Theory · 4 methods

Minimisers

What CAPS computes, as it computes it: the equation, its symbols with CAPS's defaults, when to use the method, the source file, the tests that check it, and any departure from the cited method.

L-BFGS

A quasi-Newton minimiser that approximates the inverse Hessian from the last few steps (two-loop recursion). It is the fastest of CAPS's minimisers for smooth energies of thousands of atoms.

$$d = -H_k\nabla E, \qquad H_k \text{ from the last } m \text{ pairs } \left(s_i = x_{i+1} - x_i,\; y_i = \nabla E_{i+1} - \nabla E_i\right)$$

Liu & Nocedal 1989

Symbol	Meaning	In CAPS
m	stored pairs	10
ftol	stop when the largest atomic force is below	0.5 kcal/mol/Å by default
etol	or the relative energy change is below	10⁻⁸

When to use it

The default minimiser. After growth with a contact scale below 1, run the capped push-off first (Relax does by default).

Source: core/src/relax.cpp
Tested by: Relax.EveryMinimiserLowersEnergy · Relax.CompressesOverlappingCellToTarget
Departure from the reference: none

References

Liu, D. C., Nocedal, J., "On the limited memory BFGS method for large scale optimization", Math. Program. 45, 503–528 (1989). doi:10.1007/BF01589116

FIRE

A damped dynamics minimiser: velocities are mixed towards the force direction and the time step grows while the power F·v stays positive; the motion stops when it turns uphill.

$$\begin{gathered}v \leftarrow (1 - \alpha)\,v + \alpha\,|v|\,\frac{F}{|F|} \\ \text{if } F\cdot v < 0:\quad v = 0,\;\; \Delta t \leftarrow f_\mathrm{dec}\,\Delta t,\;\; \alpha = \alpha_\mathrm{start}\end{gathered}$$

Bitzek et al. 2006

Symbol	Meaning	In CAPS
α_start	mixing	0.1
f_inc, f_dec	time-step growth and cut	1.1, 0.5
N_min	steps before growth	5

When to use it

Rough starts with large forces, where line searches struggle.

Source: core/src/relax.cpp
Tested by: Relax.EveryMinimiserLowersEnergy
Departure from the reference: standard parameters of the paper

References

Bitzek, E., Koskinen, P., Gähler, F., Moseler, M., Gumbsch, P., "Structural relaxation made simple", Phys. Rev. Lett. 97, 170201 (2006). doi:10.1103/PhysRevLett.97.170201

Conjugate gradients and steepest descent

Steepest descent follows the force; Polak–Ribière conjugate gradients mixes in the previous direction so successive steps do not undo each other.

$$d_{k+1} = F_{k+1} + \beta d_k, \qquad \beta = \max\left(0,\; \frac{F_{k+1}\cdot(F_{k+1} - F_k)}{F_k\cdot F_k}\right)$$

Polak & Ribière 1969

Symbol	Meaning	In CAPS
β	Polak–Ribière coefficient, reset to 0 when negative	—
ftol	force tolerance	0.5 kcal/mol/Å

When to use it

Comparison and teaching; L-BFGS is faster for the same tolerance.

Source: core/src/relax.cpp
Tested by: Relax.EveryMinimiserLowersEnergy
Departure from the reference: none

References

Polak, E., Ribière, G., "Note sur la convergence de méthodes de directions conjuguées", Rev. Fr. Inform. Rech. Opér. 3, 35–43 (1969)

Capped-force push-off

Overlapping atoms (from growth with lowered contact limits, or packing) make Lennard-Jones forces explode. The push-off caps them: inside the radius where the force would exceed the cap the potential becomes linear, and the cap is raised stage by stage while minimising. Optionally the push-off runs first as NVT dynamics, as in the paper: the cap is raised geometrically over a set time (the λ ramp) so chains move apart by motion before the minimisation stages.

$$\begin{gathered}F_\mathrm{LJ}(r) \to \operatorname{sign}(F)\,\min\left(|F_\mathrm{LJ}(r)|,\, F_\mathrm{cap}\right), \qquad F_\mathrm{cap} = 5, 20, 100, \dots, F_\mathrm{final}\ \text{kcal/mol/Å} \\ \text{MD:}\ \ F_{\mathrm{cap},k} = 5\left(\frac{F_\mathrm{final}}{5}\right)^{k/9}, \quad k = 0, \dots, 9, \ \text{each for } t_\mathrm{ramp}/10\end{gathered}$$

after Auhl et al. 2003

Symbol	Meaning	In CAPS
F_cap	cap of each stage	5, 20, 100, then the chosen final cap (500 by default)
t_ramp	λ ramp: the time of the MD push-off	20 ps on the Relax board; 0 = minimisation stages only
T	NVT temperature of the MD push-off (Bussi thermostat, 1 fs)	300 K

When to use it

Always before minimising a grown or packed cell; Relax does it by default.

Source: core/src/relax.cpp · core/src/field.cpp (force_cap)
Tested by: Equilibrate.PushoffTamesOverlaps · Relax.PushoffByMdRampsTheCap
Departure from the reference: the cap is raised in ten NVT segments rather than continuously; Coulomb terms are off during the push-off

References

Auhl, R., Everaers, R., Grest, G. S., Kremer, K., Plimpton, S. J., "Equilibration of long chain polymer melts in computer simulations", J. Chem. Phys. 119, 12718–12728 (2003). doi:10.1063/1.1628670