Van der Waals dimers and hydrogen bonds

With the electron force field, it should be possible to model interactions between molecules as well as interaction within molecules. We examine as test cases water dimer, hydrogen fluoride dimer, methane-water dimer, and methane-methane dimer (Figure 5.24). The interaction between two methanes is reasonably well-described ( $\mathrm{\Delta E}$ = -0.5 kcal/mol vs -0.3 kcal/mol exact, and $\mathrm{R_{CC} = 3.91\ \AA}$ versus 4.13 $\mathrm{\AA}$ exact), but in water dimer and HF dimer, the hydrogen bonds are significantly too strong and too short (for water dimer, $\mathrm{\Delta E}$ = -36 kcal/mol versus -5 kcal/mol exact, with $\mathrm{R_{OO}}$ = 2.29 $\mathrm{\AA}$ versus 2.92 $\mathrm{\AA}$ exact; for HF dimer, $\mathrm{\Delta E}$ = -54 kcal/mol versus -5 kcal/mol exact, with $\mathrm{R_{FF}}$ = 2.13 $\mathrm{\AA}$ versus 2.73 exact). The methane-water interaction falls between these two extremes, and has $\mathrm{\Delta E}$ = -3.9 kcal/mol versus -0.9 kcal/mol exact, and $\mathrm{R_{OO}}$ = 3.18 $\mathrm{\AA}$ versus 3.51 $\mathrm{\AA}$ exact.

Figure 5.24: Hydrogen bonds in eFF are too strong and too short, probably from a combination of too-small Pauli repulsion and too-large monomer dipoles.

Remarkably, electrostatics and Pauli repulsion are balanced sufficiently in the water and HF dimer cases that the angles they make relative to each other are nearly the exact values (water dimer $\mathrm{\theta_{1} = 5^{o}}$ versus $\mathrm{6^{o}}$ exact and $\mathrm{\theta_{2} = 112^{o}}$ versus $\mathrm{123^{o}}$ exact; HF dimer $\mathrm{\theta_{1} = 6^{o}}$ versus $\mathrm{7^{o}}$ exact and $\theta_{2} = 104^{o}$ versus $\mathrm{112^{o}}$ exact).

We propose that eFF hydrogen bonds are too strong and too short because the monomer dipoles are too large and the Pauli repulsion between monomers too small. Taking water dimer as a test case, we attempt to quantify the change in interaction energy and geometry we would expect if the monomer dipoles and Pauli repulsions were correct. We make a dipole moment correction by adding point dipoles onto both water molecules of a magnitude such that the net water dipole goes from 3.27 D to 1.85 D, the exact value. Moving the water molecules apart from each other with bond lengths and angles fixed, we find that the dipole correction makes the water molecules bind by only 18 kcal/mol, with an OO distance of $\sim$2.45 $\mathrm{\AA}$. It is clear that although the too-large dipole moment does cause the hydrogen bond to be stronger, it cannot be responsible for all of the too-high eFF hydrogen bond strength of 36 kcal/mol.

Figure 5.25: Interaction energy of the water dimer, with the estimated effects of changing monomer dipole moments and Pauli repulsions to be the correct values.

We estimate the Pauli correction by taking the difference between eFF and Hartree-Fock interaction curves for neon dimer and parameterizing it:

$$\log E_{\mathrm{difference}} = 8.55616 r - 15.1178 r^{2} + 8.15818 r^{3} - 1.65597 r^{4}$$

where $r$ is in bohr and the energy is in hartrees. We correct for the difference in size between a water molecule and a neon atom by scaling the distances by the cube root of the ratio of polarizabilities ( $\alpha_{neon} = 2.67 \ \mathrm{bohr}^{3}$, $\alpha_{water} = 9.92\ \mathrm{bohr}^{3}$; $\mathrm{(\alpha_{water}/\alpha_{neon})^{1/3}}$ = 1.55). With the Pauli correction added, the water dimer has a binding energy of $\sim$3 kcal/mol and an OO distance of $\sim$3.2 $\mathrm{\AA}$, closer to the expected value. It appears that a combination of correct Pauli repulsion and correct monomer dipole moments is needed to obtain a correct description of hydrogen bonding in eFF.

Unlike with conventional force fields, in eFF hydrogen bonds appear as an emergent consequence of existing electrostatic and Pauli interactions. In attempting to describe inter- and intramolecular bonding with the same set of energy expressions and parameters, we face a more difficult challenge than arises in developing conventional force fields, which usually contain explicit van der Waals and hydrogen bond terms. Advanced water force fields often combine multiple or delocalized charges with polarizable sites and explicit van der Waals and repulsive terms. Such schemes require extensive parameterization against properties of interest and use parameters and functions that are rather system-specific. In contrast, eFF accounts for electrostatic and Pauli effects in a way designed to apply generally over a broad range of molecules, and so may be useful in developing new transferable force fields with accurate descriptions of solvents.