Thole Model for Ionic Liquid Polarizability - The Journal of Physical

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Thole Model for Ionic Liquid Polarizability Yixuan Gu, and Tianying Yan J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp3105908 • Publication Date (Web): 04 Dec 2012 Downloaded from http://pubs.acs.org on December 9, 2012

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The Journal of Physical Chemistry

Thole Model for Ionic Liquid Polarizability Yixuan Gu and Tianying Yan* Institute of New Energy Material Chemistry, Tianjin Key Laboratory of Metal- and MoleculeBased Material Chemistry, Nankai University, Tianjin 300071, China.

ABSTRACT

The anisotropic ionic polarizabilities of two data sets of 216 cations (158 in training set and 58 in test set) and 80 anions (64 in training set and 16 in test set), which can be the components of ionic liquids (ILs), are fitted with Thole model against ab initio calculations. The isotropic atomic polarizabilities of H, B, C, N, O, F, S, Cl, P, Br, are fitted for cations and anions, respectively, with two different smearing functions. The ab initio anisotropic ionic polarizabilities are well fitted by Thole model with a universal set of isotropic atomic polarizabilities, which are independent of their individual chemical environment. The current study also demonstrates the good transferability of Thole model to ions of different substituents, different side chain length , and different conformations.

KEYWORDS. Anisotropic ionic polarizability; isotropic atomic polarizability; ab initio; smearing function; transferability

ACS Paragon Plus Environment

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I. Introduction. Room-temperature ionic liquids (ILs) are highly polarizable.1 Composed solely of ions, ILs are a class of organic salt whose melting point is often lower than 100℃. A fascinating feature of ILs is that they can be tailored into a tremendous variety of solvents that have desired physicochemical properties through different combinations of cations and anions.2 Therefore, such new class of liquid has stimulated many studies from past decade.3 In order to investigate the microscopic physicochemical properties of ILs with computational method using classical forcefields, much efforts have been made in order to develop accurate forcefields,4 which are mainly classified into two categories, i.e., nonpolarizable force fields, or the rigid ion models (RIMs)5-14, and polarizable ion models (PIMs).15-25 For such highly polarizable ions with anisotropic environment, a PIM is appropriate, especially for the transport properties.15,18,22,24 The reduced charge models26-28 are implemented into RIMs in order to enhance the transport properties of the simulated ions. Such model takes into account the screened electrostatic interactions, caused by the instantaneous electronic response due to electronic polarizability, in a mean field manner.29,30 Schroder showed that while the performance of the reduced charge model is excellent on the collective level, the short-range structure is relatively poor compared to the polarizable model, due to the different distance dependences for charge-charge, charge-dipole, and dipole-dipole interactions.31 Thanks to the rapidly increasing computational power, the interest in the polarizable force field keeps growing.32,33 At current stage, the PIM of ILs may be categorized to inducible atomic dipole models,15,17,19,21,22,24 fluctuating charge models,16,20 or Drude oscillator (shell) models.18,23,25 We focus in this study on the inducible dipole models, for which the initial version of the point inducible dipole model was proposed by Applequist and co-workers.34 A fascinating nature of Applequist’s model is that the anisotropic molecular polarizability tensor can be


2

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reproduced by the interactions between the isotropic atomic polarizabilities.34 The polarizable force fields based on Applequist model has been successfully implemented in the POL series water models.35-37 On the other hand, caution should be addressed on Applequist’s model because it may lead to infinite molecular polarizability as atoms approaching each other to a certain limit, then such head-to-tail induced dipoles diverge, causing the so-called “polarization catastrophe”.38 Though this event is rare, it may well occur in molecular dynamics (MD) simulation during millions of integration time steps. Such ill-behavior may be avoided in MD simulation by restricting the interactions among the induced point dipoles only to the atoms beyond dihedral.37 However, the ionic polarizabilities for the small ions, such as NO3-, BF4-, PF6-, CF3SO3-, etc., which are of specially interest in ILs, become purely additive and isotropic. When atoms come close to each other, there must be some sort of damping function that can reduce the dipole-dipole interactions from r-3 at short distance. Based on the above consideration, Thole introduced the celebrating smearing dipole model, which retains the anisotropic molecular polarizability as inherent from Applequist model, but with the short-range dipole-dipole interactions effectively damped.38 In addition, Thole adopted a universal set of parameters, with only four atomic polarizabilities (H, C, N, O) and one smearing factor independent of their individual chemical environment, and simultaneously fit the polarizabilities of 17 organic molecules, water, and gas (H2, N2, O2, CO).38 Later, van Duijnen and Swart extended Thole model to include atomic polarizabilities of S and the halogen atoms, and successfully fit 70 organic molecules and water, with training set of 52 and test set of 18 molecules, respectively.39 Recently, Duan and co-workers re-parameterized Thole model with a 420-molecule data set,40 and a benchmark data set of 481 amino acid derivatives showed that the polarizable force fields based on Thole model are significantly more accurate than the non-polarizable force fields.41,42 Thole model has been successfully implemented in TTM series of water models,43-48 AMOEBA


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water model,49 as well as the classical version of the direct reaction field (DRF) force field.50-52 PIM based on Thole model has been also developed,15 and implemented in the simulations of IL bulk,21,22 interface,53 mixture,54 as well as the coarse-grained models against PIM.55-57 So far, a complete scheme of Thole model applied to cations and anions of ILs still needs to be made. In this study, the anisotropic ionic polarizabilities of two data sets of 216 cations (158 in training set and 58 in test set) and 80 anions (64 in training set and 16 in test set), which can be the components of ILs,58 are parameterized with Thole model against ab initio calculations. The purpose of this study is twofold. On one hand, the atomic polarizabilities fitted in the current study can be readily adopted in the PIM based on Thole model of ILs.15,17,21,22,24 On the other hand, the Thole model for ILs fitted in this study can be readily adopted in analyzing the MD simulation data for the polarization response in ILs, such as Optical Kerr effect (OKE) spectroscopy,59,60 with either PIM22 or RIM.61,62

II. Methods. In both Applequist model34 and Thole model,38 every atom is a polarizable center with isotropic polarizability, and the anisotropic molecular polarizability is rebuilt from the interactions among polarizable centers. The solidness of such decomposition should be justified by, at the first place, the goodness of this model to reproduce the molecular polarizability. For a molecule consisting of N polarizable atomic centers, the induced dipole moment µi at atomic center ri is given by



µi = α i  E i + 

 Tij µ j  j =1, j ≠ i  N

∑


(1)

4

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in which α i is the atomic polarizability of atomic center ri , Ei is the external electric field at ri , Tij is the dipole field tensor at ri from the induced dipole moment µ j at r j . The second term in

the RHS of Eq. (1), i.e., Tij µ j , represents the electric field at ri generated by µ j . Eq. (1) can be re-arranged as

α i−1µi −

N

∑

Tij µ j = Ei

(2)

j =1, j ≠ i

Eq. (2) may be written explicitly as

(3)

The above expression may be briefed as Aµ = E , in which A is a 3 N × 3 N matrix. Therefore, µ = A−1 E , where A−1 is the inversed matrix of A . In a uniformly applied electric field, E1 = E 2 = L = E N = E , the induced molecular dipole moment is thus given by N



i =1

 i , j =1

N



µmol = ∑ µi =  ∑ Aij−1  E , and Eq. (3) immediately implies34 

N

α mol = ∑ Aij−1

(4)

i , j =1

where α mol is the molecular dipole tensor and Aij−1 are the 3 × 3 sub-matrices of A−1 . Eq. (4) shows that the anisotropic molecular polarizability α mol can be constructed by the isotropic atomic polarizability α i . Since α mol and molecular geometry can be determined experimentally or by ab initio calculations, the only unknowns in Eq. (4) are the atomic polarizabilities α i , which can be solved


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by nonlinear fitting procedure with high accuracy.34,38-40 The major issue is how to estimate the dipole field tensor Tij = −∇i Eijf , in which Eijf represents the fractional electric field at ri due to a spherically symmetric fractional charge density, or smearing function, ρ (r ) centered at r j with

r =| r − r j | , and

∫

∞

0

4π r 2 ρ (r )dr = 1 . Thus, Eijf =

rij

∫

3 ij

r

rij

0

4π r 2 ρ (r )dr by Gauss law, in which

rij = ri − r j and rij =| rij | . The next task is to estimate the smearing function ρ (r ) . Thole evaluated six different ρ (r ) ’s and here we pick the first two which are exactly continuous, i.e.,38

1 a3  ar  exp  −  3 A 8π  A

ρ1 (r ) =

(5a)

  r 3  1 3a ρ 2 (r ) = 3 exp  −a    A 4π   A  

(5b)

in which a is the smearing factor and A = (α iα j )

1/6

with α i and α j the isotropic atomic

polarizabilities of atoms i and j , respectively. The dipole field tensor for the above smearing functions is given by38-41 Tij = ft

3rij rij rij

5

− fe

I rij 3

(6)

in which rij rij and I are the metric tensor and unit tensor, respectively. For ρ1 (r ) , f t and f e are given by  arij f t = 1 − exp  −  A ij 

  3 1  arij   ∑    n =0 n !  Aij

  

n

  ar  , and f e = 1 − exp  − ij  A  ij  

  2 1  arij   ∑    n = 0 n !  Aij

  

n

   

(7a)

while for ρ 2 (r ) ,


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 r ij f t = 1 − exp  −a     Aij 

  

3

  r  , and f e = 1 − exp  − a  ij    Aij  

  

3

 r  1 + a  ij    Aij 

  

3

   

(7b)

It can be seen from Eqs. (7a) and (7b) that both f t and f e approach 0 as rij → 0 , so that the short range dipole-dipole interaction is effective damped. On the other hand, in Applequist model,34 the charge density shrinks to a point, i.e., ρ (r ) = δ (r ) in contrast to ρ1 (r ) and ρ 2 (r ) in Eqs. (5a) and (5b), so that Tij = −∇ i Eij = ∇ i ∇ i 1/ rij , which implies f t = f e = 1 .40,41 Therefore, the “polarization catastrophe” associated with Applequist model is elegantly solved by Thole model.38 A further comparison between ρ1 (r ) and ρ 2 (r ) shows that ρ1 (r ) is less damped at small distance and more damped at large distance than ρ 2 (r ) . As rij → ∞ , both f t and f e approach 1, and Thole model and Applequist model become equivalent. In this study, the isotropic atomic polarizabilities of 10 atoms (H, B, C, N, O, F, P, S, Cl, Br) are parameterized with Thole model against ab initio calculations for two data sets of 216 cations (158 in training set and 58 in test set) and 80 anions (64 in training set and 16 in test set), which can be the components of ILs.58 Since polarizabilities are geometrically dependent, the polarizabilities of several selected ions were calculated with different configurations. The ab initio calculations were performed with Gaussian 09,63 and the ionic polarizabilities of the

isolated ions were calculated with MP2/aug-cc-pVDZ//MP2/6-31g(d) level of theory. A Fortran genetic algorithm (GA) fitting driver64 is interfaced with Thole model incorporating ρ1 (r ) and

ρ 2 (r ) in Eqs. (5a) and (5b), respectively, to minimize the root mean square deviation (RMSD) of N ions, i.e.,

1 4N

N

4

∑∑ (α i =1 k =1

k i , calc

− α ik,Thole )

2


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in which the subscripts calc and Thole denote the polarizability calculated by ab initio calculation and Thole model, respectively. α ik , k = 1, 3 , denotes the kth principal component of the polarizability tensor of the ith ion α i , and α i4 =< α i >= Tr(α i ) / 3 = (α i1 + α i2 + α i3 ) / 3 is the isotropic ionic polarizability of the ith ion. Though α i4 is redundant, it may be considered as a weighting function that emphasizes the component of large deviation. Two independent sets of atomic polarizabilities are fitted with respect to the cations and anions, respectively.

III. Results and Discussion. III.1 Fitting results of cations and anions. In this study, isotropic atomic polarizabilities are fitted by Thole model, independent of their individual chemical environment.38,39 Also, full intramolecular atomic interactions are allowed in order to fit the anisotropic ionic polarizabilities.34,38,39 In the fitting procedure, the smearing factors of ρ1 (r ) and ρ 2 (r ) retain their original values, i.e., a1 = 2.089 and a2 = 0.572 , respectively, as in Thole model for the neutral molecules.38 By fixing the smearing factors, MD simulation may be readily conducted with mixings of ILs and neutral molecules, as has been done previously.54 Table 1 and Table 2 show the selected fitness of the imidazolium, pyridinium, triazolium, tetrazolium, pyrrolidinium, ammonium, phosphonium, animo acid cations, etc., with their derivatives, and carboxylate, sulfonamide, sulphite, sulphate, borate, amino acid anions with their derivatives, as well as inorganic anions that are commonly used in ILs. The ions in Italics are those in the test set. The complete data sets of 216 cations and 80 anions are listed in Table S1 and S2 in the Supporting Information. The RMSDs of cations in the training set are 0.69 Å3 and 0.81 Å3 for smearing functions ρ1 (r ) and ρ 2 (r ) , respectively, while for anions in training set they are 0.88 Å3 and 1.00 Å3. The test sets show similar RMSDs that are close to those of the


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training sets, i.e., 0.69 Å3 and 0.80 Å3 for cation, and 1.02 Å3 and 1.09 Å3 for anion, respectively, for smearing functions ρ1 (r ) and ρ 2 (r ) . Therefore, Thole model shows better fitness for cationic polarizabilities than that for anionic polarizabilities for both smearing functions. The reason may be attributed to the fact that most cations can be classified into a few categories, such as imidazolium, ammonium, and phosphonium, etc., with their individual derivatives, while anions are divided into more categories that have quite different ionic structures and chemical environment. Despite the above difference, the fitness with single set of atomic polarizabilities is still satisfactory with Thole scheme, as shown in Tables 1 and 2 as well as Tables S1 and S2 in the supporting information. The overall fitness of the ionic polarizabilities of Thole model, with both smearing functions

ρ1 (r ) and ρ 2 (r ) , against ab initio calculates is shown in Figure 1 and Figure 2 for the complete data sets of cations and anions, in which the open circles denote the ions in training sets, while the filled circles in test sets. The figures show anisotropic polarizabilities along the three principal axes, as well as the isotropic polarizabilities. It can be seen that the distributions of the ionic polarizabilities in the test sets are almost as good as those in the training sets, demonstrating the good transferability of the fitted Thole model for both smearing functions. It is also notable that the overall fitness of smearing function ρ1 (r ) is slightly better than that of

ρ 2 (r ) , in agreement with the RMSDs in Tables S1 and S2. Apart from that, the fitness of the highest principal component of polarizability, α 3 , is slightly lower than the other two components. The above facts may be understood as follows. Thole model introduces the smearing function to effectively screen the atomic interaction especially at intramolecular distances, which makes Thole model more additive, or, less interactive, than Applequist model. The consequences are that (1) it elegantly solves the “polarization catastrophe” associated with


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Applequist model; and (2) the anisotropic polarizability is less addressed especially for the highest principal component α 3 . Therefore, α 3 tends to be underestimated by Thole model, as implied by the larger intercept in the regression equation in Figures 1(c), 1(g), 2(c), and 2(g). The better fitness of ρ1 (r ) may also be understood from the fact that it screens the short-range atomic interactions weaker than ρ 2 (r ) . The stronger screening function makes Thole model become more additive at intramolecular atomic distance, resulting in more isotropic polarizabilities. The fitted atomic polarizabilities are listed separately for cations and anions in Table 3. It can be seen that the fitted atomic polarizabilities for anions are generally bigger than those fitted for cations, such as H, N, O, Cl, and Br. Also, the atomic polarizabilities are smaller for those of higher electronegative atoms as well as the atoms with fewer electrons. This result may be explained by different electron distribution of cations and anions. Since anions are negatively charged, the electron cloud on the anion tends to be distorted more than that on the cation, resulting in higher atomic polarizability for the same atom fitted for anions. On the other hand, it is notable that the fitted atomic polarizabilities of B and P for anions are much lower than those for cations. The reason is that B and P are bonded to the strong electronegative F in most anions. As a result, electrons of P and B are strongly attracted by F, leading to a weaker ability to be distorted, and thus smaller atomic polarizabilities.

III.2 The transferability of Thole model of ionic polarizabilities. The above analyses illustrate the good transferability of the so fitted Thole model. Since ILs are often designed with different substitution of functional groups, especially for the cations, we choose 1-substituent-3methylimidazolium and substitute the substituent with various functional groups to examine the performance of the Thole model. Figure 3 shows the comparisons between ab initio polarizabilities and Thole model of α1 , α 2 , α 3 , and < α > ,

for 10 different imidazolium


10

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derivatives, in which 5 of them are from the training set (with substituent groups -CH2CH3, CH2CN, -CH2CH2OH, -CH2CH2CH3, and -CH2COOCH3, respectively), and 5 from the test set (with substituent groups -CH2CH2F, -CF2CF3, -CH2COCH3, -CH2CH2Cl, and -CH2CH2Br, respectively). It can be seen that Thole model reproduces ab initio polarizabilities quite satisfactory for both the training and test sets, and ρ1 (r ) and ρ 2 (r ) give similar fitness. Furthermore, the results shown in Figure 3 demonstrate two factors that affect polarizability, i.e., the number of electrons and electronegativity, with more electrons and low electronegativity enhancing polarizability and vice versa. It is notable that the polarizabilities of the substituent groups -CH2CH3, -CH2CH2F, -CH2CH2Cl, and -CH2CH2Br increase with the increment of electrons. Though the sequence of electronegativity is F > Cl > Br > H, the primary factor that affects the polarizability here is the number of electrons instead of electronegativity. On the other hand, the polarizability of the substituent group -CF2CF3 is only slightly higher than that of CH2CH3, owing to the competition of the two factors mentioned above. For the substituent groups of the same number of electrons, the polarizabilities are -CH2CH2CH3 > -CH2CH2OH > CH2CH2F, which is in agreement with the group electronegativity F > OH > CH3.65 Therefore, for the same number of electrons, higher electronegativity limits the distortion of the electron cloud and results in lower polarizability. The above sequence of polarizabilities, though illustrated from ab initio polarizabilities, are well reproduced by Thole model. In order to further test the extrapolation of Thole model, we prolong the alkyl chain of 1-alkyl3-methylimidazolium, in which alkyl denotes –(CH2)nCH3, with n ranging from 0 to 11. The ionic polarizabilities in the training and test sets are connected together with a dashed line for smearing functions ρ1 (r ) and ρ 2 (r ) , while ab initio polarizabilities are demonstrated in the solid line. As shown in the insets of Figure 4, Thole model reproduces anisotropic ab initio


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polarizabilities with fairly small relative errors. Even in the test set with n=8-11, the polarizabilities calculated by Thole model are in good accordance with ab initio results. Therefore, the results in Figure 4 demonstrate good extrapolation of Thole model. Polarizability is also closely related to the ionic conformations. The induced dipole due to the polarization may affect the infrared (IR) response for the whole spectrum and alter the IR signal considerably.66 Thus, it is of importance to investigate whether Thole model matches ab initio polarizabilities of different conformations. In Figure 5, we use Thole model to calculate polarizabilties of two selected ions (BMIM+ and TFSI-) with the selected dihedral angle ranging from 0° to 180° and compare with ab initio calculations. As shown in Figure 5, Thole model of both smearing functions ρ1 (r ) and ρ 2 (r ) is in fairly good agreement with ab initio polarizabilities, especially for BMIM+ in the left column. In addition, Thole model is better for

α1 and α 2 than for α 3 of the highly principal component of ionic polarizability, and ρ1 (r ) is better than ρ 2 (r ) , especially for TFSI-. Though for α 3 the functions of both BMIM+ and TFSIdo not follow the ab initio data very closely, the relative errors are still within ±5% for the average ionic polarizabilities.

IV. Summary. In this study, the anisotropic ionic polarizabilities of 216 cations (158 in training set and 58 in test set) and 80 anions (64 in training set and 16 in test set), which can be the components of ILs, are fitted with Thole model against ab initio polarizabilities. The isotropic atomic polarizabilities of H, B, C, N, O, F, S, Cl, P, Br, are fitted for cations and anions, respectively, for two different smearing functions, ρ1 (r ) and ρ 2 (r ) in Eqs. 5(a) and 5(b), respectively. The ab initio anisotropic ionic polarizabilities are well fitted by Thole model with a universal set of isotropic atomic polarizabilities which are independent of their individual chemical environment. The slightly


12

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better fitness of ρ1 (r ) than ρ 2 (r ) may be attributed to the less screening of the intramolecular atomic interactions of ρ1 (r ) , which reproduces anisotropic polarizabilities better via the atomic interactions. For the same reason, the slightly better fitness of the principal components of polarizabilities α1 and α 2 than that of the highest principal component α 3 . The introduction of smearing function in Thole model elegantly solves the “polarization catastrophe”, but the anisotropic polarizability is less addressed, especially for the highest principal component α 3 . Overall, we find that the so fitted atomic polarizabilities have good transferability and the calculated polarizabilities by Thole model follows ab initio results reasonably well for both training and test sets. The Thole model fitted in this study is readily adopted in MD simulations of ILs with a PIM.

ASSOCIATED CONTENT Supporting Information Complete sets of the anisotropic ionic polarizabilities by Thole model against ab initio calculations, including 216 cations (158 in training set and 58 in test set) in Table S1 and 80 anions (64 in training set and 16 in test set) in Table S2. Complete reference of 63. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *

E-mail: [email protected]

Notes The authors declare no competing financial interest.


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ACKNOWLEDGMENTS This work is supported by NSFC (21073097), Natural Science Foundation of Tianjin (12JCYBJC13900), and NCET-10-0512. The computations were performed on TianHe-1(A) at National Supercomputer Center in Tianjin, China.

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Page 18 of 26

Table 1. Selected cationic polarizabilitiesa by Thole model against ab initio calculations.b Ionc

ab initiod

ρ1(r)e

ρ2(r)f

Ion

1-Ethyl-3methylimidazolium

8.25 12.23 14.69 11.72 RMSDg

8.44 12.77 15.30 12.17 0.47

8.14 12.71 15.37 12.07 0.46

1-Ethylnitrile-3methylimidazolium

1-(2,3-Dibromo propyl)-3-methyl imidazolium

15.70 18.03 24.52 19.42 RMSD

15.97 18.41 24.87 19.75 0.34

15.66 18.44 24.97 19.69 0.34

(N-methyl-imidazole) (triethylamine)BH2

N-butyronitrile pyridinium

12.66 15.21 20.64 16.17 RMSD

12.80 15.00 20.76 16.19 0.14

11.66 14.56 19.71 15.31 0.87

Ethylpyridinium

1-Ethyl-4(3,3,3trifluoropropyl) -1,2,4-triazolium

11.66 13.91 19.40 14.99 RMSD

12.04 14.20 20.44 15.56 0.64

11.56 14.13 20.28 15.32 0.48

Protonated methyl 1-aminopropionate

7.30 9.33 10.82 9.15 RMSD

7.61 9.20 10.52 9.11 0.23

7.72 9.29 11.01 9.34 0.25

1-Butyl-2-methyl pyrrolinium

13.25 15.78 19.30 16.11 RMSD

13.44 16.22 19.94 16.53 0.45

13.66 16.41 20.12 16.73 0.63

N,N,N,N-cyano methyltrimethyl ammonium

8.91 9.01 11.53 9.82 RMSD

9.60 9.69 12.55 10.61 0.81

9.60 9.65 11.97 10.41 0.61

Hexyltrimethyl phosphonium

16.84 17.45 23.40 19.23 RMSD

16.44 16.88 23.93 19.08 0.44

16.56 16.77 23.90 19.08 0.45

a

ab initio

ρ1(r)

ρ2(r)

Ion

ab initio

ρ1(r)

ρ2(r)

10.60 14.53 16.93 14.02 0.55

9.91 13.97 16.50 13.46 0.22

1-Propylimidazolium

8.80 11.57 14.85 11.74

8.82 11.84 15.66 12.11 0.46

8.46 11.88 15.53 11.96 0.42

16.80 22.60 26.40 21.93 0.27

16.73 23.00 26.56 22.10 0.43

3-Methyl-1(ethoxycarbonylmethyl) imidazolium

11.97 15.68 21.83 16.49

11.85 15.63 21.26 16.25 0.32

11.44 15.52 21.44 16.13 0.38

8.50 12.63 15.13 12.08 0.16 Triazolium & Tetrazolium 2,4,5-Trimethyl 7.38 7.97 tetrazolium 11.39 11.64 12.73 13.30 10.50 10.97 0.48 Animo acid cations Isoleucinium 11.46 11.27 12.79 12.22 13.23 13.31 12.49 12.26 0.32 Pyrrolidinium N-methyl-N-ethyl11.52 12.09 pyrrolidinium 11.79 12.49 13.47 14.41 12.26 13.00 0.75 Ammonium 2-Hydroxyethyl 4.44 4.42 ammonium 5.12 4.92 6.88 6.51 5.48 5.29 0.23 Phosphonium Triethylmethyl 13.73 13.66 phosphonium 15.88 15.93 15.88 15.93 15.17 15.17 0.05

7.87 12.43 14.80 11.70 0.48

N-hexyl pyridinium

15.24 18.41 25.77 19.81

14.82 17.75 25.76 19.44 0.43

14.34 17.79 25.58 19.23 0.63

7.68 11.46 13.59 10.88 0.45

1-Amino-3-butyl1,2,3-triazolium

11.36 13.75 18.67 14.59

11.53 13.71 18.68 14.64 0.09

11.28 13.83 18.94 14.68 0.15

11.71 12.69 13.57 12.66 0.23

Threoninium

8.68 9.78 10.53 9.66

8.44 9.29 10.02 9.25 0.43

8.63 9.59 10.81 9.68 0.17

12.31 12.82 14.68 13.27 1.02

N-ethoxyethylN-methyl pyrrolidinium

14.88 15.47 20.17 16.84 RMSD

15.17 15.68 20.56 17.14 0.31

15.46 16.03 21.15 17.54 0.73

4.53 4.96 7.20 5.57 0.19

N,N,N-trimethylN-butanesulfonic acid

15.68 16.45 20.55 17.56

15.60 16.58 21.19 17.79 0.35

15.99 17.14 21.58 18.24 0.72

13.65 15.76 15.76 15.06 0.11

Tetraethylphosphoniu m

14.98 17.80 17.80 16.86

14.78 17.79 17.79 16.79 0.11

14.75 17.71 17.71 16.72 0.15

Imidazolium 10.25 13.99 16.22 13.48 17.30 22.45 26.36 22.04 Pyridinium 8.37 12.82 15.33 12.17

Unit in Å3; bPolarizabilities along the principal axes, α i1 , α i2 , α i3 , as well as the isotropic ionic

polarizability, < α i >= Tr(α i ) / 3 = (α i1 + α i2 + α i3 ) / 3 , are shown for each ion; cIons in Italics are in the test set; dab initio calculations were performed with MP2/aug-cc-pVDZ//MP2/6-31g(d) level of theory; eThole model with smearing function ρ1 (r ) in Eq. (5a); fThole model with smearing function

ρ 2 (r ) in Eq. (5b);

g

RMSD of individual ion is defined by

2 1 4 α ik,calc − α ik,Thole ) , in which α ik,calc and α ik,Thole follow the notation in Eq. (8). ( ∑ 4 k =1


18

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


Table 2. Selected anionic polarizabilitiesa by Thole model against ab initio calculations.b Ionc

ab initiod

ρ1(r)e

ρ2(r)f

Alanine

8.20 10.14 10.74 9.69 RMSDg

8.47 9.19 9.97 9.21 0.67

8.28 9.15 10.65 9.36 0.52

Heptafluorobutanoate

8.72 10.50 11.27 10.16 RMSD

8.80 9.74 11.49 10.01 0.41

8.74 9.74 10.97 9.81 0.45

Acesulfamate

10.81 15.68 17.40 14.63 RMSD

10.18 14.82 15.45 13.48 1.25

9.78 14.28 15.37 13.15 1.52

Butylsulfonate

13.26 13.75 17.40 14.80 RMSD

12.54 12.76 16.25 13.85 0.97

12.70 12.89 16.33 13.98 0.85

Tetracyanoborate

11.70 11.70 11.70 11.70 RMSD

12.75 12.75 12.75 12.75 1.04

12.31 12.31 12.31 12.31 0.61

a

Ion

ab initio

Animo acid anion Glutamic acid 12.63 14.45 15.81 14.29 2,3Dibromopropionate

Carboxylate 12.30 13.78 17.96 14.68

Sulfonamide N-(trifluoromethyl 14.84 sulfonyl)pentafluoro 15.93 ethylsulfonamide 19.02 16.60 Sulphite & Sulphate Hydrogen sulfate 6.22 6.39 6.67 6.43 Trifluoromethyl trifluoroborate

Borate 5.46 5.46 5.47 5.46

ρ1(r)

ρ2(r)

Ion

ab initio

ρ1(r)

ρ2(r)

12.02 13.67 15.16 13.62 0.68

12.03 13.74 15.69 13.82 0.53

Valine

11.48 13.77 14.38 13.21

12.09 13.77 14.73 13.53 0.39

12.10 13.93 15.16 13.73 0.57

12.90 13.99 17.58 14.82 0.38

12.24 13.22 19.50 14.99 0.83

Glycolate

5.40 8.09 8.29 7.26

4.80 5.91 6.76 5.82 1.54

4.65 5.82 6.97 5.81 1.54

14.70 16.18 19.54 16.81 0.31

14.52 15.67 19.66 16.62 0.38

Bis((trifluoromethyl) sulfonyl)imide

13.34 13.60 17.21 14.72

12.90 13.44 17.38 14.57 0.26

12.69 13.02 17.46 14.39 0.48

5.01 5.25 5.89 5.39 1.05

4.89 5.05 6.09 5.35 1.13

Perfluorobutyl sulfonate

12.74 12.94 15.48 13.72

12.88 12.96 16.74 14.20 0.68

12.77 12.95 16.11 13.94 0.33

5.24 5.24 5.39 5.29 0.18

5.33 5.59 5.59 5.50 0.12

(Heptafluoro-n-propyl) trifluoroborate

8.92 9.06 9.88 9.29

9.30 9.40 11.09 9.93 0.73

9.59 9.76 10.83 10.06 0.78

Unit in Å3; bPolarizabilities along the principal axes, α i1 , α i2 , α i3 , as well as the isotropic ionic

polarizability, < α i >= Tr(α i ) / 3 = (α i1 + α i2 + α i3 ) / 3 , are shown for each ion; cIons in Italics are in the test set; dab initio calculations were performed with MP2/aug-cc-pVDZ//MP2/6-31g(d) level of theory; eThole model with smearing function ρ1 (r ) in Eq. (5a); fThole model with smearing function

ρ 2 (r ) in Eq. (5b);

g

RMSD of individual ion is defined by

2 1 4 α ik,calc − α ik,Thole ) , in which α ik,calc and α ik,Thole follow the notation in Eq. (8). ( ∑ 4 k =1


19


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

Page 20 of 26

Table 3. Atomic polarizabilitiesa fitted to the training sets of 158 cations and 75 anions. Cation Atom H B C N O F P S Cl Br ab

a

ρ1/Å3 0.257 1.461 1.427 1.051 0.372 0.163 2.660 3.223 2.231 3.334 2.089

Anion ρ2/Å3 0.444 1.085 1.152 0.917 0.331 0.256 1.750 2.703 2.138 3.096 0.572

ρ1/Å3 0.641 0.666 1.461 1.749 0.597 0.142 1.772 2.863 3.241 4.471 2.089

ρ2/Å3 0.829 0.434 1.126 2.305 0.669 0.247 0.965 2.380 2.903 4.161 0.572

Atomic polarizabilities of Thole model for the smearing functions ρ1 (r ) and ρ 2 (r ) in Eqs.

(5a) and (5b). bThe smearing factor a is retained to be the same as Thole’s original fitting of neutral molecules.38


20

Page 21 of 26

40 (a)α 1

40

(e)α1

20

20 y = 1.0448 x − 0.3617 R 2 = 0.9845

y = 1.0502 x − 0.6643 R 2 = 0.9903

(f)α2

40 (b)α 2

40 20

y = 1.0402 x − 0.6223 R 2 = 0.9886

40 (c)α 3

y = 1.0339 x − 0.5635 R 2 = 0.9863

(g)α3

40

3

αab initio/Å

3

20

αab initio/Å

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 y = 0.9767 x + 0.2774 R 2 = 0.9761

y = 0.9732 x + 0.2083 R 2 = 0.9718

(h)

40 (d) 20 0 0

40 20

y = 1.0180 x − 0.3659 R 2 = 0.9907

20

α ρ ( r ) /Å3 1

40 0

y = 1.0183 x − 0.3581 R 2 = 0.9864

20 40 α ρ2 ( r ) /Å3

0

Figure 1. Scatter plots of calculated vs. ab initio polarizability (in Å3) of the 158-cation training set (open circle) and the 58-cation test set (filled circle) with the corresponding regression equation. The left and right columns refer to smearing functions ρ1 (r ) and ρ 2 (r ) of Eqs. (5a) and (5b), respectively.


21


30

(a)α1

30

(e)α1

20 10 30

20 y = 0.9381x + 0.7440 R 2 = 0.9619

y = 0.9211x + 0.8067 R 2 = 0.9723

(f)α2

(b)α2

10 30

20 y = 0.9288 x + 1.1340 R 2 = 0.9640

y = 0.9067 x + 1.3110 R 2 = 0.9765

(g)α3

(c)α3

20 10 30

10 30 20

y = 0.9008 x + 1.4831 R 2 = 0.9462

y = 0.9068 x + 1.5254 R 2 = 0.9579

(d)

(h)

20

10 30 20

10 0 0

30

3

αab initio/Å

3

20

10

αab initio/Å

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

Page 22 of 26

10 y = 0.9111x + 1.2108 R 2 = 0.9766

y = 0.9219 x + 1.1023 R 2 = 0.9657

10

10

20

α ρ ( r ) /Å3 1

30

αρ

20 3

2 (r )

0 30

/Å

Figure 2. Scatter plots of calculated vs. ab initio polarizability (in Å3) of the 64-anion training set (open circle) and the 16-anion test set (filled circle) with the corresponding regression equation. The left and right columns refer to smearing functions ρ1 (r ) and ρ 2 (r ) of Eqs. (5a) and (5b), respectively.


22

Page 23 of 26

12

15

(a)α1 αab initio/Å

3

(b)α2

9 10

10 9 7

10

6 8

7 6

13

8 4

5 4

3

5

1

2

2

3

1

3

8 8 22

αab initio/Å

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

10

12

(c)α3

10

11 11 15

13

15

(d)

10 9

9

18

7

8

7

6

13 5

5

16

43

2

8 6

1

4 2 3

1

14 14

16

18

3

αThole/Å

20

22

11 11

13

3

αThole/Å

15

Figure 3. Scatter plots of Thole model vs. ab initio polarizability of 10 1-substituent-3methylimidazoliums with different substituent groups. Number 1-10 refer to imidazoliums with the substituent groups 1. –CH2CH2, 2. –CH2CH2F, 3. -CH2CN, 4. -CF2CF3, 5. -CH2CH2OH, 6. CH2CH2CH3, 7. -CH2OCH3, 8. -CH2CH2Cl, 9. -CH2COOCH3, and 10: -CH2CH2Br. Smearing functions ρ1 ( r ) and ρ 2 (r ) are represented by circles and triangles, respectively. The open symbols represent cations from training set, while the filled symbols from test set.


23

Rel.Err/%

20

10 5 0 -5 -10

25 0

4

n

8

12

20

Page 24 of 26

5 0

-5

0

4

n

8

12

15 15

10

3

40 30

(a)α1 0 10 5 0 -5 -10

4

8

12

(b)α2 0

4

8

12

5 0

25 0

4 n 8

12

-5

0

4n 8

12

20

20 10

10 30

Rel.Err/%

5

Rel.Err/%

α /Å

3

25

α /Å

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

Rel.Err/%


15

(c)α3 0

4

n

8

(d)

10 12

0

4

n

8

12

Figure 4. Polarizability vs. the length of the side alkyl-chain of 1-alkyl-3-methylimidazolium, in which alkyl denotes –(CH2)nCH3, with n ranging from 0 to 11. The solid line shows the trend of ab initio polarizabilities (filled squares) as guide to the eyes, while Thole model with smearing

functions ρ1 (r ) and ρ 2 (r ) are shown in dashed line with circles and dashed line with triangles, respectively. The ions in the training set are shown in open symbols, while those in the test set are shown in filled symbols. The insets show the relative error, which is defined as (α Thole − α ab initio ) / α ab initio × 100% .


24

Page 25 of 26

O N

N

φ

F3 C

O

φ

S

S

N

O

-5

0

60

12

Rel.Err/%

Rel.Err/%

3

α1/Å

16

0

14

0 -5

-10

14

120 180

φ/°

5

60

120 180

φ/°

Rel.Err/%

3

α3/Å

22

0

20

-5 0

60

0

60

0

60

18 18

-5

16 0

16 0

60

120

180

14

0

-5

120 180 φ/°

60

120 180

φ/°

5 Rel.Err/%

Rel.Err/%

0

120 180

φ/°

0

-5

120 180

5 3

60

5

φ/°

20

18

Rel.Err/%

22

5

-10

0

14

10

24

60 120 180 φ/°

0

-5

0

Rel.Err/%

Rel.Err/%

3

α2/Å

16

0

-5

16

0

5

5

18

CF 3 O

10

5

16

/Å

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


0

(a) φ/ °

120 180

φ/°

60

120

(b) φ/ °

180

Figure 5. Polarizabilities vs. the selected dihedral angle Ф ranging from 0° to 180° for 1-butyl-3methylimidazolium (BMIM+) in left column and bis((trifluoromethyl)sulfonyl)imide (TFSI-) in right column. The schematic structures and rotating bond of the dihedral angles of BMIM+ and TFSI- are shown on the top of left and right columns, respectively. The solid line with squares shows ab initio polarizabilities. The dashed lines with circles and triangles show Thole model with smearing functions ρ1 (r ) and ρ 2 (r ) , respectively. The insets show the relative errors. The ab initio polarizabilities are calculated with MP2/aug-cc-pVDZ//MP2/6-31g(d) level of theory,

with the dihedral angles constrained at relevant values.


25


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Table of Contents Graphic


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Thole Model for Ionic Liquid Polarizability - The Journal of Physical

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