ISAPol: distributed polarizabilities and dispersion models from a basisspace implementation of the iterated stockholder atoms procedure
Abstract
Recently, we have developed a robust, basisspace implementation of the iterated stockholder atoms (BSISA) approach for defining atoms in a molecule. This approach has been shown to yield rapidly convergent distributed multipole expansions with a welldefined basis set limit. Here we use this method as the basis of a new approach, termed ISAPol, for obtaining nonlocal distributed frequencydependent polarizabilities. We demonstrate how ISAPol can be combined with localization methods to obtain distributed dispersion models that share the many unique properties of the ISA: these models have a welldefined basis set limit, lead to very accurate dispersion energies, and, remarkably, satisfy commonly used combination rules to a good accuracy. As these models are based on the ISA, they can be expected to respond to chemical and physical changes naturally, and thus, they may serve as the basis for the next generation of polarization and dispersion models for ab initio forcefield development.
Keywords
Intermolecular interactions van der Waals Dispersion models Symmetryadapted perturbation theory Distribution algorithms Distributed polarizabilities Atoms in a molecule1 Introduction
In the last few years, the field of intermolecular interactions has seen a tangible increased level of importance. The deep level of understanding we have achieved from decades of theoretical developments has formed the basis of new models for intermolecular interactions that finally give us the promise of the longawaited accuracy and predictive power needed in application to complex molecular aggregation processes.
These intermolecular interaction models are being developed primarily from interaction energies computed using some variant of symmetryadapted perturbation theory (SAPT) and predominantly using the version of SAPT based on density functional theory, SAPT(DFT). The latter choice is based both on the favourable accuracy and on computational efficiency of SAPT(DFT). The general procedure for model development typically uses some mix of SAPT(DFT) calculations at specific, close separation dimer configurations, and an analytical multipoleexpanded form of the interaction energy suitable for the long range. The various implementations of this approach have been described elsewhere [45, 55, 77, 78, 79].
The advantage of using a theory like SAPT or SAPT(DFT) for the shortrange energies is that the resulting interaction energy has a welldefined multipoleexpanded form. Consequently, if this multipoleexpanded form can be determined analytically, there can be a rigorous match between the short and long ranges. Indeed, this has been the basis of the above philosophy for many decades (see, for example, refs. [9, 13, 30, 40, 47, 60]). Here SAPT(DFT) has an advantage over SAPT in that the multipolar molecular properties (multipole moments, polarizabilities, dispersion coefficients) can be readily derived from the underlying density functional method and usually at a comparatively low computational cost.
However, as is now well known [3, 4, 14, 16, 21, 22, 28, 34, 52, 53, 58, 66, 73, 74, 81, 83, 84], intermolecular properties must be distributed if we are to achieve high enough accuracies. The singlecentre multipole expansion, which is a useful paradigm for diatomics or triatomics, is poorly convergent for larger molecules, for which we must use multiple expansion centres. These expansion centres have usually been taken to be the locations of the nuclei in the molecule, though this need not be the case, and indeed, for some cases [19, 38] multiple, offatomic sites are chosen to obtain even faster convergence of the multipole expansion.

Uniqueness for a given choice of AIM algorithm: While the AIMs themselves are not unique, the actual atomic domains that result from a particular choice of partitioning algorithm should be unique. That is, the result should not depend on numerical parameters and should have a welldefined basis set limit. This will usually imply that the resulting distributed molecular properties are also unique.

Rapid convergence with rank: As the distributed properties will typically be used in a model for the molecular interactions, for computational reasons it is usually desirable that these models be rapidly convergent with rank. This condition implies that the atomic domains from the AIM are as close to being spherical as is possible.

Agreement with reference energies: The distributed properties should result in energies in good agreement with those from the reference electronic structure method. In our case, this will be taken to be appropriate interaction energies from SAPT(DFT).

Insensitivity to molecular conformation: We fully expect distributed properties to vary with molecular conformation, but, particularly for soft deformations, that is, those with a small change in the electronic distribution, we may expect the AIM domains and resulting molecular properties also to change only slightly.

Agreement with physical/chemical expectations: This condition is qualitative as we cannot define what the physically meaningful properties of an atom in a molecule should be. We can, however, hope that the resulting properties be in broad agreement with chemical/physical intuition.

Computational efficiency: This is important if we are to apply the distribution techniques to large systems. Ideally, we would like the algorithm to scale linearly with the size of the system.
Some of the methods used to define the properties of the atoms in a molecule can be regarded as being more mathematical or numerical, though physical properties like the van der Waals radii may be used. In these methods, the molecular properties may be partitioned in a basisspace or realspace manner, though hybrids of the two are also used. Some of the more successful of these methods include the distributed multipole analysis (DMA) of Stone [70, 72], the LoProp and MpProp approaches [21, 69], and methods based on constrained density fitting for the multipole moments [64] and for the polarizabilities [52, 66]. We will refer to the original constrained density fitting method of Misquitta & Stone [52] as the cDF method and the related ‘selfrepulsion plus local orthogonality’ method of Rob & Szalewicz [66] as the SRLO method.
Both the cDF and SRLO distribution techniques use constraints in the density fitting to allow the molecular polarizabilities to be partitioned into nonlocal, site–site polarizabilities. These are not the local polarizabilities that one might conventionally think of, but include terms that allow for nonlocal, or throughspace polarization in the molecule (see §9.2 in Ref. [73]). The methods differ in the constraints applied, with the SRLO algorithm using a constraint to reduce the charge flow terms, that is, the polarizabilities that allow for charge movement in the molecule, to nearly zero. Using appropriate localization techniques [53, 58], both the cDF and SRLO models can be made to yield effective local polarizability models. In the case of the former, we have referred to the combined method as the Williams–Stone–Misquitta, or WSM model. This model has formed the basis of much of our work so far and indeed has been used to develop intermolecular interaction models by other groups either directly [68] or by extension [43, 77, 78]. As the localization schemes in the WSM model can be applied to any of the nonlocal polarizability models, we will refer to the localized models by appending ‘L’, for example, the SRLOL model would be the SRLO nonlocal model localized using the WSM approach.
Localized, isotropic polarizabilities for the symmetrydistinct sites in the thiophene molecule computed with the cDFL model that is using cDF nonlocal polarizabilities localized using the WSM algorithm
Site  l  aDZ  aTZ  aQZ 

C1  1  7.28  7.20  6.94 
2  28.52  32.77  21.33  
3  \(\)355.10  141.68  920.37  
C2  1  10.60  10.79  11.19 
2  36.47  57.05  44.03  
3  \(\)345.20  \(\)341.59  580.76  
S  1  16.67  16.90  16.86 
2  90.78  95.12  107.16  
3  \(\)206.73  \(\)617.48  \(\)315.64  
H1  1  2.24  2.26  2.36 
2  1.51  \(\)3.42  6.14  
3  \(\)69.59  \(\)38.73  \(\)155.99  
H2  1  1.55  1.48  1.33 
2  2.88  \(\)3.66  10.09  
3  \(\)49.55  \(\)45.20  \(\)157.65 
One of these methods is the basisspace ISA, or BSISA algorithm, that we have developed and implemented in the CamCASP [56] program. We have used the BSISA algorithm to define distributed multipole models and have demonstrated that these multipoles exhibit all of the properties we have listed above. In fact, the BSISA distributed multipoles — or ISADMA models for short — surpass those from the wellestablished distributed multipole analysis (DMA) algorithm by Stone [71, 72] in the rapidity of convergence with rank and in the stability with respect to basis set. Further, we have demonstrated how the BSISA density partitioning can be used, via the distributed overlap model, to achieve robust fits to the shortrange part of the interaction energy and therefore to easily develop detailed analytic models for the intermolecular interaction [55]. Finally, in collaboration with Van Vleet and Schmidt [77, 78] data from the BSISA algorithm have been used to develop the shortrange repulsion and dispersion damping models for two general force fields: the SlaterFF and MASTIFF models.
In this paper, we extend the applicability of the BSISA algorithm to the secondorder energies and we demonstrate how we can use this method to obtain distributed frequencydependent polarization models and from these distributed dispersion models for any closedshell molecular system. We first describe this new algorithm, termed ISAPol. Next we describe a new, simplified and more flexible version of the BSISA algorithm, one that allows more accurate ISA solutions as well as additional sites and coarse graining. The ISAPol method results in what are known as nonlocal polarizabilities which describe throughspace polarization and charge movement in the system. While this is an important subject and leads to unexpected van der Waals interactions [37, 49, 51] in lowdimensional systems, we will instead focus here on the localized distributed models that lead to the conventional polarization and dispersion interactions. We describe the localization procedures in brief along with some of the important features of the methods. Then we present a wide range of results that compare the polarizabilities from ISAPol with those from cDF and SRLO and demonstrate that the new models are superior in many ways. Finally, we compare the dispersion energies from localized ISAPol models with those from SAPT(DFT). We end with an outlook on the scope and power of this method.
2 Theory
2.1 A simplified and flexible BSISA algorithm
However, in the original implementation, minimizing the \(\varDelta _\mathrm{stock(A)}\) functional tended to lead to unacceptable inaccuracies in the ISA AIM densities; in particular, the total charge of the system was often not conserved, with differences of 0.01e often encountered. Also, higher ranking molecular multipoles would not be well reproduced. Consequently, we combined the \(\varDelta _\mathrm{stock(A)}\) functional with the density fitting functional to result in a hybrid DFISA algorithm. This algorithm involved a single parameter that controlled the relative weights given to each scheme, with a 90% weighting of the DF functional being recommended. While the results were better, there were two problems: (1) the new method had a computational scaling of \(\mathcal {O}(N^3)\) and (2) despite the mixture of the density fitting and ISA functionals, there was still an overall loss in accuracy which resulted in small residual errors in the electrostatic energies computed from the DFISA algorithm compared with reference energies from SAPT(DFT).
The primary reason for the inaccuracy of the original algorithm was that the ISA atomic basis sets were constructed from the auxiliary basis used in the density fitting, and this inextricably linked the two basis sets. This placed limits on both basis sets and therefore resulted in inaccuracies both in the fitted density and in the ISA solutions. This restriction in the basis sets was required for technical reasons associated with the implementation of the BSISA algorithm in version 5.9 of the CamCASP program. It was because of these inaccuracies that we needed to use the more computationally demanding DFISA algorithm.

The main basis: used for the molecular orbitals.

The auxiliary basis: used for the density fitting. This basis may use either Cartesian or spherical GTOs.

The atomic basis sets: used for the ISA atomic expansions. This basis set must use spherical GTOs, but is otherwise independent from the above basis sets. The atomic basis sets can therefore be increased in size if needed and placed on arbitrary sites, or removed from some sites.
3 Numerical implementation
In the cDF and SRLO methods, the distribution is achieved via the auxiliary basis functions themselves [52, 66]. These methods are linked to the ISAPol algorithm by setting the probability functions \(P^{a} ({\mathbf{r}}) = 1\) and limiting the sum over k/l in Eq. (19) to include only those auxiliary functions on sites a/b. This has the advantage of simplicity, but disadvantage that the results are dependent on the auxiliary basis set [52]. In the ISAPol approach, the distributed polarizabilities are uniquely defined for a given set of probability functions \(P^{a}\), and as we know that the ISA solutions are unique [10, 36], we should expect that the ISAPol algorithm leads to unique distributed polarizabilities. We shall demonstrate this below.
3.1 Linearizing the algorithm: issues
Once the frequencydependent coefficients \(\tilde{C}_{kl}(\omega )\) have been calculated, the evaluation of \(\alpha ^{ab}_{tu}(\omega )\) using Eq. (19) for a given pair of sites a, b and angular momenta t, u scales as \(\mathcal {O}(M^2)\) where there are M auxiliary basis functions in the system. If l is the maximum angular momentum for which distributed polarizabilities have to be computed and N is the number of sites in the system, then there are \(\mathcal {O}(N^2\, l^4)\) nonlocal polarizabilities, so the total scaling of the calculation is \(\mathcal {O}(l^4\, N^2\, M^2)\). If we assume on the average m auxiliary basis functions per site, then \(M = mN\), so the computational scaling is \(\mathcal {O}(l^4\, m^2\, N^4)\), that is, it scales as the fourth power as the number of sites. While the scaling is not necessarily unfavourable, the prefactor, \(l^4\, m^2\), can easily be of the order \(10^6\), thereby making this calculation computationally burdensome, though it can be trivially parallelized over the pairs of sites a, b.
The distributed multipole integral in the auxiliary basis defined in Eq. (20) must be evaluated numerically, on a grid due to the ISA probability function \(P^{a}\). This function is defined as the ratio of the ISA shape functions which makes analytic evaluation unfeasible, but these are themselves piecewise continuous, so numerical evaluation is mandatory. As the numerical integration grid size scales with the number of atoms in the system, the evaluation of the \(Q^{a}_{t,k}\) integrals using Eq. (20) would incur a computational cost scaling as \(\mathcal {O}(l^2\, m\, n_g\, N^3 )\), where \(n_g\) is the average number of grid points per atom, that is, the scaling is \(\mathcal {O}(N^3)\) with number of atoms. As we need fairly dense grids, particularly in the angular coordinates, to converge the higher ranking multipole moment integrals, the prefactor \(l^2\, m\, n_g\) can be as large as \(10^7\). This can make the evaluation of these integrals a significant computational cost, and even though this evaluation needs to be performed only once in a calculation, it would be advantageous if the scaling could be reduced.

Integration grids: Rather than spanning all atoms in the system, the grids are based on sites in \(\mathcal {N}_{a}\).

Probability function evaluation: \(P^{a}\) includes a sum over all sites in the system, but this sum can be restricted to go over only sites in \(\mathcal {N}_{a}\).

Auxiliary basis function k: \(Q^{a}_{t,k}\) is evaluated only for those k that belong to sites in \(\mathcal {N}_{a}\) and is set to zero otherwise.
There are limitations to the use of neighbourhoods to achieve linearity in computational scaling: for heavily delocalized systems such as the \(\pi\)conjugated molecules the neighbourhoods may need to be increased in order to achieve sufficient accuracy in the polarizabilities. In this case, using neighbourhoods that are too small leads to increased charge conservation errors in the BSISA solution and to sumrule violations in the charge flow [73] contributions to the nonlocal polarizabilities.
3.2 Localization of the nonlocal polarizabilities
The main focus of this paper is not the nonlocal polarizabilities defined in Eq. (19), but rather the localized distributed polarizability models that can be derived from these using techniques described in detail in some of our previous publications [53, 58]. This is not to diminish the importance of the nonlocal polarizability models; indeed, these models are essential for heavily delocalized systems and in lowdimensional systems lead to van der Waals interactions that cannot be replicated by any local model [37, 49, 51]. However, it is the local models that are commonly used, so for very pragmatic reasons we will focus on these here.

Multipolar localization: In the twostep localization scheme that forms part of the WSM model, we first transform away the nonlocal contributions using a multipole expansion (see §9.3.3 in Ref. [73]). We have explored two schemes for this purpose: the method of LeSueur & Stone [32] and that of Wheatley & Lillestolen [83]. Of these, the latter has the advantage that the nonlocal terms are localized along the molecular bonds and should result in better convergence of the resulting model. However, either of these localization procedures leads to a degradation in the convergence of the resulting polarizability expansion.

Constrained refinement: In this step, the multipolar localized polarizability models are refined to reproduce the pointtopoint polarizabilities (see Ref. [85] and §9.3.2 in Ref. [73]) computed on a pseudorandom set of points surrounding the molecule. The idea here is to use the local polarizabilities from the first step as prior values and allows them to relax using constraints to keep them close to their original values.
In the original WSM model, we relied on nonlocal polarizabilities from the cDF algorithm as the starting point. This did not always work out well as the multipolar localized models often contained terms with unphysical values which would change by a considerable amount in the refinement stage. For this reason, the constraints we recommended [58] were weak for the dipole–dipole polarizabilities and completely absent for the higher ranking terms. The lack of constraints for the higher ranking terms was simply a recognition that our prior values were simply too unreliable. Looked at another way, the final polarizability models depended quite strongly on the kinds of constraints used.
It may seem paradoxical to use constraints of any kind if the refinement step does not alter the multipolar localized ISAPol model by much. The reason for the use of constraints is that in a mathematical optimization it is possible for parameters to alter without a meaningful change in the cost function. The constraints prevent this kind of mathematical wandering of parameters, particularly for large systems for which we rarely have enough data in the pointtopoint polarizabilities to act as natural constraints to the parameters.
4 Numerical details
All SAPT(DFT) calculations have been performed using the CamCASP 5.9 program [56] with orbitals and energies computed using the DALTON 2.0 program [24] with a patch installed from the Sapt2008 code. The Kohn–Sham orbitals and orbital energies were computed using an asymptotically corrected PBE0 [1] functional with Fermi–Amaldi (FA) longrange exchange potential [20] and the Tozer & Handy splicing scheme. Linear response calculations and ISAPol polarizabilities were performed using the same functional, but with a developer’s version of CamCASP 6.0. The kernel used in the linear response calculations is the hybrid ALDA+CHF kernel [50, 52] which contains 25% CHF (coupled Hartree–Fock) and 75% ALDA (adiabatic local density approximation). This kernel is constructed within the CamCASP code. The PW91 correlation functional [63] is used in the ALDA kernel.
The shift needed in the asymptotic correction has been computed selfconsistently using the following ionization potentials: thiophene: 0.326 a.u. [33]; pyridine: 0.3488 a.u. [55]; water: 0.4638 a.u. [33]; methane: 0.4634 a.u. [33]. The vibrationally averaged molecular geometry was used for water [39] and methane [2, 61] molecules, the pyridine geometry has been taken from Ref. [55], and the thiophene geometry has been obtained by geometry optimization using the PBE0 functional and the ccpVTZ basis [29] with the NWChem 6.6 program [76].

Methane dimer, water dimer, methane..water complex: main basis: augccpVTZ with 3s2p1d midbond set, and auxiliary basis: augccpVTZRI basis with 3s2p1dRI basis.

Pyridine dimer: main basis: SadlejpVTZ [67] with a 3s2p1d midbond set [8], and auxiliary basis: augccpVTZRI basis [82] with 3s2p1dRI basis.

The methane and water molecules: main basis: daugccpVTZ (spherical); auxiliary basis: augccpVQZRI (Cartesian) with ISAset2 with sfunctions on the hydrogen atoms limited to a smallest exponent of 0.25 a.u. atomic basis: like the auxiliary basis, but with spherical GTOs.

The pyridine molecule: main basis: daugccpVTZ (spherical); auxiliary basis: augccpVQZRI (Cartesian); atomic basis: augccpVQZRI (spherical) with ISAset2.
5 Results
Although the nonlocal polarizability models are fundamental, these are also, at present, of high complexity and are not suitable for most applications. So while we assess some features of the ISAPol nonlocal polarizability models, we will here be primarily concerned with the localized models.
5.1 Convergence with rank
The assessment of the polarizability models is complicated by the fact that there is no pure polarization energy defined in SAPT or SAPT(DFT): the secondorder induction energy in these methods contains both a polarization and a charge transfer contribution. While it is possible to separate these, for example, using regularized SAPT(DFT) [46], we inevitably then encounter the problem of damping [53, 55]. An elegant solution to the first problem is to compute the polarization energy of the molecule interacting with a point charge probe. This has the advantage that the energies can be easily displayed on a surface around the molecule, and as reference energies can be easily computed using the CamCASP program, it is relatively straightforward to make comparisons of the model and reference energies and visualize the differences on the molecular surface.
In Fig. 1, we have displayed the reference SAPT(DFT) polarization energies for the pyridine molecule interacting with a \(+1\)e point charge probe. The energies are displayed on a \(10^{3}\) isodensity surface computed using the CamCASP program. The resulting polarization energies are uncharacteristically large, due both to this choice of surface (which corresponds approximately to the van der Waals surface) and to the large size of the charge: typical local charges in atomic systems will usually be half as much. Also shown in Fig. 1 are the errors made by the damped polarization models against the reference energies. Consider first the nonlocal models: the positive errors made by the NL1 model indicate an underestimation of the polarization energy. The agreement with the reference energies gets progressively and systematically better as the maximum rank increases through 2 to 3. Results for the NL4 model (the maximum rank of the nonlocal models, and also the most accurate for the choice of damping) are not shown. The localized, anisotropic models exhibit similar errors, but the localized, isotropic models show larger variations in the errors made. In particular, these models shown an underestimation of the polarization near the hydrogen and nitrogen atoms, and a large overestimation of the polarization in the centre of the ring. This is due to the simplicity of the isotropic models: the polarizability of an anisotropic system like pyridine cannot be correctly modelled everywhere using isotropic AIM polarizabilities. As with the distributed multipole moments [57], the ISA AIMs lead to polarization models with better convergence with increasing rank and fewer artefacts in both the nonlocal and local models.
In Fig. 2 are shown similar results, this time for the models from the cDF algorithm. These differ from the ISAPol models in important ways: first of all the errors are larger, even for the nonlocal models, but perhaps more importantly, the variations in the errors are much larger for all models. It is the latter that is the bigger concern for model building, as variations in errors arise from to position and angle dependent variations in the quality of the model, leading to unreliable predictions.
6 Convergence with basis of the localized models
The next question we need to address is the basis set convergence of the ISAPol models. We will not discuss the performance of the nonlocal or local anisotropic models here as it is difficult to display the data contained in these models in a meaningful and concise manner. Instead, we will focus on the local, isotropic models.

Directly from the unrefined anisotropic model by retaining only the isotropic part of the polarizabilities.

By refining this isotropic model using the pointtopoint polarizabilities.

By refining the anisotropic model as described in Sect. 3.2 and subsequently retaining only the isotropic part of the polarizabilities.
Localized, isotropic polarizabilities for the symmetrydistinct sites in the thiophene molecule computed with the WSM algorithm starting from ISAPol nonlocal polarizabilities
Site  l  aDZ  aTZ  aQZ 

C1  1  7.39  7.40  7.36 
2  23.48  27.40  28.21  
3  458.12  553.61  579.94  
C2  1  10.53  10.62  10.64 
2  32.58  37.75  38.85  
3  654.79  766.85  806.50  
S  1  16.74  17.05  17.17 
2  97.09  109.74  115.85  
3  1449.90  1859.26  2232.79  
H1  1  2.18  2.17  2.18 
2  4.41  4.44  4.74  
3  − 17.88  − 6.67  − 0.36  
H2  1  1.53  1.47  1.46 
2  4.00  4.07  4.16  
3  − 11.64  − 2.93  5.42 
In Table 2, we present ISAPol localized, isotropic polarizabilities for the symmetrydistinct atoms in the thiophene molecule computed in three basis sets. The dipole–dipole polarizabilities (i.e. rank 1) are already reasonably well converged in the augccpVDZ basis, with the exception of the sulphur atom which needs the larger augccpVTZ basis. The quadrupole–quadrupole (rank 2) polarizabilities on the carbon and hydrogen atoms are converged in the augccpVTZ basis, but the augccpVQZ basis is needed for the sulphur atom. At rank 3, the octopole–octopole polarizabilities on the carbon atoms seem to be approaching convergence in the augccpVQZ basis, but the sulphur atom is far from convergence. The negative octopole–octopole terms on the hydrogen atoms seem to be a result of the lack of sufficient higher angular terms on these atoms and of the absence of dipole–quadrupole and quadrupole–octopole polarizabilities in this rather drastic approximation. In the augccpVQZ basis, there is only one negative term present on the H1 atom. Compare these results to those from the cDF approach shown in Table 1. The ISAPol algorithm is clearly the more systematic of the two with the AIM local polarizabilities converged or approaching convergence at all ranks.
Localized, isotropic diagonal dispersion coefficients for the symmetrydistinct sites in the pyridine, water, methane, and thiophene dimers computed with the ISAPolL model
Site..Site  \(C_{6}\)  \(C_{8}\)  \(C_{10}\)  \(C_{12}\) 

Pyridine  
C1..C1  1.249(1)  3.141(2)  1.529(4)  4.258(5) 
C2..C2  3.643(1)  6.525(2)  3.711(4)  8.643(5) 
C3..C3  2.246(1)  5.555(2)  2.097(4)  5.496(5) 
N..N  3.206(1)  6.735(2)  2.609(4)  6.196(5) 
H1..H1  3.533(0)  3.407(1)  4.384(2)  4.795(3) 
H2..H2  1.802(0)  1.758(1)  1.921(2)  1.880(3) 
H3..H3  1.689(0)  2.306(1)  3.392(2)  4.173(3) 
Water  
O..O  2.434(1)  4.899(2)  1.252(4)  2.384(5) 
H..H  0.783(0)  4.357(0)  9.061(1)  7.714(2) 
Methane  
C..C  3.184(1)  9.161(2)  3.771(4)  1.092(6) 
H..H  2.105(0)  2.132(1)  3.938(2)  4.638(3) 
Thiophene  
C1..C1  2.259(1)  5.414(2)  2.465(4)  6.726(5) 
C2..C2  3.759(1)  8.759(2)  4.254(4)  1.194(6) 
S..S  1.082(2)  3.895(3)  2.096(5)  7.904(6) 
H1..H1  2.096(0)  2.453(1)  2.864(2)  3.021(2) 
H2..H2  1.268(0)  1.630(1)  2.518(2)  3.207(3) 
In Table 3, we report the ISAPolL isotropic dispersion coefficients for the symmetrydistinct sites in the water, methane, pyridine, and thiophene molecules. Only the diagonal, that is, same site, terms are reported: the complete dispersion models for these molecules and also those for the methane..water complex are given in the S.I. Notice that while the dispersion coefficients for the carbon atoms in these molecules are of similar magnitude, they nevertheless vary considerably in accordance with what might be expected from the variations in the local chemical environment. For example, the C1 atom in pyridine and the C1 atom in thiophene both have smaller dispersion coefficients than the other carbon atoms in the molecules, which should be expected as these atoms are bonded directly to the more electronegative N and S atoms in the respective molecules. Likewise, while the dispersion terms on the hydrogen atoms are similar, those on the hydrogen atom in water are substantially smaller due to the large electronegativity of the oxygen atom in the water molecule. The ability of the ISAPolL models to provide dispersion terms from \(C_{6}\) to \(C_{12}\) which respond to the chemical environment of the atoms in the molecule could be used to develop more detailed and comprehensive models for the dispersion energy, but more extensive data sets will be needed for a full analysis.
6.1 Assessing the models using SAPT(DFT)
 Ionization potential (IP) damping [54]:where \(I_A\) and \(I_B\) are the vertical ionization energies, in a.u., of the two interacting molecules. This is the simplest of the damping models with one damping parameter for all pairs of sites (a, b) between the interacting molecules A and B.$$\begin{aligned} x_{ab}&= \left( \sqrt{2 I_A} + \sqrt{2 I_B} \right) r_{ab} = \beta _{AB} r_{ab}, \end{aligned}$$(24)
 The Slater damping from Van Vleet et al. [78]. Here the damping parameter is dependent on the pairs of interacting atoms and is given bywhere the parameter \(\beta _{ab}\) is now dependent on the sites and is defined as \(\beta _{ab} = \sqrt{\beta _{a} \beta _{b}}\), where the parameter \(\beta _{a}\) is extracted from the ISA shape function \(w^{a}\) by fitting it to an exponential of the form \(K\exp {(\beta _{a} r)}\) and \(\beta _{b}\) likewise [57, 78]. This damping function is motivated by the form of the overlap of two such Slater exponentials [78].$$\begin{aligned} x_{ab}&= \beta _{ab} r_{ab}  \frac{\beta _{ab} (2\beta _{ab} r_{ab} + 3)}{\beta _{ab} ^2 r_{ab}^2 + 3 \beta _{ab} r_{ab} + 3}, \end{aligned}$$(25)
 The scaled ISA damping model is a simplification of the Slater damping model. Here we define a scaled parameter \(\tilde{\beta }_{a}\) for each site in molecule A as follows:where \(\beta _{a}\) is defined above and \(s_A\) is the moleculespecific empirical scaling parameter. Next we define \(\beta _{ab}\) from the combination rule$$\begin{aligned} \tilde{\beta }_{a}&= s_A \beta _{a}, \end{aligned}$$(26)and \(x_{ab} = \beta _{ab} r_{ab}\). In Ref. [78], the scaling parameter is taken to be a constant \(s = 0.84\) independent of the type of molecule, but here we allow the parameter to vary according to the molecule and determine it empirically by fitting the model energies to the reference dispersion energies.$$\begin{aligned} \beta _{ab}&= \sqrt{\tilde{\beta }_{a} \tilde{\beta }_{b}}, \end{aligned}$$(27)
In Fig. 4, we display dispersion energies for the methane dimer in more than 2600 dimer configurations. Because the methane molecule has high symmetry and indeed is nearly spherical, we should expect the dispersion energy of this system to be well approximated by an isotropic dispersion model. This is indeed the case, and we see nearly perfect correlation of the ISAPolL dispersion energies with the scaled damping model with the reference energies. In this case, a scaling parameter of 0.76 was determined. On the other hand, the IP damping model which we have recommended in the past does not provide sufficient damping, and nor does the Slater model, though it is better.
Figure 5 shows data for the water dimer in more than 2000 dimer configurations. Water is a more anisotropic system than methane, and we cannot expect the isotropic models to behave as well for water dimer as for methane dimer. Once again both the IP and Slater damping models result in underdamping, though not as severely as for methane dimer. The scaled damping model with a scaling factor of 0.76 fares far better, resulting in dispersion energies for most of the dimers within \(\pm 5\)% from the reference energies. In Fig. 6, we have displayed dispersion energies for the mixed methane\(\cdots\)water system. The picture is the same, with the scaled damping model correlating very well with the reference energies.
6.2 Convergence with rank
6.3 Combination rules
Dispersion models in common intermolecular interaction models are usually constructed to satisfy combination rules, usually through a constrained fitting process (see, for example, Ref. [43]). This has the advantage of greatly reducing the number of parameters in the model, and the most commonly used geometric mean combination rule has good justification from theory, although the actual dispersion coefficients may not satisfy a combination rule accurately.
Do the ISAPol dispersion models satisfy the geometric mean combination rule? Once again this question is a complex one if we account for the angular variation in the dispersion parameters, so here we will restrict this discussion to the isotropic dispersion models only. In Fig. 9, we plot the dispersion coefficients for the thiophene molecule computed using the geometric mean combination rule against reference ISAPolL isotropic dispersion coefficients. This is performed for the augccpVnZ, \(n = \text {D,T,Q}\) basis sets. It can be seen that the ISAPolL models satisfy the combination rule very well for \(n=6,8,10,12\), that is, for all ranks of the dispersion coefficients considered in this paper. In all cases, the terms that are most in error are those involving at least one of the hydrogen atoms, but these errors are reduced as the basis set gets larger, echoing the trend to more welldefined polarizabilities shown in Table 2.
This property of the dispersion models derived from ISAPolL polarizabilities seems to hold for a variety of systems, though less well for those containing a larger fraction of hydrogen atoms. This is remarkable given that the combination rules are never imposed, and there is no reason to expect the singlepole approximation to hold or indeed for the poles on different atoms to be similar. Further work is needed to analyse exactly why this is the case, and if and when it breaks down, but this property of the ISAPolL models, if generally applicable, will be a very useful feature for the development of models of more diverse interactions.
7 Analysis and outlook
We have described and implemented the ISAPol algorithm for computing distributed frequencydependent polarizabilities and dispersion coefficients for molecular systems. This algorithm is based on a basisspace implementation [57] of the iterated stockholder atoms (ISA) algorithm of Lillestolen and Wheatley [35]. We have described a simpler and more versatile implementation of the BSISA algorithm and have implemented this algorithm in a developer’s version of CamCASP 6.0. This new algorithm allows for higher accuracies in the ISA solution and in the resulting distributed properties. Additionally, the algorithm has a computational cost that scales linearly with the system size.

Systematic convergence of the ISAPol nonlocal polarizabilities as a function of rank. This model has been demonstrated to converge more systematically than the constrained density fitting, cDF, model we have previously proposed [52], and also the related SRLO algorithm from Rob & Szalewicz [66].

The localized ISAPol polarizabilities (ISAPolL) are well defined and are usually positive definite where local models can give a good account of what are inherently nonlocal effects. In other words, for systems with relatively short electron correlation lengths, the ISAPolL models are appropriate and systematic and lead to reasonably accurate polarization energies.

We have demonstrated that the ISAPolL polarizabilities converge systematically with basis set and appear to have a welldefined basis set limit. The systematic behaviour of these distributed polarizabilities should make it possible to extrapolate the polarizabilities of the atoms in the molecule (AIMs) to the complete basis set limit. This was not possible with the WSM models [53, 58] built from cDF nonlocal polarizabilities as has been illustrated in the Introduction.

Dispersion models constructed from the ISAPolL frequencydependent polarizabilities are well defined and, when suitably damped, show exceptionally good reproduction of the SAPT(DFT) dispersion energies for a variety of anisotropic systems.

Damping of the dispersion models is achieved using the Tang–Toennies functions with atomspecific damping parameters derived using the BSISA algorithm. A single scaling parameter is used as described by Van Vleet et al. [78], though we have allowed the scaling parameter to vary with the molecule.

The isotropic dispersion coefficients from the ISAPolL algorithm have been shown to satisfy the geometric mean combination rule that is used in many empirical models for the dispersion energy, but is not imposed at any stage in developing the localized ISAPol polarizabilities. This is the case for terms from \(C_{6}\) to \(C_{12}\) and the accuracy of the combination rule improves with increase in the basis set used for the ISAPol calculation.

Anisotropy in the damping: Perhaps, the damping coefficients need to be extracted from the ISA AIM densities \(\rho ^{a}\) rather than from the ISA shape functions \(w^{a}\) as we do currently. This would have the consequence of making the damping parameters anisotropic and these may be more appropriate at modelling interactions involving sites that are themselves strongly anisotropic. This would be the case for the oxygen atom in water and for the carbon atoms in a \(\pi\)conjugated system.

Anisotropy in the dispersion coefficients: The dispersion models derived from the ISAPolL polarizabilities include anisotropy, but we have, as yet, focused only on the isotropic parts of these models. This has been performed mainly for computational reasons: most simulation codes accept only isotropic dispersion models, and the anisotropic models tend to be very complex. Recently, Van Vleet et al. [77] have demonstrated how the inclusion of atomic anisotropy can result in a rather significant improvement in the model energies, but this approach is empirical in the sense that the anisotropy parameters are determined by fitting to reference SAPT(DFT) dispersion energies. We need a way to develop practical models in a nonempirical manner.
8 Additional information
All developments have been implemented in a developer’s version of the CamCASP 6.0 [56] program which may be obtained from the authors on request. CamCASP has been interfaced to the DALTON 2.0 (2006 through to 2015), NWChem 6.6, GAMESS(US), and Psi4 1.1 programs. The supplementary information (SI) contains additional data from the systems we have investigated, but not included in this paper.
Notes
Acknowledgements
We dedicate this article to the memory of Dr János Ángyán for a friendship and for many scientific discussions, one of which led to the ISAPol method. AJM thanks Queen Mary University of London for support and the Thomas Young Centre for a stimulating environment, and also the Université de Lorraine for a visiting professorship during which part of this work was completed. We also thank Dr Rory A. J. Gilmore for assistance in calculating the SAPT(DFT) reference energies for the water..water, methane..methane, and water..methane complexes. We thank Dr Toon Verstraelen for helpful comments on the manuscript.
Supplementary material
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