Dispersion-Corrected Energy Decomposition Analysis for

ARTICLE pubs.acs.org/JPCA

Dispersion-Corrected Energy Decomposition Analysis for Intermolecular Interactions Based on the BLW and dDXDM Methods Stephan N. Steinmann,† Clemence Corminboeuf,*,† Wei Wu,‡ and Yirong Mo*,§ †

Laboratory for Computational Molecular Design, Institut des Sciences et Ing
enierie Chimiques, Ecole Polytechnique F
ed
erale de Lausanne, CH-1015 Lausanne, Switzerland ‡ State Key Laboratory for Physical Chemistry of Solid Surfaces and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, Fujian 361005, China § Department of Chemistry, Western Michigan University, Kalamazoo, Michigan 49008, United States ABSTRACT: As the simplest variant of the valence bond (VB) theory, the blocklocalized wave function (BLW) method defines the intermediate electron-localized state self-consistently at the DFT level and can be used to explore the nature of intermolecular interactions in terms of several physically intuitive energy components. Yet, it is unclear how the dispersion interaction affects such a kind of energy decomposition analysis (EDA) as standard density functional approximations neglect the long-range dispersion attractive interactions. Three electron densities corresponding to the initial electron-localized state, optimal electron-localized state, and final electron-delocalized state are involved in the BLW-ED approach; a density-dependent dispersion correction, such as the recently proposed dDXDM approach, can thus uniquely probe the impact of the long-range dispersion effect on EDA results computed at the DFT level. In this paper, we incorporate the dDXDM dispersion corrections into the BLW-ED approach and investigate a range of representative systems such as hydrogen-bonding systems, acid
base pairs, and van der Waals complexes. Results show that both the polarization and chargetransfer energies are little affected by the inclusion of the long-range dispersion effect, which thus can be regarded as an independent energy component in EDA.

’ INTRODUCTION Understanding the nature of intermolecular interactions, particularly noncovalent interactions, is essential not only for the rational design of novel materials such as self-assembling polymers and peptides and receptors (e.g., drugs) to targeted proteins but also for the development of next-generation force fields, which are indispensable in the computational simulations of nanomaterials and biosystems.1
10 While the molecular binding energy is often measurable experimentally in many ways, the identification and evaluation of various contributions of different origins to this energetic value are nevertheless neither trivial nor stringent. So far, a variety of computational approaches have been proposed to decipher the physical principles governing intermolecular interactions. All of these approaches, in general, fall into two categories, namely, the supermolecular methods11
23 and perturbation methods.24
27 For a complex (supermolecule) composed of a few monomers, the overall binding energy can be determined as the energy change from departed monomers to the supermolecule. The supermolecular methods are designed to interpret the binding energy in a number of physically meaningful energy terms. Morokuma and Kitaura made significant contribution to the field of energy decomposition analysis (EDA) by partitioning the interaction energy at the HF level of theory into the electrostatic, exchange, polarization, and charge-transfer (CT) components.11
13 The Kitaura
Morokuma (KM) scheme has been extensively applied to particularly hydrogenbonding systems. To differentiate the intra-unit charge polarization r 2011 American Chemical Society

and inter-unit electron transfer in metal
ligand complexes, Bagus et al. proposed a constrained space orbital variations (CSOV) method that allows the extension of occupied molecular orbitals (MOs) of one unit to the virtual MOs of its own and the other unit in the complex.15,28 The further application of the CSOV method at the DFT level showed that the inclusion of electron correlation normally increases the CT stabilization energy, but the physical mechanisms derived from the CSOV analyses at both HF and DFT levels are the same.29 Similarly, Stevens and Fink developed the reduced variational space self-consistent field (RVS SCF) method, where the MOs of one fragment are optimized in the field of the frozen orbitals of the other fragment.16 Both the KM and RVS SCF decomposition schemes have been extended to many-body systems.18 On the basis of the natural bond orbital (NBO) approach,30,31 Glendening formulated the natural energy decomposition analysis (NEDA) method, where the interaction energy is partitioned into electrostatic, CT, and deformation components.17,32 At the DFT level, Ziegler et al. developed the extended transitionstate (ETS) scheme, which divides the total interaction energy into electrostatic interaction, Pauli interaction, and orbital interaction energies.14,33,34 The further combination of the ETS method with the natural orbitals for chemical valence (NOCV) theory35 leads to Received: March 17, 2011 Revised: April 25, 2011 Published: May 10, 2011 5467

dx.doi.org/10.1021/jp202560d | J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A the decomposition of the electron deformation density and bond energy into various components of the chemical bond such as σ, π, δ, and so forth.36 By similarly defining an intermediate electronlocalized state, Wu et al. recently proposed a density-based EDA scheme that decomposes the binding energy into preparation, electrostatic, Pauli repulsion, polarization, CT, and BSSE energy terms.22 An EDA based on fragment-localized Kohn
Sham orbitals has also been developed by Reinhardt.21 According to valence bond (VB) theory,37
41 a molecule can be described by a set of Lewis structures where electrons are localized on bonds or atoms. The comparison between the mostcontributing Lewis structure and the true ground state, where all electrons are delocalized, highlights the CT effect (or resonance energy in conjugated systems). Thus, the primary Lewis state can best serve as the intermediate state in the study of CT. Following this strategy, some of us proposed the block-localized wave function (BLW) method, which can derive the variationally optimal wave function for the intermediate electron-localized state42,43 and subsequently be applied to the decomposition of intermolecular interactions in many-body systems,20,44,45 apart from many other applications.46
52 Later, the BLW strategy was extended to the Kohn
Sham DFT (first named BLW-DFT53 by Gao but later BLDFT54 by Gao; to reduce the confusion, we recommend the use of BLW and multi-BLW in cases of using several BLWs for the sake of simplicity and generality), and thus, the EDA computations with BLW (BLW-ED in short) can be performed at the DFT level. This BLW-ED procedure recently was reintroduced by Khaliullin et al. under the name of absolutely localized molecular orbitals energy decomposition analysis (ALMO-EDA).55 The above supermolecular methods are essentially established at the HF or DFT levels. At the HF level, an additional electron correlation term can be defined to account for the binding energy difference between the HF and higher correlated methods. Common DFT methods account for a significant amount of short-range dispersion energy but neglect long-range dispersion.56,57 As the latter may contribute considerably to the preparatory, intermediate electron-localized, and final electron-delocalized states, it is of significant interest to examine how the long-range dispersion effect affects the energy terms in EDA computations. Notably, dispersion plays a dominating role in weak interactions. The inclusion of dispersion forces in EDA is thus critical for the computational simulations of biological systems as well as for the screening or design of adsorbents for hydrogen storage and carbon capture.58 EDA schemes based on the perturbation theory have the advantage of taking the electron correlation directly into account by decomposing the overall intermolecular interaction energy into firstorder and second-order interaction energy contributions.24,25,59,60 In the symmetry-adapted perturbation theory (SAPT) method,26,61
64 the first-order polarization and exchange corrections are usually interpreted as the electrostatic and exchange energy terms, respectively, while the second-order corrections consist of induction and dispersion contributions. However, the induction energy, like the orbital interaction energy term in the ETS-EDA approach,14,33,34 is actually composed of polarization and CT contributions. As both polarization and CT interactions are essential in chemistry, ideally, these two terms had better be defined separately. An attempt to derive the CT interaction as a separate term in the SAPT framework has been recently proposed, but the scheme lacks the rigorous characteristic of the original theory.65 We also note that three-body nonadditive contributions are already highly nontrivial in SAPT66,67 and that even more complex higher-order corrections are expected.

ARTICLE

Standard DFT methods are known for the inability to describe long-range attractive dispersion interactions. A simple remedy that is gaining momentum is to add a damped pair-potential empirical term to the DFT energy a posteriori.68
72 Though accurate for the treatment of intermolecular interactions, most of these DFT-D schemes employ system-independent dispersion coefficients and thus are unable to provide deep insights into the impact of dispersion corrections on individual energy terms in the EDA. For the instance of the BLW-ED, at the same nuclear arrangement, there are three different electron densities corresponding to the initial electron-localized state, optimal electronlocalized state, and electron-delocalized states involved. It is highly desirable to adopt corrections with system-dependent dispersion coefficients into the EDA procedure. Most recently, some of us proposed a scheme, or dDXDM in short,73 on the basis of Becke and Johnson exchange-hole-dipole moment (XDM) formalism.74
77 The dDXDM correction that derives density-dependent dispersion coefficients in combination with a density-dependent damping function (dD) is thus ideal for the study of intermolecular interactions in dispersion-corrected energy terms, which are still out of reach so far. In this paper, we describe the combination of the BLW method with the dDXDM correction in the exploration of intermolecular interactions by means of EDA. In the following section, the theories will be described briefly as details on the BLW20,42,43,53,78 and dDXDM73 methods can be found in the literature. Computations on a range of exemplary systems including hydrogen-bonding systems, acid
base pairs, and van der Waals complexes with discussion will be presented afterward, followed by a final conclusion.

’ THEORY (i). Block-Localized Wave Function (BLW) Method. The independent estimates of the polarization and CT effects require a proper definition of the intermediate electron-localized state, or a resonance state in the terminology of VB theory, which can be expressed with a Heitler
London
Slater
Pauling (HLSP) function.37,38 Each HLSP function can be expanded into 2N/2
S Slater determinants (where N and S are the total number of electrons and spin quantum numbers, respectively). The major computational obstacle for ab initio VB methods, however, comes from the nonorthogonality of orbitals. However, we note that ab initio VB theory has been rejuvenated remarkably in the past two decades with a few practical programs including XMVB.79
90 The popularity of MO methods benefits from the computational efficiency due to the orthogonality constraint imposed on orbitals. Thus, a viable way to simplify the VB computations is the partial adoption of the orthogonality constraint. One successful approach in this regard is the GVB method,91,92 which retains the VB form for only one or a few focused bonds (perfect pairs) but accommodates the remaining electrons with orthogonal and doubly occupied MOs. An alternative combination of the VB and MO methods is to represent bond orbitals with nonorthogonal doubly occupied fragment-localized orbitals (or group functions93).94
105 In line with the conventional VB ideas, we have developed a BLW method where each BLW corresponds to a unique electron-localized diabatic state.20,42,43,53 The fundamental assumption in the BLW method is that the total electrons and primitive basis functions can be divided into a few subgroups (blocks), and each subgroup corresponds to a monomer in the study of intermolecular interactions. Orbitals in the same block 5468

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A

ARTICLE

are subject to the orthogonality constraint, but orbitals belonging to different subspaces are nonorthogonal. Thus, the BLW method combines the advantages of both MO and VB theories. For an interacting system composed of k monomers, the matrix of orbital coefficients in the BLW method is a direct sum of k submatrixes k

CBLW ¼ C1 x C2 x ... x Ck ¼ x Ci i¼1

ð1Þ

and at both the HF and KS DFT levels, the total electron density is a simple summation of individual parts as FBLW ¼ F1 þ F2 þ ...þ Fk ¼

k

∑ Fi i¼1

ð2Þ

With the definition of the intermediate state with BLW where electron transfer among monomers is quenched, the intermolecular binding energy ΔEb can be decomposed into a series of successive but hypothetical steps that are characteristic of the reorganization of electrons among the interacting monomers ΔEb

¼ ESuper


k

∑ EMono ðiÞ ¼ ΔEdef þ ΔEs þ ΔEpol i¼1

þ ΔECT ¼ ΔEdef þ ΔEInt

k

F0i ∑ i¼1

ð4Þ

whose corresponding energy E[FBLW0] within the DFT theory corresponds to the initial undisturbed electron-localized state. The energy difference between E[FBLW0] and separated monomers can be generally defined as the steric interaction (ΔEs). Previously, we also called this energy term as the Heitler
London energy, which is originally defined as the first-order energy from the perturbation calculation without perturbation effect106 within the MO theory and can be further decomposed into the electrostatic and Pauli exchange energy terms as in the KM11
13 and the ETS-EDA14,33,34,107,108 or other22 implementations. However, within the DFT formulation, the energy change by bringing monomers together without disturbing their individual densities involves electron correlation. The actual fraction corresponding to the “overlap dispersion” that is recovered by standard semilocal density functionals can not be separated out within our scheme, but the use of dispersionless density functionals, free from overlap dispersion, could be an alternative.109 The relaxation of the electron densities within each monomer stabilizes the supermolecule and leads to the self-consistently optimized BLW with the electron density, as shown in eq 2, and the stabilization energy is defined as the polarization energy ΔEpol ¼ E½FBLW 
E½FBLW0 

ΔECT ¼ E½F
E½FBLW  þ BSSE

ð5Þ

We note that individual polarization energies for monomers can be easily computed by restricting the relaxation of the electron density to only one monomer.

ð6Þ

The BLW code with the geometry optimization capability has been ported to the GAMESS software111 in our laboratories. The optimal electron-localized state actually corresponds to a van der Waals complex where the CT effect is absent. Because the BLW method is based on the partition of basis functions, a certain basis set artifact is expected. However, our extensive computations of the resonance energies (equivalent to the CT energy term in the current BLW-ED approach) in conjugated systems demonstrated that the BLW method, when employed with midsized basis sets, generates reasonably levelinvariant values consistent with the best evaluations based on experimental data and MO-based computations.51 (ii). dDXDM Method. The dDXDM correction adopts the Tang
Toennies (TT) damping function112

ð3Þ

The first term (ΔEdef) in the above equation refers to the energy consumption in the unfavorable geometry deformations for all monomers from their respective free and optimal states to the distorted geometries in the optimal structure of the supermolecule. If we bring the distorted monomers together to form the supermolecule without further perturbing the structures as well as the electron densities of monomers, the subsequent total electron density is the summation of individual contributions as FBLW0 ¼ F01 þ F02 þ ...þ F0k ¼

As the final step, we expand the electron movements from blocklocalized orbitals to the whole supermolecule. This expansion further stabilizes the complex and results in the electron-delocalized state with the electron density F. At this stage, the BSSE is introduced. As a consequence, we assign the BSSE correction110 completely to the CT energy term, which is defined as

Edisp; dDXDM ¼


M i
1 5

ij

∑ ∑ ∑ f2nðbRijÞRij2n2n i¼2 j¼1 n¼3 C

ð7Þ

where M is the number of atoms in the system, b is the damping factor, f2n is the damping function and Cij2n is the dispersion coefficient. According to Becke and Johnson’s XDM formalism,74,76 Cij2n relies on atomic polarizabilities and on the XDM, thereby depending on the electron density. This density-dependent XDM scheme has been implemented in Q-Chem77,113 and a simplification recently formulated.114 Uniquely, the damping of an atom pairwise dispersion correction in the dDXDM scheme depends on Hirshfeld (overlap) populations rather than on “critical” or “van der Waals” radii. The damping factor b is a function of two fitted damping parameters that are the strength of the TT damping in the medium range and the steepness factor in the short range. For details on the dispersion correction, see ref 73. (iii). BLW-dDXDM Scheme. The intermolecular interaction energy decomposition analysis based on the BLW method requires the establishment of three different states, namely, the initial electron-localized state (BLW0), optimal electron-localized state (BLW), and final electron-delocalized state (standard DFT). These three states correspond to different electron densities (FBLW0, FBLW, and F) at the same nuclear arrangement. The incorporation of density-dependent dDXDM corrections into the BLW-ED thus can generate long-range dispersioncorrected energy terms, and the comparison with uncorrected energy terms can reveal the significance of dispersion interactions on EDA results. This information is critical for the reliability of standard EDA schemes, whose results are broadly adopted for the understanding of molecular bonding natures and the development of force fields. As both the ALMO-EDA scheme,55 which is identical to the BLW-ED scheme, and the XDM correction have been implemented in the Q-Chem package,115 the dDXDM corrections were computed with this code in this work. The BSSE correction was computed without the dispersion correction. Tight convergence criteria (max DIIS error < 10
8), integral thresholds (10
12), and grid settings (99/590 Euler-
Maclaurin
Lebedev) were used. The 5469

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A

ARTICLE

Table 1. BLW-ED (kcal/mol) of a Water Dimer with the dDXDM Corrections at the B3LYP Level with the Basis Sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3) basis set BS1

BS2

BS3

Table 2. BLW-ED (kcal/mol) of a Water Trimer with the dDXDM Correction at the B3LYP Level with the Basis Sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3)

ΔEs

ΔEpol

ΔECT

ΔEInt

basis set

BLW


2.93


0.80


1.88


5.61

BS1

BLW-dDXDM


3.07


0.78


1.89


5.74

correction


0.14

0.02


0.01


0.13

BLW


2.20


0.89


1.40


4.49

BLW-dDXDM correction


2.34
0.14


0.87 0.02


1.41
0.01


4.62
0.13

BLW


2.93


0.85


1.22


5.00

BLW


6.35


4.02


5.08


15.45

BLW-dDXDM


3.24


0.82


1.23


5.29

BLW-dDXDM


7.44


3.84


5.15


16.43

correction


0.31

0.03


0.01


0.29

correction


1.09

0.18


0.07


1.02

method

Figure 1. BLW-ED for the water dimer with the B3LYP and PBE functionals and basis sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3). For each pair of bars, the right/left one refer to the BLW-ED with/without the dDXDM corrections.

regular BLW computations including geometry optimizations and vibrational frequencies were performed with the GAMESS software111 with the “in-house” modifications. In this paper, we focused on the decomposition of molecular interaction energy (ΔEInt) in terms of steric, polarization, and CT as shown in eq 3, and computations with two kinds of functionals (B3LYP and PBE) and three choices of basis sets (BS1: 6-31G(d); BS2: cc-pVTZ; and BS3: 6-311þG(d,p)) were run at the geometries either taken from the S22 test set116 (including base pairs, benzene dimers, and the benzene
water complex) or optimized at MP2/6-311þG(d,p) otherwise. Benchmark values for the S22 test set were taken from ref 117 and computed at the counterpoise-corrected level of df-MP2/ CBS (aug-cc-pVTZ/aug-cc-pVQZ) þ ΔCCSD(T)/aug-cc-pVDZ.

’ APPLICATIONS i. Hydrogen-Bonding Systems. Hydrogen-bonding interactions, which are stronger than van der Waals interactions but considerably weaker than covalent and ionic bonds, ubiquitously exist in chemical and biological systems and play an important role in the structures and reactivities of related systems. A wealth of information on hydrogen bonds can be found in the literature,118
120 and experimentally, hydrogen bonds can be probed using NMR,121 IR,122 Compton profile anisotropies,123 and so forth. Computationally, interests as well as controversies remain in the exploration of the nature of hydrogen bonds.13,124,125 Hydrogen bonds can be either purely electrostatic37,119,120 or of partial covalent nature,123,126
128 but

BS2

BS3

method

ΔEs

ΔEpol

ΔECT

ΔEInt

BLW


7.07


3.63


7.38


18.08

BLW-dDXDM


7.56


3.50


7.44


18.50

correction


0.49

0.13


0.06


0.42

BLW


4.56


4.22


6.03


14.81

BLW-dDXDM correction


5.04
0.48


4.09 0.13


6.10
0.07


15.23
0.42

Figure 2. BLW-ED for the water trimer with the B3LYP and PBE functionals and basis sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3). For each pair of bars, the right/left ones refer to the BLW-ED with/without the dDXDM corrections.

current consensus seems that hydrogen bonds are predominantly electrostatic with minor covalent character.129 As water is the most typical example of hydrogen bonds, at first, we studied the water dimer. The MP2/6-311þG(d,p) optimization results in the hydrogen bond length R(O 3 3 3 H
O) = 2.912 Å, comparable with 2.907 Å at the B3LYP level. However, computations with the BLW method result in a much lengthened distance (3.081 Å), with O
H vibrational frequencies (3811 and 3924 cm
1) very close to those of a water monomer (3809 and 3939 cm
1). These numbers suggest that the BLW optimal structure corresponds to a weakly bound (i.e., van der Waals) complex with no CT between monomers. At the MP2 geometry, the computed interaction energy is
5.00 kcal/mol, and BLW-ED indicates that the steric energy terms play the major role, suggesting that electrostatic attraction is the driving force for hydrogen bonds. Howwever, both polarization and CT effects are indispensable as well. With the incorporation of the dDXDM correction, the interaction is enhanced by 0.29 kcal/mol, but this correction dominantly goes to the steric energy term, leaving both polarization and CT energies little affected. To examine the basis set effect, we conducted similar analyses with the basis sets of 6-31G(d) and ccpVTZ. Results with all three basis sets are compiled in Table 1. Figure 1 further compares the results with both the B3LYP and PBE functionals. As expected, the basis set effect is observable from Table 1 as the total interaction energy also fluctuates, with reference to the benchmark value
4.96 kcal/mol at the CCSD(T)/CBS level. 5470

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A However, the consensus is also obvious, namely, the electrostatic interaction accounts for more than half of the interaction energy, followed by the CT interaction, which is a characteristics of chemical bonds. Figure 1 exhibits that the PBE functional generates results similar to B3LYP for the water dimer. The comparison of BLW-ED with and without the dDXDM corrections at all levels concurs that long-range dispersion mostly affects the steric energy term, which occurs in the approaching of monomers. At this point, we reiterate that this work focuses on long-range dispersion as the short-range dispersion effects are largely taken into account in the standard DFT computations. In contrast to SAPT studies, in which dispersion (short-range and long-range are together as not physically separable) is the dominating correction term to the HF interaction energy (as high as 2.0
2.4 kcal mol
1),130,131 the long-range contribution obtained with BLWdDXDM is substantially smaller (0.1
0.3 kcal mol
1). We continued to examine the water trimer, where the hydrogen bonds are slightly bent to facilitate the formation of three hydrogen bonds in a ring.9 Results are compiled in Table 2 and illustrated in Figure 2. Except the steric energy, which is about twice that in the dimer, both the polarization and CT effects stabilize the system by more than three times than those in the dimer, suggesting strong cooperative effects. As a matter of fact, all three energy terms are of comparable magnitudes in the water trimer. Like in the dimer, the largest dispersion correction is observed in the steric energy term and reaches 1.09 kcal/mol at the B3LYP/6-311þG(d,p) level. Though the dispersion correction is insignificant for the polarization and CT terms, it seems always positive for the polarization energy but negative for the CT energy. As demonstrated before,45,53,124,125,132 the BLW-ED can illustrate the electron density variations due to the polarization and CT effects by means of electron density difference (EDD) maps. Figure 3 shows the cases of the water dimer and trimer, where polarization (a1 and a2) corresponds to the difference between the optimal (FBLW) and initial (FBLW0) electron-localized densities and CT corresponds to the difference between electrodelocalized (F) and localized (FBLW) densities. The plots reveal that polarization has an impact on all atoms, where CT is more local and simply an electron donation from the oxygen lone pair to the opposite hydrogen atom. If the hydrogen-bond donor and acceptor are bulky molecules, the significance of the dispersion interaction is expected to increase. We investigated the interaction between trimethylamine (CH3)3N and a water molecule. At the MP2/6-311þG(d,p) level, the optimal hydrogen bond length is R(N 3 3 3 H
O) = 2.839 Å, compared with 2.884 Å at the B3LYP level. The noticeable discrepancy (0.045 Å) highlights the impact of the long-range dispersion effect on the geometry. If the electron transfer from the nitrogen lone pair to the water hydrogen is deactivated, the BLW optimization results in a much stretched distance of 3.170 Å. Figure 4 compares the BLW-ED results without and with the dDXDM corrections. Significantly, the long-range dispersion interaction stabilizes the complex by about 2 kcal/mol, which is missed in standard DFT computations with either the B3LYP or PBE functional, and the dDXDM corrections largely go to the steric energy term. For the polarization term, the dDXDM correction is positive and fluctuates around 0.2 kcal/mol. In contrast, long-range dispersion stabilization slightly increases with the delocalization of electrons, and the dDXDM correction is thus negative for the CT energy term, though the magnitude is only around 0.1 kcal/mol.

ARTICLE

Figure 3. EDD isosurface maps with the isovalue of 0.001 au showing (a) polarization and (b) electron-transfer effects in the water dimer (1) and trimer (2) at the B3LYP/6-311þG(d,p) level. The red/blue surfaces represent an increase/decrease in electron density.

Figure 4. BLW-ED for the (CH3)3N 3 3 3 H2O complex with the B3LYP and PBE functional and basis sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3). For each pair of bars, the right/left ones refer to the BLW-ED with/without the dDXDM corrections.

Figure 5. BLW-ED for the WC-AT base pair with the B3LYP and PBE functional and basis sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3). For each pair of bars, the right/left ones refer to the BLW-ED with/without the dDXDM corrections.

Hydrogen-bonding interactions also play a key role for the structures and functions of DNA molecules, and it has been proposed that the hydrogen bonds in the DNA base pairs basically are of electrostatic nature13,119,133 as nonpolarizable force fields have been successfully applied to simulate the DNA 5471

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A

ARTICLE

Figure 6. EDD isosurface maps with the isovalue of 0.001 au showing (a) polarization and (b) electron-transfer effects in the WC-AT base pair at the B3LYP/6-311þG(d,p) level. The red/blue surfaces represent an increase/decrease in electron density.

interactions and transitions.134,135 However, recent works revealed that the CT between the lone pairs on oxygen or nitrogen to the N
H σ antibonds in the Watson
Crick (WC) pairs are of comparable strength as electrostatic interactions.124,136
138 Here, we reinvestigated the hydrogen-bonding interactions in the WC adenine-thymine (AT) base pairs116 with the BLW energy analysis at the DFT level coupled with the dDXDM dispersion corrections. The results shown in Figure 5 illustrate the significant role played by dispersion in the DNA base pair interaction energies, which stabilizes the WC-AT pair by 2
4 kcal/mol. With the long-range dispersion correction, the overall stabilization ranges from
15.19 to
17.19 kcal/mol (with the three basis sets and two functionals), comparing well with the benchmark value of
16.74 kcal/mol at the CCSD(T)/ CBS(Δa(DT)Z) level.117 While the dispersion corrections largely go to the steric energy term, CT interactions dominate the hydrogen-bonding interactions in DNA base pairs. We note that compared with analyses at the HF level,124 polarization energies at the DFT level are quite similar, but the charge transfer energies increase considerably.29 The further consideration of the long-range dispersion effect, however, has a very minor impact on the charge transfer energy terms. To visualize the electron density changes in the process of polarization and CT, Figure 6 plots the EDD maps for the WC-AT base pair. In the interface between adenine and thymine, the N
H bond polarization from hydrogen to nitrogen is observed, whereas hydrogen bond acceptors (O and N atoms) gain electron density from adjacent atoms to prepare for the formation of hydrogen bonds, which makes these acceptors lose electron density to the proton donors. Both the polarization and CT effects are mostly local, indicating the secondary role of the π aromatic rings in the hydrogen bonding in the DNA base pairs. The strong electron donation from the proton acceptor to the opposite protons confirms the covalent nature of the hydrogen bonds in the DNA base pairs. (ii). Lewis Acid
Base Pairs. Strong dative covalent bond can be formed between a Lewis acid, which is electron-deficient, and a Lewis base, which is electron-rich, with a strength lying between the bonding and nonbonding regimes. One typical example is the reaction between boron and ammonia to give the adduct H3BNH3, which is largely stabilized by the electron transfer from the lone nitrogen pair to the vacant orbital on boron. Though BH3NH3 is isoelectronic with ethane, its bond strength (31.1 kcal/mol) is only one-third of the latter139 but still remarkably higher than the van der Waals interactions. Furthermore, the significant electron transfer from NH3 to BH3 can be verified by the high dipole moment (5.216 D) of BH3NH3.140 Extensive computational studies can be found in the literature, and there is

Figure 7. BLW-ED for the NH3
BH3 complex with the B3LYP and PBE functional and basis sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3). For each pair of bars, the right/left ones refer to the BLW-ED with/without the dDXDM corrections.

also continuing interest in the nature of the donor
acceptor interaction and correlation between CT with bonding characters such as interaction energy and bond length.44,141
155 The geometry optimization at the MP2/6-311þG(d,p) level results in a boron
nitrogen distance of 1.656 Å, which is in good agreement with the experimental data (1.6576 Å140). Optimizations at the B3LYP level generate a comparable but slightly elongated bond distance (1.668 Å). The strict localization of electrons with either BH3 or NH3 moieties, however, leads to a remarkably stretched N
B distance of 2.527 Å at the B3LYP/6311þG(d,p) level. Previously, we approximately optimized this distance with the ab initio VB method and 6-31G(d) basis set and got a value of 2.33 Å.155 On one hand, the sum of the covalent radii for nitrogen and boron (0.75 and 0.82 Å, respectively), which is very close to the N
B distance (1.656 or 1.668 Å) in the adduct H3BNH3, implies a strong bond. The BLW optimal distance (2.527 Å) and the small binding energy (4.06 or 5.22 kcal/mol with the dDXDM correction), on the other hand, are indicative of a weak interaction between the boron and nitrogen atoms in the absence of CT. There are no experimental data for the van der Waals radius for boron, but for nitrogen, its van der Waals radius is 1.55 Å.156 The shortening of the boron
nitrogen distance from the BLW minimum leads to the sharp increase of the Pauli repulsion, but this energy loss is fully compensated by the stabilizing electron-transfer interaction. BLW energy analyses at the MP2/6-311þG(d,p) geometry are shown in Figure 7, where a few features can be immediately captured; (1) dispersion corrections, which trange from
2.5 to
0.7 kcal/mol, are insignificant compared with the high interaction energies (the benchmark value at the CCSD(T)/CBS level is
44.44 kcal/mol); (2) steric energies are considerably positive due to the Pauli repulsion 5472

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A

ARTICLE

Figure 8. EDD isosurface maps with the isovalue of 0.005 au showing (a) polarization and (b) electron-transfer effects in the H3B r NH3 adduct at the B3LYP/6-311þG(d,p) level. The red/blue surfaces represent an increase/decrease in electron density.

(a separation of this term to Pauli repulsion and electrostatic interactions can be found in ref 44); and (3) basis set incompleteness is obvious from 6-31G(d) to cc-pVTZ or 6-311þG(d,p), while the latter two generate comparable results. In addition, the electron-transfer stabilization energy is relatively stable with basis sets. NH3BH3 is the ideal example to illustrate the difference between the BLW-dDXDM and SAPT results. In the BLWdDXDM scheme, the long-range dispersion energy contributes only a small fraction of the total interaction energy. In contrast, a very recent SAPT study of Hobza and co-workers157 assigned as much as 18 kcal mol
1 (of the 44 kcal mol
1 binding energy) to dispersion energy for this complex! Figure 8 shows the polarization and electron-transfer effects in terms of EDD maps. It is interesting to note that in the polarization process, the electron density of ammonia is polarized toward boron, while the electron density of boron moves away from ammonia due to the repulsion from the nitrogen lone pair electrons. The consequence of these intramolecular electron polarizations is the generation of a preparatory state (optimal BLW) for the subsequent electron transfer, which goes from the nitrogen lone pair to the vacant boron p orbital. (iii). van der Waals Complexes. While the long-range dispersion effect is insignificant and can be incorporated into the steric energy term in strongly interacting systems such as hydrogen-bonding systems and a Lewis acid
base (or electron donator
acceptor) pair, as we studied above, it possesses a dominating role in weakly interacting systems, notably in the biological processes including π
π stacking in DNA158,159 and protein folding160,161 as well as chemical processes including molecular assembling and physical absorption of absorbate to the surface of a solid. Enormous efforts have been spent on the development of sophisticated high-level quantum computational strategies to accurately evaluate the dispersion effect.117,162
165 However, it is of interests to explore how the dispersion effect influences the energy analysis using current EDA approaches. Much like in cation
π interacting systems,166 the interaction energy in π
π systems would increase systematically as the size of the π system increases. As it has been well-recognized that π stacking between adjacent nucleotides increases the stability of DNA molecules and the van der Waals attraction among the nonpolar amino acid residues is the driving force for protein folding, here, we chose six typical and well-studied examples that exhibit considerable dispersion effect in the present dDXDMcorrected BLW-ED. Figure 9 shows their structures. As the strength of dispersion interaction is proportional to the molecular size, the π
π stacking between adenine and thymine results in the highest interaction energy among the six complexes

Figure 9. van der Waals complexes: (a) π-stacked AT base pair (vdW1); (b) π-stacked uracil dimer (vdW2); (c) benzene
water (vdW3); (d) benzene
hydrogen sulfide (vdW4); (e) parallel-displaced benzene dimer (vdW5); (f) T-shaped benzene dimer (vdW6).

Figure 10. BLW-ED for the π-stacked AT complex with the B3LYP and PBE functional and basis sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3). For each pair of bars, the right/left ones refer to the BLW-ED with/without the dDXDM corrections.

Figure 11. BLW-ED for the benzene
water complex with the B3LYP and PBE functional and basis sets 6-31G(d) (BS1), cc-pVTZ (BS2), and 6-311þG(d,p) (BS3). For each pair of bars, the right one refers to the BLW-ED with the dDXDM corrections.

with a benchmark value of 11.70 kcal/mol. However, without dDXDM corrections, the base pair might be unstable. Figure 10 collects the results from various basis sets and functionals. There are two valuable features that can be drawn from the comparison. One is 5473

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A

ARTICLE

Table 3. BLW-ED (kcal/mol) of the van der Waals Complexes with the dDXDM Correction with the 6-311þG(d,p) Basis Set B3LYP system vdW1

method BLW BLW-dDXDM correction

vdW2

vdW3

vdW4

vdW5

vdW6

BLW

ΔEs

ΔEpol

PBE ΔECT

ΔEInt

ΔEs

ΔEpol

ΔECT

ΔEInt

4.08


1.06


1.59

1.43

1.52


1.23


1.76


1.47


8.50


1.12


1.69


11.31


7.72


1.31


1.82


10.85


12.58


0.06


0.10


12.74


9.24


0.08


0.06


9.38

1.05


0.99


0.80


0.74


0.65


1.07


0.91


2.63

BLW-dDXDM


6.35


1.15


0.83


8.33


6.18


1.10


0.94


8.22

correction


7.40


0.16


0.03


7.59


5.53


0.03


0.03


5.59

BLW


0.35


0.45


0.58


1.38


1.05


0.48


0.71


2.24

BLW-dDXDM correction


2.29
1.94


0.35 0.10


0.61
0.03


3.26
1.88


2.48
1.43


0.41 0.07


0.73
0.02


3.63
1.39

1.28


0.32


0.68

0.28

0.13


0.38


0.82


1.07

BLW-dDXDM


2.34


0.26


0.69


3.29


2.46


0.34


0.83


3.63

correction


3.62

0.06


0.01


3.59


2.59

0.04


0.01


2.56

4.60


0.27


0.52

3.81

2.60


0.30


0.49

1.81

BLW-dDXDM


3.16


0.35


0.58


4.09


2.85


0.40


0.52


3.77

BLW

BLW correction


7.76


0.08


0.06


7.90

5.44


0.10


0.03


5.58

BLW BLW-dDXDM

1.66
2.29


0.22
0.18


0.46
0.45

0.98
2.92

0.60
2.40


0.26
0.26


0.50
0.50


0.16
3.16

correction


3.95

0.05

0.01


3.90


3.00

0.00

0.00


3.00

that the dispersion effect is the key for the π
π interaction (a point that can be further verified in the parallel-displaced benzene dimer) and stabilizes the system by 10
12 kcal/mol, in other words, it overwhelmingly dominates the intermolecular interaction; the other is that with the increasing significance of dispersion interactions, the disparity between the B3LYP and PBE functional also becomes clear. B3LYP computations actually conclude that the complex is unstable, while PBE computations lead to marginal stabilizing interaction energies. In line with the analyses in the above subsections, the dispersion corrections mostly appear in the steric energy term. Though little affected by the strong dispersion effect, the CT effect is still obvious in both B3LYP and PBE computations and stabilizes the system by about 1 kcal/mol. The existence of the CT effect is important for the long-range electron transfer or hopping in DNA and proteins. The involvement of the directional (albeit weak) interactions of the O
H or N
H or even C
H bond with the center of an aromatic ring in the stability of protein structures has been wellrecognized, and the concept of hydrogen bonding has thus expanded to include this kind of π
H bonds.167
169 Here, we analyzed the interaction between water and benzene.170
173 As shown in Figure 11, weak electrostatic attraction exists between the O
H bond and the π cloud of benzene, as does the electrontransfer effect. However, without the dispersion corrections, the interaction energy ranges from around
1.4 kcal/mol with the B3LYP functional to around
2.1 kcal/mol with the PBE functional, compared with the benchmark value of
3.29 kcal/mol. The discrepancy is well-recovered with the dDXDM method. Hardly affected by the dispersion interaction, the CT stabilization energy accounts for about 20% of the dispersion-corrected interaction energy and seems more important than the polarization effect. For the SH
π interaction (vdW4 in Figure 9 and Table 3), our BLWdDXDM analyses indicated that both the polarization and CT effects as well as the overall interaction energy are comparable to those in the OH
π interaction, though the more destabilizing steric energy is compensated for by the higher dispersion interaction.

For some π
π stacking systems like the well-studied parallel benzene dimer,162,163 standard DFT computations even fail to locate a minimum. Thus, dispersion-corrected DFT methods are essential for the study of this kind of system. Table 3 summaries the BLW energy analyses with and without the dDXDM corrections with the basis set 6-311þG(d,p) for the six van der Waals complexes shown in Figure 9. In all of these systems, the long-range dispersion interaction is solely responsible for the structures and stability. Though polarization and CT effects insignificantly contribute to the stabilization of these complexes, the dispersion effect has little affect on both energy terms. Instead, essentially all dispersion corrections go to the steric energy term. As the steric term corresponds to the energy change in the approaching of monomers with their respective electron densities frozen, two obvious contributions, namely, the classical electrostatic interaction and the quantum Pauli exchange repulsion, have been identified. The present BLW energy analyses with the dDXDM method revealed that essentially all dispersion corrections occur in the steric energy term, suggesting that the dispersion effect can be taken out as an individual energy component. This is actually a practice that has been conducted by many researchers for long.

’ CONCLUSION We have combined the BLW-DFT method and the dDXDM approach to decompose the intermolecular interaction energies in a broad variety of complexes from strongly bound to weakly bound in terms of steric, polarization, and CT energy components with dispersion corrections. For all systems, the long-range dispersion correction dominantly appears in the steric energy term, which is presumably composed of classical electrostatic, and quantum Pauli exchange energies, whereas both polarization and CT energy terms are little affected by the long-range dispersion correction. In other words, the density redistribution due to polarization and CT is relatively small, so that the dispersion correction is essentially dominated by the frozen 5474

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A

ARTICLE

monomer densities. As such, it seems clear that the long-range dispersion effect can be well-defined as a separated energy component in energy decomposition analyses. In other words, eq 3 can be rewritten as ΔEb

¼

ESuper


k

þ ΔEpol ∑ EMono ðiÞ ¼ ΔEdef þ ΔEDFT s i¼1

þ ΔECT þ ΔEL-disp ¼ ΔEdef þ ΔEInt

ð8Þ

where ΔEL-disp refers to the long-range dispersion energy and roughly (because we are using semilocal density functional ΔEDFT s approximations for the exchange-correlation energy) corresponds to the original definition as the first-order energy from the perturbation calculation without long-rang dispersion.106 Our finding can be well-justified by the fact that within the perturbation theory, the London-type dispersion energy is a second-order energy term and, in principle, independent from both the polarization and CT energy terms. Thus, the present work endorses the practices in many EDA studies that the dispersion effect is computed as an individual energy component, and our combined BLW/dDXDM scheme provides a useful and necessary tool for broad applications particularly in the elucidation of intermolecular interactions.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected] (Y.M.); clemence.corminboeuf@ epfl.ch (C.C.).

’ ACKNOWLEDGMENT C.C. and S.N.S. acknowledge the Sandoz family foundation, the Swiss NSF Grant (200021_121577/1), and EPFL for financial support. We are grateful to Q-Chem Inc. for providing the source code and for helpful discussions with Drs. Zhengting Gan and Jing Kong. W.W. is supported by the Natural Science Foundation of China. Y.M. is grateful to the support from the Keck Foundation and Western Michigan University. ’ REFERENCES (1) Banks, J. L.; Kaminski, G. A.; Zhou, R. H.; Mainz, D. T.; Berne, B. J.; Friesner, R. A. J. Chem. Phys. 1999, 110, 741. (2) Chelli, R.; Procacci, P. J. Chem. Phys. 2002, 117, 9175. (3) Yang, Z. Z.; Wu, Y.; Zhao, D. X. J. Chem. Phys. 2004, 120, 2541. (4) Anisimov, V. M.; Lamoureux, G.; Vorobyov, I. V.; Huang, N.; Roux, B.; MacKerell, A. D. J. Chem. Theory Comput. 2005, 1, 153. (5) Maple, J. R.; Cao, Y. X.; Damm, W. G.; Halgren, T. A.; Kaminski, G. A.; Zhang, L. Y.; Friesner, R. A. J. Chem. Theory Comput. 2005, 1, 694. (6) Kaminski, G. A.; Ponomarev, S. Y.; Liu, A. B. J. Chem. Theory Comput. 2009, 5, 2935. (7) Borodin, O. J. Phys. Chem. B 2009, 113, 11463. (8) Zhao, D. X.; Liu, C.; Wang, F. F.; Yu, C. Y.; Gong, L. D.; Liu, S. B.; Yang, Z. Z. J. Chem. Theory Comput. 2010, 6, 795. (9) Gao, J.; Cembran, A.; Mo, Y. J. Chem. Theory Comput. 2010, 6, 2402. (10) Wu, J. C.; Piquemal, J.-P.; Chaudret, R.; Reinhardt, P.; Ren, P. J. Chem. Theory Comput. 2010, 6, 2059. (11) Morokuma, K. J. Chem. Phys. 1971, 55, 1236. (12) Kitaura, K.; Morokuma, K. Int. J. Quantum Chem. 1976, 10, 325. (13) Morokuma, K. Acc. Chem. Res. 1977, 10, 294. (14) Ziegler, T.; Rauk, A. Theor. Chem. Acc. 1977, 46, 1. (15) Bagus, P. S.; Hermann, K.; Bauschlicher, C. W., Jr. J. Chem. Phys. 1984, 80, 4378. (16) Stevens, W. J.; Fink, W. H. Chem. Phys. Lett. 1987, 139, 15.

(17) Glendening, E. D.; Streitwieser, A. J. Chem. Phys. 1994, 100, 2900. (18) Chen, W.; Gordon, M. S. J. Phys. Chem. 1996, 100, 14316. (19) van der Vaart, A.; Merz, K. M., Jr. J. Phys. Chem. A 1999, 103, 3321. (20) Mo, Y.; Gao, J.; Peyerimhoff, S. D. J. Chem. Phys. 2000, 112, 5530. (21) Reinhardt, P.; Piquemal, J.-P.; Savin, A. J. Chem. Theory Comput. 2008, 4, 2020. (22) Wu, Q.; Ayers, P. W.; Zhang, Y. K. J. Chem. Phys. 2009, 131, 164112. (23) Su, P.; Li, H. J. Chem. Phys. 2009, 131, 014102. (24) Hayes, I. C.; Stone, A. J. Mol. Phys. 1984, 53, 83. (25) Stone, A. J. The Theory of Intermolecular Forces; Oxford University Press: New York, 1996. (26) Jeziorski, B.; Moszynski, R.; Szalewicz, K. Chem. Rev. 1994, 94, 1887. _ (27) Rajchel, L.; Zuchowski, P. S.; Szcze
sniak, M. M.; Chazasi
nski, G. Phys. Rev. Lett. 2010, 104, 163001. (28) Bagus, P. S.; Illas, F. J. Chem. Phys. 1992, 96, 8962. (29) M
arquez, A. M.; L
opez, N.; García-Hern
andez, M.; Illas, F. Surf. Sci. 1999, 442, 463. (30) Reed, A. E.; Curtiss, L. A.; Weinhold, F. Chem. Rev. 1988, 88, 899. (31) Reed, A. E.; Weinhold, F. Isr. J. Chem. 1991, 31, 277. (32) Glendening, E. D. J. Phys. Chem. A 2005, 109, 11936. (33) Bickelhaupt, F. M.; Baerends, E. J. In Reviews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; Wiley-VCH: New York, 1999; Vol. 15, p 1. (34) te Velde, G. T.; Bickelhaupt, F. M.; Baerends, E. J.; Guerra, C. F. S.; Van Gisbergen, J. A.; Snijders, J. G.; Ziegler, T. J. Comput. Chem. 2001, 22, 931. (35) Michalak, A.; Mitoraj, M.; Ziegler, T. J. Phys. Chem. A 2008, 112, 1933. (36) Mitoraj, M. P.; Michalak, A.; Ziegler, T. J. Chem. Theory Comput. 2009, 5, 962. (37) Pauling, L. C. The Nature of the Chemical Bond, 3rd ed.; Cornell University Press: Ithaca, NY, 1960. (38) Wheland, G. W. Resonance in Organic Chemistry; Wiley & Sons: New York, 1955. (39) Cooper, D. L., Ed. Valence Bond Theory; Elsevier: Amsterdam, The Netherlands, 2002. (40) Gallup, G. A. Valence Bond Methods: Theory and Applications; Cambridge University Press: New York, 2002. (41) Shaik, S. S.; Hiberty, P. C. A Chemist’s Guide to Valence Bond Theory; Wiley-Interscience: New York, 2007. (42) Mo, Y.; Peyerimhoff, S. D. J. Chem. Phys. 1998, 109, 1687. (43) Mo, Y. J. Chem. Phys. 2003, 119, 1300. (44) Mo, Y.; Gao, J. J. Phys. Chem. A 2001, 105, 6530. (45) Mo, Y.; Gao, J. J. Phys. Chem. B 2006, 110, 2976. (46) Mo, Y.; Gao, J. J. Comput. Chem. 2000, 21, 1458. (47) Mo, Y.; Gao, J. J. Phys. Chem. 2000, 104, 3012. (48) Gao, J.; Garcia-Viloca, M.; Poulsen, T. D.; Mo, Y. Adv. Phys. Org. Chem. 2003, 38, 161. (49) Mo, Y. J. Org. Chem. 2004, 69, 5563. (50) Mo, Y.; Schleyer, P. v. R. Chem.—Eur. J. 2006, 12, 2009. (51) Mo, Y.; Hiberty, P. C.; Schleyer, P. v. R. Theor. Chem. Acc. 2010, 127, 27. (52) Mo, Y. Nat. Chem. 2010, 2, 666. (53) Mo, Y.; Song, L.; Lin, Y. J. Phys. Chem. A 2007, 111, 8291. (54) Cembran, A.; Song, L.; Y., M.; Gao, J. J. Chem. Theory Comput. 2009, 5, 2702. (55) Khaliullin, R. Z.; Cobar, E. A.; Lochan, R. C.; Bell, A. T.; HeadGordon, M. J. Phys. Chem. A 2007, 111, 8753. (56) Wang, F.-F.; Jenness, G.; Al-Saidi, W. A.; Jordan, K. D. J. Chem. Phys. 2010, 132, 134303. (57) Steinmann, S. N.; Wodrich, M. D.; Corminboeuf, C. Theor. Chem. Acc. 2010, 27, 429. 5475

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A (58) Vaidhyanathan, R.; Iremonger, S. S.; Shimizu, G. K. H.; Boyd, P. G.; Woo, T. K. Science 2010, 330, 650. (59) Diner, S.; Malrieu, J. P.; Claverie, P. Theor. Chem. Acc. 1969, 13, 1. (60) Diner, S.; Malrieu, J. P.; Jordan, F.; Gilbert, M. Theor. Chem. Acc. 1969, 15, 100. (61) Szalewicz, K.; Jeziorski, B. Mol. Phys. 1979, 38, 191. (62) Chazasi
nski, G.; Szcze
sniak, M. M. Mol. Phys. 1988, 63, 205. (63) Williams, H. L.; Chabalowski, C. F. J. Phys. Chem. A 2001, 105, 646. (64) Hesselmann, A.; Jansen, G. Chem. Phys. Lett. 2002, 357, 464. (65) Stone, A. J.; Misquittaa, A. J. Chem. Phys. Lett. 2009, 473, 201. (66) Moszynski, R.; Wormer, P. E. S.; Jeziorski, B.; van der Avoird, A. J. Chem. Phys. 1995, 103, 8058. (67) Lotrich, V. F.; Szalewicz, K. J. Chem. Phys. 1997, 106, 9668. (68) Wu, X.; Vargas, M. C.; Nayak, S.; Lotrich, V.; Scoles, G. J. Chem. Phys. 2001, 115, 8748. (69) Wu, Q.; Yang, W. J. Chem. Phys. 2002, 116, 515. (70) Grimme, S. J. Comput. Chem. 2006, 27, 1787.  ern
y , J.; Hobza, P.; Salahub, D. R. J. Comput. (71) Jurecka, P.; C Chem. 2007, 28, 555. (72) Steinmann, S. N.; Csonka, G.; Corminboeuf, C. J. Chem. Theory Comput. 2009, 5, 2950. (73) Steinmann, S. N.; Corminboeuf, C. J. Chem. Theory Comput. 2010, 6, 1990. (74) Becke, A. D.; Johnson, E. R. J. Chem. Phys. 2005, 122, 154104. (75) Becke, A. D.; Johnson, E. R. J. Chem. Phys. 2006, 124, 014104. (76) Becke, A. D.; Johnson, E. R. J. Chem. Phys. 2007, 127, 154108. (77) Kong, J.; Gan, Z.; Proynov, E.; Freindorf, M.; Furlani, T. R. Phys. Rev. A 2009, 79, 042510. (78) Mo, Y.; Bao, P.; Gao, J. Phys. Chem. Chem. Phys. 2011, 13, 6760. (79) van Lenthe, J. H.; Balint-Kurti, G. G. Chem. Phys. Lett. 1980, 76, 138. (80) van Lenthe, J. H.; Balint-Kurti, G. G. J. Chem. Phys. 1983, 76, 5699. (81) Cooper, D. L.; Gerratt, J.; Raimondi, M. Nature 1986, 323, 699. (82) Cooper, D. L.; Gerratt, J.; Raimondi, M. Chem. Rev. 1991, 91, 929. (83) Hiberty, P. C. THEOCHEM 1997, 398
399, 35. (84) Thorsteinsson, T.; Cooper, D. L. J. Math. Chem. 1998, 23, 105. (85) Wu, W.; Wu, A.; Mo, Y.; Lin, M.; Zhang, Q. Int. J. Quantum Chem. 1998, 67, 287. (86) Mcweeny, R. Int. J. Quantum Chem. 1999, 74, 87. (87) Dijkstra, F.; van Lenthe, J. H. J. Chem. Phys. 2000, 113, 2100. (88) Song, L.; Mo, Y.; Zhang, Q.; Wu, W. J. Comput. Chem. 2005, 26, 514. (89) Hiberty, P. C.; Shaik, S. J. Comput. Chem. 2007, 28, 137. (90) Song, L.; Song, J.; Mo, Y.; Wu, W. J. Comput. Chem. 2009, 30, 399. (91) Goddard, W. A. I. Phys. Rev. 1967, 157, 73. (92) Bobrowicz, F. W.; Goddard, W. A., III In Methods of Electronic Structure Theory; Schaefer, H. F., III, Ed.; Plenum: New York, 1977; pp 79. (93) McWeeny, R. Proc. R. Soc. London, Ser. A 1959, 253, 242. (94) Mulliken, R. S.; Parr, R. G. J. Chem. Phys. 1951, 19, 1271. (95) Sovers, O. J.; Kern, C. W.; Pitzer, R. M.; Karplus, M. J. Chem. Phys. 1968, 49, 2592. (96) Stoll, H.; Preuss, H. Theor. Chim. Acta 1977, 46, 11. (97) Stoll, H.; Wagenblast, G.; Preuss, H. Theor. Chim. Acta 1980, 57, 169. (98) Kollmar, H. J. Am. Chem. Soc. 1979, 101, 4832. (99) Daudey, J. P.; Trinquier, G.; Barthelat, J. C.; Malrieu, J. P. Tetrahedron 1980, 36, 3399. (100) Mehler, E. L. J. Chem. Phys. 1977, 67, 2728. (101) Mehler, E. L. J. Chem. Phys. 1981, 74, 6298. (102) F€ulscher, M. P.; Mehler, E. L. J. Comput. Chem. 1981, 12, 811. (103) Famulari, A.; Gianinetti, E.; Raimondi, M.; Sironi, M. Int. J. Quantum Chem. 1998, 69, 151.

ARTICLE

(104) Gianinetti, E.; Raimondi; Tornaghi, E. Int. J. Quantum Chem. 1996, 60, 157. (105) Gianinetti, E.; Vandoni, I.; Famulari, A.; Raimondi, M. Adv. Quantum Chem. 1998, 31, 251. (106) Tang, K. T.; Toennies, J. P.; Yiu, C. L. Int. Rev. Phys. Chem. 1998, 17, 363. (107) Lein, M.; Szabo, A.; Kovacs, A.; Frenking, G. Faraday Discuss. 2003, 124, 365. (108) Erhardt, S.; Frenking, G. Chem.—Eur. J. 2006, 12, 4620. (109) Pernal, K.; Podeszwa, R.; Patkowski, K.; Szalewicz, K. Phys. Rev. Lett. 2009, 103, 263201. (110) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (111) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. J.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347. (112) Tang, K. T.; Toennies, J. P. J. Chem. Phys. 1984, 80, 3726. (113) Proynov, E.; Gan, Z.; Kong, J. J. Chem. Phys. Lett. 2008, 455, 103. (114) Steinmann, S. N.; Corminboeuf, C. J. Chem. Phys. 2011, 134, 044117. (115) Shao, Y.; Fusti-Molnar, L.; Jung, Y.; Kussmann, J.; Ochsenfeld, C.; Brown, S. T.; Gilbert, A. T. B.; Slipchenko, L. V.; Levchenko, S. V.; O’Neill, D. P.; Distasio, R. A., Jr.; Lochan, R. C.; Wang, T.; Beran, G. J. O.; Besley, N. A.; Herbert, J. M.; Lin, C. Y.; van Voorhis, T.; Chien, S. H.; Sodt, A.; Steele, R. P.; Rassolov, V. A.; Maslen, P. E.; Korambath, P. P.; Adamson, R. D.; Austin, B.; Baker, J.; Byrd, E. F. C.; Dachsel, H.; Doerksen, R. J.; Dreuw, A.; Dunietz, B. D.; Dutoi, A. D.; Furlani, T. R.; Gwaltney, S. R.; Heyden, A.; Hirata, S.; Hsu, C.-P.; Kedziora, G.; Khalliulin, R. Z.; Klunzinger, P.; Lee, A. M.; Lee, M. S.; Liang, W.; Lotan, I.; Nair, N.; Peters, B.; Proynov, E. I.; Pieniazek, P. A.; Rhee, Y. M.; Ritchie, J.; Rosta, E.; Sherrill, C. D.; Simmonett, A. C.; Subotnik, J. E.; Woodcock, W. L., III; Zhang, W.; Bell, A. T.; Chakraborty, A. K.; Chipman, D. M.; Keil, F. J.; Warshel, A.; Hehre, W. J.; Schaefer, H. F., III; Kong, J.; Krylov, A. I.; Gill, P. M. W.; Head-Gordon, M. Phys. Chem. Chem. Phys. 2006, 8, 3172.  ern
y, J.; Hobza, P. Phys. Chem. Chem. (116) Jurecka, P.; Sponer, J.; C Phys. 2006, 8, 1985. (117) Takatani, T.; Hohenstein, E. G.; Malagoli, M.; Marshall, M. S.; Sherrill, C. D. J. Chem. Phys. 2010, 132, 144104. (118) Scheiner, S. Hydrogen Bonding: A Theoretical Perspective; Oxford University Press: New York, 1997. (119) Jeffrey, G. A. An Introduction to Hydrogen Bonding; Oxford University Press: New York, 1997. (120) Desiraju, G. R.; Steiner, T. The Weak Hydrogen Bond in Structural Chemistry and Biology; Oxford University Press: New York, 2001. (121) Tuttle, T.; Graefenstein, J.; Wu, A.; Kraka, E.; Cremer, D. J. Phys. Chem. B 2004, 108, 1115. (122) Brauer, B.; Gerber, R. B.; Kabelac, M.; Hobza, P.; Bakker, J. M.; Riziq, A. G. A.; de Vries, M. S. J. Phys. Chem. A 2005, 109, 6974. (123) Isaacs, E. D.; Shukla, A.; Platzman, P. M.; Hamann, D. R.; Barbiellini, B.; Tulk, C. A. Phys. Rev. Lett. 1999, 82, 600. (124) Mo, Y. J. Mol. Model. 2006, 12, 665. (125) Beck, J. F.; Mo, Y. J. Comput. Chem. 2007, 28, 455. (126) Tapan, K. G.; Viktor, N. S.; Patrick, R. K.; Ernest, R. D. J. Am. Chem. Soc. 2000, 122, 1210. (127) Grabowski, S. J.; Sokalski, W. A.; Leszczynski, J. J. Phys. Chem. A 2005, 109, 4331. (128) Pakiari, A. H.; Eskandari, K. J. Mol. Struct.: THEOCHEM 2006, 759, 51. (129) Gilli, G.; Gilli, P. J. Mol. Struct. 2000, 552, 1. (130) Mas, E. M.; Szalewicz, K. J. Chem. Phys. 1996, 104, 7606. (131) Rezac, J.; Hobza, P. J. Chem. Theory Comput. 2011, 7, 685. (132) Nakashima, K.; Zhang, X.; Xiang, M.; Lin, Y.; Lin, M.; Mo, Y. J. Theor. Comput. Chem. 2008, 7, 639. (133) Sponer, J.; Jurecka, P.; Hobza, P. J. Am. Chem. Soc. 2004, 126, 10142. 5476

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477

The Journal of Physical Chemistry A (134) Cheatham, T. E., III; Kollman, P. A. Annu. Rev. Phys. Chem. 2000, 51, 435. (135) Kratochvíl, M.; Sponer, J.; Hobza, P. J. Am. Chem. Soc. 2000, 122, 3495. (136) Guerra, C. F.; Bickelhaupt, F. M. Angew. Chem., Int. Ed. 1999, 38, 2942. (137) Guerra, C. F.; Bickelhaupt, F. M.; Snijders, J. G.; Baerends, E. J. Chem.—Eur. J. 1999, 5, 3581. (138) Guerra, C. F.; Bickelhaupt, F. M.; Snijders, J. G.; Baerends, E. J. J. Am. Chem. Soc. 2000, 122, 4117. (139) Haaland, A. Angew. Chem., Int. Ed. 1989, 28, 992. (140) Thorne, L. R.; Suenram, R. D.; Lovas, F. J. J. Chem. Phys. 1983, 78, 167. (141) Peyerimhoff, S. D.; Buenker, R. J. J. Chem. Phys. 1968, 49, 312. (142) Dill, J. D.; Schleyer, P. v. R.; Pople, J. A. J. Am. Chem. Soc. 1975, 97, 3402. (143) Eoeggen, I. Chem. Phys. 1992, 162, 271. (144) Mallajosyula, S. S.; Datta, A.; Pati, S. K. J. Phys. Chem. A 2006, 110, 5156. (145) Umeyama, H.; Morokuma, K. J. Am. Chem. Soc. 1976, 98, 7208. (146) Holme, T. A.; Truong, T. N. Chem. Phys. Lett. 1993, 215, 53. (147) Jonas, V.; Frenking, G.; Reetz, M. T. J. Am. Chem. Soc. 1994, 116, 8741. (148) Dapprich, S.; Frenking, G. J. Phys. Chem. 1995, 99, 9352. (149) Skancke, A.; Skancke, P. N. J. Phys. Chem. 1996, 100, 15079. (150) Bauschlicher, C. W. J.; Ricca, A. Chem. Phys. Lett. 1995, 237, 14. (151) Anane, H.; Boutalib, A.; Nebot-Gil, I.; Tomas, F. J. Phys. Chem. A 1998, 102, 7070. (152) Barrios, R.; Skurski, P.; Rak, J.; Gutowski, M. J. Chem. Phys. 2000, 113, 8961. (153) Fiacco, D. L.; Mo, Y.; Hunt, S. W.; Ott, M. E.; Roberts, A.; Leopold, K. R. J. Phys. Chem. A 2001, 105, 484. (154) Dillen, J.; Verhoeven, P. J. Phys. Chem. A 2003, 107, 2570. (155) Mo, Y.; Song, L.; Wu, W.; Zhang, Q. J. Am. Chem. Soc. 2004, 126, 3974. (156) Bondi, A. J. Phys. Chem. 1964, 68, 441. (157) Karthikeyan, S.; Sedlak, R.; Hobza, P. J. Phys. Chem. A 201110.1021/jp1112476. (158) Fiethen, A.; Jansen, G.; Hesselmann, A.; Sch€utz, M. J. Am. Chem. Soc. 2008, 130, 1802. (159) Svozil, D.; Hobza, P.; Sponer, J. J. Phys. Chem. B 2010, 114, 1191. (160) Bashford, D.; Chothia, C.; Lesk, A. M. J. Mol. Biol. 1987, 196, 199. (161) Roth, C. M.; Neal, B. L.; Lenhoff, A. M. Biophys. J. 1996, 70, 977. (162) Sherrill, C. D.; Takatani, T.; Hohenstein, E. G. J. Phys. Chem. A 2009, 113, 10146.  ez
ac, J.; Jurecka, P.; Urban, M.; (163) Piton
ak, M.; Neogr
ady, P.; R Hobza, P. J. Chem. Theory Comput. 2008, 4, 1829. (164) Riley, K. E.; Piton
ak, M.; Jurecka, P.; Hobza, P. Chem. Rev. 2010, 110, 5023.  ez
ac, J.; Hobza, P. J. Chem. Theory (165) Gr
afov
a, L.; Piton
ak, M.; R Comput. 2010, 6, 2365. (166) Vijay, D.; Sastry, G. N. Phys. Chem. Chem. Phys. 2008, 10, 582. (167) Tsuzuki, S.; Honda, K.; Uchimaru, T.; Mikami, M.; Tanabe, K. J. Am. Chem. Soc. 2000, 122, 11450. (168) Scheiner, S.; Kar, T.; Pattanayak, J. J. Am. Chem. Soc. 2002, 124, 13257. (169) Mohan, N.; Vijayalakshmi, K. P.; Koga, N.; Suresh, C. H. J. Comput. Chem. 2010, 31, 2874. (170) Li, S.; Cooper, V. R.; Thonhauser, T.; Puzder, A.; Langreth, D. C. J. Phys. Chem. B 2008, 112, 9031. (171) Prakash, M.; Samy, K. G.; Subramanian, V. J. Phys. Chem. A 2009, 113, 13845.

ARTICLE

(172) Slipchenko, L. V.; Gordon, M. S. J. Phys. Chem. A 2009, 113, 2092. (173) Ma, J.; Alfe, D.; Michaelides, A.; Wang, E. J. Chem. Phys. 2009, 130, 154303.

5477

dx.doi.org/10.1021/jp202560d |J. Phys. Chem. A 2011, 115, 5467–5477