Bond energy decomposition analysis for subsystem density functionaltheoryS. Maya Beyhan, Andreas W. Götz, and Lucas Visscher Citation: J. Chem. Phys. 138, 094113 (2013); doi: 10.1063/1.4793629 View online: http://dx.doi.org/10.1063/1.4793629 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i9 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 138, 094113 (2013)

Bond energy decomposition analysis for subsystem densityfunctional theory

S. Maya Beyhan, Andreas W. Götz,a) and Lucas Visscherb)

Amsterdam Center for Multiscale Modeling, VU University Amsterdam, De Boelelaan 1083, 1081 HVAmsterdam, The Netherlands

(Received 25 October 2012; accepted 14 February 2013; published online 7 March 2013)

We employed an explicit expression for the dispersion (D) energy in conjunction with Kohn-Sham(KS) density functional theory and frozen-density embedding (FDE) to calculate interaction energiesbetween DNA base pairs and a selected set of amino acid pairs in the hydrophobic core of a smallprotein Rubredoxin. We use this data to assess the accuracy of an FDE-D approach for the calculationof intermolecular interactions. To better analyze the calculated interaction energies we furthermorepropose a new energy decomposition scheme that is similar to the well-known KS bond formationanalysis [F. M. Bickelhaupt and E. J. Baerends, Rev. Comput. Chem. 15, 1 (2000)], but differs in theelectron densities used to define the bond energy. The individual subsystem electron densities of theFDE approach sum to the total electron density which makes it possible to define bond energies interms of promotion energies and an explicit interaction energy. We show that for the systems consid-ered only a few freeze-and-thaw cycles suffice to reach convergence in these individual bond energycomponents, illustrating the potential of FDE-D as an efficient method to calculate intermolecularinteractions. © 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4793629]

I. INTRODUCTION

Intermolecular interactions play a pivotal role in struc-ture formation of biomolecular systems and govern processeslike protein folding, molecular recognition, and stacking ofnucleobases.1–3 With the ever-increasing computational re-sources, explicit study of the corresponding interaction en-ergies by means of quantum mechanical calculations comeswithin reach. Calculation of interaction energies for biologi-cal systems with highly accurate ab initio methods is, how-ever, only possible for benchmark purposes4–8 as routineapplication is still hampered by the steep scaling of com-putational costs with system size. Density functional theory(DFT)9 scales better and is very successful in treating H-bonding but the available approximations to the exchange-correlation (XC) functional typically describe dispersion in-teractions rather poorly.10–13

One possibility to improve this behaviour is to move to-wards truly nonlocal density functionals (vdW-DF14, 15), butmost approaches rather combine a standard XC functionalwith an additional treatment of dispersion interactions (for re-cent papers see Refs. 16–19). An elegant way to do so20 is touse symmetry-adapted perturbation theory (SAPT)21 in whicha DFT description of monomers is combined with an explicittreatment of intermolecular interactions. More approximate,but popular due to their ease of implementation and compu-tational efficiency, is the class of DFT-D methods in whichan approximate expression based on the limiting C6R−6 termof the dispersion interaction is used.22–27 DFT-D is nowa-

a)Also at San Diego Supercomputer Center, University of California SanDiego, 9500 Gilman Drive MC0505, La Jolla, California 92093, USA.

b)Author to whom correspondence should be addressed. Electronic mail:l.visscher@vu.nl.

days the most widely used approach to the dispersion prob-lem in biomolecular systems and various DFT-D implementa-tions have been reported,22–24, 28–34 following pioneering workof in particular Grimme.23, 28, 35 Examples of successful treat-ments of biologically relevant systems can, e.g., be found inRefs. 16 and 36.

Even though DFT calculations are much cheaper thancorrelated wave-function based approaches, it is still very ex-pensive to carry out supermolecular calculations for biologi-cal systems. A convenient approach for modelling biologicalsystems is therefore the use of subsystem methods in whichonly a region of interest is treated at a high level of theory andthe environment is dealt with at a lower level of theory.37–42

The frozen-density method, proposed by Wesolowski andWarshel,43 is particularly attractive because this subsystemtheory is a reformulation of DFT that is in principle exact.The FDE approach is often used to calculate molecular prop-erties of solvated systems44–46 or, in its generalization to time-dependent DFT,47, 48 to describe local electronic excitationsand couplings between such excitations.49–52 In the presentcontext we are interested in the calculation of interaction en-ergies of weakly interacting systems in which the method isknown to perform rather well.53–56 This is a first step towardsa more comprehensive treatment in which also solvent effectsare included in the analysis.57–59

II. THEORY

Several methods have been devised to partition the den-sity of a complex system into two or more chemically mean-ingful components. In this paper we focus on FDE, which isthe simplest form of subsystem DFT in which two subsys-tem densities are each constrained to keep an integer number

0021-9606/2013/138(9)/094113/10/$30.00 © 2013 American Institute of Physics138, 094113-1

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094113-2 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

of electrons. An alternative is partition density functionaltheory60, 61 in which fractional occupations for a fixed num-ber of subsystems follow from the constraints that the globalpartitioning potential is universal for all subsystems. Whilethis allows for more flexibility in the partitioning when deal-ing with strongly interacting systems we prefer integer oc-cupations as this makes it easier to treat selected subsystemswith wave function based methods.62 Within subsystem DFTit is possible to define a 3-partitioning scheme in which theregion between two strongly interacting subsystems is con-tained in a model system constructed by combining two cap-ping groups.63 In this approach no orbital information fromthe other subsystems is used in the calculation of the ac-tive subsystem. Other methods that allow for partitioning ofstrongly interacting subsystems typically use either orbital ordensity matrix information for the subsystems. Energy de-composition schemes appropriate for such methods have beendevised by Fedorov and Kitaura64 and Su and Li.65

A. Frozen density embedding

FDE is a subsystem formulation66 of DFT and in mostcases the density is partitioned into two subdensities that eachcorrespond to an integer number of electrons. The total elec-tron density ρtot(r) is then given by

ρ(tot)(r) = ρ(1)(r) + ρ(2)(r). (1)

The DFT energy for this two-partitioning is written as

E[ρ(1), ρ(2)] =2∑

i=1

E(i)[ρ(i)] + Eint[ρ(1), ρ(2)], (2)

where E(i)[ρ(i)] is the KS-DFT energy of subsystem i:

E(i)[ρ(i)] = Ts[ρ(i)]

+∫

v(i)nuc(r)ρ(i)(r)dr

+1

2

∫ρ(i)(r)ρ(i)(r ′)

|r − r ′| drdr ′

+Exc[ρ(i)]. (3)

Here, v(i)nuc(r) is the electrostatic potential due to the nuclei in

subsystem i and Exc[ρ(i)] is the XC energy of subsystem i. Theelectronic interaction energy Eint between the two subsystemsis

Eint[ρ(1), ρ(2)] =

2∑i �=j

∫v(i)

nuc(r)ρ(j )(r)dr

+∫

ρ(1)(r)ρ(2)(r)

|r − r ′| drdr ′

+T nads [ρ(1), ρ(2)]

+Enadxc [ρ(1), ρ(2)], (4)

where the nonadditive contribution to the kinetic energyT nad

s [ρ(1), ρ(2)] and the nonadditive contribution to the XC en-

ergy Enadxc [ρ(1), ρ(2)] are given by

T nads [ρ(1), ρ(2)] = Ts[ρ

tot] −2∑

i=1

Ts[ρ(i)] (5)

and

Enadxc [ρ(1), ρ(2)] = Exc[ρ tot] −

2∑i=1

Exc[ρ(i)]. (6)

These terms can be evaluated once explicit density func-tionals for the XC energy, Exc[ρ], as well as for the kinetic en-ergy (KE) of the non-interacting reference KS system, Ts[ρ]are chosen. The latter is only used for the contribution tothe interaction energy, since the kinetic energy of the sub-systems can be evaluated from the KS orbitals. It is further-more convenient to combine the nuclear repulsion energy be-tween nuclei belonging to different subsystems with the twoelectrostatic terms of the electronic interaction energy, givingthe total electrostatic interaction energy EES. To obtain the or-bitals of the subsystems, a set of coupled KS-like equations isderived by minimizing the total energy functional in Eq. (2)with respect to the electron density ρ(1) of the first subsystem(1) while keeping the electron density of the other subsystem(2) frozen. The resulting equation reads[

− ∇2

2+ vKSCED

eff [ρ(1), ρ(2)](r)

]φ(1)(r) = εiφ

(1)(r), (7)

with the effective potential given by

vKSCEDeff [ρ(1), ρ(2)](r) = vKS

eff [ρ(1)](r) + vembeff [ρ(1), ρ(2)](r),

(8)where vKS

eff [ρ(1)](r) is the usual KS effective potential of sub-system 1

vKSeff [ρ(1)](r) = v(1)

nuc(r)

+∫

ρ(1)(r′)

|r − r ′ |d r′

+δExc[ρ]

δρ

∣∣∣ρ=ρ(1)(r)

, (9)

containing, respectively, the nuclear potential, the Coulombpotential, and the XC potential evaluated with only the densityof subsystem 1. The effect of subsystem 2 is described bythe embedding potential vemb

eff [ρ(1), ρ(2)] that depends on bothdensities and reads

vembeff [ρ(1), ρ(2)](r) = v(2)

nuc(r) +∫

ρ(2)(r′)

|r − r ′ |d r′

+ δExc[ρ]

δρ

∣∣∣∣ρ=ρ(tot)(r)

− δExc[ρ]

δρ

∣∣∣∣ρ=ρ(1)(r)

+ δTs[ρ]

δρ

∣∣∣∣ρ=ρ(tot)(r)

− δTs[ρ]

δρ

∣∣∣∣ρ=ρ(1)(r)

,

(10)

where v(2)nuc(r) denotes the external potential due to the nuclei

of subsystem 2 and ρ(tot) = ρ(1) + ρ(2) is the electron densityof the whole system.

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094113-3 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

If the KE density functional Ts[ρ] were exact, super-molecular KS-DFT results could be reproduced exactly bythe subsystem calculation. This requires two additional con-ditions on the fixed density ρ(2)(r): the remaining density ρ(1)

= ρ(tot) − ρ(2) should be non-negative everywhere in spaceand this density should be non-interacting vs-representable sothat it can be obtained via KS-DFT.67, 68 Since the first condi-tion is difficult to fulfill exactly by simple trial densities, oftena so-called “freeze-and-thaw” (FT) procedure69 is introducedin which the subsystem densities are adjusted in an iterativefashion. More important, however, are deficiencies in the bestknown approximations to the exact KE functional. These areparticularly apparent for ligand-metal bonds in coordinationcomplexes56, 70–72 and better approximations need to be devel-oped in order to allow for quantitative accuracy. On the otherhand, the FDE scheme is able to handle even strong hydro-gen bonds68 and can also describe weakly interacting systemswell.56, 73–76 Due to error cancellations, FDE can sometimesappear to even outperform KS-DFT with an appropriate com-bination of functionals for EXC and Ts. Often used choices areto employ the local density approximation (LDA) for both theXC77 and the KE78, 79 functional or to use the PW91 general-ized gradient approximation (GGA)80 functional in combina-tion with the PW91K KE functional.81, 82 The latter combina-tion works well for π -π stacked base pairs.76

B. Dispersion correction for the interaction energy

In an earlier study, we computed the interaction energiesof a wide range of systems to further assess the performanceof various kinetic energy functionals.56 Given the satisfac-tory reproduction of KS data for typical biomolecular inter-actions we now aim to add dispersion corrections and movetowards a quantitative description of interaction energies innon-covalently bound systems. In our view, this new FDE-D scheme should be regarded as a computationally efficientapproximation to KS-DFT, reproducing in the ideal case KS-DFT-D. The dispersion correction is in this context viewed asa correction to the XC interaction energy that has no influenceon the error in the kinetic-energy part.

This viewpoint allows for the use of dispersion correc-tions “borrowed” from KS-DFT. In this work we will employthe Grimme-2006 correction28 that is given as

Eintdisp = −s6

N1at∑

μ=1

N2at∑

ν=1

Cμν

6

R6μν

(1 + ed(1− Rμν

Rr)) (11)

in which the usual dispersion expression is multiplied by adamping function (with a fixed value d = 20) to reduce theinteraction energy when the sum of the van der Waals radii Rr

is smaller than the interatomic distance Rμν . The formula de-pends furthermore on atomic dispersion coefficients C

μν

6 andan overall scaling factor s6 that has a specific value for eachXC density functional. Note that we here define the disper-sion energy as a correction to the interaction energy, whichmeans that we need to restrict the sum over atoms to the num-ber of atoms in each fragment (Ni

at ). It is of course possibleto add also an intramonomer dispersion correction to the XCenergy of a fragment, but such a term would cancel out in thecalculation of the FDE interaction energy.

C. Bond energy decomposition

A second goal of the current work is to introduce an en-ergy decomposition scheme that can provide more insight intrends in the interaction energies. This is inspired by the well-known Morokuma scheme83 that is used to analyze bond for-mation in KS theory for two84, 85 or more subsystems.64, 86 Inthis analysis, given the KS orbitals and electron densities ρ

(1)frag

and ρ(2)frag of the two isolated fragments at the dimer geometry,

the bond energy Ebond is defined as

Ebond = �Eelstat + �EPauli + �Eoi, (12)

where �Eelstat is the electrostatic interaction calculated usingthe unperturbed densities of the fragments, �EPauli is the en-ergy change caused by orthogonalization of fragment wavefunctions, and �Eoi arises from the recombination of the or-thogonalized fragment orbitals to form the supermolecular KSwave function. The repulsive Pauli contribution can be furtherdecomposed into a change in kinetic energy (�T0) and into achange of potential energy �VPauli:

�EPauli = �T 0 + �VPauli. (13)

In this KS bond formation analysis, the orthogonalizationof the fragment wave functions thus plays a crucial role. Thisorthogonalization pushes density out of the bonding regiontowards the nuclei which raises the kinetic energy but lowersthe potential energy. The net effect is a strongly positive en-ergy that can be used to quantify the steric repulsion betweenthe fragments.

Within FDE theory fragment orbitals remain non-orthogonal and provide subsystem densities that sum to thetotal density (Eq. (1)). In order to analyze the bond energywe may thus keep the �Eelstat term, but need to replace the�EPauli and �Eoi terms that are based on the orthogonalizedorbitals. We propose to rewrite the bond energy Ebond as

Ebond =2∑

i=1

E(i)prom[ρ(i)] + Eint[ρ

(1), ρ(2)], (14)

where Eint[ρ(1), ρ(2)] is the interaction energy between the twosubsystems as given by Eq. (4) and E(i)

prom[ρ(i)] will be calledthe promotion energy of subsystem i:

E(i)prom = �E

(i)H + �E(i)

xc + �T (i)s , (15)

where �E(i)xc , �T (i)

s , and �E(i)H refer to the differences in XC,

kinetic, and Hartree energies induced by the change from thefragment ρ

(i)frag density to the fully interacting subsystem den-

sity ρ(i). This energy term is identical to the “internal strainenergy” defined by Wesolowski et al.,87 but we prefer thename promotion energy to avoid confusion with the strain en-ergy associated with the change of geometry in a bond forma-tion process.

Due to the nonlinear dependence on the density, the XCand KE component of the promotion energy need to be calcu-lated as a difference:

�E(i)xc [ρ(i)] = Exc[ρ(i)] − Exc

[ρ

(i)frag

], (16)

�T (i)s [ρ(i)] = Ts[ρ

(i)] − Ts

[ρ

(i)frag

]. (17)

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094113-4 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

The Hartree term (in which the nuclear attraction andelectron repulsion terms are summed) can also be calculatedusing the density difference �ρ(i) = ρ(i) − ρ

(i)frag, which im-

proves numerical accuracy if a numerical integration schemeis employed for its calculation:

�E(i)H [ρ(i)] =

∫v(i)

nuc(r)�ρ(i)dr

+∫

ρ(i)frag(r)�ρ(i)(r ′)

|r − r ′| drdr ′

+1

2

∫�ρ(i)(r)�ρ(i)(r ′)

|r − r ′| drdr ′. (18)

This decomposition of the bond energy does depend onthe final partitioning of the total electron density into subsys-tem densities and in principle requires an additional criteriumto make the partitioning unique. Because T nad

s is responsiblefor the errors relative to the supermolecular DFT approach,a possible strategy is to choose a partitioning in which thiscontribution to the interaction energy vanishes. This is alsothe idea behind the exact embedding scheme proposed re-cently by Manby et al.88 in which localized orbitals of sub-systems are kept orthogonal to each other by level-shiftingtechniques. One thereby eliminates the non-additive KE com-ponent of the interaction energy at the expense of increasingthe promotion energies. Another choice is to minimize thepromotion energies, and attribute the bond energy as much aspossible to an explicit interaction energy. We suspect that thecurrent procedure of applying FT cycles starting from the iso-lated fragment densities will typically give results close to thischoice.

The analysis of the bond energy in terms ofpromotion and interaction energies can provide insightin the effect of the FT procedure which is typically inter-preted as describing the environment polarization. In theapplications described below we therefore calculate thechanges in each individual component in each FT cycle. Totest this approach, we chose molecules in which hydrogenbonds and dispersion interactions dominate.89 Such systemscan be readily treated with the current realizations of FDE.We selected in this study the DNA base pairs in the BP8/05data set90 of Truhlar, which we also used in our earlier work,56

and the Rubredoxin (Rd) protein for which accurate referencedata are available from the work of Vondrášek et al.91, 92 Inorder to assess the efficiency of FDE, we will also discussbriefly the computational cost of the FDE and reference KScalculations.

III. COMPUTATIONAL DETAILS

All calculations were performed with the AmsterdamDensity Functional (ADF) program.93–95 We used the TripleZeta plus Polarization (TZP) basis set of the ADF basis setlibrary which is a triple-ζ valence/double-ζ core all-electronSlater basis augmented with one set of polarization functions.The numerical integration parameter of ADF was set to an ac-curacy of 10 digits while for the self-consistent field (SCF)procedure the ADF default setting was used. We used the

BLYP XC functional which is equivalent to the Becke96 gra-dient correction and the Lee–Yang–Parr97–99 correlation cor-rection in both the KS-DFT as well as the FDE calculations.Since ADF uses Slater-type basis functions the Coulomb con-tribution to the interaction energy (Eq. (4)) is calculated us-ing a fitted density. To be able to correct for the fit error inthis term, all FDE calculations were done using a full, super-molecular integration grid. More details on this implemen-tation of the Coulomb energy for FDE can be found in ourearlier work.56 The dispersion contribution was evaluated us-ing the 2006 Grimme correction28 for both KS-DFT and FDEcalculations. This dispersion correction is added to the totalbonding energy with the global scaling factor 1.20 for theBLYP XC functional. All the KS-DFT energies were cor-rected for the basis set superposition error (BSSE) with thecounterpoise technique.100 To setup and execute all our calcu-lations and then to retrieve the data, we used PyADF101 whichis a scripting framework for quantum chemistry implementedin the Python102 programming language.

A. BP8/05 data set

Structure of the different base pair dimers were takenfrom the supplementary data of Ref. 90. For the monomerswe thereby took the structure of the base pair in the dimer.For the FDE calculations we used the TW02103 non-additivekinetic-energy functional. We employed both a supermolecu-lar (global) expansion basis denoted as FDE(s), in which thebasis functions of all subsystems are used for the expansionof the KS orbitals of the active subsystem and a monomolec-ular basis denoted as FDE(m), in which basis functions lo-cated only on the active subsystem are used to expand itsKS orbitals. FDE(s) enables one to make a rigorous com-parison to supermolecular KS-DFT whereas FDE(m) is usedin practice and does not suffer from BSSE. Both KS andFDE(s) energies were corrected for BSSE with the counter-poise technique.100 Unless otherwise noted, we optimised thesubsystem densities from both FDE(m) and FDE(s) calcula-tions in 5 FT cycles. Throughout the text below, FT(i) with i= 1, 2· · · , 5 denotes the number of employed freeze-and-thawcycles.

B. Rubredoxin

The geometry of Rd is determined by X-ray crystal-lography (PDB code 1RB9) with high resolution (0.95 Å),and therefore, geometry optimisation was not necessary. Weadopted the approximation employed earlier by Vondrášeket al.91, 92 where they excised the hydrophobic-core aminoacid residues from the experimental structure and partitionedthe whole cluster into two distinct clusters, which are namedafter the central residues F30 and F49. They further frag-mented each cluster into chemically distinct neutral pairsof amino acids that are modelled as methylated amino acidresidues.92 The central F30 and F49 residues interact withfive (F49, K46, L33, Y13, and Y4) and seven (C39, C6,F30, K46, V5, W37, and Y4) amino acids, respectively(see Figure 1). We analyzed the interaction energies for this

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094113-5 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

FIG. 1. Interacting residues in clusters F30 and F49 of Rd.

selected set of interacting pairs using the coordinates as givenin the work of Vondrášek et al.91 We employed the BLYPXC functional and the TW02 kinetic-energy functional. Wereport here only the results of FDE(m) calculations sincewe had SCF convergence issues for most of the amino acidpairs with FDE(s) calculations. Similar convergence prob-lems with the FDE(s) scheme were encountered in our earlier

work56 where we showed that the availability of the full ba-sis set for FDE(s), allows the electron density to probe regionswhere errors in the kinetic energy functional make the embed-ding potential too attractive.104 As a consequence, the elec-tron density redistributes to yield strongly overlapping sub-system densities that an approximate functional is not able todescribe.

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094113-6 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

TABLE I. BP8/05 data set—WC G· · · C and WC A· · · T: Convergence behaviour of bond energy contributions with FDE(m). FT(i) with i = 1, 2· · · , 5 denotesthe number of freeze-and-thaw cycles with promotion energy and its components for G, C, A, and T (see Eq. (15)). WC G· · · C and WC A· · · T denote theinteraction energy between two subsystems (see Eq. (4)) and Ebond, the bond energy (see Eq. (14)). KS is BSSE corrected. All in kcal/mol.

WC G WC C WC G· · · C

�EH �Exc �Ts Eprom �EH �Exc �Ts Eprom EES Enadxc T nad

s Eint Ebond

FT(1) 2.263 −9.011 10.979 4.231 13.764 −7.447 0.700 7.017 −65.309 −23.121 49.633 −38.797 −27.550FT(2) 11.207 −9.659 6.362 7.910 16.761 −7.357 −1.247 8.157 −70.922 −23.289 49.938 −44.273 −28.207FT(3) 12.408 −9.633 5.830 8.245 17.004 −7.349 −1.404 8.251 −71.375 −23.307 49.972 −44.710 −28.214FT(4) 12.117 −9.630 5.788 8.275 17.022 −7.348 −1.418 8.255 −71.410 −23.308 49.974 −44.744 −28.214FT(5) 12.121 −9.630 5.784 8.276 17.024 −7.348 −1.419 8.257 −71.413 −23.308 49.974 −44.747 −28.214KS −25.453

WC A WC T WC A· · · T

�EH �Exc �Ts Eprom �EH �Exc �Ts Eprom EES Enadxc T nad

s Eint Ebond

FT(1) −6.461 −6.418 15.036 2.157 8.832 −4.321 −1.057 3.454 −39.184 −17.419 37.359 −19.244 −13.634FT(2) −1.921 −6.602 12.317 3.795 10.422 −4.183 −2.276 3.963 −41.918 −17.628 37.880 −21.665 −13.907FT(3) −1.483 −6.575 11.986 3.928 10.540 −4.169 −2.369 4.002 −42.126 −17.652 37.939 −21.839 −13.909FT(4) −1.447 −6.573 11.961 3.940 10.548 −4.168 −2.376 4.003 −42.141 −17.654 37.944 −21.851 −13.908FT(5) −1.447 −6.573 11.960 3.939 10.549 −4.168 −2.377 4.004 −42.142 −17.654 37.944 −21.852 −13.910KS −11.723

IV. RESULTS AND DISCUSSION

A. Convergence of bond energy contributions

1. BP8/05 data set

In Tables I and II, we show the convergence behaviour ofthe bond energy (split up according to Eqs. (14) and (15) asdescribed in Sec. II C) for the Watson-Crick (WC) hydrogen-bonded base pairs and for the A· · · T dimer as a representativeof the stacked base pairs, respectively. It can be seen that thelargest changes in all energy contributions occur during thefirst three cycles of the FT process. After FT(3), changes aresmall and at the end of FT(5) both FDE(m) and FDE(s) cal-culations are fully converged.

As expected for this type of partitioning of the density,the non-additive kinetic energy term, T nad

s , is strongly pos-itive for both the stacked and hydrogen-bonded base pairs.

It is always larger than the negative non-additive XC inter-action so that these non-classical interactions result in anoverall positive contribution to the interaction energy. Thiscounterbalances the strongly negative electrostatic attractionbetween the fragments. The promotion energy of the frag-ments will always be positive with the magnitude of the en-ergy terms a measure for the polarisation of the fragmentwhen brought from non-interacting to fully interacting. Inthis case the individual components can be negative as wellas positive. This is a difference with the Morokuma analy-sis and is a consequence of the fact that orbitals of differentsubsystems are non-orthogonal and sum to the total density.Whereas the Morokuma scheme proceeds from the orthogo-nalised fragments to the overall density via an orbital interac-tion step, this density deformation towards the bond region isalready accounted for in the FDE fragment densities and the

TABLE II. BP8/05 data set—A· · · T: Convergence behaviour of bond energy contributions with FDE(m) and FDE(s). FT(i) with i = 1, 2· · · , 5 denotes thenumber of freeze-and-thaw cycles with promotion energy and its components for A and T (see Eq. (15)). A· · · T denotes the interaction energy between twosubsystems (see Eq. (4)) and Ebond, the bond energy (see Eq. (14)). FDE(s) and KS is BSSE corrected. All in kcal/mol.

A T A· · · T

�EH �Exc �Ts Eprom �EH �Exc �Ts Eprom EES Enadxc T nad

s Eint Ebond

FDE(m)FT(1) 7.738 1.731 −9.110 0.358 16.029 2.591 −18.137 0.482 −11.391 −10.343 20.366 −1.368 −0.528FT(2) 7.147 1.515 −8.247 0.414 16.332 2.627 −18.447 0.512 −11.568 −10.377 20.427 −1.518 −0.592FT(3) 7.106 1.502 −8.189 0.419 16.344 2.629 −18.460 0.513 −11.575 −10.378 20.429 −1.524 −0.592FT(4) 7.104 1.501 −8.186 0.419 16.343 2.629 −18.460 0.511 −11.575 −10.378 20.429 −1.524 −0.594FT(5) 7.104 1.501 −8.186 0.419 16.345 2.629 −18.460 0.514 −11.575 −10.378 20.429 −1.524 −0.591FDE(s)FT(1) 8.121 2.251 −9.723 0.648 17.177 3.319 −19.744 0.752 −13.561 −11.092 22.295 −2.358 −0.958FT(2) 8.240 2.224 −9.720 0.743 17.516 3.369 −20.101 0.784 −13.851 −11.190 22.493 −2.549 −1.022FT(3) 8.206 2.214 −9.670 0.749 17.529 3.371 −20.115 0.785 −13.862 −11.193 22.499 −2.557 −1.023FT(4) 8.203 2.213 −9.668 0.748 17.527 3.370 −20.115 0.782 −13.862 −11.194 22.499 −2.557 −1.026FT(5) 8.204 2.213 −9.668 0.749 17.530 3.371 −20.115 0.786 −13.862 −11.194 22.499 −2.557 −1.023KS 3.202

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094113-7 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

TABLE III. Rubredoxin—F49· · · V5 and F30· · · Y4: Convergence behaviour of bond energy contributions with FDE(m). FT(i) with i = 1, 2· · · , 5 denotesthe number of freeze-and-thaw cycles with promotion energy and its components for F30 and F49, promotion energy for Y4 and V5 (see Eq. (15)). F30· · · Y4and F49· · · V5 denote the interaction energy between two subsystems (see Eq. (4)) and Ebond, the bond energy (see Eq. (14)). KS is BSSE corrected. All inkcal/mol.

F49 V5 F49· · · V5

�EH �Exc �Ts Eprom Eprom EES Enadxc T nad

s Eint Ebond

FT(1) −4.517 −3.069 9.073 1.487 0.434 −11.016 −7.407 18.538 0.116 2.037FT(2) −4.507 −3.069 9.077 1.501 0.480 −11.078 −7.409 18.543 0.056 2.037FT(3) −4.505 −3.069 9.076 1.502 0.482 −11.081 −7.409 18.543 0.053 2.037FT(4) −4.505 −3.069 9.077 1.503 0.484 −11.081 −7.409 18.543 0.053 2.040FT(5) −4.505 −3.069 9.076 1.503 0.483 −11.081 −7.409 18.543 0.053 2.039KS 3.314

F30 Y4 F30· · · Y4

�EH �Exc �Ts Eprom Eprom EES Enadxc T nad

s Eint Ebond

FT(1) 2.544 0.456 −2.560 0.450 0.364 −3.297 −2.863 8.532 2.371 3.156FT(2) 2.513 0.456 −2.519 0.450 0.381 −3.300 −2.859 8.525 2.365 3.197FT(3) 2.512 0.456 −2.517 0.450 0.381 −3.300 −2.859 8.524 2.365 3.196FT(4) 2.510 0.455 −2.517 0.449 0.382 −3.300 −2.859 8.524 2.365 3.196KS 3.767

resulting energy changes are therefore contained in the pro-motion energies and the non-classical components of the in-teraction energy.

For the more strongly bound WC base pairs (Table I)the deviation from the isolated fragment densities that resultsfrom the FT-procedure is larger than for stacked base pairs(Table II) but in both cases the bond energies remain ratherclose to the starting values. The fact that the electrostatic com-ponent of the interaction energies becomes more negative inthe course of FT cycles, can be interpreted as mutual polariza-tion of the fragments by the FT procedure. This polarizationmakes the interaction term more negative, but the effect onthe bond energy is minor as this is compensated by an almostequally large increase of the promotion energy.

In the FDE(m) approach charge-transfer between the sub-systems is limited by the range of the monomer basis set andthe original partitioning of the density is kept by definition. Inthe FDE(s) approach each monomer calculation is carried outin the full super molecular basis allowing for complete flex-ibility in the reproduction of the Kohn-Sham reference cal-culations. As already noted in our earlier study,56 this oftenleads to convergence problems due to the limited accuracyof the kinetic energy density function approximations. Forstacked dimers it is possible to converge the FDE(s) calcula-tions. These results are displayed in Table II and agree ratherwell with the FDE(m) calculations, with the increased flex-ibility leading to slightly more negative bond energies. Re-sults for other base pairs, including dispersion corrections, aregiven in Table IV and show the same trend of slightly morenegative FDE(s) values as compared to the FDE(m) values.For the stacked complexes convergence was fast, also with theFDE(s) expansion, except for the case of the antiparallel con-formation of the cytosine dimer for which in the first FT iter-ation the Kohn-Sham wave function was not fully convergedafter 100 iterations. This convergence problem disappeared inthe next FT iteration and fully converged densities could beobtained.

2. Rubredoxin

In Table III, we show the convergence behaviour of thebond energy (split up according to Eqs. (14) and (15) as de-scribed in Sec. II C) for the amino acid pairs F49· · · V5 asa representative of the cluster around F49 and for F30· · · Y4representing the cluster around F30. These residues are solelybound by dispersion interactions and DFT, without disper-sion correction, yields a positive (repulsive) bond energy thatshould be reproduced by the FDE approximation.

Similar to the DNA base pairs, the energies quickly con-verge in the FT cycles. In this case, the deviation relative tothe starting situation is so small that the bond energy obtainedin the first iteration is already sufficiently precise. Like in thebase pairs, the polarisation in the freeze-thaw cycles lowersthe interaction energy at the expense of increased promotionenergies. The net result is in this case even a slight increaseof the bond energy, which is likely due to numerical noise aseach step of the freeze-thaw procedure minimises the total en-ergy of the system and should hence lower the bond energy.Since all changes are much smaller than the 1 kcal/mol foundin the base pairs we may conclude that polarization effects arenot very important for this system.

B. Accuracy of bond energieswith dispersion correction

1. BP8/05 data set

Table IV shows the results for bond energies ob-tained with KS-D, FDE(s)-D, and FDE(m)-D together withthe dispersion correction contribution D as well as refer-ence values7, 105, 106 obtained with an accurate wave func-tion method. As the KS-D results show, the dispersion cor-rection brings the interaction energies in close agreementto the reference wave function results for the stacked basepairs, with errors that are mostly below 1 kcal/mol. The FDEapproximation with the TW02 functional leads to system-atic overbinding relative to the KS-D result introducing an

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094113-8 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

TABLE IV. BP8/05 data set: KS-D, FDE(s)-D (BSSE-corrected), FDE(m)-D bond energies and dispersion cor-rection D with the BLYP XC functional and the TW02 kinetic energy functional. Reference interaction energiesare obtained with basis-set extrapolated RI-MP27, 105, 106 and refer to geometries of the monomers in the dimergeometry. For hydrogen bonded complexes no SCF convergence could be reached for FDE(s). All in kcal/mol.

Complexes D KS-D FDE(s)-D FDE(m)-D Reference

StackedA· · · T − 15.0 − 11.8 − 16.0 − 15.6 − 11.6105

G· · · C − 12.0 − 16.2 − 20.8 − 20.4 − 16.9105

U· · · U − 10.3 − 9.6 − 12.1 − 11.5 − 11.67

Par C· · · C − 9.0 3.0 − 1.1 − 0.8 2.45106

Anti C· · · C − 10.3 − 10.9 − 13.9 − 13.5 − 10.47

Displ C· · · C − 8.6 − 9.3 − 11.6 − 11.4 − 9.43106

Hydrogen bondedWC A· · · T − 5.3 − 17.0 — − 19.2 − 16.9107

WC G· · · C − 6.3 − 31.8 — − 34.5 − 32.4107

error of up to 4 kcal/mol. In the more rigorous FDE(s) ap-proach this is a bit more pronounced with energies a fewtenth of a kcal/mol lower than the FDE(m) data. These resultsshow that the simple additive dispersion correction brings theFDE(m) approach in semi-quantitative agreement with refer-ence data, although the accuracy is still significantly less thancan be achieved with the “parent” KS approach. At presentthe method can serve to provide a quick first indication of themagnitude of stacking effects.

2. Rubredoxin

Table V collects the KS-D, FDE(m)-D results and dis-persion correction D as well as reference interaction ener-gies obtained with an accurate wave function method.91 Inthis case inclusion of dispersion interactions is mandatory toobtain a qualitatively correct result. With dispersion, the KSresults are in a reasonable agreement with the reference ener-gies with the correct sign and with deviations of the order ofa few kcal/mol. Given this accuracy, the differences between

TABLE V. Rubredoxin—amino acid pairs clustered around F30 and F49:KS-D (BSSE corrected), FDE(m)-D, dispersion correction D with the BLYPXC functional. Reference interaction energies are obtained with basis-set ex-trapolated RI-MP2.91 All in kcal/mol.

D KS-D FDE(m)-D Reference

F30F49 −4.15 −2.47 −2.30 −3.30K46 −3.49 −2.86 −2.58 −3.40Y4 −8.97 −5.21 −5.78 −7.00L33 −8.15 −5.32 −5.76 −5.50Y13 −5.10 −2.28 −2.72 −4.50F49K46 −5.83 −4.49 −4.36 −4.80Y4 −7.43 −3.31 −2.80 −3.10C39 −4.37 −2.34 −2.11 −2.10V5 −9.48 −6.17 −7.44 −6.70W37 −3.01 −1.74 −1.79 −2.50C6 −7.69 −3.72 −2.66 −5.00

KS-D and FDE-D are acceptable with a largest discrepancy of1.3 kcal/mol for the most strongly interacting pair (F49–V5).As noted earlier, one should not compare FDE-D directly withthe reference wave function theory results because this wouldlead to the conclusion that FDE(m)-D outperforms KS-D insome cases. This is due to error cancellation, as in the DNAbase pairs we mostly see overbinding which can fortuitouslyimprove upon a too positive KS-D value. For some of theamino acid pairs, we also note a slightly larger (less nega-tive) FDE(m)-D value as compared to KS-D, indicating thatthe systematic trend of overbinding seen for the DNA basepairs is not universal.

C. Efficiency of FDE(m)-D

In order to assess the efficiency of FDE(m)-D, we mea-sured the duration of each calculation and compared it withthat of KS-D. For the F30 cluster, the KS-D calculationstook 48–74 min, whereas the 5 FT cycles of FDE-D took33–52 min. For the full calculation, FDE(m)-D was about30% more efficient than KS-D. This difference was slightlysmaller for the F49 cluster for which FDE(m)-D was only24% more efficient than KS-D. We did not time a case withonly 3 FT-cycles, but given the linear dependence of the cal-culation time on the number of FT cycles, one may easily esti-mate that such FDE calculations would be about a factor of 3faster than KS calculations. This efficiency can be further in-creased by improving upon the costly evaluation of the fit cor-rections to the Coulomb interaction energy. The current FDEimplementation in ADF is primarily optimised for molecularproperty calculations in which evaluation of this term is notneeded.

V. CONCLUSIONS

We propose density functional theory including a disper-sion correction as approximated by the frozen-density em-bedding method (FDE-D) as an efficient method to calcu-late interaction energies between biomolecular fragments.Results for selected test cases using the dispersion correction

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094113-9 Beyhan, Götz, and Visscher J. Chem. Phys. 138, 094113 (2013)

of Grimme28 in conjunction with the BLYP XC functionaland the TW02 kinetic-energy functional indicate that a rea-sonable accuracy can be reached at a potentially much lowercost compared to supermolecular KS-DFT calculations.

In order to analyse the bonding between the fragmentsin chemically meaningful terms, we defined a bond energydecomposition inspired by the KS bond formation analysis85

but allowing for nonorthogonality of orbitals that belong todifferent subsystems. We can thereby write the bond energyin terms of promotion energies of individual subsystems andan explicit interaction energy between these (Eq. (14)). Theanalysis is thus done in terms of differences between the non-interacting fragment density and an interacting fragment den-sity that sums to the full molecular density. The promotionenergies are always positive and their magnitude can serve asa measure for the mutual polarisation of the fragments whenbrought from non-interacting to fully interacting.

For all the molecules tested in this work, we show thatthe magnitude of the promotion energies and the electrostaticattraction between the subsystem electron densities increasesin the course of the FT process which confirms the intuitivepicture of FT providing mutual polarization.

ACKNOWLEDGMENTS

This work was supported by the Netherlands Organisa-tion for Scientific Research (NWO) via the VICI and NCF(computer time) programmes. L.V. likes to thank Dr. FonsecaGuerra for fruitful discussions regarding the DNA base pairresults.

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