Generalized Microscopic Theory for the Detachment Energy of

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Generalized Microscopic Theory for the Detachment Energy of Solvated Negatively Charged Ions in Finite Size Clusters: A Step toward Bulk A. K. Pathak, A. K. Samanta,* D. K. Maity,* T. Mukherjee, and S. K. Ghosh Chemistry Group, Bhabha Atomic Research Centre, Mumbai-400085, India

ABSTRACT A new general relation is derived for the size-dependent detachment energy of solvated negatively charged ions in finite size clusters based on a microscopic theory for systems with unknown interaction potentials. The relation is tested over a large number of different kinds of finite-size hydrated clusters of the type Aq-(H2O)n (for spherical and nonspherical, singly and multiply charged ion Aq-), and an excellent agreement with experimental results is observed. More importantly, the robust scheme is shown to provide a route to obtain the bulk (infinite-size clusters) detachment energy from the results of finite-size clusters. Again, an excellent agreement with the experimental results is observed where the earlier models are shown to fail. SECTION Molecular Structure, Quantum Chemistry, General Theory

olvation energy1,2 plays a major role in connection with the distribution of elements in the geo- and hydrosphere, solubility of salts in solvent,3 stability of transition states of chemical reactions, ion hydration,4,5 conformational stability of biomolecules and the electrode-solution interface, folding or conformal transformation of proteins, DNA, RNA, and polysaccharides, association of biological macromolecules with ligands,6 transport of ions and drugs across biological membranes,7,8 or the electronic and molecular structure of molecules in solution.9 Despite such immense interest, a precise quantification of the solvation energy for finite as well as bulk systems with unknown interaction potential still remains elusive. Computer-simulation-based investigations do provide useful information on the solvation of the finite-size cluster as well as the cluster having sufficiently large size showing bulk-like properties.10 However, in many cases, the exact form of the interaction potential is not known for complex heteroclusters. Therefore, the calculation of the solvation energy for finite- as well as infinite-size (bulk) clusters becomes problematic. Although ab initio quantum-mechanicsbased approaches bypass this problem, investigations based on this approach are limited only to small-size clusters due to enormous computational cost.11,12 More importantly, simulation or quantum-mechanics-based study does not provide any explicit analytical expression for the dependence of solvation energy on the number of solvent molecules. Recently, with the advent of supersonic expansions and nozzle beam techniques, solvation in finite size heteroclusters has been amenable to experimental observation,13-19 and a renewed interest has grown in connecting finite-size cluster properties to those of the bulk (infinite-size cluster) in a theoretical arena. Work in this direction was earlier initiated by Jortner and co-workers20,21 and subsequently by Coe et al.22

S

r 2010 American Chemical Society

Theoretical considerations of the ion solvation in finite-size clusters predict that the detachment energies of a negatively charged solvated ion should scale as 1/R linearly, where R is the radius of the solvated clusters.21 In this model, the solventnumber-dependent detachment energy (both ΔEVDE(n) and ΔEADE(n); VDE and ADE refer to vertical and adiabatic detachment energies, respectively) of the cluster is expressed as13 ΔEν ðnÞ ¼ ΔEν ð¥Þ þ Aν ðnþδÞ -1=3

ð1Þ

where ν refers to ADE or VDE, ΔEν(¥) is the bulk detachment energy, and Aν is the proportionality constant. Here, δ is an equivalent solvent number corresponding to the solute and accounts for the contribution of the solute to the cluster volume. However, the extrapolation results obtained from this simple model are found to be in poor agreement with the experimental results.13,23 Dixon and co-workers23 raised an important issue concerning whether the exponent (1/3) is constant for all negatively charged clusters of different charges or not and argued that the exponent should be dependent on the nature of the ion. Accordingly, they proposed another empirical law, written as ΔEν(n) = ΔEν(¥) þ Aν(n þ δ)-p. The calculated results based on this model were better than the earlier models, but the error was still quite significant, particularly for doubly negatively charged hydrated clusters. Most importantly, the variation of the empirical parameter p from system to system was not properly defined. The possible breakdown of these laws may be due to the fact that all of these empirical laws are Received Date: January 17, 2010 Accepted Date: February 12, 2010 Published on Web Date: February 17, 2010

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DOI: 10.1021/jz100062r |J. Phys. Chem. Lett. 2010, 1, 886–890

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function gi(r,ω) of the infinite system and its finite system counterpart gi(r,ω,n). Such a relation may be derived by following the approach of Salacuse et al.24 by means of Taylor series expansion in powers of 1/n. Here, we extend the derivation from the nonpolar system to the case of an ion polar system. We consider a semigrand canonical ensemble25 where only the number of solvent molecules is allowed to fluctuate but the single negatively charged ion is kept fixed in each member of the ensemble. In this semigrand canonical ensemble, the two-particle distribution function can be expressed as ¥ X Peq ðnÞPðris , rd , ω, nÞ ð4Þ Pðris , rd , ωÞ ¼

proposed by including two or three unknown parameters in an arbitrary way which is based on the continuum theory wherein the microscopic details are not considered. However, if solute-solvent and solvent-solvent motions are strongly correlated, the continuum theory breaks down, and one needs to have a microscopic theory in order to understand the solvent number (n) dependence of ΔEVDE(n) and ΔEADE(n). There are many possibilities other than the empirical models mentioned above leading to a large number of formal asymptotic expressions based on Taylor expansion without invoking first-principles-based theory. Hoewever, the fundamental query is that, out of so many possible formal expressions, which one can be derived for any arbitrary interaction potential based on first-principles theory. In the first part of this Letter, we have answered this query by deriving an exact expression based on semigrand canonical ensemble based theory for the solvent number (n)-dependent detachment energy. The derived expression is exact in the limit where n goes to infinity and goes as 1/n, in contrast to the approximate dielectric-theory-based result or empirical-model-based results which go as n-1/3 or n-p. However, the results are available for the systems of small-size clusters; therefore, to obtain the bulk detachment energy from results of smallsize clusters, what is needed is an expression valid not only for large n but for small-size clusters as well. Therefore, in the second part of the Letter, we have modified the derived result and proposed a generalized Taylor series based on some of the boundary conditions valid for small-size clusters. The final expression of course converges to the expression derived based on first-principle theory in the large n limit. Thus, at present, our objective has been to address the important issue of obtaining the proper n dependence of the expressions. The quantities of our interest are the ΔEVDE(n) and ΔEADE(n) defined as ZZ ΔEVDE ðnÞ ¼ IVDE þ F ½U2 ðr, ωÞ -U1 ðr, ωÞg1 ðr, ω, nÞdrdω ¼ IVDE þ ΔESol VDE ðnÞ ZZ ΔEADE ðnÞ ¼ IADE þ F ½U2 ðr, ωÞg2 ðr, ω, nÞ

ð2Þ

-U1 ðr, ωÞg1 ðr, ω, nÞdrdω ¼ IADE þ ΔESol ADE ðnÞ

ð3Þ

n ¼0

where Peq(n) is the probability that in equilibrium the system contains n solvent molecules and a single ion and P(rsi ,rd,ω,n) is the pair distribution function for a system containing a single negatively charged ion and n solvent molecules. Here, rsi , rd, and ω, respectively, represent the position vector of the ith ion and the position vector and orientation of the solvent molecule. Now expanding P(rsi ,rd,ω,n) in the number P of particles around the average number of particles, nh = ¥ n=0 nPeq(n), we have 1 D2 Pðris , rd , ωÞ ¼ Pðris , rd , ω, nÞ þ ðn -nÞ2 2 Pðris , rd , ω, nÞ 2 Dn 3 1 D ð5Þ þ ðn -nÞ3 3 Pðris , rd , ω, nÞ þ ... 6 Dn where the averages of the fluctuation of the number of solvent particles are defined as ¥ X ðn -nÞm Peq ðnÞ ¼ ðn -nÞm ð6Þ n ¼0

In the homogeneous limit, one can write FF Pðris , rd , ωÞ ¼ s gi ðr, ωÞ Ω F F Pðris , rd , ω, nÞ ¼ s gi ðr, ω, nÞ Ω

where Fs, F, and Ω represent the ion density, solvent density, and angular volume, respectively, and r = rd - rsi . Using eq 7, eq 5 can be reduced to gi ðr, ωÞ ¼ gi ðr, ω, nÞ þ

where U1(r,ω) and U2(r,ω), respectively, represent the interaction potential energy between the ions with charge -q and -(q - 1) with the polar solvent molecules and gi(r,ω,n) (i = 1, 2) represents the respective static pair distribution function between the ion and solvent molecules for the system containing a finite number (n) of solvent molecules. Here, F represents the bulk density of the solvent molecules, and the symbol I represents the ionization potential of the ion Aq- in the gas Sol (n) and ΔESol phase. The quantities ΔEVDE ADE(n) in eqs 2 and 3 represent the solvation energy contribution to the detachment energy. In order to have an explicit expression for VDE and ADE of the system containing a finite number of solvent particles, what is needed is a relation between the pair distribution


ð7Þ

þ

1 ðn -nÞ2 F D2 ðFgi ðr, ω, nÞÞ 2 n nDF2

1 ðn -nÞ3 F2 D3 ðFgi ðr, ω, nÞÞ þ ... 6 n ðnÞ2 DF3

ð8Þ

One may relate the fluctuations of the solvent numbers to the compressibility defined along the et al.24 h lines of Salacuse i ðn -nÞ2 n

3

¼ Sð0Þ and ðn -nÞ ¼ Sð0Þ2 þ DSð0Þ DðβμÞ ¼ F, where n R¥ S(0) = 1 þ 4πF 0 r2dr[g(r) - 1], with g(r) = Æg(r,ω1,ω2)æω1,ω2 representing the angular average of the pair distribution function of the solvent molecules, β = 1/kT (k and T represent the Boltzmann constant and absolute temperature, respectively), and μ is the chemical potential of the solvent molecules. For as

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simplicity, relabeling by n and retaining the terms up to 1/n2, we obtain c1 ðr, ωÞ c2i ðr, ωÞ ð9Þ þ gi ðr, ω, nÞ ¼ gi ðr, ωÞ þ i n n2 where Ci1(r,ω) and Ci2(r,ω) are respectively defined as F D2 ðFgi ðr, ωÞÞ ð10Þ Ci1 ðr, ωÞ ¼ Sð0Þ 2 DF2 Ci2 ðr, ωÞ

" # Sð0Þ D2 D2 ¼ F 2 Sð0ÞF 2 ðFgi ðr, ωÞÞ 4 DF DF -

F2 D3 ðFgi ðr, ωÞÞ F 6 DF3

Now, substituting eq 9 into eqs 2 and 3, we obtain M1 M2 ΔEν ðnÞ ¼ ΔEν ð¥Þ þ υ þ 2υ n n

ð11Þ Figure 1. Plot of ΔΕVDE(n) versus (n)-1 for I- 3 nH2O systems. The results based on the new relation (eq 12) are shown by the solid line, and the solid squares (9) represent the experimental values taken from ref 18. The plot of ΔΕADE(n) versus (n þ σ)-1 (σ = 8) for SO42- 3 nH2O is shown in the inset. The results based on the new relation (eq 16) are shown by the solid line, and the open circle (O) represents the experimental values taken from ref 19.

ð12Þ

where ΔEυ ð¥Þ ¼ Iυ þ ΔESol υ ð¥Þ

ð13Þ

F D2 Sð0Þ 2 ΔEυ ð¥Þ 4 DF

ð14Þ

Mυ1 ¼ Mυ2

Here, we have assumed that σ is a positive number and greater than unity, making the above series convergent since 1/(n þ σ) is quite smaller than unity even for small-size clusters. In the large limit of n, the above expression (eq 16) should converge to the derived expression (eq 12). It implies that Aν= M1υ and Bν = M2υ. We define the parameter σ to be related as σ = qVR based on the assumption that as the solute and solvent size ratio, VR, or numerical value of the charge q becomes very large, only then do solvent molecules near the surface mainly contribute to the solvation energy. In this limit, that is, lim(q,VR) f ¥, ΔEν(n) f ΔEν(¥). The generalized eq 16 is the new result of our present work and will be used to find ΔEν(¥) by extrapolation. We will now present the results of numerical calculations carried out using the present formalism. The bulk detachment energy, ΔEν(¥), is to be evaluated by extrapolation, appearing in the two coefficients, namely, M1υ and M2υ, in terms of the gradient of density. In many cases, the exact form of the interaction potential is not known for complex heteroclusters. Therefore, the calculation of ΔEν(¥) as a function density becomes problematic, which forces us to treat the coefficients, namely, M1υ and M2υ, as unknown parameters. The procedure adopted is to find out the optimum values of M1υ and M2υ using experimental values of ΔEν(n) and the generalized eq 16, so that the calculated results are very close to the experimental values. Using these values of M1υ and M2υ, we then calculate ΔEν(n) based on eq 16 and plot it in the graph as function of 1/(n þ σ) along with experimental results. Upon extrapolation of ΔEν(n) to infinite-size clusters (n = ¥), the bulk detachment energy, namely, ΔEν(¥), is obtained. As illustrative examples, we consider singly negative and doubly negative ionic hydrated clusters, namely, I- 3 nH2O, Br- 3 nH2O, Cl- 3 nH2O, F- 3 nH2O, NO3- 3 nH2O, SO42- 3 nH2O, and C2O42- 3 nH2O. We calculate the volume ratio (VR) and find it to be 3, 2.5, 2, 1, 2.5, 4, and 4 for the systems, I- 3 nH2O, Br- 3 nH2O, Cl- 3 nH2O, F- 3 nH2O, NO3- 3 nH2O, SO42- 3 nH2O, and C2O42- 3 nH2O, respectively. We first consider the I- 3 nH2O cluster, for which experimental values of ΔEVDE(n) are available18 for n = 1-60. The

" # F2 D2 D2 F3 D3 ¼ Sð0Þ 2 FSð0Þ 2 ΔEυ ð¥Þ - F 3 ΔEυ ð¥Þ ð15Þ 4 DF DF 6 DF

The result (eq 12) is interesting in that it is fundamentally different from the usual n-1/3 (approximate dielectric-theorybased result, where molecular details are ignored), (n þ δ)-1/3 (empirical), and (n þ δ)-p (empirical) that are usually employed in this capacity. The expression (eq 12) is exact in the limit where n goes to infinity and goes as 1/n, in contrast to the approximate dielectric-theory-based result or empiricalmodel-based results which go as n-1/3 or n-p. The expression (eq 12) for the detachment energy has been derived for the finite-size system, and the errors in the calculated values of the detachment energy are expected to be more and more as the size of the cluster decreases. This trend is expected since Taylor series expansion is valid as long as 1/n is quite smaller than unity. However, the results are available for the systems of small-size clusters; again, the ionization potential I and solvation energy ΔESol ν (¥) are evaluated respectively from the expressions of ΔEν(n) and (ΔEν(n) - I) by taking the limit n f 0 in the respective expression. The error in the calculated values of these quantities ΔEν(n) and (ΔEν(n) - I) reaches a maximum in the limit n f 0 since in this limit, our derived expression for ΔEν(n) diverges. Therefore, we search for a generalized Taylor series for ΔEν(n), which is convergent even for small-size clusters, converges to the expression (eq 12) derived based on firstprinciples theory in the large n limit, and should be finite in the limit of n f 0. We therefore propose a new truncated Taylor series defined as ΔEν ðnÞ ¼ ΔEν ð¥Þ þ


Aν Bν þ n þ σ ðnþσÞ2

ð16Þ

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Table 1. Bulk Vertical Detachment ΔEVDE(¥) and Adiabatic Detachment ΔEADE(¥) Energies for I- 3 nH2O, Br- 3 nH2O, Cl- 3 nH2O, F- 3 nH2O, NO3- 3 nH2O, SO42- 3 nH2O, and C2O42- 3 nH2O Systemsa

systems

calculated ΔEADE(¥) or ΔEVDE(¥) in eV based on extrapolation using new derived relations (eq 16)b

calculated ΔEADE(¥) or ΔEVDE(¥) in eV based on extrapolation using eq 1,b

experimental ΔEADE(¥) or ΔEVDE(¥) in eVb

7.08 (4) 7.56 (7)

8.34c (13) 10.17c (25)

7.40e 8.15e

9.18 (3)

12.47 (40)

10.99 (4)

14.31 (35)

I- 3 nH2O Br- 3 nH2O Cl- 3 nH2O F- 3 nH2O

NO3- 3 nH2O

8.90e 10.6e

7.28 (2)

10.92 (46)

7.46f

SO42-.nH2O

8.07 (7)

12.3d (42)

8.65f

C2O42- 3 nH2O

6.83 (7)

11.5d (57)

7.32f

-

-

-

-

The experimental values of ΔEVDE(n) for I 3 nH2O, Br 3 nH2O, Cl 3 nH2O, and F 3 nH2O systems are taken from refs 18 and 26. The experimental values of ΔEADE(n) for NO3- 3 nH2O, SO42- 3 nH2O, and C2O42- 3 nH2O systems are taken from refs 27, 13, and 19, respectively. b ΔEVDE(¥) values for I- 3 nH2O, Br- 3 nH2O, Cl- 3 nH2O, and F-.nH2O systems and ΔEADE(¥) values for NO3- 3 nH2O, SO42- 3 nH2O, and C2O42- 3 nH2O systems. Bold values in the parentheses refer to the % of error with respect to the experimental values. c Taken from ref 23 (calculated based on eq 1). d Taken from ref 13 (calculated based on eq 1). e Taken from ref 28. f Taken from ref 29. a

calculated results of ΔEVDE(n) are plotted against 1/(n þ σ) for these systems along with experimental results in Figure 1. The best-fitted plot (see Figure 1) has a correlation coefficient (R) of 0.98, showing an excellent agreement of the calculated values with the experimental results. The calculated bulk detachment energies ΔEVDE(¥) are shown in Table 1 along with the experimentally measured value. It is clear from Table 1 that the results of ΔEVDE(¥) obtained from the expression derived here based on microscopic theory are in very good agreement with experimental results in comparison to the calculated results obtained from the empirical law given by eq 1. We then consider hydrated clusters,18,26,27 namely, Br- 3 nH2O (n = 1-15), Cl- 3 nH2O (n = 1-6), F- 3 nH2O (n = 1-4), and NO3- 3 nH2O (n = 1-5). The calculated values of ΔEVDE(¥) along with the same using empirical relation given by eq 1 for these three systems are shown in Table 1 along with experimental values, and we find that the calculated results of ΔEVDE(¥) are also in much better agreement with experimental values than the same calculated based on empirical eq 1 (see Table 1). We now consider nonspherical doubly negatively charged ionic clusters, namely, SO42- 3 nH2O and C2O42- 3 nH2O. The size of the hydrated clusters for these systems for which experimental values of ΔEADE(n) are available cover the range from n = 4 to 40.13,19 Here also, the calculated results of ΔEADE(¥) based on generalized eq 16 reported in Table 1 are found to show much better agreement than the same calculated results based on eq 1. The calculated results for the system SO42- 3 nH2O are plotted along with available experimental results in Figure 1 (inset). The ΔEADE(¥) value is also calculated for the C2O42- 3 nH2O system and shown in Table 1. The correlation coefficient (R) for all of the plots (Br- 3 nH2O, Cl- 3 nH2O, F- 3 nH2O, NO3- 3 nH2O, SO42- 3 nH2O, and C2O42- 3 nH2O) is greater than 0.99, which indicates that there is very good correspondence between the present theory and experiment. The major new aspect of the present work may be highlighted as follows. A new general relation for the size-dependent detachment energy ΔEν(n) of finite-size solvated negatively charged clusters is derived based on a microscopic theory with an unknown interaction potential. The derived expression is exact in the limit where n goes to infinity and


goes as 1/n, in contrast to the approximate dielectric-theorybased result which goes as 1/(n1/3). The result is interesting in that it is fundamentally different from the usual approximate 1/(n1/3) or 1/(n þ δ)1/3 relationships that are usually employed for this purpose. We have shown that the new extrapolation formula, when fitted to finite-size (n) cluster data, yields significantly more accurate results for bulk detachment energies of simple spherical ions (I- 3 nH2O, Br- 3 nH2O, Cl- 3 nH2O, and F- 3 nH2O) to complex nonspherical ions with singly and multiple charges (NO3- 3 nH2O, SO42- 3 nH2O, and C2O42- 3 nH2O) than the results obtained based on usual dielectric theory (eq 1). The correlation coefficient (R in this Letter) for all of the plots is greater than 0.99. The n-dependent expression for ΔEν(n) does not change from system to system, in contrast to a earlier model,23 where the exponent p varied from system to system. Our present study reveals (see Table 1) that the maximum error on the calculated bulk values ΔEν(¥) as compared to that for the experimentally measured ones is 7%, in contrast to the 57% error for the previous model. More importantly, the robust scheme proposed here provides a route to obtain ΔEVDE(¥) and ΔEADE(¥) from the knowledge of ΔEVDE(n) and ΔEADE(n) for finite and complex systems whose interparticle potentials are unknown. We have not obtained any inconsistent results of the other quantities, namely, ΔESol ν (¥) and I tested for different kinds of cluster based on our expression, in contrast to the approximate dielectric-theory-based theory. Note that this method should be equally valid for solvated clusters with any solvent molecules. At present, results are compared only with those systems for which experimental bulk detachment energy values are available.

SUPPORTING INFORMATION AVAILABLE A detailed derivation of the detachment energy based on microscopic theory is provided. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed. E-mail: dkmaity@ barc.gov.in (D.K.M.); [email protected] (A.K.S.).

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ACKNOWLEDGMENT It is a pleasure to thank Dr. N. Choudhury

(21)

for many helpful discussions. (22)

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Generalized Microscopic Theory for the Detachment Energy of

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