Modelling at the atomic level
3.3 Theory of modelling using Interatomic Potentials
Interatom ic potential m ethods for atomic m odelling attem pt to reduce the com plete problem to a problem with less variables. The com plete problem w ould o f course include each o f the electrons in each o f the atoms in the m odel. It m ight be feasible to include every electron in the m odel, but it is im possible to create a basic calculation based on electron-electron interactions because the precise location o f the electrons is not know n and cannot be measured. Therefore, an approxim ation to the electron-electron interactions m ust be form ed, and this is indeed w hat electronic calculation m ethods attem pt to address; the various m ethods each provide an approxim ation for the electron-electron interactions, or correlation. H owever, interatom ic potential m ethods assume that the electrons are spread evenly around the nucleus o f the atom and act as if they were a point-charge at the centre o f their sphere. Thus, the com plete problem may be reduced to treating each atom as a single point and using the potential param eters to describe the attractive and repulsive forces which the atom possesses.
Energy m inim isation using specified potentials is the m ethod o f choice for the m ajority of the calculations presented in the follow ing chapters. Energy m inim isation is an aspect of M olecular M echanics (M M). An understanding o f how and why the technique works is required to understand fully the softw are and the results w hich it produces.
The basic approach o f energy m inim isation calculations is to represent each atom in the system as a point or sphere with co-ordinates and charge values. M ore advanced m odels em ploy a separate centre o f m ass for the electrons o f the atom com pared to the nucleus (although the mass of the electron shell is usually set to zero). This allow s atoms to describe displacem ents o f a charge representing the valence shell from the nucleus - a representation o f polarisation. T he nucleus (core) and electron cloud (shell) are connected m athem atically by an im aginary spring which has a spring constant particular to the atom type. This fuller approach is known as the
C h apter 3 : M odellin g a t the atom ic level
3.3.1 Energy calculations in Energy minimisations
Once the system has been created, an energy calculation m ay be performed. Equations describing the energy profile of atom ic interactions are em ployed to calculate the energy o f the system in relative term s; an absolute value is not obtained in m ost cases as the equations such as the H arm onic functions are designed to provide an energy relative to the equilibrium state. The equations are known as “potentials” . Enough potentials m ust be included in the m odel to describe every significant inter-atom ic interaction.
Typical and com m on term s for the forms o f potentials[3] are shown below. The exam ples are not an exhaustive list, as m any equations w ould adequately describe the energetic relationship between atoms. H ow ever, the exam ples include the com m only used types of potentials which are know n to reproduce the attractive and repulsive forces between atoms.
B ond Stretching
M orse function Harm onic function B ond A ngle Oscillations
Harm onic function Core-Shell connection function
Spring (harmonic) D ihedral A ngle V ariations
Fourier Series Van der W aals Interactions
Lennard Jones function B uckingham function Electrostatic function Coulom b function = S D «[1 - e x p { - a ( r - r o ) } ] ^ [Eqn. 3.1] Ebond = ^ k r { r - r Q Ÿ [Eqn. 3.2] Eangle = 'Z k e ( 6 - ÔqY
Espring = S [ - |( r - ro)^ + j ( r - ro)"^]
Etorsion — ^ V/i(l H" SCOSfVUj')
E v d w =
- (-^)]
Evdw = I A ex p (-B r) - (-^) Eelectronic ~~ ^ rii [Eqn. 3.3] [Eqn. 3.4] [Eqn. 3.5] [Eqn. 3.6] [Eqn. 3.7] [Eqn. 3.8]C h apter 3 : M odellin g a t the atom ic level
In equations 3.1 to 3.8 : E represents energy, q represents charge, r represents distance, theta and om ega represent angles, k, alpha, A, B, C, D, V a n d s are constants. A subscript of 0 represents the equilibrium value. The spring potential is not always im plem ented to include the quartic term , k/4(r-roX, but this term is of considerable use in sensitive systems as a m ethod o f fine tuning the potential.
The summation o f all o f the energy contributions gives an energy value which num erically is not representative but rather can be used as a basis for com parison. Typical com parisons w ould include calculating energy changes as a function of atom ic displacem ents or calculating the difference in relative energy between different conform ations o f the sam e system.
3.3.2 Parameterisation
The constants which are utilised in the potentials are usually assigned in one o f three main ways. Em pirical fitting involves taking know n values for physical properties and using an iterative procedure to refine the potential param eters in order to reproduce the physical properties, the structural properties are obvious choices to base potential param eters on, but m any other quantities m ay be used such as dielectric and elastic constants. Ab-initio energy surface fitting is a m ore recent technique which takes the energy hypersurface generated by ab-initio calculations and extracts the values o f the potential param eters. Electron gas interaction fitting uses isolated atom pairs in the gaseous state, and calculates the interactions o f the pair across a range of distances using ab-initio m ethods. T he coefficients for potential param eters can then be extracted from the plot o f the changing interaction energy of the tw o species.
Each param eterisation m ethod has benefits and draw backs, em pirical fitting can produce excellent results for the system which the potentials w ere based upon, but can create potentials w hich are not transferable to other system s, even closely related ones. The energy surface fitting and electron gas fitting are better at producing sets of potentials which are transferable but their quality is largely dependent upon the
C h apter 3 : M odellin g a t the atom ic level
3.3.3 Geometry optimisation in Molecular Mechanics
G eom etry optim isation is a highly desirable goal in m any m odels, the geom etry of the model can be optim ised by m inim ising the energy o f the system. To undertake energy m inim isation, the atomic co-ordinates m ust be varied. In efficient m inim isations, the gradients o f the energy profile are calculated by differentiating the potential equations with respect to the Cartesian directions separately. This derivative is the Force on the atom. A tom ic co-ordinates are then altered in the direction o f the Force. An iterative procedure o f these steps theoretically leads to a point o f convergence. At convergence successive iterations do not lead to a reduction in the energy o f the system. The system is then be said to be in an “energy m inim um ”, where no further energy decreases m ay be achieved by finding the Force on the atoms. C om plicated systems may have m any m inim a, each with a different energy value. If an energy m inim um has the low est possible energy, it is said to be the “Global M inim um ”, all other m inim a are “Local M inim a” . It is often difficult to distinguish between the global m inim um o f a com plicated system and its local m inim a.
There are several m ethodologies for m inim ising energies. T he sim plest form of geom etry optim isation described above is known as “Steepest D escents” it works well (quickly m oves towards convergence) for system s w ith high initial gradients but slow s down rem arkably when the gradients becom e small. T he reason for this slow dow n is that each iteration only considers the steepest gradient it can find at each cycle and adjusts the atomic co-ordinates in that manner. A fter a particular cycle, the m inim iser may have actually pushed the atomic co-ordinates past their optim um positions thus reversing the direction o f the force. This continues and the algorithm cycles around the energy m inim um only gradually converging to it. Equation 3.9 show s how the Steepest D escents m ethod works.
C h apter 3 : M odellin g a t the atom ic level
Steepest D escents = X . - X i - g [Eqn. 3.9]
' — ;
W here X is the set o f co-ordinates for each o f the atom s in x, y and z, X is the step size o f the m inim iser (a scalar quantity), g is the Jacobian M atrix (stores all o f the negative values of the gradient vectors) and i is the cycle num ber.
An im provem ent to Steepest D escents is to use C onjugent G radients. This method converges usually much faster than Steepest D escents as each step takes into account the direction o f the force o f the previous step. As before, a direction o f force is found each cycle but now, the direction is m odified by the addition o f a fraction o f the previous direction. This has the effect o f ‘steering’ the m inim isation continually tow ards the energy m inim um (global or otherw ise) rather than suffering from the overshoot problem s of Steepest Descents. Equations 3.10 and 3.11 show how Conjugent G radients is performed.
Conjugent G radients X,+i = X. - Ài • h. [Eqn. 3.10]
w here [E q n -3.11]
Sym bols as Eqn. 3.8, h is the new conjugent gradient, for the first cycle of calculation it will be the same as g.
O ther m ethods, which can be grouped under a broad heading o f “Q uasi-N ew tonian” tend to perform poorly with high gradients but bring a system w ith sm all gradients to convergence quickly. These m ethods generally require the second derivatives of the Force as well as the first derivatives. (Com m only the second derivatives are stored in a m atrix called the Hessian matrix - for all but the sim plest o f systems, creating, storing and updating the H essian is com putationally very expensive.)
U sing second derivatives with no approxim ations, we have the “N ew ton-Raphson” (NR) m ethod. This is rarely em ployed in real calculations as it is so expensive, and
C h apter 3 : M odelling a t the atom ic level
the “Q uasi-N ew tonian” heading applies to m ethods derived from NR. The full NR m ethod is described in equation 3.12.
N R m ethod X,+i = X - [ g ' • g ] [Eqn. 3.12]
I — (
Sym bols as Eqn. 3.8 plus H represents the H essian m atrix, we have to use the inverse H essian because we actually want to divide the Jacobian m atrix by the Hessian matrix. G rant and Richards give a derivation o f the basis o f this equation. [4]
In practice, sim plifications o f and approxim ations to the full N R m ethod are em ployed. A com m on m ethod is called “Block D iagonal” and this involves considering all o f the cross term s in the H essian m atrix to be zero and only considering the diagonal elem ents o f the H essian to be non zero. This significantly reduces the cost o f several steps in the calculation.
A further m ethod o f m inim isation is som etim es em ployed. M onte Carlo modelling incorporates a random elem ent into the updating procedure o f the atom ic positions. M onte Carlo is not used in this study.