Chemical Physics Letters 388 (2004) 342–347

www.elsevier.com/locate/cplett

Theoretical studies of Na(H2O)þ19–21 and K(H2O)þ19–21 clusters:explaining the absence of magic peak for Na(H2O)þ20

Arshad Khan *

Department of Chemistry, Pennsylvania State University, DuBois, PA 15801, USA

Received 12 September 2003; in final form 24 February 2004

Published online:

Abstract

The M(H2O)þ19–21 clusters (M ¼ Na or K in cavity) consisting of broken and distorted dodecahedral cages are studied by op-

timizing geometry at the B3LYP/6-311++G** level. The stabilization energy (relative to separated H2O and Mþ) per monomer

(SEP) exhibits a maximum for K(H2O)þ20 and no such maximum for Na(H2O)þ20 cluster. While K in dodecahedral cavity carries a +1

charge, Na remains as a neutral atom, and suggests that the electron affinity (EA) of Naþ >EA of (H2O)þ20 dodecahedral cage>EA

of Kþ. On the basis of above trends in the SEP values and the charge on the metal, one can explain the absence of a magic number

peak for Na(H2O)þ20 and the presence of the magic peak for K(H2O)þ20 cluster.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

Clathrate structures of water are well-known and

often contain neutral molecules as in hydrates [1,2] or

metallic ions as in solutions or gas phase clusters [3–6].

There is an increased interest in these clathrate struc-

tures because of the role played by these clusters during

the formation of aerosol particles [7], nucleation [8] and

other atmospheric phenomena [9]. The role of neutralguest molecules within the cavity of 20-mer and larger

water clusters has been examined theoretically only in

recent years [10–13], and provides understanding of the

high stability of these clusters that exist even on the sea-

floor under a very high pressure. Although some of the

questions on stability of the clathrate structures, espe-

cially those with neutral molecules in cavity, are better

understood because of the above studies, very few effortshave been made to understand theoretically the clusters

that enclose alkali metal atoms or ions [5,14]. The ab

initio studies at the Hartree–Fock (HF/6-311G**) level

with constrained symmetry by Cioslowski and Nana-

yakkara [14] provide structures and energy values for

Li(H2O)þ20 and Na(H2O)þ20 clusters. These studies sug-

* Fax: +1-814-375-4784.

E-mail address: kub@psu.edu (A. Khan).

0009-2614/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2004.02.101

gest that the Liþ ion moves to the bottom of the dis-torted dodecahedral cage cavity and the Naþ ion moves

to the center of the cavity in the optimized structure.

Because of the constrained symmetry, the structure and

energy values are expected to be less accurate compared

to those obtained with no such symmetry restriction.

Besides, the neglect of electron correlation energy by the

HF method is also expected to introduce error in the

calculated energy values.The experimental studies by Steel et al. [5] show a

magic number mass spectral peak at M(H2O)þ20 where Mis a Li, K, Rb or Cs atom. It is puzzling that no such

magic peak is noticed at Na(H2O)þ20 cluster. It is to be

mentioned that a magic peak is a dominant mass spectral

peak beyond which the mass intensity drops abruptly.

Steel et al. also performed molecular dynamics (MD)

studies on M(H2O)þ19;20 clusters with M ¼ Na or K, andpredicted greater stability of M(H2O)þ20 clusters relative

to M(H2O)þ19. Based on this result one may expect an

increased mass intensity at Na(H2O)þ20 and K(H2O)þ20clusters relative to Na(H2O)þ19 and K(H2O)þ19 clusters

respectively. However, in their experiment an increased

mass intensity is noticed only at K(H2O)þ20 and not at

Na(H2O)þ20 cluster. Even though the above MD studies

do not report any result onM(H2O)þ21 clusters (metal anda water molecule in cavity), it seems that such studies will

A. Khan / Chemical Physics Letters 388 (2004) 342–347 343

be necessary to compare the stability of M(H2O)þ21clusters relative to M(H2O)þ19 and M(H2O)þ20 clusters,

and hence, will allow one to predict the conditions under

which a magic peak is expected. If one assumes that a

M(H2O)þ21 cluster is less stable than a M(H2O)þ20 cluster,a decrease in mass intensity will be expected from

M(H2O)þ20 to M(H2O)þ21 giving a magic peak at

M(H2O)þ20 provided that the formation of M(H2O)þ20cluster is also kinetically favored. One of the objectives of

the present study is to examine the thermodynamic sta-

bilities of M(H2O)þ19–21 clusters so that the disagreement

between the experimental and the MD results can be

better understood. As mentioned above, for the expla-nation of a magic number peak one needs to examine the

thermodynamic stability of M(H2O)þ19–21 clusters as wellas their rates of formation (kinetic factor). The thermo-

dynamic stability indicates survival possibility of a

cluster isomer once it is formed, and hence, dictates the

abundance of a cluster. However, the rate of cluster

formation may change the expected outcome of ther-

modynamic stability if a particular isomer forms fasteror slower than the other isomer even if both the isomers

may have comparable thermodynamic stability. This is-

sue has also been addressed in this Letter by examining

the charges on metal atoms in various clusters. Since the

formation rates of clusters may change depending upon

the amount of charges carried by the metal atom, both

the stabilization energy values as well as charges on the

metal atoms are examined to determine the key factorsthat are responsible for the presence or absence of magic

peaks in the experiment.

2. Method applied in calculations

The geometry optimizations of M(H2O)þ19–21 clusters

are carried out by using the 6-311++G** basis sets andapplying the Becke-3-parameter density functional the-

ory (DFT) [15,16] and Lee–Yang–Parr correlation

functional [17,18] (B3LYP). These calculations are pre-

ceded by optimization of a number of (H2O)20 dodeca-

hedral isomers obtained by rearranging non-H-bonded

H atoms (NHB H) on the cage surface and optimizing

their geometries at the HF/6-31G* level followed

by single point energy calculations at the B3LYP/6-311++G** level. Once the most stable (H2O)20 struc-

ture is obtained, a metal atom is placed in its cavity for

optimization of M(H2O)þ20 at the B3LYP/6-311++G**

level. For M(H2O)þ19 cluster a water molecule is removed

from the dodecahedron, and for M(H2O)þ21 cluster an

additional water molecule is placed within the dodeca-

hedral cavity. The geometry optimization of (H2O)20clusters is carried out at the HF/6-31G* level primarilyto save computation time. It should be pointed out that

the optimization of M(H2O)þ19–21 clusters by B3LYP/

6-311++G** method is very time consuming and re-

quire 16 days to over a month of CPU time by using

Gaussian 03 series of programs [19] running parallel on

a Linux machine. Test cases suggest that the optimiza-

tion of water clusters at the HF/6-31G* level provides

results that are comparable to those of high level abinitio calculations or experiments ([20,21] and references

therein). For dimer the HF optimized structure [20] has

O–O distance of 2.970 �A and agrees quite well with the

experimental value [22] of 2.976 �A. Similarly, for pro-

tonated dimer the HF predicted O–O distance of 2.455 �Ais close to that (2.417 �A) obtained by mp2/6-31+G*

optimization. For larger dodecahedral clusters (in hy-

drates) the experimental [1] O–O distance is 2.8�A and theHF predicted value [10] is 2.9 �A. Similarly, the experi-

mental O–O–O angle (108) is the same as that predicted

by HF calculation [10]. The energy calculations at the

B3LYP/6-311++G** level is also expected to be quite

reliable [23]. Some of these test results, especially the

energy values, for Na(H2O)þ and K(H2O)þ clusters are

presented in Table 1. These results suggest that the en-

ergy values calculated at the MP2/6-311++G** orB3LYP/6-311++G** level are very close to those calcu-

lated at the B3LYP/6-311++G**// HF/6-31G* level. The

Na–O and K–O distances are also predicted accurately

by the HF/6-31G* optimization method giving the

values of 2.21 and 2.65 �A, and are fairly close to those of

the MP2/6-311++G** values of 2.27 and 2.63 �A or

B3LYP/6-311++G** values of 2.22 and 2.61 �A respec-

tively. The average O–H distance given by each methodis around 0.96 �A. Similarly, the stabilization energy

values, SE (relative to separated H2O and Mþ ion) and

SEP (SE per water molecule and Mþ ion, Table 1), given

by the MP2 or B3LYP/6-311++G** optimization

methods are very close (usually less than 5% discrepancy)

to those predicted by the B3LYP/6-311++G**// HF/6-

31G* method. The geometry optimization is followed by

a nearest neighbor atom search. When two O atoms areat a distance of less than 3.2 �A, and the OHO (H in be-

tween O atoms) angle is larger than 146�, the H atom is

considered to be an H-bonded atom.

3. Search for the most stable dodecahedral cage

The 20-mer water cluster that contains metal in cavityis considered to have dodecahedral geometry. As dis-

cussed in previous studies, the other alternative struc-

tures [10,20,21] like fused cubes or edge-shared prisms

are not likely to form. It should be mentioned that a

dodecahedral structure has 30 H-bonded atoms along

the edges of the dodecahedron and 10 non-H-bonding

(NHB) H atoms directed away from the cage surface.

The starting dodecahedral geometry for the presentstudy is obtained by examining 8–10 optimized

dodecahedral structures with different distributions

of NHB H atoms on the cage surface. Even though

Table 1

The energy values from B3LYP/6-311++G** optimization (in parenthesis) and B3LYP/6-311++G**//HF/6-31G* level calculations are presented

Ions and molecules Energy (Hartree) Charge on metal SE (kcal/mol) SEP (kcal/mol)

Na (H2O)þ19 ()1615.244937) (+0.2) (279) (14.0)

)1615.228514 +0.2 274 13.7

Na (H2O)þ20 ()1691.718328) (+0.1) (289) (13.8)

)1691.698309 +0.1 282 13.4

Na (H2O)þ21 ()1768.196656) (0.0) (301) (13.7)

)1768.179438 +0.0 296 13.5

K (H2O)þ19 ()2052.885528) (+0.8) (259) (13.0)

)2052.869968 +0.8 254 12.7

K (H2O)þ20 ()2129.369238) (+1) (275) (13.1)

)2129.351778 +1 269 12.8

K (H2O)þ21 ()2205.824818) (+0.6) (273) (12.4)

)2205.814812 +0.7 272 12.4

K (H2O)þ ()676.249444) (+1) (18.7) (9.35)

)676.249023 +1 18.7 9.35

MP2 )675.465859 +1 19.0 9.48

Na (H2O) ()238.757004) (+0.1) (7.34) (3.67)

)238.756371 +0.0 7.21 3.60

Na (H2O)þ ()238.587199) (+1) (25.8) (12.9)

)238.586845 +1 25.8 12.9

)237.978484 MP2 +1 MP2 24.6 MP2 12.3 MP2

(H2O)20 )1529.495868 (iso1) 209 10.45

Dodecahedral )1529.490742 (iso 2) 206 10.30

Isomers (1–6) )1529.484603 (iso 3) 202 10.10

Fig. 1 )1529.473962 (iso 4) 196 9.80

)1529.470081 (iso 5) 193 9.65

)1529.456324 (iso 6) 185 9.25

H2O ()76.458531))76.458102)76.274920 MP2

Na ()162.286780)

Naþ ()162.087563))161.664286 MP2

Kþ ()599.761054))599.160709 MP2

The SE values are calculated relative to separated water molecules and metal cation (metal atom for neutral species). The SEP represents SE per

constituent water molecules plus the metal. Some of the energy values are also calculated at the MP2/6-311++G** level.

344 A. Khan / Chemical Physics Letters 388 (2004) 342–347

innumerable arrangements are possible with 10 NHB H

atoms on the cluster surface, certain trend in the sta-

bilization energy values is noticed in test cases which

help us select geometry with a large stabilization energy

(SE). Six of the representative geometries are presentedin Fig. 1 with decreasing order of SE values (Table 1) for

illustration purposes. The examination of the NHB H

atoms in these structures (1–6, Fig. 1) suggests that the

structure with fewer NHB H atoms in adjacent locations

is more stable than the one with larger number of NHB

H atoms in adjacent locations. For example, the struc-

ture 1 in Fig. 1 has three sets of 2 NHB H atoms in

adjacent locations with the remaining 4 NHB H atomsappearing singly (without having any adjacent neigh-

bor). The structure 6, on the other hand, has all of its

10 NHB H atoms in adjacent locations and is least

stable among the selected structures. The structure 2 is

similar to structure 1 except that one of the NHB H

atoms is directed toward cavity (shown with a circle)and is less stable than structure 1 by 3 kcal/mol. The

other structures in Fig. 1 have maximum of 3 (structure

3), 6 (structure 4) and 9 (structure 5) NHB H atoms in

adjacent locations. Among these dodecahedral clusters,

the structure 1 is most stable with a SE value of around

209 kcal/mol, and the structure 6 is the least stable

cluster with a SE value of around 185 kcal/mol. The

other arrangements of NHB H atoms give SE values inbetween 185 and 209 kcal/mol. Even though there is no

Fig. 1. Unfilled dodecahedral cages with different arrangement of NHB

H atoms. Among the structures examined, the structure 1 is most

stable with all the NHB H atoms directed outward and no more than 2

NHB H atoms are in adjacent locations. The structure 2 is similar to

structure 1, except, one of its NHB H atoms is directed toward cavity

(shown with a circle). The structure 6 is least stable with all the 10

NHB H atoms in adjacent locations.

A. Khan / Chemical Physics Letters 388 (2004) 342–347 345

guarantee that a global minimum structure is obtainedin this study, chances are high that the cage structure 1

may be close to the global minimum structure as every

effort has been made to arrange the NHB H atoms with

minimum number of adjacent neighbors.

Fig. 2. (H2O)19 broken cages (structures 1 and 2), (H2O)20 dodecahe-

dral cages (structures 3 and 4) and (H2O)21 dodecahedral cages

(structures 5 and 6) are shown with metal in cavity. In the (H2O)21 cage

one water molecule is also in cavity. For clarity H atoms are not

shown, and the O–O distances shorter than 3.2 �A are connected. The

shortest metal–oxygen distances are also connected.

4. Starting geometry for M(H2O)+19�21 clusters

Before geometry optimization for M(H2O)þ20 and

M(H2O)þ21 clusters, the dodecahedral cage structure 1

(Fig. 1(1)) is selected with a metal atom in its cavity. As

mentioned before, for M(H2O)þ21 cluster one water

molecule is placed within the cage cavity and for

M(H2O)þ19 cluster, the metal is placed within the cavity

of a broken cage, which is obtained by removing a water

molecule from the dodecahedral structure 1. Specialattention has been paid to the locations of NHB H at-

oms (to ensure minimum number of adjacent members)

while removing a water molecule from the cage surface.

The other alternative structures with the metal or water

molecule bonded outside of the cage cavity are ignored

as they are not likely to be stable. For example, a water

molecule bonded outside of the cage by a single H-bond

in M(H2O)þ21 cluster is likely to be readily broken. The

same water molecule within the cage cavity will be sur-

rounded by the cage and provide a shield against its

cleavage. The dissociation of any H2O molecule from

the cluster surface of such a structure will require dis-sociation of at least three H-bonds, and hence, provide

stability to M(H2O)þ21 structures.

5. Results and discussion

5.1. Structural features, SE and SEP values for

M(H2O)þ19–21 clusters

Fig. 2(1–6) represent optimized structures for

M(H2O)þ19–21 clusters. The structures 2.1 and 2.2 repre-

sent M(H2O)þ19 clusters, 2.3 and 2.4 represent M(H2O)þ20and 2.5 and 2.6 represent M(H2O)þ21 clusters with K and

Na in cavity. As mentioned above, in M(H2O)þ21 clustera metal atom as well as a water molecule (labeled as an

O) remain within the cage cavity. For clarity theH-bonds are not shown in Fig. 2, and the O atoms

346 A. Khan / Chemical Physics Letters 388 (2004) 342–347

occupying vertices of the cage are not labeled. Each line

on the cage represents an O–O distance of 3.1 �A or

shorter, and an H-bonding H (HB H) atom lies in be-

tween two such O atoms giving stability to each cage

cluster. The energy values together with the charge onthe metal are presented in Table 1. The O–O distance in

each cluster is around 2.9 �A, and the M–O distance from

the metal to the surface molecule in Na(H2O)þ19–21cluster is around 2.5 �A, and for K(H2O)þ19–21 cluster is

around 2.9 �A. Since the cavity radius for each cluster

type is around 4 �A, the above M–O distances represent

locations of the metals closer to the edge rather than the

center of the dodecahedral cage. For the M(H2O)þ21cluster of Na, the M–O distance from the metal to the O

atom of the water molecule in cavity is much shorter (2.2�A), and hence, represents a stronger bond than that

between the metal and an O atom on the cage surface.

As in Na(H2O)þ21 cluster, the distance between the metal

and the water (both in cavity) is significantly shorter

(around 2.7 �A) in K(H2O)þ21 cluster and represents a

stronger K–O bond compared to that with the surfacemolecule. While the average O–O–O angle in

Na(H2O)þ19–21 clusters ranges from 101–104�, in K

(H2O)þ19–21 clusters the average angle ranges from 99–

103�. Even though the cavity radius is only slightly

decreased in both the cluster types, the O–O–O angles

are significantly decreased compared to those present in

the dodecahedral cage (108�) with an empty cavity. As

expected, the dodecahedral cage in both the cluster typesis significantly distorted (standard deviation, SD, of

around 20� for O–O–O angle) compared to the empty

cage (SD value of 3�).The stabilization energy (SE, Table 1) relative to

separated H2O molecules and the metal ion increases

with the cluster size for each cluster type. However, the

stabilization energy per H2O molecule and metal ion

(SEP) for K(H2O)þ19, K(H2O)þ20 and K(H2O)þ21 (13.0,13.1 and 12.4 kcal/mol respectively) shows a maximum

value at K(H2O)þ20 and a relatively larger drop in SEP

value beyond this size at K(H2O)þ21. The SEP values for

Na(H2O)þ19, Na(H2O)þ20 and Na(H2O)þ21 (14.0, 13.8 and

13.7 kcal/mol respectively), on the other hand, show a

maximum value at Na(H2O)þ19 rather than at Na(H2O)þ20and no abrupt drop in SEP value beyond Na(H2O)þ20. Itis interesting that both the B3LYP/6-311++G** opti-mization method (shown in parentheses, Table 1) as well

as B3LYP/6-311++G**// HF/6-31G* method provide

very similar results and lead to the same conclusion. It

should be pointed out that the SEP value provides in-

formation about the energy requirement for the disso-

ciation of a water molecule from the cluster and hence,

indicates survival possibility of a cluster once it is

formed. Thus, if the above clusters are formed at thesame rate, on the basis of SEP values one would expect a

magic peak for K(H2O)þ20 and no such peak for

Na(H2O)þ20 clusters.

6. Charges on metal atoms in different clusters

The K atom retains a significant positive charge in

each of the K(H2O)þ19–21 cluster. The charge is maximum

(+1) for K in the dodecahedral cage and is decreased toaround +0.8 and +0.6 in 19 and 21-mer water clusters

respectively. On the other hand, the charge on Na atom

is slightly positive (+0.2, Table 1) for 19-mer cage and

+0.1 and 0 for 20 and 21-mer cages respectively. The

latter charges on Na are very close to that obtained for a

neutral Na(H2O) cluster (Table 1). Interestingly, Na in

Na(H2O)þ has +1 charge suggesting that in the process

of cage formation around, the metal loses its charge tothe cage structure. The K is highly positive in both

K(H2O)þ as well as those in which it occupies the do-

decahedral cage cavity. From this observation, one can

predict the following electron affinity (EA) trend for

metallic ions and the dodecahedral cage:

EA of Naþ > EA ofðH2OÞþ20ðdodecahedral cageÞ> EA of Kþ

That is, when a Naþ ion is placed in a dodecahedral cage

cavity, the metal ion will pick up an electron from theneutral cage forming a neutral metal atom and a posi-

tively charged cage structure around. Similarly, when

the Kþ ion is placed in a dodecahedral cavity, the metal

will retain its positive charge within the cavity.

7. Cluster formation rates, charges on the metal and

magic peak

Even though the results of the present study are in

disagreement with the earlier MD study, they are in line

with the experimental results of Steel et al. [5] which

suggest the presence of a magic peak for K(H2O)þ20 and

no such magic peak for Na(H2O)þ20 cluster. One possible

reason for the disagreement between the MD and the

present result is, the dodecahedral structure used in theMD study is less stable (comparable to structure 3 in

Fig. 1 with at least one set of 3 NHB H atoms in ad-

jacent locations) than the one used (structure 1, maxi-

mum of 2 NHB H in adjacent locations) in this study.

As mentioned above, Na remains as a neutral atom and

K remains as a Kþ ion within the dodecahedral cage. By

examining the energies of the Na(H2O) and Na(H2O)þ

clusters (Table 1) one can find that a large increase in theSE (7–26 kcal/mol) and SEP (4–13 kcal/mol) values take

place when the neutral Na metal is replaced by a posi-

tively charged Naþ ion suggesting a strong interaction

between the positively charged metal ion and the water

molecule. In other words, a neutral metal will exhibit

much smaller attraction for water molecules and will be

much less effective in nucleating water molecules

around. Based on above analysis, one can conclude that

A. Khan / Chemical Physics Letters 388 (2004) 342–347 347

a positively charged Kþ ion is expected to favor nucle-

ation of water molecules and hence, the formation of a

dodecahedral cage around it whereas a neutral Na atom

will not favor nucleation of a dodecahedral cage around.

Among the 19–21-mer water clusters, K has the maxi-mum amount of positive charge in K(H2O)þ20 and Na

has the maximum charge in Na(H2O)þ19. Hence, the

nucleation rate may favor the formation of these clusters

over the other two. Hence, both the thermodynamic

stability (SEP values, discussed above) as well as kinetics

favor the formation of K(H2O)þ20. On the other hand,

neither of these factors favors the formation of

Na(H2O)þ20, rather favors the formation of Na(H2O)þ19.Thus, a relatively large mass intensity for K(H2O)þ20 andlow mass intensity for Na(H2O)þ20 are expected relative

to their neighboring sizes, and hence, a magic number

peak is expected for K(H2O)þ20 but not for Na(H2O)þ20cluster and agrees with the experimental results.

8. Concluding comments

The M(H2O)þ19–21 clusters having broken and dis-

torted dodecahedral cages (with a metal in cavity) are

studied by optimizing geometry at the B3LYP/

6-311++G** level. The basic dodecahedral structure

required for building M(H2O)þ19–21 clusters (starting ge-

ometry) is obtained from calculations on dodecahedral

isomers at the B3LYP/6-311++G**// B3LYP/6-31G*level. Even though these dodecahedral isomers are

studied with a smaller basis set, the SE and SEP values

are expected to be quite reliable. This conclusion can be

drawn from various test results discussed above as well

as the SE and SEP values (Table 1) that are computed

by different methods.

Among the M(H2O)þ19–21 clusters, the SEP value

shows a maximum for M(H2O)þ20 cluster when M¼K(in cavity), and no such maximum when M¼Na. While

in K(H2O)þ20 cluster the +1 charge is on the K atom (in

cavity) and for Na(H2O)þ20 cluster the +1 charge is on the

cage surface with a neutral Na within the cavity.

Acknowledgements

The author acknowledges help from Sadi Khan for a

utility program, and Jeff Nucciarone and the Numeri-

cally Intensive Computing Group at the Center forAcademic Computing at Penn State for generous com-

putation time provided. Helpful discussion with Prof.

Castleman is also acknowledged.

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