Compliance constants

By Taylor series expansion, the potential energy, ${\displaystyle V}$, of any molecule can be expressed as:[5][11]

${\displaystyle V=V_{0}+G^{T}Z+{1 \over 2}Z^{T}HZ+...}$ (eq. 1)

where ${\displaystyle Z}$ is a column vector of arbitrary and fully determined displacement coordinates, and ${\displaystyle G}$ and ${\displaystyle H}$ are the corresponding gradient (first derivative of ${\displaystyle V}$) and Hessian (second derivative of ${\displaystyle V}$), respectively. The point of interest is the stationary point on a potential energy surface (PES), so ${\displaystyle G}$ is treated as zero, and by considering the relative energy, ${\displaystyle V_{0}}$ as well becomes zero. By assuming harmonic potential and regarding the third derivative term and forth as negligible, the potential energy formula then simply becomes:

${\displaystyle V={1 \over 2}Z^{T}HZ}$ (eq. 2)

Transitioning from cartesian coordinates ${\displaystyle Z}$ to internal coordinates ${\displaystyle Q}$, which are more commonly used for the description of molecular geometries, gives rise to equation 3:

${\displaystyle V={1 \over 2}Q^{T}H_{q}Q}$ (eq. 3)

where ${\displaystyle H_{q}}$ is the corresponding Hessian for internal coordinates (commonly referred to as force constants), and it is in principle determined by the frequencies of a sufficient set of isotopic molecules. Since the Hessian ${\displaystyle H_{q}}$ is the second derivative of the energy with respect to displacements and that is the same as the first derivative of the force, evaluation of this property as shown in equation 4 is often utilized to describe chemical bonds.

${\displaystyle H_{q}={\biggl (}{\partial ^{2}V \over \partial Q_{i}\partial Q_{j}}{\biggr )}_{0}}$ (eq. 4)

Nevertheless, there are several issues with this method as explained by Grunenberg,[5] 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) ${\displaystyle G_{q}}$.

${\displaystyle V={1 \over 2}{G_{q}}^{T}CG_{q}}$ (eq. 5)

This gradient ${\displaystyle G_{q}}$ is the first derivative of the potential energy with respect to the displacement coordinates, which can be expressed as shown:

${\displaystyle G_{q}=H_{q}Q}$ (eq. 6)

By substituting the expression of ${\displaystyle G_{q}}$ in eq. 5 into equation 5, equation 7 is obtained.

${\displaystyle V={1 \over 2}Q^{T}{H_{q}}^{T}CH_{q}Q}$ (eq. 7)

Thus, with the knowledge that ${\displaystyle H_{q}}$ is positive definite, the only possible value of ${\displaystyle C}$ which is the compliance matrix then must be:

${\displaystyle C={H_{q}}^{-1}}$ (eq. 8)

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.[5] 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;[13] 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[14][15] based on typical force constants calculation.

Table 1. Force constants (N/cm) of n-butane in natural internal coordinates and z-matrix coordinates

n-butane

Natural Internal Coordinates					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

Table 2. Force constants (N/cm) of cyclobutane in natural internal coordinates and z-matrix coordinates

cyclobutane

Natural Internal Coordinates						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.[13] 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.[13] Clearly for cyclobutane—and numerous other molecules, utilizing force constants therefore gives rise to inaccurate bond descriptors due to its dependence on coordinate systems.

A more accurate approach as claimed by Grunenberg[5] is to exploit compliance constants as means for describing chemical bonds as shown below.

Table 3. Compliance constants (N⁻¹) of n-butane in natural internal coordinates and z-matrix coordinates

n-butane

Natural Internal Coordinates					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

Table 4. Compliance constants (N⁻¹) of cyclobutane in natural internal coordinates and z-matrix coordinates

cyclobutane

Natural Internal Coordinates						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

All the calculated compliance constants above are given in N⁻¹ unit. For both n-butane and cyclobutane, the results are the same regardless of the choice of the coordinate systems. One aspect of compliance constants that proves more powerful than force constants in cyclobutane is because of less coupling. This compliance coupling constants are the off-diagonal elements in the inverted Hessian matrix and altogether with the compliance constants, they physically describe the relaxed distortion of a molecule closely through a minimum energy path. Moreover, the values of the compliance constants yield the same results for all the C-C bonds and the values are less compared to those obtained for n-butane. Compliance constants, thus, give results that are in accordance to what are generally known about the ring strain of cyclobutane.[13]

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,[1] 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.[2] That same year, Grunenberg reassessed the B-B bond utilizing generalized compliance constants of which he claimed better suited as a bond strength descriptor.[4]

The calculated relaxed force constants show a clear trend as the bond order between the B-B bond increases, which advocates the existence of the triple bond in Braunschweig's compound.

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 Na₂[H-GaGa-H] compound in C_2h symmetry has a compliance constant value of 0.870 aJ/Ų is in fact weaker than a Ga-Ga double bond (1.201 aJ/Ų).

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.