摘要:
Convergence from clusters to the bulk solid: an ab initio investigation Jinfeng Lai, Xin Lu* and Lansun Zheng* of clusters NanCln (n ~ 2–40) State Key Laboratory for Physical Chemistry of Solid Surface & Department of Chemistry, Xiamen University, Xiamen 361005, China Received 4th March 2002, Accepted 28th March 2002 Published on the Web 4th April 2002 The electronic structures of a series of sodium chloride clusters (NaCl)n (n ~ 2–40) cut out from the NaCl solid have been investigated by means of ab initio calculations. The calculation results demonstrated a good correlation of the topologic parameters Nd (the total amount of dangling bonds of a cut-out cluster) and b (the average dangling bonds on each in-cluster atom) with the stability of clusters as well as an evident convergence from clusters to the bulk solid.Particularly, we found that the effective charges on the Cl anions are more site dependent than size dependent. I. Introduction Clusters are intermediates in the transition between gaseous and condensed phases that make them an attractive and valuable subject of experimental and theoretical investigation of the emergence of condensed-matter properties. On the other hand, the cluster approach has been widely employed to study the properties of bulk solids and the chemisorptions and reactions on solid surfaces, and many successful applications have been made in the recent years.1–20 On both accounts, an important question is to what extent a cluster consisting of only a few atoms behaves like a respective bulk solid.To look for the answer to this question, much effort has been made by both experimentalists and theoreticians in the past decades. Some basic rules to justify the cluster approach and to guarantee a reasonable cluster modeling have been proposed by theoreticians. For example, for the cluster modeling of an ionic solid, a reasonable cluster model should at least fulfill the following principles, i.e., neutrality principle, stoichiometry principle and coordination principle. The reliability of these practical principles has been verified previously in the ab initio studies on the (MgO)x (x ~ 2–16) and (ZnO)x (x ~ 3–13) clusters.1–3 In the present work, we deal systematically with the structures and electronic properties of stoichiometric clusters NanCln for n up to 40.The main purpose of our study is to determine the cluster properties over a sufficient range of n to ascertain the convergence – e.g. of charge, structure, bond distances, binding energy – towards bulk solid properties. In the course of the study on the cluster–solid similarity, alkali halide clusters have been widely studied due to its structural simplicity.4–12 Based on SIMS (secondary ion mass spectroscopy) and other mass spectroscopic probes,21,22 which show stability islands among the alkali halide cluster cations corresponding to closed segments of the bulk lattice, it has been generally assumed that even quite small alkali halide clusters maintain the global features of bulk-like structure and ionic bonding.This view has been supported by theoretical calculations ranging in different sophistication.1–13 Ab initio investigations of clusters NanCln and KnCln (n ~ 1–32) by Ochsenfeld et al.4–7 revealed that the energetically most stable isomers of even smaller clusters show a clear preference for geometries which are fragments of the solid state lattice. Similar preference has also been found by Ayuela et al.,8–13 using the ab initio perturbed-ion (PI) model, which is formulated within the restricted Hartree–Fock (RHF) approximation, in studies of neutral stoichiometric alkali halide8–12 and (MgO)n clusters.13 In their semi-empirical SINDO1 study of the (NaCl)x (16 ¡ x ¡ 168) clusters, Jug and Geudtner19 established a quasilinear relationship between normalized binding energies or average bond distances and relative average coordination numbers.In our previous paper, we proposed three practical rules, i.e., neutrality principle, stoichiometry principle and coordination principle, for a better cluster modeling of ionic solid and justified these rules by case studies on a series of (ZnO)n (n ~ 3–13)1 and (MgO)n (n ~ 2–16)2,3 clusters. A good correlation of the stability of the metal oxide clusters with the topological parameters Nd (the total amount of dangling bonds of a cut-out cluster) and b (the average dangling bonds on each in-cluster atom) were found in these case studies. In the present work, these three principles are applied to the cluster modeling of NaCl solid and, accordingly, ab initio calculations on a series of (NaCl)n (n ~ 2–40) clusters have been performed.The size effect, as well as the site effect, of the clusters to the structures and electronic properties has been investigated systematically. II. Details of computation As a typical ionic-bonding crystal, the sodium chloride crystal has a rock-salt (cubic) structure where each bulk Na or Cl ion is coordinated by six counterions respectively and the measured interionic distance (dNa–Cl) is 2.814 A°.23 In the present work, a series of neutral, stoichiometric (NaCl)n (n ~ 2–40) clusters whose geometries are cut out from the bulk solid have been systematically investigated. Various methods and basis sets have been used to calculate the structures and electronic properties of (NaCl)n clusters.Since sodium chloride solid is generally believed to be highly ionic, the ground states of these (NaCl)n clusters considered were supposed to be in closed shells. In this way, the restricted Hartree–Fock (RHF) method has been safely employed. As alkali halides clusters are ideal candidates for an application of the ECP (effective core potential) approximation,24 we have used CEP-31G*,25 which paves the way for an efficient treatment of the larger clusters, as the basis sets for Na and Cl atoms. In addition, the possibility of correlation effects has been investigated by B3LYP,26,27 the applicability of which to ionic systems has been validated, for example, in a recent theoretical study on (MgO)n (n~1–16) and (CaO)n (n~1–12) 82 PhysChemComm, 2002, 5(12), 82–87 This journal is # The Royal Society of Chemistry 2002 Paper DOI: 10.1039/b202278hFig. 1 Geometries of the (NaCl)n (n ~ 2–16) clusters.Table 1 Electronic properties of (NaCl)n (n~2–16) clusters calculated at RHF/CEP-31G* level Cluster Symmetry N clusters20 by comparing the B3LYP predictions with the reference calculations at the MP2 and MP4 levels of theory28,29 and experimental results. What is more, to test the reliability of the effective core potential approximation, larger basis sets, i.e., the standard 6-31G* for RHF method and 6-311G* for B3LYP method, have been employed. All calculations were performed with the Gaussian98 package.30 III.Results and discussion 2a 2b 3a 4a 4b 4c 5a 5b 6a 7a 7b 8a 8b 9a 10a 10b 10c 11a 12a 12b 16a ab IIIA. Structures and electronic properties of (NaCl)n clusters The calculated structures of (NaCl)n clusters are shown in Fig. 1 where small spheres represent cations and large spheres represent anions. It should be noted that there are a number of different choices to cut out a cluster for a given cluster size n from the perfect NaCl solid, in which every Na or Cl atom is six-fold coordinated, and such procedure will inevitably give border atoms of lower coordination numbers. On this account, d and b,1 which are related to the coordination number k,19 Ncan be easily established.Table 1 presents the calculated properties of (NaCl)n clusters, including NaCl-unit RHF energy (i.e. RHF/n), bond distance, cluster dissociation energy to ions, average Mulliken charge, NaCl removal energy and cluster atomization energy. Cluster atomization energy Da NaCl removal energy D(n)(NaCl) ~ E((NaCl)n – 1) 1 E(NaCl) 2 E((NaCl)n). DD b d 2h DC‘v C2v Td C2h C2v C1 C2v 2h 2d DC1 C1 DCs C4v 2h DC2 C1 C1 C2h 2d 2d 16 4.0 214.980443 7.88 18 4.5 214.962430 6.89 22 3.67 214.990744 12.64 24 3.0 215.000390 17.91 28 3.5 214.993894 17.20 32 4.0 214.993655 17.17 30 3.0 214.999643 22.28 34 3.4 214.996573 21.86 32 2.67 215.004843 27.58 38 2.71 215.004750 32.16 38 2.71 215.003708 31.97 40 2.5 215.008158 37.50 40 2.5 215.006761 37.20 42 2.33 215.009455 42.51 48 2.4 215.009652 47.28 48 2.4 215.008798 47.05 48 2.4 215.008682 47.02 50 2.27 215.009736 52.04 52 2.17 215.011615 57.38 56 2.33 215.010651 57.07 64 2.0 215.013514 77.34 PhysChemComm, 2002, 5(12), 82–87 (n)/eVa E/n / Eh Da (n) ~ nE(Cl) 1 nE(Na) 2 E((NaCl)n).D(n)(NaCl)/ eVb 2.10 1.12 1.89 2.38 1.67 1.65 1.49 1.07 2.42 1.70 1.50 2.45 2.15 2.12 1.89 1.66 1.63 1.87 2.46 2.15 83Some interesting properties of the (NaCl)n clusters can be seen from Fig. 1 and Table 1. These are: 1. The cluster, e.g., (2a), (3a), (4a), (5a), (6a), (7a), (8a), (9a), (10a), (11a), (12a) and (16a), having the least dangling bonds (the smallest Nd) is the most stable (having the lowest NaClunit energy) among those of the same size.Such result may be due to the well-known edge effect. For a given cluster size n, the cluster, which has minimum Nd (or the highest coordination number), should have the smallest edge effect and should be the most stable. The present result fully supports the quite popular model14,31–33 that (NaCl)n clusters preferably occur as k 6 l 6 m (with km ~ 2n) fragments of the solid, especially when k ~ l ~ m. In other words, (NaCl)n clusters prefer densely packed structures as in the solid. 2. Compared to the other clusters, those clusters, as for (NaCl)2, (NaCl)3, (NaCl)5 and (NaCl)7, where no densely packed structure like in the bulk lattice is possible have reduced stability.Such a result can be easily explained by using the topological parameter Nd. To maintain a low Nd, cluster atoms with coordination number smaller than 3 should be avoided. Actually, no atoms coordinated with fewer than three counterions exist in the real NaCl solid, while cluster atoms with coordination numbers of fewer than three are unavoidable for those clusters whose sizes are 2, 3, 5 and 7. Owing to the presence of the lower coordinated cluster atoms, those clusters are unstable with higher NaCl-unit energy. As a consequence, this kind of cluster would not offer a good model of NaCl solid. In this way, it provides a simple but efficient means to exclude some cluster models from detailed consideration. 3.Along with the increase of n, there are more and more inner atoms that are fully coordinated, the corresponding b decreases and the cluster becomes more stable. In other words, the largest b corresponds to the highest NaCl-unit energy, while the smallest b corresponds to the lowest NaClunit energy, evidencing that the topological parameter b can be a measurement of the relative stability of the clusters of different size. 4. For those clusters with the same Nd, e.g., (8a, 8b) or (10a, 10b, 10c), the one with higher symmetry is more stable with lower NaCl-unit energy. This clearly demonstrates that the edge effect could be further divided into two effects, i.e., the size effect (dependent on the size of the cluster) and the shape effect (dependent on the shape of the cluster).In this regard, the topological parameters Nd and b provide a way to keep the shape effect under control. 5. The energy required to remove a NaCl molecule from the cluster (D(n)(NaCl) ~ E((NaCl)n 2 1) 2 E((NaCl)n)) is larger theoretical and experimental bulk values available (n) /eVb (eV)a Cluster D D DI (n) b (n) d 7.25 7.37 6.37 10.29 28.99 44.22 4a 6a 7.46 14.44 59.68 8a 7.49 16.56 67.45 9a 7.55 22.79 90.64 12a 7.61 31.21 121.68 16a Bulk HF Expt 7.64f 7.98h a (n) Table 2 Convergence of the structural and electronic properties of (NaCl)n (n ~ 4–16) clusters calculated at RHF/CEP-31G* level as well as the atoms. eAverage Mulliken charges Q for all inner Cl atoms.fRef. 34. gRef. 35. hRef. 36. iRef. 23. IIIB. Convergence from the clusters to the bulk solid It is of particular importance to investigate the convergence of the calculated structural and electronic properties of a cut out cluster with the increase of cluster size. However, from Table 1, it is hard to draw a clear conclusion if the shapes of the clusters are chosen arbitrarily. We then focus on the (NaCl)n (n ~ 4a, 6a, 8a, 9a, 12a, 16a) clusters, believing that the set of clusters with smallest Nd of each given size provides a suitable starting point to study the cluster size dependence of the computed properties since the shape effect is under control. From Table 2 and Fig. 2, interesting convergence, based on the NaCl-unit energy, bond distance, binding energy, as well as the effective charges in Cl atom, for this specific set of (NaCl)n clusters has been shown.1. The NaCl-unit energy of a cluster decreases monotonously with the increase of the cluster size (see Fig. 2), while the NaClunit energies of those clusters in arbitrary shapes oscillate. The more severely the shapes differ, the larger the oscillation (see Table 1). 2. The average distance between neighboring alkali and halogen atom, i.e., dNa–Cl, increases with the cluster size n. In clusters (4a), (6a), (8a), (9a) and (12a), dNa–Cl is 2.701, 2.728, 2.738, 2.758 and 2.766 A°, respectively. We note that in cluster (16a) dNa–Cl is 2.776 A°, which is similar to the measured (n) interionic distance in the bulk cubic crystal, 2.814 A°23 and to the predicted values around 2.90 A°by periodic HF calculations.34,35 The increasing Na–Cl distance could be a hint of the increasing ionicity of the in-cluster atoms.3. With the cluster size n increases, there are more and more high coordinated atoms and the effect of the ionic binding must be stronger. As a consequence, the binding energy per ion-pair also increases. Thus we note that the binding energy per ionpair (DI /n, from Table 1) increases monotonically, starting d /eVc Na–Cl/A° d 0 2.701 2.728 2.738 2.758 2.766 2.776 2.90f,2.89g 2.814i ~ nE(Cl2) 1 nE(Na1) 2 E((NaCl) Cluster dissociation energy to ions DI n). bCluster dissociation energy to NaCl Dd(n) ~ nE(NaCl) 2 E((NaCl)n).cBinding energy Db(n) ~ DI(n)/n ~ E(Cl2) 1 E(Na1) 2 E((NaCl)n)/n. dMulliken charges Q0 for different fold coordinated Cl Q3c: 20.635 3c: 20.630 4c: 20.667 3c: 20.632 4c: 20.678 3c: 20.641 4c: 20.681 5c: 20.703 3c: 20.636 4c: 20.683 5c: 20.711 3c: 20.633 4c: 20.685 5c: 20.717 84 PhysChemComm, 2002, 5(12), 82–87 for clusters (4a), (6a), (8a), (12a) while it is interesting to see that there is a destroy of the cubic-like structure from (NaCl)n to (NaCl)n 2 1 (n ~ 4a, 6a, 8a, 12a). On this account, the nonmonotonic variation of D(n)(NaCl) with n may be correlated with such a change; i.e. removing a NaCl molecule from (NaCl)n to form (NaCl)n 2 1 (that is, D(n)(NaCl)) when (NaCl)n is 4a, 6a, 8a or 12a involves a transformation from a cubic-like structure to form a non-cubic one.On the other hand, the other smaller values of D(n)(NaCl) are just associated with transformations between two cubic-like clusters or non-cubic clusters. Such a trend demonstrates that densely packed structures like fragments of the solid-state f.c.c. lattice are favored. Qe 20.635 20.642 20.655 20.666 20.672 20.680 20.985gchloride solid is highly ionic. It is natural for one to expect that with the increase of cluster size n, the magnitude of the in-cluster atomic charges will also increase and will approach to the bulk value, which should be around the formal charge ¡1,35 in the long run. From Table 2 and Fig. 2, it has been shown that although the atomic charges in clusters (NaCl)n differ significantly from the formal charge ¡1, they increase in magnitude as n increases.Similar trend has been found in (MgO)n (n v 4) by Bildyrev et al.37 However, through careful inspection of the calculated charges (see Table 2), we may find that by keeping the clusters in the same shape, the atomic charges show not only size dependence, but also noteworthy site dependence, similar to the (MgO)n case.2,37,38 That is, the average Mulliken charges of Cl atoms increase with the increase of the cluster size n, while for any given fold coordinated Cl atoms the magnitude of the Mulliken charges are always around a constant regardless of the size of the clusters. For instance, the Mulliken charges for the three-fold coordinated Cl atoms are always around 20.63, while the Mulliken charges for the four-fold and five-fold coordinated Cl atoms are always around 20.68 and 20.71, respectively.On the other hand, along with the increase of n, there are more and more inner atoms and the percent of the higher fold coordinated atoms also increases. As a consequence, the average charge of the in-cluster atoms increases with the increase of the cluster size. Thus we find the average Mulliken charges of anions in clusters (4a), (6a) and (8a) are 20.635, 20.642 and 20.655, respectively, while those of (9a), (12a) and (16a) are 20.666, 20.672 and 20.680, respectively. Fig. 2 Convergence of the structural and electronic properties of (NaCl)n (n ~ 4a, 6a, 8a, 9a, 12a, 16a) clusters.(a) RHF/n (RHF Furthermore, such convergence has also been investigated by means of B3LYP method, which is a hybrid method including a mixture of the HF exchange with the DFT exchangecorrelation. The results that incorporate the correlation effects have been shown in Table 3 and Fig. 2. From the comparisons to those from the HF calculations, we have found the same convergence as that we get from the HF calculations. Again we found that the NaCl-unit energies decrease monotonously with (n) b Ddcalculation); (b) RHF/n (B3LYP calculation); (c) binding energy (n)/n~E(Cl2) 1 E(Na1) 2 E((NaCl)n)/n; (d) bond distance Na–Cl; (e) average Mulliken charges Q for all inner Cl atoms. the increase of cluster size for the set of clusters with the fewest dangling bonds.The B3LYP-predicted bond distance also increases with the increasing of cluster size n, identical to the trend predicted by HF calculations. However, the incorporation of electron correlation leads to a longer Na–Cl distance. For example, the B3LYP predicted dNa–Cl of (2a) is 2.626 A°compared to the HF value of 2.620 A°, whereas the B3LYP bond distance of (16a) is 2.783 A°compared to the HF value of 2.776 A°. It is noteworthy that both the RHF and B3LYP predicted bond distances for the (NaCl)2 cluster (2a) agree well with the experimental value of 2.584 ¡ 0.034 A°39 measured for the gas-phase (NaCl)2 d ~DI from 7.25 eV for the cluster (4a) and through 7.37, 7.46, 7.49, 7.55 and 7.61eV for cluster (6a), (8a), (9a), (12a) and (16a), respectively, and approaching the measured lattice energy of 7.98 eV of bulk solid,36 and the predicted lattice energy of 7.64 eV by periodic HF calculation.34 4.Although the atomic charges depend largely on the model used to calculate them and they are not physically observable as well, it is still of great interest to investigate the convergence of atomic charges. As a typical ionic-bonding crystal, sodium Table 3 Convergence of the structural and electronic properties of (NaCl)n (n ~ 4–16) clusters calculated at B3LYP/CEP-31G* level Cluster /eVc /eVa (n)(eV)b Qe d D D Na–Cl/A° 0 (n) b d (n) I h 2.712 2.736 7.32 7.47 6.18 10.11 20.578 20.592 4a 6a 2.746 7.53 14.01 20.594 8a 2.766 7.57 16.07 20.602 9a 2.775 7.62 22.10 20.604 12a 2.783 7.67 30.26 20.607 16a Q3c: 20.578 3c: 20.583 4c: 20.608 3c: 20.578 4c: 20.609 3c: 20.585 4c: 20.614 5c: 20.620 3c: 20.580 4c: 20.614 5c: 20.624 3c: 20.576 4c 20.613 5c: 20.625 ~ nE(Cl2) 1 nE(Na1) 2 E((NaCl) Cluster dissociation energy to ions DI n).bCluster dissociation energy to NaCl Dd(n) ~ nE(NaCl) 2 E((NaCl)n). cBinding energy Db(n) ~ DI(n)/n ~ E(Cl2) 1 E(Na1) 2 E((NaCl)n)/n. dMulliken charges Q0 for different fold coordinated Cl D29.30 44.79 60.25 68.10 91.47 122.75 (n) a E/n / E 215.255228 215.260401 215.262836 215.264107 215.266156 215.267986 atoms. eAverage Mulliken charges Q for all inner Cl atoms.85 PhysChemComm, 2002, 5(12), 82–87Table 4 Mulliken charges of (NaCl)n (n ~ 4–40) clusters single-point calculated at RHF/CEP-31G* and B3LYP/CEP-31G* levels RHF/CEP-31G* a Cluster Qb 0 20.680 20.687 4a (26262) 6a (36262) 20.689 8a (46262) 20.695 9a (36362) 20.698 12a (46362) 20.702 16a (46462) 20.715 24 (46463) 20.722 32 (46464) 20.724 40 (46465) a Q3c: 20.680 3c: 20.681 4c: 20.698 3c: 20.679 4c: 20.699 3c: 20.680 4c: 20.705 5c: 20.716 3c: 20.678 4c: 20.704) 5c: 20.721 3c: 20.676 4c: 20.704 5c: 20.725 3c: 20.676 4c: 20.707 5c: 20.732 6c: 20.767 3c: 20.674 4c: 20.707 5c: 20.734 6c: 20.776 3c: 20.673 4c: 20.706 5c: 20.734 6c: 20.773 Mulliken charges Q0 for different fold coordinated Cl atoms.bAverage Mulliken charges Q for all inner Cl atoms. cluster. Such agreement demonstrates the applicability of the B3LYP functional to ionic systems such as NaCl. Generally speaking, the atomic charges are sensitive to the methods and basis sets employed and the inclusion of correlation effects seemly tends to decrease the magnitude of the atomic charges. From Table 2, Table 3 and Fig. 2, it can be seen that the magnitudes of the B3LYP-predicted atomic charges are constantly smaller than the corresponding HF Cluster E/n / Eh 4a 6a 2621.456437 2621.461069 8a 2621.463501 9a 2621.464784 a ~ nE(Cl2) 1 nE(Na1) 2 E((NaCl) Cluster dissociation energy to ions DI n). bCluster dissociation energy to NaCl Dd(n) ~ nE(NaCl) 2 E((NaCl)n).cBinding energy Db(n) ~ DI(n)/n ~ E(Cl2) 1 E(Na1) 2 E((NaCl)n)/n. dMulliken charges Q0 for different fold coordinated Cl Q3c: 20.685 3c: 20.684 4c: 20.725 3c: 20.682 4c: 20.727 3c: 20.686 4c: 20.726 5c: 20.771 0 Cluster E/n / Eh 4a 6a 2622.653884 2622.658441 8a 2622.660881 Table 5 RHF calculations for (NaCl)n (n ~ 4, 6, 8, 9) clusters with 6-31G* basis sets IV. Conclusion The structure and electronic properties of a series of sodium chloride clusters (NaCl)n (n ~ 2–40) have been investigated by means of the ab initio method. The main results are: (i) Whenever possible, densely packed structures like (n)/eVb (n) b D7.38 7.50 7.57 7.61 /eVb (n) b D7.23 7.35 7.42 7.46 9a 2622.662190 aatoms.eAverage Mulliken charges Q for all inner Cl atoms. Table 6 B3LYP calculations for (NaCl)n(n ~ 4, 6, 8, 9) clusters with 6-311G* basis sets atoms. eAverage Mulliken charges Q for all inner Cl atoms. D29.51 45.03 60.56 68.45 D28.92 44.13 59.37 67.11 Q3c: 20.619 3c: 20.620 4c: 20.627 3c: 20.617 4c: 20.626 3c: 20.616 4c: 20.627 5c; 20.630 3c: 20.613 4c: 20.627 5c: 20.630 3c: 20.612 4c: 20.628 5c: 20.630 3c: 20.610 4c: 20.629 5c: 20.637 6c: 20.644 3c: 20.606 4c: 20.626 5c: 20.632 6c: 20.637 3c: 20.606 4c: 20.626 5c: 20.635 6c: 20.644 ~ nE(Cl2) 1 nE(Na1) 2 E((NaCl) Cluster dissociation energy to ions DI n). bCluster dissociation energy to NaCl Dd(n) ~ nE(NaCl) 2 E((NaCl)n).cBinding energy Db(n) ~ DI(n)/n ~ E(Cl2) 1 E(Na1) 2 E((NaCl)n)/n. dMulliken charges Q0 for different fold coordinated Cl Q3c: 20.704 3c: 20.715 4c: 20.767 3c: 20.715 4c: 20.788 3c: 20.730 4c: 20.783 5c: 20.847 86 PhysChemComm, 2002, 5(12), 82–87 B3LYP/CEP-31G* Q 0 20.619 20.622 20.621 20.623 20.623 20.624 20.630 20.627 20.630 (n)/eVa I (n) /eVa (n) I (n) ones. In spite of that, it is still true that the atomic charges are more site dependent than size dependent. For instance, the Mulliken charges for the three-fold, four-fold and five-fold coordinated Cl atoms are always around 20.58, 20.61 and 20.62, respectively, while the average magnitude of the Mulliken charges increases with the increase of the cluster size n, stating from 20.578 for (4a) to 20.607 for (16a).Seeing that the clusters favor the densely packed structures, we have employed single-point calculation for (NaCl)n (n ~ 4–40), which are cubic with the nearest Na–Cl distance being fixed to the bulk value of 2.814 A°, to investigate the convergence of the larger clusters. The results are shown in Table 4. In the single-point HF calculations, the Mulliken charges for the three-fold, four-fold, five-fold and six-fold coordinated Cl atoms are always around 20.68, 20.70, 20.73 and 20.77, respectively, while the average Mulliken charges increase monotonously with the increase of the cluster size, from 20.680 for (NaCl)4 (26262) to 20.724 for (NaCl)40 (46465). In the single-point B3LYP calculations, the Mulliken charges of the three-fold, four-fold, five-fold and six-fold coordinated Cl atoms are always about 20.61, 20.63, 20.63 and 20.64, respectively, whereas the average Mulliken charge increases from 20.578 for (NaCl)4 (2 6 2 6 2) to 20.730 for (NaCl)40 (4 6 4 6 5).d D 0 d 6.18 10.03 13.91 15.96 d IIIC. Basis set effects To test the reliability of the effective core potential approximation, larger basis sets, i.e., the standard 6-31G* for RHF calculations and 6-311G* for B3LYP calculations, have been employed. The results are given in Table 5 and Table 6, respectively, which are roughly the same as those obtained by using the CEP-31G* basis set. /eVb /eVc D (n) d 5.83 9.50 13.19 15.16 Qe 20.685 20.698 20.705 20.713 Qe 20.704 20.732 20.751 20.766fragments of the solid-state f.c.c.lattice are favored. Clusters with unrealistically low coordinated atoms are of reduced stability and would not be good models to represent the solid. (ii) The topologic parameters Nd and b can be good measurements to judge the stability of clusters. For any given size, the cluster with the smallest Nd is the most stable. On the other hand, with the increase of the cluster size, the corresponding b decreases and the cluster is more stable with lower NaCl-unit energy and smoothly approaches the solid. (iii) For the set of clusters with the smallest Nd in each given size, the NaCl-unit energy, binding energy, as well as bond distance, show a convergence from the cluster to the bulk solid.The effective charges in Cl atoms are more site dependent than size dependent. That is, the Cl atom with higher coordination number accommodates more charge, while for any given degree of coordination of a Cl atom the magnitudes of the Mulliken charges are always around a constant. In this way, with the increasing of cluster size n, the number of the high fold coordinated atoms increases and the average charge of the in-cluster atoms becomes larger. 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ISSN:1460-2733
DOI:10.1039/b202278h
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年代:2002
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