- Email: [email protected]
Calculation of sequential hydrogen atom binding energies on a model lithium cluster
Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17 www.elsevier.com/locate/theochem
Calculation of sequential hydrogen atom binding energies on a model lithium cluster Paul G. Jasien * and Ryan Cross Department of Chemistry and Biochemistry, California State University San Marcos, 333 S. Twin Oaks Valley Road, San Marcos, CA 92096-0001, USA Received 22 July 2005; revised 27 August 2005; accepted 29 August 2005 Available online 2 November 2005
Abstract Density functional calculations have been performed to investigate the sequential hydrogen binding energies on a model Li18 cluster (Li18H2nC H2(Li18H2nC2, nZ0–8). Data was generated for a sample of six different isomers for each Li18H2n system. The results seem to indicate that there is a slight increase in the incremental binding energy as the cluster becomes more hydrogenated. This has been attributed to the restructuring of the cluster geometry to form regions of a regular lattice as the number of bound H atoms increase. Natural population analyses indicate that the total charge transfer to each hydrogen atom is essentially constant at approximately 0.75 e. This value seems to be independent of the number of hydrogen atoms already on the cluster or the site of hydrogen atom binding. These results are consistent with previous analyses of charge transfer in lithium hydrides. q 2005 Elsevier B.V. All rights reserved. Keywords: Lithium hydride; Lithium clusters; DFT; Binding energy
1. Introduction Since lithium is the lightest metallic element in the periodic table, investigations of its bonding have played a critical role in the development of quantum chemistry. One such role has been as a prototype to study metallic systems. The work of Koutecky et al. [1] and Gardet et al. [2] are just two of the many works that have studied moderate-sized lithium clusters with high quality quantum mechanical methods. In addition to being a model for metallic systems, binary compounds of lithium and non-metals have provided information on ionic bonding. Although, interest in the diatomic molecule lithium hydride has always been high as a benchmark for quantum mechanical methods, there is a renewed interest in polyatomic compounds of lithium and hydrogen for a number of reasons. The lithium hydride cluster provides a benchmark for reactions of hydrogen on metal surfaces. In addition, small lithium hydride clusters provide systems in which to compare the results of experimental and computational studies. Some of these recent studies have focused on the electronic absorption characteristics of nonstoichiometric lithium hydride systems [3,4]. * Corresponding author E-mail address: [email protected] (P.G. Jasien).
0166-1280/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2005.08.034
A number of purely quantum mechanical studies of small lithium hydride systems have been performed to understand the degree of ionicity in these systems and also for testing various models of ionic bonding [5–7]. Bertolus et al. [5] have investigated the bonding in some neutral LinHn clusters up to nZ7, as well as some cationic clusters, including Li14 HC 13 , to try to quantify the extent of charge transfer between the lithium and hydrogen atoms. Their conclusion was that the bonding in lithium hydride is that of a typical ionic compound. They were able to use this idea to predict the bonding with a simple ionic model from which they get reasonable structures and ‘not unreasonable’ energies. Fuentealba and Savin [7] reported on a electron localization function (ELF) study of various Li4hydrides and Li9H from which they concluded that each H atom added to the cluster localizes one of the metallic valence electrons of the cluster. This result is consistent with experimental ionization studies of Vezin et al. [8]. An older, but still relevant theoretical study on charge transfer in lithum hydrides is that of Rao and Jena [6]. In their work, they compared the quantum mechanically calculated electron density maps with those generated using completely ionic and covalent models. They concluded that although lithium hydrides are ionic, they are not completely so. Recently, research on the bonding of hydrogen with lithium has been re-energized with the search for materials that are efficient hydrogen storage devices. Such devices would play a crucial role in any transition to a hydrogen-based economy.
12
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
There have been a number of reports that have shown that there is promise for efficient hydrogen storage in lithium-based metal systems [9–12]. In general, the bonding of hydrogen to systems containing only lithium presents a problem due to the strong interaction of hydrogen with lithium metal. However, systems that use a mixed lithium-X system, where XZN, Al, B, etc. have shown more promise as a model storage system. As a first step in the study of the binding of hydrogen to lithium and mixed-metal systems, the current work investigates the sequential bonding of hydrogen on a model Li18 cluster. Instead of studying a few sparsely hydrogenated clusters, the current investigation focuses on the trends in binding energy and charge transfer on a single lithium cluster. As such, it seeks to shed some light on the question of cooperative effects in the sequential binding of hydrogen on lithium clusters. 2. Computational methods All calculations were performed using the Spartan ‘02 for Windows [13] program. The B3LYP parameterization of the electron correlation functional with non-local corrections was used [14,15] in all calculations. This method was used in conjunction with the standard 6-31G** or more extended 6-311G** basis set. Full geometry optimizations were performed at the B3LYP/6-31G** level. Optimized structures were converged to the default values for the force (3! 10K4 a.u), bond distance (1.2!10K3 a.u), and energy change (1!10K6 a.u). In most cases, this resulted in convergence in ˚. nearest neighbor atomic distances to 0.001 A Minimum energy structures were characterized through the calculation of the vibrational frequencies. These frequencies were also used to calculate the zero point energy (ZPE) for each isomer. For the small Li18Hn clusters (nR10) the calculation of the second derivatives needed for the vibrational frequency calculations were performed analytically. For larger systems, the second derivatives needed were obtained by centraldifferencing of the analytically calculated forces. In almost all cases, the minimum energy structures had no imaginary frequencies, however, there were a few cases in which there was one small (!30i cmK1) imaginary frequency. This was due to a combination of numerical accuracy limitations in the finite difference procedure and the flat potential for some modes in the flexing of the Li18 skeleton. Single point energy evaluations were done at the B3LYP/6-311G**//B3LYP/6-31G** level to test the dependence of the calculated binding energies on basis set. No attempt was made to test the effect of even larger basis sets on the calculated structures. In order to avoid possible artifacts due to differences in wavefunction type from one hydrogenated cluster to another, only systems with even numbers of hydrogen atoms were studied in this work. Therefore, all systems studied possessed a singlet electronic state and calculations used a restricted orbital wavefunction. In this way, there were no artificially introduced energy differences due to comparing results mixing restricted and unrestricted orbital wavefunctions. Even on a system as small as Li18, there are many possible surface sites for interaction of hydrogen atoms. As such, any
choice of a particular site for attachment will lead to a different calculated bonding energy than another such site. To investigate this effect, a series of calculations for hydrogen atom bonding on the cluster were performed to collect a statistical sample of the binding energies for each isomer. For each Li18H2n system (nZ1–9), a total of six different isomers (a–e, g) were studied. Starting with a Li18H2n cluster (nZ0–7), Li atom sites for the next pair of hydrogen atoms to bind were randomly selected. If an atom chosen was already bonded to one or more hydrogen atoms, another Li center was chosen at random until a suitable unoccupied site was found. The two hydrogen atoms were then placed in proximity to the chosen Li atom and the energy minimization carried out. In some cases, when obvious energetically unfavorable close contacts with other atoms would result, the positions of the added hydrogen atoms were slightly adjusted. The only case when the site of hydrogen binding was not chosen randomly occurred for the fully hydrogenated Li18H18 cluster. In this case, the number of surface positions on the precursor Li18H16 cluster was very limited, so the initial binding site for the hydrogen atoms was chosen to minimize steric interactions. All statistical analyses were performed using the SPSS 11 software package for Macintosh computers [16]. The criterion used for statistical significance was aZ0.05. This value translates into a degree of confidence in the result of 95%. Mean binding energies, atom–atom distances, and numbers of close contacts were compared using a one way analysis of variance (ANOVA) or an independent group t-test [17]. For these tests, only the p-value will be reported. A statistically significant difference will be inferred when p!0.05, i.e. a 95% degree of confidence. 3. Results and discussion 3.1. Lithium cluster The reference Li18 cluster was chosen from previous work that surveyed the structures of a number of Lix clusters and the activation energy for H2 bonding on a particular site [18].
Fig. 1. Structure of base Li18 cluster used in this work.
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
13
Table 1 Cumulative binding energies for Li18 (kcal/mol) Isomer a b c d e g average SD
2H K38.0 K37.9 K40.9 K40.8 K41.1 K40.4 K39.9 1.5
4H K75.1 K75.8 K75.7 K82.8 K79.6 K73.0 K77.0 3.5
6H
8H
K102.5 K117.2 K115.3 K118.5 K119.6 K109.1 K113.7 6.6
K158.3 K151.3 K164.2 K163.8 K155.0 K163.0 K159.3 5.3
10H K199.9 K204.9 K215.5 K211.4 K200.3 K206.1 K206.4 6.2
12H K261.8 K240.5 K257.5 K260.5 K260.1 K254.4 K255.8 8.0
14H K312.2 K300.7 K304.0 K305.9 K310.4 K297.1 K305.0 5.7
16H K351.9 K358.2 K353.8 K356.9 K362.3 K361.1 K357.4 4.0
18H K393.9 K399.3 K409.6 K400.6 K410.3 K414.1 K404.6 7.8
B3LYP/6-311G**//B3LYP/6-31G** binding energies corrected for ZPE.
A schematic representation of this cluster is shown in Fig. 1. A cluster of this size was large enough to provide a small measure of bulk surface behavior without being intractable for the computational resources at hand. Although the 18 atom cluster chosen for this study may not necessarily be the most stable cluster on the potential energy surface, it does represent a relatively tightly packed collection of Li atoms. However, it is not as tightly packed as the high symmetry structures studied by Gardet et al. [2]. One reason for choosing the current cluster for this study was its lack of high symmetry, which yields a larger number of different binding sites. This is appropriate since, this work seeks to investigate a sampling of lithium sites at which hydrogen can bind. The fact that a number of different reactive sites are used in this study mitigates somewhat the question of generality of the results presented. The calculated adiabatic ionization potential (IP) for the Li18 cluster used in this work is 3.8 eV compared with that from the work of Gardet et al. [2] of 4.06 eV and from experiment 3.31 eV [19]. The value of the work function for Li metal is 2.9 eV [20]. The atomization energy of the cluster used here is calculated to be 22.8 kcal/mol (22.0 kcal/mol with ZPE correction), compared to the values in the range 25.6– 26.3 kcal/mol for three different eighteen atom clusters studied by Gardet et al. [2]. The larger value obtained in that work is partly due to a slightly more stable cluster, but may also be attributable to differences in the density functional and atomic basis set used. A value of 23.6 kcal/mol for a Li18 cluster has been derived from experimental measurements on gas phase clusters [21] and is in line with that found in this work. In all cases, the calculated values cannot be compared with those of the bulk, since bulk properties converge slowly with cluster size [22,23]. This slow convergence is not unexpected, given
that a high percentage of the atoms found in clusters are at or near the surface and there are no true ‘bulk’ atoms. 3.2. Hydrogen binding energies Given in Table 1 are the cumulative binding energies (i.e. Li18CnH2(Li18H2n, nZ1–9) calculated at the B3LYP/6311G**//B3LYP/6-31G** level of theory. Also given in this table are the average binding energy and standard deviation for each Li18H2n system. The reported energies have been corrected for ZPE differences between the reactants and products. As expected, the total binding energy increases as the number of hydrogen atoms increases. Due to the difference in binding sites, there is not a particular bias as to which system is the most stable at any particular degree of hydrogenation. The relative ordering of isomers by stability changes for the various numbers of hydrogen atoms added. Overall, there does not seem to be a large range for the binding energies for the calculated binding energies. The largest absolute range of values is seen for the 12H and 18H systems with total binding energy ranges of 21 and 20 kcal/mol, respectively. These amount to variations of only about 8 and 5% of the total bonding energies. Smaller absolute ranges, but larger percentage ranges are seen for the 4H and 6H systems. The variations here amount to about 13 and 15% for 4H and 6H, respectively. Of greater interest in this work is the trend in incremental binding energies on going from one system to another. This reflects the energy change for the reaction Li18H2nC H2Li18H2nC2, nZ0–8. These values, along with the average and standard deviation, are given in Table 2 for each of the systems studied. From the data, it can be seen that there can be a great deal of fluctuation in incremental binding energy from
Table 2 Sequential binding energies for Li18 (kcal/mol) Isomer a b c d e g average SD
2H K38.0 K37.9 K40.9 K40.8 K41.1 K40.4 K39.9 1.5
4H K37.1 K37.9 K34.8 K42.0 K38.6 K32.7 K37.2 3.2
6H K27.4 K41.4 K39.6 K35.8 K39.9 K36.1 K36.7 5.1
8H K55.7 K34.1 K48.8 K45.2 K35.4 K53.9 K45.5 9.1
B3LYP/6-311G**//B3LYP/6-31G** binding energies corrected for ZPE.
10H K41.6 K53.6 K51.4 K47.6 K45.3 K43.1 K47.1 4.7
12H K61.9 K35.5 K42.0 K49.0 K59.8 K48.3 K49.4 10.1
14H K50.4 K60.2 K46.5 K45.4 K50.2 K42.7 K49.2 6.1
16H K39.7 K57.5 K49.8 K51.0 K51.9 K64.0 K52.3 8.1
18H K42.0 K41.1 K55.7 K43.7 K48.0 K53.0 K47.3 6.1
14
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
step to step. For instance, looking at isomer ‘a’, the sequential binding energy goes from being the lowest for 6H to being the highest for 8H. These changes are a function of the particular site at which the hydrogen atoms are bound, as well as any incremental structural changes in the overall cluster induced by this bonding. Due to the different hydrogen binding sequences the final Li18H18 clusters do not converge to the same final structure. Some examples of these different structures will be given later. Given the fluctuations in incremental binding energy, it is not possible by simple observation of the results to find a significant trend. Although it appears as if there is an overall increase in the size of the binding energy as the number of hydrogen atoms increases, this is not certain. In this case, it was deemed necessary to use more rigorous statistical tests to quantify differences in the binding energy data. Results from ANOVA tests on the various incremental binding energies reveal that there is a statistically significant difference (p!0.001) in this quantity among the groups. Closer inspection reveals that this difference is significant only between the sub-group containing 2, 4, and 6 hydrogen atoms and the sub-group containing 8–18 hydrogen atoms. Although it may appear to the eye that there is a general increase in binding energy for the sub-group containing 8–18 hydrogen atoms, this is not deemed to be statistically significant (pZ0.71). This is illustrated by the plot given in Fig. 2, which shows the mean and 95% confidence interval for the incremental binding energy. Despite the large ranges, this data clearly shows a significant difference in binding energies as you proceed from 2–6 to 8–18 hydrogen atom clusters. This plot also qualitatively illustrates the lack of difference in the 8–18 hydrogen atom group. 3.3. Basis set dependence In the previous section, only the results for the B3LYP/6311G**//B3LYP/6-31G** calculations were given. Results from the B3LYP/6-31G** calculations did not differ
Incremental Binding Energy (kcal/mol)
–20
–30
appreciably from those with the larger basis set. For clusters with 2–16 hydrogen atoms, the larger basis set calculations consistently gave slightly larger binding energies. The rms BE increase ranged from 1.9 to 2.5 kcal/mol for these systems. Only in the case of the clusters with 18 hydrogen atoms were there actually any cases for which the binding energy slightly decreased with the larger basis set. For Li18H18 clusters, the largest overall change was 2.4 kcal/mol. Trends for the 6-31G** basis set calculations gave essentially the same statistical results as reported in the previous section. 3.4. Structures The structures of the fully hydrogenated Li18H18 clusters are shown in Fig. 3. This figure clearly shows the large difference in the final structures that are possible for these isomers. Given the large number of possible structural parameters in all the systems studied, the geometric analysis will be limited to a summary of a few relevant features. Given in Table 3 is the average number of Li–H contacts per H atom for the various cluster sizes. These values give some indication of the overall degree of close packing in these systems. Although various definitions of close contact have been investigated in this work, the data reported will define a contact as either an Li–H ˚ . Since it is difficult to define an ionic distance R2.3 or R1.9 A ˚ was chosen to be an radius experimentally, the value of 2.3 A approximate value close to the sum of the Goldschmidt or Ladd ionic radii [24]. Coincidentally, this is also close to the sum of ˚ was chosen to the covalent radii of Li and H. The value of 1.9 A illustrate the sensitivity of the data to the cutoff. The last value listed in Table 3 is the average Li–H distance calculated over all Li, H pairs. ˚ , the number of Based on the close contact definition of 2.3 A Li–H contacts per H is in the range 3.2 – 3.5 per H. When all the data are analyzed together, there is not a significant difference (pZ0.55) in this value as a function of the number of ˚ , the hydrogen atoms. For the shorter cutoff distance of 1.9 A number of close contacts falls in the range 2.4–2.8 per H. Once again ANOVA indicates no significant differences (pZ0.25). Further analysis did not seem to show any obvious subgroupings, as was the case for the incremental binding energy data. Table 3 ˚) Average Li–H close contacts per H and average Li–H distance (A
–40
Cluster
Ave. close contacts per ˚) H (2.3 A
Ave. close contacts per ˚) H (1.9 A
Ave. Li–H distance ˚) (A
2H 4H 6H 8H 10H 12H 14H 16H 18H
3.42 (0.20) 3.29 (0.19) 3.22 (0.08) 3.27 (0.20) 3.37 (0.22) 3.39 (0.07) 3.36 (0.11) 3.39 (0.16) 3.48 (0.20)
2.83 (0.75) 2.54 (0.29) 2.67 (0.35) 2.69 (0.23) 2.69 (0.26) 2.57 (0.26) 2.71 (0.14) 2.59 (0.24) 2.36 (0.25)
4.42 (0.07) 4.58 (0.16) 4.64 (0.19) 4.57 (0.15) 4.51 (0.17) 4.48 (0.12) 4.49 (0.08) 4.45 (0.11) 4.34 (0.11)
–50
–60
–70 2
4
6
8
10
12
14
16
18
Number of H Atoms Fig. 2. Plot of incremental binding energy vs. number of H atoms.
SD in parentheses.
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
15
Fig. 3. Structures of Li18H18 final structures. Top row: isomers a, b, and c. Bottom row: isomers d, e, and g. H atoms are light and Li atoms are dark colored.
show some noticeable semblance of the cubic structure in small regions. The more closely packed structure of isomer ‘c’ is further illustrated by the fact that the average Li–H distance is ˚ for ‘c’ while this quantity is 4.36, 4.49, 4.42, 4.35, and 4.16 A ˚ for structures a, b, d, e, and g, respectively. 4.30 A Interestingly, there does not seem to be a strong correlation between binding energy and this parameter (Table 1). 3.5. Partial charge analysis Previous studies by Fuentealba and Savin [7] and Rao and Jena [6] investigated the charge distribution in lithium hydride clusters. In this work, the natural bond charges as defined by Reed, Curtis, and Weinhold [25] have been used for the analysis of the lithium hydride clusters. Although the absolute value of the charge in a cluster of this type is not experimentally defined, the trends in charge as the cluster 4.9 4.8
Average Li-H Distance (Å)
Analysis of the data via ANOVA does seem to indicate a statistically significant difference (pZ0.02) for the overall Li– H distance (averaged over all Li, H pairs) in the clusters. However, there is not a straightforward relationship between average distance and cluster size. This can be seen in Fig. 4, which is a plot of the 95% confidence intervals for the average Li–H distances in the various clusters. The plot shows a slight increase and then a decrease in the average Li–H distances as the cluster becomes more hydrogenated. Since, the 2H and 18H clusters seem to deviate most from the average, further statistical testing was done excluding these individually. The results showed that removal of the 2H data still showed significant differences in the average Li–H distances (pZ0.04). However, removal of the 18H data showed no statistically significant differences in the 2H–16H data for the average Li–H distances (pZ0.13). Further testing of the total binding energy vs. various geometric parameters did not seem to reveal any significant correlations. Given the lack of correlation of calculated binding energy or structural parameters with number of hydrogen atoms seems to indicate that each addition of two hydrogen atoms seems to be independent of the other additions. The only evidence contrary to this is the apparent difference in incremental binding energies for clusters with 2–6 and those with 8–18 hydrogen atoms. Visual inspection of Fig. 3 shows an example of the geometric differences between the fully hydrogenated clusters. These structures show that even in clusters as small as these, different degrees of the bulk crystal structure already begin to appear. (Lithium hydride crystallizes in the same bodycentered cubic lattice as sodium chloride). Structure ‘c’ shows an appreciable region where this bulk structure occurs. There are 3!3 and 3!4 ion assemblies on the front and top face, respectively. Such a high degree of organization is not seen in any of the other structures. However, the other clusters
4.7 4.6 4.5 4.4 4.3 4.2 4.1 2
4
6
8
10
12
14
16
18
Number of H Atoms Fig. 4. Plot of average Li–H distance vs. number of H atoms.
16
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
Table 4 Total natural charge on hydrogen atoms Isomer
2H
4H
6H
8H
10H
12H
14H
16H
18H
a b c d e g
1.57 1.54 1.58 1.52 1.54 1.56
3.10 3.04 3.06 3.05 3.08 3.11
4.56 4.60 4.51 4.54 4.63 4.55
6.07 6.08 6.02 6.07 6.05 6.13
7.59 7.72 7.52 7.54 7.51 7.60
8.90 9.14 9.05 8.96 9.02 9.08
10.43 10.54 10.49 10.43 10.48 10.49
11.84 11.95 11.98 11.77 11.86 11.89
13.41 13.34 13.42 13.33 13.36 13.33
Results from B3LYP/6-31G** calculations.
size changes are meaningful and will be examined. The following analysis will use the total charge on hydrogen atoms as a measure of overall charge transfer, as opposed to trying to analyze individual atom charges. Given that the hydrogen atoms are found in various bonding sites, it is expected that the individual charge will depend on the nearness and number of Li atoms. Table 4 presents the total charge on hydrogen atoms for each of the clusters studied. From this data it can be seen that the total H atom charges differ insignificantly from one isomer to another. Closer examination of the raw data showed that the maximum charge of a hydrogen atom in any cluster was K0.84 e and the minimum was K0.69 e. Those hydrogen atoms having a charge over K0.8 e generally had more (i.e. 4–5) nearest neighbor Li atoms from which to withdraw charge. Despite some variation in the individual H atom charges in the clusters, the total charge transfer as a function of the number of hydrogen atoms was remarkably linear for all clusters. In fact, linear fits of total charge vs. number of H atoms (with the y-intercept constrained to be zero) gave a value of R2 of greater than 0.999 for each of the six clusters. The average slope of all six lines was found to be K0.748(0.003) e. This seems to indicate that independent of the particular site of H atom bonding or the shape of the cluster, there is a transfer of about 0.75 e for each H atom. Once again, it should be noted that this value of 0.75 e is not exactly quantifiable, however, the fact that this value is constant across clusters is. As was concluded in the work of Bertolus et al. [5], the lithium hydride clusters are considerably ionic in nature. In addition, the current results are in qualitative agreement with the work of Fuentealba and Savin [7] on smaller lithium hydride clusters in which they concluded that there was a transfer of one electron per H atom added to the metal cluster. The idea that there is a constant transfer of electrons independent of the cluster size or number of hydrogen atoms is also consistent with the older work of Rao and Jena [6] in analyzing the smaller systems of Li9H9 and Li16H. Given that the average charge on hydrogen atoms is essentially constant. It may be possible to rationalize the slight increase in incremental binding energy purely on a simple electrostatic argument. As the clusters become more hydrogenated, they have a greater tendency to form small regions of a regular lattice with alternating Li and H ions. These regions allow for more favorable contacts between positive and negatively charged atoms. This may account for the slight increase in incremental bonding energy with increasing
hydrogenation, which, although not excessively large, is statistically significant. 4. Conclusion Calculations of the sequential binding energy of hydrogen atoms on a model Li18 cluster seem to indicate that there is a slight increase in incremental bonding energy as the cluster becomes more hydrogenated. This slight increase has been attributed to the restructuring of the cluster geometry to form regions of a regular lattice as the number of bound H atoms increases. Natural population analysis indicates that the total charge transfer to each hydrogen atom is essentially constant at 0.75 e. This seems to be independent of the number of hydrogen atoms already on the cluster or the site of hydrogen atom binding. These results are consistent with previous analyses of charge transfer in lithium hydrides [6,7]. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.theochem.2005. 08.034.
References [1] J. Koutecky, V. Bonacic-Koutecky, I. Boustani, P. Fantucci, W. Pewestorf, in: J. Jortner, A. Pullman, B. Pullman (Eds.), Large Finite Systems, Reidel, Dordrecht, 1987 p. 303. [2] G. Gardet, F. Rogemond, H. Chermette, J. Chem. Phys. 105 (1996) 9933. [3] B. Vezin, Ph. Dugourd, C. Bordas, D. Rayane, M. Broyer, V. BonacicKoutecky, J. Pittner, C. Fuchs, J. Gaus, J. Koutecky, J. Chem. Phys. 102 (1995) 2727. [4] V. Bonacic-Koutecky, J. Pittner, Chem. Phys. 225 (1997) 173. [5] M. Bertolus, V. Brenner, P. Millie, J. Chem. Phys. 115 (2001) 4070. [6] B.K. Rao, P. Jena, J. Phys. C: Solid State Phys. 19 (1986) 5167. [7] P. Fuentealba, A. Savin, J. Phys. Chem. A 105 (2001) 11531. [8] B. Vezin, Ph. Dugourd, D. Rayane, P. Labastie, J. Chevaleyre, M. Broyer, Chem. Phys. Lett. 206 (1993) 521. [9] J. Chen, N. Kuriyama, Q. Xu, H.T. Takeshita, T. Sakai, J. Phys. Chem. B 105 (2001) 11214. [10] P. Chen, Z. Xiong, J. Luo, J. Lin, K.L. Tan, Nature 420 (2002) 302. [11] Y.H. Hu, E. Ruckenstein, Ind. Eng. Chem. Res. 43 (2004) 2464. [12] F.E. Pinkerton, G.P. Meisner, M.S. Meyer, M.P. Balogh, M.D. Kundrat, J. Phys. Chem. B 109 (2005) 6. [13] Spartan ’02 for Windows, Wavefunction, Inc., Irvine, CA. [14] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [15] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [16] SPSS 11for Macintosh Computers, SPSS, Inc., Chicago, 2002.
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17 [17] M. Triola, Elementary Statistics, seventh ed., Addison-Wesley, Reading (MA), 1998. [18] R. Cross, Undergraduate Thesis, Department of Chemistry, California State University, San Marcos (2004). [19] Ph. Dugourd, D. Rayane, P. Lebastie, B. Vezin, J. Chevaleyre, M. Broyer, Chem. Phys. Lett. 197 (1992) 433. [20] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 74th ed., CRC Press, Boca Raton, 1993, p. 12–105.
17
[21] C. Brechignac, H. Busch, P. Cahuzac, J. Leygnier, J. Chem. Phys. 101 (1994) 6992. [22] G. Pacchioni, P. Fantucci, J. Mol. Catal. 12 (1981) 213. [23] E.L. Meutterties, T.N. Rhodin, E. Band, C.F. Brucker, W.R. Pretzer, Chem. Rev. 79 (1979) 91. [24] F.A. Cotton, G. Wilkinson, Advanced Inorganic Chemistry, third ed., Wiley, New York, NY, 1972, p. 52. [25] A.E. Reed, L.A. Curtiss, F.J. Weinhold, J. Chem. Phys. 83 (1985) 735.
Calculation of sequential hydrogen atom binding energies on a model lithium cluster Paul G. Jasien * and Ryan Cross Department of Chemistry and Biochemistry, California State University San Marcos, 333 S. Twin Oaks Valley Road, San Marcos, CA 92096-0001, USA Received 22 July 2005; revised 27 August 2005; accepted 29 August 2005 Available online 2 November 2005
Abstract Density functional calculations have been performed to investigate the sequential hydrogen binding energies on a model Li18 cluster (Li18H2nC H2(Li18H2nC2, nZ0–8). Data was generated for a sample of six different isomers for each Li18H2n system. The results seem to indicate that there is a slight increase in the incremental binding energy as the cluster becomes more hydrogenated. This has been attributed to the restructuring of the cluster geometry to form regions of a regular lattice as the number of bound H atoms increase. Natural population analyses indicate that the total charge transfer to each hydrogen atom is essentially constant at approximately 0.75 e. This value seems to be independent of the number of hydrogen atoms already on the cluster or the site of hydrogen atom binding. These results are consistent with previous analyses of charge transfer in lithium hydrides. q 2005 Elsevier B.V. All rights reserved. Keywords: Lithium hydride; Lithium clusters; DFT; Binding energy
1. Introduction Since lithium is the lightest metallic element in the periodic table, investigations of its bonding have played a critical role in the development of quantum chemistry. One such role has been as a prototype to study metallic systems. The work of Koutecky et al. [1] and Gardet et al. [2] are just two of the many works that have studied moderate-sized lithium clusters with high quality quantum mechanical methods. In addition to being a model for metallic systems, binary compounds of lithium and non-metals have provided information on ionic bonding. Although, interest in the diatomic molecule lithium hydride has always been high as a benchmark for quantum mechanical methods, there is a renewed interest in polyatomic compounds of lithium and hydrogen for a number of reasons. The lithium hydride cluster provides a benchmark for reactions of hydrogen on metal surfaces. In addition, small lithium hydride clusters provide systems in which to compare the results of experimental and computational studies. Some of these recent studies have focused on the electronic absorption characteristics of nonstoichiometric lithium hydride systems [3,4]. * Corresponding author E-mail address: [email protected] (P.G. Jasien).
0166-1280/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2005.08.034
A number of purely quantum mechanical studies of small lithium hydride systems have been performed to understand the degree of ionicity in these systems and also for testing various models of ionic bonding [5–7]. Bertolus et al. [5] have investigated the bonding in some neutral LinHn clusters up to nZ7, as well as some cationic clusters, including Li14 HC 13 , to try to quantify the extent of charge transfer between the lithium and hydrogen atoms. Their conclusion was that the bonding in lithium hydride is that of a typical ionic compound. They were able to use this idea to predict the bonding with a simple ionic model from which they get reasonable structures and ‘not unreasonable’ energies. Fuentealba and Savin [7] reported on a electron localization function (ELF) study of various Li4hydrides and Li9H from which they concluded that each H atom added to the cluster localizes one of the metallic valence electrons of the cluster. This result is consistent with experimental ionization studies of Vezin et al. [8]. An older, but still relevant theoretical study on charge transfer in lithum hydrides is that of Rao and Jena [6]. In their work, they compared the quantum mechanically calculated electron density maps with those generated using completely ionic and covalent models. They concluded that although lithium hydrides are ionic, they are not completely so. Recently, research on the bonding of hydrogen with lithium has been re-energized with the search for materials that are efficient hydrogen storage devices. Such devices would play a crucial role in any transition to a hydrogen-based economy.
12
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
There have been a number of reports that have shown that there is promise for efficient hydrogen storage in lithium-based metal systems [9–12]. In general, the bonding of hydrogen to systems containing only lithium presents a problem due to the strong interaction of hydrogen with lithium metal. However, systems that use a mixed lithium-X system, where XZN, Al, B, etc. have shown more promise as a model storage system. As a first step in the study of the binding of hydrogen to lithium and mixed-metal systems, the current work investigates the sequential bonding of hydrogen on a model Li18 cluster. Instead of studying a few sparsely hydrogenated clusters, the current investigation focuses on the trends in binding energy and charge transfer on a single lithium cluster. As such, it seeks to shed some light on the question of cooperative effects in the sequential binding of hydrogen on lithium clusters. 2. Computational methods All calculations were performed using the Spartan ‘02 for Windows [13] program. The B3LYP parameterization of the electron correlation functional with non-local corrections was used [14,15] in all calculations. This method was used in conjunction with the standard 6-31G** or more extended 6-311G** basis set. Full geometry optimizations were performed at the B3LYP/6-31G** level. Optimized structures were converged to the default values for the force (3! 10K4 a.u), bond distance (1.2!10K3 a.u), and energy change (1!10K6 a.u). In most cases, this resulted in convergence in ˚. nearest neighbor atomic distances to 0.001 A Minimum energy structures were characterized through the calculation of the vibrational frequencies. These frequencies were also used to calculate the zero point energy (ZPE) for each isomer. For the small Li18Hn clusters (nR10) the calculation of the second derivatives needed for the vibrational frequency calculations were performed analytically. For larger systems, the second derivatives needed were obtained by centraldifferencing of the analytically calculated forces. In almost all cases, the minimum energy structures had no imaginary frequencies, however, there were a few cases in which there was one small (!30i cmK1) imaginary frequency. This was due to a combination of numerical accuracy limitations in the finite difference procedure and the flat potential for some modes in the flexing of the Li18 skeleton. Single point energy evaluations were done at the B3LYP/6-311G**//B3LYP/6-31G** level to test the dependence of the calculated binding energies on basis set. No attempt was made to test the effect of even larger basis sets on the calculated structures. In order to avoid possible artifacts due to differences in wavefunction type from one hydrogenated cluster to another, only systems with even numbers of hydrogen atoms were studied in this work. Therefore, all systems studied possessed a singlet electronic state and calculations used a restricted orbital wavefunction. In this way, there were no artificially introduced energy differences due to comparing results mixing restricted and unrestricted orbital wavefunctions. Even on a system as small as Li18, there are many possible surface sites for interaction of hydrogen atoms. As such, any
choice of a particular site for attachment will lead to a different calculated bonding energy than another such site. To investigate this effect, a series of calculations for hydrogen atom bonding on the cluster were performed to collect a statistical sample of the binding energies for each isomer. For each Li18H2n system (nZ1–9), a total of six different isomers (a–e, g) were studied. Starting with a Li18H2n cluster (nZ0–7), Li atom sites for the next pair of hydrogen atoms to bind were randomly selected. If an atom chosen was already bonded to one or more hydrogen atoms, another Li center was chosen at random until a suitable unoccupied site was found. The two hydrogen atoms were then placed in proximity to the chosen Li atom and the energy minimization carried out. In some cases, when obvious energetically unfavorable close contacts with other atoms would result, the positions of the added hydrogen atoms were slightly adjusted. The only case when the site of hydrogen binding was not chosen randomly occurred for the fully hydrogenated Li18H18 cluster. In this case, the number of surface positions on the precursor Li18H16 cluster was very limited, so the initial binding site for the hydrogen atoms was chosen to minimize steric interactions. All statistical analyses were performed using the SPSS 11 software package for Macintosh computers [16]. The criterion used for statistical significance was aZ0.05. This value translates into a degree of confidence in the result of 95%. Mean binding energies, atom–atom distances, and numbers of close contacts were compared using a one way analysis of variance (ANOVA) or an independent group t-test [17]. For these tests, only the p-value will be reported. A statistically significant difference will be inferred when p!0.05, i.e. a 95% degree of confidence. 3. Results and discussion 3.1. Lithium cluster The reference Li18 cluster was chosen from previous work that surveyed the structures of a number of Lix clusters and the activation energy for H2 bonding on a particular site [18].
Fig. 1. Structure of base Li18 cluster used in this work.
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
13
Table 1 Cumulative binding energies for Li18 (kcal/mol) Isomer a b c d e g average SD
2H K38.0 K37.9 K40.9 K40.8 K41.1 K40.4 K39.9 1.5
4H K75.1 K75.8 K75.7 K82.8 K79.6 K73.0 K77.0 3.5
6H
8H
K102.5 K117.2 K115.3 K118.5 K119.6 K109.1 K113.7 6.6
K158.3 K151.3 K164.2 K163.8 K155.0 K163.0 K159.3 5.3
10H K199.9 K204.9 K215.5 K211.4 K200.3 K206.1 K206.4 6.2
12H K261.8 K240.5 K257.5 K260.5 K260.1 K254.4 K255.8 8.0
14H K312.2 K300.7 K304.0 K305.9 K310.4 K297.1 K305.0 5.7
16H K351.9 K358.2 K353.8 K356.9 K362.3 K361.1 K357.4 4.0
18H K393.9 K399.3 K409.6 K400.6 K410.3 K414.1 K404.6 7.8
B3LYP/6-311G**//B3LYP/6-31G** binding energies corrected for ZPE.
A schematic representation of this cluster is shown in Fig. 1. A cluster of this size was large enough to provide a small measure of bulk surface behavior without being intractable for the computational resources at hand. Although the 18 atom cluster chosen for this study may not necessarily be the most stable cluster on the potential energy surface, it does represent a relatively tightly packed collection of Li atoms. However, it is not as tightly packed as the high symmetry structures studied by Gardet et al. [2]. One reason for choosing the current cluster for this study was its lack of high symmetry, which yields a larger number of different binding sites. This is appropriate since, this work seeks to investigate a sampling of lithium sites at which hydrogen can bind. The fact that a number of different reactive sites are used in this study mitigates somewhat the question of generality of the results presented. The calculated adiabatic ionization potential (IP) for the Li18 cluster used in this work is 3.8 eV compared with that from the work of Gardet et al. [2] of 4.06 eV and from experiment 3.31 eV [19]. The value of the work function for Li metal is 2.9 eV [20]. The atomization energy of the cluster used here is calculated to be 22.8 kcal/mol (22.0 kcal/mol with ZPE correction), compared to the values in the range 25.6– 26.3 kcal/mol for three different eighteen atom clusters studied by Gardet et al. [2]. The larger value obtained in that work is partly due to a slightly more stable cluster, but may also be attributable to differences in the density functional and atomic basis set used. A value of 23.6 kcal/mol for a Li18 cluster has been derived from experimental measurements on gas phase clusters [21] and is in line with that found in this work. In all cases, the calculated values cannot be compared with those of the bulk, since bulk properties converge slowly with cluster size [22,23]. This slow convergence is not unexpected, given
that a high percentage of the atoms found in clusters are at or near the surface and there are no true ‘bulk’ atoms. 3.2. Hydrogen binding energies Given in Table 1 are the cumulative binding energies (i.e. Li18CnH2(Li18H2n, nZ1–9) calculated at the B3LYP/6311G**//B3LYP/6-31G** level of theory. Also given in this table are the average binding energy and standard deviation for each Li18H2n system. The reported energies have been corrected for ZPE differences between the reactants and products. As expected, the total binding energy increases as the number of hydrogen atoms increases. Due to the difference in binding sites, there is not a particular bias as to which system is the most stable at any particular degree of hydrogenation. The relative ordering of isomers by stability changes for the various numbers of hydrogen atoms added. Overall, there does not seem to be a large range for the binding energies for the calculated binding energies. The largest absolute range of values is seen for the 12H and 18H systems with total binding energy ranges of 21 and 20 kcal/mol, respectively. These amount to variations of only about 8 and 5% of the total bonding energies. Smaller absolute ranges, but larger percentage ranges are seen for the 4H and 6H systems. The variations here amount to about 13 and 15% for 4H and 6H, respectively. Of greater interest in this work is the trend in incremental binding energies on going from one system to another. This reflects the energy change for the reaction Li18H2nC H2Li18H2nC2, nZ0–8. These values, along with the average and standard deviation, are given in Table 2 for each of the systems studied. From the data, it can be seen that there can be a great deal of fluctuation in incremental binding energy from
Table 2 Sequential binding energies for Li18 (kcal/mol) Isomer a b c d e g average SD
2H K38.0 K37.9 K40.9 K40.8 K41.1 K40.4 K39.9 1.5
4H K37.1 K37.9 K34.8 K42.0 K38.6 K32.7 K37.2 3.2
6H K27.4 K41.4 K39.6 K35.8 K39.9 K36.1 K36.7 5.1
8H K55.7 K34.1 K48.8 K45.2 K35.4 K53.9 K45.5 9.1
B3LYP/6-311G**//B3LYP/6-31G** binding energies corrected for ZPE.
10H K41.6 K53.6 K51.4 K47.6 K45.3 K43.1 K47.1 4.7
12H K61.9 K35.5 K42.0 K49.0 K59.8 K48.3 K49.4 10.1
14H K50.4 K60.2 K46.5 K45.4 K50.2 K42.7 K49.2 6.1
16H K39.7 K57.5 K49.8 K51.0 K51.9 K64.0 K52.3 8.1
18H K42.0 K41.1 K55.7 K43.7 K48.0 K53.0 K47.3 6.1
14
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
step to step. For instance, looking at isomer ‘a’, the sequential binding energy goes from being the lowest for 6H to being the highest for 8H. These changes are a function of the particular site at which the hydrogen atoms are bound, as well as any incremental structural changes in the overall cluster induced by this bonding. Due to the different hydrogen binding sequences the final Li18H18 clusters do not converge to the same final structure. Some examples of these different structures will be given later. Given the fluctuations in incremental binding energy, it is not possible by simple observation of the results to find a significant trend. Although it appears as if there is an overall increase in the size of the binding energy as the number of hydrogen atoms increases, this is not certain. In this case, it was deemed necessary to use more rigorous statistical tests to quantify differences in the binding energy data. Results from ANOVA tests on the various incremental binding energies reveal that there is a statistically significant difference (p!0.001) in this quantity among the groups. Closer inspection reveals that this difference is significant only between the sub-group containing 2, 4, and 6 hydrogen atoms and the sub-group containing 8–18 hydrogen atoms. Although it may appear to the eye that there is a general increase in binding energy for the sub-group containing 8–18 hydrogen atoms, this is not deemed to be statistically significant (pZ0.71). This is illustrated by the plot given in Fig. 2, which shows the mean and 95% confidence interval for the incremental binding energy. Despite the large ranges, this data clearly shows a significant difference in binding energies as you proceed from 2–6 to 8–18 hydrogen atom clusters. This plot also qualitatively illustrates the lack of difference in the 8–18 hydrogen atom group. 3.3. Basis set dependence In the previous section, only the results for the B3LYP/6311G**//B3LYP/6-31G** calculations were given. Results from the B3LYP/6-31G** calculations did not differ
Incremental Binding Energy (kcal/mol)
–20
–30
appreciably from those with the larger basis set. For clusters with 2–16 hydrogen atoms, the larger basis set calculations consistently gave slightly larger binding energies. The rms BE increase ranged from 1.9 to 2.5 kcal/mol for these systems. Only in the case of the clusters with 18 hydrogen atoms were there actually any cases for which the binding energy slightly decreased with the larger basis set. For Li18H18 clusters, the largest overall change was 2.4 kcal/mol. Trends for the 6-31G** basis set calculations gave essentially the same statistical results as reported in the previous section. 3.4. Structures The structures of the fully hydrogenated Li18H18 clusters are shown in Fig. 3. This figure clearly shows the large difference in the final structures that are possible for these isomers. Given the large number of possible structural parameters in all the systems studied, the geometric analysis will be limited to a summary of a few relevant features. Given in Table 3 is the average number of Li–H contacts per H atom for the various cluster sizes. These values give some indication of the overall degree of close packing in these systems. Although various definitions of close contact have been investigated in this work, the data reported will define a contact as either an Li–H ˚ . Since it is difficult to define an ionic distance R2.3 or R1.9 A ˚ was chosen to be an radius experimentally, the value of 2.3 A approximate value close to the sum of the Goldschmidt or Ladd ionic radii [24]. Coincidentally, this is also close to the sum of ˚ was chosen to the covalent radii of Li and H. The value of 1.9 A illustrate the sensitivity of the data to the cutoff. The last value listed in Table 3 is the average Li–H distance calculated over all Li, H pairs. ˚ , the number of Based on the close contact definition of 2.3 A Li–H contacts per H is in the range 3.2 – 3.5 per H. When all the data are analyzed together, there is not a significant difference (pZ0.55) in this value as a function of the number of ˚ , the hydrogen atoms. For the shorter cutoff distance of 1.9 A number of close contacts falls in the range 2.4–2.8 per H. Once again ANOVA indicates no significant differences (pZ0.25). Further analysis did not seem to show any obvious subgroupings, as was the case for the incremental binding energy data. Table 3 ˚) Average Li–H close contacts per H and average Li–H distance (A
–40
Cluster
Ave. close contacts per ˚) H (2.3 A
Ave. close contacts per ˚) H (1.9 A
Ave. Li–H distance ˚) (A
2H 4H 6H 8H 10H 12H 14H 16H 18H
3.42 (0.20) 3.29 (0.19) 3.22 (0.08) 3.27 (0.20) 3.37 (0.22) 3.39 (0.07) 3.36 (0.11) 3.39 (0.16) 3.48 (0.20)
2.83 (0.75) 2.54 (0.29) 2.67 (0.35) 2.69 (0.23) 2.69 (0.26) 2.57 (0.26) 2.71 (0.14) 2.59 (0.24) 2.36 (0.25)
4.42 (0.07) 4.58 (0.16) 4.64 (0.19) 4.57 (0.15) 4.51 (0.17) 4.48 (0.12) 4.49 (0.08) 4.45 (0.11) 4.34 (0.11)
–50
–60
–70 2
4
6
8
10
12
14
16
18
Number of H Atoms Fig. 2. Plot of incremental binding energy vs. number of H atoms.
SD in parentheses.
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
15
Fig. 3. Structures of Li18H18 final structures. Top row: isomers a, b, and c. Bottom row: isomers d, e, and g. H atoms are light and Li atoms are dark colored.
show some noticeable semblance of the cubic structure in small regions. The more closely packed structure of isomer ‘c’ is further illustrated by the fact that the average Li–H distance is ˚ for ‘c’ while this quantity is 4.36, 4.49, 4.42, 4.35, and 4.16 A ˚ for structures a, b, d, e, and g, respectively. 4.30 A Interestingly, there does not seem to be a strong correlation between binding energy and this parameter (Table 1). 3.5. Partial charge analysis Previous studies by Fuentealba and Savin [7] and Rao and Jena [6] investigated the charge distribution in lithium hydride clusters. In this work, the natural bond charges as defined by Reed, Curtis, and Weinhold [25] have been used for the analysis of the lithium hydride clusters. Although the absolute value of the charge in a cluster of this type is not experimentally defined, the trends in charge as the cluster 4.9 4.8
Average Li-H Distance (Å)
Analysis of the data via ANOVA does seem to indicate a statistically significant difference (pZ0.02) for the overall Li– H distance (averaged over all Li, H pairs) in the clusters. However, there is not a straightforward relationship between average distance and cluster size. This can be seen in Fig. 4, which is a plot of the 95% confidence intervals for the average Li–H distances in the various clusters. The plot shows a slight increase and then a decrease in the average Li–H distances as the cluster becomes more hydrogenated. Since, the 2H and 18H clusters seem to deviate most from the average, further statistical testing was done excluding these individually. The results showed that removal of the 2H data still showed significant differences in the average Li–H distances (pZ0.04). However, removal of the 18H data showed no statistically significant differences in the 2H–16H data for the average Li–H distances (pZ0.13). Further testing of the total binding energy vs. various geometric parameters did not seem to reveal any significant correlations. Given the lack of correlation of calculated binding energy or structural parameters with number of hydrogen atoms seems to indicate that each addition of two hydrogen atoms seems to be independent of the other additions. The only evidence contrary to this is the apparent difference in incremental binding energies for clusters with 2–6 and those with 8–18 hydrogen atoms. Visual inspection of Fig. 3 shows an example of the geometric differences between the fully hydrogenated clusters. These structures show that even in clusters as small as these, different degrees of the bulk crystal structure already begin to appear. (Lithium hydride crystallizes in the same bodycentered cubic lattice as sodium chloride). Structure ‘c’ shows an appreciable region where this bulk structure occurs. There are 3!3 and 3!4 ion assemblies on the front and top face, respectively. Such a high degree of organization is not seen in any of the other structures. However, the other clusters
4.7 4.6 4.5 4.4 4.3 4.2 4.1 2
4
6
8
10
12
14
16
18
Number of H Atoms Fig. 4. Plot of average Li–H distance vs. number of H atoms.
16
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17
Table 4 Total natural charge on hydrogen atoms Isomer
2H
4H
6H
8H
10H
12H
14H
16H
18H
a b c d e g
1.57 1.54 1.58 1.52 1.54 1.56
3.10 3.04 3.06 3.05 3.08 3.11
4.56 4.60 4.51 4.54 4.63 4.55
6.07 6.08 6.02 6.07 6.05 6.13
7.59 7.72 7.52 7.54 7.51 7.60
8.90 9.14 9.05 8.96 9.02 9.08
10.43 10.54 10.49 10.43 10.48 10.49
11.84 11.95 11.98 11.77 11.86 11.89
13.41 13.34 13.42 13.33 13.36 13.33
Results from B3LYP/6-31G** calculations.
size changes are meaningful and will be examined. The following analysis will use the total charge on hydrogen atoms as a measure of overall charge transfer, as opposed to trying to analyze individual atom charges. Given that the hydrogen atoms are found in various bonding sites, it is expected that the individual charge will depend on the nearness and number of Li atoms. Table 4 presents the total charge on hydrogen atoms for each of the clusters studied. From this data it can be seen that the total H atom charges differ insignificantly from one isomer to another. Closer examination of the raw data showed that the maximum charge of a hydrogen atom in any cluster was K0.84 e and the minimum was K0.69 e. Those hydrogen atoms having a charge over K0.8 e generally had more (i.e. 4–5) nearest neighbor Li atoms from which to withdraw charge. Despite some variation in the individual H atom charges in the clusters, the total charge transfer as a function of the number of hydrogen atoms was remarkably linear for all clusters. In fact, linear fits of total charge vs. number of H atoms (with the y-intercept constrained to be zero) gave a value of R2 of greater than 0.999 for each of the six clusters. The average slope of all six lines was found to be K0.748(0.003) e. This seems to indicate that independent of the particular site of H atom bonding or the shape of the cluster, there is a transfer of about 0.75 e for each H atom. Once again, it should be noted that this value of 0.75 e is not exactly quantifiable, however, the fact that this value is constant across clusters is. As was concluded in the work of Bertolus et al. [5], the lithium hydride clusters are considerably ionic in nature. In addition, the current results are in qualitative agreement with the work of Fuentealba and Savin [7] on smaller lithium hydride clusters in which they concluded that there was a transfer of one electron per H atom added to the metal cluster. The idea that there is a constant transfer of electrons independent of the cluster size or number of hydrogen atoms is also consistent with the older work of Rao and Jena [6] in analyzing the smaller systems of Li9H9 and Li16H. Given that the average charge on hydrogen atoms is essentially constant. It may be possible to rationalize the slight increase in incremental binding energy purely on a simple electrostatic argument. As the clusters become more hydrogenated, they have a greater tendency to form small regions of a regular lattice with alternating Li and H ions. These regions allow for more favorable contacts between positive and negatively charged atoms. This may account for the slight increase in incremental bonding energy with increasing
hydrogenation, which, although not excessively large, is statistically significant. 4. Conclusion Calculations of the sequential binding energy of hydrogen atoms on a model Li18 cluster seem to indicate that there is a slight increase in incremental bonding energy as the cluster becomes more hydrogenated. This slight increase has been attributed to the restructuring of the cluster geometry to form regions of a regular lattice as the number of bound H atoms increases. Natural population analysis indicates that the total charge transfer to each hydrogen atom is essentially constant at 0.75 e. This seems to be independent of the number of hydrogen atoms already on the cluster or the site of hydrogen atom binding. These results are consistent with previous analyses of charge transfer in lithium hydrides [6,7]. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.theochem.2005. 08.034.
References [1] J. Koutecky, V. Bonacic-Koutecky, I. Boustani, P. Fantucci, W. Pewestorf, in: J. Jortner, A. Pullman, B. Pullman (Eds.), Large Finite Systems, Reidel, Dordrecht, 1987 p. 303. [2] G. Gardet, F. Rogemond, H. Chermette, J. Chem. Phys. 105 (1996) 9933. [3] B. Vezin, Ph. Dugourd, C. Bordas, D. Rayane, M. Broyer, V. BonacicKoutecky, J. Pittner, C. Fuchs, J. Gaus, J. Koutecky, J. Chem. Phys. 102 (1995) 2727. [4] V. Bonacic-Koutecky, J. Pittner, Chem. Phys. 225 (1997) 173. [5] M. Bertolus, V. Brenner, P. Millie, J. Chem. Phys. 115 (2001) 4070. [6] B.K. Rao, P. Jena, J. Phys. C: Solid State Phys. 19 (1986) 5167. [7] P. Fuentealba, A. Savin, J. Phys. Chem. A 105 (2001) 11531. [8] B. Vezin, Ph. Dugourd, D. Rayane, P. Labastie, J. Chevaleyre, M. Broyer, Chem. Phys. Lett. 206 (1993) 521. [9] J. Chen, N. Kuriyama, Q. Xu, H.T. Takeshita, T. Sakai, J. Phys. Chem. B 105 (2001) 11214. [10] P. Chen, Z. Xiong, J. Luo, J. Lin, K.L. Tan, Nature 420 (2002) 302. [11] Y.H. Hu, E. Ruckenstein, Ind. Eng. Chem. Res. 43 (2004) 2464. [12] F.E. Pinkerton, G.P. Meisner, M.S. Meyer, M.P. Balogh, M.D. Kundrat, J. Phys. Chem. B 109 (2005) 6. [13] Spartan ’02 for Windows, Wavefunction, Inc., Irvine, CA. [14] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [15] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [16] SPSS 11for Macintosh Computers, SPSS, Inc., Chicago, 2002.
P.G. Jasien, R. Cross / Journal of Molecular Structure: THEOCHEM 756 (2005) 11–17 [17] M. Triola, Elementary Statistics, seventh ed., Addison-Wesley, Reading (MA), 1998. [18] R. Cross, Undergraduate Thesis, Department of Chemistry, California State University, San Marcos (2004). [19] Ph. Dugourd, D. Rayane, P. Lebastie, B. Vezin, J. Chevaleyre, M. Broyer, Chem. Phys. Lett. 197 (1992) 433. [20] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 74th ed., CRC Press, Boca Raton, 1993, p. 12–105.
17
[21] C. Brechignac, H. Busch, P. Cahuzac, J. Leygnier, J. Chem. Phys. 101 (1994) 6992. [22] G. Pacchioni, P. Fantucci, J. Mol. Catal. 12 (1981) 213. [23] E.L. Meutterties, T.N. Rhodin, E. Band, C.F. Brucker, W.R. Pretzer, Chem. Rev. 79 (1979) 91. [24] F.A. Cotton, G. Wilkinson, Advanced Inorganic Chemistry, third ed., Wiley, New York, NY, 1972, p. 52. [25] A.E. Reed, L.A. Curtiss, F.J. Weinhold, J. Chem. Phys. 83 (1985) 735.