Universal Pairwise Interatomic van der Waals Potentials Based on Quantum Drude Oscillators

Abstract

Repulsive short-range and attractive long-range van der Waals (vdW) forces play an appreciable role in the behavior of extended molecular systems. When using empirical force fields, the most popular computational methods applied to such systems, vdW forces are typically described by Lennard-Jones-like potentials, which unfortunately have a limited predictive power. Here, we present a universal parameterization of a quantum-mechanical vdW potential, which requires only two free-atom properties—the static dipole polarizability α₁ and the dipole–dipole C₆ dispersion coefficient. This is achieved by deriving the functional form of the potential from the quantum Drude oscillator (QDO) model, employing scaling laws for the equilibrium distance and the binding energy, and applying the microscopic law of corresponding states. The vdW–QDO potential is shown to be accurate for vdW binding energy curves, as demonstrated by comparing to the ab initio binding curves of 21 noble-gas dimers. The functional form of the vdW–QDO potential has the correct asymptotic behavior at both zero and infinite distances. In addition, it is shown that the damped vdW–QDO potential can accurately describe vdW interactions in dimers consisting of group II elements. Finally, we demonstrate the applicability of the atom-in-molecule vdW–QDO model for predicting accurate dispersion energies for molecular systems. The present work makes an important step toward constructing universal vdW potentials, which could benefit (bio)molecular computational studies.

Introduction

Van der Waals (vdW) forces play an indisputably important role in determining the structure and dynamics of many biomolecular, solid-state, and polymeric systems.¹⁻⁵ The accurate description of vdW interactions requires sophisticated quantum-mechanical treatment using the adiabatic-connection fluctuation–dissipation theorem (ACFDT) in density-functional theory or high-level quantum chemistry methods, such as coupled cluster or quantum Monte Carlo.^4,5 However, the prohibitive computational cost of these methods precludes their applicability to extended (bio)molecular systems. Therefore, practical simulations of large and complex systems are often done using classical force fields such as AMBER,⁶ CHARMM,⁷ or GROMACS.⁸

For the description of vdW forces, these popular force fields resort to the seminal Lennard-Jones (LJ)⁹ (or an improved Buckingham¹⁰) potential as a practical shortcut. Two parameters, well depth D_e and equilibrium position R_e, fully specify the LJ potential. However, these parameters can be determined unambiguously only for relatively simple vdW-bonded systems, such as noble-gas dimers or crystals. Moreover, the LJ potential is notorious for its lack of flexibility and very limited quantitative accuracy.^11,12 On the other hand, the celebrated Tang–Toennies (TT) potentials¹³⁻¹⁶ are derived from first principles and yield high accuracy for dimers including noble gases and group II elements. To achieve such an accuracy, the TT potentials employ 5 to 9 parameters depending on the exact flavor.¹⁶ Setting these parameters requires knowledge of R_e and D_e for each vdW-bonded dimer,^13,15 which prevents a generalization of the TT models to the whole periodic table. Moreover, like the LJ potential, the most recent conformal Tang-Toennies-Sheng (TTS) potential¹⁵ is prone to large errors for dispersion coefficients (see Figure 1a) despite its high accuracy close to equilibrium distances. Hence, a vdW potential combining wide transferability across the periodic table, high accuracy, and minimal parameterization is not yet available.

Figure 1.

(a) Errors in the dispersion coefficients arising from the LJ and TTS potentials. The TTS dispersion coefficients are obtained as C_2n = C_2n* × D_eR_e²ⁿ,¹⁵ and the reference dispersion coefficients C_2n^ref are given in Table 1. (b) vdW–QDO potential for neon dimer benchmarked to the TTS potential and the reference CCSD(T) potential.⁵⁵ Vertical dotted lines indicate the equilibrium distance as predicted by the CCSD(T) (black) and vdW–QDO potential (red). For comparison, the LJ potentials in two different parameterizations (see our discussion in the text) are also displayed in blue and green.

Here, we develop a universal conformal pairwise vdW potential, which can be parameterized for all chemical elements based solely on two nonbonded atomic properties—static dipole polarizability α₁ and dipole–dipole dispersion coefficient C₆. Our potential is consistently derived within the framework of the quantum Drude oscillator (QDO) model¹⁷ using the Heitler–London perturbation theory,^18,19 and it is devoid of adjustable parameters. This is achieved by building connections between atomic scaling laws,²⁰⁻²² the microscopic law of corresponding states,²³⁻²⁶ and symmetry-adapted perturbation theory (SAPT)^27,28 for intermolecular interactions. The derived exchange repulsion term in our potential obeys correct physical limits both at R → 0 and R → ∞, and the predicted C₆ dispersion coefficients are significantly more accurate compared to the other conformal LJ and TTS¹⁵ potentials. The designed vdW–QDO potentials are twice as accurate as the LJ potentials when averaged over 15 noble-gas dimers. In addition, the vdW–QDO potential augmented by a damping function can accurately describe the binding curves of dimers consisting of (closed-shell) group II atoms. Moreover, the vdW–QDO potential can be applied to molecular systems, when coupled with the atom-in-molecule (AIM) approach.²⁹ We demonstrate this by accurately reproducing the dispersion energy for dispersion-dominated molecular dimers from the S66 × 8 data set.³⁰

We derive the vdW potential in the QDO framework, which is a coarse-grained model for the electronic response³¹⁻³⁶ and proved to be accurate and insightful in many applications across various fields.^{17,21,36−45} Within the QDO model, the response of valence electrons is described via a quasi-particle (drudon or Drude particle) with a negative charge −q and mass μ, harmonically bound to a positively charged pseudonucleus of charge q with a characteristic frequency ω. Coupled QDOs are also extensively used in the development of vdW density functionals,^{29,40,46−106} quantum mechanical,^17,36 and polarizable force fields^39,47−50 as well as recent machine learning force fields.^51,52

The QDO model has already been used to build interatomic vdW potentials for water or noble-gas dimers and crystals.^{17,36,38,39,53} However, within the corresponding studies, the repulsive term was added in an ad hoc manner, either by fitting Born–Mayer^10,54 exponents to ab initio repulsive walls^17,36,38,39 or by directly adding the Hartree–Fock exchange energy.⁵³ Therefore, such potentials cannot be generalized beyond the systems for which direct first-principles simulations are possible. In contrast, here, we suggest a consistent treatment of both Pauli (exchange) repulsion and vdW dispersion within the QDO framework. To our knowledge, this is the first vdW potential of such a type, which does not directly utilize the reference binding energy of dimers or the Hartree–Fock exchange energy curve but nevertheless provides relatively good accuracy.

Results

Model Construction

The long-range vdW dispersion energy for two identical QDOs is given by the usual multipolar series^17,35

1

where the dispersion coefficients are related to the oscillator parameters via the closed-form expressions¹⁷

2

where α₁ = q²/μω² is the QDO dipole polarizability and k_e = 1/4πε₀. Tang and Toennies showed^13,14 that including the three leading dispersion terms is sufficient to obtain the accurate vdW potential. Therefore, we also truncate the series of eq 1 to the C₁₀ term.

The exchange repulsion is introduced into the model according to refs (20) and (21), where multipole contributions to the exchange energy of a homonuclear dimer were derived by considering two identical drudons as bosons, assuming that they represent closed valence shells of atoms with zero total spin. Consequently, the total wave function of a dimer is represented by a symmetrized product

3

where and are, respectively, the ground-state wave functions of drudons centered at nuclei A and B separated by R. Within the Heitler–London perturbation theory,^18,19 the exchange energy of two identical vdW-bonded QDOs for distances near equilibrium and larger is well approximated by the exchange integral²⁰

4

The evaluation of eq 4 with the multipole expansion of Coulomb coupling V̂_C between the two QDOs results in multipole contributions to the exchange energy.^20,21 In dipole approximation, this yields

5

where S is the overlap integral. Higher-order multipole contributions (l > 1) to the exchange repulsion energy J_ex^(l) have the same leading-term dependence on internuclear distance R, with the only difference in a proportionality coefficient, i.e., J_ex^(l) ∝ k_eq²S/R.²¹ Therefore, we introduce an effective exchange repulsion energy as

6

with the proportionality coefficient A to be determined self-consistently, in what follows. In this way, we effectively include multipole contributions to all orders. Importantly, our E_ex^eff has a 1/R dependence, which properly describes the infinite repulsive wall at short distances. Thus, in contrast to the Born–Mayer or Duman–Smirnov⁵⁶⁻⁵⁸ functional forms for exchange repulsion possessing a finite value of E_ex at R → 0, eq 6 is in agreement with the orbital overlap model for Pauli repulsion.^59,60 Moreover, our E_ex^eff does not rely on empiricism as it explicitly depends only on the QDO parameters (vide infra), whereas the existing Pauli repulsion models require fitting to some ab initio data.^{17,36,37,58,60−62}

In order to determine the coefficient A in eq 6, we employ the force balance condition at the equilibrium distance, , which yields

7

To evaluate the equilibrium distance R_e in our model, we use the quantum-mechanical relation between the atomic (static) dipole polarizability and vdW radius²⁰

8

where the proportionality coefficient Φ is given by²²

9

with α_fsc = e²/4πε₀ℏc ≈ 1/137.036 as the fine-structure constant. The relation given by eqs 8 and 9 turned out to be valid for real atoms. Especially, it is very accurate for noble gases, where the mean absolute relative error (MARE) ⟨|R_vdW – R_vdW^ref|/R_vdW^ref⟩ is about 1%.^20,22 Since by definition R_vdW is a half of the equilibrium distance R_e in a homonuclear vdW-bonded dimer,^20,63 accurate equilibrium distances can be obtained via

10

With α₁ and C₆ being fixed, there are two unknown quantities in eq 7, A and μω, since C₈ and C₁₀ are solely expressed in terms of C₆ and μω via eq 2.

As shown in ref (45), the product μω can be obtained from the force balance in the dipole approximation

11

with R_e substituted from eq 10. The solution of this transcendental equation allows us to determine the three oscillator parameters {q, μ, ω} given only {α₁, C₆}. Let us call this parameterization scheme vdW–OQDO, similar to the recently suggested optimized quantum Drude oscillator (OQDO) scheme.⁴⁵ The details of the procedure and the corresponding values of {q, μ, ω} can be found in the Supporting Information.

Solving eqs 11 and 7 together, one can obtain

12

and the total vdW potential

13

The vdW–QDO potential for neon is displayed by the red curve in Figure 1b, which shows excellent agreement with the TTS potential¹⁵ as well as with the CCSD(T) calculations⁵⁵ across the whole range of distances from 0.7R_e (∼4 bohr) to infinity. Inclusion of C₈ and C₁₀ dispersion coefficients together with the suggested approach to treat exchange repulsion energy allows us to predict the correct depth and shape of the potential without losing the accuracy in predicting the equilibrium distance, which is inherited from the dipole approximation. In addition, we compare our potential to the LJ potential, for which we use two different parameterizations: LJ1 derived from thermodynamical properties and LJ2 designed to reproduce reference values of R_e and D_e (see the Supporting Information for more details). We note that the present vdW–QDO potential (eq 13) performs accurately over the whole range of distances, whereas the LJ1 potential (blue curve in Figure 1b) underestimates the energy in the potential minimum region and the LJ2 potential overestimates the long-range energy (green curve), although both being reasonably accurate in the repulsive region. This imbalance and lack of flexibility of the LJ potential, which is observed for all noble gases, is one of the main issues limiting its quantitative predictive power.^11,12 Moreover, the LJ potential severely overestimates the C₆ coefficient (Figure 1a), which is responsible for the correct long-range energy. The proposed vdW–QDO potential overcomes these difficulties without increasing the number of parameters. Moreover, our potential recovers correct bonding behavior using only a free atom property α₁ and the asymptotic interaction parameter C₆, which do not contain information about the interaction between atoms at short distances.

Application to Noble-Gas Dimers

With the accurate Ne₂ potential curve in hand, its counterparts for all other noble-gas dimers can be derived using the conformality of their potentials,^15,66,67 which is a microscopic manifestation of the law of corresponding states.²³⁻²⁵ Namely, for the vdW potential of other noble-gas dimers, we write

14

where U^Ne_QDO(x) = V^Ne_QDO(xR^Ne_e)/V^Ne_QDO(R^Ne_e) is the dimensionless potential (shape) of the Ne₂ dimer

15

with the numerical values of the starred (unitless) parameters and their definitions presented in Table 2.

Table 2. Dimensionless Parameters in eq 15^a.

parameter	definition	numerical value
A*	Akeq2/ReDe	1508.917
γ*		3.912
C6*	C6/DeRe6	1.1779
C8*	5C6/DeRe6(γ*)2	0.3848
C10*		0.1540

^aThe neon dimer parameters used in the second column are D_e(Ne₂) = 3.586 meV = 13.178 × 10^–5 a.u. [eq 16] and R_e(Ne₂) = 5.875 bohr [eq. 10]. The QDO parameters for the Ne₂ dimer are q = 1.18865, μ = 0.37164, and ω = 1.19326 (all in a.u.).

Thus, only R_e and D_e for every dimer are required to obtain their vdW potential. For R_e, the accurate scaling law (eq 10) is already established, whereas an analogous scaling law for D_e of noble-gas dimers is not yet known. Substituting R = R_e to eq 13 and using eq 7 to eliminate yields

16

with . Analyzing reference CCSD(T) data for D_e from ref (15), we found that eq 16 truncated at the first two terms can accurately predict D_e for all noble-gas dimers

17

In Figure 2, the D_e values by eq 17 are compared to the reference CCSD(T) data. The bar chart shows that eq 17 is accurate for homo- and heteronuclear dimers of He–Xe with all errors below 1 meV. For dimers with Rn, the error is larger, with Rn₂ being underbound by 4.5 meV or 13%. The larger errors for Rn dimers likely stem from the fact that the reference coupled-cluster calculation⁷⁴ is less reliable than the corresponding calculations for the lighter dimers He₂–Xe₂.^55,68−71 For example, the D_e of the Xe₂ dimer reported in ref (74) is by 7.5% larger than the one of ref (71), which is a state-of-the-art calculation. Thus, a similar or even larger overestimation of D_e should be expected for Rn₂,⁷⁴ where relativistic effects are more pronounced. Accounting for that, the estimated error of eq 17 for Rn₂ would not exceed 5.5%. We conclude that the suggested scaling law (eq 17) allows one to accurately evaluate D_e for all noble-gas dimers given only {α₁, C₆} without involving any adjustable parameters.

Figure 2.

Potential well depth D_e by eq 17 compared to the reference CCSD(T) values¹⁵ for 21 noble-gas dimers. Mean error (ME) and mean absolute error (MAE) are displayed.

To extend the developed potential to heteronuclear dimers, combination rules for the potential parameters can be used. The simplest ones are given by

18

and known as the Lorentz–Berthelot rules, which are often used for the LJ potential and implemented in many molecular simulation packages.⁶⁻⁸ However, the Lorentz–Berthelot rules are not accurate.^20,75−77 Therefore, instead of mixing R_e and D_e, we use mixing rules for α₁ and C₆ since our potential is fully parameterized by these two quantities. With the effective mixed values {α₁^AB,C₆^AB}, we can set three oscillator parameters {q, μ, ω} through the same vdW–OQDO parameterization procedure as for homonuclear dimers, which is described in the Supporting Information. By doing so, even the heteronuclear dimer AB is effectively represented by two identical oscillators, which still allows us to apply the formalism for exchange repulsion, eqs 3 and 4, developed for homonuclear dimers. For the C₆ dispersion coefficient, the very accurate combination rule arising from the London formula is already well-known^13,78

19

To combine polarizabilities, we employ the robust mixing rule for vdW radii which was established in ref (20), where it was shown that accurate equilibrium distances in noble-gas dimers (MARE = 1%) are delivered by

20

similar to the homonuclear case (eq 8). Thus, the effective polarizability α₁^AB can be simply represented by

21

Thereby, combining eqs 14, 15, 17, 19, and 21 with the vdW–OQDO parameterization scheme (see the Supporting Information), we obtain vdW–QDO potentials for all 21 noble-gas dimers. They are shown in Figure 3 with excellent agreement to both TTS potential and reference CCSD(T) calculations for homo- and heteronuclear dimers of Ne, Ar, Kr, and Xe on panels (a) and (c), as well as for He–Ar, He–Kr (b), and Rn–Xe (d). The other Rn dimers are challenging for our model due to the discrepancies in D_e, as discussed above. In the case of He dimers, to a large extent the error is caused by the underestimated R_e of He₂, with 5.35 bohr predicted by eq 10 against the reference value of 5.608 bohr.⁶⁸ In addition, the actual error in potential for He dimers is small in magnitude (despite being seemingly large visually due to the scale of the y-axis in Figure 3b).

Figure 3.

vdW–QDO potentials (solid lines) for (a) homonuclear and (b–d) heteronuclear noble-gas dimers benchmarked to the TTS potential¹⁵ (dashed lines) and the reference CCSD(T) calculations (circles).^55,68−73

To evaluate the accuracy of our potential quantitatively, we introduce the normalized area difference metric Δ_S between tested and reference potentials as

22

The essential physical meaning of Δ_S is illustrated in Figure 4b which shows that this single unitless number represents a measure of the difference between tested and reference potentials. The integration limits are set to 0.8R_e and 2.0R_e in order to evaluate accuracy close to the minimum region, whereas the long-range accuracy can be evaluated separately in terms of dispersion coefficients. Previously, Δ-gauge was used in benchmarks of various density-functional codes in calculations of equations of state for solids.^79,80 The TTS potential¹⁵ was chosen as the reference potential, and we benchmark the vdW–QDO and LJ1 potentials with respect to it (a similar benchmark for LJ2 can be found in the Supporting Information). The computed Δ_S-matrices for 15 He–Xe dimers are displayed in Figure 4c–d (Rn dimers are omitted since there are no LJ parameters for Rn available in the literature). We note that the vdW–QDO potential has twice better accuracy than the LJ one with ⟨Δ_S^QDO⟩ = 9.0% compared to ⟨Δ_S^LJ1⟩ = 18.4% and ⟨Δ_S^LJ2⟩ = 17.3% when averaged over 15 dimers. Helium is the obvious outlier for vdW–QDO with the highest Δ_S = 31.1%, whereas for all other dimers, Δ_S is below 13.6%. In contrast, the LJ potential shows much broader variations in Δ_S spanning from 3.7% for Ar–Kr to 43.2% for Xe₂. Although the LJ potential shows better Δ_S values than vdW–QDO for some of the dimers (He₂, Ne₂, He–Ne, Ne–Ar, and Ar–Kr), overall the performance of the vdW–QDO potential is more accurate and robust. Generally, Figure 4 supports the above conclusions about the accuracy of the vdW–QDO potential based on Figure 3. We note that the predictions of LJ potential become worse for heteronuclear dimers composed of small and large atoms (e.g., He–Ar, He–Kr, and Ne–Kr) than for atoms with a relatively close size (Ne–Ar, Ar–Kr) (Figure 4c). In contrast, the evenly accurate predictions of the vdW–QDO model (Figure 4d) suggest that the combination rules eqs 19 and 21 employed in this work are more accurate and robust than the Lorentz–Berthelot rules (eq 18).

Figure 4.

(a) Errors in dispersion coefficients predicted by vdW–QDO (dark colors) and TTS (light colors) potentials. (b) Schematic illustration of the Δ_S metric calculation. (c) Heatmaps showing Δ_S (in %) for LJ1 (left) and vdW–QDO (right) potentials with respect to the reference TTS potential. The left and right colorbars have the same scale.

To evaluate the quality of the potentials in the long-range limit, in Figure 4a we compare the dispersion coefficients predicted by vdW–QDO and TTS potentials to the reference ab initio values (Table 1). Such a comparison is fair, since both potentials are built as conformal ones, unlike the earlier TT-2003 potentials,¹³ which directly utilize reference C_2n coefficients for every element being therefore not strictly conformal. To recover the TTS dispersion coefficients, we used the reported scaling law C_2n = C_2n* × D_eR_e²ⁿ with C₆* = 1.3499, C₈* = 0.4147, and C₁₀* = 0.1716.¹⁵ For vdW–QDO dispersion coefficients, from eqs 14, 15 and 17, one can obtain

23

where C₆* = 1.1779, C₈* = 0.3848, and C₁₀* = 0.1540 (see Table 2). We found that both potentials severely underestimate C₈ and C₁₀ and demonstrate a similar magnitude of these errors increasing with the atomic number. However, for C₆, the vdW–QDO potential performs much better, showing a homogeneous overestimate of 12–13%, whereas the TTS potential again possesses an increasing magnitude of error, reaching its maximum of 41% for Rn. While C₈ and C₁₀ are important to deliver accurate potential near the equilibrium, in the asymptotic limit the quality of the potential is fully determined by the leading C₆ coefficient. Therefore, we can conclude that our conformal vdW–QDO potential shows physically more reasonable long-range behavior than the conformal TTS potential.

Table 1. Reference Values of Static Dipole Polarizability α₁ (in a.u.), Dispersion Coefficients C₆, C₈, C₁₀ (in a.u.), and Dimer Potential Well Parameters R_e (in bohr) and D_e (in meV) for Noble-Gas Dimers^a.

	α164	C6(64)	C8(65)	C10(65)	Reref (in Å)15	Re (in Å)	Deref (in K)15	De (in K)
He2	1.38	1.46	14.123	183.79	5.608 (2.97)	5.35 (2.83)	0.948 (10.99)	1.634 (19.0)
Ne2	2.67	6.38	90.265	1532.8	5.83 (3.09)	5.87 (3.11)	3.632 (42.15)	4.049 (47.0)
Ar2	11.1	64.3	1621.5	49,033	7.11 (3.76)	7.20 (3.81)	12.319 (142.95)	12.00 (139.3)
Kr2	16.8	129.6	4040	150,130	7.589 (4.02)	7.64 (4.04)	17.310 (200.87)	16.94 (196.6)
Xe2	27.3	285.9	12,004	588,210	8.273 (4.38)	8.19 (4.33)	24.126 (279.97)	24.64 (285.9)
Rn2	33.54	420.6*	19,263*	1,067,000*	8.37 (4.43)	8.43 (4.46)	34.885 (404.81)	30.38 (352.5)

^aFor R_e and D_e, the values in Å and Kelvin, respectively, are extra given in parentheses. Also, we compare their reference values (columns 5 and 7) to the predictions of eqs. 10 and 17 (columns 6 and 8). The values for C₆, C₈, and C₁₀ labeled with the star (*) are taken from ref (13) instead of refs (64) and (65).

In contrast to the TTS potentials,^13,15 the above vdW–QDO model does not employ any damping of the dispersion energy. However, for noble-gas dimers, the damping of the dispersion energy is not essential, and interatomic vdW potential can be effectively described even without a damping function, as was shown above. This provides additional reasoning why the scaling law for vdW radius (eq 8), which was originally derived without considering dispersion damping,²⁰ works so well. To check the effect of the damping function on the results, we derived the damped vdW–QDO potential, where the QDO damping function reads

24

It was found that for noble-gas dimers the obtained damped vdW–QDO potential curves are practically the same as the undamped ones within the considered range of distances (see Figure S2 in the Supporting Information). Interestingly, the damping function (eq 24) differs from the TTS damping function⁸¹ just in the upper summation limit [for the Tang–Toennies(TT) it is 2n] and in the physical meaning of the unitless variable (z = bR for the TT damping function, with b stemming from the Born–Mayer repulsion term Ae^–bR). Note that, due to the distinct form of the Pauli repulsion in the vdW–QDO and TT models, the QDO damping function contains only even powers of R up to 2n (see the derivation of the QDO damping function and more detailed discussion in the Supporting Information).

We also observe that for both the TT and vdW–QDO models exchange repulsion and dispersion terms separately do not agree with their SAPT counterparts, whereas the total potentials show very close agreement with the sum of the SAPT terms (see the Supporting Information). This finding is in line with the statement of ref (58) that the generalized Heitler–London theory delivers a more compact expansion of interaction energy than the SAPT.

Application to Group II Dimers

Another class of vdW systems where TTS potentials work well consists of Me₂ dimers, with Me = Mg, Ca, Sr, Ba, Zn, Cd, and Hg. Although such systems are not purely vdW-bonded, it was demonstrated that their interatomic potentials can be also well described by the TT potential.^14,82−87 Moreover, the potentials of the group II dimers also obey the principle of corresponding states, albeit the underlying potential shape is distinct from that of noble-gas dimers.⁸⁴⁻⁸⁶ The only exception is the Be–Be dimer, which has been a longstanding puzzle for quantum chemistry. The potential curve of Be₂ has a remarkably different shape in the long-range region, compared to other alkaline-earth elements.⁸⁵ Since this dimer does not obey the principle of corresponding states, it is excluded from our consideration here. In what follows, we show that the vdW–QDO potential is also capable of describing the potential curves of the group II dimers upon several modifications.

First, in contrast to noble gases, for Me₂ dimers, the damping function (eq 24) cannot be omitted due to much larger polarizabilities and hence dispersion coefficients than those of noble gases (see Table 3). As a result, without the damping function, a pronounced divergence of the vdW–QDO potential would already occur at near-equilibrium distances. Second, the scaling laws of eqs 10 and 17 are not so accurate for the group II dimers since the bonding in Me₂ is not purely of vdW type.^84,85 Therefore, the reference values of R_e and D_e (Table 3) were used for the vdW–QDO potential.

Table 3. Reference Ab Initio Values of the Dipole Polarizability α₁ (in a.u.), Dipolar Dispersion Coefficient C₆ (in a.u.), and Dimer Potential Well Parameters R_e (in bohr) and D_e (in meV) of the Group II Dimers^a.

	α1	C6	Re (in Å)	De (in K)
Mg2	71.3b	627b	7.35 (3.89)c	53.81 (624.4)c
Ca2	157.1b	2121b	8.13 (4.30)c	130.18 (1511)c
Sr2	197.2b	3103b	8.88 (4.70)c	129.69 (1505)c
Ba2	273.5b	5160b	9.43 (4.99)c	169.36 (1965)c
Zn2	38.67d	359e	7.23 (3.83)f	28.64 (332.4)f
Cd2	46d	686e	7.32 (3.87)f	40.91 (474.8)f
Hg2	33.9d	392e	6.95 (3.68)g	48.60 (564.0)g

^aFor R_e and D_e, the values in Å and Kelvin, respectively, are extra given in parentheses. The used data sources are the following.

^bReference (88).

^cReference (89).

^dReference (90).

^eReference (86).

^fReference (91).

^gReference (92).

Following Tang et al.,⁸⁵ we choose the strontium (Sr₂) dimer as the reference system to get the shape of the potential curve and then rescale it onto other dimers. Similar to the case of noble gases, the vdW–QDO potential shape for the Sr₂ dimer was derived as (see the Supporting Information)

25

with the numerical values of its parameters presented in Table 4. The QDO parameters {q, μ, ω} for the Sr₂ dimer were set using α₁, C₆, and R_e following the damped vdW–OQDO procedure, as explained in the Supporting Information. Altogether, the three physical quantities are employed to parameterize the vdW–QDO potential for Sr₂, compared to the five {R_e, D_e, C₆, C₈, and C₁₀} in the case of the TT potential.⁸⁵ For other Me₂ dimers, we need only R_e and D_e from Table 3 to perform the rescaling

26

Table 4. Dimensionless Parameters in eq 25^a.

parameter	definition	numerical value
Ad*	Adkeq2/ReDe	58.051
γ*		2.992
C6*	C6/DeRe6	1.6209
C8*	5C6/DeRe6(γ*)2	0.9053
C10*		0.6194

^aThe strontium dimer parameters used in the second column are D_e(Sr₂) = 129.7 meV = 4.766 × 10^–3 a.u. and R_e(Sr₂) = 8.88 bohr.⁸⁹ The QDO parameters for the Sr₂ dimer are q = 1.5433, μ = 1.0671, and ω = 0.1064 (all in a.u.).

The results in Figure 5 show that vdW–QDO potentials are in excellent agreement with both ab initio and experimentally derived potentials (when they are available). This is a remarkable result since the binding energies of group II dimers are up to 5 times larger than those of the heaviest noble gases. Furthermore, the shape of their potentials is distinct to the one of noble gases (Figure 6). Thus, the vdW–QDO functional form is robust and well-suited to describe vdW potentials across various types of systems. In contrast, the LJ potential cannot be employed to describe group II dimers with any combination of parameters since its energy well is too narrow for such binding curves (Figure 6).

Figure 5.

Interatomic potentials of (a) Mg₂, Ca₂, Sr₂, and Ba₂ and of (b) Zn₂, Cd₂, and Hg₂ dimers. The vdW–QDO potentials are shown by solid lines, circles mark coupled-cluster calculations,^89,91,92 and crosses display experimental potential curves.⁹³⁻⁹⁵

Figure 6.

Dimensionless shapes of the vdW–QDO potential curves for Ne₂ (red) and Sr₂ (green) compared with the shapes of the LJ (dashed blue) and TTS (dashed black) potentials.

Application to Molecular Dimers

Finally, we can show that the developed vdW–QDO potential is also applicable to molecular systems, with an example of eight dispersion-dominated molecular dimers from the S66 × 8 benchmark data set.³⁰ They are hydrocarbons, including benzene as well as aliphatic and cyclic molecules (see the list in the Supporting Information). Such systems were chosen to diminish an influence of the electrostatic term not included in the current vdW–QDO potential.

We compute the energy of the vdW interaction between molecules A and B at the given intermolecular separation as

27

where summation goes over the atoms i and j of the molecules A and B, respectively, and R_ij is the interatomic distance. Interaction energy between a pair (i, j) is given by the damped vdW–QDO potential

28

To set the QDO parameters {q, μ, ω}, we apply the vdW–OQDO procedure (see the Supporting Information) coupled with the AIM approach to each pair of atoms (i, j). Following the Tkatchenko–Scheffler (TS) method,²⁹ the AIM polarizabilities and dispersion coefficients are defined by

29

where V^free_i and V^AIM_i are the corresponding Hirshfeld volumes. To compute them, single-point density-functional theory (DFT-PBE0)^96,107 calculations for every dimer were performed at their equilibrium geometry. Then, the effective polarizability α₁^ij and dispersion coefficient C₆^ij for a pair (i, j) were defined using the combination rules of eqs 19 and 21. Finally, the vdW–OQDO parameterization procedure (see the Supporting Information) was applied to map {α₁^ij,C₆^ij} onto {q, μ, ω}. After performing pairwise summation in eq 27 and repeating the whole procedure for all 8 intermolecular separations, the vdW–QDO interaction curves for dimers are obtained and compared to the reference CCSD(T) interaction curves.³⁰ For comparison, interaction energies of dimers at PBE0 + TS,²⁹ PBE0 + many-body dispersion (MBD),⁴⁰ and density functional tight-binding (DFTB3)⁹⁷ + MBD levels of theory were also calculated. All DFT calculations in this work were performed using FHI-aims code⁹⁸ with the “tight” atomic basis sets.

We showcase the obtained results with an example of a neopentane dimer (Figure 7b). One can see that PBE0 + TS overbinds the neopentane dimer significantly and underestimates the equilibrium separation by 5%. Inclusion of many-body effects at the PBE0 + MBD level improves the results of PBE0 + TS for energy, but the predicted equilibrium separation is still 95% of the reference value. On the other hand, the DFTB3 + MBD method clearly lacks repulsion and attraction at short-range and long-range distances, respectively, although at the equilibrium distance the two errors largely cancel each other. As for the vdW–QDO potential, we note that the minimum of the intermolecular interaction curve is very close to the reference CCSD(T) energy, although our method overestimates the equilibrium separation by 10%. When considering the overall interaction curve, however, the vdW–QDO potential deviates significantly from the reference, being too repulsive at shorter distances. These conclusions remain valid also for the other seven dimers, as supported by the corresponding results in the Supporting Information.

Figure 7.

(a) Dispersion (blue) and exchange (red) contributions to the interaction energy of neopentane dimer (shown as inset) calculated by SAPT–DFT⁹⁹ (solid lines) and the damped vdW–QDO potential of eq 27 (dotted lines). In addition, the electrostatic term from SAPT–DFT is displayed in black. (b) Binding energy curves of neopentane dimer as calculated by different methods: CCSD(T)³⁰ (black); PBE0 + TS (blue) and PBE0 + MBD (green); DFTB3 + MBD (magenta); damped vdW–QDO potential (cyan); SAPT-corrected vdW–QDO potential (eq 30) (red). (c) Errors in the interaction energy of eight dispersion-dominated dimers for the five methods considered. Yellow filling depicts the “chemical accuracy” region of ±1 kcal/mol error.

To get insights into such a behavior, we compared the exchange and dispersion terms in vdW–QDO potential (eq 27) to their counterparts from SAPT–DFT calculations⁹⁹ (see Figure 7a). The dispersion part of the vdW–QDO potential provides excellent agreement with E_disp⁽²⁾ from SAPT–DFT, which illustrates the well-known fact that vdW dispersion interaction between small molecules can be effectively described by a pairwise potential, despite being essentially many-body in its nature.^4,40,100 In contrast, for the vdW–QDO exchange repulsion term we observe a noticeable deviation from the E_exc⁽¹⁾ contribution of SAPT–DFT. This might indicate that exchange repulsion between molecules is not accurately described by the commonly used pairwise approach and hence exchange repulsion requires many-body treatment as well.¹⁰¹

SAPT–DFT calculations⁹⁹ also indicate that electrostatic contributions are small but not negligible even for dispersion-dominated dimers (Figures 7a and S6–S8 in the Supporting Information). To check a possible effect of the inclusion of accurate many-body exchange and electrostatic interactions on our results, we consider the corrected vdW–QDO potential, where the first-order SAPT–DFT energy was added to the dispersion energy from the vdW–QDO model

30

This approach is similar to the Hartree–Fock plus dispersion plus first-order correlation [HFDc⁽¹⁾] scheme of Podeszwa et al.¹⁰² with the important difference that here the dispersion energy is calculated from the QDO model, whereas in the HFDc⁽¹⁾ method, this energy is computed from SAPT. The corrected vdW–QDO curve in Figure 7b delivers a much better description of the interaction energy than the original vdW–QDO method. Notably, at larger distances, V^corr_QDO also shows a good agreement with PBE0 + MBD energy.

The overall statistics for the eight molecular dimers is shown in Figure 7c in terms of the error in the equilibrium energy obtained by five methods with respect to the reference CCSD(T) values.³⁰ Purely analytical vdW–QDO predictions are scattered within the 2 kcal/mol range, which is remarkable considering the approximations made in the model. Including the first-order SAPT energy reduces errors by roughly 1 kcal/mol (chemical accuracy).

This test demonstrates that the vdW–QDO approach is not limited to atomic dimers and can be generalized to molecular complexes, although in that case, the accuracy is currently lower. Nevertheless, considering the approximations made, the fact that the vdW–QDO method (even without SAPT corrections) properly predicts binding of the considered weakly bound molecular complexes from a set of AIM quantities (α₁ and C₆) is already reassuring.

Summary and Outlook

We developed a universal pairwise vdW potential devoid of empiricism and parameterized by only two atomic nonbonded parameters. The developed vdW–QDO potential combines the strengths of the LJ and TTS models. Similarly to the former, the vdW–QDO potential is fully determined by only two parameters. At the same time, our potential is comparable in accuracy to the TTS potential for noble-gas dimers, being twice as accurate as the LJ one. Moreover, the vdW–QDO potential has advantages that are present neither in LJ nor in TTS models. First, the two parameters {α₁, C₆} are readily available for the whole periodic table,^64,90 being computed by highly accurate ab initio methods. This makes our potential widely applicable, as demonstrated for atomic dimers of group II elements as well as organic molecular dimers. Second, the conformal vdW–QDO potential has better long-range behavior than the LJ and conformal TTS potentials. This is crucial for applications to extended systems where errors in the long-range vdW energy accumulate over many atomic pairs.

The key idea behind the presented potential is the synergy between vdW scaling laws, the coarse-grained QDO formalism for exchange repulsion, and the principle of corresponding states. In its current form, the vdW–QDO potential does not explicitly include the electrostatic contribution arising from charge penetration between atoms at short distances, which is non-negligible according to the SAPT. Although the short-range electrostatic interaction can be introduced into the QDO model in the form of a Coulomb integral,⁵³ this requires the introduction of an additional “electrostatic charge” parameter³⁶ into the vdW–QDO potential. Therefore, given the good accuracy of the current vdW–QDO potential, we decided to dispense with the explicit modeling of short-range electrostatics at this stage of model development. In its current version, the vdW–QDO potential can be incorporated into classical force fields as a nonempirical replacement for the LJ potential. For polar systems, an additional electrostatic/polarization term is needed in a force field (like it is done in the case of the LJ potential) to complete the description of noncovalent interactions.

We consider the present vdW–QDO potential as an important step toward a new generation of universal vdW potentials to be used in force fields and for biomolecular applications. To extract atom-in-molecule parameters, we currently employ ab initio calculations, which is a certain limitation. However, active development of machine-learning models^51,52,103 paves the way to obtain atom-in-molecule partitioning without costly electronic-structure calculations. Moreover, there is an increasing trend in creating extensive molecular data sets such as QM7-X,¹⁰⁴ which include information about atom-in-molecule volumes. This enables the direct applicability of the vdW–QDO potential to arbitrary molecular systems. Finally, the vdW–QDO potential can be generalized to include polarization and electrostatic contributions, which are the subject of our ongoing studies. Eventually, this effort should deliver a general coarse-grained force field for all noncovalent interactions entirely based on the model system of coupled QDOs.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.3c00797.

• vdW-OQDO parameters for atomic dimers, derivation of the QDO damping function and the damped vdW–QDO potential, and results for eight molecular dimers (PDF)

The authors declare no competing financial interest.