Interatomic potential curve

The dispersive and exchange-repulsive interactions between atoms and molecules can be calculated using quantum mechanics, though such calculations are far from trivial, requiring electron correlation and large basis sets. For a force field we require a means to model the interatomic potential curve accurately (Figure 4.32), using a simple empirical... 

The shapes of the interatomic potential curves are approximations chosen for mathematical convenience. Such potential functions are generally used in discussions on a variety of properties of molecules and lattices optical absorption and luminescence, laser action, infrared spectroscopy, melting, thermal expansion coefficients, surface chemistry, shock wave processes, compressibility, hardness, physisorption and chemisorption rates, electrostriction, and piezoelectricity. The lattice energies and the vibration frequencies of ionic solids are well accounted for by such potentials. On heating, the atoms acquire a higher vibrational energy and an increasing vibrational amplitude until their amplitude is 10-15% of the interatomic distance, at which point the solid melts. 

A piezoelectric solid (e.g., quartz) acquires an electrical dipole moment upon mechanical deformation and, conversely, if it is subjected to an electric field E it becomes distorted by an amount proportional to the field strength E. The dipole moment disappears without the mechanical force. Piezoelectricity is only possible in lattices that do not have an inversion center. Electrostriction is also mechanical distortion in an electric field (strain proportional to E ) but ionic lattices that have a center of symmetry also show this effect. Figure 4.25 is a schematic representation of the source of these effects using the interatomic potential curve. A ferroelectric material is not only piezoelectric but its lattice has a permanent electric dipole moment (below its Curie temperature), which most other piezoelectric materials (such as quartz) do not have. 

Figure 7.9 The contributions of the electron kinetic and potential energy expectation values to the interatomic potential curve for H2. The internuclear term is also added to ensure the potential goes to zero at large R - The inset plot (bottom right) shows the electron density integrated over planes perpendicular to the bond axis with the basis set decay factor optimized at the potential minimum the dotted curve shows the density that would be obtained using the decay factor for an isolated H atom.

Fig. 5.37 Comparison of the calculated phonon dispersion curve for Al with the experimental values measured using neutron diffraction. (Figure redrawn from Michin Y, D Farkas, M ] Mehl and D A Papaconstantopoulos 1999. Interatomic Potentials for Monomatomic Metals from Experimental Data and ab initio Calculations. Physical Review 359 3393-3407.)...

When two atoms approach each other so closely that their electron clouds interpenetrate, strong repulsion occurs. Such repulsive van der Waals forces follow an inverse 12th-power dependence on r (1/r ), as shown in Figure 1.13. Between the repulsive and attractive domains lies a low point in the potential curve. This low point defines the distance known as the van der Waals contact distance, which is the interatomic distance that results if only van der Waals forces hold two atoms together. The limit of approach of two atoms is determined by the sum of their van der Waals radii (Table 1.4). 

Figure 1.3. Real-time femtosecond spectroscopy of molecules can be described in terms of optical transitions excited by ultrafast laser pulses between potential energy curves which indicate how different energy states of a molecule vary with interatomic distances. The example shown here is for the dissociation of iodine bromide (IBr). An initial pump laser excites a vertical transition from the potential curve of the lowest (ground) electronic state Vg to an excited state Vj. The fragmentation of IBr to form I + Br is described by quantum theory in terms of a wavepacket which either oscillates between the extremes of or crosses over onto the steeply repulsive potential V[ leading to dissociation, as indicated by the two arrows. These motions are monitored in the time domain by simultaneous absorption of two probe-pulse photons which, in this case, ionise the dissociating molecule.

The effectiveness of these forces differs and, furthermore, they change to a different degree as a function of the interatomic distance. The last-mentioned repulsion force is by far the most effective at short distances, but its range is rather restricted at somewhat bigger distances the other forces dominate. At some definite interatomic distance attractive and repulsive forces are balanced. This equilibrium distance corresponds to the minimum in a graph in which the potential energy is plotted as a function of the atomic distance ( potential curve , cf. Fig. 5.1, p. 42). 

Figure 2.2 Illustrative plot of the Lennard-Jones-Devonshire interatomic potential showing the force and the modulus curve for the pair interaction. Positive values indicate repulsion and negative values indicate attraction...

Fig. 6.11 The structural-energy differences between bcc and fee (full curves) and hep and fee (dashed curves) as a function of the relative atomic volume, fl/Q f°r sodium, magnesium, and aluminium. The curves in the upper panel (a) were predicted by Moriarty and McMahan (1982) using their first principles interatomic potentials. The curves in the middle and lower panels (b) and (c) were predicted by Pettifor and Ward (1984) using three terms (<1, + 2 + 3) and one term 3 respectively in their analytic interatomic potentials.

The form of the potential for the system under study was discussed in many publications [28,202,207,208]. Effective pair potentials are widely used in theoretical estimates and numerical calculations. When a many-particle interatomic potential is taken into account, the quantitative description of experimental data improves. For example, the consideration of three-body interactions along with two-particle interactions made it possible to quantitatively describe the stratification curve for interstitial hydrogen in palladium [209]. Let us describe the pair interaction of all the components (hydrogen and metal atoms in the a. and (j phases) by the Lennard Jones potential cpy(ry) = 4 zi [(ff )12- / )6], where Sy and ai are the parameters of the corresponding potentials. All the distances ry, are considered within c.s. of radius r (1 < r < R), where R is the largest radius of the radii of interaction Ry between atoms / and /). 

Lennard-Jones potential U f) as a function of interatomic distance r. The characteristic parameters and cr determine this potential curve see Eq. (9). 

The shapes of the calculated Cs-rare gas potential curves show an avoided crossing of the curves associated with the 7S and 605/2,111=15 (5d Z) states (Figure 1). Such a situation characterizes a strong coupling between the two states and the shape of the calculated potential curves is then very sensitive to the calculation method. The comparison between theoretical predictions and experimental results is expected to be of particular interest in this range of interatomic distances. 

It must be kept in mind, however, that the two-dimensional potential curves of Fig. 1 do not consider the possibility of a change in the interatomic distances r, in the molecule, due to the adsorption process. A chemisorption, in particular, implies an increase of the interatomic distance of those atoms which enter into covalent bonds with widely separated centers of the surface 14). The excitation process in the free molecule often leads to a loosening of the bonds manifested in decreased vibrational frequencies and, consequently, to increased interatomic distances. In the limit, for molecules like iodine, the absorption of photons near the band maximum corresponds to a dissociation into atoms separated on the surface by a distance definitely larger than that when they were bound in the ground state. 

To keep sight of this intramolecular change, the two-dimensional potential curves plotted as functions of the distance B from the adsorbent must be supplemented by a third coordinate giving the interatomic distance in diatomic molecules or between two key atoms in a polyatomic one. The corresponding three-dimensional diagram for a diatomic molecule has been represented by de Boer and Custers 2). It is advisable, however, to keep the usual two-dimensional curves, imagining them to be drawn in space as contour lines on the respective surfaces of potential energy. 

So, while the 1960s started with the belief that SCF could give a complete potential curve for closed-shell atoms, at the end of the decade it was known that the inclusion of interatomic correlation is essential for obtaining the dispersion attraction. The new decade saw the light with the two papers just mentioned [29,28] proving this quantitatively. 

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