Analysis of charge transfer effects in molecular complexes based … Chem Phys 2008 128... · 2013-01-29 · merous systems, ranging from small water clusters to nanodroplets, bulk

Analysis of charge transfer effects in molecular complexes basedon absolutely localized molecular orbitals

Rustam Z. Khaliullin,1,2,a� Alexis T. Bell,1,3 and Martin Head-Gordon1,2

1Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA2Department of Chemistry, University of California, Berkeley, California 94720, USA3Department of Chemical Engineering, University of California, Berkeley, California 94720, USA

�Received 10 December 2007; accepted 27 March 2008; published online 14 May 2008�

A new method based on absolutely localized molecular orbitals �ALMOs� is proposed to measurethe degree of intermolecular electron density delocalization �charge transfer� in molecularcomplexes. ALMO charge transfer analysis �CTA� enables separation of the forward and backwardcharge transfer components for each pair of molecules in the system. The key feature of ALMOCTA is that all charge transfer terms have corresponding well defined energetic effects that measurethe contribution of the given term to the overall energetic stabilization of the system. To simplifyanalysis of charge transfer effects, the concept of chemically significant complementaryoccupied-virtual orbital pairs �COVPs� is introduced. COVPs provide a simple description ofintermolecular electron transfer effects in terms of just a few localized orbitals. ALMO CTA isapplied to understand fundamental aspects of donor-acceptor interactions in borane adducts,synergic bonding in classical and nonclassical metal carbonyls, and multiple intermolecularhydrogen bonds in a complex of isocyanuric acid and melamine. These examples show that theALMO CTA results are generally consistent with the existing conceptual description ofintermolecular bonding. The results also show that charge transfer and the energy lowering due tocharge transfer are not proportional to each other, and some interesting differences emerge which arediscussed. Additionally, according to ALMO CTA, the amount of electron density transferredbetween molecules is significantly smaller than charge transfer estimated from various populationanalysis methods. © 2008 American Institute of Physics. �DOI: 10.1063/1.2912041�

I. INTRODUCTION

Molecular complexes represent a broad class of systemswith interesting chemical and physical properties controlledby the intermolecular interactions. Similarly, molecular inter-actions determine many important properties of liquids, so-lutions, and molecular solids. They govern physisorption invan der Waals systems and control self-assembly and self-organization processes in supramolecular systems.1,2 Hydro-gen bonding plays an important role in the chemistry of nu-merous systems, ranging from small water clusters tonanodroplets, bulk water, and solvated biomolecules.3–7

Metal-ligand interactions give rise to a wide variety of metalcomplexes with different physical properties, chemical be-havior, and numerous practical applications.8,9 Many chemi-cal reactions involve nondissociative molecular adsorption,formation of �-complexes, and solvent-active siteinteractions.10–12 These interactions directly affect reactionenergetics and pathways.

The strength of intermolecular binding is inextricablyconnected to the fundamental nature of the interactions be-tween molecules.13 Intermolecular complexes can be stabi-lized through electrostatic effects �for example, charge-charge, charge-dipole, and charge-induced dipoleinteractions� and donor-acceptor type orbital interactions

such as forward and backdonation of electron density be-tween the molecules. Depending on the extent of these inter-actions, the intermolecular binding can vary in strength fromjust several kJ/mol �van der Waals complexes� to severalhundred kJ/mol �metal-ligand bonds in metal complexes�.Understanding the contributions of various interaction modesmay enable one to tune the strength of the intermolecularbinding to a target range by making appropriate chemicalmodifications.

The demand for a physically reasonable description ofthe intermolecular interaction components has resulted in nu-merous energy decomposition analysis �EDA� schemes overthe years.14–26 Most decomposition methods represent the to-tal interaction energy as a sum of a frozen density interactionenergy, a polarization energy, and charge transfer energyterms. The frozen density term is calculated as the interactionenergy of the unrelaxed electron densities on the moleculesallowing only for Pauli repulsion. The polarization term isdue to the relaxation �or polarization� of the electron cloudsof the molecules in the field of each other. Quantum me-chanically, it can be described as the energy lowering due tothe intramolecular relaxation of the molecular orbitals. Fi-nally, the charge transfer can be pictured as the electron flowto and from each molecule in the system due to intermolecu-lar relaxation �or mixing� of the molecular orbitals. Althoughquantitative values for charge transfer effects vary from oneEDA scheme to another, all energy decomposition methods

a�Author to whom correspondence should be addressed: Electronic mail:[email protected].

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give unequivocal evidence that they have significant weightin almost all types of intermolecular interactions. In this pa-per, we introduce a quantitative scale for measuring the de-gree of intermolecular electron density delocalization in mo-lecular complexes and demonstrate a direct connection ofsuch charge transfer effects to the energetic stabilization ofthe system.

To study electron density delocalization in molecularcomplexes, the system is usually partitioned into molecularsubsystems in either three dimensional space or in Hilbertspace. Methods such as the quantum theory of atoms inmolecules,27,28 the electron localization function,29–31 andHirshfeld analysis32 are examples of topological methodsthat study the spatial distribution of scalar fields derivedfrom reduced electron density matrices. In this work, we areconcerned with methods based on partitioning Hilbert space.Existing examples include such popular methods as naturalpopulation analysis,33 Mulliken population analysis�MPA�,34,35 Löwdin population analysis,36 and manyothers.37–40 In each method in this category, one-electron ba-sis functions �orbitals� that span the supermolecular Hilbertspace are assigned to individual molecules and the formalcharge on a molecule is calculated as a sum of the populationof all orbitals associated with the given molecule and itsnuclear charges. To obtain the orbital population and thus themolecular formal charges, each method needs to specify aprescription for constructing basis functions and a procedurefor treating the basis function overlap. In population analysis�PA� methods, the intermolecular charge transfer �Q can beinferred from the formal molecular charges �see Appendix�.

In this paper, we present another way of measuring �Qin molecular complexes described by single-determinantwavefunctions �Hartree–Fock or density functional theory�DFT��. We demonstrate that, unlike �Q inferred from PAmethods, the intermolecular electron density transfer in ourapproach has a corresponding well defined energy of stabili-zation.

To isolate intermolecular charge transfer from electrondensity penetration effects �i.e., spatial overlap of the unre-laxed electron densities on the molecules� and from polariza-tion effects �i.e., intramolecular charge reorganization in thefield of the neighboring molecules�, it is convenient to use anintermediate partially optimized wavefunction that �i� al-ready includes frozen density interactions �spatial overlap�,�ii� is variationally optimized to self-consistency to includepolarization effects, and �iii� gives zero charge transfer be-tween molecules according to some preselected populationanalysis method. Employing an intermediate state �wave-function� to isolate intra- and intermolecular electron densityreorganization effects makes such analysis closely related toEDA methods as the latter also require an intermediate stateto separate the charge transfer energy from the rest of theenergy terms. In this work, the reference intermediate state�wavefunction� is constructed from the variationally opti-mized absolutely localized molecular orbitals �ALMOs�,41–47

which have been recently used for energy decompositionmethods by Mo et al.25 �BLW EDA� and by Khaliullin et al.�ALMO EDA�.26 Unlike conventional MOs, which are gen-erally delocalized over all molecules in the system, the

ALMOs are expanded in terms of the atomic orbitals �AOs�of only a given molecule.25,26,41,42,44,45 It can be shown thatthe MPA gives zero charge transfer between molecules forany one-electron density matrix constructed from ALMOs.Therefore, the state constructed from variationally optimizedALMOs �i.e., polarized ALMOs�25,26 satisfies conditions�i�–�iii�. Its main advantage over other ALMO states is afully self-consistent treatment of polarization effects. Thus,the polarized ALMO state is particularly useful to studycharge transfer effects in molecular complexes. The self-consistent field procedure for the variational optimization ofthe ALMOs �i.e., for finding the polarized ALMO state� iscalled SCF for molecular interactions, or SCF MI, and itsmathematical and algorithmic details has been described bymany authors.41,42,44,45

It has been recently shown26 that the charge transfer en-ergy term in the ALMO EDA can be separated into forwardand backdonation components using a perturbative Roothaanstep �RS� approximation starting from the optimized ALMOreference. In this paper, we show that a similar formulationcan also be used to separate �Q into bonding and backbond-ing components for each pair of molecules in the complex.The charge decomposition analysis �CDA� method ofDapprich and Frenking48 also defined forward donation andbackbonding components of intermolecular charge transfer.The simplicity and the ease of implementation of CDA havemade this method popular for estimating the amount of theelectron delocalization in donor-acceptor complexes. How-ever, as mentioned in the original CDA paper,48 this defini-tion of charge transfer should not be confused with “physicalinterpretation of chemical bonding.” As shown in theAppendix, some terms in CDA do not have a clear physicalmeaning. We examine the results obtained with the CDAmethod and compare them to the results of our ALMO-basedcharge transfer analysis �ALMO CTA�.

In addition to quantifying the amount and energetics ofintermolecular charge transfer, it is often useful to have asimple description of orbital interactions in intermolecularcomplexes. The natural bond orbital �NBO� analysis ofdonor-acceptor complexes has gained wide popularity be-cause of its ability to provide such an orbital interactionpicture.49 In this paper, we demonstrate how the ALMO ap-proach can be used to construct a conceptually simple orbitalinteraction model. The polarized ALMOs obtained from theSCF MI procedure and used as a reference basis set in thedecomposition analysis do not directly show whichoccupied-virtual orbital pairs are of most importance informing intermolecular bonds. We demonstrate that, by per-forming rotations of the polarized ALMOs within a mol-ecule, it is possible to find a “chemist’s basis set” that repre-sents bonding between molecules in terms of just a fewlocalized orbitals. This orbital interaction model validates ex-isting conceptual descriptions of intermolecular bonding. Forexample, in the modified ALMO basis, donor-acceptor bond-ing in ammonia borane �H3N–BH3� is represented as thelone orbital pair on the nitrogen atom donating electrons tothe empty orbital on the boron atom, and the description ofsynergic bonding in metal complexes agrees well withsimple Dewar–Chatt–Duncanson model.50,51

184112-2 Khaliullin, Bell, and Head-Gordon J. Chem. Phys. 128, 184112 �2008�


Summarizing, in this paper, we propose a new scale tomeasure intermolecular charge transfer effects in molecularcomplexes. We also show how to separate the total chargetransfer term into forward and backdonation components foreach pair of molecules in the system. Finally, we propose amethod to construct chemically significant orbitals localizedon molecules that provide a simple chemical description ofintermolecular interactions. In addition, we demonstrate thateach charge transfer term defined in this work has a corre-sponding energy lowering that measures the contribution ofthe given term to the overall energetic stabilization of thesystem. Thus, the ALMO-based method proposed here uni-fies energy decomposition analysis, charge transfer decom-position analysis, and orbital-orbital interaction analysis formolecular systems.

To demonstrate the validity and usefulness of the newALMO-based charge transfer analysis based on the abso-lutely localized molecular orbitals �ALMO CTA�, we appliedit to a series of problems of chemical interest:

• donor-acceptor interactions in borane adducts;

• synergic bonding in classical and nonclassical metalcarbonyls; and

• multiple intermolecular hydrogen bonds in a complexof isocyanuric acid and melamine.

We show that the ALMO CTA results are generally consis-tent with the existing conceptual pictures of intermolecularbonding and agree with the ALMO energy decompositionanalysis. We hope that the ALMO approach reported in thispaper will become a useful theoretical tool for studying andanalyzing binding mechanisms in molecular complexes.

II. THEORY

A. Summary of the ALMO energy decompositionanalysis

Energy decomposition methods based on ALMOs havebeen proposed by Mo et al.25 �BLW EDA� and by Khaliullinet al. �ALMO EDA�.26 In these methods, the overall bindingenergy is decomposed into frozen density component �FRZ�,polarization �POL�, and charge transfer �CT� terms

�ETOTAL = �EFRZ + �EPOL + �ECT. �1�

The frozen density term �FRZ� is defined as the SCF energychange that corresponds to bringing infinitely separated mol-ecules into the complex geometry without any relaxation ofthe MOs on the monomers �apart from changes associatedwith satisfying the Pauli principle�.

�EFRZ � ESCF�RFRZ� − �x

ESCF�Rx� , �2�

where ESCF�Rx� is the SCF energy of the variationally opti-mized one-electron density matrix of the isolated molecule x

with its nuclei fixed at the complex geometry and RFRZ is thedensity matrix of the complex constructed from the unre-laxed nonorthogonal occupied MOs of the molecules �seeEq. �8� below�.

The polarization energy is defined as the energy lower-ing due to intramolecular relaxation of each molecule’s ab-solutely localized MOs in the field of all other molecules inthe system. The intramolecular relaxation is constrained toinclude only variations that keep MOs localized on theirmolecule:

�EPOL � ESCF�RPOL� − ESCF�RFRZ� , �3�

where RPOL is the density matrix constructed from the fullyoptimized �polarized� ALMOs according to Eq. �8�. Accord-ing to the Mulliken PA, the ALMO expansion explicitly ex-cludes charge transfer from one molecule to another. Math-ematical details of the SCF procedure for finding the

polarized state RPOL are given elsewhere.41

The remaining portion of the total interaction energy, theCT energy term, is calculated as the energy difference be-

tween the state formed from the fully relaxed ALMOs, Rpol,and the state constructed from the fully optimized delocal-

ized MOs, RSCF.

�ECT � ESCF�RSCF� − ESCF�RPOL� , �4�

where �ECT includes the energy lowering due to electrontransfer from occupied orbitals on one molecule to virtualorbitals of another molecule as well as the further energychange caused by induction �or repolarization� that accom-panies such an occupied-virtual mixing. In the ALMO EDAof Khaliullin et al., the energy lowering of the occupied-virtual excitations �mixing� is described with a single nonit-erative RS correction starting from the converged ALMOsolution.26,41 The remaining higher order �HO� relaxationterm includes all induction effects that accompany occupied-virtual charge transfer and is generally small,

�ECT = �ECTRS + �ECT

HO. �5�

Most importantly, for the purpose of the energy decomposi-tion analysis, the mathematical form of the RS energy ex-pression allows one to decompose the occupied-virtual mix-ing term into forward and backdonation components for eachpair of molecules in the complex,

�ECTRS = �

x,y�x

��Ex→yRS + �Ey→x

RS � . �6�

The HO energy contribution cannot be naturally divided intoforward and backdonation terms.26

Thus, the four states—RFRZ, RPOL, RRS, and

RSCF —define energy terms in the ALMO EDA. The Rpol

state is the lowest energy state that can be constructed fromcompletely localized molecular orbitals. Naturally, the elec-

tron density relaxation from RPOL to RSCF provides a measurefor intermolecular charge transfer effects �Q in the system.

B. Notation

We denote polarized ALMOs as �xp. The first indexindicates that this orbital is localized on �or assigned to� mol-ecule x, and the second index refers to the number of theorbital within the molecular subset. Latin letters x–z are forthe molecular indices, letters i–k are used to label occupied

184112-3 Absolutely localized molecular orbitals J. Chem. Phys. 128, 184112 �2008�


orbitals, a–c denote virtual orbitals, and p– t correspond togeneric �occupied and virtual� orbitals. O and V refer to thenumber of occupied and virtual basis functions in the system,respectively, N is the total number of MOs in the system. ox,vx, and nx correspond to the number of occupied, virtual, andtotal MOs on molecule x, respectively.

ALMOs are not orthogonal from one molecule to thenext and their overlap is denoted as �xp,yq= ��xp �yq. Thebiorthogonal basis function are denoted with superscripts,

�xp = �yq

N

�yq��−1�yq,xp. �7�

The one-electron density operator constructed from thepolarized ALMOs represents the projector onto the occupiedsubspace and is given by

R � RPOL = �xi,yj

O

�xi��o−1�xi,yj��yj = �

xi

O

�xi��xi , �8�

where ��o−1� is the inverse of the O�O overlap matrix of

occupied basis functions and should not be confused with��−1�. Biorthogonal functions in the occupied subspace areconstructed as follows:

�xi = �yj

O

�yj��o−1�yj,xi. �9�

To simplify our notation, we denote RPOL with R throughoutthe paper. It is also convenient to define projector onto the

virtual subspace Q, the orthogonal complement of R,

Q = 1 − R = �xa,yb

V

�xa��v−1�xa,yb��yb = �

xa

V

�xa��xa ,

�10�

where projected virtual orbitals �ya span the virtual sub-space and are orthogonal to the occupied orbitals,

�ya = �1 − R��ya = �ya − �xi

O

�xi�xi,ya. �11�

These definitions use the V�V overlap matrix of projectedvirtual orbitals,

��v�xa,yb = ��xa�yb . �12�

The biorthogonal basis in the virtual subspace is defined ac-cording to the following equation:

�xa = �yb

V

�yb��v−1�yb,xa. �13�

C. ALMO charge transfer analysis

The electron density reorganization from an initial state

Rs to a final state Rf is defined as the amount of the electrondensity that, in the final state, occupies the initially unoccu-pied subspace. Such a reorganization term is given by thefollowing equation:

�Q�s→f� = Tr�QsRfQs� = − Tr�RsRfRs − Rs� , �14�

where Qs=1− Rs is the projector onto the initially unoccu-pied �virtual� subspace. Note that positive values of �Qcorrespond to the negative charge being transferred.

As mentioned in the ALMO EDA section, intermolecularcharge transfer effects are measured as the change from thepolarized ALMO state to the fully converged SCF state,

�QCT � �Q�POL→SCF�. �15�

In agreement with ALMO EDA, the charge transfer due tooccupied-virtual mixing ��QCT

RS� and the accompanyinghigher order relaxation terms ��QCT

HO� are defined as follows:

�QCTRS � �Q�POL→RS�, �16�

�QCTHO � �QCT − �QCT

RS. �17�

These definitions of �QCTRS and �QCT

HO enable one to assign anenergy lowering to each intermolecular electron transferterm. Since the Mulliken PA gives zero charge transfer be-tween molecules, �QFRZ and �QPOL terms do not contributeto intermolecular electron density transfer.

In the ALMO EDA, the single RS perturbation theory isused to account for the energy lowering due to the occupied-virtual mixing �ECT

RS. We will show now how the correspond-ing charge transfer term �QCT

RS is calculated.

The RS is the search for the unitary operator U thatminimizes the energy functional

E�U� = Tr�FURU†� , �18�

where R= RPOL and F is the Fock operator constructed fromthe polarized ALMOs. The post RS �transformed� operatorsare given by

R� � RRS = URU† �19�

and

F� = U†FU . �20�

Then, the RS energy lowering can be written as follows:

�ECTRS = Tr�RF�R − RFR� . �21�

By using definition of �QCTRS �Eq. �16�� and the general ex-

pression for charge reorganization �Eq. �14��, we have

�QCTRS = − Tr�RR�R − R� . �22�

There is a direct similarity between the RS energy �Eq. �21��and charge �Eq. �22�� expressions. In the ALMO EDA, theenergy term can be readily decomposed into bonding andbackbonding components for each pair of molecules in thecomplex �Eq. �6��. Here, we show how a similar approachcan be used to isolate bonding and backbonding componentsof the charge transfer term. To do that, it is convenient torewrite Eqs. �21� and �22� in terms of the excitation operator

X. First, we demonstrate how the energy equation can berewritten �original derivation is given by Liang andHead-Gordon52� and then show how similar results can beobtained for the charge expression.



Equation �20� can be written as FU= UF� and then pre-

and postmultiplied by R to obtain

�RFR��RUR� + �RFQ��QUR�

= �RUR��RF�R� + �RUQ��QF�R� . �23�

The last term in the right-hand side of Eq. �23� is zero sincethe minimization of the energy functional �Eq. �18�� is

equivalent to finding U that nullifies the virtual-occupied

block of the transformed Fock operator: QF�R=0. Equation

�23� is then premultiplied by �RUR�−1

�RUR�−1�RFR��RUR� + �RUR�−1�RFQ��QUR� = RF�R .

�24�

The obtained expression for RF�R is plugged into Eq. �21�,and the terms inside the trace are rearranged as follows:

�ECTRS = Tr��RFQ��QUR��RUR�−1� . �25�

Defining a new operator as

X � QUR�RUR�−1, �26�

we have the following equation for the Roothaan step energylowering:

�ECTRS = Tr�RFQX� . �27�

It can be shown that the newly defined operator X rep-

resents a Cayley generator of unitary transformation U,53

U = �1 + X − X†��1 + X†X + XX†�−1/2, �28�

and is directly related to the first-order transformation of thezero-order orbitals

U = 1 + X − X† + O�X2� . �29�

Operator X satisfies the projection relation X= QXR by defi-nition. Therefore, the ALMO occupied orbitals are mixedonly with the zero-order virtual orbitals upon first-order

transformation �Eq. �29��. Thus, operator X is referred to asexcitation operator and its matrix elements represent excita-tion amplitudes from the occupied to the virtual orbitals.

The excitation amplitudes can be found by solving thefollowing single RS quadratic equation:53

QFR + QFQXR − QXRFR − QXRFQXR = 0. �30�

The operator X obtained from this equation gives the RSenergy lowering that is equvalent to the result of single Fockmatrix diagonalization or to the infinite-order single excita-tion perturbation theory result.41,52,53

We can express �QCTRS in terms of the excitation operator

X. First, equation R�U= UR is pre- and postmultiplied by R:

�RR�R��RUR� + �RR�Q��QUR� = RUR , �31�

and then the resulting Eq. �31� is postmultiplied by �RUR�−1:

RR�R = R − �RR�Q��QUR��RUR�−1. �32�

Next, the obtained expression for RR�R is substituted intoEq. �22� and is simplified by using the definition of the ex-citation operator �Eq. �26��:

�QCTRS = Tr�RR�QX� . �33�

Equation �33� gives the total amount of charge transfer

from the occupied subspace to the virtual subspace of R

� Rpol when this state is relaxed to R�� RRS. Equation �27�gives the energy lowering associated with this charge trans-fer.

In the next step, we partition the total charge transferinto the pairwise contributions assigned to electron flowfrom an individual occupied orbital to an individual virtualorbital. To do this, we introduce a set of partition operators,

each corresponding to an occupied �Pxi� or to a virtual �Pxa�orbital. These operators must satisfy the resolution of iden-tity property

�xi

O

Pxi = R , �34�

�xa

V

Pxa = Q , �35�

and the idempotency property

Pxi2 = Pxi, �36�

Pxa2 = Pxa. �37�

Using these partition operators, we define charge transferfrom orbital xi to orbital ya and its corresponding energy asfollows:

�Qxi→yaRS = Tr�PxiR�PyaX� , �38�

�Exi→yaRS = Tr�PxiFPyaX� . �39�

Charge transfer from molecule x to molecule y and its energyare then expressed as

�Qx→yRS = �

i

ox

�a

vy

Tr�PxiR�PyaX� , �40�

�Ex→yRS = �

i

ox

�a

vy

Tr�PxiFPyaX� . �41�

Properties �34� and �35� ensure that the charge and energycomponents sum up to the correct total values:

�QCTRS = �

x,y�Qx→y

RS = �xi

O

�ya

V

�Qxi→yaRS , �42�

�ECTRS = �

x,y�Ex→y

RS = �xi

O

�ya

V

�Exi→yaRS . �43�



It should be noted that �QCTRS and �ECT

RS do not depend on

the choice of partition operators �Pxq�. Their values are de-fined by the natural and unambiguous partitioning of the

Hilbert space into occupied �R� and unoccupied �Q� sub-spaces. However, when one attempts to define forward andbackbonding components of these terms, it is impossible toavoid ambiguities associated with the partitioning of the oc-cupied and virtual spaces into molecular �or orbital� sub-spaces. In other words, there is no unique definition for op-

erators Pxq. This problem is closely related to the problem ofdescribing charge distribution among nonorthogonal sub-spaces associated with orbitals, atoms, or molecules in PAmethods. The most widely used partition operators in PA areMulliken and Löwdin operators.34–36,54 Mulliken partitionoperators are not Hermitian which sometimes leads to un-physical negative orbital populations and the population ofsome orbitals being higher than two. It can be shown, how-ever, that the deviations from the physical �0, 2� interval aresecond order in the overlap between the orbitals.40 Such de-viations can be considered acceptable in many applicationsespecially for partitioning electron density in molecular com-plexes since overlap effects between molecules in such com-plexes are generally smaller than, for example, overlap be-tween atoms in molecules. Orbital populations obtained fromthe Löwdin partitioning are always within the physicalinterval but the basis set functions cannot be made strictlylocalized in this approach.

A Mulliken-type partitioning defined by the followingequations:

Pxi = �xi��xi , �44�

Pxa = �xa��xa �45�

is the best choice for the ALMO decomposition analysissince �i� the polarized reference gives zero intermolecularcharge transfer according to the Mulliken PA and �ii� thecharge transfer energy terms within a molecule, �Ex→x

RS , arezero if the Mulliken-type partitioning is used because thevariational optimization of ALMOs nullifies intramolecularoccupied-virtual elements of the contravariant-covariant ma-

trix of the Fock operator ��xiF�xa �see Eq. �47� below�.Matrix equations for intermolecular charge transfer compo-nents and their energy on the basis of the polarized ALMOswith the Mulliken-type partitioning are as follows:

�Qx→yRS = �

i

ox

�a

vy

��xiR��ya��yaX�xi = Tr�R�xy�� X�yx�� ,

�46�

�Ex→yRS = �

i

ox

�a

vy

��xiF�ya��yaX�xi = Tr�F�xy�X�yx�� ,

�47�

where R�xy�� and F�xy� are ox�vy matrices and X�yx� is avy �ox matrix.

The same partitioning of the charge transfer energy termis used in the ALMO EDA.26 Our tests show that the

Löwdin-type partitioning �i.e., symmetric orthogonalizationof the occupied and projected virtual orbitals� produces re-sults that are close to the Mulliken-type analysis. However,intrafragment components are slightly larger in the Löwdin-type approach due to partial delocalization of ALMOs aftersymmetric orthogonalization.

D. Significant complementary occupied-virtual pairs

Equations �46� and �47� clearly show that in the canoni-cal polarized ALMO, basis charge transfer from molecule xto molecule y is represented as each occupied orbital on xdonating electrons to each virtual orbital on y. In general,there are no occupied-virtual pairs in this representation thatcan be neglected. However, orbital rotations within the occu-pied subset and the virtual subset of a molecule leave �Qx→y

RS

and �Ex→yRS unchanged. Thus, we can use this freedom to find

such unitary matrices Ux�o� and Ux

�v� that rotate orbitals withinthe occupied subset of molecule x and the virtual subset ofmolecule y, respectively, to perform singular value decom-position of X�yx�:

X�yx� = Uy�v�x�yx�Ux

�o�†, �48�

where the vy �ox matrix x�yx� contains min�vy ,ox� singularvalues of X�yx� on the diagonal and zeros off the diagonal.Because of the diagonal structure of x�yx�, in the new repre-sentation, charge transfer from x to y is described as eachoccupied orbital on x donating electrons to only one�complementary� virtual orbital on y. If ox�vy, then�ox−vy� occupied orbitals on x will not participate in theintermolecular bonding between x and y at all:

�Qx→yRS = Tr��Ux

�o�†R�xy�� Uy�v��Uy

�v�†X�yx�Ux�o��

= Tr�r�xy�� x�yx�� , �49�

�Ex→yRS = Tr��Ux

�o�†F�xy�Uy�v��Uy

�v�†X�yx�Ux�o��

= Tr�f�xy�x�yx�� . �50�

Orbitals constructed by performing unitary transformationsUx

�o� within the occupied subset of molecule x and Uy�v� within

the virtual subset of molecule y are called complementaryoccupied-virtual pairs �COVPs� for the x→y intreaction.Construction of the COVPs greatly simplifies the picture ofintermolecular orbital interactions. A set of all complemen-tary occupied-virtual pairs describes charge transfer betweena pair of molecules exactly. We will show in Sec. III that themajor contribution to bonding between two molecules �interms of both energy and amount of electrons transferred�comes from just a few �one or two� COVPs, which we willcall significant complementary occupied-virtual pairs. Suchpairs provide approximate simplified representation of inter-molecular orbital interactions which is useful for a concep-tual description of intermolecular bonding.

E. Counterpoise correction

The basis set superposition error �BSSE� is not intro-duced when the absolutely localized MOs are used to calcu-

late the intermediate state RPOL because the constrained MO



optimization prevents electrons on one molecule from bor-rowing the AOs of other molecules to compensate for incom-pleteness of their own AOs. However, the BSSE enters thecharge transfer contribution since both the BSSE and chargetransfer result from the same physical phenomenon of delo-calization of fragment MOs. All charge transfer terms re-ported in this work ��ECT, �QCT, and their components� arecounterpoise corrected.

The charge transfer obtained from the fully convergedSCF calculations are corrected using the standard counter-poise correction method of Boys and Bernardi.55 In thismethod, full SCF calculations are performed for each mol-ecule with the ghost orbitals in place of all orbitals of theother molecules. All charge transfer terms calculated with thesingle RS method are BSSE corrected using similar singleRS counterpoise calculations. In this modification of theoriginal Boys–Bernardi method, calculations are also per-formed separately for each molecule and the ghost orbitalsalso replace orbitals on the rest of the molecules. However,instead of full SCF calculations, only one RS is performedby solving the amplitude equation �Eq. �30�� starting fromthe fully converged orbitals of the isolated molecule with noghost orbitals.

The difference between the Roothaan step counterpoisecorrection, �EBSSE

RS ��QBSSERS �, and the variational �Boys–

Bernardi� correction, �EBSSE ��QBSSE�, can be treated ashigher order BSSE and is generally very small �see Sec. III�.

III. RESULTS AND DISCUSSION

The ALMO CTA and ALMO EDA algorithms wereimplemented in the Q-CHEM software package.56 We alsoimplemented the Dapprich–Frenking CDA and PA methodsto compare them to the ALMO decomposition analysis. Weverified that our implementation of CDA reproduces litera-ture results.48 Charge transfer terms for the Mulliken PA andCDA are calculated according to Eqs. �A9� and �A12�, re-spectively. All charge transfer terms presented in this paperare BSSE corrected. The BSSE is presented in the tables justto show the degree of the basis set completeness.

When �QCTRS is partitioned into pairwise molecule-to-

molecule contributions, small spurious negative intramolecu-lar, �Qx→x, terms are often observed. Such negative termsarise from overlap effects and are common in all methodsthat use Mulliken-type �biorthogonal basis� treatment of theorbital overlap and disappear if Löwdin-type �symmetric or-thogonalization� partitioning is used. These negative in-trafragment transfer terms can, in principle, be removed by

excluding intramolecular excitation terms from the X opera-tor when solving Eq. �30�. However, the spurious terms areusually significantly smaller that the dominant intermolecu-lar terms and do not complicate the analysis. In order to keepthe theory simple, we did not attempt to remove them in thiswork.

A. Simple two-orbital system: Minimal basis H2

Before using ALMO CTA on real chemical systems, wewould like to apply it to a simple system of two closed shellions H− �fragment A� and H+ �fragment B� that interact with

each other to form a closed shell molecule, H2. This examplewill help us to illustrate several important concepts and toclarify the difference between the PA methods and ALMOCTA. For the sake of simplicity, we consider a restrictedminimal basis set case, so each fragment in the system hasonly one molecular orbital. The process of the H2 moleculeformation can be represented by three states schematicallyshown in Fig. 1.

�1� In the initial state of infinitely separated ions, the or-bital on H−, �A, is doubly populated and the orbital onH+, �B, is empty �Fig. 1�a��.

�2� In the intermediate state, the ions are brought to theirequilibrium geometry in the H2 molecule but their den-sities are not allowed to relax �Fig. 1�b��. The molecu-lar orbital overlap in this state is �AB= ��A �B.

�3� In the final state, RSCF, the density is relaxed to self-consistency �Fig. 1�c��.

Our main goal is to quantify the amount of charge trans-fer when the molecule is formed. To make this example moreillustrative, we compare ions A and B to two cities A and B.Figure 1 may then be regarded as showing three populationsnapshots in the history of the cities:

�1� City A is populated and city B is not built yet�Fig. 1�a��.

�2� City B is built in the close proximity of A �Fig. 1�b��but the new houses are still empty.

�3� People moved from city A to new houses in city B, andthe final population density is uniform �Fig. 1�c��.

In this analogy, finding the degree of charge transfer isequivalent to answering the question “How many peoplemoved after city B was built?.” The answer to the demo-graphic question is relatively simple: All people who live inthe new houses did move �it is, of course, assumed thatpeople who are still in A did not move from one apartment toanother�. Mathematically, the answer is the integral of thefinal population density over the white area in Fig. 1�b�. Notethat it would be incorrect to integrate over the area of city Bin an attempt of finding the answer �in fact, we do not evenhave to define the area of city B yet�.

Therefore, to find the amount of electron transfer �Q, it

is necessary to integrate the final electron distribution RSCF,over the previously “unoccupied” Hilbert subspace which is

spanned by the projected orbital B, �B:

�B = �1 − R��B =1

�1 − �AB2

��B − �A�AB� , �51�

FIG. 1. �Color online� A simple system of two subsystems.



�Q = ��BRSCF�B = 1 − �AB. �52�

Equation �52� is exactly our main working equation�Eq. �14�� applied to a two-orbital system with one occupiedand one virtual orbital. For the H2 molecule with the inter-atomic distance 0.700 Å and with the STO-3G basis setfunctions on the atoms ��AB=0.686�, the charge reorganiza-tion from the intermediate state to the final state is 0.314e.There is no electron reorganization going from the initial tothe intermediate state since the electron densities are keptfixed.

In a PA, one determines the amount of electron densitythat “belongs” to each hydrogen atom in the final state. To dothis, it is necessary to specify how the Hilbert space is par-tioned among the overlaping reference atomic orbitals. Thisproblem is similar to finding the final population of each cityin Fig. 1�c� and requires integration over the area of the city.In Mulliken PA, population of orbitals are calculated as di-agonal elements of the density matrix in the “natural”contravariant-covariant �or biorthogonal� representation. Inthe H2 case, the initial and the final population of orbital Bare 0e and 1e, respectively:

popB�RFRZ� = ��BRFRZ�B = 0, �53�

popB�RSCF� = ��BRSCF�B = 1, �54�

�B =1

1 − �AB2 ��B − �A�AB� , �55�

�Q�MPA� = popB�RSCF� − popB�RFRZ� = 1. �56�

Therefore, it is said that the formal charges on atoms in theH2 molecule are zero and that one electron is transferredfrom H− to H+ when the H2 molecule is formed �see Eq. �A9�in Appendix�.

In PA, electron transfer effects are inferred from inte-grating over the orbital �B not over the unoccupied sub-

space defined by projected orbital orbital �B. This is the

main reason for the difference between the CTA and PA re-sults. This difference is not surprising since these two meth-ods are designed to solve different problems �transfer versuspopulation�. CTA measures charge transfer effects with welldefined energies, while PA values for charge transfer are con-sistent with the formal charges on atoms. A detailed compari-son of ALMO CTA and PA is presented in the Appendix.

B. Donor-acceptor interactions in H3B–COand H3B–NH3

Donor-acceptor bonding is a central concept inchemistry.57–59 Borane compounds such as ammonia borane�H3B–NH3� and borane carbonyl �H3B–CO� are textbookexamples of donor-acceptor complexes. Donor-acceptorbonding in these compounds has been the subject of numer-ous experimental60,61 and theoretical studies.19,25,48,62–66

ALMO energy decomposition analysis25,26,66 as well as ap-plication of other EDA methods19,63 have shown that theelectronic structure of ammonia borane is correctly describedby a simple Lewis structure with the nitrogen atom donatingits lone electron pair into the empty orbital of the boronatom. The donor-acceptor bonding in borane carbonyl hasmore complicated character with significant contribution ofback-donation of the electron density from BH3 toCO.26,48,63,65

The standard Kohn–Sham DFT with the B3LYP func-tional and 6-31�+, + �G�d , p� basis set was used to obtain thecomplex geometries. Decomposition analysis was performedusing the same functional for a series of five basis sets fromthe 6-31G family with increasing number of polarized anddiffuse functions on atoms �Table I�. The ALMO decompo-sition method demonstrates very good stability: Both energyand charge components change only slightly as the basis setsize increases. As shown before,26 the total interaction ener-gies are approximately the same for H3B–NH3 andH3B–CO but the fundamental nature of donor-acceptor in-teractions is very different �Table I�. While H3B–NH3 isbetter stabilized by noncharge transfer interaction ��EFRZ

TABLE I. B3LYP ALMO CTA and EDA results for borane complexes. Geometry is optimized at B3LYP /6-31�+, + �G�d , p� level. Basis sets used for ALMOanalysis are �1� 6-31G, �2� 6-31G�d , p�, �3� 6-31�+, + �G�d , p�, �4� 6-31�+, + �G�df , pd�, and �5� 6-31�2+ ,2+ �G�df , pd�. All terms are BSSE corrected.

Basis

H3B–CO H3B–NH3

Scale �Q �me� �E �kJ/mol� �Q �me� �E �kJ/mol�

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

FRZ 0 232 274 317 313 315 0 −5 45 103 106 106POL 0 −127 −164 −203 −222 −227 0 −67 −89 −128 −132 −133RS�D→D�a −3 −5 −7 −7 −8 0 −5 −8 −5 −5 −2 0RS�D→BH3�a 46 45 46 40 41 −140 −145 −143 −126 −123 55 58 58 26 53 −113 −124 −134 −131 −130RS�BH3→D�a 196 173 180 178 178 −132 −128 −131 −129 −128 5 5 8 8 8 −6 −9 −11 −11 −11RS�BH3→BH3� −5 −4 −5 −8 −8 0 −1 −1 −1 −1 −2 0HO-CT −21 −15 −20 −22 −22 −9 −11 −8 −6 −6 −1 0 2 2 2 0 −2 −3 −3 −3TOTAL 213 195 195 182 181 −177 −173 −168 −170 −170 53 54 61 59 59 −190 −178 −172 −172 −172BSSE 7 5 1 1 1 15 11 3 5 5 10 9 3 3 3 16 13 6 7 6RS-BSSE 6 4 1 1 1 14 10 3 5 5 8 7 3 3 3 15 12 6 7 6MPA TOTAL 739 835 1005 1019 1000 N/A 319 374 543 559 435 N/AMPA Q�BH3� 53 −90 −324 −283 −416 −243 −299 −538 −551 −743

aD stands for CO in BH3–CO and for NH3 in BH3–NH3.



and �EPOL� than H3B–CO, the charge transfer effects in thelatter are stronger due to backdonation from H3B to CO.

�QCTRS terms �Table I� show no evidence of significant

backdonation in H3B–NH3, which is in excellent agreementwith the energy terms. The electronic structure of this com-plex is well described by the single RS—the higher orderrelaxation terms are very small. By contrast, the backdona-tion term in H3B–CO is stronger than forward donation andthe occupied-virtual mixing is accompanied by strong induc-tion �i.e., �QCT

HO is not negligible�.It is remarkable that the ratio of the backbonding and

forward donation terms in borane carbonyl is significantlydifferent on the charge and energy scales ��Qback /�Qforw

4, whereas �Eback /�Eforw 1�. In the perturbation theory,which describes the charge transfer correction to the zero-order polarized state �Eqs. �46� and �47��, �Q terms are pro-portional to the product of the single excitation amplitudeswith the density matrix elements ��Q�Ria� Xai�, whereas �Eterms are proportional to the product of the amplitudes withthe Fock matrix elements ��E�FiaXai�. Therefore, the ratioof the relative strength of forward and backward chargetransfer on the charge and energy scale is determined by theratio of the corresponding Fock and density matrix elements:

�Qback/�Qforw

�Eback/�Eforw�

Fforw

Fback

Rback�

Rforw�. �57�

For H3B–CO, the ratios of the matrix elements correspond-ing to the most significant charge transfer terms �see COVPsbelow� are Fforw /Fback 2 and Rback� /Rforw� 2. This exampleillustrates the value of the charge scale introduced in thiswork and shows that �-bonding �backdonation� in boranecarbonyl is approximately four times weaker than �-bonding�forward donation� if measured in terms of energy/per elec-tron transferred.

The absolute values of �Q show that intermolecularcharge transfer effects are significantly smaller than thoseinferred from the Mulliken population analysis �Table I�. TheALMO CTA intermolecular electron transfer does not exceed0.06e in H3B–NH3 and 0.20e in H3B–CO, whereas MPApredicts transfer of approximately 0.50e in H3B–NH3 and1.00e in H3B–CO. Moreover, MPA predicts that the BH3

fragment is negatively charged, in disagreement with ALMOCTA which gives stronger donation from BH3. As shown in

the Appendix, the major difference between ALMO CTA andMPA is in the treatment of virtual orbitals. ALMO CTA usesprojected virtual orbitals to ensure that all measured chargetransfer effects are only due to relaxation of the localizedoccupied orbitals into the virtual subspace. Such delocaliza-tion has a well defined energy since �E can be assigned onlyto charge transfer to the strongly orthogonal virtual subspace.In PA methods, which are originally constructed to measureformal charges on the fragments, virtual orbitals are treatedon the same footing with occupied orbitals. Therefore, for-mal charges and �Q inferred from PA methods include boththe effect of relaxation of the occupied orbitals �i.e., truedelocalization� as well as the effect of occupied-virtual over-lap �see the H2 example above and the Appendix for a de-tailed comparison of these two methods�. It is also not un-common for MPA to predict charges of the wrong sign evenin very simple systems.67

Of course, construction of the projected virtual orbitalsin ALMO methods unavoidably destroys absolute locality ofthe virtual orbitals �Eq. �11��. We verified, however, that thedegree of such a delocalization is not significant. Moreover,ALMO CTA calculations with very compact STO-3G basisset reproduce the results presented in Table I. Therefore, allcharge transfer terms in ALMO CTA have properly localizeddonor and acceptor centers. We conclude that despite thecommon view on the bonding in borane carbonyl, CO is abetter Lewis acid than BH3.

We also performed the CDA proposed by Dapprich andFrenking.48 CDA is based on the same principles as MPAand, therefore, also produces large charge transfer terms�Table II�. As demonstrated in the Appendix, some of theCDA terms do not have clear physical meaning. For ex-ample, intramolecular virtual-virtual terms do not correspondto any physical effect and can be of the same order of mag-nitude as the forward donation and backbonding terms�Table II�.

It is interesting to note that small charge transfer terms�i.e., similar to ALMO CTA� are obtained for H3B–NH3 inthe work of Mo and Gao.68 To calculate the amount of elec-tron density transferred from NH3 to BH3, these authors ap-plied natural population analysis to the polarized referenceand to the fully relaxed wavefunction and took the differenceof the charges on NH3. In this approach the polarized refer-

TABLE II. Dapprich–Frenking CDA for borane complexes. Geometry is optimized atB3LYP /6-31�+, + �G�d , p� level. Basis sets used for CDA analysis are �1� 6-31G, �2� 6-31G�d , p�, �3�6-31�+, + �G�d , p�, �4� 6-31�+, + �G�df , pd�, and �5� 6-31�2+ ,2+ �G�df , pd�. All terms are in me and are notBSSE corrected.

ScaleBasis

H3B–CO H3B–NH3

1 2 3 4 1 2 3 4

2*�Q�occ�D�→vir�BH3��a 537 538 458 432 345 411 383 376

2*�Q�occ�BH3�→vir�D��a 301 340 342 300 −4 31 −97 −154

2*�Q�occ�D�→occ�BH3��a −349 −218 −381 −437 −417 −328 −642 −688

2*�Q�vir�D�→vir�BH3��a −4 −3 −34 −31 −4 −13 −50 −61

�Q�vir�D�→vir�D��a 201 194 202 207 15 19 55 63�Q�vir�BH3�→vir�BH3�� 280 357 348 324 136 186 188 191

aD stands for CO in BH3–CO and for NH3 in BH3–NH3.



ence state also plays the key role but this method incorrectlyproduces nonzero charge transfer for the polarized state. Itwould also fail to separate �Q into forward and backdona-tion components for systems in which both effects are im-portant �e.g., H3B–CO�.

Absolutely localized complementary occupied-virtualpairs for borane complexes at the equilibrium geometry wereconstructed at the B3LYP /6-31�+, + �G�d , p� level of theory.The most significant orbitals for forward donation and back-bonding are shown in Fig. 2. In agreement with the classicaldonor-acceptor theory, the donating occupied orbitals onNH3 and CO fragments closely resemble sp3-hybridized lonepairs while the virtual orbitals are sp3-hybridized empty or-bitals on the boron atom. These pairs recover more than 95%of the overall charge transfer in this direction. The backbond-ing complementary orbital pairs in H3B–CO are doubly de-generate and each recover 48%–49% of backdonation �onlyone of the two degenerate orbitals is shown in Fig. 2�.Figure 2 indicates that backdonation occurs from the B–H�-bonding orbitals to the antibonding �* orbital of the COmolecule.

C. Bonding in carbonyl complexes

The CO ligand is one of the most important ligands inchemistry of transition metals. The metal-carbonyl bondingis a classic example of the Dewar–Chatt–Duncanson model

of synergic bonding in metal complexes.50,51 According tothis model, the carbonyl ligand donates an electron pair fromthe occupied antibonding orbital on the carbon atom �so-called 5� orbital� into the metal unoccupied orbitals, and themetal donates the electron density back from its occupiedorbitals into the carbonyl �*-antibonding orbitals.69,70 As aresult of the metal-ligand interaction, metal-carbonyl com-plexes exhibit C–O stretching frequencies that are shiftedfrom the frequency of the free CO molecule. In “classical”carbonyl complexes, the C–O stretching frequency is lowerthan in isolated CO. This shift is interpreted as a result ofstrong backdonation into the antibonding �* orbitals of theCO molecule which weakens the C–O bond.70 In anotherclass of metal-carbonyl complexes, called “nonclassical” car-bonyls, the C–O stretching mode is close to or even higher inenergy than that in free CO.71–74 This unusual behavior isattributed to donation from the CO antibonding 5� orbitalwithout compensating backdonation into the CO antibonding�* orbitals.48,71,73,74

The relative strength of forward donation and backbond-ing effects in metal-carbonyl complexes can be made quan-titative with ALMO CTA. In this work, we applied ALMOdecomposition to typical nonclassical complexes-AgCO+

and AuCO+, and to a classical carbonyl-W�CO�6. Complexgeometries were optimized using the B3LYP density func-tional with the effective core potential LANL2DZ basis forthe metal atoms and the 6-31+G�d� basis for all other atoms.The EDA calculations are also done at theB3LYP /LANL2DZ /6-31+G�d� level and the results aresummarized in Table III.

As evident from the computed bond lengths and vibra-tional frequencies �Table III�, the C–O bond, indeed, be-comes shorter and stronger in nonclassical carbonyls,AgCO+ and AuCO+. The opposite trend is observed for clas-sical W�CO�6. In agreement with the widely accepted under-standing of bonding in carbonyls, the backbonding chargetransfer energy components are smaller relative to the for-ward donation terms in nonclassical carbonyls, whereas�EM→CO

RS in the classical carbonyl is noticeably larger than�ECO→M

RS .The same trends are observed on the charge scale. The

only exception is that forward- and backdonation terms onthe charge scale in AuCO+ are of the same magnitude, whileforward donation is twice as large as backdonation on theenergy scale �for explanations, see the H3B–CO exampleabove�. To verify that our method properly separatesforward-donation and backbonding terms in AuCO+ we ap-plied ALMO CTA to AuCN and compared results for thesetwo complexes �Table III�. CO is known as a strong, wellbalanced �-donor and �-acceptor, while the CN− ligand is agood �-donor and very poor �-acceptor. ALMO CTA cor-rectly predicts large forward-donation and negligible back-bonding for AuCN.

Based on the results of ALMO CTA and ALMO EDA,we conclude that backbonding in nonclassical carbonyls isnot negligible even though this effect is smaller than forwarddonation. Backdonation to the CO virtual antibonding orbit-als does not fully compensate for forward donation from theoccupied CO orbitals in nonclassical carbonyls. The back-

FIG. 2. �Color� Significant COVPs in borane complexes calculated at theB3LYP /6-31�+, + �G�d , p� level. Isovalue surface of 0.15 a.u. Occupied or-bitals are represented with saturated colors. Faint colors represent comple-mentary virtual orbitals. D stands for CO in BH3-CO and for NH3 inBH3-NH3.



bonding term in the classical tungsten carbonyl is signifi-cantly stronger that the forward donation term. The relativestrength of these terms explain blue and red shifts in vibra-tional spectra of coordinated CO in nonclassical and classicalcarbonyls.74

This explanation is supported by the more detailedCOVP analysis which shows that in both classes of carbonylcomplexes almost 100% of forward donation is recoveredwith a single COVP. The shape of the occupied orbital of thisCOVP �Fig. 3� is exactly the shape of the HOMO of the COmolecule �5� orbital�. The backdonation to the CO antibond-ing doubly degenerate �* orbitals recovers 100% of thetransfer in classical carbonyls and 77% in nonclassical car-bonyls. The remaining 23% of backdonation in nonclassicalcomplexes is due to the transfer into the �* orbital of CO�Fig. 3�.

D. Multiple intermolecular bonds in a complexof isocyanuric acid and melamine

Isocyanuric acid �1,3,5-triazine-2,4,6-trione� andmelamine �1,3,5-triazine-2,4,6-triamine� are thought to be amajor cause of two outbreaks of pet food-associated renalfailure in cats and dogs.75 Although neither of these chemi-cals is individually toxic, it is believed that their potencymay increase when they are present together.76 Isocyanuricacid �ICA� and melamine �MA� can form networks of hydro-gen bonds, creating a tilelike planar structure through mo-lecular self-assembly.1 Three hydrogen bonds formed in theplanar ICA-MA complex are shown in Fig. 4 �complex 1�.We apply ALMO decomposition to isolate electron transfereffects associated with individual hydrogen bonds in thecomplex. To verify accuracy of the decomposition, we alsocompare ALMO CTA results for the planar complex and thecomplex in which ICA and MA molecules lie in perpendicu-lar planes �complex 2� and in which only one hydrogen bondbetween the interacting molecules is formed.

Geometry optimization and EDA calculations were per-

formed using the B3LYP density functional with the6-31�+, + �G�d , p� basis set for all atoms in the system. Theresults are summarized in Table IV.

In the planar complex, charge transfer from MA to ICAis 19.2 me and charge transfer in the opposite direction is8.0 me. The energy lowering corresponding to these delocal-ization effects is proportional to the amount of charge trans-fer and are 32.3 kJ /mol and 15.3 kJ /mol, respectively. In-tramolecular effects and the higher order relaxation aresmall. Charge and energy terms in each directions can befurther decomposed into molecular orbital contributions.COVP decomposition shows that 93% of �EMA→ICA

RS and95% of �QMA→ICA

RS result from electron donation from thenitrogen’s lone pair in MA to the N–H �-antibonding orbitalin ICA �Fig. 5�. Two COVPs contribute to this effect. Occu-pied orbitals in these COVPs are different linear combina-tions of sp2-hybridized lone pair orbitals on nitrogen atomsin the aromatic ring of MA. Charge transfer in the oppositedirection is primarily due to interactions of the lone pairs onoxygen atoms in ICA with N–H �-antibonding orbitals inMA �Fig. 5�. Again, two COVPs, symmetric and antisym-metric �Fig. 5�, contribute to this effect. They account for64% and 71% of �EICA→MA

RS and �QICA→MARS , respectively.

The remaining charge transfer in this direction is covered bythe other 31 COVPs, none of which exceeds 12% and 10%of �EICA→MA

RS and �QICA→MARS , respectively.

The ALMO decomposition results for complex 1 can becompared to the decomposition for complex 2, in which onlyone hydrogen bond is formed between the interacting frag-ments. The difference in �EFRZ and �EPOL between complex1 and complex 2 can be explained by purely electrostaticeffects: Oxygen and hygrogen atoms of the peripheral hydro-gen bonds have partial charges of the opposite sign and arecloser to each other in complex 1. This leads to strongerelectrostatic attraction between MA and ICA and also tostronger mutual polarization in complex 1 compared to com-plex 2. As expected, charge transfer from ICA to MA incomplex 2 is negligible and charge transfer in the oppositedirection is very close to that in complex 1. COVP decom-

TABLE III. ALMO CTA and EDA results for carbonyl complexes. Geometry optimization and decomposition analysis are performed atB3LYP /LANL2DZ /6-31+G�d� level. All terms are BSSE corrected.

Scale

Ag+–CO Au+–CO �CO�5W–CO Au+–CN−

�Q �me� �E �kJ/mol� �Q �me� �E �kJ/mol� �Q �me� �E �kJ/mol� �Q �me� �E �kJ/mol�

frz 0.0 22.0 0.0 180.3 0.0 138.4 0.0 −388.4pol 0.0 −60.8 0.0 −159.1 0.0 −68.7 0.0 −205.4RS�M→M� −0.4 0.0 5.3 0.0 −2.8 0.0 14.0 0.0RS�M→L�a 12.0 −14.1 55.5 −66.9 245.9 −141.7 16.7 −31.2RS�L→M�a 27.8 −37.0 58.9 −132.8 37.3 −101.0 205.7 −259.3RS�L→L�a −12.9 0.0 −6.5 0.0 8.2 0.0 −46.7 0.0HO-CT 5.9 −3.1 33.5 −17.2 −27.7 −11.8 −5.9 −0.8TOTAL 32.3 �93.1 146.8 �195.8 260.9 �184.8 183.6 �885.2BSSE 0.9 1.9 0.8 2.4 2.2 6.6 1.1 2.6RS-BSSE 0.8 1.8 0.7 2.3 2.0 6.4 1.0 2.5d�C–O /N� �Å�b 1.126 1.126 1.151 1.166�C–O /N� �cm−1�b 2305 2302 2056 2273

aL stands for CO in the carbonyl complexes and for CN− in the Au+–CN− complex.bd�C–O�=1.137 Å and �C–O�=2203 cm−1 in the uncoordinated CO molecule. d�C–N�=1.183 Å and �C–N�=2125 cm−1 in the uncoordinated CN− anion.



position for MA→ ICA terms in both complexes are almostidentical. Small electron transfer effects from ICA to MA incomplex 2 do not have dominant terms in the COVP decom-position.

Based on these results we conclude that in the planarcomplex delocalization effects for the central N¯H hydro-gen bond is approximately four to five times stronger�19.2 me and 32.3 kJ /mol� than for the peripheral O¯Hbond �4.0 me and 7.7 kJ /mol�.

IV. CONCLUSIONS

Energy decomposition methods based on absolutely lo-calized orbitals �ALMO� have been succesful in studying

and analyzing binding mechanisms in numerous molecularcomplexes.25,26,66–68,77–79 The ALMO energy decompositionanalysis �EDA� �Refs. 25 and 26� enables accurate separationof the total intermolecular interaction energy into intramo-lecular �frozen density and polarization� terms and intermo-lecular �charge transfer� energy terms.

In this paper, we have extended decomposition analysismethod based on absolutely localized molecular orbitals�ALMO�. First, we proposed a quantitative scale to measureintermolecular charge transfer effects. All terms on this scalesuch as forward-donation, backdonation, and higher orderrelaxation have well defined energetic effects consistent withthe previously defined energy terms in ALMO EDA. Second,we introduced concept of chemically significant complemen-tary occupied-virtual orbital pairs which provides a simpledescription of electron transfer effects in intermolecular in-teractions.

The newly proposed generalized ALMO decompositionmethod has been tested successfully on systems involvinghydrogen bonding and donor-acceptor interactions. For casessuch as borane complexes, metal carbonyls, and complexeswith hydrogen bonding ALMO decomposition analysis re-

FIG. 3. �Color� Significant COVPs in metal-carbonyl complexes calculatedat the B3LYP /LANL2DZ /6-31�+�G�d� level. Isovalue surface of 0.1 a.u.Occupied orbitals are represented with saturated colors. Faint colors repre-sent complementary virtual orbitals.

FIG. 4. Complex of isocyanuric acid and melamine �complex 1�.

TABLE IV. ALMO CTA and EDA results for complexes of isocyanuric acid�ICA� with melamine �MA�. Geometry optimization and decompositionanalysis are performed at B3LYP /6-31�+, + �G�d , p� level. All terms areBSSE corrected.

Scale

Complex 1 Complex 2

�Q �me� �E �kJ/mol� �Q �me� �E �kJ/mol�

FRZ 0.0 2.8 0.0 18.8POL 0.0 −29.2 0.0 −18.2RS�MA→MA� −2.5 0.0 −3.1 0.0RS�MA→ ICA� 19.2 −32.3 18.4 −30.0RS�ICA→MA� 8.0 −15.3 0.4 −1.2RS�ICA→ ICA� −1.1 0.0 0.1 0.0HO-CT 3.1 −2.9 1.6 −1.7TOTAL 26.8 �76.1 17.4 �32.4BSSE 1.1 4.7 1.1 3.3RS-BSSE 1.1 4.6 1.0 3.2



FIG. 5. �Color� Significant COVPs in complexes of isocyanuric acid �ICA� and melamine �MA� calculated at the B3LYP /6-31�+, + �G�d , p� level. Isovaluesurface of 0.07 a.u. Occupied orbtials are represented with saturated colors. Faint colors represent complementary virtual orbitals.



sults are broadly consistent with existing understanding ofintermolecular bonding. However, according to ALMOcharge transfer analysis the amount of electron density trans-ferred between molecules is significantly smaller than chargetransfer estimated from various population analysis methods.Additionally, charge transfer and the associated energy low-ering are by no means directly proportional to each other, asmight sometimes be naively assumed. For example inH3B–CO, the magnitude of charge backdonated by BH3 ex-ceeds that forward donated by CO by a factor of approxi-mately four, although forward and back-donation yield simi-lar energy lowering.

ACKNOWLEDGMENTS

This work was supported by the Director, Office of BasicEnergy Sciences, Chemical Sciences Division of theU.S. Department of Energy under Contract No. DE-AC03-76SF00098, by the National Science Foundation underGrant No. CHE-0535710, and by BP through the MethaneConversion Cooperative.

APPENDIX:

Comparison of ALMO CTA to population analysisand charge decomposition analysis

As we show in Sec. II, charge transfer effects measuredin the ALMO CTA methods are different from charge trans-fer inferred from the population analysis methods. The majordifference between ALMO CTA and population analysis isthe way the Hilbert space of the molecular complex is parti-tioned. The population analysis methods are specifically de-signed to calculate formal charges on molecules in molecularcomplexes �see below�. In these methods, the entire Hilbertspace spanned by nonorthogonal reference orbitals is divided

into molecular subspaces with partition operators Px�pa�:

Px�pa� = �

p

nx

�xp��xp � �Px�pa��†. �A1�

Although these operators satisfy the idempotency and reso-lution of identity conditions

�Px�pa��2 = Px

�pa�, �A2�

�x

Px�pa� = 1, �A3�

they do not represent true projectors since they are generallynon-Hermitian. Therefore, we say that the Hilbert space isvaguely partitioned into molecular subspaces in the PA meth-ods.

In contrast, ALMO CTA is constructed to measure theamount of charge reorganization in the final electronic statewith respect to some reference state. For the final electronicstate, this method calculates the amount of the electron den-sity that occupies the initially unoccupied subspace �seeEq. �14��. Therefore, ALMO CTA first of all partitions theentire Hilbert space of the reference state into the stronglyorthogonal occupied and virtual subspaces with true projec-

tors R and Q. Only then, if one is interested in forward do-nation and backbonding components of the total chargetransfer term, both the occupied and virtual subspaces arevaguely partitioned into molecular subspaces with operators

Px�o� and Px

�v�:

Px�o� � �

i

ox

�xi��xi = �i

ox

Pxi, �A4�

Px�v� � �

a

vx

�xa��xa = �a

vx

Pxa. �A5�

Non-Hermitian idempotent operators Px�o� and Px

�v� are not

true projectors. In this sense, they are similar to Px�pa�. How-

ever, they sum up to projectors R and Q which do define theoccupied and virtual subspaces:

�x

Px�o� = R , �A6�

�x

Px�v� = Q . �A7�

It is important to note that the proper definition of thesesubspaces allows one to assign the energetic components tocharge transfer terms because the corresponding energy low-ering is unambiguously defined only for charge transfer fromthe occupied subspace to the unoccupied subspace�Eq. �21��. The lack of well defined occupied and unoccupiedsubspaces in the population analysis makes these methodsinapplicable for studying charge transfer. Thus, ALMO CTAgives the correct intermolecular electron transfer and the en-ergy associated with it, whereas PA methods produce theproper formal charges on molecules.

The CDA method48 attempts to quantify the degree ofelectron delocalization in molecular complexes. As we showbelow, CDA uses partitioning operators similar to theMulliken population analysis operators �Eq. �A1�� and, there-fore, suffers from the same shortcomings as the PA methodswhen it is applied to study electron density relaxation. CDAemploys the frozen density state as a reference state, whereasALMO CTA uses the polarized state. The latter is a moreappropriate reference for studying intermolecular electrondensity transfer since the results are not contaminated byintramolecular relaxation effects. Another serious drawbackof CDA is that some terms obtained in this method do nothave clear physical interpretation.

A detailed mathematical description of the PA methodsand CDA is presented below.

Population analysis

Any one-electron density operator Rf �usually con-structed from delocalized occupied molecular orbitals� canbe the subject of population analysis �PA� in some localizedbasis set, �xp. The electronic population of molecule x isusually defined as the expectation value of �a generally non-

Hermitian� operator �Px�pa�RfPx

�pa��:



popx�Rf� = �p

nx

��xpRf�xp = Tr�Px�pa�RfPx

�pa�� , �A8�

where �xp denotes basis orbitals and Px�pa� is the partition

operator associated with the Hilbert subspace of molecule x�Eq. �A1�� that must satisfy the idempotency �Eq. �A2�� andresolution of identity �Eq. �A3�� conditions. The formal

charge of molecule x is obtained by summing popx�Rf� andthe nuclear charges of the atoms in the molecule.

If nonorthogonal canonical orbitals of isolated mol-

ecules, which define the RFRZ state, are used as basis func-tions, Eq. �A8� represents the Mulliken population analysis�MPA�.34,35 If the basis orbitals are symmetrically orthogo-nalized then the partiton operators become Hermitian, andEq. �A8� describes Löwdin population analysis.36 Weinhold’sanalysis49 utilizes a basis set of so-called natural bond orbit-als that are orthogonal by construction �expanded in terms ofthe underlying basis set of occupancy-weighted symmetri-cally orthogonalized natural atomic orbitals� and can also bedescribed by Eq. �A8�. It is important to state that there is noclearly “best” method for allocating a charge distributionamong nonorthogonal subspaces associated with orbitals, at-oms or molecules. For example, Mulliken analysis can pro-duce orbital populations outside physical range �0,2�,whereas in Löwdin analysis the basis functions are notstrictly localized after the symmetric orthogonalization isperformed �the same is true for natural bond orbitals�.

Formal charges on molecules obtained from populationanalysis do not provide full information about electron reor-ganization and binding in the system. For example, in a com-plex with synergic bonding, the analysis of molecular formalcharges does not capture full complexity of charge transfereffects in the complex. In population analysis methods, thetotal charge transfer is calculated as the population of allvirtual orbitals in the final state:

�Q�pa� � �x

popx�v��Rf�

= �x

Tr�Vx�pa�RfVx

�pa��

= − �x

�Tr�Ox�pa�RfOx

�pa�� − ox� , �A9�

where partition operators Ox�pa� and Vx

�pa� �Ox�pa�+ Vx

�pa�

= Px�pa�� are associated with the occupied and virtual orbitals

of molecule x, respectively,

Ox�pa� = �

i

ox

�xi��xi , �A10�

Vx�pa� = �

a

vx

�xa��xa . �A11�

Note that operators Ox�pa� and Vx

�pa� are not projectors onto theoccupied and virtual subspaces. Their sums over all mol-ecules do not give proper projectors either �compare them to

operators Px�o� and Px

�v� defined in the context of ALMO CTAin Eqs. �A4� and �A5��.

Charge decomposition analysis „CDA…

For systems comprising more than two molecules, it isoften important to know not only how much charge is trans-ferred to a molecule but also where it came from, i.e., to findforward donation and backbonding components for each pairof molecules in the system. The PA methods are unable toprovide this level of detail. In the CDA method of Dapprichand Frenking,48 the amount of charge transferred from theoccupied orbitals on molecule x to the virtual orbitals onmolecule y is defined as follows:

�Qocc�x�→vir�y��CDA� = �

i

ox

�a

vy

��yaRf�xi�xi,ya

= Tr�Vy�MPA�Rf�Ox

�MPA��†� . �A12�

The basis functions used in CDA are converged canonicalorbitals of isolated fragments �FRZ basis� and a Mullikenpartitioning �denoted with the MPA superscript� is used.

The CDA has proved useful for comparative studies ofcharge transfer but there are several problems with adoptingsuch a definition for �Qocc�x�→vir�y�

�CDA� . First, the CDA definition

of the charge transfer components is not consistent with thedefinition of �Q�MPA� given by Eq. �A9� even though thestarting point for deriving the CDA expression is theMulliken PA,

�x,y

�Qocc�x�→vir�y��CDA� � �Q�MPA�. �A13�

Second, it is very hard to assign physical meaning tooften large negative terms corresponding to mixing occupiedorbitals on one fragment with occupied orbitals on another:

�Qocc�x�→occ�y��CDA� = �

i

ox

�j

oy

��yjRf�xi�xi,yj

= Tr�Oy�MPA�Rf�Ox

�MPA��†� . �A14�

The original assignment of these terms to polarization effectsdoes not agree with a widely accepted view that polarizationeffects correspond to occupied-virtual orbital mixing withina molecule. Even though the reference orbitals are not opti-mized for intrafragment relaxation in CDA, intrafragmentoccupied-virtual charge transfer terms are always zero in thismethod. This is because orbitals within a molecule can al-ways be orthogonalized thus giving zero � matrix elementsin Eq. �A12� for �Qocc�x�→vir�x�

�CDA� .

Third, large intermolecular virtual-virtual terms��Qvir�x�→vir�y�

�CDA� � and �generally small� intramolecular virtual-

virtual ��Qvir�x�→vir�x��CDA� � do not have a clear physical meaning

either.Finally, CDA does not produce correct charge transfer

results in the limit of orthogonal basis orbitals because of thedirect inclusion of the overlap matrix into each charge trans-fer term. Therefore, zero charge transfer is produced for any



density matrix, Rf, for all methods in which virtual orbitalsare orthogonal to the occupied orbitals, and for all systems inwhich charge transfer occurs between spatially separatedfragments.

Most importantly population analysis and CDA methodsdo not define occupied and virtual subspaces for thereference state and, therefore, produce electron densityreorganization effects without corresponding well definedenergies of charge transfer.

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