Structural relaxation

This section describes how to find the zero-Kelvin equilibrium atomic structure, given a starting structure with non-zero forces and/or stresses. CONQUEST can employ a variety of algorithms to minimise energy with respect to atomic positions, including: stabilised quasi-Newton method (SQNM); L-BFGS; conjugate gradients (CG); and damped molecular dynamics (both MDMin and FIRE approaches). The minimisation of energy or enthalpy with respect to cell vectors is restricted to conjugate gradients at present, though L-BFGS will be implemented.

Setting AtomMove.WriteXSF T for all flavours of optimisation will dump the trajectory to the file trajectory.xsf, which can be visualised using VMD and XCrysDen. Setting AtomMove.AppendCoords T will append the structure at each step to UpdatedAtoms.dat in the format of a CONQUEST structure input.

For the SQNM, L-BFGS and conjugate gradients relaxations, the progress of the calculation can be monitored by searching for the word GeomOpt; grepping will print the following:

$ grep GeomOpt Conquest_out
GeomOpt - Iter:    0 MaxF:   0.00329282 H:  -0.14168571E+03 dH:   0.00000000
GeomOpt - Iter:    1 MaxF:   0.00331536 H:  -0.14168995E+03 dH:   0.00424155
GeomOpt - Iter:    2 MaxF:   0.00350781 H:  -0.14168997E+03 dH:   0.00001651
GeomOpt - Iter:    3 MaxF:   0.00504075 H:  -0.14169161E+03 dH:   0.00164389
GeomOpt - Iter:    4 MaxF:   0.00725611 H:  -0.14169172E+03 dH:   0.00010500
GeomOpt - Iter:    5 MaxF:   0.01134145 H:  -0.14169329E+03 dH:   0.00157361
GeomOpt - Iter:    6 MaxF:   0.01417229 H:  -0.14169385E+03 dH:   0.00056077
GeomOpt - Iter:    7 MaxF:   0.01434628 H:  -0.14169575E+03 dH:   0.00190304
GeomOpt - Iter:    8 MaxF:   0.01711197 H:  -0.14170001E+03 dH:   0.00425400
GeomOpt - Iter:    9 MaxF:   0.02040556 H:  -0.14170382E+03 dH:   0.00381110
GeomOpt - Iter:   10 MaxF:   0.01095167 H:  -0.14170752E+03 dH:   0.00370442

In this example, MaxF is the maximum single force component, H is the enthalpy and dH is the change in enthalpy.

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Ionic relaxation

To optimise the ionic positions with respect to the DFT total energy, the following flags are essential:

AtomMove.TypeOfRun sqnm
AtomMove.MaxForceTol 5e-4
AtomMove.ReuseDM T

The parameter AtomMove.TypeOfRun can take the values sqnm, lbfgs or cg for iterative optimisation. All three algorithms are robust and relatively efficient in most instances; SQNM [SR1] is recommended in most cases, though if the initial forces are large it may be worth performing quenched MD to reduce them (see below) before applying SQNM. The parameter AtomMove.MaxForceTol specifies the force convergence criterion in Ha/bohr, i.e. the calculation will terminate when the largest force component on any atom is below this value. The parameter AtomMove.ReuseDM specifies that the density matrix (the K-matrix for diagonalisation or L-matrix for O(N) calculations) from the previous step will be used as an initial guess for the SCF cycle after propagating the atoms; this should generally decrease the number of SCF cycles per ionic step. When using CG, the line minimiser can be chosen: safe uses a robust though sometimes slow line minimiser; backtrack uses a simple back-tracking line minimiser (starting with a step size of 1 and reducing if necessary to ensure the energy goes down); adapt uses an adaptive back-tracking line minimiser (which increases the starting step size if the energy goes down on the first step). In many cases the back-tracking line minimiser is more efficient, though the efficiency of the adaptive approach varies with problem.

If the self-consistency tolerance is too low, the optimisation may fail to converge with respect to the force tolerance; this may necessitate a tighter minE.SCTolerance for diagonalisation (also possibly minE.LTolerance for O(N) calculations). A grid which is too coarse can also cause problems with structural relaxation to high tolerances.

For large initial forces or problematic cases where the relaxation algorithms fail to find a downhill search direction, it may be worth trying quenched molecular dynamics, which propagates the equations of motion following a simple NVE approach, but resets the velocities to zero when the dot product of force and velocity is zero.

AtomMove.TypeOfRun md
AtomMove.QuenchedMD T
AtomMove.MaxForceTol 5e-4
AtomMove.ReuseDM T

The FIRE algorithm [SR2] is a variant of quenched MD that has been shown to outperform conjugate gradients in some circumstances.

AtomMove.TypeOfRun md
AtomMove.FIRE T
AtomMove.MaxForceTol 5e-4
AtomMove.ReuseDM T

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Simulation cell optimisation

The simulation cell can be optimised with respect to enthalpy with fixed fractional coordinates (AtomMove.OptCellMethod 1) using the following input:

AtomMove.TypeOfRun cg
AtomMove.OptCell T
AtomMove.OptCellMethod 1
AtomMove.ReuseDM T
AtomMove.EnthalpyTolerance 1E-5
AtomMove.StressTolerance 0.1

Note that stress is in GPa and enthalpy is in Ha by default.

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Combined optimisation

For simple crystals, the fractional ionic coordinates vary trivially with changes in the simulation cell lengths; however for more complicated systems such as molecular crystals and amorphous materials, it is necessary simultaneously relax the ionic positions and simulation cell lengths (recalling that CONQUEST only allows orthorhombic unit cells). This can be done by setting AtomMove.OptCellMethod 2 or AtomMove.OptCellMethod 3

AtomMove.TypeOfRun cg
AtomMove.OptCell T
AtomMove.OptCellMethod 2
AtomMove.ReuseDM T
AtomMove.MaxForceTol 5e-4
AtomMove.EnthalpyTolerance 1E-5
AtomMove.StressTolerance 0.1

Note that stress is in GPa and enthalpy is in Ha by default.

The enthalpy will generally converge much more rapidly than the force and stress, and that it may be necessary to tighten minE.SCTolerance (diagonalisation) or minE.LTolerance (order(N)) to reach the force and stress tolerance, if it is even possible. For combined optimisation, we recommend using AtomMove.OptCellMethod 2, which uses a simple but robust double-loop minimisation: a full ionic relaxation (using either cg or sqnm) followed by a full simulation cell relaxation (using cg). While this may be less efficient than optimising all degrees of freedom simultaneously, it is much more robust. It is also possible to optimise cell vectors and atomic positions simultaneously, using AtomMove.OptCellMethod 3, but this should be monitored carefully, as it can be unstable.

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SR1

Bastian Schaefer, S. Alireza Ghasemi, Shantanu Roy, and Stefan Goedecker. Stabilized quasi-newton optimization of noisy potential energy surfaces. J. Chem. Phys., 142(3):034112, 2015. doi:10.1063/1.4905665.

SR2

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.

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