# 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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