates (S3) and nally our own generalized compliance coordinates, 6b consisting of a redundant set of stretching coordinates (S4). For a better comparison with Schnöckel's data, all geometry optimizations and Cartesian force constants were computed at the BP87/dz level of theory. The transformation of the rigid Hessian matrices (S1, S2, S3) were done using Fogarasi's and Pulay's fctint code, 8 while the relaxed force constants (S4) were computed using our COM-PLIANC 3.0 code, freely available from our site http://www.oc.tubs.de/Grunenberg (Fig. 1).
As expected (see Table 1), our relaxed diagonal boron-boron (f BB ¼ 5.9 mdynÅ À1 ) and boron-carbon (f BC ¼ 5.1 mdynÅ À1 ) force constants comprise a lower boundary for all other possible coordinate systems. While the effect is quite small for Schnöckel's (S1) and Pulay's (S2) coordinate systemsboth f BB values are 6.0 mdynÅ À1the f BB value expressed in z-matrix variables (S3) of 7.3 mdynÅ À1 again is "pretending" a stronger BB bond. The same is true for the boron-carbon bond. 9 Most important nevertheless, the true coupling between the boronboron bond and the boron-carbon bond, aer the elimination of all numerical artifacts, is more or less negligible (f BB/BC ¼ À0.003).
In combination with a quick analysis of three model systems of archetypical B-B single, double and triple bonds and a comparison with the relaxed force constant of 6.3 mdynÅ À1 computed by Fischer and Braunschweig 10 for the real diboryne (B 2 IDip 2 ) allows a unique interpretation of the B-B bond under question as a triple bond. 11

Conclusions
(1) Rigid coupling force constants, as applied by Schnöckel and Köppe, are ill-dened and hence invalid as bond strength descriptors. The numerical values depend on the denition of all other coordinates. Relaxed force constants, on the other hand, do not depend on the coordinate system. They address the question "which force has to be applied against a specic internal coordinate in order to achieve a given displacement, while all other forces thereby introduced are allowed to relax". The displacements of all other coordinates caused by these forces are given by the compliance coupling constants, which are the off-diagonal terms of the compliance matrix.
(2) If rigid force constants are employed for the description of individual bond strength anyhow, the bonding situation is always described as being too strong, since the values of rigid force constants are necessarily higher than the values of the corresponding relaxed force constants. This is of course also true for non-covalent interactions, 12 even if there are still some misunderstandings on this aspect. 13 (3) The true coupling between the boron-boron bond and the boron-carbon bond, aer the elimination of all numerical artifacts, is negligible (f BB/BC ¼ À0.003).
Applying the method of generalized compliance constants (GCC), the calculation of relaxed force constants for covalent and non-covalent coordinates is now a straightforward task. It is somehow disturbing that, nearly 15 years aer our original publication, 14 the use of ill-dened, rigid force constants as bond strength descriptors is still prevalent.