Thomas Crawford

Thomas Crawford

Superatoms

Senior Comprehensive Paper

4/11/07

Abstract

Superatoms are clusters of ions which, as a whole, can exhibit properties of specific chemical families, such as halides and alkaline earth metal ions. The unique mimicking characteristics of superatomic clusters are due to their ionization potentials, which act similarly to electron shell closings. Al13- was found to exhibit properties of the halides, whereas Al14++ exhibits alkaline earth metal properties, and have great electronic and geometric stability. Clusters of M@Au14 (M= Zr, Hf) and M@Au14 (M=Sc, Y) (where @ symbolizes a single atom encapsulated within a compound) were found to have very large HOMO-LUMO gaps, with the latter set of clusters having an electron affinity greater than that of elemental chlorine. [As@Ni12@As20]3- is a cluster whose superatomic core is protected by a cage, possibly allowing for rather unstable superatomic clusters or atoms to be stabilized. These superatoms will help pave new avenues of discovery across many fields in science.

I. Introduction

Though superatoms have been developed and createdonly recently, theyoffer a new path of study in inorganic materials production. Superatoms are defined as “clusters of atoms which seem to exhibit some of the properties of elemental atoms.”1As opposed to when two separate substances come together and form a compound whose properties are different than their constituents, such as sodium and chlorine, these clusters can come together to form compounds which can mimic the properties of elements. The purpose of this paper is to bring to light recent developments of a phenomenon which is currently being pursued by many different fields in science.

The beginning of the creation of superatomic clusters began in the 1960’s and 1970’s and were considered only as small molecules with no observed special properties or periodicity. Independent studies in the early 1980’s began to develop the image of a superatom as having afree-electron structure withthe positive ionic cores spread throughout the cluster, within an equally distributed, finite volume, ignoring the ionic bonding characteristics of the cluster. 2,3,4The nuclei and core electrons are spread out over a predefined potential well, with valence electrons filling up electronic shells,5 and the electron energy levels are grouped in a fashion similar to the nuclear shell model.6Thejellium model treats the cluster of atoms as a jelly-like blob, hence the name jellium.7Clusters on the order of thousands of molecules began to be developed, and little to no change in properties was observed between the gigantic and the tiny clusters.2

The creation and separation of metal clusters can be achieved by a variety of means. When necessary, the clusters are separated by means of mass analyzers as a wide variety of cluster sizes are produced.2 In general, the production of metal clusters relies on the vaporization of a metal source and the subsequent cooling of the vapor to allow for aggregation. Experimental procedures for cluster creation are discussed on page 23, section V.2

These superatoms fill their electron shells in the exact same manner as would normal atoms. However, studies have focused on ionization potentials of the metal cluster, as these potentials act similarly to filled electron shells (Figure 1). This is due to the fact that these ionization potentials have characteristic local maxima that match electron shell closures.3

Figure 1. Graph which shows the degeneracy of the states within the jellium model on the momentum scale, which are subshells. Shows how many electrons are required for an ionization potential subshell closure along with the corresponding subshell. 8

Metal cluster chemistry follows a different method of defining the principal quantum number n, called the nuclear convention, whereas the most general and well known method for chemists is the atomic convention. In the nuclear convention, the shells created are denoted by both an n and l. For l, the lowest possible state is n = 1. No physical analog exists between the atomic and nuclear principle number, n. l is also not bound by n, so states which would not normally be seen in the atomic orbital diagram appear in the nuclear orbital diagram.9,10 Figure 2 displays the difference between the atomic orbital and the nuclear orbital energy diagram.

Figure 2. A. Energy diagram for atomic orbital. B. Energy diagram for nuclear orbital. Not to scale.11

The occupancy for each level is determined by the equation

ms(2l+1) = ml(Equation 1)

or

2(2l+1) = ml(Equation 2)9

Similar to the stability characteristic of the filling of atomic orbitals, the nucleus may also have stability at certain filled shells. Figure 3 provides a look at how the degeneracy of some states varies depending on the shape of the potential well for the jellium cluster, specifically three-dimensional harmonic, intermediate and square-well potentials. Also, it is important to note that depending on the well’s shape state, the spaces between shell closings, as well as ordering of potentials may vary significantly. 2

Figure 3. From left to right, occupation of various energy levels in a three dimensional spherical, intermediate, and square potential well. Note that depending on the shape of the potential well, the ordering of states may be modified. 2

According to the Aufbau principle, or the build-up principle, each electron introduced to the atom should enter into the lowest energy filled orbital. The general order of filling proceeds as 1s22s22p63s23p64s23d10… and so on until there are no more electrons added to the atom. And each time an orbital is filled, the atom exhibitsa certain stability which is mainly found in the noble gases. Therefore, there are “magic numbers” which dictate the point at which one shell is completely filled. For examples, helium has a full 1s orbital and neon has a full 2s and 2p orbital. These magic numbers correspond to each filled shell:Thus the magic number for helium would be 2 for 2 electrons filling the 1s orbital; and the number for neon would be 8, for the filled 1s, 2s, and 2p orbitals. The sequence for these numbers is 2, 10, 18, 36, 54 and 86, corresponding to filled shells (Figure 4).12

Figure 4. Image of the concept of the Aufbau or building up principle.12

To summarize, superatoms rely heavily on the idea of magic numbers. That is, at certain points of cluster size, some clusters are more stable than other cluster sizes that preceded or anteceded the original. This is due to the fact that at some point, there is a closing of an electron shell, a shell in terms of ionization potential, or a structural shell. These closures occur at a specific number of atoms or electrons and are termed ‘magic numbers’ for their marked stability. The numbers vary, however, depending on the property focused on, either the structure or electron count. One point to remember is that there are no set list magic numbers; that is, the list is not arbitrarily made.These numbers are recorded and observed because they constantly arise from spectroscopy or experimental data for which there is a great abundance. For example, focusing on the number of electrons in shell closures will show that the noble gases are the most stable, as they have completed an electron shell. The quantum number shell closures would occur at every noble gas, being 2, 10, 18, 36, 54 and 86 electrons.Focusing on ionization potentials will lead to a different picture of stability, with stability being reached with every subshell closures.Closures of this kind occur at 2, 8, 18, 20, 34, 40, 58, 68, 90, 92, 106, 132, 138, and 156 electrons, which, following the nuclear convention, would translate to 1s1p1d2s1f2p1g2d1h3s2f1i3p2g electron configurations2. Structural closures vary greatly, though, as one must take into account a vast assortment of high-order geometric shapes.Icosahedra and cuboctahedra shell closures occur at 13, 55 and 147 atoms, whereas a tetrahedral structure needs 4, 10 and 20 atoms.8 For example, Icosahedra and cuboctahedra structures can be determined by the equation

N(K) = 1/3(10K3+ 15K2+ 11K1+ 3) (Equation 3)8

where N is the number of atoms, and K is the shell index. So, for the first appearance of an icosahedra or cuboctahedra, K = 1, and when used in the equation, the number of atoms needed is 13. For K = 2, the number required is 55. Table 1 displays a list of the numbers required for a complete shell in various structures. 8

Table 1: List of number of atoms needed to create a variety of geometric figures.8

II. Characteristics

Depending on the number of neutrons and/or protons in the nucleus, specific combinations give rise to a structure within the nucleus which exhibits an enhanced stability. For instance, helium has 2 protons and 2 neutrons, which are both magic numbers. Oxygen has 8 neutrons or 8 protons, which are also both magic numbers. In the case of these two, the magic number is reached when a geometric figure with a high degree of order is created by the protons or neutrons - for instance, a bipyramidal figure or an icosohedron.13, 14

Also, to add to the image of cluster stability, it is important to look at the effects of different subshells, sizes, and shapes. When adding in all intermediates to figure 1, a pattern emerges, which appears to be alternating bands of light and dark. However, more than just a graphical pattern, as depicted in Figure 5, at certain points, subshells build up to create a bunched state, known as a supershell.8

Figure 5. Bunching of subshells into states as depicted by the alternation of dark and light bands.8

This creation of an electronic shell was also observed in the 1984 Knight et al. paper, which stated that these structures can be seen when there is either a large dip in ionization energy with an increasing cluster size or a sharp increase or decrease in mass spectra peak intensity, as seen in Figure 6.8, 15

Figure 6. Graph of change in electronic energy versus number of sodium atoms. Each labeled peak represents a shell closure.15

Another characteristic of the supershell is its enhancementthrough a phenomenon called supershell beating. Due to the fact that components of an atom exhibit a wave-particle duality, the possibility exists of the potential for constructive and destructive interference that could occur between different orbital patterns. Specifically, interference existsbetween square and triangular wave orbitals inside a well of potential.4The constructive interference creates a large amplitude curve in the area of binding energies, whereas the destructive interference only provides a small amount of total interference. When plotted as a shell correction factor δF (being the difference between the cluster energy and the average part) versus N1/3(where N is the number of atoms in the cluster), a sine-wave like pattern begins to emerge. When extrapolated to fit different temperatures, and in this figure at absolute zero, a pattern is clearly visible with steps occurring at the magic number sequence, 2, 8, 20, 34, 58, 92 etc. (Figure 7).4

Figure 7. Shell Correction Factor (energy difference) versus N1/3. 4

Finally, the shape of the cluster must be taken into account. As stated previously, there exists an enhanced stability to some clusters of a specific size. Upon the addition of an extra atom to a closed-shell cluster, a new shell is created, with a size increase generally proportional to the cluster element’s interatomic distance. Clusters with an enhanced stability will have a central atom which may pose a problem with even-numbered clusters. Simply put, more elaborate schemes are necessary to create an even-numbered stable cluster. Clusters with high-order geometry, such as icosahedrons or cuboctohedrons, are also a great source of stability. According to a paper by Martin, et al., the first point at which a complete encapsulated icosahedral complex is formed from a total of 13 atoms (which can be viewed as Al@Al12, where @ symbolizes a single atom encapsulated within a compound), which is of great import due to topics addressed later in this paper. 8

To bolster the experimental data, a mathematical system known as DFT, or Density-Functional Theory, is employed to explain the observed experimental results. DFT was created to discern the structure of multi-component systems, especially for the condensed matter phase. Previously, the most common method to describe many-bodied systems looked at the wave-function properties of the atoms, which relied on Cartesian coordinates for every electron in the system to be properly described. Instead, DFT simply looks at the density as a function of three variables, easing the work load.16

Within DFT, there are two other commonly used forms of the construct -Kohn-Sham DFTand Local Density Approximation(LDA). First, Kohn-Sham is used to simplify the problem of many bodies interacting in a given cluster. Instead of looking at the many-bodied cluster as having interacting electrons and a static potential, Kohn-Sham defines the cluster as having non-interacting electrons within an effective potential. This effective potential includes Coulomb interactions between the electrons and the effective potential. LDA comes into play when these Coulomb interactions have to be determined, as defining the interactions otherwise requires a great deal of work. With LDA any energy exchange within the system is given an exact number easily matching the results of the interactions had they been obtained experimentally.16The LDA model is actually derived from the jellium model, as it treats the interaction energy in every point in the cluster’s space as an interaction energy of a non-interacting cloud of electrons.17

In relation to other methods of calculations, DFT is a more efficient system than previous means of determining structure, reactivity, and other properties. The Hartree-Fock method focuses on the wave functions of the electrons by using a complicated set of equations. These equations are processed until all possible combinations of wave functions are agreed upon within a certain criteria defined by the user. Another set of calculations are semi-empirical methods and ab initio methods. Semi-empirical methods rely mainly on using estimated integrals based on spectroscopy or physical properties of the atom and work within a series of parameters that set specific integrals within the calculations to zero. Ab initio methods attempt to calculate all secular determinant integrals. Again, while the results prove to be rather close to experimental results, it takes a large amount of computational calculations which require a good deal of time and computing power.18

III. Current Research

Walter Knight’s group was among the first to begin research on superatoms. Knight et al. published a paper in 1984 which focused on producing clusters of sodium atoms. The results showed that there were large peaks in mass spectra which occurred at NaN where N = 8, 20, 40, 58 and 92 (Figure 8).Other peaks were found to occur at 18, 34, 68, and 70, but were weaker due to the sensitivity of the parameters from the equation used to calculate these potentials. 15

Figure 8. Mass spectrum of Sodium ions. Note that each step shows the amount of cluster ions and not electron count, so each peak noted above is one step above the observed magic numbers. Peaks for 18, 34, 68, and 70 actually were found, but were weaker than the rest of the peak absorbencies.8, 15

They realized that the peaks occurred at those intervals because the structure exhibited a free-electron model with the 3s valence electrons were delocalized throughout the cluster.15This pioneering work helped spark the recent development into the field of metal cluster chemistry.

Castleman’s group has recently published a paper on aluminum clusters following the example of Leuchtner, et al. whosepaper proposed that a special inertness of certain anionic aluminum clusters, specifically clusters Al13- , Al27-, and Al37-exists. They were testing the reactivity of aluminum clusters to oxygen etching; that is, clusters were formed and then stripped away oxidatively of single aluminum atoms, one by one. Al13- , Al27-, and Al37- clusters could not be reduced further. As stated previously, the jellium model allowed for stable clusters in accordancewith the magic number rule. In this case, Aluminum, having 13 electrons (of them, three valence electrons), provides a total of 39 valence electrons in the Al13 cluster, one electron short of the jellium magic number of 40. Figure 1 illustrates that an electron ionization potential shell closure occurs at 40 electrons, which fills the 2p6 electron in the series 1s21p61d101f142s22p6. The addition of an extra electron, making the compound Al13-, satisfies this closed electron rule, causing it to become a very stable compound. In this respect, the Al13- cluster is similar to a halide, in that it has 1 valence electron to donate toward a compound.9

Apart from having characteristics of halide ions, these aluminum clusters do not form polyhalide-like compounds. For example, polyiodides form according to the equation I2n+1- in chains, with an I2 molecule as the center and either I- or I3- attaching to this center. I5- forms a chain with a V-shape, having two iodides attached to a single iodide between the two arms. As it was predicted that Al13- can act as a halide, it would seem that the clustercould also form these polyhalide chains.3However, such is not the case. Upon the addition of I2, the cluster favors the breaking of the I-I bond and the formation of an Al-I bond. By looking at the atomization energies of these two compounds, one can see that Al-I and I2, show that Al-I has a total energy of 3.83 eV, whereas I2 only has a 2.21 eV atomization energy. In addition, the compound Al-I3 exhibits an atomization energy of 8.75 eV, bringing the Al-I atomization energy to 2.92 eV per iodide atom. On an interesting note, though, while the Al-I bonding may not be structurally similar to polyhalide molecules, it maintains a certain stability for a total odd number of atoms in the molecule, for these Al-I clusters are most stable with an even number of iodine ions.3,19 The charge density of the HOMO and the addition of the 40th electron to the Al13 cluster were analyzed and it was found that the extra negative charge was delocalized throughout the cluster, further adding to this image of aluminum having a superhalide character.19In addition, another paper reported the electron affinity of the Al13- cluster as being 3.57 eV, which is just a few hundredths of an eV under the electron affinity of Cl, 3.61 eV.20

Upon the addition of more iodides, the Al13In- clusterexhibited an interesting pattern. It seemed that when Al13In-, with the addition of an even number n of iodine, showed a greater stability than odd number n (See figure 9B). When the Al13In- cluster has an odd number of n, the reaction produces an active site,an area of charge density, opposite from where the odd numbered atom is placed, as seen in Figure 9A. Also, when this cluster has an odd number of I, it bonds in a sigma-like fashion; when there is an even number, the bond exhibits a pi-like character. In each case of n up to 13, the cluster maintains an almost perfect icosohedral structure.3,19