Interatomic potentials

In interatomic potential methods, the various interactions between atoms that contribute to the lattice energy o f the structure are represented by mathematical functions. The parameters can be determined two ways. The most straightforward method is to calculate the parameters directly from quantum mechanics calculations. The second method, known as empirical fitting, is to adjust the parameters by a least squares method to give the best possible fit o f calculated structures and properties to those either found by experiment or determined by quantum mechanical calculations. Only potential energy contributions to the lattice energy are considered, with kinetic energy terms being ignored, so effectively the structures found correspond to those at absolute zero, and temperature effects are neglected.

The total lattice energy o f a periodic crystal, U(r), is calculated by summing the potential terms that arise from forces between all the atoms at positions r, as in equation 2.1. The two-body term, V2, is calculated for each possible combination o f ion pairs in

the lattice and depends on their separations. These are the major contribution for ionic or semi-covalent structures. The three-body term, V3, may also be considered, which

depends on the positions o f three atoms. Such three body terms are commonly used to account for bond-bending forces; thus considering covalent bonding effects. Higher order terms such as four-body terms are only used in our models that include organic molecules e.g. TPA^. The total energy can thus be expressed as the summation o f these two and three body terms over all the atoms in the structure:

U(r) = X ,., V, (r,rj ) + ) (2-1 )

The two-body terms consist o f a number o f contributions:

L J L ' - I L 2 J

where r is the separation o f the ion pair being considered, qi and q% are their charges, and A, r, C, k2 and m are all adjustable parameters. The first term accounts for the

coulombic interactions between ion pairs. The second term has two components that are collectively known as the Buckingham potential; a repulsive exponential term to

Chapter 2 Methodology o f interatomic potential computer models

account for the overlap o f charge, and an attractive dipole interaction due to Van der Waal’s forces. The third term accounts for the polarisation o f electrons around each ion due to the charge on the nucleus o f adjacent atoms. A shell model [4] is used to approximate the electron cloud to a massless, displaceable, spherieal shell that is connected to the ion core inside it by a spring o f force constant kj, as shown in Figure

2.1.

Figure 2.1 The shell model

A three-body potential, is also used here to include bond bending, and account for the degree o f covalency in the zeolite structure:

y>(nr,r,) = U { O - 0 , Y

where k is a fitted foree constant, 0 is the angle between the three atoms and Go is its

ideal value.

The Buckingham, spring, and three-body potentials are relatively short-range, and are negligible for separations greater than the order o f 10 to 16 À (particularly the three- body term), allowing the implementation o f a cut-off beyond which interactions are ignored to reduce the computational cost o f these ealculations. However, the electrostatic term is long-range, and as such converges much more slowly with separation. Hence, the Ewald summation is used to sum these coulombie terms [5]. This uses a method by which each point charge in the reciprocal lattice is associated with a

Chapter 2 M ethodology o f interatomic potential computer models

screening charge o f opposite sign, which often has a Gaussian distribution. This means that the interactions between the screened point charges converge much more quickly, and can be summed directly in real space. The energy o f the charge distribution array must then be accounted for to find the total coulombic potential o f the original point charge system. Thus the interaction energy o f a lattice o f charge distributions o f opposite sign to the screening charge distribution (the same signs as the original point charges).is found by summing their Fourier transforms in reciprocal space, and then converting back to real space. The sum o f the coulombic potential terms in the original lattice is simply the sum o f these two energies.

A Morse potential is sometimes used to represent a particular covalent bond between particular species. In the calculations presented here it is used to represent the 01-H bond, where 01 is used to denote the bridging oxygen in the Bronsted acid groups and is given by:

y ^ < o . „ = D , l l - e x f ( r a ( r - r , ) ) y (2.4)

where a is a fitting parameter, ro is the equilibrium bond length, and Dg is the dissociation energy to separate the two atoms, and the last term is to subtract the coulomb interaction energy, since it is effectively included in the Morse potential.

Previous studies have developed a set o f transferable potentials to reproduce the various bonds and interactions between the atoms in aluminosilicate structures in a number o f stages. A model o f a-quartz first derived the interactions between Si and O - fitting them to reproduce its elastic constants and dielectric constants [6]. These were then

combined with potentials representing A l-0 interactions in aluminium oxide [7] to model a range o f aluminosilicates [8], The Morse potential parameters used to describe

the interactions between a bridging oxygen, O l, and the adjacent hydrogen ion were originally derived for sodium hydroxide, and fitted to ab initio data, due to a lack o f experimental data [9]. These framework and acid site potentials were subsequently collated and the O-H Morse potential modified for aluminosilicates [10]. Interactions between extra-framework Na^ and Ca^^ cations and the framework are described by coulombic interactions (both ions being modelled by formal charges) and a Buckingham

Chapter 2 M ethodology o f interatomic potential computer models

potential to represent the interactions between them and the shells o f the framework oxygen [11]. These potentials have since been used to reproduce a variety o f experimental results, including the structure, vibrational frequencies and acidity o f acid sites in zeolites [1,2, 1 2].

The potentials used are defined by the parameters listed in Tables 2.1, 2.2, 2.3, 2.4 and 2.5. O l is used to denote the bridging oxygen in the hydroxyl group o f the Bronsted acid sites.

Table 2.1 Electrostatic charges used to determine electrostatic term

Species _{qi (e)} Si core +4.000 A1 core +3.000 O core +0.869 O shell -2.869 O l core -1.426 H core +0.426 Na core +1 .0 0 0 Ca core +2 .0 0 0

Table 2.2 Buckingham potential parameters

Interacting species A (eV ) _{P(Â )} C (eV

A")

A1 c o r e .,.. O shell 1460.300 0.299 0 .0 0 0 O l c o re... A1 core 1142.676 0.299 0 .0 0 0 0 s h e ll.. . O shell 22764.000 0.149 27.880 Si core ,. . O shell 1283.907 0.321 10.662 O l c o re ... Si core 983.557 0.321 10.662 0 1 c o re ... 0 shell 22764.000 0.149 27.880 H co re... O shell 311.970 0.250 0 .0 0 0 Na c o re ... O shell 1226.840 0.307 0 .0 0 0 Ca c o re .,.. O shell 1090.40 0.3437 0 .0 0 0

Chapter 2 M ethodology o f interatomic potential computer models

Table 2.3 Spring potential parameters

Atom k2 (eV A'^)

O 74.920

Table 2.4 Three body potential parameters

Interacting species k (eV rad'^) _{8oC )} A1 core ... 0 sh e ll... O shell 2.097 109.47 A1 core ... Ol core ... 0 shell 2.097 109.47 Si core ... 01 core ... 0 shell 2.097 109.47 Si core ... Ol core... O shell 2.097 109.47

Table 2.5 Morse potential parameters

Species De (eV) a (A ') To (A) H core ... O l core 7.053 2.199 0.949