Paula Mori-Sánchez, Aurora Costales , Miguel A. Blanco , A. Martín Pendás , and Víctor Luaña
Departamento de Química Física y Analítica.
Universidad de Oviedo, 33006-Oviedo, Spain
 Present address: Department of Physics, Michigan Technological University, Houghton, Michigan 49931-1295, USA.
 Present address: Department of Chemistry, McMaster University, Hamilton, Ontario L8S 4M1, Canada.
Metals have played a prominent role in the solid state physics theory from the end of 19th century onwards. The metallic state is one of the most important states of matter, being present not only on metals but rather in the very high pressure phases of otherwise insulating systems. In fact, it has been conjectured that all solids under sufficiently high pressure either metallize or descompose.
In the last fifty years, qualitative and quantitative models of metallic state have been developed, including different methodologies to calculate the crystalline wavefunction. The result has been the growth of an extraordinary theoretical baggage that explains the electronic, and hence, thermal, magnetic and mechanic properties of metals. Nevertheless, the dominant interpretation of metallic bonding lays in very rough approximations to the exact quantum mechanical solutions.
The Atoms In Molecules (AIM) theory by R.F.W. Bader and co-workers  is the rigorous theory of bonding based on the postulates of quantum mechanics. It has been widely used in small organic and organometallic molecules and only recently extended to solid state systems [17, 18, 19]. The very active work being done in this area includes the topological analysis of both theoretical and experimental high quality crystalline densities of ionic, covalent and van der Waals compounds. Complete topological studies concerning metallic systems, on the other hand, are specially difficult and very few works have been published [4, 5].
Table 1: The description of the experimental geometries of the eight metallic phases selected in this study includes: space group, atomic positions, Z molecules per unit and lattice parameter.
Most chemical elements are metallic solids under atmospheric pressure. Although a single crystalline form is known for some metals, most of them undergo structural changes induced by temperature and/or pressure and numerous high-pressure or temperature polymorphs have been reported. With few exceptions, metals crystallize with one or more of these three structures (see Table 1): the Hexagonal and Cubic Close-Packed structures (HCP and FCC respectively) and the Body-Centred Cubic (BCC) structure (the high-temperature form of metals which are closed-packed at lower temperatures).
Our basic aim in this report is to contribute to the knowledge of metallic bonding in the light of the AIM new paradigm. To this end we have selected for detailed study eight prototypical metallic systems: the BCC phases of Li, Na and -Fe; the FCC phases of Al, Cu and -Fe; and the HCP phases of Be and Mg. Next section describes the method used to obtain the wavefunction and the topological properties on each system. Section 3 describes and discusses the results. Some final remarks are collected on section 4.
The HF-LCAO CRYSTAL95 method (Pisani and Dovesi, 1980 ) has been used to determine the total electron density for each crystal. In the CRYSTAL95 code the linearized Hartree-Fock-Roothaan (HFR) equations in the Bloch function representation are solved in the reciprocal space, by using a basis set made of a small number of atomic orbitals per atom expressed as a linear combination of Gaussian Type Orbitals (GTOs). The accuracy of the bielectronic Coulomb and exchange series calculation is limited and truncated according to overlap-like criteria defined in the input. These aproximations severely limit the number and spatial extension of the GTOs employed. Therefore, high quality standard molecular basis sets cannot be adopted without modifications, and the valence basis functions must be variationally reoptimized for every crystal. Aiming at obtaining a basis set library of homogeneous and reasonable quality we have performed systematic optimizations of diffuse valence shell exponents in several extended basis sets. This task is extremely slow and expensive in metallic systems, where much attention must be devoted to the careful selection of the computational parameter that controls the Fermi level and density matrix determination at each state of the self-consistent procedure. Even with those limitations CRYSTAL95 is one of the best available codes able to get the all electron crystalline density that we need for our AIM topological analyses.
We have finally used optimized double- or triple- basis sets in our definitive CRYSTAL95 calculations. To check our results, we have examined the E(V) equation of state, as well as the band structure and the total density of states (DOS) at the experimental geometry of each system. As a general rule, our results agree with the experimental data available.
The topological properties of the CRYSTAL95 computed electron densities have been obtained using the CRITIC code .
Let us start by analyzing the topologies of BCC Li, Na and -Fe at the experimental geometry, which are described in Table 2. We have found three different topological schemes, i.e. three different arrangements of critical points (CPs). It is useful to consider n, b, r and c the total number of nuclei, bonds, rings and cages in the unit cell, respectively, and , , and the total number of simmetrically different CPs of each type. We can classify the topological structures by giving the values of: .
Table 2: Li, Na and Fe BCC topological structures. Graph plots have been designed with tessel  and rendered with POVRay . Big and small spheres illustrate the atomic and bond CPs positions, respectively. Bond densities () are given in e/bohr and bond Laplacians () in e/bohr. Notice NNM ocurrence in Li (white) and Na (green) topologies. In the Li structure there are sandwich configurations among metallic atoms and the shady rings of interstitial NNMs. In the Na structure, however, first neighbors are linked through bridge internuclear NNMs. On the other hand, the Fe structure is NNM-free and shows the expected bonds between first and second neighbors.
-Fe shows the simplest structure, with closed-shell () Fe-Fe bonds between first and second neighbors. 14 bond, 24 ring and 12 cage CPs complete a consistent set, 1(2)2(14)1(24)1(12), that fulfils the Morse's relation, n-b+r-c=0. Li and Na topologies are much more peculiar exhibiting maxima of the electron density at positions well separated from the nuclei, i.e. non-nuclear maxima (NNMs). Such maxima behave as attractors in the gradient vector field giving rise to pseudoatoms (N) surrounded by zero-flux surfaces.
In the Na structure, 2(10)1(16)1(12)1(6), second neighbor bonds have dissapeared and first neighbors are not linked each other but through bridge midway NNMs. The N-Na bonds are closed-shell interactions with and values smaller than those found in -Fe. The Li topology, 2(14)2(36)3(38)1(16), is even more exotic. Diatomic interactions are less significant and delocalized multi-centered bonds appear in the structure. Metallic atoms are octahedrically coordinated to 6 ring CPs located in the middle of square interstitial NNM arrangements over the unit cell faces. Each couple of these unusual Li-r bonds in the second neighbor directions generates a sandwich configuration with a low and positive bond Laplacian. At the same time, each NNM is linked to 4 nearest-neighboring NNMs by shared-shell () bonds producing a connected peripheral anionic network spread through the whole crystal. The ocurrence of NNM in both Li and Na theoretical densities was also predicted by Mei and co-workers .
The most significant feature of metallic topologies is the very low value found for densities (less than 0.06 e/bohr) and Laplacians (less than 0.01 a.u.) at the bond CPs, in all crystals studied. This is a consequence of the great planarity and diffuse nature of the electron density in the interatomic region. The ratio of the minimum density value at a cage point and the maximum density at a bond point, , provides a measurement of the density planarity and, therefore, can be understood as a topological metallicity index. The closer this index is to 1, the smaller the bond curvatures (, , and ) and the density differences () among valence CPs will be. In such situations very shallow and highly delocalized NNMs are usually found. Our calculations clearly show zero-pressure internuclear NNMs in Na, and interstitial NNMs in Li, , Be and Mg.
On the other hand, the metallicity index increases with hydrostatic pressure and, accordingly, the eight phases certainly present interstitial NNMs if sufficiently compressed. In all topological structures with NNMs, valence is lower than 0.01 a.u.
The flatness of the valence density is the source of another new and significant topological behavior that we have observed. Small changes of the crystalline volume are enough to change the density and produce new topological structures, that is, topological polymorphs. In this way, the three graphs in table 2 are recovered for each BCC metal by varying computational conditions or crystal size.
The described features clearly separate metallic from ionic and covalent topologies. As a simple example, the replacement of central Na by a F atom in the metallic BCC lattice to produce the ionic B2 phase of NaF drastically changes the topology. Bond densities and Laplacians are slightly increased and the metallicity index is greatly decreased (see table 3). The diamond phase of carbon (C-diamond), a prototypical example of covalent crystal, exhibits a network of strong shared bonds with densities larger than in both ionic and metallic systems, and large negative Laplacians.
Table 3: NaF (B2 form) and C-diamond topological structures, as typical ionic and covalent systems respectively, to be compared with the metallic topologies in a previous table.
The ratio between the parallel and perpendicular curvatures at a bond CP, , is a parameter to measure the directional character of the bonding. We have found a gradation from the very directional bonds with ratios close to zero, like in shared-shell bonds (N-N in Li or C-C in C-diamond) up to the big values typical of sandwich interactions (Li-r, for instance), being the usual closed-shell bonds (Na-F, F-F, Fe-Fe, etc.) somewhere in the middle.
Analogous results have been obtained in the FCC and HCP phases. In relation to FCC phase, -Fe, Cu and Al present the three different topological structures described in Table 4. Cu shows the simplest structure, 1(4)1(24)1(32)2(12), with closed-shell bonds among the nearest neighbors in accordance with Aray and co-workers' previous results . Each bond splits into one ring and two curved tense bonds, both between the first neighbors, in the Al 1(4)1(48)2(56)2(12) topology. This structure is highly unstable and easily converted to either Cu or -Fe structures. We have again found interstitial NNMs in the -Fe scheme. Each tetrahedral cage in the 1(4)1(48)2(56)2(12) structure collapses with the nearest ring and bond CPs generating a NNM in the original cage position and 4 very near closed-shell n-N bonds. In this way, a fluorite-like structure (Fe) is produced, with interstitial NNMs playing the anionic role. These very small NNMs are not linked to one another.
Table 4: -Fe, Al and Cu FCC chemical graphs. Unlike Al and Cu, Fe shows a fluorite-like crystal structure, with small interstitial NNMs playing the anionic role.
Figure 1: The CRYSTAL95 electron density map of Be (110) plane for the density region 0.03-0.05 e/bohr with 0.0013 e/bohr intervals. A NNM can be easily seen in the middle of each nearest tetrahedral sites (*).
In the HCP phase of Be and Mg, we have found NNMs at the center of trigonal bipyramids of metallic atoms, in good agreement with the experimental density maps reported by Iversen and co-workers [6, 7] in Be, and Kubota and co-workers  in Mg. The electronic isodensity map of the Be (110) plane depicted in Fig. 1 clearly shows a NNM between two nearest tetrahedral sites. In experimental geometries, both Be and Mg share the same chemical graph (see Fig. 2) with N-N and n-N bonds between layers as well as basal n-N bonds. N-N bonds are shared-shell interactions whereas n-N bonds are ionic interactions between closed shells. Complete topological graphs are, however, different. The Mg structure, 2(4)3(16)2(14)1(2), is simpler, with two types of ring CPs and only one type of cages in the octahedral holes. In the Be structure, 2(4)3(16)3(20)2(8), interactions between adjacent layers are stronger and most of ring and cage CPs are found in the interlayer space, explaining its extremely low c/a ratio (1.57) quite different from the ideal value (1.63).
Figure 2: HCP Mg and Be chemical graph. Notice that the HCP structure may be described as a set of vertices sharing trigonal bipyramids and interstitial NNMs have been found at their centers.
In a topological structure, nuclei compete for attracting the electron density and define their attraction domains, i.e. atomic basins or quantum atoms. A consequence of the great topological variability of metallic densities is the diversity of atomic shapes found in experimental geometries.
Basins are topologically equivalent to polyhedra, in the sense that
they have faces, edges and vertices, which satisfy Euler's relationship
vertices=2). Each face is the attraction surface of a bond CP, each ring the attraction line of a ring CP, and vertices are minima of the electron density. Simple metals with low topological index (Al, Cu, -Fe) present typical ideal polyhedric basins with linear edges and planar faces. As an example, we show in Table 5 the atomic basin of Cu in the FCC phase. Each Cu atom is linked to twelve nearest neighbors forming a regular cubooctahedral basin with vertices in the octahedral and tetrahedral cages. Appearance of NNMs in the remainder crystals is a ionization process which produces electrides formed by convex metallic cations and more delocalized concave anionic NNMs. Table 5 shows how metallic atoms become denser and rounder during ionization and merge into Fermi-gas behavior. Li clarly represents the final limiting situation, where metallic atoms are spherical cores of highly compact and correlated density, linked indirectly through ionic bonds to much more disperse and asymmetrical NNMs. This picture is singularly close to the traditional model of a uniform density electron gas submerged in a set of compensanting cationic cores.
Table 5: Atomic basins and topological charges () of some simple metals. The sequence can be understood as a ionization process towards the Fermi-gas behavior.
Analogous sequences have been determined in each crystal by studying the topological evolution through increasing compression from the large distance limit.
We have determined how the topology of the electron density changes by varying both the crystal volume and shape. The huge variability of metallic densities produces a rich topological polymorphism.
As a first step, we have employed the procrystal approximation to the crystalline density (the total density is built adding up radial atomic densities) in order to obtain fast maps of metallic topologies in the three, HCP, FCC and BCC phases.
Figure 3: Structural CRYSTAL95 diagrams of Li, Na and Fe BCC, to be compared with the Li procrystal prediction.
In Fig. 3 we have represented the polymorphism of Li at the procrystal level in terms of the only free control parameter: the crystal reduced volume . We have characterized 21 topological structures according to the 21 crystal size ranges contained in the figure. Each range is biunivocally related to one stability region of a topological structure in the nuclear configuration space, and structural transitions among them are catastrophes in Thom's sense . Most representative chemical graphs are shown in Table 6. All the topologies, excepting the large distance limit one, exhibit either internuclear or interstitial NNMs. For reduced volumes between 0.72 and 0.52, particular bonds appear connecting Li atoms to ring or N-N bond CPs. These structures are highly unstable due to the required coincidence of the monodimensional and bidimensional attraction basins of the connected CPs. That is reason behind having 14 different structures in such a small range of the lattice parameter. If we further compress the crystal, the topological structure is held unchanged until, finally, the core shells start to interact following an equivalent process than valence shells.
Table 6: Most representative procrystal Li graphs. Internuclear or interstitial NNMs (N, green) as well as bonds-to-bonds CPs or bonds-to-rings CPs are usually found. Linked atoms in each structure are listed below.
The subsequent analysis of CRYSTAL95 densities ratifies the validity of the promolecular approximation. In general, HF maps are contained in the procrystal ones and only the less stable structures are modified, the self-consistency influencing most the bond CP regions. The CRYSTAL95 polymorphism map of BCC Li (also displayed in Fig. 3) shows the displacement of each polymorph to larger values of the reduced volume and the occurrence of three new topologies, as compared to the procrystalline map. The CRYSTAL95 map contains only 12 polymorphs instead of the 21 ones found with the procrystal model, but this can be simply a consequence that much smaller distances were available for analysis with this model. Definitively, self-consistence favours topological change by transferring density towards the valence region.
Compared to the large complexity of BCC Li procystal map, the HCP Be map shows about 11 polymorphs when both a and c lattice parameters are varied, and the much open FCC phases of Al, Cu and -Fe present just one and the same polymorph.
The CRYSTAL95 maps for HCP Be and Mg have 12 and 6 polymorphs, respectively, when c/a ratio is held fixed to the experimental value and the cell volume alone is varied from 0.74 to 1.8 times the equilibrium volume. Our equivalent CRYSTAL95 exploration of FCC Al and -Fe revealed three different polymorphs, whereas Cu only presented the large distance limit topology already found with the procrystal model.
The location of structural frontiers depends, on the other hand, on the nature of the system. Dotted lines in Fig. 3 connect identical catastrophes among different BCC crystals, so their parallelism indicate the analogy of the different topological sequences found in each metal. The sucessive appearance of the different schemes on compressing is progressively postponed in going from Li to Na and -Fe. This behavior coincides with the supossed increase in covalence from Li to Fe.
The same happens in FCC metals in the Cu-Fe-Al series, according to the inverse variation of electronic conductivity. In the HCP phase, Be exhibits richer topological polymorphism and higher ionic character than Mg. Among the 9 structures found in Be and the 5 in Mg none shows internuclear NNMs, although the procrystalline model does in Be, and most polymorphs accomodate from 1 up to 3 types of interstitial NNMs in the bipyramidal space. Nucleus-ring (n-r) bonds are again found both in Be and Mg. As an illustration of the whole HCP topologies obtained, we present in Table 7 all the Mg topological structures in terms of the basic Mg units.
Table 7: Mg HCP topological polymorphism in terms of the basic bipyramidal Mg units. Open and full circles represent Mg atoms and interstitial NNMs, respectively. Triangles and crosses point out ring and cage CPs, respectively.
Very interesting is the fact that, in spite of the many differences between the phases and crystals analyzed in this work, all of them follow the same common general sequence, that we have summarized in Table 8. Starting from the large distance limiting topology, a couple of bound internuclear NNMs appear between pairs of first neighbors which are later collapsed to produce a single internuclear NNM. As the hydrostatic pressure continues increasing, isolated internuclear NNMs get bound each other forming an increasingly more tensed connected network. Finally, the accumulated tension in the N-N bonds moves the NNMs from internuclear to interstitial positions. Is in these circumstances when the most novel topological phenomena appear, such as bonds-to-bonds CPs or bonds-to-rings CPs.
Table 8: General common sequence that summarizes how topological structures evolve in relation to the internuclear distance in all the phases studied. From left to right: (A) large distance limit topology; (B) topology with internuclear NNM's; (C) connected network of internuclear NNM's; and (D) interstitial NNM's.
A recent controversy has surrounded the ocurrence of NNMs in the theoretical densities of molecules and crystals and in Si, Be and Mg experimental density maps reconstructed from X-ray form factors by using the maximum entropy method (MEM). Some authors plead for their real existence [9, 6, 7, 8,10] and others argue NNMs to be an artifact of the MEM technique [12, 13]. With the object of going deeply into this controversy we have studied the general electron density evolution in relation to the internuclear distance in molecules and crystals . Some of our results will be presented here.
Let us start with the simplest example, a homodiatomic molecule A(A-A'), with internuclear separation . According to the promolecular model, the molecular density is , where is the in vacuo spherical atomic density of atom A. The second parallel () and perpendicular () derivatives of the electron density at the internuclear midpoint depend on the second () and first () radial derivatives of the atomic densities, respectively. As is negative definite and is bounded, the midpoint is a maximum (with three negative curvatures) if it is located in a nonconvex region. In fact, we have found that all the homodiatomics formed by nonconvex atoms (Z=3-6, 16-18) display ranges with NNMs at this promolecular level. Let us call this ranges stability windows.
Electron-nuclei and electron-electron interactions introduce limited nondestructive effects on the promolecular densities. We have done HF and CISD GAMESS  calculations, using valence triple- basis sets extended with polarization and diffuse functions, in all homodiatomic molecules up to the third period. A decrease of and an increase (in absolute value) of is encountered, favouring NNMs even in monotonically convex atoms (Z=3-9, 11, 14-18, at least). This is the situation in the N molecule. As we have depicted in Fig. 4, is positive definite and there is not NNM stability window at the promolecular level, but it arises both at HF and CISD levels. As a general rule, SCF and correlation effects make minima deeper and the actual stability windows wider and slightly deplaced towards smaller distances, as compared with the promolecular results.
Figure 4: Promolecular in comparison with HF and CISD values in N molecule.
We have summarized our results in Fig. 5, where we show NNM windows for all the diatomics studied. It is clearly seen that the position of the promolecular minima of is a very good indication of the location of a plausible NNM. All nonconvex and almost all convex atoms (excepting Ne, Mg and Al) show NNMs. And even more, the position of the windows is a periodic property, depending on Z and, therefore, atomic in nature. The basic organizing principle underlying this fact is the atomic shell structure. As two identical atoms interact, the Laplacian at the bond point follows the electronic shell structure, allowing us to speak of bond between valence shells, internal shells, etc. As every atomic shell is associated with one negative and one positive Laplacian region, a succession of closed-shell interactions, shared-shell interactions and NNM stability ranges are found for each electronic shell as both atoms approach from infinite. Equilibrium molecular geometries and the positions of the windows are decoupled, so that molecules at equilibrium in their ground states will only bear NNMs if their internuclear distances occur inside the windows, like in Li, Na, B and P. C-C distance in the C molecule is a frontier; there are not NNMs at greater distances, like in ethane, but they appear at slightly small separations, as happens with acetylene.
Figure 5: Stability windows of NNMs for the first, second, and third period homodiatomics. Full circles () indicate the position of the minima of the valence promolecular . Question marks indicate molecules lacking NNMs. Error bars point out the NNM windows at the HF (left) and CISD (right, only for selected systems) levels for each system. Dotted bars correspond to regions with two internuclear NNMs. Open squares () show the experimental equilibrium geometries of the diatomics, and open circles () the first neighbor distances in non-molecular solids. The double arrows include those diatomics with NNMs at the promolecular level.
In homoatomic clusters there is a competence between interatomic lines and groups of 3 or closer atoms to accomodate NNMs. If the first effect dominates, internuclear NNMs are produced mainly due to the nonconvexity of the atomic radial density. In the other case, the cooperative contribution of several neighbors to an inner point of appropriate geometry is the reason behind interstitial NNMs. In our study of A (equilateral triangle), A (tetrahedron) and A (trigonal bipyramid configuration) clusters, we have observed a general NNM displacement from internuclear to interstitial positions while compressing. Promolecular models take the main features of this process. Internuclear NNMs appear at the internuclear distances predicted by the molecular analog, and interstitial NNMs extend internuclear stability windows towards upper and lower distances. On the contrary, systems that most likely lack internuclear NNMs, like Mg or Al, start developing interstitial NNMs by instability at the center of the clusters.
Homoatomic crystals behave locally as big clusters, subject to the same general principles we have presented. They will display interstitial NNMs if sufficiently compressed. Some of them, moreover, will present zero-pressure interatomic distances inside the stability windows of NNMs, either internuclear or interstitial. Full circles in Fig. 5 show distances between first neighbors in non-molecular homoatomic solids. Our analysis predicts NNM ocurrence in Li and Na (very difficult systems to deal with) but also in Be, supporting Iversen and co-workers [6, 7] X-ray difraction data analysis and our CRYSTAL95 calculations previously discussed. The silicon crystal present a frontier distance to the internuclear NNM window, and the crystal geometry does not favour interstitial NNMs, so the real ocurrence of NNMs is uncertain.
Heteronuclear combinations, on the other hand, behave quite differently, and NNMs could only appear under exceptional circumstances. Let us go back to the simplest case, a heterodiatomic molecule AB. At the promolecular level, the CP along the internuclear axes is not fixed by symmetry and occurs when , where is the CP position measured from nucleus A and r the internuclear distance. This condition is independent from the one that ensures a negative parallel curvature, . Given the narrow atomic nonconvex regions, the simultaneous fulfilment of both conditions at a point is most unlikely, independent from r. We have actually been unable to find NNMs in any of the 45 first and second period diatomic molecules we have studied.
The situation changes, however, in heteroatomic clusters or crystals containing homoatomic groups in narrow contact. Our preceding reasonings are then equally applicable. We have only found NNMs in GaP and MgCu intermetallic phases, out of nearly 60 heteroatomic crystals examined. In both systems, NNMs are located at the center of homoatomic clusters. In the GaP structure, we have found two types of NNMs, respectively located at the center of P and Ga tetrahedrons. The Friauf-Laves phase of MgCu, presents NMMs at the center of rectangular base bipyramids with two Mg apical atoms and Cu atoms at basal positions. The ocurrence of an internuclear NNM between two Mg atoms does not violate our original general principles. At the central cluster point, the out-of-plane derivative of the Cu fragment reduces the Mg parallel derivative, producing the transformation of the usual bond CP into a NNM.
With this, we can finally conclude that there is not a direct relationship between NNMs and the intrinsically metallic properties. NNMs are a normal step in the chemical bonding of homonuclear groups, if analyzed in the appropiate range of internuclear distances. For most elements, however, this range occurs far away from the stable geometry under normal thermodynamic conditions. The experimental search of these objects may thus be guided through the most appropriate systems and geometries.
One of the most important sides of AIM theory is that it gives a unique and universal description of chemical bonding, carrying out Lewis' initial wish. We have proved in this work that AIM can be applied to metallic systems to reproduce and quantify many of the classical ideas about metallic bonding, but also to show genuinely new phenomena and concepts. As for the principal conclusions of our work we can highlight the following:
This work was done under finantial support of the Spanish Dirección General de Investigación Científica y Técnica (DGICYT), Project No. PB96-0559. P.M.S thanks the Spanish Ministerio de Educación y Cultura for the award of a FPU/FPI research grant. A.C. wants to thank the US Defense Department for the grant she is presently enjoying. M.A.B and A.M.P are supported by postdoctoral grants from the Spanish Ministerio de Educaci'on y Cultura.
Topological analysis of the electron density in simple metals
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