Optimizers

OpenMECP provides several geometry optimizers. All optimizers work on the MECP effective gradient described in the MECP Algorithm chapter.

BFGS (Initial Steps)

The Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method provides a stable warm-up for the first switch_step iterations (default: 3). It builds up curvature information in the approximate Hessian before DIIS takes over.

When to use BFGS exclusively: Set switch_step = 999 (or any value ≥ max_steps) to run pure BFGS throughout — most stable but slower.

Relevant keywords: hessian, bfgs_rho, max_step_size

GDIIS — Geometry DIIS (Default)

use_gediis = false (default)

The Geometry Direct Inversion in the Iterative Subspace method constructs the next geometry as an optimal linear combination of previous Newton-step corrections:

$${\mathbf{e}}_i={\mathbf{H}}^{-1}\cdot ({\mathbf{g}}_i+{\mathbf{f}}_i)$$

$${\mathbf{B}}_{jk}=⟨{\mathbf{e}}_j\mid {\mathbf{e}}_k⟩$$

$$\left(\begin{array}{cc}\mathbf{B}& 1\\ 1^{\top }& 0\end{array}\right)\left(\begin{array}{c}\mathbf{c}\\ \lambda \end{array}\right)=\left(\begin{array}{c}0\\ 1\end{array}\right)$$

$${\mathbf{x}}^{\ast }=\sum _i{c}_i{\mathbf{x}}_i,{\mathbf{x}}_{\text{GDIIS}}={\mathbf{x}}^{\ast }-{\mathbf{H}}^{-1}\cdot P({\mathbf{x}}^{\ast })$$

GDIIS minimizes residual Newton-step forces and delivers robust quadratic convergence. Proven reliable for ~10-step convergence on production systems.

Performance: ~2–3× faster than pure BFGS.

GEDIIS — Energy-Informed DIIS

use_gediis = true, use_hybrid_gediis = false

Extends GDIIS by adding an energy-based diagonal bias to the B-matrix:

$$B_{ii}+=\alpha \cdot E_i^2,{\text{RHS}}_i=-E_i$$

where $E_i=|E_{1,i}-E_{2,i}|$ is the energy gap at step $i$. The diagonal term biases interpolation toward geometries with small energy gaps (i.e., close to the crossing seam). The constraint is $c_i>0$ — GEDIIS only interpolates, never extrapolates.

Best for: systems where the crossing seam needs to be approached from a geometry far from it.

Sequential Hybrid GEDIIS/GDIIS

use_gediis = true, use_hybrid_gediis = true

Implements the 3-phase switching strategy of Li & Frisch (JCTC 2006):

Default thresholds: gediis_switch_rms = 0.005 Ha/Å, gediis_switch_step = 0.001 Å.

Performance: ~2–4× faster than pure GDIIS on most systems.

GDIIS_blend

use_gediis = "blend"

A per-step blend of a GDIIS step and an EDIIS step, governed by a trust radius:

$${\mathbf{x}}_{\text{new}}=w\cdot {\mathbf{x}}_{\text{EDIIS}}+(1-w)\cdot {\mathbf{x}}_{\text{GDIIS}}$$

When use_hybrid_gediis = false, no EDIIS component is added (pure GDIIS with trust radius control). When use_hybrid_gediis = true, the blend weight $w$ is determined by gediis_blend_mode:

The trust radius automatically contracts when energy increases and expands when energy decreases, preventing wild steps.

Requirement: A direct Hessian update method (direct_psb, bofill, powell, or bfgs_powell_mix). inverse_bfgs is incompatible.

Optimizer Switching

The switch_step parameter controls when the warm-up BFGS steps hand off to DIIS:

switch_step = 3     # Default: 3 BFGS steps, then DIIS
switch_step = 0     # Pure DIIS from step 1 (fastest, needs good geometry)
switch_step = 10    # 10 BFGS steps, then DIIS (better for difficult cases)
switch_step = 999   # Pure BFGS (most stable, slowest)

The smart_history option retains the most useful geometry history points rather than using a simple FIFO queue, which can accelerate convergence.

Performance Comparison