BaekA Multistate MECI

BaekA is OpenQP's adaptive penalty-function algorithm for minimum-energy conical intersections involving two or more electronic states. Select it as an algorithm of the ordinary meci(...) driver; it is not a separate three-state calculation type.

Baek, Lee, Filatov, and Choi introduced and validated the independent-gap adaptive construction for a minimum-energy three-state conical intersection. OpenQP exposes the same mathematical construction for N >= 2 states. The two-state case is the one-gap specialization, while support for more than three states is an OpenQP generalization. The cited paper validates the three-state case; neither non-three-state case is directly validated by those benchmarks. See the method reference.

Canonical .oqp Commands

The same public command covers two, three, and more states:

mrsf(nstate=5)/bhhlyp/6-31g*
meci(S0,S1,algorithm=baeka)
geom="guess.xyz"

mrsf(nstate=5)/bhhlyp/6-31g*
meci(S0,S1,S2,algorithm=baeka)
geom="guess.xyz"

mrsf(nstate=6)/bhhlyp/6-31g*
meci(S0,S1,S2,S3,algorithm=baeka)
geom="guess.xyz"

A two-state meci(S0,S1) call defaults to auglag, the augmented Lagrangian, so write algorithm=baeka when BaekA is intended for two states. With three or more states OpenQP infers baeka, because the other MECI algorithms are two-state only; keeping the option explicit makes the chosen method clearest.

All selected states must be consecutive response roots in the same physical spin manifold. OpenQP normalizes their order. Thus meci(S0,S1,S2,algorithm=baeka) and meci(T0,T1,T2,T3,algorithm=baeka) have valid shapes, whereas skipping an intervening root or mixing singlets and triplets is rejected. A crossing between different spin manifolds is an MECP problem.

A three-state production-style route with every public BaekA control explicit is:

mrsf(nstate=5)/bhhlyp/6-31g*
meci(S0,S1,S2,algorithm=baeka,sigma=1.0,alpha=0.02,delta_beta=0.025,beta_schedule="10,10,25,25,100,100,1000,1000,3000",gap=1e-4,maxit=100,energy_shift=1e-6,rmsd_grad=1e-4,coordsys=auto,trust=0.15,trust_max=0.5)
geom="guess.xyz"

The concise route selects the native OpenQP optimizer automatically. Do not add lib=oqp. MRSF also supplies its internal high-spin working reference automatically; labels in meci(...) are the physical target surfaces.

Objective and Independent Constraints

For N energy-ordered states, define the N - 1 adjacent gaps d_a = E_(a+1) - E_a. BaekA minimizes the mean state energy plus one penalty for each independent gap:

F = (1/N) sum_a E_a + sigma sum_a P(d_a)
P(d) = d^2 / (d + alpha)

Using adjacent gaps removes redundant pairwise conditions: making all N - 1 adjacent gaps zero makes every selected state degenerate. The objective gradient uses the corresponding adjacent gradient differences. alpha and the energy gaps have Hartree units; sigma and its additive increments are dimensionless.

BaekA tests degeneracy with the outer span

E_last - E_first <= gap

not with the largest individual adjacent gap. The default gap=1e-4 Hartree is the tight outer-span criterion used for this method.

Adaptive Sigma State Machine

sigma is the current dimensionless penalty weight. The update depends on the current optimization state:

1. After an ordinary, nonstationary macro-step, apply the small additive update sigma <- sigma + delta_beta.
2. If the projected stationary tests pass but the outer span is still above gap, consume the next entry beta from beta_schedule and apply the larger additive update sigma <- sigma + beta. The strengthened objective is re-formed immediately at the same geometry.
3. If the projected stationary tests and outer-span test all pass, the BaekA search is converged.

The beta_schedule entries are additive increments. They are neither multipliers nor gap thresholds, and they are consumed only by case 2. Exhausting the schedule while another large jump is required is an error rather than a silent continuation with an invented value.

BaekA does not use pen_incre

pen_incre is the historical multiplicative control for the older two-state penalty implementation. BaekA instead uses the dedicated additive controls delta_beta and beta_schedule, which lower to pen_delta and pen_jump. The earlier simplified native TCI prototype multiplied sigma by pen_incre; that behavior is not the exact BaekA algorithm.

Projected Stationary Tests

The N - 1 adjacent gap-gradient vectors define the constraint subspace. OpenQP obtains its independent numerical basis with an SVD, then decomposes the BaekA objective gradient into:

• a component parallel to the gap-constraint subspace, divided by the current sigma; and
• a perpendicular component along the intersection seam.

A local stationary point requires ||g_parallel||_2 / sigma and ||g_perpendicular||_2 to be at or below the legacy-named rmsd_grad threshold. These are Euclidean norms, not RMS values. It also requires the objective change to be at or below energy_shift. Because sigma evolves, OpenQP recombines the previous geometry's sigma-independent mean-energy and penalty terms using the current same sigma before forming this delta_F; a mere change of penalty weight therefore cannot masquerade as geometry progress or destroy a valid same-objective comparison.

The complete BaekA convergence test is the same-sigma delta_F, the two projected gradient tests, and the outer energy span. Generic max_grad, rmsd_step, and max_step values may still appear in optimization diagnostics, but they are not the BaekA termination criteria. Concise .oqp therefore rejects those three controls when they are written explicitly in a BaekA call.

Public BaekA Controls

.oqp control	Legacy key	Default	Meaning
sigma	pen_sigma	1.0	Current initial dimensionless penalty weight.
alpha	pen_alpha	0.02 Hartree	Positive energy smoothing parameter in P(d).
delta_beta	pen_delta	0.025	Small dimensionless additive increment after an ordinary nonstationary step.
beta_schedule	pen_jump	10,10,25,25,100,100,1000,1000,3000	Ordered larger additive increments used only when locally stationary with an open outer span.
gap	energy_gap	1e-4 Hartree	Maximum permitted outer span E_last - E_first.

alpha must represent a positive energy. In concise .oqp, omitting it uses 0.02 Hartree and explicitly writing alpha=0 is rejected. The traditional schema retains a global pen_alpha=0.0 sentinel because the older two-state penalty algorithm uses that value differently; under meci_search=baeka, an omitted/sentinel pen_alpha=0.0 means the fixed BaekA default 0.02 Hartree. It does not request a gradient-derived automatic value. A gradient RMS has different units and is not a valid replacement for energy-valued alpha.

Common native controls remain available: maxit, energy_shift, rmsd_grad, coordsys, trust, and trust_max.

Use the public gap spelling for BaekA. The internal compatibility spelling energy_gap represents the same setting, so supplying both is rejected as a duplicate rather than choosing one silently.

gap_weight is fixed at 1.0 for BaekA. Adjust sigma instead; introducing a second independent weight would change both the published objective and its projected convergence interpretation.

State Selection

• Supply at least two distinct, consecutive states of one multiplicity.
• Set nstate high enough to contain the highest requested state. Computing a few extra roots can help when inspecting nearby-state character.
• Start from a geometry reasonably close to the desired intersection seam.
• The optimizer follows selected response-root indices. It does not track electronic character or relabel roots during the geometry search.
• auglag is the two-state default, and penalty, ubp, and hybrid remain available. Select algorithm=baeka when the additive independent-gap formulation is intended, including for a two-state calculation.

Python and Legacy .inp Configuration

The concise list of physical labels lowers to an ordered internal state list. The Python workflow accepts the same public names:

job.workflow.meci(
    algorithm="baeka",
    states=[1, 2, 3],
    sigma=1.0,
    delta_beta=0.025,
    beta_schedule=[10, 25],
    gap=1e-4,
)

The equivalent legacy .inp control surface is:

[input]
runtype=meci
method=tdhf
functional=bhhlyp
basis=6-31g*

[scf]
type=rohf
multiplicity=3

[tdhf]
type=mrsf
nstate=6

[optimize]
lib=oqp
meci_search=baeka
states=1,2,3,4
maxit=100
pen_sigma=1.0
pen_alpha=0.02
pen_delta=0.025
pen_jump=10,10,25,25,100,100,1000,1000,3000
energy_gap=1e-4

[oqp]
coordsys=auto
trust=0.15
trust_max=0.5

[optimize] states is authoritative for BaekA and must contain two or more distinct, ascending, consecutive internal roots. The older istate/jstate pair remains the traditional two-state spelling for the other MECI algorithms.

tci Compatibility

Existing tci(S0,S1,S2,...) and sectioned runtype=tci inputs remain supported with their pre-existing three-state adaptive-penalty algorithm and multiplicative pen_incre update. They are not aliases for BaekA. New BaekA inputs should use the explicit general form:

meci(S0,S1,S2,algorithm=baeka)

The shipped C2H4_BHHLYP-MRSFTDDFT_TCI_OQP.inp remains a regression for that legacy behavior. The separate C2H4_BHHLYP-MRSFTDDFT_BAEKA_OQP.inp regression exercises the new MECI algorithm. Their small iteration limits keep continuous integration short; they are smoke tests, not converged scientific templates.