Atomic positions

Much surface work is concerned with the local atomic structure associated with a single domain. Some surfaces are essentially bulk-temiinated, i.e. the atomic positions are basically unchanged from those of the bulk as if the atomic bonds in the crystal were simply cut. More coimnon, however, are deviations from the bulk atomic structure. These structural adjustments can be classified as either relaxations or reconstructions. To illustrate the various classifications of surface structures, figure A1.7.3(a ) shows a side-view of a bulk-temiinated surface, figure A1.7.3(b) shows an oscillatory relaxation and figure A1.7.3(c) shows a reconstructed surface. 

The three-dimensional synnnetry that is present in the bulk of a crystalline solid is abruptly lost at the surface. In order to minimize the surface energy, the themiodynamically stable surface atomic structures of many materials differ considerably from the structure of the bulk. These materials are still crystalline at the surface, in that one can define a two-dimensional surface unit cell parallel to the surface, but the atomic positions in the unit cell differ from those of the bulk structure. Such a change in the local structure at the surface is called a reconstruction. 

The most popular of the scanning probe tecimiques are STM and atomic force microscopy (AFM). STM and AFM provide images of the outemiost layer of a surface with atomic resolution. STM measures the spatial distribution of the surface electronic density by monitoring the tiumelling of electrons either from the sample to the tip or from the tip to the sample. This provides a map of the density of filled or empty electronic states, respectively. The variations in surface electron density are generally correlated with the atomic positions. 

The integrand in this expression will have a large value at a point r if p(r) and p(r+s) are both large, and P s) will be large if this condition is satisfied systematically over all space. It is therefore a self- or autocorrelation fiinction of p(r). If p(r) is periodic, as m a crystal, F(s) will also be periodic, with a large peak when s is a vector of the lattice and also will have a peak when s is a vector between any two atomic positions. The fiinction F(s) is known as the Patterson function, after A L Patterson [14], who introduced its application to the problem of crystal structure detemiination. 

In all of these stmctures the atomic positions are fixed by the space group syimnetry and it is only necessary to detennine which of a small set of choices of positions best fits the data. According to the theory of space groups, all stmctures composed of identical unit cells repeated hi three dimensions must confomi to one of 230 groups tliat are fomied by coinbinmg one of 14 distinct Bmvais lattices with other syimnetry operations. 

Referring to figure Bl.8.5 the radii of the tliree circles are the magnitudes of the observed structure amplitudes of a reflection from the native protein, and of the same reflection from two heavy-atom derivatives, dl and d2- We assume that we have been able to detemiine the heavy-atom positions in the derivatives and hl and h2 are the calculated heavy-atom contributions to the structure amplitudes of the derivatives. The centres of the derivative circles are at points - hl and - h2 in the complex plane, and the three circles intersect at one point, which is therefore the complex value of The phases for as many reflections as possible can then be... 

Diffraction is based on wave interference, whether the wave is an electromagnetic wave (optical, x-ray, etc), or a quantum mechanical wave associated with a particle (electron, neutron, atom, etc), or any other kind of wave. To obtain infonnation about atomic positions, one exploits the interference between different scattering trajectories among atoms in a solid or at a surface, since this interference is very sensitive to differences in patii lengths and hence to relative atomic positions (see chapter B1.9). 

It is relatively straightforward to detemiine the size and shape of the three- or two-dimensional unit cell of a periodic bulk or surface structure, respectively. This infonnation follows from the exit directions of diffracted beams relative to an incident beam, for a given crystal orientation measuring those exit angles detennines the unit cell quite easily. But no relative positions of atoms within the unit cell can be obtained in this maimer. To achieve that, one must measure intensities of diffracted beams and then computationally analyse those intensities in tenns of atomic positions. 

With XRD applied to bulk materials, a detailed structural analysis of atomic positions is rather straightforward and routine for structures that can be quite complex (see chapter B 1.9) direct methods in many cases give good results in a single step, while the resulting atomic positions may be refined by iterative fitting procedures based on simulation of the diffraction process. 

Table 81.21.1. Surface stmctural detemiination methods. The second colunni indicates whether a technique can be considered a diffraction method, in the sense of relying on wave interference. Also shown are statistics of surface stmctural detemiinations, extracted from the Surface Stmcture Database [14], up to 1997. Counted here are only detailed and complete stmctural determinations, in which typically the experiment is simulated computationally and atomic positions are fitted to experiment. (Some stmctural detemiinations are perfomied by combining two or more methods those are counted more than once in this table, so that the colunnis add up to more than the actual 1113 stmctural detemiinations included in the database.)...

So it is essential to relate the LEED pattern to the surface structure itself As mentioned earlier, the diffraction pattern does not indicate relative atomic positions within the structural unit cell, but only the size and shape of that unit cell. However, since experiments are mostly perfonned on surfaces of materials with a known crystallographic bulk structure, it is often a good starting point to assume an ideally tenuinated bulk lattice the actual surface structure will often be related to that ideal structure in a simple maimer, e.g. tluough the creation of a superlattice that is directly related to the bulk lattice. 

Broadening of spots can result from themial diffuse scattering and island fomiation, among other causes. The themial effects arise from the disorder in atomic positions as they vibrate around their equilibrium sites the sites themselves may be perfectly crystalline. 

Since ED by a surface is a complicated process, there is no routine method available to directly and accurately extract atomic positions from the experimental data. Direct holographic methods have been proposed [24], but have not yet... 

DeWit A G J, Bronckers R P N and Fluit J M 1979 Oxygen adsorption on Ou(110) determination of atom positions with low energy ion scattering Surf. Sc/. 82 177-94... 

Our intention is to give a brief survey of advanced theoretical methods used to detennine the electronic and geometric stmcture of solids and surfaces. The electronic stmcture encompasses the energies and wavefunctions (and other properties derived from them) of the electronic states in solids, while the geometric stmcture refers to the equilibrium atomic positions. Quantities that can be derived from the electronic stmcture calculations include the electronic (electron energies, charge densities), vibrational (phonon spectra), stmctiiral (lattice constants, equilibrium stmctiires), mechanical (bulk moduli, elastic constants) and optical (absorption, transmission) properties of crystals. We will also report on teclmiques used to study solid surfaces, with particular examples drawn from chemisorption on transition metal surfaces. 

Step 1 A short conventional MD simulation (typically extending over a few lOOps) is performed to generate an ensemble of protein structures x 6 71 (each described by N atomic positions), which characterizes the initial conformational substate. The 2-dimensional sketch in Fig. 9 shows such an ensemble as a cloud of dots, each dot x representing one snapshot of the protein. 

In our implementation of SMD, modified versions of VMD and Sigma communicate with each other using a customized, lightweight protocol. Sigma sends atomic positions resulting from each molecular dynamics time step to VMD for display. When the user specifies restraints on parts of the displayed model, VMD sends them to Sigma, where they are converted into potential-well restraints added to the force field [21]. 

The most well-known and at the same time the earliest computer model for a molecular structure representation is a wire frame model (Figure 2-123a). This model is also known under other names such as line model or Drciding model [199]. It shows the individual bonds and the angles formed between these bonds. The bonds of a molecule are represented by colored vector lines and the color is derived from the atom type definition. This simple method does not display atoms, but atom positions can be derived from the end and branching points of the wire frame model. In addition, the bond orders between two atoms can be expressed by the number of lines. 

One drawback to a molecular dynamics simulation is that the trajectory length calculated in a reasonable time is several orders of magnitude shorter than any chemical process and most physical processes, which occur in nanoseconds or longer. This allows yon to study properties that change w ithin shorter time periods (such as energy finctnations and atomic positions), but not long-term processes like protein folding. 

One cannot simply optimi/e the position of each atom in sequence and say the job is done. Any change in an atomic position brings about a small change in the forces on all the other atoms. Optimization has to be repeated until the lowest rnoleeular potential energy is found that satisfies all the forees on all the atoms. The final location of an atom will, in general, be at a position that is some small distance from the position it would have if it were not influenced by the other atoms in the molecule. 

Choose an initial set of atom positions. The same techniques used for molecular dynamics simulations are applicable. 

Atomic positions (connectivity) must be the same in all resonance structures only the electron posi tions may vary among the various contributing structures... 

Structural formula J is a Lewis structure of nitro methane K is not even though it has the same atomic positions and the same number of electrons... 

If the atomic positions at a particular time, t, are rj (t), the velocities, Vj (t), are the first derivative of the positions with respect to time (equation 25). 

---