Polarizability, Ionization Potential, and Softness of Water and

May 31, 2012 - Yan-Long Ma , Ru-Jin Zhou , Xing-Ye Zeng , Ya-Xiong An , Song-Shan Qiu , Li-Jun Nie. Journal of Molecular Structure 2014 1063, 226-234 ...
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Polarizability, Ionization Potential, and Softness of Water and Methanol Clusters: An Interrelation Kartick Gupta,† Tapan K. Ghanty,* and Swapan K. Ghosh Theoretical Chemistry Section, Chemistry Group, Bhabha Atomic Research Centre, Mumbai 400 085, India ABSTRACT: The properties of methanol clusters [(CH3OH)n, n = 1−12] have been studied by using ab initio electronic structure calculations with reference to the aggregation number dependence of several reactivity descriptors, such as ionization potential, electron affinity, polarizability, hardness, and binding energy. A good correlation between the dipole polarizability and the ionization potential of these hydrogen-bonded molecular clusters is shown to exist. The softness parameter has also been shown to correlate strongly with the dipole polarizability of these molecular clusters. Similar good correlations are also demonstrated to exist for water clusters [(H2O)n, n = 1−20]. This work can thus be useful for calculating the polarizability of larger methanol or water clusters in terms of the corresponding ionization potential. Ghosh and co-workers have shown18 that the calculated dipole polarizability has good correlation with the ionization potential and softness parameter for lithium and sodium metal clusters. We do not find, however, any such correlation reported for molecular clusters in the literature to the best of our knowledge. In this work, we have investigated the correlation of several reactivity descriptors such as ionization potential (IP), electron affinity (EA), hardness, binding energy, and dipole polarizability with the aggregation number of methanol clusters, (CH3OH)n, n = 1−12. The static dipole polarizability is also correlated with the ionization potential and softness of the methanol clusters. Ab initio molecular orbital methods have been used here to compute the reactivity descriptors, binding energy, and static polarizability of these clusters. The correlations of polarizability with IP and softness have also been investigated here for other hydrogen bonded molecular clusters, such as water clusters. In what follows, we first present the theoretical background and computational approach in section 2, followed by discussion of the results in section 3. Finally, we present a few concluding remarks in section 4.

1. INTRODUCTION The importance of methanol as a solvent and its anomalous behavior like water have given rise to a lot of studies on the structure of liquid methanol as well as its properties. The structure and properties of methanol clusters are different from that of water clusters due to the presence of a hydrophobic methyl group that causes a disturbed hydrogen bonded network as compared to that in water clusters. The investigation of liquid methanol and its clusters is an active topic of research.1,2 The enthalpies, geometries, and IR spectra of methanol clusters, (CH3OH)n have been calculated by Pires and DeTuri.3 Boyd and Boyd4 have reported that the dissociation energy of methanol clusters (per molecule of methanol) increases as the aggregation number, n, increases up to 5−6, beyond which it remains almost constant. Alsenoy et al.5 have calculated the polarizabilities of methanol clusters at the B3LYP/6-311++G(d,p) level of theory and partitioned them into molecular contributions using the Hirshfeld method. Recently, we have demonstrated good linear correlations for the first and second excited state polarizabilities as a function of the aggregation number of methanol clusters.6 Several studies have been carried out to show the correlation between softness of systems with polarizability.7−12 However, in spite of all these investigations, progress on applications of such correlations to molecular clusters is limited. There are, however, several studies in the literature to establish a relationship between the polarizability and ionization potential for atomic systems.13−16 Correlation of polarizability with electronegativity has also been demonstrated to exist for atoms.17 The static dipole polarizability is a measure of the distortion of the electron density under the effect of an external static electric field. However, ionization potential is a measure of the binding energy of the outermost electron bound by the nuclear attractive field of the system. Consequently, an inverse relationship between these two parameters is expected to exist. © XXXX American Chemical Society

2. THEORETICAL BACKGROUND AND COMPUTATIONAL DETAILS The ground-state energy of an atomic or molecular system, within the framework of density functional theory (DFT), can be expressed as a functional of its electron density ρ(r) as,19 E[ρ] = F[ρ] +

∫ drν(r)ρ(r)

(1)

Received: August 10, 2011 Revised: May 31, 2012

A

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where ν(r) is the external potential characterizing the manyelectron system and arises from the potential due to the nuclei (and also external field), and F[ρ] is a universal functional of density, consisting of the electronic kinetic energy and electron−electron interaction energy components. The chemical potential (μ) and the global hardness (η) of the system are defined20−22 as the first and second derivatives, respectively, with respect to the number of electrons, N, at fixed external potential, viz., ⎛ ∂E ⎞ ⎜ ⎟ ⎝ ∂N ⎠ν(r)

(2)

⎛ ∂ 2E ⎞ η = ⎜ 2⎟ ⎝ ∂N ⎠ν(r)

(3)

μ=

The derivatives can be computed numerically by using the finite field method. The mean polarizability is obtained from the diagonal elements of the polarizability tensor as α=

3. RESULTS AND DISCUSSION Electronic structure calculations have been carried out on molecular clusters of two species, viz., the methanol clusters and water clusters. The geometries of the methanol molecular clusters have been taken from the minimum energy geometries reported by Pires and De Turi,3 whereas the same for the geometries of the water clusters are taken from the minimum energy geometries as reported by Maheshwary et al.25 The chemical potential (μ), hardness (η), softness (S), and static polarizability (α) of the molecular clusters of methanol (n = 1− 12), calculated by different methods/basis sets by employing Koopmans’ theorem and the ΔSCF method are reported in Tables 1 and 2. The chemical potential signifies the direction of electron transfer from one system to another21 and is equal to the negative of the electronegativity in the Mulliken scale.27 Hardness parameter provides a measure of the resistance to the change in the chemical potential due to the change in the number of electrons. The hardness of a system is highest at the equilibrium nuclear configuration and is directly linked to the stability of a species,28 as reflected in the principle of maximum hardness.29 It is found that both IP and hardness decrease monotonically with the number of the methanol molecules in the cluster (n) and finally approach toward the bulk value for large n [Figure 1a,b]. The binding energy per molecule increases (becoming more negative) with the aggregation number of methanol in the same fashion although the bulk value is reached with a smaller number of methanol molecules in the cluster [Figure 1c]. In contrast to the variation of IP and hardness, the EA and the polarizability per molecule are found to increase with n [Figure 2a,b]. Static dipole polarizability is a measure of the distortion of the electronic density and the response of the system under an external static electric field. However, ionization potential is a measure of how tightly an electron is bound by the field due to the nuclei in a many-electron system. Dimitrieva and Plindov were the first to report a relationship between the polarizability and the ionization potential using a statistical model.16 Fricke has shown that the polarizability of neutral atoms correlates very well with their first ionization potential (in logarithmic scale) within the groups of elements with the same angular momentum of the outermost electrons.14 Politzer et al.13 have reported an inverse relationship between α and IP for several

(4)

The working formulas in DFT for calculating these parameters are as follows: μ ≈ −(IP + EA)/2

(5)

η ≈ (IP − EA)/2

(6)

S ≈ 1/(IP − EA)

(7)

where IP and EA denote the first ionization energy and electron affinity of the system, respectively. These quantities, which are linked with the response of a system to the change in the number of electrons, have been found to play an important role in providing an understanding of the chemical binding in molecular systems. In this context, for an atomic or molecular aggregate or cluster, An, formed from the corresponding atomic or molecular species, A, the quantity of primary concern is the binding energy per atom or molecule, which is defined as BE = [E(A n) − nE(A)]/n

(8)

where E(An) and E(A) denote, respectively, the total energy of the cluster and the atomic or molecular species. An important response property of a system is concerned with the perturbation due to an external electric field. The static response properties of a system can be obtained by considering the expansion of the field-dependent energy, E(F), as a series in the components of a uniform electric field F, viz., 1 E(F) = E(0) + ∑ μi Fi + ∑ αijFF i j + ··· 2 ij i (9) where E(0) is the energy of the system in the absence of the electric field, μ is the dipole moment vector, and α is the dipole polarizability tensor. The components of the polarizability tensor are obtained as the second-order derivatives of energy with respect to the Cartesian components (i, j = x, y, z) of the electric field, viz., ⎛ ∂ 2E ⎞ ⎟⎟ αij = ⎜⎜ ⎝ ∂Fi ∂Fj ⎠ F = 0

(11)

The basis set dependence of polarizability is well-known, and the Sadlej basis sets23 used here are specially designed for accurate calculation of polarizability. The reactivity descriptors such as ionization potential, electron affinity, polarizability, chemical potential, hardness, and softness have been computed here by employing the B3LYP/6-311++G(d,p), B3LYP/Sadlej, HF/Sadlej, and MP2/Sadlej methods. The IP, EA, and related reactivity descriptors of the clusters have been computed using Koopmans’ theorem24 as well as the ΔSCF method. The latter involves the calculation of energy of the cation, neutral and anion species for obtaining the IP and EA values. All the calculations in this work have been performed with the GAMESS electronic structure program26 using the basis sets mentioned.

The global hardness η is a measure of the charge capacity of the system, which has also been found to dictate its overall stability. The inverse of the hardness, η, defines the global softness as S ≈ 1/2η

1 (αxx + αyy + αzz) 3

(10) B

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Table 1. Chemical Potential (μ), Hardness (η), and Softness (S) of Methanol Clusters (CH3OH)n, n = 1−12), as Computeda by Different Methods/Basis Sets Employing Koopmans’ Theorem/ΔSCF Method aggregation number, n

chemical potential, μ (eV)

hardness, η (eV)

softness, S (eV)−1

1

−5.098 −5.615 −4.470 −4.666 −5.294 −4.100 −4.551 −5.504 −4.297 −4.448 −5.462 −4.249 −4.374 −5.460 −4.219 −4.306 −5.526 −4.995 −4.306 −5.362 −4.154 −4.207 −5.462 −4.198 −4.180 −5.332 −4.219 −4.195 −5.332 −4.100 −4.153 −5.303 −4.059 −4.118 −5.407 −4.034

5.792 6.673 5.404 5.190 6.279 4.899 5.126 6.588 5.244 5.014 6.575 5.198 4.888 6.559 5.158 4.799 6.468 5.934 4.737 6.412 5.005 4.647 6.504 5.106 4.583 6.388 5.071 4.576 6.374 4.935 4.493 6.324 4.875 4.471 6.242 4.845

0.0863 0.0749 0.0925 0.0963 0.0796 0.1020 0.0975 0.0759 0.0953 0.0997 0.0760 0.0962 0.1023 0.0762 0.0969 0.1042 0.0773 0.0843 0.1055 0.0780 0.0999 0.1076 0.0769 0.0979 0.1091 0.0783 0.0986 0.1092 0.0784 0.1013 0.1113 0.0791 0.1026 0.1118 0.0801 0.1032

2

3

4

5

6

7

8

9

10

11

12

Table 2. Polarizability (a.u.) of the Molecular Clusters of Methanol Clusters (CH3OH)n, n = 1−12), as Computed by Different Methods/Basis Sets aggregation number (n)

HF/ Sadlej

B3LYP/6-311+ +G(d,p)

B3LYP/ Sadlej

MP2/ Sadlej

1 2 3 4 5 6 7 8 9 10 11 12

17.59 36.00 55.02 74.15 92.54 119.72 129.70 148.04 167.02 185.96 204.39 237.81

19.10 39.70 61.10 82.30 102.75 123.42 144.20 164.50 185.80 207.01 227.52 248.64

21.84 43.92 66.53 88.92 110.73 133.01 154.50 177.23 198.82 220.31 242.85 264.49

21.33 42.81 64.87 86.74 108.10 129.79 150.82 172.88 194.00 215.13 237.03 258.23

a

B3LYP/6311++G** (ΔSCF), HF/Sadlej (Koopmans), and MP2/ Sadlej (ΔSCF) data in successive rows for each cluster size, n.

neutral atoms with a linear correlation coefficient of 0.954. Ghanty and Ghosh10a have demonstrated an inverse correlation between α1/3 and the hardness for several systems. In the present work, the polarizability (α) and the IP of the methanol clusters as calculated by the B3LYP method are shown in Figure 3a,b, and a very good correlation between the cube root of α (actually, α1/3/n) and inverse of IP (actually IP−1/n) is shown to exist, which might be useful in providing a route to estimate the polarizability of these clusters. Similar good correlation is also found to exist between these parameters as obtained by employing the MP2/Sadlej method [Figure 3c]. For a comparison, we have also presented the results for water clusters, as computed by the B3LYP method and 6-311+ +G(d,p) basis set employing ΔSCF method [Figure 4], and a good correlation between the α and IP is found to exist. This is an important observation because this is the first report on this type of relationship for hydrogen-bonded clusters of larger size,

Figure 1. Variation of the (a) ionization potential (IP), (b) hardness, and (c) binding energy (BE) with aggregation number (n) of methanol clusters, as computed by employing the B3LYP method and 6-311++G(d,p) basis set along with the ΔSCF method.

indicating the possibility of prediction of polarizability from their ionization potential. This would also help in connecting the local ionization energies with local polarizability, as suggested by Politzer et al.15 Although the global softness parameter is directly related to the polarizability of the system, applications of these C

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Figure 2. Variation of the (a) electron affinity (EA) and (b) average polarizability with aggregation number (n) of methanol clusters, as computed by employing the B3LYP method and 6-311++G(d,p) basis set along with the ΔSCF method.

correlations to molecular systems have not been explored, particularly for hydrogen-bonded molecular clusters. Figure 5a−c shows a good linear correlation between the average of cubic root of polarizability (α1/3/n) and average softness (S/n) of the methanol clusters as obtained by employing (a) the B3LYP/Sadlej/Koopman, (b) B3LYP/6-311++G(d,p)/ΔSCF, and (c) MP2/Sadlej/ΔSCF methods, respectively. Figure 6 shows a good linear correlation between the average of the cubic root of polarizability (α1/3/n) and average softness (S/n) of water clusters as computed by the B3LYP method and 6311++G(d,p) basis set by employing the ΔSCF method. In general, it is well-known that for any neutral chemical species, the value of electron affinity is much less compared to its ionization potential. This is quite clear from the calculated results reported in Table 1 and Figures 1−2 that the electron affinity values of the methanol clusters are indeed much less as compared to the corresponding ionization potential values. Hence, the global softness can be approximated in terms of the inverse of the ionization potential alone. Thus, the correlation of polarizability with both softness and ionization potential is quite good. In fact, for systems like alkali metal clusters, it has earlier been demonstrated18 that the polarizability correlates very well with the corresponding ionization potential or softness. The softness of any system representing the charge capacity8 of a species is expected to be proportional to its size.10 Since, for a sphere of radius R, the polarizability scales as R3, a relationship between the cube root of polarizability and softness of the molecular cluster is expected to exist. It can be assumed that the electron density cloud on the hydrogen-bonded molecular cluster is also approximately spherical in shape, similar to that for atoms, a class of molecules, and atomic clusters. On the basis of the Fermi−Amaldi model, Dmitrieva and Plindov16 have shown that the IP is inversely proportional to the radius of the atomic systems. Consequently, an inverse relationship between the cube root of polarizability and IP is

Figure 3. Relationship between the polarizability (α) and ionization potential (IP) of the methanol clusters, as computed by using (a) the B3LYP method with Sadlej basis set, employing Koopmans’ theorem, (b) the B3LYP method and 6-311++G(d, p) basis set, employing the ΔSCF method, and (c) the MP2 method and Sadlej basis set, employing the ΔSCF method.

Figure 4. Relationship between the polarizability (α) and ionization potential (IP) of water clusters, as computed by using the B3LYP method and 6-311++G(d,p) basis set along with the ΔSCF method.

expected to exist. We have shown here that these relationships can be validated for hydrogen-bonded molecular clusters such as methanol and water clusters as well. It may be worthwhile to delineate some of the issues discussed in other approaches to linear response properties of molecules in the presence of other molecules. Our work has thus relevance to the work of York and Yang.30 This type of approach can also be used for obtaining polarizable force field D

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the exact ground-state molecular or fragment density instead of the neutral atom densities; hence, it recovers the exact charge distribution in the absence of a perturbing field. The approach uses a simple semiempirical form for the hardness matrix that allows non-Coulombic contributions to be modeled by an overlap term. Classical polarizable force fields have also been parametrized from ab initio calculations for ions by Tabacchi, Mundy, and Hutter,31 where the choice of reference states has been restricted to subsystems of spherical symmetry, e.g., atoms or isolated ions. Riccardo and Piero32 have discussed several shortcomings that may arise when Tabacchi’s model is applied to complex molecular systems.

4. CONCLUSIONS In this work, several reactivity descriptors such as ionization potential, electron affinity, hardness, softness, polarizability, and binding energy for hydrogen-bonded methanol and water clusters have been calculated and variations of these quantities with cluster size have been investigated. Ionization potential and hardness of methanol and water clusters have been found to decrease nearly exponentially with cluster size, n. The electron affinity, softness, and polarizability per molecule are found to increase with n. The binding energy per molecule increases (becoming more negative) with the aggregation number for methanol clusters, although the bulk value is reached with a smaller number of methanol molecules in the cluster, as compared to water cluster. It is demonstrated that the aggregation number-dependent polarizability strongly correlates with the ionization potential, softness, and hardness of the methanol and water clusters. These correlations have not previously been reported for any hydrogen-bonded molecular cluster. This type of correlation between the polarizability and ionization potential will be useful to estimate the polarizability of larger clusters from the corresponding experimental/ calculated ionization potential values.

Figure 5. Relationship between the polarizability (α) and softness (S) of methanol clusters, as computed by using (a) the B3LYP method with Sadlej basis set, employing Koopmans’ theorem, (b) the B3LYP method and 6-311++G(d,p) basis set, employing the ΔSCF method, and (c) the MP2 method and Sadlej basis set, employing the ΔSCF method.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. † On leave from Department of Chemistry, Ramananda College, Bishnupur 722122, West Bengal, India.

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Figure 6. Relationship between the polarizability (α) and softness (S) of water clusters computed by using the B3LYP method and 6-311+ +G(d,p) basis set, employing the ΔSCF method.

ACKNOWLEDGMENTS K.G. gratefully acknowledges the support of the Indian Academy of Sciences, Bangalore, Indian National Science Academy and National Academy of Sciences, Allahabad, for the award of a Summer Research Fellowship and the Governing Body, Ramananda College, Bishnupur for granting leave of absence. The work of S.K.G. is also supported by INDO-EU project MONAMI and J. C. Bose Fellowship from DST, India.

for performing computer simulation, which can supplement the conventional classical force fields. Thus, for a system of interacting molecules, the chemical potential equalization (CPE) approach of York and Yang30 provides a model for the linear density response of each molecule in the field of other molecules, with the possibility of charge transfer. The CPE parameters are shown to be transferable for obtaining intermolecular interactions as demonstrated in the case of water dimers. Here, one considers an expansion of the energy around

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