Reparameterization of the chemical-potential equalization model with DFTB3: A practical balance between accuracy and transferability

Abstract

To improve the performance of the third-order density-functional tight-binding method (DFTB3) for non-covalent interactions involving organic and biological molecules, a chemical-potential equalization (CPE) approach was introduced [J. Phys. Chem. A, 116, 9131 (2012)] and parameterized for the H, C, N, O, and S chemical elements [J. Chem. Phys., 143, 084123 (2015)]. Based largely on equilibrium structures, the parameterized DFTB3/CPE models were shown to exhibit improvements in molecular polarizabilities and intermolecular interactions. With more extensive analyses, however, we observe here that the available DFTB3/CPE models have two critical limitations: (1) they lead to sharply varying potential energy surfaces, thus causing numerical instability in molecular dynamics (MD) simulations, and (2) they lead to spurious interactions at short distances for some dimer complexes. These shortcomings are attributed to the employed screening functions and the overfitting of CPE parameters. In this work, we introduce a new strategy to simplify the parameterization procedure and significantly reduce free parameters down to four global (i.e., independent of element type) ones. With this strategy, two new models, DFTB3/CPE(r) and DFTB3/CPE(r^†) are parameterized. The new models lead to smooth potential energy surfaces, stable MD simulations, and alleviate the spurious interactions at short distances, thus representing consistent improvements for both neutral and ionic hydrogen bonds.

I. INTRODUCTION

Density-functional tight-binding (DFTB) is a semiempirical quantum mechanical (SQM) method^1–8 that has emerged as an essential and widely-used tool in chemical and biochemical research.^3,4,9–12 The method is an approximation to the density functional theory (DFT)^13–17 and is derived as a Taylor expansion of the DFT energy in terms of the electron density fluctuation relative to a reference system, which is usually taken to be the sum of neutral atoms.^18–22 With a well-optimized parameter set, DFTB can outperform other SQM methods derived from the Hartree–Fock (HF) formalism and quantitatively or qualitatively reproduce DFT results with a double-zeta plus polarization quality basis set.^4,23,24 Due to several approximations (see below), DFTB is 2–3 orders of magnitude faster than standard DFT, making it readily applicable in molecular dynamics (MD) simulations of chemical and biological systems, especially in the framework of the hybrid quantum mechanical/molecular mechanical (QM/MM) approach.^11,25

Besides the truncated Taylor series expansion, the main approximations in DFTB include the two-center integral approximation in calculating matrix elements of the Hamiltonian, the monopole approximation in treating the charge–charge interaction,²⁰ and the use of a confined minimal atomic basis.^1,3,26,27 While the integral approximation affects the transferability of DFTB to different covalent interactions, the monopole approximation and confined minimal atomic basis limit its performance for non-covalent interactions.²⁸ The monopole approximation usually causes an underestimation of attractive non-covalent interactions like hydrogen bonds, and the use of confined minimal atomic bases leads to an underestimation of Pauli repulsion and electronic polarizability.^29–36

To improve the performance of the third-order DFTB model (DFTB3)^21,22 for non-covalent interactions, especially those involving charged and polarizable species, an extended polarization correction for DFTB3 using a chemical-potential equalization (CPE)^37,38 approach was introduced.³⁹ The CPE approach augments DFTB3 with an auxiliary response density to improve the electronic polarizability of systems without increasing the size of the atomic basis set. The combination of DFTB3 with the CPE approach, namely, the DFTB3/CPE model, was parameterized for the H, C, N, O, and S chemical elements.⁴⁰ In that work, static-structure-based benchmark results showed encouraging improvements in predicting molecular polarizabilities and intermolecular interactions. In this work, with more extensive analysis, we observe that the available CPE parameters have two critical limitations: (1) they lead to sharply varying potential energy surfaces, thus causing numerical instability in molecular dynamics (MD) simulations, and (2) they lead to spurious interactions at short distances for some dimer complexes. To alleviate these shortcomings, we introduce a new parameterization strategy to essentially eliminate element-specific parameters and emphasize fitting entire potential energy curves rather than just based on the interaction energies of minimum-energy structures. We demonstrate that the new strategy leads to DFTB3/CPE models that are numerically stable and exhibit modest but robust improvements over DFTB3 for hydrogen-bonding interactions involving both neutral and charged species.

II. METHODOLOGY

A. Theoretical background

The detailed derivation and formula of DFTB methods coupled with the CPE approach were already discussed elsewhere.^39,40 In this section, as a reminder, the necessary background of the DFTB3/CPE method is summarized below for introducing the key quantities and notations used in the subsequent presentations and discussions. The DFTB3 energy is written as

$$\begin{array}{ll}\hfill E^{\text{dftb}3}& =\sum _f^{\text{occ.}}{n}_f⟨{\mathrm{\Psi }}_f|\widehat{H}[{\rho }_0]|{\mathrm{\Psi }}_f⟩+\frac{1}{2}\sum _{ij}{\gamma }_{ij}{q}_i{q}_j+\frac{1}{3}\sum _{ij}{\mathrm{\Gamma }}_{ij}{q}_i^2{q}_j+\sum _{i>j}{E}_{ij}^{\text{rep}},\hfill \end{array}$$ (1)

where $\widehat{H^0}[{\rho }_0]$ is the reference (zero-th order) Hamiltonian,^18,19 $\left|{\mathrm{\Psi }}_f\right.⟩$ are the Kohn–Sham molecular orbitals (MOs); q_i is the Mulliken point charge⁴¹ on atom i; γ_ij is a kernel function represents the interaction between the two spherical charges on atom i and atom j, and Γ_ij is the derivative of γ_ij with respect to the atomic charge q_i;^20–22 $E_{ij}^{\text{rep}}$ is a pairwise repulsive potential between atom i and atom j.

In DFTB, the electron density fluctuation is approximated as the sum of atomic Slater-type spherical charge densities²⁰

$$\delta {\rho }^{\text{dftb}}=\sum _i{q}_i\delta {\rho }_i^{\text{dftb}}\approx \frac{1}{\sqrt{4\pi }}\sum _i{q}_i\frac{{\tau }_i^3}{8\pi }{e}^{-{\tau }_i|\mathbf{r}-{\mathbf{R}}_i|},$$ (2)

where r represents the Cartesian coordinate of the electron, R_i is the Cartesian coordinate of atom i, and the exponential parameter τ_i is determined from the chemical hardness U_i as^20–22,42

$${\tau }_i=\frac{16}{5}\left(U_i+\frac{\partial U_i}{\partial q_i}{q}_i\right).$$ (3)

The chemical-potential equalization approach improves the DFTB3 model by introducing an additional response density, $\delta {\rho }^{\text{cpe}}={\sum }_i{c}_i\delta {\rho }_i^{\text{cpe}}$.^39,43 For each atom, the atom-centered response density $\delta {\rho }_i^{\text{cpe}}$ is represented as p type functions,

$$\delta {\rho }_i^{\text{cpe}}(\mathbf{r})=2{\zeta }_i^2{\left(\frac{{\zeta }_i^2}{\pi }\right)}^{3/2}(k-K_i)e^{-{\zeta }^2|\mathbf{r}-{\mathbf{R}}_i{|}^2},$$ (4)

where k and K are the Cartesian components of r and R, respectively, ζ_i is a basis-set exponent. The basis-set exponent depends empirically on the partial charge as

$${\zeta }_i=Z_i{e}^{B_i{q}_i},$$ (5)

where parameters Z_i and B_i are element-specific parameters. The CPE energy contribution is a sum of CPE-DFTB Coulomb interactions and CPE–CPE Coulomb interactions, thus the combined DFTB3/CPE energy is given by

$$\begin{array}{ll}\hfill E^{\text{dftb}3\text{/cpe}}& =\sum _f^{\text{occ.}}{n}_f⟨{\mathrm{\Psi }}_f|\widehat{H}[{\rho }_0]|{\mathrm{\Psi }}_f⟩+\frac{1}{2}\sum _{ij}{\gamma }_{ij}{q}_i{q}_j\hfill \\ \hfill & +\frac{1}{3}\sum _{ij}{\mathrm{\Gamma }}_{ij}{q}_i^2{q}_j+\sum _{i>j}{E}_{ij}^{\text{rep}}+{\mathbf{c}}^T\cdot \mathbf{M}\cdot \mathbf{q}+\frac{1}{2}{\mathbf{c}}^T\cdot \mathbf{N}\cdot \mathbf{c},\hfill \end{array}$$ (6)

where c and q are sets of coefficients for the CPE response density and DFTB Mulliken charges, respectively; M and N are matrices representing CPE-DFTB and CPE–CPE Coulomb interactions, respectively.^39,40 The CPE response density (or c) is determined in a self-consistent manner with the DFTB3 density, making the DFTB3/CPE model potentially more transferable than pure molecular mechanical corrections to SQM methods.^{30,40,44–46} In the CPE-DFTB Coulomb interaction matrix element

$$M_{ij}=f(R_{ij})\iint \frac{\delta {\rho }_i^{\text{cpe}}\left(\mathbf{r}\right)\delta {\rho }_j^{\text{dftb}}\left({\mathbf{r}}^{\prime }\right)}{\left|\mathbf{r}-{\mathbf{r}}^{\prime }\right|}\mathrm{d}\mathbf{r}\mathrm{d}{\mathbf{r}}^{\prime },$$ (7)

f(R_ij) is an empirical screening function introduced to account for the missing kinetic energy component and dampens CPE-DFTB interaction at short-range distances,

$$\begin{array}{cc}\hfill f(R_{ij})=& \left\{\begin{array}{cc}0,\hfill & \text{if}{R}_{ij}<R^l\hfill \\ 1,\hfill & \text{if}{R}_{ij}>R^u\hfill \\ 1-10x^3+15x^4-6x^5,\hfill & \text{if}{R}^l\le R_{ij}\le R^u\hfill \end{array}\right.\hfill \end{array}$$

where,

$$R^l=R_i^l+R_j^l,R^u=R_i^u+R_j^u,$$

and

$$x=\frac{R^u-R_{ij}}{R^u-R^l}.$$ (8)

Here, $R_i^l$ and $R_i^u$ are empirical parameters fitted for each chemical element.

B. Reparameterization

For each chemical element, DFTB3/CPE requires four extra parameters compared to the standard DFTB3: Z and B for the basis-set exponent ζ of the atom-centered response density [Eq. (5)]; R^l and R^u for the empirical screening function [Eq. (8)]. With the increase in the number of chemical elements, the number of fitted parameters can be rather large.

The number of parameters can be reduced to three per element by considering ζ being independent from atomic charge, setting all B = 0. Both charge-independent, referred to as DFTB3/CPE(ζ), and charge-dependent, referred to as DFTB3/CPE(q) models, were considered and parameterized.⁴⁰ Due to the similarity in representations of the atomic electron density fluctuation, δρ^dftb in Eq. (2), and the atomic response density, δρ^cpe in Eq. (4), one can further reduce the number of parameters by empirically assuming that the value of Z for a chemical element can be determined by scaling its corresponding Hubbard U by a global scaling factor s_Z as

$$Z_i=s_Z{U}_i.$$ (9)

This model was referred to as DFTB3/CPE(U).⁴⁰ For a set of five chemical elements H, C, N, O, and S, the number of additional parameters DFTB3/CPE(U), DFTB3/CPE(ζ), DFTB3/CPE(q) is 11, 15, and 20, respectively.

In the previous work,⁴⁰ these models were parameterized automatically by minimizing a cost function that measures the difference between DFTB3/CPE computed non-covalent interactions or polarizabilities and reference datasets calculated at a higher level of theory such as CCSD(T). In addition, the D3(BJ)^47,48 parameters were reoptimized with these models. Nonetheless, based on our tests (presented in the Results and Discussion section), the available DFTB3/CPE models have two critical limitations: (1) sharply varying potentials, causing numerical instability in molecular dynamics (MD) simulations; and (2) spurious attractions in the short range of some interactions.

In this work, we find that to obtain a smooth potential, the difference between R^u and R^l in Eq. (8) should be greater than 2.5 a.u. The shortcoming of the available parameter sets can be relieved by applying an additional restraint condition,

$$R_{ij}^u=\left\{\begin{array}{c}{R}_i^u+R_j^u,\text{\hspace{0.17em}if\hspace{0.17em}}{R}_i^u+R_j^u-R_{ij}^l\ge 2.5\hfill \\ R_{ij}^l+2.5,\text{if\hspace{0.17em}}{R}_i^u+R_j^u-R_{ij}^l<2.5.\hfill \end{array}\right.$$ (10)

This modification is tested below for improving the DFTB3/CPE(q) and DFTB3/CPE(ζ) models.

Alternatively, we note that the number of fitted parameters can be reduced significantly by assuming that the lower limit R^l and upper limit R^u can be determined from the corresponding covalent radii R^cov and non-covalent radii R^ncov for each chemical element using the following relationships:

$$R_i^l=R_i^{\mathit{cov}}+a^l\text{\hspace{0.17em}and\hspace{0.17em}}{R}_i^u=R_i^{\mathit{ncov}}+a^u,$$ (11)

where a^l and a^u are two global parameters. While the use of the covalent radii, R^cov, for determining R^l helps to avoid the additional CPE contributions interfering with the well-parameterized covalent interactions in DFTB3, the use of the non-covalent radii, R^ncov, for determining R^u aims to enhance the DFTB3 performance for non-covalent interactions. Values of the covalent radii R^cov and the non-covalent radii R^ncov for C, H, N, O, and S chemical elements were taken from Refs. 49 and 50, respectively.

For the CPE basis-set exponent parameters, a similar strategy as the DFTB3/CPE(U) model is used: the global variable s_Z is optimized to determine the value of Z from the Hubbard U via Eq. (9). Similarly, we argue that the B parameters can be determined by scaling the Hubbard charge derivative, ∂U/∂q, following the relationship:

$$B_i=s_B\frac{\partial U_i}{\partial q_i},$$ (12)

where s_B is a global parameter. This approximation is based on the similarity in the charge dependence of the DFTB exponential parameter τ and that of the CPE basis-set exponent parameter ζ in Eqs. (3) and (5), respectively.

With the approximations described for $R_i^l$, $R_i^u$, Z_i and B_i [Eqs. (9), (11) and (12)], the number of free parameters in the CPE model is reduced to be merely 4. To determine the values of these global parameters, a^l, a^u, s_Z, and s_B, instead of brute-force optimization, we employ a tuning procedure based on the global fit of a training set including only six potential energy curves shown in Fig. 4. It is worth clarifying that the other datasets (HB375, IHB100, HB300SPX, C15, and I9), which are mentioned in the Results and Discussion section, are not included in the training. With a^l = 0.5, a^u = 2.0, s_Z = 3.2, s_B = −0.7, and two exceptional cases of (1) B_H = 0.8 for hydrogen, and (2) Z_S = 1000, B_S = 0.0, $R_S^l=2000$, $R_S^u=3000$ for sulfur, we are able to reproduce the training potential energy curves. The exceptional value of B_H ensures that the polarization of hydrogen reduces rapidly with the increase of atomic charge, as the proton should have zero polarizability. The large values of Z_S = 1000, R^l = 2000, and R^u = 3000 aim to turn off DFTB3-CPE interactions for sulfur: as explained below, this is to avoid the underestimated repulsion of sulfur with other atoms at short distances. The corresponding Z, B, R^l, and R^u calculated from these optimized global parameters are listed in Table I. In short, compared to the previous work, our current reparameterization strategy lies in two key aspects: (1) only four global parameters need to be optimized instead of 3–4 times the number of chemical elements, and (2) the training is based on global potential energy curves instead of only equilibrium structures.

FIG. 4.

Comparison of potential energy profiles for representative neutral hydrogen bonds: (A) O⋯H bond in water dimer, (B) N⋯H bond in water and ammonia complex, and (C) N⋯H bond in ammonia dimer; and for representative repulsive interactions: (D) O⋯O in water dimer, (E) N⋯O in water and ammonia complex, and (F) N⋯N in ammonia dimer.

TABLE I.

Element-specific parameters (in atomic units) in the DFTB3/CPE(r) and DFTB3/CPE(r^†) models. Note that these parameters are computed based on the four optimized global parameters (a^l, a^u, s_Z, and s_B) and standard Hubbard parameters, their charge derivatives in DFTB3/3OB, and the covalent and non-covalent radii for elements.^a,49,50

	CPE parameters
	Z	B	Rl	Ru
H	1.34	0.8000	1.10	4.27
C	1.17	0.1044	1.92	5.21
N	1.38	0.1074	1.84	4.93
O	1.59	0.1102	1.69	4.87
Sb	1000	0.0	2000	3000

	D3(BJ) parameters
	a1	a2	s8
CPE(r)c	0.5719	3.6017	0.5883
CPE(r†)	0.38	3.60	0.00

^a Parameters Z, B, R^l, R^u are the same for DFTB3/CPE(r) and DFTB3/CPE(r^†) models, only D3(BJ) parameters are further tuned for DFTB3/CPE(r^†).

^b Numeric values are chosen to switch off the CPE contribution from sulfur atoms. See discussion in the text.

^c Parameters from Ref. 52.

C. Computational details

All DFTB calculations were carried out using CHARMM⁵¹ version 46a1.51 with the 3OB parameter set,^31,33 the γ^h hydrogen correction, and the D3(BJ) dispersion correction. The D3(BJ) parameters of uncorrected DFTB3 were taken from Ref. 52, the parameters of DFTB3/CPE(ζ) and DFTB3/CPE(q) were taken from Ref. 40. Here, we refer to CPE(ζ) and CPE(q) with the modified screening parameters listed in Eq. (10) as CPE(ζ′) and CPE(q′), respectively; other parameters were taken from the original DFTB3/CPE(ζ) and DFTB3/CPE(q) models in Ref. 40.

The new DFTB3/CPE model developed in this work is referred to as DFTB3/CPE(r), which uses the previously optimized D3(BJ) parameters; with the D3(BJ) parameter further optimized to minimize the root-mean-square error (RMSE) to the CCSD(T)/CBS binding energy in the S66x8 dataset⁵³ (including displacement factors of 0.90, 1.00, 1.10, 1.25, and 1.50), the model is referred to as DFTB3/CPE(r^†). For reference values, all G4⁵⁴ calculations were carried out using Gaussian 16⁵⁵ with the zero-point and thermal corrections excluded.

III. RESULTS AND DISCUSSION

A. Screening function and numerical stability

As mentioned in Sec. II B, one critical limitation of previous CPE parameters is the sharply varying potential energy surfaces at some distances. The smoothness of DFTB3/CPE potential energy surfaces lies in the choice of the screening function f(R_ij, R^l, R^u) and its arguments: the lower limit R^l and the upper limit R^u. The automated optimization protocol used in the previous work led to small gaps between R^l and R^u in many cases: for instance, in the DFTB3/CPE(q) model, R^l = R^u = 0.3796 a.u. for hydrogen, and R^l = 3.4832 a.u., R^u = 3.6050 a.u. for oxygen; in the DFTB3/CPE(ζ) model, R^l = 5.8490 and R^u = 5.8496 for nitrogen, and R^l = 3.1834 a.u., R^u = 3.1836 a.u. for sulfur. The small differences in R^l and R^u make the screening function, and subsequently, the interaction energies vary sharply at distances close to R^l and R^u. For example, as shown in Fig. 1, the CPE(q) screening function increases sharply from 0 to 1 within an O⋯H distance interval of less than 0.065 Å, making it a discontinuous-like function in the context of MD simulations (see below). By contrast, with the additional condition to ensure a significant gap between R^l and R^u [Eq. (10)], the CPE(q′) screening function gradually increases with O⋯H distance over an interval of 1.32 Å.

FIG. 1.

Comparison of CPE(q) and CPE(q′) screening functions for the O⋯H interaction.

For an explicit illustration of the effect on the potential energy surface, Fig. 2 compares potential energy profiles along the O–H intermolecular distances for the H₂O⋯H₂O and the H₂O⋯H₂S complexes and along the N–H intermolecular distances for the NH₃⋯H₂O complex calculated by different DFTB3/CPE models. Here, DFTB3/CPE with four parameter sets CPE(ζ), CPE(q), CPE(ζ′), and CPE(q′) are compared to the uncorrected DFTB3 and the composite G4 method. As far as the numerical stability is concerned, DFTB3/CPE(ζ′) and DFTB3/CPE(q′) are as reliable as the uncorrected DFTB3 and G4 methods. On the contrary, the DFTB3/CPE(ζ) energy profile is discontinuous-like in the case of H₂O⋯H₂S complex, and DFTB3/CPE(q) shows discontinuous-like behavior in all three cases.

FIG. 2.

Comparison of potential energy profiles calculated by DFTB3, DFTB3/CPE(ζ), DFTB3/CPE(q), DFTB3/CPE(ζ′) and DFTB3/CPE(q′) along (A) O–H in H₂O⋯H₂O, (B) N–H in NH₃⋯H₂O, and (C) O–H in H₂O⋯H₂S complexes.

To further demonstrate the effect of the screening function on the numerical stability of MD simulations, we carry out the total energy conservation test for DFTB3/CPE(q) and DFTB3/CPE(q′) models. The microcanonical ensemble (NVE) MD simulations are performed for a small water cluster for 10 ps with a time step of 0.5 fs at 300 K. Figure 3 shows that DFTB3/CPE(q′) is able to maintain the variation in total energy (kinetic energy plus total electronic + nuclear energy as potential energy) to be lower than ±0.4 kcal/mol during the MD simulation. By contrast, DFTB3/CPE(q) exhibits significant energy drift as much as 11.5 kcal/mol. Evidently, significant gaps between the lower limit R^l and upper limit R^u are needed to ensure smooth DFTB3/CPE potential energy surfaces and numerically stable MD simulations. Indeed, the new models DFTB3/CPE(r) and DFTB3/CPE(r^†) exhibit the same level of energy conservation as the DFTB3/CPE(q′) model (see Fig. S1 in the supplementary material).

FIG. 3.

Drift in total energy during NVE simulation for a water cluster (H₂O)₃₂ with the DFTB3/CPE(q) and DFTB3/CPE(q′) models. For the results of DFTB3/CPE(r) and DFTB3/CPE(r^†), see Fig. S1 in the supplementary material.

B. Performance for C, H, N, and O

The performance of the new DFTB3/CPE models for describing non-covalent interactions between compounds containing O, N, C, and H is assessed using various potential energy profiles of neutral hydrogen bonds, repulsive interactions, and ionic hydrogen bonds. Here, DFTB3/CPE energy profiles calculated with the new CPE models, DFTB3/CPE(r) and DFTB3/CPE(r^†), are compared with ones calculated by the standard DFTB3, DFTB/CPE models with the old CPE parameters⁴⁰ except for the screening function, CPE(ζ′) and CPE(q′), and the reference G4 method.

1. Neutral hydrogen bonds and repulsive interactions

The changes in attractive and repulsive potential energy profiles of three representative systems, including water dimer (H₂O⋯H₂O), water and ammonia (NH₃⋯H₂O), and ammonia dimer (NH₃⋯NH₃), are analyzed. Figures 4(a)–4(c) compare attractive hydrogen-bonding interaction energy curves for these complexes, while Figs. 4(d)–4(f) compare the repulsive interactions involving these complexes in which the heavy atoms point toward one another.

The standard DFTB3 model systematically underestimates the hydrogen-bond binding energy, but to different degrees in the three examples. While the underestimation is slight for the O⋯H hydrogen bond in the case of water dimers, the underestimation is more significant, 2–3 kcal/mol, in the case of the N⋯H hydrogen bonds. The small deviation in water dimer is due to the fact that it was explicitly included in the parameterization of the 3OB model with an empirical γ-damping correction. Evidently, the empirical correction does not work as well for the N⋯H hydrogen bond. As discussed previously,⁴ this mainly reflects the lack of multipolar representation of the electron density in the DFTB3 model.

Overall, the CPE correction with all parameters increased the hydrogen-bond binding energy. The large contributions in CPE(q′) and CPE(ζ′) lead to a significant overbinding for water and ammonia complex with DFTB3/CPE(q′) and the ammonia dimer with DFTB3/CPE(ζ′). The newly developed DFTB3/CPE(r) and CPE(r^†) models show more consistent improvements over DFTB3, with very small differences between CPE(r) and CPE(r^†).

Similar to the case of neutral hydrogen bonds, the standard DFTB3 model systematically underestimates the O⋯O, N⋯O, and N⋯N repulsive interactions, especially at short-range distances. The CPE correction with all parameter sets further reduces the repulsion, making the underestimation larger in magnitude. Again, as compared to DFTB3/CPE(r) and DFTB3/CPE(r^†), the larger contributions in DFTB3/CPE(q′) and DFTB3/CPE(ζ′) cause more noticeable changes in the potential energy curves: a spurious local minimum is observed with DFTB3/CPE(q′) for the N⋯O repulsive potential energy profile and with DFTB3/CPE(ζ′) for the N⋯N repulsive potential energy profile. In contrast, these spurious interactions are largely alleviated with the new DFTB3/CPE(r) and DFTB3/CPE(r^†) models, due in part to the fact that these potential energy profiles were included in the training set in the parameterization process.

2. Ionic hydrogen bonds

To further evaluate the effects of the CPE correction, we analyze the potential energy profile for six ionic hydrogen bonds, including three cationic and three anionic complexes. Figures 5(a)–5(c) show a comparison of the energy curves for positively charged ${\text{H}}_2\text{O}\cdots {[{\text{H}}_3\text{O}]}^{+}$, ${\text{H}}_2\text{O}\cdots {[{\text{NH}}_4]}^{+}$, and ${\text{NH}}_3\cdots {[{\text{NH}}_4]}^{+}$ complexes, and Figs. 5(d)–5(f) show a comparison for negatively charged H₂O⋯[HO]⁻, NH₃⋯[HO]⁻, and ${\text{NH}}_3\cdots {[{\text{NH}}_2]}^{-}$ complexes.

FIG. 5.

Comparison of potential energy profiles for representative ionic hydrogen bonds: (a) ${\text{H}}_2\text{O}\cdots {[{\text{H}}_3\text{O}]}^{+}$, (b) ${\text{H}}_2\text{O}\cdots {[{\text{NH}}_4]}^{+}$, (c) ${\text{NH}}_3\cdots {[{\text{NH}}_4]}^{+}$, (d) H₂O⋯[HO]⁻, (e) NH₃⋯[HO]⁻, and (f) ${\text{NH}}_3\cdots {[{\text{NH}}_2]}^{-}$ complexes.

Ionic hydrogen bonds are generally stronger than neutral ones, with the binding energy varying from 20 to 30 kcal/mol. DFTB3 deviations in binding energy for ionic hydrogen bonds are larger than those for neutral hydrogen bonds. Another noticeable trend with the DFTB3 models, either with or without CPE correction, is that they usually underestimate the hydrogen-bond binding energy of cations but overestimate the binding energy for anions. The CPE correction increases the binding energy of these complexes, therefore reducing the deviation in the case of cations but increasing the overbinding of anions. It is worth noting that in the case of the NH₃⋯[HO]⁻ complex, DFTB3/CPE(ζ′) and DFTB3/CPE(q′) exhibit unexpected oscillations in the potential profiles in the range of 2.5–5.0 Å. In the case of the ${\text{NH}}_3\cdots {[{\text{NH}}_2]}^{-}$ complex, DFTB3/CPE(q′) increases the binding energy by as much as 13 kcal/mol, making it the largest deviation among the tests. The new DFTB3/CPE(r) and DFTB3/CPE(r^†) models, on the other hand, show similar improvements for cations and a smaller amount of overbinding for anions. Overall, the new DFTB3/CPE models lead to smooth potential energy profiles closer to the G4 results.

C. Performance for sulfur

Unlike previous parameterizations,⁴⁰ we find it better to switch off the CPE contribution completely for S atoms in our new parameterization. In this section, we analyze representative sulfur interactions and discuss why switching off CPE contributions on sulfur atoms is necessary. Figure 6 compares contributions from CPE with old and new parameter sets on sulfur-containing hydrogen bonds in H₂O⋯H₂S, H₂S⋯H₂O, and H₂S⋯H₂S complexes, and on sulfur-containing repulsive interactions in H₂S⋯H₂O, H₂S⋯NH₃, and H₂S⋯H₂S complexes.

FIG. 6.

Comparison of potential energy profiles for three representative sulfur-containing hydrogen bonds: (a) O–H in H₂O⋯H₂S, (b) S–H in H₂S⋯H₂O, and (c) S–H in H₂S⋯H₂S complexes; and for three repulsive interactions: (d) S–O in H₂S⋯H₂O, (e) S–N in H₂S⋯NH₃, and (f) S–S in H₂S⋯H₂S complexes.

Despite considerable effort in DFTB parameterization for sulfur,³³ accurately describing non-covalent interactions for sulfur-containing compounds remains a challenge with DFTB methods. The first “caveat,” in the form of an artificial local minimum in S⋯N, S⋯O, and S⋯S repulsive interaction energy profiles, was reported by Petraglia and Corminboeuf⁵⁶ for DFTB3 and the MIO parameter set.²⁰ The issues were caused by the use of a confined minimal atomic basis in DFTB, which led to an underestimation of Pauli repulsion. These shortcomings were reduced in the 3OB parametrization for the DFTB3 model by increasing the confinement radii for sulfur.³³ However, the spurious local minima remain as shown in Figs. 6(d)–6(f), albeit with a smaller magnitude.

The CPE correction increases molecular polarizability and results in increasing attractive interactions and reducing repulsive interactions, as already shown above in Figs. 4 and 5. Figure 6 further shows that DFTB3/CPE(ζ′) and DFTB3/CPE(q′) increase the overbinding of sulfur-containing hydrogen bonds and make the artificial local minimum in S⋯N, S⋯O, and S⋯S repulsive interaction profiles more noticeable. Even with the DFTB3/CPE(r) and CPE(r^†) models, in which the additional polarizability on the S atom is effectively turned off, non-covalent interactions are still affected as the polarizabilities of other atoms are enhanced. Nevertheless, the effects are smaller, making the new DFTB3/CPE(r) and CPE(r^†) models less problematic compared to the previous DFTB3/CPE models.

D. Performance for comprehensive databases: HB375, IHB100, and HB300SPX

To assess the transferability of the various DFTB3/CPE models, a set of comprehensive databases consisting of HB375, IHB100, and HB300SPX^57,58 is tested. The HB375 set consists of 375 neutral hydrogen bonds and non-hydrogen bonds, the IHB100 set consists of 100 ionic hydrogen bonds in organic molecules. The HB300SPX set consists of hydrogen bonds to phosphorus, sulfur, and halogens; here, we only consider sulfur-containing hydrogen bonds. The CCSD(T)/CBS binding energy is used for these test sets as the reference. Figure 7 shows a comparison of performance of the DFTB3, DFTB3/CPE with different parameters for these test sets along with the xTB/GFN2 model.⁵⁹

FIG. 7.

Errors of DFTB and xTB/GFN2 calculations for the HB375, the IHB100, and the HB300SPX test sets.

Overall, all CPE parameters show improvements over the standard DFTB3 model for C, H, N, and O interactions in both the HB375 and IHB100 test sets. The improvements are modest for neutral systems, with RMSE reduced by 0.19–0.44 kcal/mol, and more significant for ionic hydrogen bonds, with RMSE reduced by 0.35–0.63 kcal/mol. On the contrary, CPE worsens DFTB3 performance in the case of the HB300SPX test set, with the RMSE values slightly increasing by 0.07–0.15 kcal/mol when compared to the performance of DFTB3/3OB. Among the DFTB3/CPE models, DFTB3/CPE(r^†) shows a slightly smaller RMSE compared to DFTB3/CPE(r) for HB375 and IHB100 and the same RMSE for HB300SPX. It also has a similar RMSE as the DFTB3/CPE(ζ′) or DFTB3/CPE(q′) models for HB375 and outperforms them in the two other tests. Similar trends are observed for the HB375×10⁵⁷, IHB100×10⁵⁷, and HB300SPX×10⁵⁸ datasets. The detailed statistical results are listed in Tables S1–S30 in the supplementary material. In addition, errors of DFTB3 and DFTB3/CPE models for the C15 and I9 datasets,⁴⁰ which were used to train the DFTB3/CPE(ζ) and DFTB3/CPE(q) models previously, are listed in Tables S31 and S32.

Considering the smoother potential energy curves and smaller errors for repulsive interactions discussed above, we conclude that DFTB3/CPE(r^†) is the most satisfactory choice, offering a better balance between accuracy and transferability for the description of intermolecular interactions. Overall, DFTB3/CPE(r^†) has comparable performance as the xTB/GFN2 model,⁵⁹ despite the use of a monopole model for charge distributions; for ionic hydrogen bonds (IHB100), DFTB3/CPE(r^†) leads to modest but systematic improvements over xTB/GFN2 (see Tables S11–S20 in the supplementary materials). With further extension to multipolar electrostatics (Vuong et al., work in progress), we anticipate that further systematic improvements will be realized.

IV. CONCLUSIONS AND OUTLOOK

To improve the treatment of non-covalent interactions involving polarizable species within the minimal basis set framework for DFTB3, a chemical potential equalization (CPE) model was introduced³⁹ and parameterized.⁴⁰ In this work, we further evaluate the performance of the available DFTB3/CPE models in terms of accuracy and numerical stability. We have identified their limitations in MD simulations and observed spurious interactions for complexes at short distances. To alleviate these shortcomings, an additional restraint condition that modifies the screening function was introduced, along with a set of approximate schemes to determine the key CPE parameters using global parameters and element-specific Hubbard parameters. These revisions lead to smooth potential energy surfaces and a significantly reduced number of parameters in the DFTB3/CPE model. Only four global parameters are now required, and their values are determined based on six potential energy curves for non-covalent interactions.

The new DFTB3/CPE models (CPE(r) and CPE(r†), which differ only in the D3(BJ) parameters) exhibit balance between accuracy and numerical stability. They offer a modest improvement over DFTB3/3OB for neutral hydrogen bonds, with RMSE reduced from 1.65 kcal/mol for DFTB3 to 1.34 kcal/mol with the HB375 test set, and from 4.04 kcal/mol for DFTB3 to 3.45 kcal/mol with the IHB100 test set, which includes ionic species. Overall, the new DFTB3/CPE models offer more systematic and consistent improvements than the previous CPE models⁴⁰ and can be used in stable MD simulations.

Through the development of the new DFTB3/CPE models, we demonstrate that, besides employing large test sets to evaluate performance, it is crucial to examine the model for a wide range of orientations and distances that include both attractive and repulsive interactions.

Our benchmarks also underline two remaining limitations of the current DFTB3 model and point to directions for improvement. First, hydrogen bonding interactions, especially those involving N atoms, remain underestimated. The underestimation is attributed to the monopole approximation of electron density and needs to be corrected by a multipolar charge density model, as shown with the xTB/GFN2 method.^59,60 Using CPE to correct the underestimation will make the method less transferable and increase the underestimation of repulsive interactions; for instance, the N–N and N–O repulsive interactions are strongly underestimated with DFTB3/CPE(q) because the parameters were optimized to best fit attractive N–H hydrogen bonds.⁴⁰

Second, in addition to the cases of N–N and N–O repulsive interactions, our analysis of the sulfur-containing interactions also highlights that the current DFTB3 model still suffers severely from the underestimation of Pauli repulsion, which consequently causes underestimation for N–N, O–O, N–O, S–N, S–O, and S–S interactions. We note that the 3OB parameter set was optimized to balance the underestimations of Pauli repulsion with the underestimations of attractive interaction due to the use of the monopole approximation and minimum basis set. Therefore, correcting the polarizability using the CPE model while keeping the electronic parameters (e.g., compression radii) untouched will lead to an imbalance in the total interaction energy; this is explicitly illustrated in Tables S33 and S34 and Figs. S5 and S6 in the supplementary material. In our on-going work, reoptimization of the electronic parameters along with a better description of Pauli repulsion, multipolar charge expansion, and chemical-potential equalization correction are being pursued to improve the description of non-covalent interactions within the DFTB framework in a physically sound fashion.

SUPPLEMENTARY MATERIAL

The supplementary material includes the comparison of energy drifts in NVE simulations for a water cluster (H₂O)₃₂ with the DFTB3/CPE(q), DFTB3/CPE(q′), DFTB3/CPE(r), and DFTB3/CPE(r^†) models, tables of statistical results of DFTB3, DFTB3/CPE and xTB/GFN2 models for the HB375×10, IHB100×10, HB300SPX×10, C15 and I9 sets. It also includes the comparison of polarizabilities for molecules in the QM7b and AlphaML showcase databases using various DFTB models and B3LYP/6-31G(d).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Van-Quan Vuong: Conceptualization (equal); Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Qiang Cui: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.