MECP Algorithm

OpenMECP implements the MECP optimization algorithm of Harvey et al. (Theor. Chem. Acc. 99, 95–99, 1998) with several modern enhancements.

Problem Statement

Given two electronic states with energies $E_1(\mathbf{R})$ and $E_2(\mathbf{R})$ and gradients ${\mathbf{f}}_1(\mathbf{R})$ and ${\mathbf{f}}_2(\mathbf{R})$, find the geometry ${\mathbf{R}}^{\ast }$ that simultaneously satisfies:

1. Degeneracy: $E_1({\mathbf{R}}^{\ast })=E_2({\mathbf{R}}^{\ast })$
2. Minimum energy: ${\mathbf{R}}^{\ast }$ is the lowest-energy point on the crossing seam

Effective Gradient

Harvey et al. construct an effective gradient that encodes both conditions. First, define the normalized gradient difference:

$$\hat{\mathbf{x}}=\frac{{\mathbf{f}}_1-{\mathbf{f}}_2}{|{\mathbf{f}}_1-{\mathbf{f}}_2|}$$

The effective gradient has two orthogonal components:

f-vector — drives the energy gap to zero:

$$\mathbf{f}=(E_1-E_2)\hat{\mathbf{x}}$$

g-vector — minimizes the average energy in the intersection seam:

$$\mathbf{g}={\mathbf{f}}_1-(\hat{\mathbf{x}}\cdot {\mathbf{f}}_1)\hat{\mathbf{x}}$$

The total effective gradient is:

$${\mathbf{G}}_{\text{eff}}=\mathbf{f}+\mathbf{g}$$

Convergence Criteria

All five criteria must be satisfied simultaneously:

Optimization Strategy

OpenMECP uses a hybrid strategy combining stability and speed:

Step 1–switch_step:  BFGS (builds Hessian curvature information)
Step switch_step+1:  GDIIS / GEDIIS (quadratic-convergence DIIS acceleration)

The default switch_step = 3 provides a 3-step BFGS warm-up before engaging DIIS. See Optimizers for a full description of each algorithm.

Two-State QM Calculation

At each MECP optimization step, OpenMECP:

1. Writes the QM input files for state A and state B from the current geometry.
2. Runs the QM program for state A, reads $E_1$ and ${\mathbf{f}}_1$.
3. Runs the QM program for state B, reads $E_2$ and ${\mathbf{f}}_2$.
4. Evaluates the MECP effective gradient ${\mathbf{G}}_{\text{eff}}$.
5. Updates the geometry using the selected optimizer.
6. Checks convergence; writes checkpoint file if requested.

Unit Conventions