Quantum Monte Carlo Study of Water Dimer Binding Energy and

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Quantum Monte Carlo Study of Water Dimer Binding Energy and Halogen–# Interactions D. ChangMo Yang, Dong Yeon Kim, and Kwang S. Kim J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b04072 • Publication Date (Web): 16 Aug 2019 Downloaded from pubs.acs.org on August 19, 2019

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The Journal of Physical Chemistry

Quantum Monte Carlo Study of Water Dimer Binding Energy and Halogen–π Interactions D. ChangMo Yang,∗ Dong Yeon Kim, and Kwang S. Kim∗ Center for Superfunctional Materials, Department of Chemistry, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea E-mail: [email protected]; [email protected]

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Abstract Halogen–π systems are involved with competition between halogen bonding and πinteraction. Using the diffusion quantum Monte Carlo (DMC) method, we compare the interaction of benzene with diatomic halogens (X2: Cl2/Br2) with the typical hydrogen bonding in the water dimer, taking into account explicit correlations of up to three bodies. The benzene Cl2/Br2 binding energies (13.07 ± 0.42 / 16.62 ± 0.02 kJ/mol) attributed to both halogen bonding and dispersion are smaller than but comparable to the typical hydrogen bonding in the water dimer binding energy (20.88 ± 0.27 kJ/mol). All the above values are in good agreement with those from the coupled-cluster with single, double, and non-iterative triple excitations (CCSD(T)) results at the complete basis set limit (benzene Cl2/Br2: 12.78 / 16.17 kJ/mol; water dimer: 21.0 kJ/mol).

Introduction The field of molecular engineering is essentially built upon the precise control of molecular interactions. 1–5 Accurate calculations of intermolecular forces are vital in the assembly of molecules for a desired molecular architecture. Within such an assembly, there must exist a fine adjustment of the cooperation and competition 6–8 between different types of noncovalent molecular interactions 9–11 such as hydrogen bonding, 12,13 π-interactions, 14–17 halogen bonding, 18,19 and ionic interactions. 20 These non-covalent interactions have been studied extensively using high-level ab initio methods including CCSD(T) calculations at the complete basis set (CBS) limit. However, such extensive calculations are not easily feasible for large systems including complex π-systems. Density functional theory (DFT) with dispersion correction (DFT-D) 21,22 has been widely used for such systems. However, DFT-D often fails to obtain reliable binding energies in the particular case of halogen–π (X π) systems where electrostatic and dispersion interactions are complementing. 23–25 Here we consider the quantum Monte Carlo method (QMC) as an alternative, whose computing time scales cubically in the number of electrons. 26–29 As with other Markov chain Monte Carlo methods, 2


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QMC is near-embarrassingly parallelizable and works to its full potential in high-performance computing clusters. With QMC, we investigate the binding energy of water dimer as typical hydrogen bonding which reflects significant electrostatic interactions with small dispersion correction and the binding energies of benzene X2 (X=Cl/Br) complexes as a competition between π-interaction and halogen bonding (Figure 1).

(a)

(b)

(H2O)2

Benzene ― X2

Figure 1: Minimum energy structures of the water dimer and beznene X2. Before discussing the results, we address the computational issue of halogen bonding. The halogen bond is a noncovalent interaction between a halogen atom with an electrophilic region and a Lewis base with a nucleophilic region. 30,31 The halogen bond plays an important role in molecular, 32 biomolecular, 33 and material 34 architectures. In general chemistry, halogen atoms have been considered negatively charged due to their large electronegativity. However, they have both positively charged atom-end site and negatively charged atom-side site due to the σ-hole effect. 35–38 Though the main contribution to the halogen bonding is considered to be the electrostatic energy, the geometries are determined not only by electrostatics but also by other factors such as polar flattening, dispersion, and exchange repulsion. 39 When halogen atoms interact with π-systems, the resulting system has X π interactions which frequently appear in ligand-protein complexes, 40–42 supra-molecular systems, 43–48 and halogen-adsorbed carbon compounds such as graphene (or graphite) 49–55 and nanotubes. 56,57 In the case when the π-system is benzene, the beznene X2 structures show strongly halogen3



bonded conformers, for which most of the density functionals show noticeable deviations from the CCSD(T)/CBS binding energies. The halogen bond shows highly anisotropic electron density around halogen atoms and the DFT results are very sensitive to basis sets as well as density functionals. The unsatisfactory performance of many density functionals could be mainly due to inadequate correction for exchange and dispersion. This is evidenced from the better performance by the dispersioncorrected hybrid functionals such as PBE0 58 using Tkatchenko–Scheffler (TS) 22 or Grimme’s D3, 21 and double hybrid functionals such as B2GP-PLYP-D3. 59 So far, no QMC study on the benzene X2 system has been reported. QMC calculations are not readily applicable to new systems whose energies are of subchemical sensitivity (0.1 kcal/mol) without well-reasoned intuition about the distribution of electrons of those systems. Considerable amount of groundwork is required to establish a workflow that can yield quantitatively useful results. We previously performed CCSD(T)/CBS on benzene X2 system. 23,25 Here we report QMC results of the benzene X2 system. To assess the validity of our QMC calculation, we address the QMC result for the water dimer which has already been extensively studied using high-level ab initio calculations, 60–66 as a benchmark study for the validity of the remainder of our QMC calculations.

Computational details The results of this work come from a sequence of calculations, namely DFT-D, variational Monte Carlo (VMC), and diffusion Monte Carlo (DMC) methods. DFT calculations with the PBE0-D3 are used to obtain the molecular orbitals (MOs), which are passed onto VMC as input data. Within VMC, the Slater determinant of these MOs is augmented with parametrized explicit correlations whose shape can be continuously distorted by adjusting the variational parameters. Once the set of parameters giving the lowest possible energy (or the smallest variance) is found after several iterations of VMC runs, DMC is run to give a

4


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more accurate value of the ground state energy.

Geometries and orbitals In practice, the preparation of optimized geometries and orbitals is delegated to computational methods other than QMC that have well-defined schemes to calculate forces and to account for fermionic antisymmetry. For all of the systems reported in this study, we have used the geometries that are available from previous calculations. 23,67,68 The MO-generating methods need not be the same method that was used to optimize the geometries. It is generally desirable to use a method that minimizes the margin for variational improvement in the VMC calculation that takes in the MOs. The form of these MOs also affects the results of the DMC calculation through the nodal structure of their Slater determinant. Energy consistent pseudopotentials and corresponding valence basis sets were used for all of the atoms. 69 We will return to the discussion of valence-only (or frozen-core) calculations after a summary of the features of the QMC algorithm that are of interest in this work.

Variational Monte Carlo: optimizing explicit correlations We leave the detailed explanation of the VMC and DMC algorithms to the review literature. 26,70,71 A VMC calculation that is supplied only with the Slater determinant of e Hartree–Fock (HF) orbitals as the trial wave function ΨT ({rj }N j=1 ) yields an energy estimate

of EVMC = EHF . If the orbitals are imported from a higher-level calculation that partially accounts for correlation, it is inferred from the lowering of the calculated ground state energy Ne e that the shape of ΨT ({rj }N j=1 ) tends closer to the true ground state wave function Ψ0 ({rj }j=1 ).

Our QMC calculations are carried out with QMCPACK, a highly optimized open-source package for parallel computing environments. 72 The trial wave function used in the calculations has the form ΨT (R) = D↑ D↓ e−J(R) 5


(1)


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where D↑ and D↓ are the determinants of spatial orbitals for the two electron species. Electron Nnuc e coordinates {rj }N j=1 are abbreviated to R, and the dependence on nuclear coordinates {RI }I=1

as constant parameters is implied. 73 J(R) is called the Jastrow function, with the constraint of symmetry under exchange of like-pairs of electrons. In this work, we divide J(R) into terms that represent nucleus-electron (J1 ), electronelectron (J2 ), and nucleus-electron-electron (J3 ) correlations as J(R) = J1 + J2 + J3 =

XX I

+

(1)

uI,σ (rIj ) +

I

XX

(2)

uσ,σ0 (rjk )

σ,j σ 0 ,k

σ,j

XXX

(2)

(3)

uI,σ,σ0 (rIj , rIk , rjk )

(3)

σ,j σ 0 ,k

where I is the index of the nuclei, (σ, σ 0 ) ∈ {↑, ↓} are the indices of the electron spin species, and (j, k) are the indices of the electrons within a particular species. 74,75 Each term in Eq. 3 contains a collection of variational parameters whose values are adjusted over several optimizing runs of VMC, using the linear method. 76 The determination of a suitable set of variational parameters that apply to all of the electrons only reduces the variance of the calculated energy without altering the nodal structure of ΨT (R). Assigning individual Jastrow factors to each molecular orbital has been shown to move the nodes closer to those of Ψ0 (R), 77 though this is not done in this work. Any remaining discrepancy is to be dealt with in the DMC stage, where the excited state components of ΨT (R) are further suppressed in favor of the ground state.

Fixed-node diffusion Monte Carlo The DMC algorithm implemented in most of the production-level packages today, including QMCPACK, is the so-called importance sampled version in which the imaginary-time

6


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Schrödinger equation in Φ is transformed to

−

∂ ˆ 0 − ET f (R, t) f (R, t) = H ∂t 1 ≡ − ∇2 f (R, t) 2

(4)

+ ∇ · [V(R)f (R, t)] − S(R)f (R, t)

(5)

with f (R, t) = Φ(R, t)ΨT (R) and ∇ ≡ (∇1 , ∇2 , · · · , ∇Ne ). This has the form of the driftdiffusion equation and V(R) ≡ ∇ ln |ΨT (R)| is accordingly called the drift velocity. S(R) ≡ ET − EL (R) is called the branching factor and takes the role of the source or the sink of f (R, t). EL (R) is the same local energy defined with VMC and ET is the energy shift that approximates the true ground state energy E0 that is as yet not precisely known. In a system of electrons, a direct interpretation of f (R) as a probability density leads to the fermion sign problem, where the signal-to-noise ratio of the calculated observables falls exponentially with run time. As an approximate workaround, the positive and negative regions of f (R) are sampled separately. The DMC method remains variational even with this so-called fixed-node (FN) approximation, 78,79 and is known to recover about 95% of the correlation energy. 26 Several DMC runs using different values of τ are needed to make an extrapolation of the energy in the limit τ → 0. The sensitivity of the DMC estimate of the energy to τ depends on how closely the optimized ΨT (R) from VMC approximates Ψ0 (R), vanishing in the exact limit. To reduce this time-step error, QMCPACK adopts some modifications to the algorithm stated so far. One is to replace the drift velocity with an average expression ¯ V(R) =

−1 +

¯ to This limits the drift displacement Vτ

q

1 + 2V (R)2 τ

V (R)2 τ √

V(R) .

(6)

2τ for large values of V 2 τ which occur near the

7



nodes. Another modification is to attenuate the branching factor if the local energy differs from the running average energy Eest by more than some cutoff Ecut . A variant of this so-called ZSGMA modification is implemented in QMCPACK, and is shown to remedy the size-consistency breakage caused by the FN approximation. 80 Any remaining size-consistency error can be further reduced by comparing the energy Ebond of the bonded system against the energy Efar of the system with the molecular fragments relocated far away from each other. In this work, we choose this distance to be at least 10 Å.

Pseudopotentials Aside from the test case of the all-electron water dimer, we use energy-consistent scalarrelativistic pseudopotentials (PPs) proposed by Burkatzki, Filippi, and Dolg (BFD) 69 throughout this work. The valence basis sets are based on the data from BFD, but with additional diffuse functions copied from the all-electron basis set of the corresponding elements. 81–83 For example, the VTZ(BFD) basis set of carbon, which has the form [3s3p2d1f ], becomes [4s4p3d2f ] after adding the augmenting terms [1s1p1d1f ] which are part of the all-electron basis set aug-cc-pVTZ. We shall refer to the resulting valence basis set as aug-VTZ(BFD), and analogously for other cardinalities. It is possible to further optimize the exponents of the augmenting terms to give the lowest possible configuration interaction (CISD) energies of the singly charged anions of their associated elements, 81 but this only results in CISD energy shifts smaller than 0.8 kJ/mol and is virtually inconsequential in the DMC stage.

Results and Discussion Water dimer The binding energies of two water molecules were calculated as a test case. The experimental geometry of the water dimer shows the interoxygen distance RO-O of 2.976 Å and the angle ψ of 57◦ ± 10◦ between the interoxygen separation vector and the molecular plane 8


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of the proton-accepting monomer. 84 The binding energy is 5.44 ± 0.7 kcal/mol. 85 After more reliable zero-point energy (ZPE) correction, the binding energy is now considered to be 5.02±0.7 kcal/mol. 86 The calculated RO-O falls between 2.91 and 2.92 Å at the CCSD(T)/CBS limit, 67,68,87 but the ZPE and thermal energy correction elongate this by 0.033 Å, resulting in the best estimate of 2.958 Å. 87 Nevertheless, the distance change is not substantial and the resulting binding energy is also small. For convenience of comparison, we use the CCSD(T)/aug-cc-pVQZ geometry using RO-O of 2.92 Å (Figure 1a). The binding energies were calculated from both the all-electron Hamiltonian and the frozen-core Hamiltonian. CCSD/aVQZ+J3 B3LYP-D3/aVQZ+J3 PBE0-D3/aVQZ+J3

-152.84

Ebond (a.u.)

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


-152.86

-152.88

0

0.01

τ (a.u.)

0.02

Figure 2: Total FN-DMC energy comparison of MO-generating methods for the water dimer, with linear fits. VMC optimizations of Jastrow correlations of up to 3-body terms were carried out beforehand. The all-electron calculations were done using the aug-cc-pVnZ basis sets (n = D, T, Q, abbreviated as aVnZ). Figure 2 shows a cursory comparison of total FN-DMC energies with MOs from CCSD/aVQZ, B3LYP 88 -D3/aVQZ, and PBE0-D3/aVQZ calculations at τ = 0.005, 0.01, 0.015, and 0.02 au. The quotable FN-DMC energies are the extrapolated values at τ = 0. Varying τ within this range changes the energy by about 85 kJ/mol for all three methods. Since the FN-DMC method is variational, the clear separation of CCSD energies from the others justifies the preference of DFT calculations over CCSD for MO generation. Energies from B3LYP-D3 and PBE0-D3 are relatively close to each other, with B3LYP-D3 giving slightly smaller time-step errors and PBE0-D3 giving slightly smaller 9



statistical errors. We proceed with the remainder of the FN-DMC calculations using PBE0-D3 to generate the MOs (and the fermionic nodes).

-18

-21.00

De (kJ/mol)

-21.76(58)

-20 -22

-21.97(23)

-24

-22.19(23)

0

CCSD(T)/CBS DMC/aVDZ+J3 DMC/aVTZ+J3 DMC/aVQZ+J3

0.01

τ (a.u.)

0.02

Figure 3: Time-step dependence of the all-electron water dimer binding energies, with the ZSGMA branch cutoff scheme. The numbers in the plot are the extrapolated values at τ = 0 for the data points of their respective colors.

CCSD(T)/CBS DMC/aug-VDZ(BFD)+J3 DMC/aug-VTZ(BFD)+J3 DMC/aug-VQZ(BFD)+J3

-18 De (kJ/mol)

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

-20

-20.64(11)

-22

-20.88(27) -21.00

-24

-21.12(18)

0

0.01

0.02

τ (a.u.)

Figure 4: Time-step dependence of the frozen-core water dimer binding energies, with the T-moves treatment for PPs and the ZSGMA branch cutoff scheme. The extrapolated values at τ = 0 are written for the data points of their respective colors. Following the ZSGMA scheme mentioned previously, the differences De = Ebond − Efar are calculated and shown in Figure 3. Each data point represents the difference between the averages over approximately 5 × 109 independent samples. It is noted that the allelectron binding energies are slightly larger than those from CCSD(T)/CBS and from earlier all-electron FN-DMC calculations which were done with Hartree–Fock and B3LYP nodes. 62 10


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For the frozen-core calculations, we compare the aug-VnZ(BFD) valence basis sets. The τ -dependence of De is shown in Figure 4. Compared to the all-electron FN-DMC energies, these binding energies are in better agreement with the CCSD(T)/CBS value. This suggests that the averaging of core degrees of freedom as part of the design of PPs stabilizes the energies against time step bias. Also, the time step bias that is more prominent in total energies (see Figure 2) is largely canceled out as a result of the subtraction to obtain De . This makes it permissible to forgo extrapolation to τ = 0 and report energies at sufficiently small single τ values such as 0.005 or 0.01 au. 64,65,89

Benzene and diatomic halogens The interaction of benzene and X2 has been previously studied using CCSD(T)/CBS calculations. 25 However, no QMC calculations have been reported yet for this system, despite some QMC results for the interactions of benzene with H2, H2O, and benzene. 29,90,91 Table 1: Basis set superposition error-corrected binding energies (kJ/mol) for benzene Cl2/Br2.a Method CCSD(T) PBE-D3 BLYP-D3 PBE0-D3 HSE06-D3 B2GP-PLYP a

aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ 9.45/11.79 11.23/14.03 12.12/15.27 19.20/24.39 16.83/21.29 16.60/21.13 20.30/24.93 18.20/22.23 17.87/22.10 14.69/19.95 13.03/17.54 13.02/17.50 15.31/20.92 13.57/18.28 13.55/18.22 13.02/16.73 13.07/16.50 13.30/16.69

CBS 12.78/16.17 16.43/20.90 17.67/21.85 13.02/17.48 13.54/18.17 13.48/16.93

The DFT results are from Ref. 25. The CBS calculations were obtained based on the formalism of Ref. 92.

We used the all-electron geometries of Cl2/Br2 adsorption on benzene optimized at the CCSD(T)/aVTZ level (Figure 1b). 23 The energies used for CBS extrapolation for CCSD(T) as well as other DFT methods are shown in Table 1. The QMC calculations were done with the aug-VTZ(BFD) valence basis sets. Figures 5 and 6 show the binding energies of benzene Cl2/Br2 obtained from both the unaugmented and augmented BFD basis sets, for comparison. Each data point except the DMC/aug-VTZ(BFD) values at τ = 0.005 au 11



-10 -11.80(91) -12.78 -13.07(42)

De (kJ/mol)

-11 -12 -13

CCSD(T)/CBS DMC/VTZ(BFD)+J3 DMC/aug-VTZ(BFD)+J3

-14 -15 0

0.005

0.01

τ

0.015

0.02

(Ha-1)

Figure 5: Time-step dependence of benzene Cl2 binding energies, with the T-moves treatment for PPs and the ZSGMA branch cutoff scheme. The extrapolated values at τ = 0 are written according to color.

-15

-16.17 -16.47(111)

De (kJ/mol)

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

-16.62(2)

-16 -17

CCSD(T)/CBS DMC/VTZ(BFD)+J3 DMC/aug-VTZ(BFD)+J3

-18 0

0.005

0.01

τ

0.015

0.02

(Ha-1)

Figure 6: Time-step dependence of benzene Br2 binding energies, with the T-moves treatment for PPs and the ZSGMA branch cutoff scheme. The extrapolated values at τ = 0 are written according to color.

12


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represents the difference between the averages over 1010 independent samples. The data points for DMC/aug-VTZ(BFD) at τ = 0.005 au represent differences between averages over 1011 samples. Despite the fortuitous extrapolations of FN-DMC energies based on VTZ(BFD) being closer to the CCSD(T)/CBS energies than the FN-DMC energies based on aug-VTZ(BFD), the statistical and the overall time-step errors are comparable in size and are only about an order of magnitude smaller than |EDMC |. This tendency is more severe in benzene Br2 than in benzene Cl2. Unlike the case with the frozen-core water dimer, we may not expect an estimate of EDMC at some small τ to be accurate without extrapolation to τ = 0.

Conclusion We have studied the benzene Cl2/Br2 system involved in X π interactions in comparison with the typical hydrogen bonding of the water dimer. QMC calculations accounting for explicit correlations of up to 3 bodies were carried out. The FN-DMC binding energies show good agreement with the CCSD(T)/CBS values within error bars. In spite of this, the difference between the microscopic processes within X π interactions and within hydrogen bonding is clearly apparent in the time-step dependence of DMC energies. We expect further elucidation of the nature of X π interaction to be attained from a more extensive investigation of halogens with carbon allotropes. Up to chemical accuracy (1 kcal/mol), FN-DMC is shown to be insensitive to the choice of basis sets and could be treated as a black box in most studies involving interatomic forces and chemical reactions. 26 More caution is needed in preparing ΨT in the study of noncovalent interactions, including X π interactions, where the desirable accuracy is subchemical (0.1 kcal/mol). 64 The significance of our present work is in that it can be extended to more complicated systems held together by these interactions, and that it opens up an alternate route to study, among other things, halogen intercalation phenomena between

13



graphene layers. Much of the labor in obtaining accurate energies for these systems would thus be focused on testing a sizable collection of relevant basis sets and orbital-generating preliminary calculations, rather than on the FN-DMC calculation itself.

Supporting Information Available All of FN-DMC binding energies plotted in Figures 3 through 6.

Author Information • D. ChangMo Yang: • Dong Yeon Kim: • Kwang S. Kim:

0000-0002-7476-9838 0000-0001-7798-1108 0000-0002-6929-5359

Note The authors declare no competing financial interest.

Acknowledgement K.S.K. was supported by KISTI (KSC-2018-CHA-0057, KSC-2018-CRE-0077) and D.C.Y. was supported by KISTI (KSC-2016-S1-0034, KSC-2019-CRE-0034).

References (1) Drexler, K. E. Molecular engineering: An approach to the development of general capabilities for molecular manipulation. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, 5275– 5278. (2) Qiu, S.; Zhu, G. Molecular engineering for synthesizing novel structures of metal–organic frameworks with multifunctional properties. Coord. Chem. Rev. 2009, 253, 2891–2911. 14


Page 14 of 26

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


(3) Lee, J. Y.; Hong, B. H.; Kim, W. Y.; Min, S. K.; Kim, Y.; Jouravlev, M. V.; Bose, R.; Kim, K. S.; Hwang, I.-C.; Kaufman, L. J. et al. Near-field focusing and magnification through self-assembled nanoscale spherical lenses. Nature 2009, 460, 498–501. (4) Ashkenasy, G.; Cahen, D.; Cohen, R.; Shanzer, A.; Vilan, A. Molecular Engineering of Semiconductor Surfaces and Devices. Acc. Chem. Res. 2002, 35, 121–128. (5) Tiwari, J. N.; Sultan, S.; Myung, C. W.; Yoon, T.; Li, N.; Ha, M.; Harzandi, A. M.; Park, H. J.; Kim, D. Y.; Chandrasekaran, S. S. et al. Multicomponent electrocatalyst with ultralow Pt loading and high hydrogen evolution activity. Nat. Energy 2018, 3, 773–782. (6) Lee, E. C.; Kim, D.; Jurečka, P.; Tarakeshwar, P.; Hobza, P.; Kim, K. S. Understanding of Assembly Phenomena by Aromatic–Aromatic Interactions: Benzene Dimer and the Substituted Systems. J. Phys. Chem. A 2007, 111, 3446–3457. (7) Lee, J. Y.; Hong, B. H.; Kim, D. Y.; Mason, D. R.; Lee, J. W.; Chun, Y.; Kim, K. S. Tuning Molecular Self-Assembly Toward Intriguing Nanomaterial Architectures. Chem.: Eur. J. 2013, 19, 9118–9122. (8) Cho, Y.; Cho, W. J.; Youn, I. S.; Lee, G.; Singh, N. J.; Kim, K. S. Density Functional Theory Based Study of Molecular Interactions, Recognition, Engineering, and Quantum Transport in π Molecular Systems. Acc. Chem. Res. 2014, 47, 3321–3330. (9) Whitesides, G. M.; Simanek, E. E.; Mathias, J. P.; Seto, C. T.; Chin, D.; Mammen, M.; Gordon, D. M. Noncovalent Synthesis: Using Physical-Organic Chemistry To Make Aggregates. Acc. Chem. Res. 1995, 28, 37–44. (10) Georgakilas, V.; Otyepka, M.; Bourlinos, A. B.; Chandra, V.; Kim, N.; Kemp, K. C.; Hobza, P.; Zboril, R.; Kim, K. S. Functionalization of Graphene: Covalent and NonCovalent Approaches, Derivatives and Applications. Chem. Rev. 2012, 112, 6156–6214.

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(11) Georgakilas, V.; Tiwari, J. N.; Kemp, K. C.; Perman, J. A.; Bourlinos, A. B.; Kim, K. S.; Zboril, R. Noncovalent Functionalization of Graphene and Graphene Oxide for Energy Materials, Biosensing, Catalytic, and Biomedical Applications. Chem. Rev. 2016, 116, 5464–5519. (12) Bernstein, J.; Davis, R. E.; Shimoni, L.; Chang, N.-L. Patterns in Hydrogen Bonding: Functionality and Graph Set Analysis in Crystals. Angew. Chem. Int. Ed. 1995, 34, 1555–1573. (13) Lee, H. M.; Kumar, A.; Kołaski, M.; Kim, D. Y.; Lee, E. C.; Min, S. K.; Park, M.; Choi, Y. C.; Kim, K. S. Comparison of cationic, anionic and neutral hydrogen bonded dimers. Phys. Chem. Chem. Phys. 2010, 12, 6278–6287. (14) Kim, K. S.; Tarakeshwar, P.; Lee, J. Y. Molecular Clusters of π-Systems: Theoretical Studies of Structures, Spectra, and Origin of Interaction Energies. Chem. Rev. 2000, 100, 4145–4186. (15) Tkatchenko, A.; Alfè, D.; Kim, K. S. First-Principles Modeling of Non-Covalent Interactions in Supramolecular Systems: The Role of Many-Body Effects. J. Chem. Theory Comput. 2012, 8, 4317–4322. (16) Lee, E. C.; Hong, B. H.; Lee, J. Y.; Kim, J. C.; Kim, D.; Kim, Y.; Tarakeshwar, P.; Kim, K. S. Substituent Effects on the Edge-to-Face Aromatic Interactions. J. Am. Chem. Soc. 2005, 127, 4530–4537. (17) Sherrill, C. D. Energy component analysis of π-interactions. Acc. Chem. Res. 2012, 46, 1020–1028. (18) Desiraju, G. R.; Ho, P. S.; Kloo, L.; Legon, A. C.; Marquardt, R.; Metrangolo, P.; Politzer, P.; Resnati, G.; Rissanen, K. Definition of the halogen bond (IUPAC Recommendations 2013). Pure Appl. Chem. 2013, 85, 1711–1713.

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Page 16 of 26

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


(19) Kolář, M. H.; Hobza, P. Computer Modeling of Halogen Bonds and Other σ-Hole Interactions. Chem. Rev. 2016, 116, 5155–5187. (20) Yoon, J.; Kyung Kim, S.; Jiten Singh, N.; S. Kim, K. Imidazolium receptors for the recognition of anions. Chem. Soc. Rev. 2006, 35, 355–360. (21) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104. (22) Tkatchenko, A.; Scheffler, M. Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data. Phys. Rev. Lett. 2009, 102, 073005. (23) Kim, D. Y.; Madridejos, J. M. L.; Ha, M.; Kim, J.-H.; Yang, D. C.; Baig, C.; Kim, K. S. Size-dependent conformational change in halogen–π interaction: from benzene to graphene. Chem. Commun. 2017, 53, 6140–6143. (24) Ha, M.; Kim, D. Y.; Li, N.; Madridejos, J. M. L.; Park, I. K.; Youn, I. S.; Lee, J.; Baig, C.; Filatov, M.; Min, S. K. et al. Adsorption of Carbon Tetrahalides on Coronene and Graphene. J. Phys. Chem. C 2017, 121, 14968–14974. (25) Youn, I. S.; Kim, D. Y.; Cho, W. J.; Madridejos, J. M. L.; Lee, H. M.; Kołaski, M.; Lee, J.; Baig, C.; Shin, S. K.; Filatov, M. et al. Halogen–π Interactions between Benzene and X2/CX4 (X = Cl, Br): Assessment of Various Density Functionals with Respect to CCSD(T). J. Phys. Chem. A 2016, 120, 9305–9314. (26) Foulkes, W. M. C.; Mitas, L.; Needs, R. J.; Rajagopal, G. Quantum Monte Carlo simulations of solids. Rev. Mod. Phys. 2001, 73, 33–83. (27) Grossman, J. C. Benchmark quantum Monte Carlo calculations. J. Chem. Phys. 2002, 117, 1434–1440. 17



(28) Tanaka, S., Roy, P.-N., Mitas, L., Eds. Recent Progress in Quantum Monte Carlo; ACS Symposium Series; American Chemical Society, 2016; Vol. 1234. (29) Dubecký, M.; Mitas, L.; Jurečka, P. Noncovalent Interactions by Quantum Monte Carlo. Chem. Rev. 2016, 116, 5188–5215. (30) Cavallo, G.; Metrangolo, P.; Milani, R.; Pilati, T.; Priimagi, A.; Resnati, G.; Terraneo, G. The Halogen Bond. Chem. Rev. 2016, 116, 2478–2601. (31) Parisini, E.; Metrangolo, P.; Pilati, T.; Resnati, G.; Terraneo, G. Halogen bonding in halocarbon–protein complexes: a structural survey. Chem. Soc. Rev. 2011, 40, 2267–2278. (32) Mukherjee, A.; Tothadi, S.; Desiraju, G. R. Halogen Bonds in Crystal Engineering: Like Hydrogen Bonds yet Different. Acc. Chem. Res. 2014, 47, 2514–2524. (33) Riley, K. E.; Hobza, P. Noncovalent interactions in biochemistry. WIREs: Comput. Mol. Sci. 2011, 1, 3–17. (34) Metrangolo, P.; Meyer, F.; Pilati, T.; Resnati, G.; Terraneo, G. Halogen Bonding in Supramolecular Chemistry. Angew. Chem. Int. Ed. 2008, 47, 6114–6127. (35) Politzer, P.; Murray, J. S. Halogen Bonding: An Interim Discussion. ChemPhysChem 2013, 14, 278–294. (36) Politzer, P.; Lane, P.; Concha, M. C.; Ma, Y.; Murray, J. S. An overview of halogen bonding. J. Mol. Model. 2007, 13, 305–311. (37) Politzer, P.; Murray, J. S.; Clark, T. Halogen bonding: an electrostatically-driven highly directional noncovalent interaction. Phys. Chem. Chem. Phys. 2010, 12, 7748–7757. (38) Politzer, P.; S. Murray, J.; Clark, T. Halogen bonding and other σ-hole interactions: a perspective. Phys. Chem. Chem. Phys. 2013, 15, 11178–11189. 18


Page 18 of 26

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


(39) Stone, A. J. Are Halogen Bonded Structures Electrostatically Driven? J. Am. Chem. Soc. 2013, 135, 7005–7009. (40) Wallnoefer, H. G.; Fox, T.; Liedl, K. R.; Tautermann, C. S. Dispersion dominated halogen–π interactions: energies and locations of minima. Phys. Chem. Chem. Phys. 2010, 12, 14941–14949. (41) Matter, H.; Nazaré, M.; Güssregen, S.; Will, D.; Schreuder, H.; Bauer, A.; Urmann, M.; Ritter, K.; Wagner, M.; Wehner, V. Evidence for C—Cl/C—Br· · · π Interactions as an Important Contribution to Protein–Ligand Binding Affinity. Angew. Chem. Int. Ed. 2009, 48, 2911–2916. (42) Imai, Y. N.; Inoue, Y.; Yamamoto, Y. Propensities of Polar and Aromatic Amino Acids in Noncanonical Interactions: Nonbonded Contacts Analysis of Protein–Ligand Complexes in Crystal Structures. J. Med. Chem. 2007, 50, 1189–1196. (43) Li, B.; Zang, S.-Q.; Wang, L.-Y.; Mak, T. C. W. Halogen bonding: A powerful, emerging tool for constructing high-dimensional metal-containing supramolecular networks. Coord. Chem. Rev. 2016, 308, 1–21. (44) Dumele, O.; Trapp, N.; Diederich, F. Halogen Bonding Molecular Capsules. Angew. Chem. Int. Ed. 2015, 54, 12339–12344. (45) Sedlak, R.; Kolář, M. H.; Hobza, P. Polar Flattening and the Strength of Halogen Bonding. J. Chem. Theory Comput. 2015, 11, 4727–4732. (46) Jungbauer, S. H.; Huber, S. M. Cationic Multidentate Halogen-Bond Donors in Halide Abstraction Organocatalysis: Catalyst Optimization by Preorganization. J. Am. Chem. Soc. 2015, 137, 12110–12120. (47) Pan, F.; Beyeh, N. K.; Rissanen, K. Concerted Halogen-Bonded Networks with N-Alkyl

19



Ammonium Resorcinarene Bromides: From Dimeric Dumbbell to Capsular Architectures. J. Am. Chem. Soc. 2015, 137, 10406–10413. (48) Langton, M. J.; Robinson, S. W.; Marques, I.; Félix, V.; Beer, P. D. Halogen bonding in water results in enhanced anion recognition in acyclic and rotaxane hosts. Nat. Chem. 2014, 6, 1039–1043. (49) Kim, N.; Kim, K. S.; Jung, N.; Brus, L.; Kim, P. Synthesis and Electrical Characterization of Magnetic Bilayer Graphene Intercalate. Nano Lett. 2011, 11, 860–865. (50) Jung, N.; Crowther, A. C.; Kim, N.; Kim, P.; Brus, L. Raman Enhancement on Graphene: Adsorbed and Intercalated Molecular Species. ACS Nano 2010, 4, 7005–7013. (51) Yang, J. W.; Lee, G.; Kim, J. S.; Kim, K. S. Gap Opening of Graphene by Dual FeCl3-Acceptor and K-Donor Doping. J. Phys. Chem. Lett. 2011, 2, 2577–2581. (52) Wu, B.-R.; Yang, C.-K. Electronic structures of graphane with vacancies and graphene adsorbed with fluorine atoms. AIP Adv. 2012, 2, 012173. (53) Şahin, H.; Ciraci, S. Chlorine Adsorption on Graphene: Chlorographene. J. Phys. Chem. C 2012, 116, 24075–24083. (54) Xu, C.; Brown, P. A.; Lu, J.; Shuford, K. L. Electronic Properties of Halogen-Adsorbed Graphene. J. Phys. Chem. C 2015, 119, 17271–17277. (55) Rudenko, A. N.; Keil, F. J.; Katsnelson, M. I.; Lichtenstein, A. I. Adsorption of diatomic halogen molecules on graphene: A van der Waals density functional study. Phys. Rev. B 2010, 82, 035427. (56) Ulbricht, H.; Zacharia, R.; Cindir, N.; Hertel, T. Thermal desorption of gases and solvents from graphite and carbon nanotube surfaces. Carbon 2006, 44, 2931–2942. (57) Girão, E. C.; Liebold-Ribeiro, Y.; Batista, J. A.; Barros, E. B.; Fagan, S. B.; Filho, J. M.; Dresselhaus, M. S.; Souza Filho, A. G. Functionalization of single-wall carbon nanotubes 20


Page 20 of 26

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


through chloroform adsorption: theory and experiment. Phys. Chem. Chem. Phys. 2010, 12, 1518–1524. (58) Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158–6170. (59) Karton, A.; Tarnopolsky, A.; Lamère, J.-F.; Schatz, G. C.; Martin, J. M. L. Highly Accurate First-Principles Benchmark Data Sets for the Parametrization and Validation of Density Functional and Other Approximate Methods. Derivation of a Robust, Generally Applicable, Double-Hybrid Functional for Thermochemistry and Thermochemical Kinetics. J. Phys. Chem. A 2008, 112, 12868–12886. (60) Xu, J.; Deible, M. J.; Peterson, K. A.; Jordan, K. D. Correlation Consistent Gaussian Basis Sets for H, B–Ne with Dirac–Fock AREP Pseudopotentials: Applications in Quantum Monte Carlo Calculations. J. Chem. Theory Comput. 2013, 9, 2170–2178. (61) Gillan, M. J.; Manby, F. R.; Towler, M. D.; Alfè, D. Assessing the accuracy of quantum Monte Carlo and density functional theory for energetics of small water clusters. J. Chem. Phys. 2012, 136, 244105. (62) Benedek, N. A.; Snook, I. K.; Towler, M. D.; Needs, R. J. Quantum Monte Carlo calculations of the dissociation energy of the water dimer. J. Chem. Phys. 2006, (63) Gurtubay, I. G.; Needs, R. J. Dissociation energy of the water dimer from quantum Monte Carlo calculations. J. Chem. Phys. 2007, (64) Dubecký, M.; Jurečka, P.; Derian, R.; Hobza, P.; Otyepka, M.; Mitas, L. Quantum Monte Carlo Methods Describe Noncovalent Interactions with Subchemical Accuracy. J. Chem. Theory Comput. 2013, 9, 4287–4292. (65) Dubecký, M.; Derian, R.; Jurečka, P.; Mitas, L.; Hobza, P.; Otyepka, M. Quantum

21



Monte Carlo for noncovalent interactions: an efficient protocol attaining benchmark accuracy. Phys. Chem. Chem. Phys. 2014, 16, 20915–20923. (66) Řezáč, J.; Dubecký, M.; Jurečka, P.; Hobza, P. Extensions and applications of the A24 data set of accurate interaction energies. Physical Chemistry Chemical Physics 2015, 17, 19268–19277. (67) Řezáč, J.; Hobza, P. Describing Noncovalent Interactions beyond the Common Approximations: How Accurate Is the “Gold Standard,” CCSD(T) at the Complete Basis Set Limit? J. Chem. Theory Comput. 2013, 9, 2151–2155. (68) Miliordos, E.; Xantheas, S. S. An accurate and efficient computational protocol for obtaining the complete basis set limits of the binding energies of water clusters at the MP2 and CCSD(T) levels of theory: Application to (H2O)m, m = 2-6, 8, 11, 16, and 17. J. Chem. Phys. 2015, 142, 234303. (69) Burkatzki, M.; Filippi, C.; Dolg, M. Energy-consistent pseudopotentials for quantum Monte Carlo calculations. J. Chem. Phys. 2007, 126, 234105. (70) Hammond, B. L.; Lester Jr., W. A.; Reynolds, P. J. Monte Carlo Methods in Ab Initio Quantum Chemistry; World Scientific Lecture Notes in Chemistry; World Scientific, 1994; Vol. 1. (71) Nightingale, M. P., Umrigar, C. J., Eds. Quantum Monte Carlo Methods in Physics and Chemistry; NATO Science Series C; Springer, 1999; Vol. 525. (72) Kim, J.; Baczewski, A. D.; Beaudet, T. D.; Benali, A.; Bennett, M. C.; Berrill, M. A.; Blunt, N. S.; Borda, E. J. L.; Casula, M.; Ceperley, D. M. et al. QMCPACK : an open source ab initio quantum Monte Carlo package for the electronic structure of atoms, molecules and solids. J. Phys.: Condens. Matter 2018, 30, 195901.

22


Page 22 of 26

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


(73) Born, M.; Oppenheimer, R. Zur Quantentheorie der Molekeln. Ann. Phys. 1927, 389, 457–484. (74) Esler, K.; Kim, J.; Ceperley, D.; Shulenburger, L. Accelerating Quantum Monte Carlo Simulations of Real Materials on GPU Clusters. Comput. Sci. Eng. 2012, 14, 40–51. (75) Drummond, N. D.; Towler, M. D.; Needs, R. J. Jastrow correlation factor for atoms, molecules, and solids. Phys. Rev. B 2004, 70, 235119. (76) Umrigar, C. J.; Toulouse, J.; Filippi, C.; Sorella, S.; Hennig, R. G. Alleviation of the Fermion-Sign Problem by Optimization of Many-Body Wave Functions. Phys. Rev. Lett. 2007, 98, 110201. (77) Bouabça, T.; Braïda, B.; Caffarel, M. Multi-Jastrow trial wavefunctions for electronic structure calculations with quantum Monte Carlo. J. Chem. Phys. 2010, 133, 044111. (78) Moskowitz, J. W.; Schmidt, K. E.; Lee, M. A.; Kalos, M. H. A new look at correlation energy in atomic and molecular systems. II. The application of the Green’s function Monte Carlo method to LiH. J. Chem. Phys. 1982, 77, 349–355. (79) Reynolds, P. J.; Ceperley, D. M.; Alder, B. J.; Jr, W. A. L. Fixed-node quantum Monte Carlo for molecules. J. Chem. Phys. 1982, 77, 5593–5603. (80) Zen, A.; Sorella, S.; Gillan, M. J.; Michaelides, A.; Alfè, D. Boosting the accuracy and speed of quantum Monte Carlo: Size consistency and time step. Phys. Rev. B 2016, 93, 241118. (81) Kendall, R. A.; Jr, T. H. D.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96, 6796–6806. (82) Woon, D. E.; Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98, 1358– 1371. 23



(83) Wilson, A. K.; Woon, D. E.; Peterson, K. A.; Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. IX. The atoms gallium through krypton. J. Chem. Phys. 1999, 110, 7667–7676. (84) Odutola, J. A.; Dyke, T. R. Partially deuterated water dimers: Microwave spectra and structure. J. Chem. Phys. 1980, 72, 5062–5070. (85) Curtiss, L. A.; Frurip, D. J.; Blander, M. Studies of molecular association in H2O and D2O vapors by measurement of thermal conductivity. J. Chem. Phys. 2008, 71, 2703–2711. (86) Mas, E. M.; Bukowski, R.; Szalewicz, K.; Groenenboom, G. C.; Wormer, P. E. S.; van der Avoird, A. Water pair potential of near spectroscopic accuracy. I. Analysis of potential surface and virial coefficients. J. Chem. Phys. 2000, 113, 6687–6701. (87) Kim, K. S.; Mhin, B. J.; Choi, U.; Lee, K. Ab initio studies of the water dimer using large basis sets: The structure and thermodynamic energies. J. Chem. Phys. 1992, 97, 6649–6662. (88) Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648–5652. (89) Dubecký, M. Noncovalent Interactions by Fixed-Node Diffusion Monte Carlo: Convergence of Nodes and Energy Differences vs Gaussian Basis-Set Size. J. Chem. Theory Comput. 2017, 13, 3626–3635. (90) Beaudet, T. D.; Casula, M.; Kim, J.; Sorella, S.; Martin, R. M. Molecular hydrogen adsorbed on benzene: Insights from a quantum Monte Carlo study. J. Chem. Phys. 2008, 129, 164711. (91) Azadi, S.; Cohen, R. E. Chemical accuracy from quantum Monte Carlo for the benzene dimer. J. Chem. Phys. 2015, 143, 104301. 24


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(92) Min, S. K.; Lee, E. C.; Lee, H. M.; Kim, D. Y.; Kim, D.; Kim, K. S. Complete basis set limit of Ab initio binding energies and geometrical parameters for various typical types of complexes. J. Comp. Chem. 2008, 29, 1208–1221.

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Graphical TOC Entry 𝑸𝒖𝒂𝒏𝒕𝒖𝒎 𝑴𝒐𝒏𝒕𝒆 𝑪𝒂𝒓𝒍𝒐 𝜳𝑻 𝓡 = 𝑫↑ 𝑫↓ 𝒆−𝑱(𝓡)

𝑯𝟐 𝑶

𝑩𝒆𝒏𝒛𝒆𝒏𝒆 − 𝑿𝟐

𝟐

26


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Quantum Monte Carlo Study of Water Dimer Binding Energy and

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