Compliance constants are the elements of an inverted Hessian matrix. The calculation of compliance constants provides an alternative description of chemical bonds in comparison with the widely used force constants explicitly ruling out the dependency on the coordinate system. They provide the unique description of the mechanical strength for covalent and non-covalent bonding. While force constants (as energy second derivatives) are usually given in aJ/Å or N/cm, compliance constants are given in Å/aJ or Å/mdyn.
Hitherto, recent publications[1] that broke the wall of putative chemical understanding and presented detection/isolation of novel compounds with intriguing bonding characters can still be provocative at times.[2] [3] [4] The stir in such discoveries arose partly from the lack of a universally accepted bond descriptor. While bond dissociation energies (BDE) and rigid force constants have been generally regarded as primary tools for such interpretation, they are prone to flawed definition of chemical bonds in certain scenarios whether simple[5] or controversial.[6] [7]
Such reasons prompted the necessity to seek an alternative approach to describe covalent and non-covalent interactions more rigorously., a German chemist at the TU Braunschweig and his Ph.D. student at the time, Kai Brandhorst, developed a program COMPLIANCE[8] (freely available to the public), which harnesses compliance constants for tackling the aforementioned tasks. The authors use an inverted matrix of force constants, i.e., inverted Hessian matrix, originally introduced by W. T. Taylor and K. S. Pitzer.[9] The insight in choosing the inverted matrix is from the realization that not all elements in the Hessian matrix are necessary—and thus redundant—for describing covalent and non-covalent interactions. Such redundancy is common for many molecules,[10] and more importantly, it ushers in the dependence of the elements of the Hessian matrix on the choice of coordinate system. Therefore, the author claimed that force constants albeit more widely used are not an appropriate bond descriptor whereas non-redundant and coordinate system-independent compliance constants are.[11]
By Taylor series expansion, the potential energy,
V
V=V0+GTZ+{1\over2}ZTHZ+...
where
Z
G
H
V
V
G
V0
V={1\over2}ZTHZ
Transitioning from cartesian coordinates
Z
Q
V={1\over
| TH | |
| 2}Q | |
where
Hq
Hq
Hq=l({\partial2V\over\partialQi\partialQj}r)0
Nevertheless, there are several issues with this method as explained by Grunenberg, including the dependence of force constants on the choice of internal coordinates and the presence of the redundant Hessian which has no physical meaning and consequently engenders ill-defined description of bond strength.
Rather than internal displacement coordinates, an alternative approach to write the potential energy of a molecule as explained by Decius[12] is to write it as a quadratic form in terms of generalized displacement forces (negative gradient)
Gq
V={1\over
| TCG | |
| 2}{G | |
| q |
This gradient
Gq
Gq=HqQ
By substituting the expression of
Gq
V={1\over
| TCH | |
| 2}Q | |
Thus, with the knowledge that
Hq
C
C=
| -1 | |
| {H | |
| q} |
Equation 7 offers a surrogate formulation of the potential energy which proves to be significantly advantageous in defining chemical bonds. Specially, this method is independent on coordinate selection and also eliminates such issue with redundant Hessian that the common force constant calculation method suffers with. Intriguingly, compliance constants calculation can be employed regardless of the redundancy of the coordinates.
To illustrate how choices of coordinate systems for calculations of chemical bonds can immensely affect the results and consequently engender ill-defined descriptors of the bonds, sample calculations for n-butane and cyclobutane are shown in this section. Note that it is known that the all the four equivalent C-C bonds in cyclobutane are weaker than any of the two distinct C-C bonds in n-butane; therefore, juxtaposition and evaluation of the strength of the C-C bonds in this C4 system can exemplify how force constants fail and how compliance constants do not. The tables immediately below are results that are calculated at MP2/aug-cc-pvtz level of theory[13] [14] based on typical force constants calculation.
| Z-matrix Coordinates | ||||||||
|---|---|---|---|---|---|---|---|---|
| 1-2 | 2-3 | 3-4 | 1-2 | 2-3 | 3-4 | |||
| 1-2 | 4.708 | 1-2 | 4.708 | |||||
| 2-3 | 0.124 | 4.679 | 2-3 | 0.124 | 4.679 | |||
| 3-4 | 0.016 | 0.124 | 4.708 | 3-4 | 0.016 | 0.124 | 4.708 | |
| Z-matrix Coordinates | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1-2 | 2-3 | 3-4 | 4-1 | 1-2 | 2-3 | 3-4 | 4-1 | |||
| 1-2 | 4.173 | 1-2 | 4.914 | |||||||
| 2-3 | 0.051 | 4.173 | 2-3 | -0.459 | 4.906 | |||||
| 3-4 | 0.155 | 0.051 | 4.173 | 3-4 | -0.864 | 0.813 | 5.504 | |||
| 4-1 | 0.051 | 0.155 | 0.051 | 4.173 | 4-1 | 0.786 | -0.771 | -0.976 | 5.340 | |
Tables 1 and 2 display a force constant in N/cm between each pair of carbon atoms (diagonal) as well as the coupling (off-diagonal). Considering natural internal coordinates on the left, the results make chemical sense. Firstly, the C-C bonds are n-butane are generally stronger than those in cyclobutane, which is in line with what is expected.[15] Secondly, the C-C bonds in cyclobutane are equivalent with the force constant values of 4.173 N/cm. Lastly, there is little coupling between the force constants as seen as the small compliance coupling constants in the off-diagonal terms.
However, when z-matrix coordinates are used, the results are different from those obtained from natural internal coordinates and become erroneous. The four C-C bonds all have distinct values in cyclobutane, and the coupling becomes much more pronounced. Significantly, the force constants of the C-C bonds in cyclobutane here are also larger than those of n-butane, which is in conflict with chemical intuition. Clearly for cyclobutane—and numerous other molecules, using force constants therefore gives rise to inaccurate bond descriptors due to its dependence on coordinate systems.
A more accurate approach as claimed by Grunenberg is to exploit compliance constants as means for describing chemical bonds as shown below.
| Z-matrix Coordinates | ||||||||
|---|---|---|---|---|---|---|---|---|
| 1-2 | 2-3 | 3-4 | 1-2 | 2-3 | 3-4 | |||
| 1-2 | 0.230 | 1-2 | 0.230 | |||||
| 2-3 | -0.010 | 0.233 | 2-3 | -0.010 | 0.233 | |||
| 3-4 | 0.002 | -0.010 | 0.230 | 3-4 | 0.002 | -0.010 | 0.230 | |
| Z-matrix Coordinates | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| 1-2 | 2-3 | 3-4 | 4-1 | 1-2 | 2-3 | 3-4 | 4-1 | |||
| 1-2 | 0.255 | 1-2 | 0.255 | |||||||
| 2-3 | -0.006 | 0.255 | 2-3 | -0.006 | 0.255 | |||||
| 3-4 | -0.010 | -0.006 | 0.255 | 3-4 | -0.010 | -0.006 | 0.255 | |||
| 4-1 | -0.006 | -0.010 | -0.006 | 0.255 | 4-1 | -0.006 | -0.010 | -0.006 | 0.255 | |
Diboryne or a compound with boron-boron triple bond was first isolated as a N-heterocyclic carbene supported complex (NHC-BB-NHC) in the Braunschweig group, and its unique, peculiar bonding structure thereupon catalyzed new research to computationally assess the nature of this at that time controversial triple bond.
A few years later, Köppe and Schnöckel published an article arguing that the B-B bond should be defined as a 1.5 bond based on thermodynamic view and rigid force constant calculations. That same year, Grunenberg reassessed the B-B bond using generalized compliance constants of which he claimed better suited as a bond strength descriptor.
| NHC-HBBH-NHC | 1.5 | single | |
|---|---|---|---|
| NHC-HBBH-NHC | 3.8 | double | |
| NHC-BB-NHC | 6.5 | triple |
Grunenberg and N. Goldberg[16] probed the bond strength of a Ga-Ga triple bond by calculating the compliance constants of digallium complexes with a single bond, a double bond, or a triple bond. The results show that the Ga-Ga triple bond of a model Na[H-GaGa-H] compound in C symmetry has a compliance constant value of 0.870 aJ/Å is in fact weaker than a Ga-Ga double bond (1.201 aJ/Å).
Besides chemical bonds, compliance constants are also useful for determining non-covalent bonds, such as H-bonds in Watson-Crick base pairs.[17] Grunenberg calculated the compliance constant for each of the donor-H⋯acceptor linkages in AT and CG base pairs and found that the central N-H⋯N bond in CG base pair is the strongest one with the compliance constant value of 2.284 Å/mdyn. (Note that the unit is reported in a reverse unit.) In addition, one of the three hydrogen bonding interactions in a AT base pair shows an extremely large compliance value of >20 Å/mdyn indicative of a weak interaction.