Open-shell pair interaction energy decomposition analysis (PIEDA):Formulation and application to the hydrogen abstraction in tripeptidesMandy C. Green, Dmitri G. Fedorov, Kazuo Kitaura, Joseph S. Francisco, and Lyudmila V. Slipchenko Citation: J. Chem. Phys. 138, 074111 (2013); doi: 10.1063/1.4790616 View online: http://dx.doi.org/10.1063/1.4790616 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i7 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, 074111 (2013)

Open-shell pair interaction energy decomposition analysis (PIEDA):Formulation and application to the hydrogen abstraction in tripeptides

Mandy C. Green,1 Dmitri G. Fedorov,2 Kazuo Kitaura,3 Joseph S. Francisco,1

and Lyudmila V. Slipchenko1

1Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA2NRI, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono,Tsukuba, Ibaraki 305-8568, Japan3Graduate School of System Informatics, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe,Hyogo 657-8501, Japan

(Received 13 November 2012; accepted 24 January 2013; published online 21 February 2013)

An open-shell extension of the pair interaction energy decomposition analysis (PIEDA) withinthe framework of the fragment molecular orbital (FMO) method is developed. The open-shellPIEDA method allows the analysis of inter- and intramolecular interactions in terms of electro-static, exchange-repulsion, charge-transfer, dispersion, and optional polarization energies for molec-ular systems with a radical or high-spin fragment. Taking into account the low computational costand scalability of the FMO and PIEDA methods, the new scheme provides a means to character-ize the stabilization of radical and open-shell sites in biologically relevant species. The open-shellPIEDA is applied to the characterization of intramolecular interactions in capped trialanine uponhydrogen abstraction (HA) at various sites on the peptide. Hydrogen abstraction reaction is the firststep in the oxidative pathway initiated by reactive oxygen or nitrogen species, associated with ox-idative stress. It is found that HA results in significant geometrical reorganization of the trialaninepeptide. Depending on the HA site, terminal interactions in the radical fold conformers may becomeweaker or stronger compared to the parent molecule, and often change the character of the non-covalent bonding from amide stacking to hydrogen bonding. © 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4790616]

I. INTRODUCTION

Non-covalent interactions govern structure and functionof biological macromolecules such as DNA and proteins, thestate of liquids and colloids, and adsorption processes. Formany years, there has been an inexhaustible effort to un-derstand and model non-covalent interactions. For smallersystems, very accurate high-level quantum-mechanical stud-ies of molecular interactions have been conducted.1–10 Thebasis set superposition error (BSSE) can be eliminatedwith the counterpoise (CP) correction,11 with self-consistentfield for molecular interactions,12–14 or avoided with modelpotentials such as the effective fragment potential (EFP)method.15–20

Non-covalent interactions are typically thought of con-sisting of Coulomb, polarization, charge-transfer, and disper-sion attractive forces that are counterpoised by a repulsiveterm arising due to the quantum nature of electronic wavefunctions. Decomposition of non-covalent interactions intothese components is important for creating physically mean-ingful models for their description, such as new-generationforce fields. Decomposition of non-covalent interactions maybe achieved by either supermolecular or perturbative energydecomposition schemes. The former typically treats fragmentpairs variationally, while the latter uses a perturbation ap-proach based on the fragment electronic states. The Green’sfunction formalism21 provides some insight into the connec-tion of the two approaches.

Perturbative treatment of fragment interactions has beendeveloped in the symmetry-adapted perturbation theory(SAPT) by Moszynski, Jeziorski, and Szalewicz22, 23 in whichthe interaction energy is expressed in orders of an intermolec-ular interaction operator V and a many body perturbation the-ory (MBPT) operator W. Recent developments in this areainclude the use of SAPT with density functional theory24, 25

(DFT), which lowers its scaling to O(N5) (compared to theO(N7) scaling of full SAPT)26, 27 and a combination of SAPTideas with XPol variational method by Herbert.28 Efficientimplementation of SAPT algorithms enabled applications tosystems containing hundreds of atoms.29–31

In variational Hartree-Fock (HF) energy decompositionanalysis (EDA) methods,32–45 pioneered by Morokuma andKitaura,32 the total energy is typically represented as a sumof a frozen density interaction energy (resulting in Coulomband exchange-repulsion terms), a polarization energy, and acharge-transfer energy. Dispersion energy may be obtained asa correlated part of the interaction energy. The frozen den-sity term is calculated as the interaction of the unrelaxedelectron densities on the interacting fragments (frozen den-sity refers to the electron density of isolated fragments). Thepolarization term originates from the deformation of electronclouds of the interacting fragments in the fields of each other,while the charge transfer arises from the electron flow be-tween the fragments. Quantum mechanically, polarization andcharge transfer terms can be described as lowering of en-ergy due to the intra- and inter-fragment relaxation of the

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074111-2 Green et al. J. Chem. Phys. 138, 074111 (2013)

molecular orbitals, respectively. The main differences in thevariational EDA schemes come from the manner in whichthe intermediate self-consistent energies, corresponding to thevariationally optimized antisymmetrized wave functions con-structed from molecular orbitals localized on the individualfragments, are determined.

Most of the EDA and perturbation schemes have been de-signed to analyze interactions in non-covalently bound frag-ments (intermolecular interactions), with a notable exceptionof a density-based EDA by Wu et al.44 that is applicable to co-valent systems and intramolecular interactions. To analyze in-tramolecular interactions in biological systems, Fedorov andKitaura46 have extended the EDA method to covalently boundfragments by introducing the pair interaction energy decom-position analysis (PIEDA) within the fragment molecular or-bital (FMO) framework.47–51 Recently, Tanaka et al.52 havedeveloped an entropic contribution to pair interaction ener-gies, taking into account configuration averaging in a classicalmodel.

FMO is one of a number of fragment-based methods53–61

recently reviewed.62 Most of these methods are developed forclosed shell wave functions and only relatively few permitopen-shell treatment.63, 64 In the FMO formalism, one per-forms fragment calculations in the electrostatic field of otherfragments, mutually self-consistent with each other. PIEDAbased on the FMO formalism has been applied to studies ofprotein-DNA binding,65 protein-ligand binding,66 investigat-ing intramolecular interactions in synthetic γ -peptides,67 andin quantitative structure activity relationship (QSAR)-relatedstudies.68

Alternatively to PIEDA, for FMO there is the configura-tion analysis for fragment interactions (CAFI)69 and the frag-ment interaction analysis based on local second order pertur-bation theory (MP2) (FILM).70

BSSE corrections have been attempted in the FMOframework.71, 72 However, because of the absence of a gen-eral way to introduce BSSE corrections in PIEDA with itscomplicated evaluation of components and in view of a criti-cal assessment of the CP method,73–75 we did not use the CPmethod in the present study.

In this work, we develop an extension of the PIEDAscheme to open-shell fragments. This extension allows one toquantify inter- and intramolecular interactions in systems con-taining open-shell fragments or molecules, such as radicalsor high-spin species. The new technique is applied to a sim-ple tripeptide trialanine upon hydrogen abstraction (HA) fromvarious sites, to characterize and compare the intramolecularnon-covalent interactions in the peptide.

Radicals in biological systems play a dual role as bothharmful and beneficial species.76 The interactions specific toradical chemistry are difficult to measure, and as a resultthey are poorly understood. Radical reactions are not site-specific; many product variants are generated, hindering sys-tematic experimental investigation. New methods are requiredto investigate radical mechanisms and to understand their ef-fects in large biological systems. Ab initio methods suitablefor systematic investigation of open-shell systems are typi-cally limited to small molecules on the order of tens of atoms.Therefore, biological systems that often contain thousands of

atoms have previously been beyond the reach of computa-tional methods appropriate for open-shell molecules. Com-bining FMO with open-shell PIEDA provides a new meansto investigate the inter- and intramolecular properties of largeopen-shell systems.

Oxidative stress is a physiological condition resultingfrom overproduction of reactive oxygen species (ROS) andreactive nitrogen species (RNS), which damage biologicalstructures such as lipids in membranes, DNA, and proteins.Oxidative stress is linked to the pathogenesis of many dis-eases in biological systems such as cancer, cardiovascular dis-ease, atherosclerosis, hypertension, ischemia/reperfusion in-jury, diabetes mellitus, multiple neurodegenerative diseases,rheumatoid arthritis, and aging.76 ROS and RNS are usuallyopen-shell radical molecules such as hydroxyl radical, su-peroxide, and nitric oxide, although they can be non-radicalspecies such as hydrogen peroxide.77–79 Damage to biologicalmolecules, in this case via protein oxidation, occurs by the di-rect attack of ROS and RNS. HA reactions are the first step inthe oxidative pathway initiated by radical ROS and RNS. Inthis work, we consider products of a prototypical HA reactionin which hydroxyl radical attacks a capped trialanine peptide:OH• + AAAH → HOH + AAA•. Hydrogen abstraction canoccur at any atom in the molecule bonded to a hydrogen atom.In this study, we consider the product variants to HA at theα-carbons, the terminal carbons, and nitrogen to characterizeand compare the intramolecular non-covalent interaction be-tween the radical sites. The extension of the PIEDA schemeto open-shell fragments allows one the quantification of themolecular interactions in the open-shell fragments trialaninepeptide radicals.

The structure of the paper is as follows: Sec. II providestheoretical details on FMO, PIEDA, and open-shell PIEDAmethods. Section III summarizes computational details.Section IV describes application of PIEDA analysis to par-ent and HA-trialanine radicals, while the main conclusionsare summarized in Sec. V.

II. THEORY

Development of the restricted open-shell PIEDA (RO-PIEDA) scheme is based on a recent spin-restricted extensionof FMO to open-shell systems80, 81 and the original (closed-shell) PIEDA.46 In this work, the unrestricted extension ofFMO82 is not used. A brief overview of FMO and PIEDA isprovided below. Correlation is included using MP2. Alterna-tively, the coupled cluster (CC) approach can be used instead.

A molecular system in FMO is divided into N fragments,also referred to as monomers. In the first step, each monomeris calculated using the restricted Hartree-Fock (RHF) wavefunction in the Coulomb field exerted by all remainingmonomers. In the open-shell FMO, one of the fragments isopen-shell; it is calculated using restricted open-shell Hartree-Fock (ROHF wave function). The monomer calculations arerepeated until self-consistency is reached. In the second step,each dimer (pair of monomers) is computed in the Coulombfield exerted by all remaining monomers. This correspondsto the two-body FMO expansion (FMO2). The PIE is then

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074111-3 Green et al. J. Chem. Phys. 138, 074111 (2013)

obtained by subtracting monomer energies from dimer energy

�EintIJ = (E′

IJ − E′I − E′

J ) + T r(�DIJ VIJ )

+(Ecorr

IJ − EcorrI − Ecorr

J

), (1)

where I and J are fragments. In FMO, RHF or ROHF cal-culations are performed in the embedding potential actingupon fragments I (VI) or fragment pairs IJ (VIJ), yielding themonomer EI and dimer EIJ energies. E′

I and E′IJ are the in-

ternal energies, calculated as E′IJ = EIJ − T r(DIJ VIJ ), i.e.,

subtracting the contribution of the embedding potential. �DIJ

is the difference of the dimer DIJ and two monomer DI and DJ

electron densities. EcorrI and Ecorr

IJ are the monomer and dimercorrelation energies, respectively.

PIEDA decomposes �EintIJ into the electrostatic (ES),

exchange-repulsion (EX), charge-transfer with higher ordermixed terms (CT + mix), and dispersion (DI) contributionsas follows:

�EintIJ = �EES

IJ + �EEXIJ + �ECT+mix

IJ + �EDIIJ . (2)

For covalently connected fragments, the pair interactionenergy and its components are corrected for the correspondingbond detached atom (BDA) energies. This is accomplished bydoing an appropriate model calculation, for example, for a de-tached C–C bond, one calculates CH3–CH3, divided into twofragments. The corresponding interaction energy between thetwo methyl fragments is taken to be the BDA energy correc-tion, which is subtracted from the PIE of two connected frag-ments in real systems.46

The four components in Eq. (2) can be obtained for open-shell PIEDA following the recipe employed in closed-shellPIEDA,46 because of the general nature of their definition per-mitting the use of ROHF instead of RHF. Very briefly, we re-capitulate the component definitions. The ES component iscalculated as the Coulomb interaction between the electronicdensities and nuclei of the two fragments. The EX compo-nent is calculated as the cost of mixing occupied molecularorbitals of the two fragments (the dimer energy is calculated

for the orthogonalized set of molecular orbitals, from whichthe electrostatic interaction is subtracted). The energy of CT+ mix is calculated by subtracting the other two (ES and EX)components from the total pair interaction energy at the RHF(or ROHF) level. The dispersion component is calculated asthe difference in the electron correlation energy of a dimer andthe two monomers. Analogously to the original PIEDA imple-mentation, the polarization energy in the open-shell PIEDAcan be obtained after performing calculations of fragment free(i.e., isolated) states.

Throughout this work, we used the hybrid orbitalprojection operator approach83 to detaching covalent bonds.The default values for the integral accuracy for convergencethresholds were used. Restricted open-shell PIEDA was im-plemented in the GAMESS electronic structure package.84, 85

III. COMPUTATIONAL DETAILS

In order to avoid artifacts due to terminal interactions,86

the N-terminal of trialanine is capped with an acetyl and theC-terminal is capped with methylamide functional groups.Hydrogen abstraction in the capped trialanine is consideredat three Cα sites (Cα1, Cα2, Cα3), N′ and C′ terminal caps, andat the N3 position (see Figure 1). The N′ and C′ caps havetheir methyl functional groups in pseudo Cα positions in thebackbone. All considered HA products are doublets.

Geometry optimizations of parent and HA-trialanines intheir strand and fold conformations are performed at the re-stricted and unrestricted MP287/6-31G* (Refs. 88 and 89) lev-els of theory, for closed and open-shell systems, respectively.Cartesian coordinates of the optimized structures are providedin the supplementary material.90 For radical products, single-point energy calculations using ROHF reference were alsoperformed. FMO2 single point energies are obtained at thesame level of theory ((RO)MP2/6-31G*), followed by PIEDAcalculations. ROHF wave function was used for a HA radicalfragment in the self-consistency cycle of FMO and PIEDA;

FIG. 1. (a) Capped trialanine with considered HA sites (orange circles) and its (b) unfolded (β-strand) and (c) folded (β-turn) conformations.

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074111-4 Green et al. J. Chem. Phys. 138, 074111 (2013)

FIG. 2. Fragmentation schemes used in FMO2 and PIEDA calculations.Scheme D represents a default FMO fragmentation of a polypetide. Customschemes A-C are adapted to avoid artifacts due to fragmenting near radicalsites.

ROMP2 was employed for subsequent monomer and dimercalculations including the HA fragment.

By default, fragmentation for proteins in FMO is per-formed at the bonds between an α-carbon and a carbonyl.This fragmentation scheme, containing four fragments, is em-ployed for the parent trialanine (scheme D, see Figure 2).Three other fragmentation schemes A, B, and C shown inFigure 2 contain three fragments each. These schemes areemployed for describing radical HA products to avoid frag-menting the bond near a radical center. In particular, schemeA is used for HA on Cα1, scheme B for HA on Cα2, andscheme C for HA on Cα3. The N3

• product was fragmentedwith schemes B and D; while the N′cap• product was frag-mented with schemes A and D; and the C′cap• product wasfragmented with schemes C and D. In the three latter cases,the fragmentation schemes with a larger radical-containingfragment (A, B, or C) provided consistently more accurateresults than scheme D. Thus, presentation of the N3

•, N′cap•,and C′cap• results will be limited to fragmentation schemesB, A, and C, respectively. For consistency, FMO and PIEDAcomputations with the same fragmentation schemes are per-formed for the parent trialanine as well.

All calculations were performed in the GAMESS elec-tronic structure package.84, 85

FIG. 3. 1-4 terminal intramolecular interactions in parent trialanine in thestrand and turn conformations using four different fragmentation schemes.

IV. RESULTS AND DISCUSSION

A. Intramolecular interactions in parent trialanine

The goal of this section is to characterize intramolecularnon-covalent interactions in the parent closed-shell trialanineand to provide a reference for comparing the interactions inradical HA products. The same fragmentation schemes areused for parent trialanines and HA products to ensure thatthere are no artifacts due to differences in fragmentation.

Interactions between terminal amide planes are de-composed by PIEDA and shown in Figure 3. We also callthe terminal interactions as “1-4 interactions” based on thenumbering in the D scheme even though they are formally 1-3interactions in schemes A-C. The 1-4 interactions are non-covalent in all fragmentation schemes. Not surprisingly, theterminal interactions between amide planes 1 and 4 are muchstronger in the turn conformation than in the strand isomer.Terminal interactions in the turn are attractive by −7 to−9 kcal/mol; they are much weaker (−0.4 to +2.0 kcal/mol),depending on the fragmentation, in the strand isomer. Theleading contribution to the interaction energy in both the turnand strand is the electrostatic term that determines a signand magnitude of the total interaction energy. However, theother contributions in the turn (dispersion, charge-transfer,and exchange) are also non-negligible. Strong electrostaticinteraction between terminal planes in the turn can beexplained by dipole-dipole interactions between the peptidebonds typical for β-structures.

For 1-4 terminal interactions, fragmentation schemes Band D produce almost identical results (see Figure 3). This isbecause these schemes have identical terminal fragments. Thetotal interaction energies and their components are somewhatdifferent between schemes A, C, and B/D, because one ofthe interacting fragments in schemes A and C is larger thanthe terminal fragments in schemes B and D. Interestingly,electrostatic interactions in schemes A and C become lessattractive (or even repulsive for the strand), suggesting someelectrostatic repulsion between central and terminal parts ofthe peptide. The decrease in electrostatics is partly compen-sated by the increase in charge-transfer and dispersion terms,which is understandable as one of the fragments is larger.

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074111-5 Green et al. J. Chem. Phys. 138, 074111 (2013)

TABLE I. Electronic energy differences �E (kcal/mol) for hydrogen ab-straction reaction for selected sites in capped-trialanine with full ab initioand FMO methods.

Conformer Radical site FMO �E Full �E FMO �E error

Strand Cα1_A − 24.91 − 25.23 0.31Strand Cα2_B − 24.15 − 24.48 0.33Strand Cα3_C − 23.66 − 23.99 0.33Strand N′cap_A − 12.14 − 12.47 0.32Strand C′cap_C − 15.23 − 15.62 0.38Strand N3_B 0.59 − 0.19 0.78Turn Cα1_A − 14.20 − 14.30 0.11Turn Cα2_B − 17.25 − 17.58 0.33Turn Cα3_C − 16.17 − 16.56 0.39Turn N′cap_A − 11.99 − 12.47 0.48Turn C′cap_C − 15.88 − 16.27 0.38Turn N3_B − 0.23 − 0.76 0.53

B. Hydrogen-abstracted trialanines

1. Validation of open-shell FMO for HA-trialanine

We consider radical products resulting from hydrogen ab-straction reaction from trialanine

OH• + AAAH → HOH + AAA•, (3)

where AAAH is the parent capped trialanine; AAA• is a HA-radical product. Hydrogen abstraction on the backbone of thetripeptide can occur at three Cα sites (Cα1, Cα2, Cα3). Ad-ditionally, HA sites at the N′ and C′ terminal caps (N′cap•

and C′cap•, respectively), which have their methyl functionalgroups in pseudo Cα positions, and the site at the N3 positionare considered (see Figure 1). The terminal and N3 sites areselected due to their enhanced 1-4 intramolecular interactions,as analyzed in detail below.

Electronic energy differences, �E, for the HA reaction(Eq. (3)) at different sites are reported in Table I. The HA re-actions are considered separately for strand and turn confor-mations of trialanine; geometries of the reactants and prod-ucts are optimized within each conformation. The accuracyof FMO2 is compared against full ab initio calculations, atthe same level of theory (MP2/6-31G*). Negative �E valuesmean that the HA reaction is exothermic. As seen from datain Table I, the Cα sites are the most thermodynamically favor-able for HA out of all the sites on the protein backbone.

For the Cα sites in both the strand and turn conforma-tions, the relative difference in HA energies between the fullab initio and FMO methods is less than 0.4 kcal/mol. FMO-ab initio discrepancies for the terminal sites on the caps arealso very small (<0.5 kcal/mol).

Different from hydrogen abstraction reactions from car-bon sites, HA from the nitrogen site N3 is almost energeticallyneutral, with the full ab initio method producing a HA en-ergy change of −0.2 kcal/mol in the strand and −0.8 kcal/molin the turn. A radical site on nitrogen is more challengingfor the description by FMO, which produces relative errorsof 0.5–0.8 kcal/mol, probably due to the larger amount ofcharge-transfer between fragments. Generally, compared tofull ab initio, FMO slightly under-stabilizes HA radical prod-

TABLE II. Relative stability of turn versus strand conformers(Eturn−Estrand) for parent trialanine and HA products and stabilization energy(Estab = (Eturn−Estrand)HA − (Eturn−Estrand) parent) of the turn conformerdue to hydrogen abstraction (kcal/mol).

Radical site Eturn−Estrand Estab

Parent − 6.40a 0.00Cα1_A 4.13 10.72Cα2_B 0.05 6.90Cα3_C 0.78 7.48N′cap_A − 6.43 0.16C′cap_C − 7.35 − 0.65N3_B − 7.67 − 0.82

aFMO relative stabilization energy of the turn using fragmentation scheme D. The corre-sponding energies using other fragmentation schemes are: −6.59 kcal/mol (scheme A),−6.85 kcal/mol (scheme B), −6.70 kcal/mol (scheme C).

ucts compared to the parent trialanine; however, the discrep-ancies do not exceed 1 kcal/mol.

In the parent trialanine, the turn conformer, with favor-able dipole-dipole intramolecular interactions, is more stablethan the strand by ∼6 kcal/mol (see Table II). As followsfrom Table II, HAs at α-carbon sites destabilize the turn by7–11 kcal/mol and make the strand conformers energeticallymore preferable. However, HAs at other sites produce muchsmaller changes (typically within 1 kcal/mol) in stability ofradical products. This is the key observation of this study.Destabilization (and potential unfolding) of the turn structuresby HA at α-carbons may be significant for understanding theinfluence of oxidative stress on secondary and tertiary struc-tures of proteins. The reasons of destabilization of the turnconformers upon HA are analyzed below using the developedopen-shell PIEDA.

2. Intramolecular interactions uponhydrogen abstraction

Section IV B 2 describes changes in the intramolecularinteractions upon hydrogen abstraction in the turn conform-ers. In strand conformers, where 1-4 interactions are alreadyvery weak (see Figure 3), almost no changes occur upon hy-drogen abstraction.

Decomposition of 1-4 intramolecular interactions in theturn conformers of radical Cα

• trialanine products by open-shell PIEDA analysis is shown in Figure 4. The absoluteenergies are shown on the left, followed by the relative en-ergies on the right. The relative energies are determined asa difference between the 1-4 intramolecular energy in a Cα

•

product and the corresponding energy in a parent trialaninewith the same type of fragmentation. Note that the geometriesof HA-products are optimized; thus, Figure 4 shows adiabaticrelative energy differences. The presence of radical centerson Cα1 and Cα3, i.e., the centers that are involved in 1-4 ter-minal interactions, dramatically decreases total 1-4 interac-tion energies. Additionally, the 1-4 interactions change theirnature from being predominantly electrostatic in the parenttrialanine to dispersion-dominated in Cα1 and Cα3 radicals.

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074111-6 Green et al. J. Chem. Phys. 138, 074111 (2013)

FIG. 4. Absolute and relative (with respect to parent trialanines) 1-4 terminalinteractions in Cα HA products in the turn conformation.

On the contrary, there is a slight increase in the 1-4 interac-tion energy in the Cα2 product (where the radical center is notexplicitly involved in terminal interactions), mainly occurringdue to enhanced electrostatic attraction and diminished ex-change repulsion.

The intriguing changes in the terminal interaction ener-gies upon HA can be rationalized by geometrical changes inradical products (see Figure 5 and Table III). Compared tothe parent trialanine, Cα1

• and Cα3• radicals have more open

structure, with larger separation between terminal planes.This happens because of structural reorganizations in theproximity of the radical centers. In particular, HA at the Cα

and the cap sites causes a change in hybridization of the car-

bon atom from sp3 to sp2 resulting in a trigonal planar localgeometry instead of a tetrahedral one. The changes in a localmolecular geometry near the radical sites alter the backbonedihedral angles and the overall backbone shape. As a result,HA at Cα1 and Cα3 sites causes more separation between thefirst and fourth amide planes and a loss of favorable dipole-dipole interactions. On the contrary, HA at the Cα2 site leadsto less separation and increased terminal interactions throughthe formation of a hydrogen bond (see Table III).

In order to distinguish effects of geometry changes andchanges in the electronic structure upon HA, we comparedthe terminal interaction energies in the parent and radicalmolecules at the fixed geometry of the parent trialanine. Thesevertical energy differences are shown in Figure 6. Indeed, aradical center at Cα2 does not affect the terminal interactionenergies, while the changes due to radicals in Cα1 and Cα3

are very minor (of the order of 0.2–0.3 kcal/mol) and origi-nate from electrostatics.

Thus, we conclude that the main changes in the termi-nal interactions upon HA at α-carbons originate from geo-metrical rearrangements induced by local changes of bond-ing and hybridization at the radical centers. The geometricalrearrangements initiated by change of the electronic struc-ture at radical center destabilize the turn with respect to thestrand. However, this destabilization may be counteractedby restructuring of neighboring groups and formation of H-bond. Evolution of energetic balance in turn and strand struc-tures upon HA in larger peptides will be a topic of futurework.

Hydrogen abstraction at the terminal and N3 sites in-creases attraction between the terminal planes (see Figure 7).

FIG. 5. Structures of the parent capped trialanine and HA radical products in the turn conformation. In each peptide, the fourth carbonyl is perpendicular to theimage, and the fourth amide plane is horizontal. The hydrogen bond interaction between C4 = O and N1–H is shown with a dotted black line.

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074111-7 Green et al. J. Chem. Phys. 138, 074111 (2013)

TABLE III. Characteristic distances and angles between terminal planes inparent trialanine and HA products.

Radical H1. . . O4 N1–H1–O4 Interactionsite distance (Å) angle (deg) type

Parent (turn) 3.862 91.3 Amide stackingCα1

• 6.383 73.5 NoneCα3

• 5.214 113.3 NoneCα2

• 2.476 136.9 Hydrogen bondN3

• 1.9967 158.3 Hydrogen bondN′cap• 2.228 140.5 Hydrogen bondC′cap• 2.588 132.5 Hydrogen bond

The strongest effect is observed in case of the N3 radical site.HA at the N3 site causes the molecular geometry near nitrogento change from trigonal planar to bent. The backbone shapealternation due to HA at N3 results in a more compact struc-ture of the turn conformer and formation of hydrogen bondbetween terminal planes, with an O. . . H separation of 1.997 Å(see Table III). PIEDA analysis is consistent with this con-clusion: the increase in the terminal interaction energy occursdue to more attractive electrostatics in HA products. However,similar to the case of HA at α-carbons, the main effect origi-nates from the structural changes produced by HA. Analysisof relative vertical interaction energies, i.e., when the radicalis considered at the geometry of the parent trialanine, revealsonly a minor (<0.3 kcal/mol) destabilization of the terminalinteractions in the HA product due to less favorable electro-statics (see Figure 8).

The cap sites are unique in that HA there does not alterthe backbone shape. However, a decrease in sterical repul-sion between the terminal groups when one of them becomesCH2 (instead of CH3) results in a reduction in the separationbetween the first and fourth amide planes and formation ofH-bond (see Table III). This is again consistent with strongerelectrostatic interactions in the N′cap• radical compared to theparent trialanine.

Different from other HA sites, HA at terminal caps hasa non-negligible effect on terminal interaction energies evenwhen geometrical rearrangements are excluded. As follows

FIG. 6. Relative terminal (1-4) intramolecular interactions in the Cα HAproducts in the turn conformation calculated at the geometry of the parentturn trialanine.

FIG. 7. Absolute and relative (with respect to parent trialanines) 1-4 terminalinteractions in N3 and terminal caps HA products in the turn conformation.

from Figure 8, both N′cap• and C′cap• radicals have weakerterminal interactions compared to the parent molecule. How-ever, the effect is more significant for N′cap• (1.7 kcal/mol)than for C′cap• (0.2 kcal/mol). Interestingly, while changes incharge-transfer, dispersion, and exchange-repulsion contribu-tions are very similar between N′cap• and C′cap•, changes inelectrostatic energies are of opposite signs. Terminal electro-static interactions stabilize C′cap• and destabilize N′cap• withrespect to the parent trialanine. Changes in electrostatic inter-actions can be rationalized by radical-induced changes in thedipole moments associated with the terminal groups. The π

electrons of sp2-hybridized carbons in C′cap• and N′cap• rad-icals participate in π -conjugation of adjacent amide groups.As shown in Figure 9, the resonance structure stabilizes un-charged form of the amide group in N′cap•, while the reso-nance structure in C′cap• possesses charges on nitrogen andoxygen. Thus, the radical center in N′cap• effectively de-creases the dipole moment of the amide group, but the rad-ical at the C′cap• increases the corresponding dipole moment.This is consistent with the observed decrease in 1-4 elec-trostatic (dipole-dipole) interactions in N′cap• product and aslight increase in case of the C′cap•.

FIG. 8. Relative terminal (1-4) intramolecular interactions in the N3 and ter-minal caps HA products in the turn conformation calculated at the geometryof the parent turn trialanine.

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074111-8 Green et al. J. Chem. Phys. 138, 074111 (2013)

FIG. 9. Representative resonance structures in the parent trialanine and N′cap• and C′cap• radicals. Zwitterionic resonance structures affect the dipole momentof the peptide groups. Radical site in N′cap• stabilizes neutral form and effectively decreases the dipole moment of the terminal peptide group; radical site inC′cap• stabilizes zwitterionic form and enhances the dipole moment.

V. CONCLUSIONS

The PIEDA scheme for analysis of inter- and intramolec-ular interactions has been extended to open-shell systems.This development allows one to characterize non-covalentinteractions (in terms of electrostatic, exchange-repulsion,charge-transfer, dispersion, and optional polarization terms)in systems containing radicals or radical sites or high-spinspecies, as well as to compare interaction energies in open-shell and related closed-shell molecules. The open-shellPIEDA was applied to investigate non-covalent interactionsin capped trialanine upon hydrogen abstraction at varioussites.

Hydrogen abstraction in capped trialanine was character-ized with respect to geometrical changes, energetic stabilities,and intramolecular non-covalent interactions as a function ofa hydrogen abstraction site. HA at α-carbons is the most en-ergetically favorable in strand conformers. However, geomet-rical changes due to HA at α-carbons destabilize the turn con-formers (with respect to the strand conformers). HA at othersites (middle nitrogen N3 and terminal caps) has a negligi-ble effect on the stability of the turn with respect to the strandconformers. As a result, in the turn conformers HA at the capsis energetically comparable to HA at α-carbons.

The strand conformer of the parent trialanine has veryweak terminal interactions, while the terminal planes in theturn conformer exhibit non-covalent attractions dominated bydipole-dipole forces. If considered at a rigid geometry of the

parent trialanine, HA products experience only minor changesin terminal interactions, with exclusion of HA at the termi-nal caps. However, strong structural rearrangements associ-ated with HA at any site result in significant changes in ter-minal interaction energies. In particular, HA at Cα1 and Cα3

sites weakens terminal interactions, but HA at the other con-sidered sites strengthens them. Hydrogen abstraction at Cα2,N3, and terminal cap sites results in formation of hydrogenbonds between terminal planes.

The 1-4 interactions are a model for inter-strand inter-actions in a beta sheet as they are critical for turn formationand protein folding.91, 92 Protein misfolding and aggregationare the most likely cause of various neurological and sys-tematic diseases, classified together as amyloids because theaggregates all have the same underlying beta structure. Ox-idative stress initiates accumulation of amyloid protein ag-gregates in several diseases,93–95 but the mechanism and anunderstanding for why the aggregates form beta structures re-main elusive.96 PIEDA provides specifics for the altered in-tramolecular interactions and as such is instrumental in exam-ining for how beta structures respond to radical damage. Aswe found in this study, radical damage local to the 1-4 interac-tions, where the unpaired electron resonates with the 1st and4th amide planes, results in formation of a hydrogen bond in-stead of amide stacking. Thus, amyloid aggregates may adoptthe beta structures to compensate for radical damage by delo-calizing the unpaired electron into the local amide planes andthe hydrogen bonds.

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074111-9 Green et al. J. Chem. Phys. 138, 074111 (2013)

Being highly scalable, FMO and PIEDA can be used toinvestigate the properties of much larger biological systemsthan considered here. In our future work, we will employopen-shell PIEDA to gain insight into the origin of radical sta-bilization in proteins and examine electro/nucleophilic sub-stituents in changing the preferred radical site. Another im-portant effect to be considered in future studies is how solventalters interaction patterns and stabilization of radical centers.This can be done by using a combination of open-shell FMOwith either explicit (for example, treated with EFP97–99) or im-plicit (polarizable continuum model, PCM100–103) solvation.

ACKNOWLEDGMENTS

L.V.S. acknowledges support from National ScienceFoundation (NSF) (Grant No. CHE-0955419) and PurdueUniversity. D.G.F. and K.K. have been supported in part bythe Next Generation Super Computing Project, NanoscienceProgram (MEXT, Japan) and Strategic Programs for Innova-tive Research (SPIRE, Japan).

1J. C. Faver, M. L. Benson, X. He, B. P. Roberts, B. Wang, M. S. Marshall,M. R. Kennedy, C. D. Sherrill, and K. M. Merz, J. Chem. Theory Comput.7(3), 790–797 (2011).

2L. Gráfová, M. Pitonák, J. Rezác, and P. Hobza, J. Chem. Theory Comput.6(8), 2365–2376 (2010).

3P. Jurecka, J. Sponer, J. Cerny, and P. Hobza, Phys. Chem. Chem. Phys.8(17), 1985–1993 (2006).

4M. S. Marshall, L. A. Burns, and C. D. Sherrill, J. Chem. Phys. 135(19),194102 (2011).

5L. F. Molnar, X. He, B. Wang, and J. K. M. Merz, J. Chem. Phys. 131(6),065102 (2009).

6C. A. Morgado, P. Jurecka, D. Svozil, P. Hobza, and J. Sponer, Phys.Chem. Chem. Phys. 12(14), 3522–3534 (2010).

7J. Rezác, K. E. Riley, and P. Hobza, J. Chem. Theory Comput. 7(8), 2427–2438 (2011).

8K. E. Riley, M. Pitonák, P. Jurecka, and P. Hobza, Chem. Rev. 110(9),5023–5063 (2010).

9M. O. Sinnokrot, E. F. Valeev, and C. D. Sherrill, J. Am. Chem. Soc.124(36), 10887–10893 (2002).

10L. A. Burns, A. Vázquez-Mayagoitia, B. G. Sumpter, and C. D. Sherrill,J. Chem. Phys. 134(8), 084107 (2011).

11S. F. Boys and F. Bernardi, Mol. Phys. 19, 533 (1970).12R. Z. Khaliullin, M. Head-Gordon, and A. T. Bell, J. Chem. Phys. 124(20),

204105–204111 (2006).13E. Gianinetti, M. Raimondi, and E. Tornaghi, Int. J. Quantum Chem.

60(1), 157–166 (1996).14S. Iwata, J. Chem. Phys. 135(9), 094101 (2011).15M. S. Gordon, M. A. Freitag, P. Bandyopadhyay, J. H. Jensen, V. Kairys,

and W. J. Stevens, J. Phys. Chem. A 105(2), 293–307 (2001).16M. S. Gordon, L. V. Slipchenko, H. Li, and J. H. Jensen, Annu. Rep.

Comp. Chem. 3, 177–193 (2007).17L. V. Slipchenko and M. S. Gordon, J. Phys. Chem. A 113(10), 2092–2102

(2009).18Q. A. Smith, M. S. Gordon, and L. V. Slipchenko, J. Phys. Chem. A

115(18), 4598–4609 (2011).19Q. A. Smith, M. S. Gordon, and L. V. Slipchenko, J. Phys. Chem. A

115(41), 11269–11276 (2011).20L. V. Slipchenko and M. S. Gordon, J. Comput. Chem. 28(1), 276–291

(2007).21K. Yasuda and D. Yamaki, J. Chem. Phys. 125(15), 154101 (2006).22B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev. 94(7), 1887–

1930 (1994).23R. Moszynski, Mol. Phys. 88(3), 741–758 (1996).24A. J. Misquitta, B. Jeziorski, and K. Szalewicz, Phys. Rev. Lett. 91(3),

033201 (2003).25A. J. Misquitta, R. Podeszwa, B. Jeziorski, and K. Szalewicz, J. Chem.

Phys. 123(21), 214103–214114 (2005).

26A. Hesselmann, G. Jansen, and M. Schutz, J. Chem. Phys. 122(1), 014103(2005).

27R. Podeszwa, R. Bukowski, and K. Szalewicz, J. Chem. Theory Comput.2(2), 400–412 (2006).

28J. M. Herbert, L. D. Jacobson, K. Un Lao, and M. A. Rohrdanz, Phys.Chem. Chem. Phys. 14(21), 7679–7699 (2012).

29E. G. Hohenstein and C. D. Sherrill, J. Chem. Phys. 132(18), 184111(2010).

30E. G. Hohenstein and C. D. Sherrill, J. Chem. Phys. 133(10), 104107(2010).

31E. G. Hohenstein, R. M. Parrish, C. D. Sherrill, J. M. Turney, H. F. Schae-fer III, J. Chem. Phys. 135(17), 174107 (2011).

32K. Kitaura and K. Morokuma, Int. J. Quantum Chem. 10(2), 325–340(1976).

33W. J. Stevens and W. H. Fink, Chem. Phys. Lett. 139(1), 15–22 (1987).34W. Chen and M. S. Gordon, J. Phys. Chem. 100(34), 14316–14328

(1996).35P. S. Bagus, K. Hermann, and J. C. W. Bauschlicher, J. Chem. Phys. 80(9),

4378–4386 (1984).36P. S. Bagus and F. Illas, J. Chem. Phys. 96(12), 8962–8970 (1992).37E. D. Glendening, J. Phys. Chem. A 109(51), 11936–11940 (2005).38E. D. Glendening and A. Streitwieser, J. Chem. Phys. 100(4), 2900–2909

(1994).39G. K. Schenter and E. D. Glendening, J. Phys. Chem. 100(43), 17152–

17156 (1996).40R. Z. Khaliullin, E. A. Cobar, R. C. Lochan, A. T. Bell, and M. Head-

Gordon, J. Phys. Chem. A 111(36), 8753–8765 (2007).41Y. R. Mo, J. L. Gao, and S. D. Peyerimhoff, J. Chem. Phys. 112(13), 5530–

5538 (2000).42A. van der Vaart and K. M. Merz, J. Phys. Chem. A 103(17), 3321–3329

(1999).43P. F. Su and H. Li, J. Chem. Phys. 131(1), 014102 (2009).44Q. Wu, P. W. Ayers, and Y. K. Zhang, J. Chem. Phys. 131(16), 164112

(2009).45P. Su, H. Liu, and W. Wu, J. Chem. Phys. 137(3), 034111 (2012).46D. G. Fedorov and K. Kitaura, J. Comput. Chem. 28(1), 222–237

(2007).47D. G. Fedorov, T. Nagata, and K. Kitaura, Phys. Chem. Chem. Phys.

14(21), 7562–7577 (2012).48K. Kitaura, E. Ikeo, T. Asada, T. Nakano, and M. Uebayasi, Chem. Phys.

Lett. 313(3–4), 701–706 (1999).49D. G. Fedorov and K. Kitaura, in Modern Methods for Theoretical Phys-

ical Chemistry of Biopolymers, edited by E. B. Starikov, J. P. Lewis, andS. Tanaka (Elsevier, Amsterdam, 2006), pp. 3–38.

50D. G. Fedorov and K. Kitaura, J. Phys. Chem. A 111(30), 6904–6914(2007).

51D. G. Fedorov and K. Kitaura, The Fragment Molecular Orbital Method:Practical Applications to Large Molecular Systems (CRC, Boca Raton,2009).

52S. Tanaka, C. Watanabe, and Y. Okiyama, Chem. Phys. Lett. 556, 272–277(2013).

53P. Otto and J. Ladik, Chem. Phys. 8, 192 (1975).54J. Gao, J. Phys. Chem. B 101(4), 657–663 (1997).55D. W. Zhang, Y. Xiang, and J. Z. H. Zhang, J. Phys. Chem. B 107(44),

12039–12041 (2003).56H. R. Leverentz and D. G. Truhlar, J. Chem. Theory Comput. 5, 1573

(2009).57M. S. Gordon, J. M. Mullin, S. R. Pruitt, L. B. Roskop, L. V. Slipchenko,

and J. A. Boatz, J. Phys. Chem. B 113(29), 9646–9663 (2009).58L. Huang, L. Massa, I. Karle, and J. Karle, Proc. Natl. Acad. Sci. U.S.A.

106, 3664 (2009).59P. Söderhjelm, J. Kongsted, and U. Ryde, J. Chem. Theory Comput. 6,

1726 (2010).60S. D. Yeole and S. R. Gadre, J. Chem. Phys. 132, 094102 (2010).61X. He and K. M. Merz, J. Chem. Theory Comput. 6, 405 (2010).62M. S. Gordon, D. G. Fedorov, S. R. Pruitt, and L. V. Slipchenko, Chem.

Rev. 112(1), 632–672 (2011).63J. Korchowiec, F. L. Gu, and Y. Aoki, Int. J. Quantum Chem. 105, 875

(2005).64M. Kobayashi, T. Yoshikawa, and H. Nakai, Chem. Phys. Lett. 500(1–3),

172–177 (2010).65H. Mori and K. Ueno-Noto, J. Phys. Chem. B 115, 4774 (2011).66T. Sawada, D. G. Fedorov, and K. Kitaura, J. Am. Chem. Soc. 132(47),

16862–16872 (2010).

Downloaded 27 Feb 2013 to 128.210.126.199. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

074111-10 Green et al. J. Chem. Phys. 138, 074111 (2013)

67W. H. James III, E. G. Buchanan, C. W. Müller, J. C. Dean, D. Kosenkov,L. V. Slipchenko, L. Guo, A. G. Reidenbach, S. H. Gellman, and T. S.Zwier, J. Phys. Chem. A 115(47), 13783–13798 (2011).

68M. P. Mazanetz, O. Ichihara, R. J. Law, and M. Whittaker, J. Cheminf. 3,2 (2011).

69Y. Mochizuki, K. Fukuzawa, A. Kato, S. Tanaka, K. Kitaura, and T.Nakano, Chem. Phys. Lett. 410, 247 (2005).

70T. Ishikawa, Y. Mochizuki, S. Amari, T. Nakano, H. Tokiwa, S. Tanaka,and K. Tanaka, Theor. Chem. Acc. 118, 937 (2007).

71T. Ishikawa, T. Ishikura, and K. Kuwata, J. Comput. Chem. 30, 2594(2009).

72Y. Okiyama, K. Fukuzawa, H. Yamada, Y. Mochizuki, Y. Nakano, and S.Tanaka, Chem. Phys. Lett. 509, 67 (2011).

73D. W. Schwenke and D. G. Truhlar, J. Chem. Phys. 82(5), 2418–2426(1985).

74D. W. Schwenke and D. G. Truhlar, J. Chem. Phys. 86(6), 3760 (1987).75D. W. Schwenke and D. G. Truhlar, J. Chem. Phys. 84(7), 4113 (1986).76M. Valko, D. Leibfritz, J. Moncol, M. T. D. Cronin, M. Mazur, and J.

Telser, Int. J. Biochem. Mol. Biol. 39(1), 44–84 (2007).77B. Halliwell and J. M. Gutteridge, Biochem. J. 219(1), 1–14 (1984).78H. Wiseman and B. Halliwell, Biochem. J. 313(1), 17–29 (1996).79I. Fridovich, J. Exp. Biol. 201(8), 1203–1209 (1998).80S. R. Pruitt, D. G. Fedorov, K. Kitaura, and M. S. Gordon, J. Chem. The-

ory Comput. 6(1), 1–5 (2009).81S. R. Pruitt, D. G. Fedorov, and M. S. Gordon, J. Phys. Chem. A 116(20),

4965–4974 (2012).82H. Nakata, D. G. Fedorov, T. Nagata, S. Yokojima, K. Ogata, K. Kitaura,

and S. Nakamura, J. Chem. Phys. 137(4), 044110 (2012).83T. Nakano, T. Kaminuma, T. Sato, Y. Akiyama, M. Uebayasi, and K.

Kitaura, Chem. Phys. Lett. 318(6), 614–618 (2000).84M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J.

H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. J. Su, T. L. Windus,M. Dupuis, and J. A. Montgomery, J. Comput. Chem. 14(11), 1347–1363(1993).

85M. S. Gordon and M. W. Schmidt, in Theory and Applications of Compu-tational Chemistry, edited by C. E. Dykstra, G. Frenking, K. S. Kim, andG. E. Scuseria (Elsevier, 2005), pp. 1167–1189.

86W. Chin, F. Piuzzi, I. Dimicoli, and M. Mons, Phys. Chem. Chem. Phys.8(9), 1033–1048 (2006).

87C. Moller and M. S. Plesset, Phys. Rev. 46, 618–622 (1934).88W. J. Hehre, R. Ditchfield, and J. A. Pople, J. Chem. Phys. 56(5), 2257

(1972).89P. C. Hariharan and J. A. Pople, Theor. Chim. Acta 28(3), 213–222 (1973).90See supplementary material at http://dx.doi.org/10.1063/1.4790616 for

Cartesian coordinates for the strand, turn, and hydrogen abstractionproducts.

91S. S. Zimmerman and H. A. Scheraga, Proc. Natl. Acad. Sci. U.S.A74(10), 4126–4129 (1977).

92A.-S. Yang, B. Hitz, and B. Honig, J. Mol. Biol. 259(4), 873–882(1996).

93A. Nunomura, G. Perry, G. Aliev, K. Hirai, A. Takeda, E. K. Balraj, P. K.Jones, H. Ghanbari, T. Wataya, S. Shimohama, S. Chiba, C. S. Atwood, R.B. Petersen, and M. A. Smith, J. Neuropathol. Exp. Neurol. 60(8), 759–767 (2001).

94A. Nunomura, G. Perry, M. A. Pappolla, R. P. Friedland, K. Hirai, S.Chiba, and M. A. Smith, J. Neuropathol. Exp. Neurol. 59(11), 1011–1017(2000).

95D. J. Bonda, X. Wang, G. Perry, A. Nunomura, M. Tabaton, X. Zhu, andM. A. Smith, Neuropharmacology 59(4–5), 290–294 (2010).

96M. Stefani and C. Dobson, J. Mol. Med. 81(11), 678–699 (2003).97T. Nagata, D. G. Fedorov, K. Kitaura, and M. S. Gordon, J. Chem. Phys.

131(2), 024101 (2009).98T. Nagata, D. G. Fedorov, T. Sawada, K. Kitaura, and M. S. Gordon, J.

Chem. Phys. 134, 034110 (2011).99T. Nagata, D. G. Fedorov, T. Sawada, and K. Kitaura, J. Phys. Chem. A

116(36), 9088–9099 (2012).100D. G. Fedorov and K. Kitaura, J. Phys. Chem. A 116(1), 704–719

(2012).101D. G. Fedorov, K. Kitaura, H. Li, J. H. Jensen, and M. S. Gordon, J. Com-

put. Chem. 27(8), 976–985 (2006).102H. Li, D. G. Fedorov, T. Nagata, K. Kitaura, J. H. Jensen, and M. S.

Gordon, J. Comput. Chem. 31, 778 (2010).103T. Nagata, D. G. Fedorov, H. Li, and K. Kitaura, J. Chem. Phys. 136(20),

204112 (2012).

Downloaded 27 Feb 2013 to 128.210.126.199. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions