Solvation Effects on Chemical Shifts by Embedded Cluster Integral

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Solvation Effects on Chemical Shifts by Embedded Cluster Integral Equation Theory Roland Frach, and Stefan M. Kast J. Phys. Chem. A, Just Accepted Manuscript • Publication Date (Web): 07 Nov 2014 Downloaded from http://pubs.acs.org on November 7, 2014

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The Journal of Physical Chemistry

Solvation Effects on Chemical Shifts by Embedded Cluster Integral Equation Theory Roland Frach, Stefan M. Kast* Physikalische Chemie III, TU Dortmund, Otto-Hahn-Str. 6, 44227 Dortmund, Germany

Corresponding Author *E-mail: [email protected].

ACS Paragon Plus Environment

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ABSTRACT. The accurate computational prediction of nuclear magnetic resonance (NMR) parameters like chemical shifts represents a challenge if the species studied is immersed in strongly polarizing environments such as water. Common approaches to treating a solvent in the form of, e.g., the polarizable continuum model (PCM) ignore strong directional interactions such as H-bonds to the solvent which can have substantial impact on magnetic shieldings. We here present a computational methodology that accounts for atomic-level solvent effects on NMR parameters by extending the embedded cluster reference interaction site model (EC-RISM) integral equation theory to the prediction of chemical shifts of N-methylacetamide (NMA) in aqueous solution. We examine the influence of various so-called closure approximations of the underlying three-dimensional RISM theory as well as the impact of basis set size and different treatment of electrostatic solute-solvent interactions. We find considerable and systematic improvement over reference PCM and gas phase calculations. A smaller basis set in combination with a simple point charge model already yields good performance which can be further improved by employing exact electrostatic quantum-mechanical solute-solvent interaction energies. A larger basis set benefits more significantly from exact over point charge electrostatics, which can be related to differences of the solvent’s charge distribution.

KEYWORDS. 3D RISM theory, nuclear magnetic resonance (NMR), chemical shielding, Nmethylacetamide (NMA), aqueous solution.


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INTRODUCTION Chemical shifts are key data obtained from nuclear magnetic resonance (NMR) spectroscopy to determine chemical structure. With increasing molecular size, conformational freedom, and chemical complexity the task to assign spectral lines to nuclei becomes ever more difficult, which gives rise to increasing demand for accurate computational prediction. Empirical group contribution assignment is not sufficient if, for instance, detailed conformational information has to be deduced from spectra, even if more elaborate NMR techniques are employed. Quantumchemical calculations are in principle capable of filling the gap since NMR parameters are observables that can be inferred from a compound’s wave function. Since the first practical application of quantum-chemically computed chemical shifts by Schindler and Kutzelnigg1 ab initio and density functional theory (DFT) calculations have provided meaningful support for the interpretation of complex spectra for example of small molecules2,3 or even of peptides.4 While certainly much more time consuming than common empirical NMR spectra simulation programs,5 quantum chemistry allows for the investigation of molecular conformations and their surroundings on NMR parameters under controlled conditions and without parametrization. However, accurate quantum-chemical predictions of measured chemical shifts (and of course other properties such as coupling constants) suffer from several sources of error.6 Besides limitations of basic quantum-mechanical factors such as basis sets and levels of theory, these inaccuracies result from uncertainties about conformational ensemble averages that have to be considered7,8 as well as from the necessity to describe solute-solvent interactions in a condensed phase environment appropriately. In particular, the presence and properties of a solvent directly influences chemical shielding constants by perturbing the vacuum electron density of a solvated molecule,9 and indirectly by modulating the conformational distribution of the molecule of


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interest.10 Therefore, quantum-chemical predictions of chemical shifts require not only an adequate description of the solute alone, but also of the surrounding solvent and its interaction with the solute. Continuum solvation models based on the dielectric response such as the polarizable continuum model (PCM)11 are commonly used for the description of polarization effects and typically improve chemical shift calculations when directional interactions are of minor importance.12 Beyond chemical shift predictions, combinations of conformational analysis with PCM-based NMR coupling constant calculations can guide the interpretation of complex experiments.13 However, PCM (and other self-consistent reaction field approaches) ignore directional interactions, higher-order electric multipoles, or dynamic solvation effects and can therefore not fully account for observable solvent contributions.14 As an alternative, explicit solvent molecules can be added, for instance within quantum-mechanical/classical-mechanical (QM/MM) models,15 by fragmentation schemes, like the adjustable density matrix assembler (ADMA),16,17 or by combining both, as in the automatic fragmentation QM/MM18 model that can be combined with classical or ab initio molecular dynamics (MD) simulations8 to deal with the conformational sampling problem. Explicit solvent calculations dramatically enhance chemical shift predictions,8 but are not routinely used because of their computational demand particularly when coupled with simulations. As alternative to continuum models, integral equation theories can be used for describing solvent distributions around solute molecules on a molecular level, since the granular solvent properties are preserved in this case. One computationally manageable way is to employ the reference interaction site model (RISM)19,20 integral equation theory, more specifically in the form of the three-dimensional (3D RISM)21,22 approximation, that yields solvent-site distribution functions around arbitrarily shaped solutes. One way to couple this representation of the solvent structure


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with quantum-chemical calculations is to map the resulting solvent-charge distribution onto discrete, embedding point charges that represent an additional component of the solute’s electronic Hamiltonian. In this way, the solvent distribution polarizes the solutes, which, in turn, exhibits a solvent-modulated electrostatic potential on the solvent sites giving rise to a change of the solvent distribution by solving the 3D RISM equations for the modified solute-solvent potential. Repeating these steps until self-consistency between solute’s electronic and solvent structure is obtained establishes the “embedded cluster reference interaction site model” (ECRISM).23 We have developed this approach for predicting thermodynamic solvation properties since RISM theory yields analytical expressions for the excess chemical potential that, by adding the electronic energy, allows for the determination of the solute’s free energy, G, in solution. ECRISM utilizes common molecular solvent models such that the granular and directional nature of solute-solvent interactions is preserved. The approach was successfully applied to conformational equilibria and pKa shifts in water,23 to tautomeric equilibria within the SAMPL2 predictions contest,24 and to pKa predictions in dimethyl sulfoxide (DMSO).25 In all cases examined so far, we find close agreement between theory and experiment. Since in the past the EC-RISM methodology was only tested for the sum of electronic energy and chemical potential, we had so far no means to determine whether the promising results were the consequence of a fortuitous cancellation of error. As a stringent test of the quality of the electronic structure component, we here extend the approach to magnetic shielding calculations. This facilitates the computation of isotropic chemical shifts δi = σ ref − σ i

(1)

by subtracting the isotropic nuclear magnetic shielding σi of the nucleus of interest i from its


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reference value σref. To this end, the EC-RISM procedure is easily augmented by adding a GIAO26 (gauge invariant atomic orbital) nuclear magnetic shielding determination step to the final electronic structure calculation after self-consistency has been reached. As a proof of principle, we apply the procedure to the prediction of the chemical shifts of all nuclei (except for oxygen for which the presently applied level of theory does not yield satisfactory results with nearly an order of magnitude larger deviations as shown by Auer27) of the small bioorganic building block N-methylacetamide (NMA) in aqueous solution (for nuclear designators, see Figure 1) and examine the impact of various approximations to the 3D RISM theory as well as the effect of varying the basis set size and the accuracy of treating solute-solvent electrostatic interactions.

Figure 1. Structure of N-methylacetamid (NMA) with nuclear designators used throughout.

COMPUTATIONAL METHODS Theory. We only briefly summarize the theory behind the EC-RISM approach, for further details see earlier work.23,24,25 The 3D RISM equations for modeling solvation thermodynamics aim at predicting total correlation functions between solute and solvent sites γ, hγ(r), on a spatial grid defined by vectors r, with gγ(r) = hγ(r)+1 defining the corresponding pair distribution


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functions. The product of solvent site density ργ and hγ is given by the convolution product ρ γ hγ (r ) = ∑ c γ ' (r ) ∗χ γγ ' (| r |)

(2)

γ'

between molecule-site direct correlation function cγ(r) and a precomputed pure solvent site-site susceptibility χγγ’(|r|). These integral equations are “closed” in a sense of supplementing them with an additional “closure” relation that we approximate in this work by our previously developed “partial series expansion” of order n (PSE-n)28 that gradually approaches the “hypernetted chain closure” (HNC)29,30 approximation for increasing order, ∑k (t γ* (r )) i / i! − 1 ⇔ t γ* > 0 hγ (r ) =  i =0 * ⇔ t γ* ≤ 0, exp(t γ (r )) − 1

(3)

where t γ* (r ) = − βu γ (r ) + hγ (r ) − c γ (r ) defines the renormalized indirect correlation function. The interaction between solute and solvent sites is specified by uγ(r), multiplied by a reciprocal temperature β. Order 1 (PSE-1) is also known under the acronym “Kovalenko-Hirata” (KH) closure in the literature.31 In agreement with findings by others,32 we anticipate better predictive quality with increasing order, notably for the numerically stable orders 3 and 4 for which close correspondence with HNC results can be expected. Converged solution to this set of nonlinear equations yield pair distribution functions along with an analytical prediction of the excess chemical potential.28 The solute-solvent interaction is modeled by the sum of apolar (repulsion/dispersion) and electrostatics terms, the latter given by the electrostatic potential (ESP) originating from the wave function under the effect of a polarizing environment. Here, we can choose between a simple point charge representation of the solute’s ESP that determines the electrostatic solute-


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solvent interaction, and the formally exact ESP that arises directly from the electronic wave function and the nuclear charges in the context of EC-RISM calculations. The latter approach is significantly more costly since it involves calculations of the ESP on a large number of grid points, in contrast to the point charge approximation that requires only a selected number of points close to the solute surface. In practice we employ a similar methodology as we developed to account for the ESP difference between atomic mono- and multipoles in earlier work on 3D RISM theory for polarizable solute models.33 Within EC-RISM, we account for the polarization by discretizing the solvent charge density, ρ q (r ) = ∑ q γ ρ γ g γ (r ) ,

(4)

γ

to produce a set of background charges, q (ri ) = ∫ ρ q (r ) dr ≈ ρ q (r ) Vi , Vi

(5)

that are added to the molecular electronic Hamiltonian. The resulting molecular ESP (from fitting to atom-centered point charges or directly from the quantum-chemical calculation) enters the closure relation (3) which yields, upon solution to the 3D RISM/closure equations, a new set of background charges. This cycle is iterated to self-consistency between solute-solvent distribution and electronic wave function. For the latter, we here employ the GIAO approach in order to extract isotropic magnetic shieldings. Computational Details. GIAO calculations were applied to both optimized vacuum and solution phase geometries, in the latter case by using PCM calculation results also for EC-RISM postprocessing, as was shown earlier to be a reasonable approximation.24 For comparison, GIAO


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results were also obtained for PCM wave functions. Following on one hand standard procedures for the quantum-chemical part, we employed the B3LYP hybrid functional34,35,36 and the 631+G(d) basis set37 within the Gaussian 0338 program package (Rev. D.02) throughout, using also its implementation of the CHELPG procedure39 for the determination of point charges. On the other hand, we also tested the performance of the larger aug-cc-pVTZ40,41 basis set in conjunction with B3LYP. Except for the nuclear magnetic shielding calculation in vacuo the minimum energy structure studied was generated by geometry optimization with the “integral equation formalism” (not to be confused with integral equation theory) IEF-PCM model11 in water using default atomic parameters. The PCM free energy in solution was calculated as the sum of electrostatic and nonelectrostatic terms. Only the trans conformation depicted in Figure 1 was taken into account since the cis form obtained from rotating around the N(H)-C(O) bond was found to be energetically disfavored by 1.9 kcal mol-1 by B3LYP/6-31+G(d)/PCM optimization, therefore essentially not contributing to a thermal ensemble at ambient temperature. The chemical shielding estimation was done by the secondary standard methodology δi = σ sec,ref − σ i + δsec, ref ,

(6)

where the calculated nuclear magnetic shielding reference in water, σsec,ref, is shifted by the chemical shift δsec,ref of the reference compound to common primary standards. Therefore we use the nuclear magnetic shielding values of TMS (tetramethylsilane) for 1H, NH3 for dioxane for

13

15

N and 1,4-

C in aqueous solution, thus avoiding discrepancies due to inconsistent solvent

models. The nuclear magnetic shielding of TMS, NH3 and 1,4-dioxane in aqueous solution were used as secondary standards for 1H,

15

N and

13

C with chemical shifts of -0.094 ppm42, -19.4


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ppm43 and 66.6 ppm44 to TMS in CDCl3 and neat NH3. The solvation model for the reference calculations were used throughout corresponding to the NMA solvation model. The proton chemical shifts of both methyl groups were estimated by averaging over the corresponding 1H chemical shifts. To describe repulsion-dispersion interactions within the EC-RISM solvation model between the solvent and the solute we used the 12-6 Lennard-Jones potential with the “General Amber Force Field” GAFF (version 1.4, March 2010)45,46 parameters, and parameters from Makrodimitri et al.47 for the silicon of TMS (see Table 1). Procedural parameters for EC-RISM calculations at 298.15 K follow closely earlier studies;24,25 in particular, we used 1043 grid points with 0.3 Å spacing and a convergence threshold of 10-5 kcal mol-1. As in earlier work,24 we describe water by a modified variant of the SPC/E48 model. All computational predictions were compared with experimental data taken from Exner et al.49 Table 1. Lennard-Jones parameters of all solute species. NMA C(1) H(1) N(H) H(N) C(O) O(C) C(2) H(2) TMS Si C H NH3 N H 1,4-dioxane C O H

σ/Å 3.400 2.471 3.250 1.069 3.400 2.960 3.400 2.650 σ/Å 3.385 3.400 3.650 σ/Å 3.250 1.069 σ/Å 3.400 3.000 2.471

-1

ε / cal mol 109.4 15.7 170.0 15.7 86.0 210.0 109.4 15.7 -1 ε / cal mol 585.0 109.4 15.7 -1 ε / cal mol 170.0 15.7 -1 ε / cal mol 109.4 170.0 15.7


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RESULTS AND DISCUSSION Statistics. With raw shielding data summarized in supporting Tables S1-S2 we computed isotropic chemical shifts listed in Tables 2 and 3 and displayed in Figure 2. Note that we indicate the point charge approach by the superscript “q”, whereas the exact electrostatics approach is denoted by superscript “φ” throughout. As expected, the presence and adequate treatment of the solvent has most substantial impact on the amide proton H(N), the amide carbon C(O), and the nitrogen nucleus N(H), whereas only modest influence on the methyl groups is found. Looking first at a comparison of all solvation methods under investigation for the smaller 631G+(d) basis set, we find that the chemical shifts of both methyl group carbons deviate only by around -1.54 to -1.99 ppm for C(1) and -3.58 to -2.22 ppm for C(2). For this basis set the 1H nuclei of the methyl groups show a similar trend with a deviation of around -0.06 to 0.20 ppm for H(1) and -0.34 to 0.16 ppm for H(2). Regarding the aug-cc-pVTZ results, the methyl group chemical shifts can also be considered quite insensitive to the solvation model, whereas the deviation range for the C(1) and C(2) nuclei is slightly greater. This insensitivity to the presence of a solvent was also captured in other calculations49 for NMA, suggesting that interaction with solvent molecules is not significant for these nuclei. In contrast, chemical shifts of the amide nuclei depend heavily on the solvation modeling methodology employed. The EC-RISM model cannot account for partial electron transfer between solute and solvent, as would be possible only if explicit water molecules were considered. Technically, this would require sampling of solvent configurations, implying dramatically larger effort. However, based on conclusions drawn by Exner and co-workers in their ab initio molecular dynamics study8 of NMA in water, electron transfer does not significantly influence chemical shifts.


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Table 2. Chemical shifts (in ppm) calculated by GIAO/B3LYP/6-31+G(d) in the gas phase (with vacuum-optimized structures) and using the solvation models PCM and EC-RISM (PSE-[1-4] and HNC) with optimized PCM structures compared with experimental values (deviations are shown in parenthesis). Superscript “q” indicates the point charge model for electrostatic solutesolvent interactions while superscript “φ” refers to the exact quantum-mechanical electrostatic potential. Statistical quality is measured by the root mean square deviation (RMSD) and the mean signed error (MSE).

literature vacuum PCM q

PSE-1

φ

PSE-1

q

PSE-2

φ

PSE-2

q

PSE-3

φ

PSE-3

q

PSE-4

φ

PSE-4 HNC

q

HNC

φ

C(1)

H(1)

N(H)

H(N)

C(O)

C(2)

H(2)

28.65

2.63

114.24

7.73

177.24

24.32

1.90

27.06 (-1.59) 26.81 (-1.84) 27.09 (-1.56) 26.76 (-1.89) 27.10 (-1.55) 26.71 (-1.94) 27.10 (-1.55) 26.69 (-1.96) 27.11 (-1.54) 26.68 (-1.97) 27.11 (-1.54) 26.66 (-1.99)

2.56 (-0.07) 2.57 (-0.06) 2.81 (0.18) 2.80 (0.17) 2.82 (0.19) 2.82 (0.19) 2.83 (0.20) 2.82 (0.19) 2.83 (0.20) 2.82 (0.19) 2.83 (0.20) 2.82 (0.19)

89.82 (-24.42) 103.16 (-11.09) 118.15 (3.91) 114.36 (0.12) 121.16 (6.92) 116.18 (1.94) 122.16 (7.92) 116.69 (2.45) 122.54 (8.30) 116.83 (2.59) 122.75 (8.51) 116.84 (2.60)

4.20 (-3.53) 6.18 (-1.55) 6.55 (-1.18) 6.54 (-1.19) 6.74 (-0.99) 6.74 (-0.99) 6.82 (-0.91) 6.82 (-0.91) 6.84 (-0.89) 6.86 (-0.87) 6.86 (-0.87) 6.89 (-0.84)

158.73 (-18.51) 164.25 (-12.99) 168.17 (-9.07) 167.99 (-9.25) 168.78 (-8.46) 168.57 (-8.67) 168.99 (-8.25) 168.76 (-8.48) 169.06 (-8.18) 168.82 (-8.42) 169.10 (-8.14) 168.85 (-8.39)

20.74 (-3.58) 21.00 (-3.32) 21.93 (-2.39) 22.04 (-2.28) 22.00 (-2.32) 22.09 (-2.23) 22.02 (-2.30) 22.11 (-2.21) 22.03 (-2.29) 22.11 (-2.21) 22.03 (-2.29) 22.10 (-2.22)

1.56 (-0.34) 1.70 (-0.20) 2.02 (0.12) 2.01 (0.11) 2.05 (0.15) 2.04 (0.14) 2.06 (0.16) 2.05 (0.15) 2.06 (0.16) 2.05 (0.15) 2.06 (0.16) 2.05 (0.15)

RMSD

MSE

11.75

-7.99

6.64

-5.60

3.91

-2.76

3.70

-2.03

4.28

-2.21

3.56

-1.65

4.46

-2.03

3.54

-1.54

4.54

-1.96

3.53

-1.51

4.58

-1.92

3.52

-1.50


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Table 3. Chemical shifts (in ppm) calculated by GIAO/B3LYP/aug-cc-pVTZ in the gas phase (with vacuum-optimized structures) and using the solvation models PCM and EC-RISM (PSE[1-4] and HNC) with optimized PCM structures compared with experimental values (deviations are shown in parenthesis). Superscript “q” indicates the point charge model for electrostatic solute-solvent interactions while superscript “φ” refers to the exact quantum-mechanical electrostatic potential. Statistical quality is measured by the root mean square deviation (RMSD) and the mean signed error (MSE). Data for EC-RISM closure approximations PSE-4φ and HNCφ and the large basis set have been omitted due to divergence of the corresponding 3D RISM solutions.

literature vacuum PCM q

PSE-1

φ

PSE-1

q

PSE-2

φ

PSE-2

q

PSE-3

φ

PSE-3

q

PSE-4

φ

PSE-4 HNC

q

HNC

φ

C(1)

H(1)

N(H)

H(N)

C(O)

C(2)

H(2)

28.65

2.63

114.24

7.73

177.24

24.32

1.90

24.08 (-4.57) 23.57 (-5.08) 23.89 (-4.76) 25.06 (-3.59) 23.91 (-4.74) 25.00 (-3.65) 23.91 (-4.74) 24.97 (-3.68) 23.92 (-4.73) 24.95 (-3.70) 23.92 (-4.73) 24.94 (-3.71)

2.65 (0.02) 2.67 (0.04) 2.73 (0.10) 2.90 (0.27) 2.75 (0.12) 2.91 (0.28) 2.75 (0.12) 2.91 (0.28) 2.75 (0.12) 2.92 (0.29) 2.75 (0.12) 2.92 (0.29)

97.84 (-16.40) 113.34 (-0.90) 125.48 (11.24) 124.37 (10.13) 128.14 (13.90) 123.10 (8.86) 129.04 (14.80) 119.72 (5.48) 129.38 (15.14)

4.97 (-2.76) 6.84 (-0.89) 7.00 (-0.73) 7.23 (-0.50) 7.18 (-0.55) 7.42 (-0.31) 7.24 (-0.49) 7.50 (-0.23) 7.27 (-0.46) 7.54 (-0.19) 7.28 (-0.45) 7.57 (-0.16)

168.40 (-8.84) 174.29 (-2.95) 178.91 (1.67) 181.04 (3.80) 179.72 (2.48) 181.74 (4.50) 180.00 (2.76) 181.96 (4.72) 180.10 (2.86) 182.02 (4.78) 180.16 (2.92) 182.05 (4.81)

17.81 (-6.51) 18.41 (-5.91) 19.49 (-4.83) 20.76 (-3.56) 19.60 (-4.72) 20.85 (-3.47) 19.63 (-4.69) 20.87 (-3.45) 19.65 (-4.67) 20.87 (-3.45) 19.65 (-4.67) 20.88 (-3.44)

1.70 (-0.20) 1.83 (-0.07) 1.99 (0.09) 2.16 (0.26) 2.01 (0.11) 2.18 (0.28) 2.02 (0.12) 2.19 (0.29) 2.02 (0.12) 2.19 (0.29) 2.02 (0.12) 2.19 (0.29)

129.57 (15.33) -

RMSD

MSE

7.73

-5.61

3.19

-2.25

5.01

0.40

4.52

0.97

5.91

0.94

4.21

0.93

6.22

1.13

3.34

0.49

6.34

1.20

-

-

6.41

1.24

-

-


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Figure 2. Nucleus-specific deviations between calculated chemical shifts δcalc of NMA and experimental values49 δexp from vacuum and various solution-phase calculations, employing GIAO/B3LYP with the basis sets 6-31+G(d) (left) and aug-cc-pVTZ (right). Superscript “q” indicates the point charge model for electrostatic solute-solvent interactions while superscript “φ” refers to the exact quantum-mechanical electrostatic potential.

The largest discrepancy to experimental values is observed for vacuum calculations. The PCM implicit solvent approach improves the results, but, for the smaller basis set, only to an extent where the deviations to experimental values are still off at least by -1.55 ppm for H(N), -11.09 ppm for N(H) and -12.30 ppm for C(O). For all these nuclei the EC-RISM solvation model combined with B3LYP/6-31+G(d) level of theory further improves the chemical shift


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predictions, leading to deviations of 8.51 ppm for N(H) and -9.25 ppm for C(O) in the worst case. Furthermore the EC-RISM approach with the HNC closure for this basis set reduces the amide proton shift error down to -0.84 ppm, which is nearly half the deviation of the best PCM result. This is a clear indication that directional solvent interactions that are properly resolved by EC-RISM theory are immediately responsible for the experimentally observed data. As also pointed out by Bader50 and Exner et al.,49 this solvent directionality is crucial for capturing nuclear magnetic shieldings of the amide groups. Our results illustrate that a continuum description is insufficient in this case.

Figure 3. Root mean square deviation (RMSD) and mean signed error (MSE) of the calculated chemical shifts over all nuclei investigated with respect to experimental values.49 Methodology and color codes correspond to Figure 2.


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Changing the basis set to the much more expensive aug-cc-pVTZ basis dramatically improves the amide nuclei chemical shifts for the PCM solvation model. The chemical shift errors for the carbonyl carbon within the EC-RISM solvation model also decrease with the larger basis, but this choice apparently worsens results for the amide nitrogen N(H) in comparison with 631+G(d), to a lesser extent also for non-amide carbons and hydrogens. On the other hand, in agreement with findings by others,51 a triple-ζ basis is obviously required to describe the amide carbonyl C(O) chemical shift properly. Similarly, also the amide proton H(N) chemical shift prediction is improved down to a deviation of only -0.16 ppm for EC-RISM(HNC) calculations with exact electrostatics. Exact electrostatics appears to be most essential for the amide nitrogen N(H) predictions for both basis sets examined (we will rationalize the origin of this effect below), but this trend is also a general pattern as reflected by two statistical descriptors, the root mean square deviation (RMSD) and the mean signed error (MSE). Figure 3 reveals that the deviations to experimental chemical shifts, estimated with the EC-RISM solvation model in combination with all closure relations and both solute-solvent interaction schemes, are smaller than every PCM result with one exception, the RMSD of the B3LYP/aug-cc-pVTZ/PCM). Its value of 3.19 ppm represents the overall smallest RMSD, closely followed by the B3LYP/aug-cc-pVTZ/EC-RISM(PSE-3φ) result with an RMSD that is only 0.15 ppm larger. (Note that, due to convergence problems for the reference compound ammonia, the EC-RISM(PSE-4φ) and EC-RISM(HNCφ) chemical shift statistics are not available, but we can expect a continuation of the trend towards smaller RMSD for exact electrostatics.) However, RMSD data should be interpreted with caution. The RMSD is only statistically meaningful if the data set is large enough to obey symmetric deviations from the mean. For our


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very small sample size comprising of only a single compound with a few nuclei, the RMSD has to be viewed in conjunction with the MSE since we are then in a better position to reveal systematic in contrast to random deviations. Taking both RMSD and MSE metrics together shows that the larger basis set with exact electrostatics, B3LYP/aug-cc-pVTZ/EC-RISM(PSE3φ), performs best for the integral equation model. The much larger MSE of the PCM model compared with EC-RISM (-2.25 vs. 0.49) indicates more pronounced systematic error whereas the EC-RISM approach appears to be more balanced. Similarly, data for the larger basis set in comparison with the smaller one indicate less random scatter for the former. Enhancing the level of theory for both, the quantum-chemical and the solvation part of the model allows for a systematic improvement. In this way, the EC-RISM approach appears to be promising although we are well aware that no general conclusion can be drawn from the small sample size analyzed in this work. In summary, within the sequence of EC-RISM closure approximations, we can reaffirm earlier observations that PSE-3 represents the best compromise between numerical stability and accuracy; PSE-1 (KH) is clearly inferior. While this statement holds most notably for exact electrostatics, it also holds for the point charge approximation at least for the small basis set, with the exception of the amide nitrogen that always benefits from an exact treatment. In order to adequately describe the shielding of this critical nucleus, a large basis sets appears to be necessary, however at the prize to also compute exact electrostatic interactions since the point charge model is clearly insufficient in this case. Physical rationalization. The impact of the treatment of electrostatics particularly for N(H) has to be translated into solvation patterns since the structure of polarizing environment determines the chemical shift. Its deviation from experimental values vastly increases for the point charge


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EC-RISM approximation with higher order closure relation, resulting in a large overall RMSD. As already pointed out by Bader,50 the amide nitrogen chemical shift of NMA is very sensitive to surrounding charges, in particular to those located in the amide bond plane.

Figure 4. Solvent charge density difference ∆ρq(r) between exact quantum-mechanical electrostatics and the point charge solute-solvent interaction approximation within the ECRISM/HNC solvent model for the B3LYP/6-31+G(d) level of theory. The color code covers ±0.429 e/Å3; for NMA the extreme values are -0.429 and 0.345 e/Å3, respectively.

In order to rationalize the effect of different treatments of solute-solvent electrostatics, we computed the difference between the solvent charge densities of the point charge approach and the exact electrostatics approach, ∆ρ q (r ) = ρ q (r, HNC φ ) − ρ q (r , HNC q ) .

(7)

Figure 4 illustrates this difference in the amide bond plane for the converged EC-


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RISM/B3LYP/6-31+G(d) results. Largest discrepancies are found along the N-H bond axis and around the carbonyl oxygen. Bader50 showed that a localized extra charge moved along the N-H bond axis barely influences the 15N chemical shift, whereas the impact of charges moved along the N-C(O) and N-C(2) bond is large. Corresponding charge density differences to the point charge approach are indeed found around the carbonyl oxygen, indicating the relevance of lone electron pairs that are only captured by the exact electrostatics approach.

Figure 5. Correlation between NMA dipole moment and calculated chemical shifts for increasing PSE order (analogous trend for all lines) along with linear regression curves (data is found in supporting Tables S4-S6), from top to bottom: N(H), H(N), C(O).


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As far as the influence of the closure approximation is concerned, we find in general improvement with increasing order. This trend is markedly visible for the amide nuclei, in particular for N(H). Higher PSE orders induce more pronounced solvent structuring.28 Regarding deviation from experimental values, the chemical shifts tend to move to lower fields with increasing PSE-n order, indicating that NMA is polarized more strongly by enhancing solvent structure. Figure 5 shows that the chemical shifts of core NMA amide nuclei depend almost linearly on its dipole moment, which increases in parallel with the PSE order. Among the amide atoms, the chemical shift of the C(O) shows less dependence on the dipole moment with a slope of 2.57 ppm/D to 3.62 ppm/D, but particularly chemical shifts of H(N) (0.81-1.03 ppm/D) as well as of N(H) (7.52-14.06 ppm/D) change dramatically (relative to their experimental value) in response to enhanced solvent structure.

CONCLUDING REMARKS In summary, this proof-of-principle study has succeeded to demonstrate the basic applicability of the EC-RISM methodology to NMR chemical shift determination and its predictive advantages compared with the PCM model. We have shown that EC-RISM with sufficient quality of the closure approximation is capable of capturing the essential physical effects of a polarizing environment on electronic structure. This is particularly true when the full quantum-mechanical electrostatic potential is applied for the solute-solvent interaction description. As a general suggestion, the combination of the small 6-31+G(d) basis with the PSE-3 closure and exact electrostatics appears to be a reasonable compromise between computational speed and predictive power.


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This motivates us to pursue in future work the necessary steps for further development that have not been mentioned here, such as the examination of the quantum-chemical level of theory, basis set dependence, impact of dispersion/repulsion models, consistent EC-RISM-optimal geometries, conformational ensembles, predicting coupling constants, and so forth. From a practical point of view, a regression analysis for a large data set of compounds could be advantageous since this would eliminate the need for a sophisticated choice and adequate treatment of reference compounds. An interesting and straight-forward extension would be the combination with ADMA-type fragment-based quantum-chemical calculations16,17 in order to scale up the methodology to large systems such as proteins. But even in the present form, EC-RISM has the potential to be a useful complement of the computational chemistry arsenal to tackle solventphase NMR problems.


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ASSOCIATED CONTENT Supporting Information. Tables containing chemical shielding constants for all compounds and dipole moments of NMA for all levels of theory along with linear regression results; structural data of optimized geometries. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION Corresponding Author *Email: [email protected] Notes The authors declare no competing financial interests. ACKNOWLEDGMENTS Figures 1 and 4 and the table of content figure have been created with the Molcad software (http://www.molcad.de/, Molcad GmbH, Darmstadt, Germany). We thank the Stiftung Mercator for financial and the IT and Media Center (ITMC) of the TU Dortmund for computational support. Discussions with Benedikt Piorr, Klaus Jurkschat, Hans-Robert Kalbitzer, and Thomas Exner are also greatly acknowledged.


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(50) Bader, R. Utilizing the Charge Field Effect on Amide

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N Chemical Shifts for Protein

Structure Validation. J. Phys. Chem. B 2009, 113, 347-358. (51) Helgaker, T.; Jaszuński, M.; Ruud, K.; Ab Initio Methods for the Calculation of NMR Shielding and Indirect Spin-Spin Coupling Constants. Chem. Rev. 1999, 99, 293-352.


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TOC GRAPHICS


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Structure of N-methylacetamid (NMA) with nuclear designators used throughout. 37x19mm (300 x 300 DPI)


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Nucleus-specific deviations between calculated chemical shifts δcalc of NMA and experimental values47 δexp from vacuum and various solution-phase calculations, employing GIAO/B3LYP with the basis sets 6-31+G(d) (left) and aug-cc-pVTZ (right). Superscript “q” indicates the point charge model for electrostatic solutesolvent interactions while superscript “φ” refers to the exact quantum-mechanical electrostatic potential. 116x76mm (300 x 300 DPI)



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Root mean square deviation (RMSD) and mean signed error (MSE) of the calculated chemical shifts over all nuclei investigated with respect to experimental values.47 Methodology and color codes correspond to Figure 2. 121x85mm (300 x 300 DPI)


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Solvent charge density difference ∆ρq(r) between exact quantum-mechanical electrostatics and the point charge solute-solvent interaction approximation within the EC-RISM/HNC solvent model for the B3LYP/631+G(d) level of theory. The color code covers ±0.429 e/Å3; for NMA the extreme values are 0.429 and 0.345 e/Å3, respectively. 69x66mm (300 x 300 DPI)



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Correlation between NMA dipole moment and calculated chemical shifts for increasing PSE order (analogous trend for all lines) along with linear regression curves (data is found in supporting Tables S4-S6), from top to bottom: N(H), H(N), C(O). 119x178mm (300 x 300 DPI)


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Solvation Effects on Chemical Shifts by Embedded Cluster Integral

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