Structural relaxation: Theory
Structural relaxation involves optimisation of the ionic coordinates, optimisation of the simulation cell, or both, with respect to the DFT total energy or the enthalpy if the cell is not fixed.
Ionic relaxation
L-BFGS:
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Conjugate gradients
The most naive geometry optimisation algorithm is steepest descent: we calculate the gradient of the DFT total energy (i.e. the force) and propagate the system in the direction of the steepest gradient (the direction of the force vector) until the energy stops decreasing. We choose the direction (largest gradient in this case) and perform a line search. This will be sufficient if the potential energy surface is well-behaved, but in most cases convergence will require many iterations. Conjugate gradients is a well-established method the improves upon steepest descent in the choice of search direction. Without going into too much detail, we choose a new search direction that is orthogonal to all previous search directions using the conjugacy ratio \(\beta\). At iteration \(n\), it is given by,
This is the Fletcher-Reeves formulation; note that \(\beta_0 = 0\). We can then construct the search direction at step \(n\), \(D_n\),
and peform the line minimisation in this direction. This process is repeated until the maximum force component is below some threshold.
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Quenched MD
The system is propagated in the direction of steepest descent as determined by the DFT forces, and the velocity is scaled down as the system approaches its zero-temperature equilibrium configuration.
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FIRE Quenched MD
The system is propagated using the modified equation of motion [Tb1],
which has the effect of introducing an acceleration in a direction that is steeper than the current direction of motion. If the power \(P(t) = \mathbf{F}(t)\cdot\mathbf{v}(t)\) is positive then the system is moving “downhill” on the potential energy surface, and the stopping criterion is when it becomes negative (moving “uphill”).
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Cell optimisation
When optimising the cell with fixed fractional ionic coordinates, the same conjugate gradients method is used as above, but minimising the enthalpy with respect to the cell vectors.
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Combined optimisation
The ionic and cell degrees of freedom can be relaxed simultaneously by combining all of their coordinates into a single vector and optimising them with respect to the enthalpy of the system. However, this atomic forces and total stresses having numerical values of the same order of magnitude, and changes in ionic coordinates and cell vectors being of the same order of magnitude. Using the method of Pfrommer et al. [Tb2], the latter can be enforced by using fractional coordinates for the ionic positions, and fractional lattice vectors of the form \(h = (1 + \epsilon)h_0\) where h is the matrix of lattice vectors, \(h_0\) is the matrix for some reference configuration and epsilon is the strain matrix. The “fractional” force on the i th atom is then \(\mathbf{F}_i = g\mathbf{f}_i\) where \(\mathbf{f}_i\) is the DFT-calculated force multiplied by the metric tensor \(g = h^Th\). The “fractional” stress is,
where \(\sigma\) is the DFT-calculated stress, \(p\) is the target pressure and \(\Omega\) is the volume. The resulting vector is optimised using the same conjugate gradients algorithm as before, minimising the enthalpy.
E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, and P. Gumbsch. Structural Relaxation Made Simple. Phys. Rev. Lett., 97:2897, 2006. doi:10.1103/PhysRevLett.97.170201.
B. G. Pfrommer, M. Côté, S. Louie, and M. L. Cohen. Relaxation of Crystals with the Quasi-Newton Method. J. Comput. Phys., 131:233, 1997. doi:10.1006/jcph.1996.5612.
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