Calculation of the Molecular and Atomic Properties of Selected Anions

ARTICLE pubs.acs.org/JPCB

Calculation of the Molecular and Atomic Properties of Selected Anions in Water Asbjørn Holt and Gunnar Karlstr€om* Department of Theoretical Chemistry, Chemical Center, Lund University, S-221 00 Lund, Sweden

Jose Manuel Hermida-Ramon Departamento de Química Física, Facultade de Químicas, University of Vigo, Lagoas-Marcosende s/n 36200 Vigo, Spain

bS Supporting Information ABSTRACT: The polarizability and other properties have been studied for F-, Cl-, Br-, and HCOO- in water using a combined quantum chemical statistical mechanics simulation model that explicitly takes into account the Pauli repulsion as well as the electrostatic coupling between the QM system and the classical surroundings. It is shown that the surrounding molecules significantly reduce both the polarizability and the size of the anions. For the formate ions, local properties have been computed.

1. INTRODUCTION Ions play a key role in aqueous chemistry. They are important in the interactions between macromolecules and in a large range of biochemical situations. The molecular properties of ions change depending on the surroundings of the ion in question. This has been known and studied for a long time, with one of the first works published in 1924.1 For the anions there has been a particular focus on the behavior of the polarizability, and it has been observed for a long time that the polarizability of the anions reduces significantly when going from the free ion in the gas phase to anions in, for example, crystalline or liquid state.2 The origin of this reduction has been explained by the fact that nuclear charge in the anions is screened so that the excess electron in the anion does not experience a strong attraction from the nucleus. This makes the excess electron very flexible when responding to an external perturbing field, thus making the polarizability large. In the condensed phase, however, there is an external pressure on the excess electrons due to the Pauli repulsion from the electrons of the surrounding molecules. This repulsion confines the excess electron to a more limited volume, and the reduction of the molecular size leads to a reduction of the polarizability. It can be noted that this occurs not only for anions but also to a lesser extent for other molecules such as water.4 Various approaches have been used in order to compute the polarizabilities of anions, and for a more exhaustive historical account on this topic the reader is referred to the excellent work of Shannon and Fischer.3 In general, the polarizabilities are derived either from experimental results on crystals (as for instance done by Shannon and Fischer3 and earlier by Pyper r 2011 American Chemical Society

et al.5) or via simulations of molecular systems, often coupled with some quantum chemical methodology.6-9,15 In most cases, the behavior of the polarizability has only been studied for monatomic systems. There might be several reasons for this, but the two most probable are the biological importance of the halide anions (in particular of Cl-) and that the effect should be larger for the monatomic systems which cannot polarize by transferring charges internally in the molecule. Some selected polarizabilities of halide anions from the literature are presented in Table 1. As can be seen from this table, there is really no consensus about the size of the polarizabilities of the halide anions, other than the fact that they are significantly smaller that the gas-phase value (17.15 a30 for F- and 37.43 a30 for Cl-, see ref 16). It is also clear that the Amber 10 force field is far more conservative than any of the other force fields and derived polarizabilities. An important aspect of the explanation of the reduced polarizability of the anions is the coupling of the Pauli repulsion to the polarization of the molecule. This coupling is not trivial and requires that quantum chemical methodology is used if it is to be correctly accounted for. A discussion of this coupling can be found elsewhere.17 In order to be able to predict the polarizability (and also other molecular properties) of molecules and ions in a condensed phase, we must use methods that explicitly take this into account. One way of doing this is to perform quantum Received: November 11, 2010 Revised: December 21, 2010 Published: January 20, 2011 1098

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Table 1. Polarizabilities of Solvated Ions Reported in the Literature (a30) F-

Cl-

ref 5

8.8

25.4

ref 3 ref 15

8.74 12.05

31.38 26.78

Table 3. Parameters Used in the QMSTAT Simulationsa F-

Br34.2 35.51

Cl-

Br-

C

H

d

-0.6 -0.5

-0.5

-0.8

β4

15.0

25.0

0.0

52.96

β6

10.0

20.0

100.0

0.0

β10

200.0 1000.0 6000.0

O

4493.1

ref 6

19.28/20.08

DO,X

40.0

200.0

200.0

148.9

50.0

134.0

ref 7

26.99

DH,X

8.0

15.0

30.0

60.3

0.033

0.047

AO,X

0.0

0.0

0.0

1.9  108 1.9  108 1.9  108

0.0 0.0 -0.5 -0.5

ref 9

8.44

Amber 1010

2.159

12.889

19.435

Amoeba 0911

9.110

26.993

38.128

AH,X R

Dang-Chang12-14

7.000

24.901

32.190

no. of orbitals 18

19

35

29

no. of water

88

87

87

molecules

Table 2. Geometry of the Formic Acid Anion (Å) x

y

88

0.0 1.9  109 1.9  108 1.9  108 -0.25 -0.33

a

For an explanation of the symbols, see ref 23.

z

C

0.00000

0.04200

0.00000

H O

0.00000 1.11041

-1.07680 0.55490

0.00000 0.00000

O

-1.11041

0.55490

0.00000

chemical calculations on small clusters containing water and the molecule of interest. One problem with this will be that there is no certainty that the calculations will be done on statistically relevant configurations. In order to ensure this, one must turn to statistical mechanics simulations of the system of interest. Moreover, only the properties of the total system studied are welldefined and the properties of the considered molecule are strictly not accessible. There are two ways to get a proper sampling of the studied system and still take into account the coupling exchange repulsive coupling between the studied molecule and the surrounding: (1) using the Car-Parrinello methodology18 or (2) using some QM/MM scheme that takes this coupling into account. One example of this is the so-called QMSTAT method first developed by Moriarty and Karlstr€om,19 and later extended to include a pseudopotential20 that couples the Pauli repulsion between the QM and MM system to the solution of the Schr€odinger equation21 of the QM system (a detailed description of QMSTAT can be found elsewhere22,23). This in turn couples the Pauli repulsion to the description of the polarization of the system. In this study, we will use the QMSTAT model to compute the molecular and atomic properties of anions in an infinitely diluted solution. It is important to note, as pointed out before, that if the aim is to compute molecular properties, care must be taken not to include other molecules into the quantum chemical calculation, as only the global polarizability of the system is an observable. This is particularly important for cluster calculations. One way around this problem might be to use localized polarizabilities,24-28 but as these are not observables they must be treated with some care. In this work, we will study the halide anions from F to Br and the formate ion. For the formate ion we will not only study the molecular properties but also the atomic properties. The approach used to compute the molecular and atomic properties will be outlined in the coming sections.

2. MODEL DESCRIPTION In the present study, we use the QMSTAT method. Since a detailed explanation of this model is provided elsewhere,22,23 this

section will be limited to a summary of the model. In the model, a quantum chemical part is surrounded by a number of classical solvent (water) molecules and both these regions are in turn surrounded by a dielectric continuum. The interaction between the classical molecules is described by a NEMO approach,29 and the interaction with the dielectric continuum is given by the image charge approximation.30,31 An effective Hamiltonian operator is constructed for the quantum part Heff ¼ H0 þ Velec þ Vind þ Vnonelec

ð1Þ

where H0 is the Hamiltonian for the quantum core in the gas phase (the unperturbed Hamiltonian) and Velec is the perturbation from the permanent charge distribution of water, represented as a set of point charges. To facilitate the evaluation of the electrostatic interaction between the charge density of quantum part and the solvent point charges, the quantum core charge density is multicenter multipole expanded in a set of multipoles (up to quadrupolar order) in several centers. The same expansion is used to evaluate the perturbation from the induced dipoles on water, Vind. Because Vind depends on the charge density of the quantum core, the problem is nonlinear and is solved with the usual generalized self-consistent reaction field method, that is, through iteration.32,33 In addition to the electrostatic contributions, we also include a nonelectrostatic perturbation, Vnonelec. It models the effect on the quantum system from the antisymmetry requirements between solvent and solute, which introduces the Pauli repulsion. Three additional terms in the solute-solvent interaction are added to the total energy: a dispersion energy that depends on a set of distributed 1/r6 terms, and two extra repulsion terms needed to model the very short-range repulsion, where Vnonelec is not adequate. The latter terms depends on the solute-solvent wave function overlap raised to the power of 4, 6, and 10. In addition to this, a very steep short-range 1/r21 potential was added for the simulation of the formate ion. This latter shortrange repulsion should only be viewed as a soft cutoff used for simulation purposes, that is, to avoid regions where a polarization catastrophe can occur. In addition to this repulsion, the electric field is multiplied by a damping function: (1 - e-(ar))3, where a is a fitted parameter and r is the distance between the quantum and classical sites. This is also done to avoid a polarization catastrophe. A detailed description of the various interaction terms 1099

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Table 4. Basis Set Behaviour of the Polarizability for the Halogen Ions in Vacuum at the HF-SCF Level of Theory (a30) F-

a

Cl-

Br-

basis set

UCHF

perturbation

UCHF

perturbation

UCHF

perturbation

ANO-RCC-MB

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

ANO-RCC-VDZ

0.5269

0.7277

2.4017

3.5414

3.4553

5.0675

ANO-RCC-VDZP

1.5971

1.7958

10.1043

10.0075

11.8472

9.9169

ANO-RCC-VTZ

3.2534

3.7810

16.3954

16.7163

23.4749

23.3209

ANO-RCC-VTZP

3.2534

3.7810

16.3954

16.7163

23.5516

23.3324

ANO-RCC-VQZP

4.3873

5.3101

20.5307

22.3485

29.0751

31.0360

ANO-rcc uncontracted

6.0938

8.0985

23.8421

27.9929

31.3810

35.3697

ANO-L-MB ANO-L-VDZ

0.0000 0.6345

0.0000 0.8956

0.0000 2.7765

0.0000 4.1934

ANO-L-VDZP

1.7552

2.0225

10.4231

10.6815

ANO-L-VTZP

4.6101

5.7690

19.8133

21.4704

ANO-L-VQZP

5.8446

7.6679

23.1600

26.7419

ANO-L uncontracted

6.1165

8.1880

23.7930

27.9848

Pol

6.3958

8.1718

23.7641

27.3870

STO-3G

0.0000

0.0000

0.0000

0.0000

3-21G 4-31G

0.1572 0.3093

0.1926 0.4104

1.7411 1.2303

2.5160 1.6971 3.3142

6-31G

0.3361

0.4504

3.2383

cc-pVDZ

0.7594

0.8770

5.9640

5.9747

cc-pVTZ

1.7430

1.9342

11.0935

10.6043

cc-pVQZ

2.7139

3.0917

15.0177

cc-pV5Z

3.4631

4.0480

aug-cc-pVDZ

4.2156

5.6192

aug-cc-pVTZ aug-cc-pVQZ

5.3733 6.1572

7.0944 8.2225

aug-cc-pV5Z

6.5410

8.8437 9.83

36.3941

15.0211 16.2721

27.1743

28.3652

18.9670

21.4438

26.5269

29.3660

22.1041 23.9950

25.5067 28.4835

32.2436 33.7668

36.9150 39.3781

28.5217 30.704

34.2447

40.3021

d-aug-cc-pVDZa t-aug-cc-pVDZa

10.59

31.282

q-aug-cc-pVDZa

10.6

31.288

x-aug limita

10.6

31.290

d-aug-cc-pVTZa

10.31

31.206

t-aug-cc-pVTZa q-aug-cc-pVTZa

10.67 10.67

31.475 31.476

x-aug limita

10.67

31.480

Taken from ref 16.

€ and their origin can be found in a recent paper of Ohrn and Karlstr€om.23 Once the interaction between the solute and the solvent is defined, a Monte Carlo simulation is performed to solve the statistical mechanical problem. The solute is embedded by a number of classically described water molecules, and the whole system is kept inside a spherical dielectric cavity of variable radius. After equilibration, a Monte Carlo simulation is performed and every nth step, data are collected and stored for subsequent analysis. All water molecules are moved in each Monte Carlo step. In order to obtain the properties of the quantum system in solution, two kinds of perturbations have been evaluated. First, all stored configurations in the simulation have been employed to calculate the mean electrostatic potential, field, and field gradients. These have been used to evaluate the electrostatic perturbation produced by the solvent in the quantum core. Second, the stored configurations have also been employed to obtain an

average nonelectrostatic perturbation, which depends on the overlap between the orbitals of the solute and the orbitals of the solvent molecules. Both perturbations are introduced in the Hamiltonian used to calculate the wave function employed to obtain the polarizabilities and electrical properties of the quantum system in solution.

3. COMPUTATIONAL DETAILS The simulation parameters used in the simulation are presented in Table 3. These are employed together with a compact basis of natural orbitals19,34,35 to solve the Hartree-Fock equation for each step of the Monte Carlo simulation. The number of natural orbitals in the basis is given in Table 3. The basis set of Sadlej (also called the POL basis)36-38 has been used to construct the compact basis set used by QMSTAT and in the description of the repulsion between the QM system and the classical water molecules. Parameters in Table 3 are fitted to 1100

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Table 5. Computed Isotropic Polarizability of the Halogen Ions at the Various Level of Theory (a30) F-

Cl-

Br-

UCHF in vacuo

6.3958

23.7641

-

confined Ψ in aqua

5.2617 5.0988

20.4210 20.3754

-

Perturbation,HF-SCF in vacuo

8.1718

27.3870

36.3941

confined Ψ

6.6473

22.8957

28.4870

in aqua

6.4227

22.8254

28.2893

in vacuo

10.5191

30.2558

39.2924

confined Ψ

8.1904

24.5026

29.7773

in aqua

7.8769

24.4165

29.5506

Perturbation, MP2

reproduce the BSSE-corrected MP2 potential energy surface computed using the Sadlej basis set. This basis set is optimized to give the best possible description of the polarizability and is thus well suited for the study performed here. The simulations were carried out at 300 K and a pressure of 1 atm using the nPT ensemble. After relaxation, a total of 452 400 Monte Carlo steps were taken, and each 100th step was sampled and stored. The geometry of the formate ion is presented in Table 2. The number of explicit water molecules present during the simulations is given in Table 3. We have computed the molecular polarizabilities using both the uncoupled Hartree-Fock (UCHF) approximation25 and finite-field perturbation theory (FFPT). The localized properties were computed using both the MPPROP39 and the LOPROP28 method. The methodology used to compute the molecular and atomic properties using MP2 has been described elsewhere.40 All calculations have been performed using the MOLCAS quantum chemical package,41,42 where the QMSTAT method is implemented in a development version.

4. RESULTS AND DISCUSSION Before discussing the results of the simulations, we will shortly focus on the basis set behavior of the polarizability for the anions in vacuo. This has been computed at the HF-SCF level of theory for a large number of basis sets, and are presented in Table 4 for the halide ions (a similar study for the formate ion is given in Table 1 in the Supporting Information). The first observation from this table is that the polarizability grows rapidly with the size of the basis set. This is most pronounced for the monatomic ions in Table 4. It is clear that the smaller basis sets give a too small polarizability and that the diffuse functions are required if the basis set are to give reasonable polarizabilities for the molecule in vacuo. This was of course expected, since the increased number of basis functions available gives an increased flexibility to describe the shift in the charge distribution. From Table 4 we also see that the POL basis set,36-38 although small, gives polarizabilities of the same size as the largest aug-cc-pVXZ and ANO basis sets. This is because the POL basis set has been optimized to give the best possible description of the response to an electric field, while the other basis sets have primarily been developed to give the best possible description of the energy. As the energy is dependent on having a good description of the core

electrons, a large basis set optimized for this purpose will not in the first instance focus on the diffuse functions that are required for calculations of the polarizability. The results in Table 4 also highlight another problem, namely that not only are the polarizabilities of an ion in water difficult to compute but also the calculation of the polarizability in the gas phase is also nontrivial due to the basis set dependence. The observations done in this basis set study are by no means novel, and similar studies have for instance been done by Woon and Dunning,16 who also included the effect of electron correlation into their study and did basis set limit extrapolation. The major difference between the most accurate results presented in Table 4 and the results obtained by Woon and Dunning16 is due to electron correlation as shown in their work. However, as the use of computational methods to compute molecular properties increases, it serves as a reminder that care must be taken when using quantum chemical methodology to obtain these, also in the selection of the basis set, and that perhaps the approach of Sadlej should be followed, and extended, when the calculation of other properties than the polarizability is considered. It is comforting to note that very accurate results can be obtained using the relatively small POL basis set used in this work (for instance, the polarizability of F- is just 2.5 a30 smaller than the basis set limit, using a small number of basis functions). We will now turn our focus to the results of the simulations, starting with the halogen ions. The computed polarizabilities are presented in Table 5 for the UCHF, HF-SCF, and MP2 levels of theory. From this table, it is clear that there is a significant reduction of the polarizability for all the ions and methods. The reductions are between 19 and 25% and largest for the MP2 polarizabilities. Furthermore, we see that the polarizabilities computed with only the nonelectrostatic perturbation on the anion (called “confined Ψ”) and all the perturbations on the anion (in aqua) are of the same magnitude. This means that the effect of the electrostatic perturbation on polarizability of the molecule are small, and lead us to the conclusion that we are still in the linear polarization regime. If this was not the case, we would expect significant deviation between the confined Ψ and the in aqua method. As mentioned in the Introduction, the halogen ions have previously been the subject of a number of studies, with some results presented in Table 5. These numbers vary quite a lot, but are in most cases larger than the values computed by our method. The values that are closest to our result are the ones from the Dang-Chang force field and the results of Pyper et al.,5 which both quite well resemble our findings (on the basis of the MP2 polarizabilities). As the reduction of the polarizability of the ions in condensed phase is related to the size of the ion, it is interesting to see how this size varies. One measure of size on a molecular level is the expectation value of r2, or the second-order moment (in order to avoid confusion, we will use this term instead of the term quadrupole moment, which is usually traceless). The secondorder moments of the ions are presented in Table 6 for all the halogen ions. From this table, we see that the second-order moments of the ions are reduced when going from the in vacuo calculations to the calculations of ions in water. This suggests that the polarizability is reduced due to a reduction of the size of the molecule. From Table 6 it is also clear that the reduction of the size of the molecule is more dramatic for the large Br- ion than for the two smaller ions. This reduction of the size can be important to take into account when constructing potential functions describing negatively charged ions, as the trace of the 1101

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Table 6. Expectation Values of r2 Computed for the Halogen Ions Expanded around [0,0,0] (ea30) F-

Cl-

Br-

Table 8. Computed Molecular Polarizability of the Formate Ion for the Various Methods (a30) xx

yy

zz

Æisoæ

UCHF

HF-SCF in vacuo

-7.1344

-17.2053

-23.9156

in vacuo

26.1325

23.0417

18.4280

22.5341

confined Ψ in aqua

-4.9035 -4.8453

-12.1324 -12.1247

-16.6320 -16.6093

confined Ψ in aqua

23.8492 17.9949

20.7167 14.4552

16.1155 12.5366

20.2271 14.9956

Perturbation, HF-SCF

MP2 in vacuo

-5.8555

-13.1927

-18.2296

in vacuo

31.8413

28.5641

20.2985

26.9013

confined Ψ

-5.2226

-12.3391

-16.8304

confined Ψ

27.6346

24.8430

16.9286

23.1354

in aqua

-5.1373

-12.3293

-16.8031

in aqua

23.0158

18.3752

14.2974

18.5628

in vacuo

40.2166

36.9383

24.2255

33.7935

confined Ψ

32.5859

29.6231

18.7398

26.9829

in aqua

26.5089

20.5655

15.3430

20.8058

Perturbation, MP2

Table 7. Average Induced Dipole Moment Computed for the Halogen Ions Expanded around [0,0,0] (ea0) F-

Cl-

Br-

HF-SCF confined Ψ

0.1006

0.0487

0.0976

in aqua

0.0616

0.1008

0.1934

confined Ψ

0.1224

0.0554

0.1061

in aqua

0.0718

0.1062

0.1970

Table 9. Computed Molecular Dipole Moment of the Formate Ion for the Various Methods (Expansions Are Done around [0,0,0]) (ea0)

MP2

x

second-order moments enters the expression of the repulsion energy.43 It is also interesting to note that the second-order moment is reduced when electron correlation is introduced. This means that when allowing the electrons to explicitly interact, the sizes of the anions are reduced, while at the same time the polarizabilities of the ions are increased. The average induced dipole moment of halide anions is presented in Table 7. The results there show that the monatomic anions have a nonzero average induced dipole moment. This is in agreement with the observation in previous studies of both Wick € and Karlstr€om,17 and emphasizes the and Xantheas44 and Ohrn importance of using polarizable force fields when studying anions. The results for the molecular polarizability, dipole moment, and second-order moment of the formate ion are presented in Tables 8, 9, and 10, respectively. It should be noted that the calculation using the confined Ψ approach is done in the full basis, whereas the calculation using the in aqua approach is done in the compact basis set used in the QMSTAT calculation. Starting with the polarizability, we see that this is again reduced significantly when compared with the in vacuo results. This is true for all the components of the polarizability and all methods of computing it. The reduction of the polarizability is rather large for the formate ion, with the largest reduction in the plane component parallel to the symmetry axis. The difference between the confined Ψ and the in aqua method is probably due to the fact that the reduced QMSTAT basis was used when computing the latter. The somewhat reduced number of basis functions should slightly reduce the polarizability. This does not automatically lead to the conclusion that the in aqua method is less accurate than the confined Ψ method, as the very diffuse basis functions excluded in the reduced basis set most probably do not represent physically important aspects of the ion in water. Furthermore, it cannot be excluded that electrostatic perturbation plays a larger role for this anion than that for the

y

z

|μ|

HF-SCF 0.0000

-1.3299

0.0000

1.330

-0.008381

-1.3107

-0.009928

1.311

in aqua

0.04069

-1.5092

-0.01600

1.510

in vacuo

0.00000

-1.1814

0.0000

1.181

confined Ψ in aqua

0.01491 0.01987

-1.1533 -1.4390

-0.01342 -0.01928

1.153 1.439

in vacuo confined Ψ

MP2

monatomic anions. The dipole moment of the molecule is presented in Table 9, and if we keep in mind that the total dipole moment of a molecule in the presence of a perturbing field is ! μ permanent þ ! μ induced ð2Þ μ tot ¼ ! we see that the only sizable change in the dipole moment occurs when the system is perturbed by the external electrostatic field. This means that there is no effect of the nonelectrostatic perturbation on the molecular dipole moment. In terms of representing reality, the correct dipole moment of the formate ion in water would be the in aqua result. The effective induced dipole moment of the formate ion is roughly 0.2 ea0, or 14% of the total dipole moment. This is a significant change in the dipole moment and confirms that a polarizable force field must be used in order to correctly study this molecule using Monte Carlo or molecular dynamics simulations. The last molecular property presented in this study is the second-order moment of the formate ion presented in Table 10. For the formate ion, we observe the same trend as for the halogen ions, that is, a reduction of the trace of the second-order moment when going from the in vacuo molecule to the solvated molecule. In both the dipole moment and second-order moment of formate ion, there are components that are expected to be zero due to symmetry that are nonzero. The sizes of these apparent breaks of symmetry can be seen as an estimate of the uncertainty of the method, and would become smaller if the number of Monte Carlo steps were 1102

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Table 10. Computed Molecular Second-Order Moment of the Formate Ion for the Various Methods (Expansions Are Done around [0 0.0000, 0.3816 Å, 0.0000]) (ea20) xx

xy

xz

yy

yz

zz

Tr(Q)

HF-SCF in vacuo

-21.7267

0.0000

0.0000

-14.9965

0.0000

-13.8981

-16.8738

confined Ψ in aqua

-20.9586 -21.2150

-0.0474 -0.0685

0.0192 0.0413

-14.6289 -14.1166

-0.0195 -0.0015

-13.4162 -13.4567

-16.3346 -16.2628

MP2 in vacuo

-21.9376

0.0000

0.0000

-16.0746

0.0000

-14.4894

-17.5005

confined Ψ

-20.8434

-0.0665

0.0287

-15.4481

-0.0264

-13.7862

-16.6926

in aqua

-21.3300

-0.0738

0.0478

-14.5714

-0.0019

-13.7367

-16.5460

Table 11. Atomic Polarizability Moments of the Formate Ion Computed Using the LOPROP Method the MP2 Level of Theory (a30) xx

xy

xz

yy

yz

zz

Æisoæ

In Vacuo C 16.8931 -0.0006 -0.0000 15.8486 H

3.8318 -0.0001

O

9.7452

O

9.7463 -1.4570

0.0016 10.2213 14.3210

0.0000

9.1131

0.0006

3.4790

5.4747

1.4562 -0.0000

5.9882

0.0002

5.2626

6.9987

5.9884

0.0002

5.2626

6.9991

0.0000

Confined Ψ C 12.2083

0.2613 -0.2253 10.0763

0.0985

6.2802

9.5216

H O

2.8340 8.7614

0.1105 -0.0381 1.6425 -0.0407

0.0142 0.1108

2.7276 4.8360

4.3833 6.5715

O

8.7820 -1.5967

5.8415 -0.0152

4.8959

6.5065

0.0675

7.5882 6.1171

In Aqua C 10.6153

0.1549 -0.1167

8.1304

0.0292

5.9984

8.2480

H

2.0459

0.0997 -0.0579

5.3025 -0.0220

2.2447

3.1977

O

6.8922

1.8382 -0.0515

3.6468

0.0385

3.5672

4.7021

O

6.9555 -1.7173 -0.0552

3.4858

0.0021

3.5327

4.6580

Table 12. Atomic and Molecular Polarizabilities for the Formate Ion from Commonly Used Force Fields (a30) 10

Amber 10 Amoeba 0911 ref 5

C

H

O

O

molecular polarizability

4.157 9.002

1.127 3.347

2.929 5.648

2.929 5.648

11.142 23.646 28.4

increased. Keeping in mind the rather slow convergence of the Monte Carlo procedure (O((N)1/2), where N is the number of steps), and the small size of these apparent symmetry violations compared to the changes of the other components of the dipole and second-order moments, we are confident that a sufficient number of Monte Carlo steps have been taken to get quantitatively consistent results. The atomic polarizabilities of the formate ion computed with the LOPROP method at the MP2 level of theory are presented in Table 11. The other localized properties (charges, dipole moments, second-order moments, and polarizabilities) computed by the MPPROP or LOPROP method at the HF-SCF method can be found in the Supporting Information. The atomic polarizabilities in Table 11 show a reduction of the polarizability of all sites in the molecule. This applies for all components,

except the xy-component, which cancels out for the total polarizability. The main polarization contribution to the xycomponent is probably the transfer of charge in the CdO bond, and this is not likely to be very much affected by the surrounding molecules. The fact that all the atomic polarizabilities are reduced is related to the small size of the ion. Although the formate ion is a multiatomic molecule, all atoms in the molecule are in contact with the surrounding water molecules and are perturbed by them. If one or several of the atoms were not in contact with the water molecules, then we might expect that these atoms would not experience such a large reduction of the polarizability (although we must keep in mind that these polarizabilities are coupled and that the reduction of one local polarizability will influence the others). The polarizabilities of the atoms (and the total molecular polarizability) presented here can be compared with the polarizabilities used by other force fields presented in Table 12. Care should be taken when comparing nonobservables such as localized polarizabilities; however, we see that the atomic polarizabilities used by the Amoeba 09 force field are in the same range as the polarizabilities we have computed for the confined Ψ and in aqua method, although slightly larger. Furthermore, we see that both the Amoeba 09 and the molecular polarizability of Pyper et al.5 are in the same range as our result. The Amber 10 force field on the other hand again significantly underestimates the polarizabilities, and this gives rise not only to too low atomic polarizabilities but also to a molecular polarizability that is about half of the polarizability that we obtain for the formate ion in water.

5. CONCLUSIONS In this work, we have presented a consistent methodology to compute the polarizability and other properties of ions solvated by water from first principles. The halide anions F-, Cl-, and Brhave been studied along with the formate ion, and it is found that the polarizabilities of these molecules are significantly reduced. This is in line with the results from previous studies based on fitting to experimental data. It has also been shown that the second-order moments of the molecules are reduced. ’ ASSOCIATED CONTENT

bS

Supporting Information. Basis set behaviour of the polarizability for formate ion; atomic charges, atomic dipole moments, and atomic polarizabilities for the formate ion. This material is available free of charge via the Internet at http://pubs. acs.org.

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The Journal of Physical Chemistry B

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