DFT-SAPT Intermolecular Interaction Energies Employing Exact

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DFT-SAPT intermolecular interaction energies employing exact-exchange Kohn-Sham response methods Andreas Hesselmann J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01233 • Publication Date (Web): 22 Mar 2018 Downloaded from http://pubs.acs.org on March 22, 2018

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Journal of Chemical Theory and Computation

DFT-SAPT intermolecular interaction energies employing exact-exchange Kohn-Sham response methods

Andreas Heÿelmann∗ Lehrstuhl für Theoretische Chemie, Universität Erlangen-Nürnberg, Egerlandstr. 3, D-91058 Erlangen, Germany E-mail: [email protected]

Abstract Intermolecular interaction energies have been calculated by symmetry-adapted perturbation theory based on density-functional theory monomer properties (DFT-SAPT) employing response functions from time-dependent exact-exchange (TDEXX) kernels. Combined with a new asymptotic correction scheme for the xc potentials of the monomers, which is similar in its performance to standard asymptotic correction methods, it is shown that this DFT-SAPT[TDEXX] method delivers highly accurate intermolecular interaction energies for the S22, S66 and IonHB benchmark data bases by Hobza al..

et

A corresponding DFT-SAPT approach employing the adiabatic TDEXX kernel in

the response calculations has also been tested. This DFT-SAPT[ATDEXX] method performs almost as well as DFT-SAPT[TDEXX] for dispersion-dominated dimer systems, but less accurately for hydrogen-bonded dimers. Compared to this, the DFTSAPT[TDEXX] yields a balanced description of the interaction energies for various ∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment

Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 58

interaction-type motifs, similar to the standard DFT-SAPT method that utilises the ALDA xc kernel to compute the response functions.

Introduction Commonly, there exist two quantum chemistry approaches to describe intermolecular interaction energies, the supermolecular method and intermolecular perturbation theories The supermolecular method describes the interaction energy between two subsystems

B

113

.

A and

via the equation

super ∆Eint (AB) = E(AB) − E(A) − E(B)

where

E(AB)

monomers.

denotes the energy of the dimer and

E(A)

and

E(B)

(1)

are the energies of the

The simple scheme of Eq. (1) has the advantage that the interaction energy

can practically be computed with any quantum chemical method. However, care needs to be exercised regarding potential errors related to the size consistency property and also with respect to the basis set superposition error (BSSE). The former problem is usually solved by computing

∆Eint

with size consistent quantum chemical methods, e.g., many-body

perturbation theory or coupled-cluster methods

1,14,15

. The latter problem can conventionally

be solved by using the Boys-Bernadi counterpoise correction

16,17

. Another shortcoming of

supermolecular methods, that can not easily be resolved, is that Eq. (1) does not give access to the underlying interaction contributions that are responsible for the interactions between the monomers, namely electrostatic, polarisation, dispersion or exchange-repulsion forces

18

.

While these terms can be extracted from the supermolecular energy by using energy decomposition methods

13

, a more rigorous and qualitatively accurate decomposition of

can be obtained with the aid of intermolecular perturbation theory methods

7,10,11,13,19

∆Eint . The

to date most widely used variant of these is the symmetry-adapted perturbation theory (SAPT) method which has been developed by Jeziorski

2

et al. 4,7,11,19,20 .


In this method the

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interaction energy is described by the sum

(1)

(1)

7

(2)

(2)

(2)

(2)

SAPT ∆Eint = Epol + Eexch + Eind + Eexch−ind + Edisp + Eexch−disp + . . .

where

(1)

Epol :

induction,

electrostatic,

(2)

Edisp :

(1)

Eexch :

dispersion and

rst-order exchange,

(2)

Eexch−disp :

(2)

Eind :

induction,

(2)

Eexch−ind :

(2)

exchange-

exchange-dispersion interaction energy contribu-

tion. The superscripts in Eq. (2) denote the order in the intermolecular interaction. While third-order SAPT contributions have been derived by Patkowski expansion of

∆Eint

et al. 21 ,

the perturbation

in Eq. (2) is conventionally truncated at second order. Since higher order

terms can become signicant in strongly interacting systems, these are typically estimated by adding the energy dierence

22,23

SAPT (1+2)

super δ (HF) = ∆Eint (HF) − ∆Eint

to the expansion in Eq. (2). Here, (HF) interaction energy and

super ∆Eint (HF)

SAPT (1+2)

∆Eint

(HF)

(3)

denotes the supermolecular Hartree-Fock

(HF) is the SAPT interaction energy where each of

the rst and second order terms in Eq. (2), excluding the intermolecular correlation energy terms, is computed on the HF level. When the SAPT method is combined with a supermolecular HF calculation through Eq. (2), the method may actually be interpreted as a hybrid supermolecular and intermolecular perturbation theory approach.

In order to improve the accuracy of the supermolecular

HF energy, both intramolecular and intermolecular correlation eects need to be accounted for. The rst correlated SAPT method that has been developed is the many-body SAPT (MB-SAPT) approach in which a double perturbation theory ansatz is used to expand the interaction energy both with respect to the intermolecular potential as well as the uctuation potentials of the monomers

7,2430

. Innite order summations of intramolecular correlation

eects have also been realised through coupled-cluster techniques (CC-SAPT), see Refs. 31 36. It was found that both MB-SAPT and CC-SAPT methods can deliver highly accurate

3



interaction energies

30,37

Page 4 of 58

.

However, many-body and coupled-cluster SAPT methods become unfeasible for large dimer systems due to the steep scaling behaviour of many-body perturbation and coupledcluster approaches with respect to the molecular size. In order to describe the interactions between extended systems using the SAPT method, therefore the SAPT0 method, in which intramolecular correlation interactions are neglected altogether, has become a popular alternative

3740

. It was observed in previous works, however, that intramolecular correlation

eects to each of the terms in Eq. (2) can be signicant and therefore may have a large impact on the accuracy of total interaction energies, see Refs. 7,2429. Consequently, the SAPT0 method, while being useful for analysing the binding mechanism in extended weakly bonded systems, can not be regarded as an accurate approach to describe intermolecular interactions. A more rigorous and, regarding the polarisation energy terms, potentially exact SAPT method for large systems is the DFT-SAPT method

4150

.

In this method the rst-order

SAPT terms are described by DFT (density functional theory) densities and density matrices

42

and the second-order terms by static and frequency-dependent response functions

computed with time-dependent density functional theory (TDDFT) response methods

44,45

.

Provided that the exact exchange-correlation (xc) potentials and kernels of the monomers were known, the DFT-SAPT method would be capable to yield 'exact' results for the polarisation energy terms, namely

(1)

(2)

Epol , Eind

and

(2)

Edisp .

This follows by construction: the external

potential to the KS pseudoparticle system is chosen such that the density is identical to the (exact) density of the many-body interacting system

51

. A corresponding relation holds

true also for the time-dependent case (within the linear response regime), i.e., knowledge of the time-dependet KS xc potential and/or kernel suces to determine exact response properties of the system

52

. In practice, however, both the xc potential and kernel need to be

approximated. Quasi-exact xc potentials, however, can be obtained from inversions of the KS equations using accurate ab-initio densities as input

4

5360


. XC potentials obtained in this

Page 5 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


way were employed in DFT-SAPT calculations of rare gas dimers

46,61

and it has been found

that for the helium dimer the total interaction potential better reproduced the quantum Monte-Carlo reference interaction potential than the CCSD(T) (coupled-cluster singles doubles with peturbative triples) potential

46

. The xc kernel chosen in this study was the ALDA

(adiabatic local density approximation) kernel, indicating that, for the systems considered, the form of the kernel has much less an impact on the accuracy for (exchange-)induction and dispersion energies than the potential. A similar nding was also made in a work by Gisbergen et al. regarding polarisabilities and excitation energies for a number of atoms and small molecules

62

(see also a related work by Petersilka et al.

63

). Almost exact xc potentials

from the ORPA (orbital-optimised random-phase approximation) method

64

combined again

with the ALDA xc kernel were used in a DFT-SAPT study of the interaction energies of a set of small dimers

65

taken from Ref. 42. It was shown that the agreement of the ORPA based

DFT-SAPT energies with CCSD(T) reference results was better than for the corresponding many-body SAPT method (employing second- and third-order intramonomer correlation contributions). In addition, Ref. 65 also presents DFT-SAPT interactions energies for the asymptotically corrected PBE0AC model xc potential for the corresponding dimers that exhibited a similar accuracy as the ORPA based approach.

This also supports the ndings

made for the individual interaction energy contributions using the PBE0AC potential in DFT-SAPT calculations

42,44,45

.

While the aforementioned DFT-SAPT studies were done in combination with local or hybrid (combining the ALDA with a Hartree-Fock response kernel) xc kernels, we would like to point out that KS response methods can principally also be systematically improved to achieve higher accuracies in the description of the electronic response in more challenging systems, e.g., extended conjugated molecules

6668

.

Insuciencies of KS response methods

involving local xc kernels, e.g. the underestimation of charge-transfer (CT) excitations or the overestimation of molecular polarisabilities of polyacene molecules, can mostly be attributed to the exclusion of exact exchange interactions. This can partially be remedied with the use

5



of hybrid-kernels

48

or range-separated kernels

using exact-exchange KS response methods

69,70

7175

Page 6 of 58

or could even completely be overcome by

.

In this regard it has recently been argued by Holzer

et al.

SAPT method using response functions calculated by the

that an alternative DFT based

GW

method through the Bethe-

Salpeter equation could be more suitable to describe intermolecular interaction energies than the original DFT-SAPT method, since standard TDDFT response methods are unable to describe long-range CT excitations

76

. In fact, a comparison of the interaction energies

for a range of small dimers from DFT-SAPT employing the PBE0AC xc potentials and (hybrid-)ALDA kernels and the corresponding

GW -SAPT

method indicated that the latter

approach can deliver higher accuracies when compared to accurate coupled-cluster reference values

76

(see, however, the Erratum to this work). Numerous studies of the performance of

the DFT-SAPT method employing combinations of asymptotically corrected xc potentials with standard local (or hybrid) xc kernels demonstrate, however, that the accuracy of the method is fairly good in many dierent types of dimer systems intermolecular interactions in extended supramolecular complexes

70,7785 86

even for describing

. Therefore, the question

arises whether the use of more advanced KS response approaches employing more accurate xc kernels for the description of the (dynamic) response properties of the monomers would inuence the performance of the DFT-SAPT method? In this work a new DFT-SAPT approach is presented and tested that is based on monomer response functions computed with exact-exchange Kohn-Sham response methods. The KohnSham orbitals and eigenvalues employed in this method, as in our earlier studies

78,80

, have

been computed with the localised PBE0AC (LPBE0AC) xc potentials for the monomers. Unlike in our previous works, however, the derivative-discontinuity shifts required for the asymptotic correction have been estimated by a new scheme that avoids the knowledge of the ionisation energy in advance of the KS monomer calculations. The results for this alternative DFT-SAPT method will be compared to the standard DFT-SAPT method, which utilises the ALDA xc kernel, for a number of benchmark systems, namely the S22 dimers

6


87

, the S66×8

Page 7 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dimers

88

and the IonHB dimers

89

.

It will be shown that, for the systems considered, the

average performance of the two DFT-SAPT schemes is fairly similar and that both methods can reproduce coupled-cluster reference interaction energies with a high accuracy. One can therefore deduce that deciencies of the ALDA xc kernel regarding long-range excitations are insignicant for the description of the response functions of the monomers contained in the data bases examined. This work is organised as follows: section

Method describes the methods used, including

a description of the calculation of the xc potential, the exact-exchange kernel and how the dispersion and exchange-dispersion energies are computed within the density-tting DFTSAPT implementation. Section calculations and section

Computational details

Results

describes technical details of the

compiles the results. Finally, section

Summary

gives a

summary and outlook.

Method Exchange-correlation potential The localised and asymptotically corrected PBE0 exchange-correlation (xc) potential, termed as LPBE0AC

78

, was used to compute the monomer eigenfunctions. The PBE0 xc potential

is a hybrid potential constructed from the PBE (Perdew-Burke-Ernzerhof ) exchange (PBEx) and correlation (PBEc) potential

PBE0 (r, r′ ) = vxc

with

γ

90

and the nonlocal exact exchange potential

91,92

:

3 δExPBE 1 −γ(r, r′ ) δEcPBE δ(r − r′ ) + + δ(r − r′ ) 4 δρ(r) 4 |r − r′ | δρ(r)

(4)

denoting the one-particle density matrix. To convert the PBE0 xc potential into a

local Kohn-Sham xc potential, the optimised eective potential (OEP) approach

7


93100

has


Page 8 of 58

been employed by solving the integral equation

∫

LPBE0AC ′ dr′ χ0 (r, r′ )vxc (r ) = −2

∑

ϕia (r)

ia

where

ϕi,j,k,... /ϕa,b,c,...

PBE0 ⟨ϕi |vxc |ϕa ⟩ εia

denote occupied/unoccupied molecular orbitals and

(5)

χ0

is the static

Kohn-Sham response function given by

′

χ0 (r, r ) = −4

occ ∑ virt ∑ ϕia (r)ϕia (r′ )

In Eqns. products

(6)

εia

a

i

(5) and (6) a short-hand notation is used for the occupied-unoccupied orbital

ϕi ϕa → ϕia

and the orbital energy denominator

εa − εi → εia .

Note also that

here and in the following the molecular orbitals are assumed to be real valued. Moreover, a closed-shell formalism will be used throughout. Utilising the auxiliary basis set OEP approach described in Ref. 99 (see also Refs. 95 97,101,102), Eq. (5) can be transformed into a linear equation system (LES) that can be solved eciently even for large molecular systems. However, in order to resolve the numerical instabilities which are common to nite basis-set OEP methods

99,100,102104

, the regularisation

method described in Ref. 105 has been used to solve the LES. This has been done by adding the Coulomb-metric matrix scaled by a factor of

3 × 10−4

to the coecient matrix of the

LES. The scaling factor was determined by minimising the deviations of the thus obtained xc potentials to the xc potentials determined with the balanced auxiliary basis sets of Ref. 99 for a number of small molecules. Moreover, both the charge and the HOMO conditions have been employed in the solution of the OEP equation, see Ref. 99. The charge condition ensures that the xc charge density integrates to

−1

like the exact xc potential

106

:

∫ dr ρxc (r) = −1

8


(7)

Page 9 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and guarantees that the xc potential correctly decays with a Coulombic behaviour of the asymptotic region. The HOMO condition constraints the matrix element

−1/r in

LPBE0AC ⟨ϕHOMO |vxc |ϕHOMO ⟩

of the highest occuppied orbital to be identical to the matrix element of the corresponding nonlocal potential

LPBE0AC PBE0 ⟨ϕHOMO |vxc |ϕHOMO ⟩ = ⟨ϕHOMO |vxc |ϕHOMO ⟩

(8)

A straight application of Eq. (8) would, however, yield HOMO orbital eigenvalues which reproduce the HOMO eigenvalues from a corresponding PBE0 calculation.

It is known,

however, that the HOMO orbital energies from continuum functionals (including also hybrid functionals) deviate from exact negative (vertical) ionisation potentials, unlike KS methods employing exact xc potentials

107,108

. This property, however, is a crucial condition to cor-

rectly describe the asymptotic decay of the molecular density in the asymptotic region

50,61

and can, in addition to the correct Coulombic shape (Eq. (7)), have a strong impact on the quality of intermolecular interaction energy contributions calculated within the DFTSAPT approach

4245

. Commonly this deciency of continuum functionals can be resolved

by shift-and-slice asymptotic correction schemes

107,109,110

. In these the bulk potential, de-

scribed by the continuum functional, is shifted by the derivative discontinuity

∆xc

and then

is connected to an asymptotically correct potential in the intermediate region in a seamless way. The derivative discontinuity

∆xc

is dened by the the dierence between the ionisation

potential and the negative HOMO energy of the continuum functional. The disadvantage of this shift-and-slice asymptotic correction schemes is, however, that ionisation potentials (IPs) need to be determined in advance by performing, e.g.,

∆SCF

calculations for the neutral molecule and its cation. In order to avoid this step, Dixon et al.

111,112

have derived an asymptotic correction method in which the IP is determined approx-

imately for a given functional using a linear t of the HOMO orbital eigenvalues to accurate IP data for a range of molecules. Since this is based on the observation that a strong linear

9



Page 10 of 58

correlation between Kohn-Sham HOMO eigenvalues for standard (hybrid-)GGA functionals and IPs exists, in this work an alternative approach will be used that is particularly suitable for hybrid xc potentials of the form of Eq. (4). Namely, in this work the HOMO eigenvalue will be constrained to be identical to

3 1 PBE0 εLPBE0AC = εEXX HOMO HOMO + εHOMO 4 4

(9)

to be employed in the HOMO constraint in the OEP method described above. Here,

εEXX HOMO

is the HOMO energy corresponding to the (nonlocal) exact exchange potential, i.e., is given by

1 2 NL εEXX HOMO = ⟨i| − ∇ + vnuc |i⟩ + ⟨i|vCoul |i⟩ + ⟨i|vx |i⟩ 2 where

vxNL

vnuc

denotes the nuclear attraction potential,

is the nonlocal exact exchange potential.

vCoul

is the Coulomb potential and

The orbital index

denote the index of the highest occupied orbital

(10)

i

in Eq. (10) here shall

calculated with the LPBE0AC potential.

This means that the scheme which underlies Eq. (9) does not require an additional selfconsistent calculation with the EXX or PBE0 potential, but can be carried out solely by using the LPBE0AC orbitals. The prefactors in Eq. (9) have been determined by minising the deviation to experimental IPs for Neon and H2 O, yieling: a.u. (H2 O) as obtained by Eq. (9) compared to

0.793

obtained by experimental measurements, respectively

(−)0.795 a.u.

a.u. and

113

0.464

(Ne) and(−)0.468

a.u. for Ne and H2 O

.

Table 1 contains the ionisation energies as determined by Eq. (9) for the molecules contained in the S22 benchmark data base by Hobza et al.

87

. For comparison, the second column

in the table contains ionisation energies as determined by the

∆SCF method using

the PBE

xc functional to describe the energies of the neutral molecule and the cation. It can be seen that with the interpolation scheme of Eq. (9), a fairly good agreement between (negative) LPBE0AC HOMO energies and the small mean absolute deviation of

∆PBE

0.02

ionisation energies is obtained, indicated by the

Hartree (∼0.5 eV) displayed in the last line of the

10


Page 11 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


table. In some cases, however, the deviation to the

∆PBE

values is signicantly larger than

the average deviation, e.g., for ethylene (0.06 Hartree) and HCN (0.04 Hartree).

Exact exchange Kohn-Sham kernel The nonadiabatic exact-exchange Kohn-Sham response kernel has been derived by Görling

114,115

and is given by

∫ fx (r1 , r2 , ω) =

−1 dr3 dr4 χ−1 0 (r1 , r3 , ω)hx (r3 , r4 , ω)χ0 (r4 , r2 , ω)

with

χ0 (r1 , r2 , iω) = −4

occ ∑ virt ∑ i

a

(11)

εia ϕia (r1 )ϕia (r2 ) εia + ω 2

(12)

describing the dynamic Kohn-Sham response function at an imaginary frequency frequency-dependent and nonlocal function

hx

in Eq. (11) is given by

iω .

The

75,114,115

[2] hx = h[1] x + hx ) ∑ [ −4εia εjb + 4ω 2 ( NL NL h[1] (r , r , iω) = (ij |ab ) + δ ⟨i|ˆ v − v ˆ |j⟩ − δ ⟨a|ˆ v − v ˆ |b⟩ 1 2 ab x ij x x x x (ε2ia + ω 2 )(ε2jb + ω 2 ) ij,ab ] −4εia εjb − 4ω 2 (ib|ja) ϕia (r1 )ϕjb (r2 ) (13) + 2 (εia + ω 2 )(ε2jb + ω 2 ) [ ] ∑ 4εja NL h[2] (r , r , iω) = ⟨i|ˆ v − v ˆ |a⟩ ϕ (r )ϕ (r ) + ϕ (r )ϕ (r ) 1 2 x ij 1 ja 2 ja 1 ij 2 x x ε (ε2 + ω 2 ) ij,a ia ja [ ] ∑ −4εib NL + ⟨i|ˆ vx − vˆx |a⟩ ϕab (r1 )ϕib (r2 ) + ϕib (r1 )ϕab (r2 ) (14) εia (ε2ib + ω 2 ) i,ab

The quantities

⟨r|ˆ vxNL − vˆx |s⟩

in Eqs.

matrix elements of the nonlocal

vˆxNL

(13) and (14) denote the dierences between the

and local exact exchange potential

vˆx .

Note that in

order to combine the kernel with general xc potentials, both the local and nonlocal exchange potentials need to be computed in terms of the orbitals that correspond to this general xc potential. The integrals

(rs|tu) in Eqs.

(13) and (14) denote two-electron Coulomb-repulsion

11



Page 12 of 58

integrals in the chemist's notation. In order to calculate the exchange kernel of Eq. (11) eciently, we follow the resolutionof-the-identity (RI) technique of Ref. 116 and introduce an orthonormalised auxiliary basis set

{gp , gQ , . . . }

labeled by the indices

P, Q, R, · · · .

When the auxiliary function set is or-

thonormalised over the Coulomb norm, then one can write

−1 Fx (ω) = X−1 0 (ω)Hx X0 (ω)

dening a matrix with the matrix elements

∫

(15)

dr1 dr2 gP (r1 )fx (r1 , r2 , ω)gQ (r2 )

and, corre-

spondingly

[X0 (ω)]P Q = ⟨e gP |χ0 (ω)|e gQ ⟩ with

∫ geP (r) =

The matrix

Hx

(16)

gP (r′ ) dr |r − r′ | ′

in Eq. (15), being the auxiliary basis set representation of

hx

of Eq. (13),

can be computed by (using the Einstein-summation convention over repeated indices)

[ ] [ [1] ] 1 Hx P Q = CP,ia λia (ω) 4CR,ia CR,jb − CS,ib CS,ja − CT,ij CT,ab λjb (ω)CQ,jb + 4 [ ] [ ] + CP,ia λia (ω) − ⟨a|∆vx |b⟩ λib (ω)CQ,ib + CP,ia λia (ω) − ⟨i|∆vx |j⟩ λja (ω)CQ,ja (17)

with

∆vx = vˆxNL − vˆx

and

εia + ω2

(18)

CP,ia = (P |ia)

(19)

λia (iω) = −4

12

ε2ia


Page 13 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


where

(P |ia)

denes a Coulomb-repulsion integral over the auxiliary function

occupied-virtual product density tion

ϕi ϕa .

Note that in Eq. (17) the second part

[2]

hx

gP

and the

of the func-

hx , depending on occupied-virtual orbital transformation terms ⟨i|∆vx |a⟩, is omitted.

It

has been found in previous works that these contributions usually have only a small impact on excitation energies

7375,117

or electron correlation energies

116,118

(when the

fx

response

kernel is used in conjunction with the adiabatic-connection uctuation-dissipation theorem approach).

Therefore, in this work the additional contribution

[2]

hx

to the kernel will be

neglected, assuming that the KS and HF orbitals can be transformed into each other via an occupied-occupied and virtual-virtual unitary transformation.

Dispersion energies In this section the computation of the dispersion energies using density tting techniques will briey be reviewed. For further details the references

47,48,78,119,120

The dispersion interaction energy between two systems the Casimir-Polder formula

(2) Edisp

with

1 =− 2π

χA and χB

∫

A

and

dω

is generally given by

1,121

∫

∞

B

may be consulted.

dr1 dr2 dr3 dr4 χA (r1 , r3 , iω)χB (r2 , r4 , iω)

0

1 1 |r1 − r2 | |r3 − r4 |

being the coupled response functions of the two monomers.

(20)

Using the

auxiliary function basis set formalism introduced above, these can be calculated by solving the Dyson-type equation

X(iω) = X0 (iω) + X0 (iω)W(iω)X(iω)

with

X

(21)

dening the coupled response function in the auxiliary function respresentation (cf.

Eq. (16)) and

W dening the interaction matrix (second functional derivative of the electron-

13



Page 14 of 58

electron interaction energy) that is given by

⟨ ⟩ 1 ⟨ ⟩ WP Q (iω) = P Q + P |fxc (iω)|Q r12

(22)

Since in this work the auxiliary functions are orthornormalised with respect to the Coulomb norm, the Coulomb kernel matrix in Eq. (22) is just identical to the unit matrix. With this,

W(iω) = 1 + Fxc (iω)

(23)

in this work. The solution to Eq. (21) is then given by

( )−1 X(iω) = 1 + X0 (iω)W(iω) X0 (iω)

(24)

involving just the inversion of a matrix of the dimension of the auxiliary function space. The dispersion energy in the density tting approximation can now be computed by

(2) Edisp

1 =− 2π

The integration over the frequency

ω

∫

∞

( ) dω Tr XA (iω)XB (iω)

in Eq. (25) is performed numerically in this work using

a 10-point Gauss-Legendre quadrature grid

122

.

Three dierent approximations to the xc kernel

ALDA fxc (r1 , r2 ) =

fxc

have been used in this work, namely:

LDA δ 2 Exc δ(r1 − r2 ) δρ(r1 )δρ(r2 )

TDEXX ATDEXX (r1 , r2 , 0) (r1 , r2 ) = fxc fxc ∫ −1 TDEXX (r1 , r2 , ω) = dr3 dr4 χ−1 fxc 0 (r1 , r3 , ω)hx (r3 , r4 , ω)χ0 (r4 , r2 , ω)

ALDA fxc

(25)

0

(26)

(27) (28)

is the adiabatic and local density approximation kernel, dened by the second func-

tional derivative of the LDA xc energy.

fxATDEXX is the adiabatic but nonlocal exact-exchange

14


Page 15 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Kohn-Sham kernel which is given by Eqs. (11), (13) and (14), but neglecting the frequency-

fxTDEXX

dependent terms. Finally,

is the complete exact-exchange Kohn-Sham kernel that is

both frequency-dependent and nonlocal, see also the previous section. Note that in previous works using the density-tting DFT-SAPT method always the ALDA response kernel (Eq. (26)) was employed. Results for this standard DFT-SAPT method will be presented in this work, too, for comparison.

Exchange-dispersion energies The coupled exchange-dispersion energy within the single-determinant and tions can be computed by

(2) Eexch−disp

= −2 ∑(

∑

AB Tia,jb

) ∑ ∑( ′ ′ ′ ′ ′ − (ia |jb )Sa′ b Sab′ + (i a|jb )Sib − 2(ia|jb )Si′ b Si′ b′ + a′ b′

∑

(i′ a|jb)Sij ′ Si′ j ′ +

i′ j ′

∑

B

(i′ a|j ′ b)Sij ′ Si′ j + 2

i′ j ′ A

A

(−2ω jb Saj ′ + ω j ′ b Saj )Sij ′ +

∑

B.

The integrals

(rs|tu)

The terms

A

ω

and

B

ω

potentials of both monomers. The coupled amplitudes

AB Tia,jb

i′ j ′ B

(−2ω ia Si′ b + (29)

A

and indices

j, b

labeling

in Eq. (29) are 2-external Coulomb-repulsion

integrals in the chemist's notation and the quantities the MO's of the two monomers.

(ia|j ′ b)Si′ j ′ Si′ j +

b′

labeling occupied/virtual MO's of monomer

indices of monomer

∑

i′

] ∑B ∑A − ω ia′ Sa′ b Saj − ω jb′ Sib Sab′ a′

i/a

∑

j′

ω i′ a Sib )Si′ j

with indices

i′ ,b′

)

(ia′ |j ′ b)Saj − 2(ia′ |jb)Saj ′ Sa′ j ′ −

a′ ,j ′

2

approxima-

25

[

ia,jb

S2

Srs

denote overlap integrals between

in Eq. (29) are the eective electric

Tia,jb

in Eq. (29) are dened as

∫ 1 ∑ ′ ′ ′ ′ B (i a |j b ) dω χA =− ia,i′ a′ (iω)χjb,j ′ b′ (iω) 8π i′ a′ ,j ′ b′

15


(30)


with

χA (r1 , r2 , iω) =

∑

Page 16 of 58

ϕia (r1 )χA ia,i′ a′ (iω)ϕi′ a′ (r2 )

(31)

ia,i′ a′ and for

T AB

χB

correspondingly. While the computational expense to calculate the amplitudes

can be signicantly reduced with the aid of the density-tting implementation described

in Ref. 78, the eort for computing energy (cf.

(2)

Eexch−disp

previous section).

(2)

Eexch−disp

is still much larger than for the dispersion

In order to further reduce the computational cost of the

term, therefore in Ref. 86 the coupled amplitudes in Eq. (30) have been replaced

by the uncoupled amplitudes scaled by a parameter dispersion energies for the S22×5 dimer systems

123

α

that was tted to coupled exchange-

:

(ia|jb) AB AB Teia,jb =α ≈ Tia,jb εia + εjb

(32)

It was found that the coupled exchange-dispersion energy could be reproduced with a very good accuracy with a scaling factor of

α = 0.686

due to a strong linear correlation between

the coupled and uncoupled exchange-dispersion energies (with t of

r2 = 0.9993).

Since the linear

α was performed for coupled amplitues calculated with the ALDA xc kernel (Eq. (26)),

in this work we have repeated the t also for exchange-dispersion energies calculated with the ATDEXX and TDEXX response kernels for a selection of systems from the S22 data base, see table 2. As can be seen therein, analogously to the ndings of our earlier study, the ATDEXX and TDEXX exchange-dispersion energies show a strong linear correlation to the uncoupled ones, yielding scaling factors of the

(2)

Eexch−disp

α = 0.696

and

α = 0.698,

values calculated with the ALDA xc kernel a value of

respectively. For

α = 0.715 is obtained for

the subset of dimers in table 2. While this value diers slightly from the result obtained by the t to the full S22×5 data base, for consistency (since DFT-SAPT interaction energies for the dierent response kernels shall be compared with each other hereafter) in this work only the scaling factors from table 2 will be utilised to approximate the coupled amplitudes through Eq. (32). Note also that dierences in the linear t of Eq. (32) may also be caused by

16


Page 17 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dierences of the xc potential of this work and the one used in Ref. 86 for the KS monomer calculations.

Computational details Intermolecular interaction energies with the DFT-SAPT method employing the LPBE0AC xc potential and ATDEXX, TDEXX and ALDA xc response kernels have been performed for the S22, S66×8 and IonHB benchmark data bases by Hobza

et al. 8789 .

The S22 data

base contains 22 dierent dimers including 7 hydrogen bonded structures, 8 dimers stabilised mainly by dispersion interactions and 7 structures with mixed attractive electrostatic and dispersion energy contributions

87

. Due to the fact that several interaction motifs, including

single hydrogen bonds and aromatic-aliphatic interactions, are strongly underrepresented or even missing in the S22 set of dimers, Hobza et al. have developed a larger set of 66 dierent dimers which give a more balanced description of dierent intermolecular types

88

comprising

23 hydrogen bonded structures, 23 dispersion dominated structures and 20 structures which can not be assigned to the former groups and are termed as Other in Ref. 88. The IonHB data base contains 15 dierent structures involving ionic hydrogen bonds

89

. This data base

has originally been designed to develop advanced corrections of hydrogen bonding and dispersion for semiempirical quantum chemical methods, see Ref. 89.

Both, in case of the

S66 and the IonHB data base each equilibrium dimer structure (obtained by MP2 geometry optimisations) has been altered by scaling the monomer-monomer distance with factors of

r/requi = 0.9, 0.95, 1.0, 1.05, 1.1, 1.25, 1.5

and

2.0

yielding, for each of the 66 respectively

15 dimers, a coarse sample of the potential energy curve. With this, unlike in the S22 benchmark set, it is possible to also assess the performance of a given quantum chemical method at nonequilibrium geometries. The reference interaction energies for the S22, S66 and IonHB data bases were calculated on the CCSD(T) level using basis set extrapolation techniques to estimate the complete basis set limit. These reference energies were taken from Refs. 124

17



Page 18 of 58

(S22), 88 (S66) and 89 (IonHB). (Exchange-)induction and (exchange-)dispersion energies have been calculated with the response methods described in the previous section. Further technical details, including a description of the estimation of the higher order interaction energy terms using the

δ (HF)

correction, are as described in Ref. 86. DFT-SAPT interaction energies have been performed using the aug-cc-pVTZ and augcc-pVQZ basis sets by Dunning

125128

using a dimer centered basis set

129

. The dispersion

and exchange-dispersion energies have been extrapolated to the complete basis set (cbs) limit with the two-point extrapolation formula by Bak et al.

130

. The cbs extrapolated DFT-SAPT

total interaction energies were then obtained by adding all other contributions calculated with the aug-cc-pVQZ basis set. The results for all total interaction energies and individual interaction energy contributions are compiled in the supporting information. Density-tting of two-electron integrals was used throughout, unless otherwise noted, utilising the aug-cc-pVX Z-JKFit tting basis sets

131

in the SCF monomer calculations and

the corresponding aug-cc-pVX Z-MP2Fit tting basis sets

132

in the SAPT calculations. The

aug-cc-pVX Z-MP2Fit basis sets have also been used as auxiliary basis sets in the OEP approach described in the previous section. Core electrons were kept frozen in the calculation of the (exchange-)dispersion energy. All calculations have been performed using a developers version of the Molpro program

133,134

,

except for the calculations of the CCD+ST(CCD) dispersion energies which have been done using the Psi4 quantum chemistry package

135

.

Results Dispersion energies In order to assess the accuracies of the dispersion energies of the dierent DFT-SAPT approaches of this work, we consider the nine dierent dimer systems oriented in 23 dierent

18


Page 19 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


conformations from Ref. 136. This set of dimer systems comprises both weakly bonded and hydrogen-bonded complexes and has also been used to test the performance of the DFTSAPT method in our earlier studies

42,44,45

.

As a reference we have computed CCD+ST(CCD) (coupled-cluster doubles including noniterative singles and triples excitations) dispersion energies as derived by Williams

31

.

Due to the high computational cost of this method the natural orbital truncation approach of Hohenstein

et al.

has been utilised, see Refs. 137,138. For this, the occupation number

−8 threshold value was set to 10 . The CCD+ST(CCD) dispersion energies obtained within this approximation are shown in the third column in table 3. In Ref. 137 it was observed that the error of the

(22)

Edisp (T )

triples term that is caused by the natural orbital (NO) truncation

scheme can be signicantly reduced by scaling this term by the ratio of the uncoupled dispersion energies computed exactly and within the NO approach. The correspondingly corrected CCD+ST(CCD) dispersion energies are displayed in the fourth column in table 3 and will serve as a reference in the following. While one can observe that these are all lower than the uncorrected values, the deviation between the corrected and uncorrected CCD+ST(CCD) dispersion energies is quite small, see bottom line in the table.

This indicates that the

threshold chosen for discarding virtual NO's in the calculations is adequate. The fth to seventh columns in table 3 show the DFT-SAPT dispersion energies as obtained with the LBPE0AC xc potential and the three dierent response kernels TDEXX, ATDEXX and ALDA. It can be seen that the TDEXX and ATDEXX values are fairly close to each other which indicates that the nonadiabatic contributions to the dispersion energy as described with TDEXX are relatively small. Compared to this, the dispersion energies from the ALDA xc kernel are systematically lower by -0.02 to -0.03 kcal/mol than the (A)TDEXX values on average. The total absolute errors of the dierent DFT-SAPT methods to the CCD+ST reference results are fairly small and range between 0.027 and 0.038 kcal/mol, see last but one line in table 3. While with this there is hardly any dierence in the performances of the exact-

19



Page 20 of 58

exchange response methods on the one hand and the ALDA response method on the other hand, it is, however, remarkable that with only one exception (the rst structure of the water dimer) both the ATDEXX and TDEXX dispersion energies are more close to the reference dispersion energies than the ones obtained with the ALDA xc kernel, see table 3.

This

indicates that an improved performance of DFT-SAPT employing exact-exchange response kernels over the standard approach might be expected also for total interaction energies, as will be investigated in the following sections.

S22 dimer systems The results for the DFT-SAPT interaction energies using the various xc response kernels are presented in table 4. In addition, the table also contains CCSD(T) reference interaction energies from Ref. 124 for comparison. As can be seen, all DFT-SAPT approaches perform fairly well, yielding mean absolute errors (mae) of 0.14 to 0.20 kcal/mol to the CCSD(T) reference energies, see last but one line in the table. A comparison between the ATDEXX and TDEXX interaction energies indicates that almost throughout the results obtained with the nonadiabatic kernel are more accurate. A decompostion of the total errors into the individual interaction groups exhibits that this dierence in the performance of the two methods mainly originates from the comparably large mae of 0.27 eV for the hydrogen bonded set of dimers for ATDEXX, see diagram in gure 1. In contrast to this, the dierences between the DFTSAPT interaction energies with the TDEXX and ALDA xc kernels are fairly small for all dimers of the S22 systems. The error decomposition displayed in gure 1 shows, however, that the DFT-SAPT[TDEXX] interaction energies are systematically more accurate than the DFT-SAPT[ALDA] interaction energies for the dispersion-dominated and mixed-type systems, supporting the results of the previous section for the bare dispersion energies. The errors for the individual group of complexes furthermore shows that (in contrast to the ATDEXX results) the errors of the interaction energiesfrom DFT-SAPT[TDEXX] are almost independent on the interaction energy type. In total both the DFT-SAPT[TDEXX]

20


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and DFT-SAPT[ALDA] methods yield fairly accurate interaction energies for the S22 dimer systems with mae's of 0.14 and 0.18 kcal/mol, which is close to the uncertainty of the reference values. It should be noted in addition that the results presented here for the DFT-SAPT[ALDA] interaction energies for the S22 dimers are clearly better than the ones reported in Ref. 86, see the last column in table 4. Since practically all technical details were chosen identical to our earlier work

86

, the reason for this can be attributed to the better performance of the

LPBE0AC xc potential used in the monomer calculations (see Method section) compared to the potential that was used in Ref. 86. While in Ref. 86, too, the PBE0 xc potential has been employed in the calculations, the asymptotic correction that was used to construct the xc potential was the GRAC scheme by Grüning

et al. 109

using ionisation energies computed

with the PBE functional. Moreover, the exact exchange contribution has been computed approximately in Ref. 86, utilising the Becke-Roussel-Johnson model exchange potential

139,140

.

A comparison of the DFT-SAPT[ALDA] interaction energies for the parallel displaced and T-shaped structures of the benzene dimer of this work and the corresponding values of our previous work

78

, which were obtained using LPBE0AC xc potentials constructed, too, util-

ising the GRAC asymptotic correction scheme

109

but an exact exchange potential in the

PBE0 hybrid scheme, has shown a fairly good coincidence.

S66×8 dimer systems

Total interaction energies The performance of the dierent DFT-SAPT approaches for the S66×8 dimer systems is summarised in the diagrams in gure 2.

In the upper diagram in the gure, the mean

absolute errors (mae's) to the CCSD(T) reference energies is broken down into the dierent monomer distances, referenced by the scaling factors displayed on the abscissa axis in the diagram.

Since the total magnitude of the interaction energies for the 66 dimer systems

generally decrease with the distance of the two monomers, as expected also the average errors

21



Page 22 of 58

for the various DFT-SAPT methods become smaller for larger scaling factors. However, even for the dimer structures with the smallest distances the average errors for the DFT-SAPT methods do not exceed 0.3 kcal/mol, which is clearly better than for MP2, a number of dispersion corrected DFT functionals as well as Kohn-Sham based RPA approaches, see e.g. Refs. 65,141,142. A comparison of the performances of the dierent DFT-SAPT methods again shows that the DFT-SAPT[TDEXX] and DFT-SAPT[ALDA] methods are very similar in performance. Stronger dierences between the two methods can only be observed for the short-distance dimers, where the average error of the DFT-SAPT[ALDA] method is almost 0.1 kcal/mol larger than with the DFT-SAPT[TDEXX] method, see upper diagram in gure 2. Interestingly, here it can be seen that the corresponding DFT-SAPT[ATDEXX] method performs best among the three approaches, yielding a mae of about 0.15 kcal/mol only. The lower diagram in gure 2 shows the average errors with respect to the dierent interaction types of the S66 data base. Similar to the ndings for the S22 dimer systems, see the last section, the DFT-SAPT[TDEXX] and DFT-SAPT[ALDA] methods are almost equal in performance for the hydrogen-bridged systems while in case of the dispersion-domated systems and the "Other" group of complexes the DFT-SAPT[ALDA] method yields a comparably larger error of about 0.12 kcal/mol while the errors obtained by DFT-SAPT[TDEXX] are 0.1 and 0.05 kcal/mol for the two interaction types, respectively, see gure 2.

The

most balanced description for the dierent interaction motifs is obtained with the DFTSAPT[TDEXX] method which yields mae's of 0.1 (HydBond), 0.1 (Disper) and 0.05 kcal/mol (Other) for the three interaction-type groups. In total all three DFT-SAPT methods, including the DFT-SAPT[ATDEXX] method, exhibit a high accuracy for the S66 dimer systems with average errors of about 0.1 kcal/mol to the CCSD(T) reference interaction energies. Individual interaction energy curves for a selection of two dimers from the hydrogen bonded group of complexes, three dimers from the dispersion-dominated group of complexes and two dimers from the mixed-type complexes are presented in the gures 3-5. In all poten-

22


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tials in the diagrams it can be seen that the deviations to the CCSD(T) interaction potentials become fairly small for the largest monomer distances. This indicates that the long-range interaction contributions, mainly represented by the electrostatic and dispersion interaction terms, are very accurate for the dierent DFT-SAPT methods. The most remarkable result that is highlighted in the potentials in gures 3-5 is, however, that all three DFT-SAPT approaches exhibit also a reasonable accuracy in the short-distance region. This indicates that the

S 2 -approximation,

which only takes into account single electron pair exchanges in the

exchange interaction energy terms, is still accurate enough at

r/requi = 0.9

scaled distances

for the dimer systems considered.

IonHB dimer systems The results for the interaction energies for the S22 and S66×8 dimer systems presented in the previous sections have revealed that the accuracies of the DFT-SAPT[ALDA] and DFTSAPT[TDEXX] methods are generally very similar to each other. In this section we now investigate whether a more clear-cut behaviour in the performance of both approaches can be found for strongly interacting dimer systems involving charged monomers. Such systems can be regarded as challenges for intermolecular perturbation theory methods in general, because the interactions are typically much stronger than for neutral dimer complexes. In this respect it should be noted, however, that strong errors in the interaction energy due to a truncation of the interaction energy at second order can be corrected by the is added to the

(1+2)

Eint

δ (HF)

contribution, which

energies to capture third and higher order polarisation interactions

on the Hartree-Fock level, see Eq. (3). To test the various DFT-SAPT methods for charged dimer systems, interaction energies for the ionic hydrogen-bonded benchmark systems from Hobza as IonHB data base

89

et al.,

hereafter refered to

, were calculated. The average interaction energy at the equilibrium

structures for this data base is larger by almost a factor of four compared to the S66 data

avg base (Eint (IonHB)

= −19.5

kcal/mol compared to

23

avg Eint (S66) = −5.5


kcal/mol). The mean


Page 24 of 58

absolute errors for the dierent DFT-SAPT methods at the dierent intermolecular distances are shown in gure 6.

A comparison with the corresponding diagram for the S66 dimer

systems in gure 2 indicates generally larger average errors for the IonHB dimers than for the S66 dimers. Due to the much larger total interaction energies of the former dimer systems, however, the relative errors are similar for DFT-SAPT for the two benchmark systems, see also the supporting information. A comparison of the performances of the dierent DFTSAPT approaches shows that at the equilibria and towards larger monomer distances (except for the largest distance of

r/requi = 2.0) the DFT-SAPT[ALDA] interaction energies are more

accurate than the (A)TDEXX DFT-SAPT methods, see gure 6. Note, however, that the average errors of about 0.3 kcal/mol of DFT-SAPT[ATDEXX] and DFT-SAPT[TDEXX] are signicantly lower than the errors which were obtained with the SAPT0 and SAPT2 methods for this benchmark data base, see Ref. 143. Figure 7 displays individual potential energy curves for two examples from the IonHB dimer systems, namely for the methylammonium-methanol and imidazolium-formaldehyde dimers. One can see that in both cases the DFT-SAPT interaction energies reproduce the CCSD(T) reference energies with a very good accuracy in the long-range region and even in the short-range part of the potentials. At the equilibria of the two dimers deviations of to

+0.5

+0.2

kcal/mol to the reference values are observed for the various DFT-SAPT methods.

Comparison to other quantum chemistry methods The gures 8 and 9 show the mean absolute deviations of the interaction energies from dierent quantum chemistry methods to the extrapolated CCSD(T) reference energies for the S22 and S66 dimer systems. The dierently coloured bars in the diagrams in the two gures distinguish the errors of the dierent method types, namely of various coupled-cluster (CC) methods compiled from Refs. 88,124,144, of various DFT methods compiled from Refs. 124,141,142,145, of various many-body perturbation theory methods (MBPT) compiled from Refs. 88,120,124,144, of various RPA methods compiled from Refs. 65,146151 and those of

24


Page 25 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


dierent SAPT methods compiled from Refs. 37,86. The acronyms of the dierent underlying methods are shown in the legend in the header of the gures and are ordered from the best performing to the worst performing method. A more detailed assignment of the benchmark results to the literature reference is given in the supporting information.

Note that the

diagram in gure 8 contains both the DFT-SAPT[ALDA] benchmark for the S22 systems of this work (denoted as DFTSAPT-ALDA1 in the legend) as well as our earlier benchmark using a dierent xc potential as described above (denoted as DFTSAPT-ALDA2 in the legend)

86

.

As one can see in gure 8, both the performance of the DFT-SAPT method (comparing the mae's of DFTSAPT-ALDA1 and DFTSAPT-ALDA2) as well as the performance of dispersion corrected DFT methods depends strongly on the underlying xc functional or potential which is used.

Compared to this, the inuence of the xc kernel that is used in

the DFT-SAPT method has a much weaker impact on the interaction energies, see gures 8 and 9. The results from the DFT-SAPT method employing the LPBE0AC xc potential used in this work are among the best benchmark results reported in the literature for the S22 and S66 dimers, having errors in the range of 0.1 to 0.2 kcal/mol which approaches the accuracy of the reference interaction energies themselves, see the labelled bars in the two gures. For the S66 dimers only the coupled-cluster based SCS-MI-CCSD method exhibits a better performance than the DFTSAPT-TDEXX method of this work, see gure 9. It can be concluded that the DFT-SAPT method employing the LPBE0AC xc potential and exact-exchange kernels describes intermolecular interaction energies for a wide range of dierent interaction motifs with a very high accuracy at a moderate cost compared to wave function correlation methods. Namely, the scaling behaviour of density-tting DFT-SAPT, including the new variants presented in this work, is While this is larger than the scaling of

N3

N 5 with respect to the molecular size N .

that can be achieved with standard dispersion-

corrected DFT methods, the computational cost of DFT-SAPT is similar to that of the cheapest wave function methods including second-order perturbation theory and random-

25



phase approximation electron correlation methods.

Page 26 of 58

A comparison of the timings for the

calculation of the interaction energies for the DFT-SAPT and the MP2 method for large supramolecular complexes has been given in Ref.

86

.

Summary A new DFT-SAPT method which employs Kohn-Sham monomer response functions calculated using the exact-exchange KS kernel has been presented and tested in this work. The results for the interaction energies for this new method have been compared to results obtained by a corresponding DFT-SAPT method which utilises the ALDA xc kernel in the response calculations. In both cases a new asymptotically corrected xc potential has been used which is based on derivative discontinuity shifts that were estimated by ionisation energies obtained by interpolating the HOMO energies from the PBE0 and exact-exchange KS methods. It was found that with this new model potential the interaction energies of DFT-SAPT[ALDA] for the S22 dimer systems strongly improved as compared to our earlier results

86

.

The performance of the dierent DFT-SAPT methods has been tested for the S22, S66×8 and IonHB dimer systems.

A comparison of the results for DFT-SAPT[ALDA]

and DFT-SAPT[TDEXX] to accurate CCSD(T) interaction energies indicated that both methods deliver a high accuracy, irrespective of the type of interaction.

It was found

that for uncharged and charged hydrogen-bonded dimers the ALDA and TDEXX methods yield similar total interaction energies, while in case of dispersion-dominated and mixedtype complexes the TDEXX approach is generally more accurate.

We have also tested

a DFT-SAPT approach employing the adiabatic exact-exchange kernel.

While with this

method the monomer response calculations are more ecient, because the kernel matrix elements need not be calculated at each grid point of the frequency integration grid, the corresponding DFT-SAPT[ATDEXX] interaction energies are slightly less accurate than the

26


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DFT-SAPT[TDEXX] ones. The accuracy of the dispersion energies yielded by the dierent response methods has been tested for a selection of small dimer systems for which highly accurate coupled-cluster calculations could be performed for comparison.

It has been shown that the dispersion

energies from the (A)TDEXX response methods are almost systematically smaller and also closer to the reference energies than the ones obtained with the ALDA xc kernel. Compared to this, a replacement of the ALDA xc kernel by GGA xc kernels has the opposite eect, namely yields slightly larger dispersion energies. This has been demonstrated for two rare gas dimers and the water and CO dimer by Misquitta

et al. 152 .

This work may have shed some light on the role of the DFT response kernel on the performance of the DFT-SAPT method for the description of intermolecular interaction energies. It has been demostrated that for small to intermedium-sized dimers (of sizes as large as the adenine-thymine base pair) the deviations between interaction energies obtained by standard TDDFT (employing local xc kernels) and exact-exchange DFT (employing nonadiabatic and nonlocal xc kernels) response methods are small and that with both methods highly accurate interaction energies can be obtained, provided that accurate xc potentials are used in the underlying KS calculations of the monomers. It can be anticipated, however, that the DFT-SAPT[TDEXX] method will more clearly outperform the standard DFT-SAPT[ALDA] method for systems where standard TDDFT response methods fail, e.g., for systems involving extended conjugated molecules

66,153155

. This will be explored in a future work.

Supporting Information Available Total intermolecular interaction energies and interaction energy contributions for the S22, S66×8 and IonHB dimer systems for the dierent DFT-SAPT methods. available free of charge via the Internet at

http://pubs.acs.org/.

27


This material is


Page 28 of 58

Acknowledgement Financial support of this work through the DFG (Deutsche Forschungsgemeinschaft) priority Program No. SPP1807 ("Control of London dispersion interactions in molecular chemistry") is gratefully acknowledged.

References (1) McWeeny, R.

Methods of molecular quantum mechanics ;

Academic Press, 1996.

(2) Hayes, I. C.; Stone, A. J. An intermolecular perturbation theory for the region of moderate overlap.

Mol. Phys. 1984, 53,

83.

(3) Jeziorski, B.; Kolos, W. On symmetry forcing in the perturbation theory of weak intermolecular interactions.

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Comput. Mol. Sci. 2012, 2, (136) Heÿelmann, A.;

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(137) Hohenstein, E.; Sherrill, C. D. Ecient evaluation of triple excitations in symmetryadapted perturbation theory via second-order Møller-Plesset perturbation theory natural orbitals.

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(138) Parrish, R. M.; Hohenstein, E. G.; Sherrill, C. D. Tractability gains in symmetryadapted perturbation theory including coupled double excitations: CCD+ST(CCD) dispersion with natural orbital truncations.

J. Chem. Phys. 2013, 139,

174102.

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(144) Burns, L. A.; Marshall, M. S.; Sherrill, C. D. Appointing silver and bronze standards for noncovalent interactions: A comparison of spin-component-scaled (SCS), explicitly correlated (F12), and specialized wavefunction approaches.

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(145) Grimme, istry,

S.

GMTKN30

kinetics,

and

-

a

database

non-covalent

for

general

interactions.

main

2011;

,

group see

thermochemhttp://toc.uni-

muenster.de/GMTKN/GMTKN30/GMTKN30main.html.

(146) Toulouse, J.; Zhu, W.; Savin, A.; Jansen, G.; Angyan, J. G. Closedshell ring coupled cluster doubles theory with range separation applied on weak intermolecular interaction.

J. Chem. Phys. 2011, 135,

084119.

(147) Eshuis, H.; Furche, F. Basis set convergence of molecular correlation energy dierences within the random phase approximation.

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084105.

(148) Heÿelmann, A. Random-phase-approximation correlation method including exchange interactions.

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012517.

(149) Heÿelmann, A.; Görling, A. On the Short-Range Behavior of Correlated Pair Functions from the Adiabatic-Connection Fluctuation-Dissipation Theorem of DensityFunctional Theory.


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(150) Ren, X.; Rinke, P.; Scuseria, G. E.; Scheer, M. Renormalized second-order perturbation theory for the electron correlation energy: Concept, implementation, and benchmarks.

Phys. Rev. B 2013, 88,

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(153) Champagne,

B.;

Perpete,

E.

A.;

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Jacquemin,

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Gisbergen,

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J.

A.;

Baerends, E. J.; Soubra-Ghaoui, C.; Robins, K. A.; Kirtman, B. Assessment of Conventional Density Functional Schemes for Computing the Dipole Moment and (Hyper)polarizabilities of Push-Pull Pi-Conjugated Systems.

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4755.

(154) Körzdörfer, T.; Sears, J. S.; Sutton, C.; Bredas, J.-L. Long-range corrected hybrid functionals for -conjugated systems: Dependence of the range-separation parameter on conjugation length.

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204107.

(155) Heÿelmann, A. Polarisabilities of long conjugated chain molecules with density functional response methods: The role of coupled and uncoupled response.

2015, 142, 164102.

45


J. Chem. Phys.


Page 46 of 58

Tables Table 1: for

Ionisation energies for the molecules from the S22 set of dimers.

∆PBE

were obtained by

∆SCF

The values

calculations (def2-TZVP basis set) and the values for

LPBE0AC correspond to the negative HOMO orbital eigenvalues (aug-cc-pVTZ basis set). The last line in the table contains the mean absolute deviation of the (negative) Kohn-Sham orbital energies to the

∆SCF

results. Values are in a.u. molecule

∆PBE

LPBE0AC

H2 O

0.466

0.468

NH3

0.401

0.391

CH4

0.519

0.504

HCN

0.502

0.459

C2 H 2

0.418

0.378

C2 H 4

0.391

0.349

HCOOH

0.409

0.428

CHONH2

0.371

0.394

C6 H 6

0.340

0.313

C5 H5 ON

0.306

0.289

C5 H6 N2

0.297

0.281

C8 H 7 N

0.279

0.259

Pyrazine

0.333

0.367

Phenole

0.307

0.292

Uracile

0.343

0.345

Adenine

0.321

0.318

mae

0.020

46


Page 47 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: Uncoupled and coupled exchange-dispersion energies for some systems from the S22 data base (aug-cc-pVTZ basis set). The last two lines in the table display the linear tting coecients of Eq. (32) and the correlation coecients describing the linear correlation between the coupled and uncoupled values. Energies are in kcal/mol. dimer

uncoupled

ATDEXX

TDEXX

ALDA

(H2 O)2

0.75

0.57

0.57

0.58

(NH3 )2

0.56

0.44

0.44

0.45

(CH4 )2

0.11

0.09

0.09

0.09

(HCOOH)2

3.88

2.72

2.73

2.74

(CHONH2 )2

2.89

2.06

2.07

2.08

C2 H 4 · · · C2 H 2

0.29

0.21

0.21

0.21

(C2 H4 )2

0.38

0.30

0.30

0.31

0.69564

0.69792

0.714614

0.9997

0.9997

0.9996

α r2

47



Page 48 of 58

Table 3: Dispersion energies from CC-SAPT and DFT-SAPT for the dimer systems from Ref. 136.

The triples contributions of the CCD+ST(CCD) dispersion energies have been

computed within a natural orbital representation to truncate the virtual space.

see Refs.

137,138.

The fourth column contains the CCD+ST(CCD) dispersion energies using the (22) 137 −8 scaled Edisp (T) contributions as described in . An occupation tolerance of 10 was used to discard virtual natural orbitals. All energies were calculated using the aug-cc-pVQZ basis set. The last but one line shows the mean absolute errors and the last line the percental deviations to the CCD+ST(CCD) reference results. (energies in kcal/mol)

dimer

struct.

CC-SAPT (2) (22) ϵdisp (CCD)+Edisp [S(CCD)] (22) (22) +Edisp [T(CCD)] +Edisp [T(CCD)](est.)

DFT-SAPT

LPBE0AC TDEXX ATDEXX ALDA

He2

5.5 a0

-0.045

-0.045

-0.050

-0.050

-0.052

Ne2

6.0 a0

-0.126

-0.126

-0.128

-0.128

-0.132

Ar2

7.0 a0

-0.641

-0.645

-0.632

-0.632

-0.658

NeAr

6.5 a0

-0.278

-0.280

-0.279

-0.279

-0.289

NeHF

1

-0.417

-0.421

-0.432

-0.431

-0.438

2

-0.230

-0.232

-0.237

-0.236

-0.245

ArHF

(H2 )2

(HF)2

(H2 O)2

3

-0.207

-0.208

-0.210

-0.210

-0.218

1

-0.512

-0.515

-0.515

-0.514

-0.522

2

-0.305

-0.307

-0.305

-0.305

-0.317

3

-0.280

-0.281

-0.276

-0.275

-0.286

1

-1.113

-1.119

-1.198

-1.197

-1.224

2

-0.656

-0.661

-0.704

-0.703

-0.732

3

-0.380

-0.382

-0.398

-0.397

-0.411

1

-1.938

-1.951

-2.034

-2.027

-2.061

2

-1.725

-1.738

-1.838

-1.833

-1.874

3

-1.449

-1.457

-1.509

-1.503

-1.525

4

-1.116

-1.127

-1.135

-1.132

-1.172

1

-8.335

-8.378

-8.251

-8.215

-8.383

2

-2.727

-2.749

-2.745

-2.733

-2.792

3

-0.944

-0.953

-0.967

-0.963

-0.983

4

-2.302

-2.324

-2.321

-2.313

-2.378

5

-1.664

-1.677

-1.722

-1.714

-1.696

6

-1.420

-1.436

-1.433

-1.428

-1.490

mae

0.009

0.027

0.028

0.038

|∆|

0.71

2.65

2.59

4.70

48


Page 49 of 58 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4: Intermolecular interaction energies for the S22 complexes (in kcal/mol). The mean absolute errors (mae) and relative deviations (|∆|) to the CCSD(T) reference values are shown in the last two lines of the table. The last column contains the DFT-SAPT[ALDA] interaction energies from Ref. 86 computed with the LPBE0AC(BRJ) xc potential for comparison. All values are extrapolated to the complete basis set limit. type

dimer

CCSD(T)

DFT-SAPT LPBE0AC ATDEXX

HydBond

(NH3 )2 (C2h ) (H2 O)2 (Cs ) (HCOOH)2 (C2h ) (CHONH2 )2 (Uracil)2 (C2h ) C5 H5 ON-C5 H6 N2 AT (WC)

Disper

(CH4 )2 (D3d ) (C2 H4 )2 (D2d ) Bz-CH4 (C3 ) (Bz)2 (C2h ) (Pyrazine)2 (Cs ) (Uracil)2 (C2 ) Indole-Bz (stacked) AT (stacked)

Mixed

C2 H4 -C2 H2 (C2v ) Bz-H2 O (Cs ) Bz-NH3 (Cs ) Bz-HCN (Cs ) Bz-Bz (C2v ) Indole-Bz (T-shaped) (Phenole)2

TDEXX

LPBE0AC(BRJ) ALDA

ALDA

−3.17 −5.02 −18.80 −16.12 −20.69 −17.00 −16.74

−3.04 −4.79 −18.58 −15.90 −20.29 −16.70 −16.27

−3.06 −4.79 −18.74 −16.03 −20.48 −16.89 −16.50

−3.13 −4.88 −18.66 −15.93 −20.37 −16.88 −16.50

−3.19 −5.00 −17.71 −15.32 −19.04 −15.96 −15.26

−0.53 −1.50 −1.45 −2.62 −4.20 −9.74 −4.59

−0.52 −1.48 −1.43 −2.57 −4.46 −9.83 −4.23

−0.53 −1.50 −1.45 −2.66 −4.55 −10.06 −4.39

−0.56 −1.54 −1.55 −3.00 −4.83 −10.20 −4.84

−0.58 −1.56 −1.55 −3.06 −4.33 −9.43 −4.72

−11.66 −1.51 −3.29 −2.32 −4.55 −2.71 −5.62 −7.09

−10.99 −1.60 −3.24 −2.28 −4.48 −2.64 −5.44 −6.57

−11.33 −1.61 −3.27 −2.30 −4.51 −2.69 −5.53 −6.68

−11.69 −1.65 −3.38 −2.40 −4.67 −2.83 −5.72 −6.78

−10.87 −1.60 −3.25 −2.33 −4.33 −2.58 −5.15 −6.29

0.20 3.15

0.14 2.30

0.18 4.24

0.48 5.64

mae |∆|

49



Figures ATDEXX TDEXX ALDA

mae [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 50 of 58

0.2

0.1

0

all

HydBond

Disper

Mixed

Figure 1: S22 dimer systems: mean absolute errors of DFT-SAPT interaction energies using the ATDEXX, TDEXX and ALDA xc kernels to the CCSD(T)/cbs reference energies. The labels on the abscissa axis correspond to the hydrogen bridged systems subgroup (HydBond), the dispersion dominated systems subgroup (Disper) and the mixed complexes.

50


Page 51 of 58

mae [kcal/mol]

ATDEXX TDEXX ALDA 0.2

0.1

0

total 0.90

0.95

1.00

1.05

1.10

1.25

1.50

2.00

0.2

ATDEXX TDEXX ALDA mae [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.1

0

total

HydBond

Disper

Other

Figure 2: S66×8 dimer systems: mean absolute errors of DFT-SAPT interaction energies employing the ATDEXX, TDEXX and ALDA xc kernels to the extrapolated CCSD(T) values. In the upper diagram the numbers on the abscissa refer to scaling factors for the intermolecular distance at the minimum, see text. In the lower diagram the errors are broken down by the three dierent interaction types: hydrogen bonded interactions (HydBond), dispersion interactions (Disper) and other interaction types. Both diagrams show the total averaged errors for all 528 systems for comparison.

51



0

0

DFT-SAPT[ATDEXX] DFT-SAPT[TDEXX] DFT-SAPT[ALDA] CCSD(T)

-1


-2 -4

Eint [kcal/mol]

-6

Eint [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 52 of 58

-2

-3

-8 -10 -12 -14 -16

-4

-18

-5

Figure 3:

1

1.2

1.4 r/requi

1.6

1.8

2

-20

1

1.2

1.4 r/requi

1.6

1.8

Interaction energy curves for two dimers from the hydrogen bonded group of

complexes of the S66×8 database.

Left: water dimer representing a system with a single

O· · · H hydrogen bond. Right: acetic acid dimer representing a system with a cyclic hydrogen bond.

52


2

Page 53 of 58



0

Eint [kcal/mol]

0

Eint [kcal/mol]

-1

-2

-1

-2

-3

-3 1

1.2

1.4 r/requi

1.6

0

-1

Eint [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.8

2

1

1.4 r/requi

1.2

1.6

1.8


-2

-3

-4

-5 1

1.4 r/requi

1.2

1.6

1.8

2

Figure 4: Interaction energy curves for three dimers from the dispersion-dominated group of complexes of the S66×8 database. Top left: parallel displaced benzene dimer representing a

π -stacked

system. Top right: cyclopentane dimer representing a system with aliphatic dis-

persion interactions. Bottom: pentane-uracile dimer representing a system with interactions.

53


π -aliphatic

2


0

0

-1



Eint [kcal/mol]

-1

Eint [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 54 of 58

-2

-3

-4

-3 1

Figure 5:

-2

1.2

1.4 r/requi

1.6

1.8

2

1

1.2

1.4 r/requi

1.6

1.8

Interaction energy curves for two dimers from the mixed interaction group of

complexes of the S66×8 database.

Left:

methanol-benzene dimer representing a system

with OH· · · π interactions. Right: ethene-N-methylacetamide dimer which is characterised both by hydrogen bond and CH· · · π interactions.

54


2

Page 55 of 58

ATDEXX TDEXX ALDA

mae [kcal/mol]

0.5 0.4 0.3 0.2 0.1 0

total 0.90

0.95

1.00

1.05 1.10 r/requi

1.25

1.50

2.00

Figure 6: IonHB dimer systems: mean absolute errors of DFT-SAPT interaction energies using the ATDEXX, TDEXX and ALDA xc kernels to the CCSD(T)/cbs reference energies. The labels on the abscissa axis correspond to the hydrogen bridged systems subgroup (HydBond), the dispersion dominated systems subgroup (Disper) and the mixed complexes.

-8 -10

-6


-8

-12

Eint [kcal/mol]

Eint [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-14 -16 -18


-10

-12

-14

-20 -16

-22

Figure 7:

1

1.2

1.4 r/requi

1.6

1.8

2

1

1.2

1.4 r/requi

1.6

1.8

Interaction energy curves for two dimers from the IonHB database.

methylammonium· · · methanol dimer. Right: imidazolium· · · formaldehyde dimer.

55


2

Left:


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

DW-CCSD(T**)-F12 CCSD(T**)-F12b SCS(MI)-CCSD SAPT2+(CCD)δMP2 ACFDT[ALDA] B2PLYP-D DFTSAPT-TDEXX CCSD(T**)-F12a RPAX2 MP2C DFTSAPT-ALDA1 MP2.5 NLDFT DFTSAPT-ATDEXX SAPT2+3(CCD)δMP2 BLYP-D3

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

SCS-CCSD EXX-RPA M06-D3 B2PLYP-D3 DSD-BLYP-D3 LC-ωPBE-D3 MPW1B95-D3 B2GPPLYP-D3 MPWB1K-D3 revPBE-WXhole PW6B95-D3 RPA+rSE B3LYP-D3 M062x-D3 TPSSh-D3 revPBE38-D3

33 34 35 36 37 38 39 40 41 42 43 44 45 46

SOSEX srRPA-SO2 XYG3 DFTSAPT-ALDA2 PBE-D3 SOSEX-rSE PBE0-D3 CAM-B3LYP-D3 MP3 rsRPA SCS-MP2 MP2 CCSD RPA[PBE]

1 0.9

CC methods DFT methods MBPT methods RPA methods SAPT methods

0.8 0.7

mae [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 56 of 58

0.6 this work 0.5 0.4 0.3 0.2 0.1 0

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

method Figure 8: S22 dimer systems: mean absolute errors of coupled-cluster (red bars), DFT (green bars), many-body perturbation theory (blue bars), random-phase approximation (yellow bars) and dierent SAPT methods (orange bars) to the extrapolated CCSD(T) interaction energies from Ref. 124.

56


Page 57 of 58

1 2 3 4 5 6 7 8 9 10 11

SCS-MI-CCSD DFTSAPT-TDEXX MP2C DFTSAPT-ATDEXX MP2.5 DFTSAPT-ALDA1 RPAX2 SCS-CCSD LC-wPBE-D3 NLDFT M06-2X-D3

12 13 14 15 16 17 18 19 20 21 22

RPA+rSE-diag M06-L-D3 BLYP-D3 DW-MP2 SCS-MI-MP2 SOSEX B97-D3 DSD-BLYP B2PLYP-D3 rPT2 B2GP-PLYP-D3

23 24 25 26 27 28 29 30 31 32

TPSS-D3 RPA+rSE revPBE-D3 MP2 MP3 RPA B3LYP-D3 CCSD PBE-D3 SCS-MP2

1 0.9

CC methods DFT methods MBPT methods RPA methods SAPT methods

0.8 0.7

mae [kcal/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.6 this work

0.5 0.4 0.3 0.2 0.1 0

0

1

2

3

4

5

6

7

8

9

10

11 12

13

14

15

16

17

18

19

20

21 22

23

24

25

26

27

28

29

30

31 32

33

method Figure 9:

S66 dimer systems (equilibrium structures):

mean absolute errors of coupled-

cluster (red bars), DFT (green bars), many-body perturbation theory (blue bars), randomphase approximation (yellow bars) and dierent SAPT methods (orange bars) to the extrapolated CCSD(T) interaction energies from Ref. 88.

57



TOC Graphic

58


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DFT-SAPT Intermolecular Interaction Energies Employing Exact

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