Approximate Hessian for accelerating ab initio structure relaxation by force fitting

We present a method to approximate the Hessian matrix of the Born-Oppenheimer energy landscape by using a simple force field model whose parameters are fitted to on-the-flight ab-initio results. The inversed Hessian matrix is used as the preconditioner of conjugate gradient algorithms to speed up the atomic structure relaxation, resulting in a speedup factor of 2 to 5 on systems of bulk, slab, sheets, and atomic clusters. Because the force field model employed is simple and general, the parameter fitting is straightforward; the method is applicable to a variety of complicated systems for minimum structure relaxation. In the metal cluster new structure search, the new method yields better structures than the one obtained before with conventional algorithms.

Figure 1

The relaxation of ${\mathrm{Cd}}_4$${\mathrm{Se}}_4$, ${\mathrm{C}}_{60}$, ${\mathrm{Si}}_{20}$, and ${\mathrm{Cd}}_{32}$${\mathrm{Se}}_{32}$ systems by the conventional CG algorithm and the proposed PCG algorithm. The $x$ axis is the number of relaxation steps, while the $y$ axis is $E-E_f$ (in units of eV), where $E$ is the energy of the current step and $E_f$ is the energy of the finally sought structure.

Figure 2

The relaxation of three ${\mathrm{Cd}}_4$${\mathrm{Se}}_4$ systems by conventional CG, by PCG with the Hessian generated with nonfitting original LJ parameters taken from the initial atomic configuration (nPCG) and from the final atomic configuration (nPCG-$R_f$) (for testing purposes), and by the proposed force-fitted PCG. The maximal initial atomic displacement in (a) is 0.05 Å, in (b) is 0.1 Å, $$ and in (c) is 1.0 Å. In (c), the Hessian matrix for the PCG algorithm has been updated three times during the iterations (at the 5$\text{th}$, 12$\text{th}$, and 23$\text{rd}$ step). The $x$ axis is the number of relaxation steps, while the $y$ axis is $E-E_f$ (in units of eV), where $E$ is the energy of the current step and $E_f$ is the energy of the finally sought structure.

Figure 3

Hessian matrix of ${\mathrm{Cd}}_4$${\mathrm{Se}}_4$ and ${\mathrm{Si}}_{20}$. (a) is the accurate 24$\times$24 Hessian matrix for ${\mathrm{Cd}}_4$${\mathrm{Se}}_4$ calculated by DFT and (b) is its approximated one by force-fitted LJ. (c) is the accurate 60$\times$60 Hessian matrix for ${\mathrm{Si}}_{20}$ and (d) is its approximated one by the fitted LJ. The value of each element in the matrix is depicted by the color of the bar at the left.

Figure 4

The relaxation of ${\mathrm{Pt}}_{38}$ and ${\mathrm{Pt}}_{54}$ clusters by CG and PCG. The $x$ axis is the number of relaxation steps, while the $y$ axis is $E-E_f$ (in units of eV), where $E$ is the energy of the current step and $E_f$ is the energy of the finally sought structure.

Figure 5

Electron charge density of HOMO of ${\mathrm{Pt}}_{38}$ found by (a) CG at 200 steps and (b) PCG.

Figure 6

The relaxation of the system (${\mathrm{C}}_{128}$${\mathrm{Au}}_4$) of Au clusters adsorbed on a monolayer graphene by CG and PCG. The $x$ axis is the number of relaxation steps, while the $y$ axis is $E-E_f$ (in units of eV), where $E$ is the energy of the current step and $E_f$ is the energy of the finally sought structure.