Dispersion without Many-Body Density Distortion: Assessment on Atoms and Small Molecules

Abstract

The “fixed diagonal matrices” (FDM) dispersion formalism [Kooi, D. P.; et al. J. Phys. Chem. Lett. 2019, 10, 1537.] is based on a supramolecular wave function constrained to leave the diagonal of the many-body density matrix of each monomer unchanged, reducing dispersion to a balance between kinetic energy and monomer–monomer interaction. The corresponding variational optimization leads to expressions for the dispersion energy in terms of the ground-state pair densities of the isolated monomers only, providing a framework to build new approximations without the need for polarizabilities or virtual orbitals. Despite the underlying microscopic real space mechanism being incorrect, as in the exact case there is density relaxation, the formalism has been shown to give extremely accurate (or even exact) dispersion coefficients for H and He. The question we answer in this work is how accurate the FDM expressions can be for isotropic and anisotropic C₆ dispersion coefficients when monomer pair densities are used from different levels of theory, namely Hartree–Fock, MP2, and CCSD. For closed-shell systems, FDM with CCSD monomer pair densities yield a mean average percent error for isotropic C₆ dispersion coefficients of about 7% and a maximum absolute error within 18%, with a similar accuracy for anisotropies. The performance for open-shell systems is less satisfactory, with CCSD pair densities performing sometimes worse than Hartree–Fock or MP2. In the present implementation, the computational cost on top of the monomer’s ground-state calculations is O(N⁴). The results show little sensitivity to the basis set used in the monomer’s calculations.

1. Introduction

The attractive London dispersion interaction between atoms and molecules is weaker than covalent bonding forces, but while the latter decay exponentially with the separation R between the monomers, dispersion interactions decay only polynomially in 1/R. Because of this dominating long-range character, dispersion plays a crucial role in various chemical systems and processes, such as protein folding, soft solid state physics, gas–solid interfaces, etc. An accurate, computationally efficient, and fully nonempirical treatment of dispersion forces remains an open challenge, and it is the objective of several ongoing efforts (see, e.g., refs (1−3) for recent reviews and benchmarks).

We have recently introduced a class of variational wave functions that capture the long-range interactions between two quantum systems without deforming the diagonal of the many-body density matrix of each monomer.⁴ The variational take on dispersion is certainly not new, as, for example, variational calculations of the dispersion coefficients have been performed in the context of (Hylleraas) variational perturbation theory⁵ and variational calculations of the dispersion energy at finite intermonomer distance have been carried out in the framework of symmetry-adapted perturbation theory (SAPT) using orthogonal projection.⁶ The distinctive feature of our approach is the reduction of dispersion to a balance between kinetic energy and monomer–monomer interactions only, providing an explicit expression for the dispersion energy in terms of the ground-state pair densities of the isolated monomers.

Although the supramolecular wave function constructed in this “fixed diagonal matrices” (FDM) approach can never be exact, as density distortion is prohibited, it provides a variational expression for the dispersion energy when accurate pair densities for the momoners are used, at a computational cost given essentially by the ground-state monomer calculations. The FDM approach has been found to yield exact results for the dispersion coefficients up to C₁₀ for the H–H case (and up to C₃₀ for the second-order coefficients), and very accurate results (0.17% error on C₆) for He–He and He–H.⁴ This is achieved by reshuffling the contributions of kinetic energy and potential energy inside each monomer, as shown in Table 1 of ref (7). Another way to look at it is the following: dispersion between two systems in their ground states is a competition between a distortion of the fragments’ ground states (which raises the energy with respect to E₀^A + E₀) and the interfragment interaction that can lower the energy of the two systems together. As proven by Lieb and Thirring,⁸ the rise in energy due to the distortion of the fragments’ ground states can be always made quadratic with respect to a set of variational parameters, with the interfragment interaction being linear. With our FDM constraint we force the quadratic rise in energy of the isolated fragments to be of kinetic energy origin only, since only the off-diagonal elements of the monomer’s reduced density matrices are allowed to change. For the special case of two ground-state one-electron fragments, this can be shown to give the same result for the dispersion coefficients as second-order Rayleigh–Schrödinger perturbation theory.⁷

This FDM construction is fundamentally different from approaches to incorporate dispersion based on the adiabatic connection (AC) and the fluctuation dissipation theorem (see, for example, ref (9) and references therein). In these methods the interacting system is connected to a noninteracting one with the same density via the AC formalism: the monomer’s pair density changes as the electron–electron interaction is turned on. A different AC approach in which only the monomer–monomer interaction is turned on has been introduced very recently in ref (10): the main difference from our FDM formalism is that in our case we keep the densities (and pair densities) of the monomers equal to their isolated ground-state values, while in ref (10) the density is kept equal to the one of the complex for all coupling strength values.

The FDM expressions for the dispersion energy in terms of the ground-state pair densities of the isolated monomers offer a neat theoretical framework to build new approximations, by using pair densities from different levels of accuracy, including exchange–correlation holes from density functional theory. This idea is similar in spirit to the exchange dipole moment (XDM) of Becke and Johnson,^11,12 with the main difference that in our case we do not need the static atomic polarizabilities, as everything can be expressed in terms of ground–state monomer densities and exchange–correlation holes, with a clear supramolecular variational wave function behind our expressions.⁷ Another related approach is the “weighted” exchange-hole (WXhole) idea,¹³ in which an approximation for the frequency dependence of the response function is used in order to perform the frequency integration. Again, this is not needed in the FDM approach, which is based on ground-state quantities only.

Besides a route for DFT-based approximations, the FDM idea has the advantage of providing not only energetics but also a simplified supramolecular wave function that can be used in other contexts, for example, in a QM/MM framework to model interactions between the QM and MM parts, following the ideas of refs (14−16) and using results for the functional derivative of the FDM dispersion energy reported in ref (7).

Before considering the use of the FDM framework to build DFT-based approximations or other kinds of models, one should ask the fundamental question: how accurate can this approach be if we use accurate monomer’s ground-state pair densities, beyond the simple H and He cases? In other words, how accurate can the FD construction be if we eliminate the other sources of error? The aim of this work is exactly to answer to this question by exploring the performance of the FDM expression for the dispersion C₆ coefficient for atoms and molecules using different levels of theory for the monomer calculations, studying the convergence and basis set dependence of the results. We test the approach on 459 pairs of atoms, ions, and small molecules, using Hartree–Fock (HF), second-order Møller–Plesset perturbation theory (MP2), and coupled cluster with singles and doubles (CCSD) ground-state pair densities. We should keep in mind that the FDM expression is guaranteed to be variational, yielding a lower bound to C₆, only when we use exact pair densities of the monomers. As we shall see, for closed-shell systems, this is almost always the case with CCSD pair densities, which yield in general good results, slightly underestimating C₆, although there are exceptions. With HF pair densities, as it was already found in a preliminary result for the Ne–Ne case in ref (4), C₆ is, in the vast majority of cases, overestimated.

The paper is organized as follows. In section 2 we illustrate our working equations, including the expressions for the isotropic C₆ coefficients and for the anistropies, with the computational details reported in section 3. The results are discussed in section 4, and conclusions and perspectives are in section 5.

2. Theory

We consider two systems A and B separated by a (large) distance R having isolated ground-state wave functions and , where x denotes the spin-spatial coordinates (r, σ) and denote the whole set of the spin-spatial coordinates of electrons in system A/B. The FDM framework is defined by the following constrained minimization problem^4,7

1

where T̂ is the usual kinetic energy operator acting on the full set of variables x̲_A, x̲_B, and

With T₀^A/B we denote the ground-state kinetic energy expectation values of the two separated systems and

2

where ρ₀^A(B) are the ground-state one-electron densities of the two systems. The constraint Ψ_R → |Ψ₀|²,|Ψ₀^B|² means that the search in eq 1 is performed over wave functions Ψ_R(x̲_A, x̲_B) that leave the diagonal of the many-body density matrix of each fragment unchanged with repect to the ground-state isolated value. We work in the polarization approximation, in which the electrons in A are distinguishable from those in B. The constrained-search formulation, eq 1, makes dispersion a simple competition between kinetic energy and monomer–monomer interaction, as all the other monomer energy components cannot change by construction. This also guarantees that no electrostatic or induction contributions appear in eq 1.

For the minimizer of eq 1 we use the variational ansatz of ref (4)

3

where the function J_R correlates electrons in A with those in B and is written in the form

4

where c_ij,R are parameters, which are determined variationally. The functions b_i^A/B(r) for now are an arbitrary set of “dispersals,” used as basis to expand J_R. The constraint Ψ_R → |Ψ₀|², |Ψ₀^B|² is enforced by imposing⁴

5

6

Thanks to this constraint, the expectation of the external potential and that of the electron–electron interactions inside each monomer cancel out in the interaction energy, whose variational minimization takes a simplified form.^4,7 If we peform the multipolar expansion of the monomer–monomer interaction, we can, accordingly, expand c_ij,R in a series of inverse powers of R:

7

which leads to explicit expressions for the dispersion coefficients. In this paper, we focus on the leading C₆ coefficient of the term −C₆R^–6 in the dispersion interaction energy, which is determined by the variational parameters c_ij⁽³⁾ in eq 7, denoted simply c_ij in the rest of this work.

As detailed in the Supporting Information of ref (4), the variational equation for C₆ corresponding to our wave function is given in terms of the matrices τ_ij^A/B, S_ij, and P_ij^A/B (which determine the kinetic correlation energy)

8

9

10

with similar expressions for system B, and of the matrix w_ij (which determines the monomer–monomer interaction)

11

with e = x, y, z, and h_e = (1, 1, −2) when the intermolecular axis is parallel to the z-axis. The vectors d_i^A and D_i determine the dipole–dipole interaction terms:

12

13

with, again, similar expressions for monomer B. In eqs 10 and 13P₀^A/B is the ground-state pair density of the two monomers, with usual normalization to N(N – 1).

In our previous work⁴ the matrices S_ij^A/B + P_ij were diagonalized through a Löwdin orthogonalization among the b_i, transforming the matrices w_ij and τ_ij^A/B accordingly. The variational coefficients c_ij were then determined via the solution of a Sylvester equation:^4,17

14

Here we diagonalize τ_ij^A/B with S_ij + P_ij^A/B as a metric through a generalized eigenvalue problem, again transforming accordingly w_ij, so the indices indicate from now on matrix elements with the transformed b_i. The advantage is that this eigenvalue problem needs to be solved only once for each monomer, while the Sylvester equation, eq 14, needs to be solved for each pair AB. This way we can directly obtain the variational coefficients c_ij as

15

16

and

17

The dispersion coefficient C₆ then takes the simpler (and computationally faster) form

18

For molecules, eq 18 gives access to the orientation-dependent C₆^AB coefficient, where the kinetic energy terms τ_i are clearly rotationally invariant, and the dependence on the relative orientation of the monomers enters through w_ij, as shown by eqs 11–13. In order to compare with values from the literature, it is often necessary to compute the orientation-averaged isotropic C̅₆^AB coefficients, which can be obtained by performing the orientation average directly on each w_ij², yielding

19

The is the spherically averaged interaction term given by

20

To also assess the accuracy for the orientation dependence, we consider the case of linear molecules, for which one usually defines anisotropic dispersion coefficients by writing the dispersion coefficient C₆ as¹⁸

21

where P_n denotes Legendre polynomials and denotes spherical harmonics. The anisotropic dispersion coefficients Γ₆^AB and Δ₆ can be obtained from our formalism as

22

23

A similar expression holds for Γ₆^BA, but with the roles of A and B exchanged.

On top of the monomer calculations, the diagonalization to compute C₆ scales formally as n_A³ + n_B³, where n_A/B is the number of dispersals b_i^A/B needed to converge, which, however, seems so far independent of system size. We should however also mention the cost of computing the matrix elements: the most expensive part is the first step of the two-step contraction to obtain P_ij, which scales as , while the second step scales as , as expensive as obtaining S_ij and τ_ij, where N_orb is the number of spatial orbitals used in the monomer calculations.

3. Computational Details

3.1. Choice of the Dispersals b_i(r)

For the dispersals b_i^A/B(r) of eq 4, we have chosen multipoles centered in r₀ = (x₀, y₀, z₀):

24

For atoms the obvious choice for r₀ is the position of the nucleus; for molecules, in this first exploration, we have set r₀ at the center of nuclear mass. We include all b_i, such that s_i + t_i + u_i < n_max, where n_max^A/B is a parameter, which is set equal to 22 in all our calculations, which yields in general reasonably converged results (see section 3.5 for a more detailed discussion on convergence). We should remark that the choice of eq 24 is dictated mainly by the immediate availability of integrals: our goal here is to investigate whether the method is worth investing in further implementation and optimization. The question on how to determine the best possible b_i(r) is open, with different strategies discussed in ref (7).

In our previous work,^4,7 the atomic (spherically symmetric) case was also studied in detail. In those calculations a different basis set was used instead of eq 24, namely, a spherical multipole

25

where now the parameters l_i, m_i, and n_i need to be chosen. This set of spherical multipoles includes the previous (Cartesian) multipoles from eq 24, but also includes cases where l_i + n_i are odd, which are not included in the Cartesian multipoles. The spherical multipole expansion was found to converge faster (sometimes much faster), but it requires nonstandard integrals. Since our purpose here is to test the overall accuracy of the method, we have preferred to use easily available integrals.

3.2. Matrix Elements

We denote the spatial orbitals used in the monomer calculations by ϕ_a(r) with indices a, b, c, and d. The spin-summed one-body reduced density matrix (1-RDM) is written as γ_ab:

26

normalized here to N. The method only depends on the spatial diagonal ρ₀(r) = γ(r, r). The 2-RDM is written as Γ_ab,cd, again spin-summed, corresponding to

27

with normalization N(N – 1). The method only depends on the spatial diagonal (pair density), P₀(r₁, r₂) = Γ(r₁, r₂; r₁, r₂).

To compute the matrix elements of section 2, we need the 1-RDM and 2-RDM of the monomers and the integrals of the dispersals b_i(r) with the spatial orbitals, which, with the choice of eq 24, are all of the kind

28

For every monomer we need to calculate S_ij of eq 9, τ_ij of eq 8, and d_i of eq 12 from the 1-RDM and P_ij of eq 10 and D_i of eq 13 from the 2-RDM. We first write all the matrix elements by assuming that the constraint of eqs 5 and 6 is satisfied, which amounts to assuming

29

When this does not hold, we make the appropriate modifications in terms of p_i; see eqs 36–39 below.

We then have for the matrix S_ij of eq 9

30

and for τ_ij of eq 8

31

The components of the vector d_i of eq 12 are given by the dipole moment in directions e = x, y, z. For example, for the x-direction

32

while for y and z we get analogous expressions with and , respectively. For convenience, we also define (with analogous expressions for the y- and z-directions)

33

The matrix P_ij of eq 10, which is a sort of overlap mediated by the pair density, is given by

34

For the components of the vector D_i of eq 13 we have, for example, in the x-direction

35

with similar expressions with I_0,1,0^ab and I_0,0,1 for the other two components. When p_i of eq 29 is not zero, we need to modify the matrix elements according to

36

37

38

39

with analogous expressions for the components y and z of d_i and D_i and for d_x,0 defined in eq 33.

3.3. Implementation

The expression for the dispersion coeffcients of eqs 18 and 19, with the computational details just described, has been written in Python and interfaced with PySCF¹⁹ and HORTON.²⁰ The Python package is open-source and is available on Github (https://github.com/DerkKooi/fdm). The reduced density matrices of the monomers are obtained from PySCF, and the multipole moment integrals are calculated using HORTON. The monomer densities and pair densities have been computed at three different levels of theory—Hartree–Fock, MP2, and CCSD—where for open-shell systems we used restricted open-shell Hartree–Fock (ROHF). The geometries of the molecules were optimized using the ORCA program package²¹ using the MP2 level of theory with the def2-TZVPPD basis set.

3.4. Choice of Basis Set for Monomer Calculations

We have extensively explored the dependence on the basis set used for the monomer pair density calculations for all but the largest molecules, finding that, in general, going beyond a def2-TZVPP (or equivalent) quality does not particularly improve the overall results, with few singular exceptions. The mean absolute percentage errors (MAPEs) for dispersion coefficients of molecules obtained with the def2-QZVPP basis set differ from the def2-TZVPP ones by 1.5–2.2%, with Hartree–Fock being the least and CCSD the most sensitive. When diffuse functions are incorporated into the basis set, the MAPE difference between def2-TZVPP and def2-TZVPPD basis sets ranges from 2.6 to 3.5%, with Hartree–Fock being the least sensitive and MP2 the most sensitive. These differences are less than half the MAPE with respect to the reference values. As a representative example, in Figure 1 we show the C̅₆^AA for the molecules considered here with HF, MP2, and CCSD pair densities using different basis sets compared with calculations done using the def2-TZVPP basis set, which is our choice for all the results presented in section 4.

Figure 1.

Isotropic C̅₆^AA dispersion coefficients for molecules calculated using Hartree–Fock, MP2, and CCSD pair densities with different basis sets compared with calculations done using def2-TZVPP basis set. Coefficients calculated with the MP2 pair density are the most sensitive to the basis set used, while Hartree–Fock and CCSD methods produce quite robust results. The largest outlier for all the methods used is the CS₂ molecule, which is further discussed in section 3.5 and in Figure 2.

We should also remark that, since Hartree–Fock pair densities usually lead to an overestimation of C₆, if one uses a smaller double-ζ basis set the performance in this case usually improves as the smaller basis makes the overestimation less profound. The results obtained using a correlated pair density, however, become worse if we go below triple-ζ quality.

All our results obtained with different basis sets are available in the Supporting Information.

3.5. Convergence with Respect to Number n_max of Dispersals

In all our calculations we have fixed n_max = 22, which yields in general well-converged results for the vast majority of cases, and it is also a value for which the multipole integrals are numerically stable. However, we should remark that there are a few cases in which the convergence with the number of b_i has not been satisfactorily reached. As a typical example for how the vast majority of systems behave, we show in the left panel of Figure 2 the convergence of C̅₆^AA for CH₄ with respect to n_max, for both HF and CCSD pair densities, with and without diffuse functions in the basis set for the monomer calculation. We see that the result is well converged and that the addition of diffuse functions has little effect, with CCSD underestimating the C₆ coefficient. There are however three molecules (SO₂, CS₂, and CO₂) where the values between n_max = 20 and n_max = 22 deviate more than 1%. The worst case is CS₂, shown in the right panel of Figure 2: we see that even at n_max = 28 the dispersion coefficient of CS₂ is not converged and that CCSD overestimates C₆. The inclusion of diffuse functions, in this case, improves both the convergence profile and the accuracy.

Figure 2.

Isotropic C̅₆^AA coefficients for CH₄ (left) and CS₂ (right) as a function of n_max, related to the number of dispersals used to expand the function J_R(r, r′) of eq 4, for Hartree–Fock and CCSD pair densities of the monomers in different basis sets. Reference values from DOSD measurements are shown with a dotted line. The case of CH₄ is representative for what we have observed for the vast majority of systems. CS₂ is the worst case found: it is clearly not well converged and needs diffuse functions in the basis set.

4. Results

Dispersion coefficients were computed for five data sets:

• (1)C₆^AA for 23 atoms and ions
• (2)C₆^AB for 253 mixed pairs consisting of atoms and ions
• (3)isotropic C̅₆^AA for a set of 26 molecules
• (4)isotropic C̅₆^AB for a set of 157 mixed molecule pairs
• (5)anisotropic Γ₆^AB and Δ₆ (where applicable) for three diatomics and interacting with noble gas atoms

In all cases we compare our results with reference values obtained from dipole oscillator strength distribution (DOSD) data computed²² or constructed from measurements and theoretical constraints.^18,23−37

4.1. Dispersion Coefficients for Atoms and Ions

The results for set 1, using HF, MP2, and CCSD pair densities (def2-TZVPP basis set, with effective core potential (ECP) for fifth and sixth row elements) for the monomers are presented in Table 1 and compared with accurate reference data.²² The MAPEs for Hartree–Fock, MP2, and CCSD monomer pair densities are 62.1, 17.7, and 16.2%, respectively. The same results are also illustrated in Figure 3. Notice that the result for H in Table 1 has a small residual error of 1.2% due to the basis set used, since the results for C₆^AA (as well as C₈ and C₁₀^AA) from our wave function are exact when the exact hydrogenic orbital is used.⁴

Table 1. Dispersion Coefficients C₆^AA for a Set of Atoms and Ions Computed Using Hartree–Fock, MP2, and CCSD Pair Densities for Monomers with the def2-TZVPP Basis Set, with Effective Core Potential (ECP) for Fifth and Sixth Row Elements^a.

species	ref 22	HF	MP2	CCSD
H	6.50	6.42	6.42	6.42
Li	1395.80	1024.59	1013.58	981.77
Na	1561.60	1458.17	1400.47	1211.02
K	3906.30	4636.05	3919.56	3034.83
Rb	4666.90	6493.38	5207.21	3833.89
Cs	6732.80	11244.92	7894.26	6008.81
Cu	249.56	466.54	393.98	312.73
Ag	342.29	741.72	441.06	392.27
Be+	68.80	40.00	39.36	38.95
Mg+	154.59	120.87	115.17	109.78
Ca+	541.03	565.94	425.62	383.40
Sr+	775.72	1040.23	667.53	623.11
Ba+	1293.20	2284.40	1306.28	1348.81
Be	213.41	443.51	273.87	161.69
Mg	629.59	1257.52	750.44	523.40
Ca	2188.20	5035.02	2441.13	1809.39
Sr	3149.30	7882.73	3508.83	2750.55
Ba	5379.60	15037.42	6184.94	5892.73
He	1.46	1.62	1.43	1.43
Ne	6.38	6.79	5.91	6.19
Ar	64.30	96.28	54.60	58.57
Kr	129.56	211.12	110.30	122.45
Xe	285.87	537.65	221.15	275.55

	HF	MP2	CCSD
MAPE (%)	62.1	17.7	16.2
AMAX (%)	179.5	57.9	43.4

^aFor each species, the dispersion coefficient closest to the reference value is in bold font. The mean absolute percentage error (MAPE) as well as the maximum absolute percent deviation (AMAX) for the data set are reported.

Figure 3.

Results for C₆^AA for 23 atoms and ions (see Table 1). The solid line depicts one-to-one correspondence of the model with the reference data obtained from ref (22). The mean absolute percentage errors for Hartree–Fock, MP2, and CCSD monomer pair densities are 62.1, 17.7, and 16.2%, respectively.

For test set 2, the different pairs are formed by selecting A and B from the species listed in Table 1. The results for the dispersion coefficients C₆^AB computed using different pair densities for the monomers, again with the def2-TZVPP basis set, are compared to accurate reference values²² in Figure 4. The MAPEs for Hartree–Fock, MP2, and CCSD are slightly better, being 52.3, 12.1, and 11.9%, respectively. All the values obtained are available in the Supporting Information.

Figure 4.

Dispersion coefficients C₆^AB for 253 pairs formed by selecting A and B from the species listed in Table 1, computed using Hartree–Fock, MP2, and CCSD pair densities for the monomers with the def2-TZVPP basis set. The solid line depicts one-to-one correspondence of the model with the reference data obtained from ref (22). The mean absolute percentage errors for Hartree–Fock, MP2, and CCSD monomer pair densities are 52.3, 12.1, and 11.9%, respectively.

These results for atoms and ions are not extremely promising, in particular because they do not always improve with the accuracy of the theory used to treat the monomers. From Figure 4, it is evident that the use of the Hartree–Fock pair densities leads to an overestimation of the dispersion coefficients. However, for some systems (Li, Na, Be⁺, Mg⁺) the FDM method combined with correlated pair densities considerably underestimates the dispersion coefficient, and in those cases Hartree–Fock pair densities yield better results than MP2 and CCSD ones. As it should, the CCSD pair density tends to produce a lower bound for the dispersion coefficient, but with some exceptions (e.g., Ag, Cu, Ba⁺).

The picture improves considerably if we look at closed-shell species only: if we consider the 15 noble gas pairs, the MAPEs for HF, MP2, and CCSD pair densities are 41.9, 12.2, and 4.3%, respectively. Also, if we consider the subset of our data set formed by the 45 pairs of the noble gas and alkali elements used by Becke and Johnson¹¹ in their original paper on the exchange-hole dipole moment (XDM) dispersion model (see their Table 1), we obtain MAPEs for MP2 and CCSD equal to 9.6 and 7.7%, respectively, lower than the one of XDM (11.4%), while with Hartree–Fock pair densities our MAPE is 27.3%.

Overall, these first results indicate that the constrained FDM ansatz can work well for closed-shell species, while being less reliable for open-shell cases. As we shall see in section 4.2, the results for the isotropic dispersion coefficients for closed-shell molecules are reasonably accurate and robust, confirming these first findings.

4.2. Isotropic Dispersion Coefficients for Molecules

Isotropic molecular dispersion coefficients C̅₆^AA were computed for 26 molecules consisting mainly of first and second row elements. Our results are compared with reference values calculated from DOSD²³⁻³⁶ in Table 2 and are also illustrated in Figure 5. The MAPE using Hartree–Fock pair density with def2-TZVPP basis is 52.5%. This comes down to 13.7 and 8.6% when MP2 and CCSD pair densities, respectively, are used. This is in line with the results for the noble gas atoms: there is now a clear systematic improvement with the level of theory of the monomer pair densities, with CCSD yielding good results with the lowest variance.

Table 2. Isotropic Dispersion Coefficients C̅₆^AA for a Set of Molecules Calculated Using def2-TZVPP Basis Set^a.

species	ref	HF	MP2	CCSD
H2	12.123	16.42	15.76	11.60
C2H6	381.924	542.19	411.68	346.17
C2H4	300.235	472.25	310.09	273.50
C2H2	204.129	372.13	192.44	192.34
H2O	45.323	55.57	38.88	40.55
H2S	216.828	358.57	193.01	199.08
NH3	8923	114.66	77.58	77.52
SO2	293.926	473.00	281.11	287.00
SiH4	343.933	462.54	346.18	308.09
N2	73.323	135.53	58.37	70.57
HF	1927	21.70	16.67	17.66
HCl	130.427	201.66	108.00	115.47
HBr	216.627	372.17	187.65	205.25
H2CO	165.238	207.04	135.75	135.13
CH4	129.624	183.70	128.52	120.00
CH3OH	22234	288.75	223.07	196.93
CS2	871.126	2017.72	906.90	975.32
CO	81.439	122.69	66.98	75.13
CO2	158.739	232.44	153.00	153.45
Cl2	389.230	676.48	384.86	368.85
C3H6	662.135	993.49	769.43	590.50
C3H8	768.124	1092.09	919.72	688.07
C4H8	1130.235	1699.13	1527.79	1023.72
C4H10	1268.224	1821.45	1714.29	1137.31
C5H12	1905.024	2733.69	2872.35	1695.39
C6H6	1722.729	3116.90	2148.87	1630.94

	HF	MP2	CCSD
MAPE (%)	52.1	13.7	8.6
AMAX (%)	131.6	50.8	18.2

^aFor each species, the dispersion coefficient closest to the reference value is in bold font.

Figure 5.

Isotropic dispersion coefficients C̅₆^AA for molecules calculated using Hartree–Fock, MP2, and CCSD pair densities and reported in Table 2. The solid line depicts one-to-one correspondence of the model with the reference values.

For test set 4, we have computed isotropic dispersion coefficients C̅₆^AB for 157 mixed molecule pairs selected from Table 2. The results are illustrated in Figure 6 where they are compared, again, with reference values from DOSD measurements²³⁻³⁶ and are available in the Supporting Information. The MAPEs using Hartree–Fock, MP2, and CCSD pair densities are 57.1, 7.9, and 7.2%, respectively.

Figure 6.

Isotropic dispersion coefficients C̅₆^AB for 157 mixed molecule pairs selected from Table 2 calculated using Hartree–Fock (MAPE 57.1%), MP2 (MAPE 7.9%), and CCSD (MAPE 7.2%) monomer pair densities (def2-TZVPP basis set). The solid line depicts one-to-one correspondence of the model with the reference values.

From these calculations we can confirm that, for closed-shell molecules, the Hartree–Fock pair density leads to consistent overestimation of the dispersion coefficients. The use of a correlated pair density (MP2 or CCSD) improves the results considerably, with CCSD providing better accuracy and lowest scattering of the results. For CCSD, 11 of the 26 C̅₆^AA’s deviate from the reference value by more than 10%, compared to 16 for MP2 and 26 for Hartree–Fock. With the exception of H₂CO, all the CCSD values are within 13% of the reference value.

Regarding the FDM results with Hartree–Fock pair densities, we should stress that in this work we are testing the formalism as derived by our trial FDM wave function when we assume that an exact description of the monomers is used.^4,7 However, if we use Hartree–Fock wave functions for the monomers, we could also revise the formalism to take into account that, in Hartree–Fock theory, part of the intramonomer electron–electron interaction is described in terms of the off-diagonal elements of the one-body reduced density matrix (1-RDM), which do change in the FDM and are quadratic in the variational parameters, such as the kinetic energy. This and other flavors of approximations using Kohn–Sham orbitals and DFT exchange–correlation holes will be extensively tested in future works.

4.3. Anisotropic Dispersion Coefficients

To test the applicability of the method for the orientation dependence (anisotropy) of the dispersion coefficients, we performed calculations for the diatomics H₂, N₂, and CO and for these diatomics interacting with noble gas atoms. The resulting Γ₆^AB values are listed in Table 3, and the resulting Δ₆ values are listed in Table 4. For both Γ₆^AB and Δ₆ CCSD (MAPE 6.9 and 7.4%, respectively) performs better than MP2 (MAPE 40.3 and 58.9%), which in turn performs better than Hartree–Fock (MAPE 111.1 and 210.2%). MP2 performs better for the pairs involving H₂ than for the pairs involving N₂ and CO.

Table 3. Anisotropic Dispersion Coefficients Γ₆^AB for Diatomics H₂, N₂, and CO and for the Diatomics Interacting with Noble Gas Atoms Calculated Using def2-TZVPP Basis Set^a.

pair	ref	HF	MP2	CCSD
H2–H2	0.100618	0.1416	0.1099	0.1021
H2–N2	0.110918	0.1350	0.1040	0.0972
N2–H2	0.096618	0.1884	0.0474	0.1251
N2–N2	0.106818	0.1809	0.0442	0.1211
H2–He	0.092418	0.1288	0.1013	0.0947
H2–Ne	0.090118	0.1240	0.0981	0.0920
H2–Ar	0.097118	0.1343	0.1046	0.0977
H2–Kr	0.098618	0.1369	0.1059	0.0990
H2–Xe	0.100518	0.1397	0.1078	0.1006
N2–He	0.102718	0.1738	0.0429	0.1192
N2–Ne	0.099918	0.1672	0.0412	0.1164
N2–Ar	0.107418	0.1800	0.0446	0.1214
N2–Kr	0.108718	0.1827	0.0452	0.1223
N2–Xe	0.110418	0.1856	0.0461	0.1234
CO–CO	0.09437	0.1013	0.0600	0.0956
CO–H2	0.094937	0.1030	0.0616	0.0970
H2–CO	0.097637	0.1350	0.1047	0.0979
CO–N2	0.093937	0.1014	0.0598	0.0954
N2–CO	0.107737	0.1808	0.0446	0.1216
CO–He	0.09337	0.0997	0.0591	0.0947
CO–Ne	0.091637	0.0975	0.0580	0.0933
CO–Ar	0.094237	0.0975	0.0600	0.0955
CO–Kr	0.094337	0.1016	0.0603	0.0958
CO–Xe	0.094437	0.1021	0.0608	0.0961

	HF	MP2	CCSD
MAPE (%)	111.1	40.3	6.9
AMAX (%)	213.7	67.3	23.1

^aFor each species, the coefficient closest to the reference value is in bold font. The MAPE and AMAX are calculated for the product C₆Γ₆^AB.

Table 4. Anisotropic Dispersion Coefficients Δ₆^AB for the Diatomics H₂, N₂, and CO Calculated Using def2-TZVPP Basis Set^a.

pair	ref	HF	MP2	CCSD
H2–H2	0.010818	0.0214	0.0128	0.0110
H2–N2	0.011418	0.0269	0.0053	0.0126
N2–N2	0.012118	0.0346	0.0021	0.0151
CO–CO	0.009037	0.0104	0.0037	0.0092
CO–H2	0.009437	0.0142	0.0066	0.0096
CO–N2	0.010337	0.0188	0.0028	0.0118

	HF	MP2	CCSD
MAPE (%)	210.2	58.9	7.4
AMAX (%)	427.0	85.9	19.8

^aFor each species, the coefficient closest to the reference value is in bold font. The MAPE and AMAX are calculated for the product C₆Δ₆^AB.

5. Conclusions and Perspectives

The “fixed-diagonal matrices” (FDM) idea^4,7 provides a framework to build new approximations for the dispersion energy in terms of the ground-state pair densities (or the exchange–correlation holes) of the monomers, without the need for polarizabilities. The underlying supramolecular wave function describes a simplified physical mechanism for dispersion, in which only the kinetic energies of the monomers can change. While this is not what happens in the exact case, where all the terms in the isolated monomer Hamiltonian change with respect to their ground-state values, for one-electron fragments the FDM still provides the exact second-order Rayleigh–Schrödinger dispersion energy.^4,7 The purpose of this work was to investigate how accurate the FDM description can be for systems beyond the simple H and He cases when a good pair density of the monomers is used, focusing on the C₆ dispersion coefficients. In the present implementation the computational cost of the step needed on top of the monomers’ ground-state calculation is .

We have found that for closed-shell species FDM yields rather accurate isotropic dispersion coefficients when CCSD (or even MP2) monomer pair densities are used, with a mean absolute percentage error (MAPE) for CCSD for the whole closed-shell data set (all noble gas atoms and molecule pairs, summarized in Figure 7) of 7.1% and a maximum absolute error (AMAX) within 18.2%. FDM on top of CCSD ground states also predicts the anisotropy of dispersion coefficients, which on a limited set of pairs involving diatomics and noble gas atoms yields satisfactory results for the anisotropy Γ₆^AB (MAPE 6.9%, AMAX 23.1%) and the anisotropy Δ₆ (MAPE 7.4%, AMAX 19.8%).

Figure 7.

Isotropic dispersion coefficients for the whole closed-shell data sets (all molecules and noble gas atoms) calculated using Hartree–Fock (MAPE 55.3%, AMAX 131.6%), MP2 (MAPE 8.9%, 50.78%), and CCSD (MAPE 7.1%, AMAX 18.2%) monomer pair densities (def2-TZVPP basis set). The solid line depicts one-to-one correspondence of our FDM method with the reference values.

From this study it also emerged that the basis set used in the monomer calculations has little effect on the computed dispersion coefficients, as the results are essentially converged at the triple-ζ level. Although not competitive with linear-response (LR) CCSD based methods at the complete basis set (CBS) limit, which can achieve accuracies^40,41 of 1–3%, FDM combined with CCSD pair densities seems to have similar accuracies (for closed-shell systems) of LR CCSD based methods when triple- or double-ζ basis sets are used for the latter.^40,42 From the data available, the FDM dispersion coefficients calculated by using CCSD pair densities also outperform XDM for both atoms¹¹ and molecules.⁴³ The DFT-D4 dispersion model⁴⁴ has a notably lower MAPE for a test set consisting of closed-shell molecules, i.e., test sets 3 + 4, but a higher AMAX (MAPE 4.3%, AMAX 29.1%). Other methods such as TS⁴⁵ and LRD⁴⁶ have also a similar or slightly better performance than FDM with CCSD. The TD-DFT results for all atoms and ions reported by Gould and Bucko⁴⁷ can achieve an accuracy between 1 and 5%, and the more refined MCLF method of Manz et al.⁴⁸ has again a MAPE of 4.5% for a set of closed-shell molecules. An advantage over methods like DFT-D4 is that FDM also predicts the anisotropy of dispersion coefficients and gives access not only to energetics but also to a wave function that can be used in various frameworks, for example, in a QM/MM setting to model the interaction between the QM and MM parts.¹⁴⁻¹⁶ We should also remark that the performance of our method is less satisfactory for open-shell atoms and ions.

The main motivation for this work is to provide a solid basis for constructing DFT and other approximations based on a microscopic real-space mechanism for dispersion, given by a simple competition between kinetic energy and intermonomer interaction. Before making approximations for the monomers’ description, it was important to assess how accurate the method can be when good pair densities are used. Considering that the FDM is parameter-free and does not use the polarizabilities as input, the results for closed-shell systems are satisfactory, indicating that the simplified physical mechanism behind it, although not exact, is a reasonable approximation. In our view, this is also conceptually interesting, as it indicates that it is possible to describe reasonably well the overall rise in energy of the monomers with kinetic-energy-only effects.

In future works we will investigate possible ways to improve the results for open-shell fragments, and we will work on building approximations based on model exchange–correlation holes from density functional theory but also on revisiting the formalism in the Hartree–Fock framework by taking into account the effects of the change in the off-diagonal elements of the 1-RDM on the monomer’s Fock operators. We should also stress that the choice of the basis in which to expand the density constraint, i.e., the dispersals b_i(r) of eq 4, is arbitrary and that the current choice of eq 24 is far from optimal. For some cases, the convergence with the number of dispersals is slow, and too-high multipole moment integrals may become numerically unstable. We will thus also explore more closely the determination of an optimal choice for the dispersals as we have preliminary indications⁷ that with a proper choice it is possible to use just a few of them to obtain well-converged results.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.1c00102.

• Computed C₆ coefficients for the pairs of atoms and ions calculated using various basis sets; isotropic C̅₆^AA and C̅₆ dispersion coefficients for molecule pairs; anisotropic Γ₆^AB and Δ₆ for three diatomics and for the diatomics interacting with noble gas atoms; convergence of HF and CCSD pair densities (XLS)

• Descriptions of spreadsheets in Supporting Information (PDF)

The authors declare no competing financial interest.