Conical intersections are the regions of the potential energy surface characterized by degeneracy between two or more electronic states with the same symmetry. For a two-state intersection, the intersection consists of an -dimensional hypersurface (the *seam space*) within which the two states are degenerate, where is the number of internal coordinates of the molecule. (The remaining two dimensions for a *branching space * in which the degeneracy is lifted by any infinitesimal displacement.) Radiationless transitions between the two electronic states are likely to occur in an around a conical seam. The first step in any study of nonadiabatic excited-state dynamics is often an exploration of the geometries and energies of the lowest-energy point within the seam space, which is the so-called *minimum-energy crossing point * (MECP). In some sense, the MECP is to internal conversion and photochemical processes what the transition state is to single-surface chemical reactions.

The two-dimensional branching space between electronic states and is spanned by a pair of vectors that are usually denoted and [416]:

(9.2) | ||||

(9.3) |

While is available analytically for any electronic structure method that has analytic excited-state gradients, analytic implementations of the nonadiabatic coupling vector are not routinely available. For this reason, several algorithms have been developed to optimize MECPs without the need to evaluate , and three such algorithms are available in Q-Chem. The first of these is a penalty-constrained optimization algorithm developed by Levine *et al.* [417], in which the objective function that is optimized to locate the MECP is

(9.4) |

where is a fixed parameter to avoid singularities and is a Lagrange multiplier for a penalty function meant to drive the energy gap to zero. Optimization of proceeds iteratively for increasingly large values of the parameter . The second MECP optimization algorithm is a simplification of the first one that we call the “direct” method. Here, the gradient of the objective function is

(9.5) |

where

(9.6) |

is the mean energy gradient (with being the nuclear gradient for state ), and

(9.7) |

is the normalized difference gradient. Finally,

(9.8) |

projects the gradient difference direction out of the mean energy gradient in Eq. eq:direct_MECP. The third and final MECP optimization algorithm that is available in Q-Chem is the branching-plane updating method developed by Maeda *et al.* [418]. This algorithm uses a gradient that is similar to that in Eq. eq:direct_MECP but projects out not just in Eq. eq:MECP_projector but also a second vector that is orthogonal to it.

None of these three methods requires evaluation of nonadiabatic couplings, and all three can be used to optimize MECPs at the CIS, SF-CIS, TDDFT, SF-TDDFT, and SOS-CIS(D0) levels. The direct algorithm can also be used for EOM-XX-CCSD methods. Note, however, that all linear-response methods (a category that includes CIS, TDDFT, and EOM-XX-CCSD) incorrectly describe the topology of any conical intersection that involves the reference (usually, ground) state. Specifically, it can be shown in such cases that only a one-dimensional branching space is obtained [419]. If conical intersections involving the ground state are to be described correctly, one must use a different reference state, which can be accomplished with spin-flip (SF) methods [420, 421, 422]. In particular, using a high-spin triplet reference state, the ground-state singlet () appears as an excitation (possibly with a negative excitation energy) alongside the excited-state singlets, , so there is no topology problem in describing an / conical intersection. Accurate MECP geometries, as compared to multireference configuration-interaction benchmarks, have been reported [421]. It should be noted, however, that the component of the triplet state also shows up as an excitation in SF methods, and while Q-Chem attempts to identify this triplet state automatically (based on a threshold for ), severe spin contamination can sometimes hamper one’s ability to distinguish this state from the singlet excited states [421].

In Q-Chem 4.3, analytic derivative couplings for CIS and TDDFT have been implemented 10.3. Thus, CIS and TDDFT MECP optimizations can use this new feature to reduce the optimization cycles.[422] Note that the *$derivative_coupling* section is not required in MECP optimization jobs, and *$rem* variable MECP_METHODS must be set to BRANCHING_PLANE.

MECP_OPT

Determines whether we are doing MECP optimizations.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TRUE

Do MECP optimizations.

FALSE

Don’t do MECP optimizations.

RECOMMENDATION:

None.

MECP_METHODS

Determines which method to be used.

TYPE:

STRING

DEFAULT:

BRANCHING_PLANE

OPTIONS:

BRANCHING_PLANE

Use the branching-plane updating method.

MECP_DIRECT

Use the direct method.

PENALTY_FUNCTION

Use the penalty-constrained method.

RECOMMENDATION:

The direct method is stable for small molecules or molecules with high symmetries, but the branching-plane updating method is more efficient for larger molecules. However, the latter does not work if the two states have different symmetries. If using branching plane updating method, GEOM_OPT_COORDS must be set to 0 in the

$remsection (i.e., this algorithm is available in Cartesian coordinates only). The penalty-constrained method converges slowly and is suggested only when the other methods do not work.

MECP_STATE1

Determines the first state for crossing.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

None

OPTIONS:

[,]

find the th excited state with the total spin of ; means the SCF ground state.

RECOMMENDATION:

is ignored for restricted calculations; for unrestricted calculations, can only be 0 or 1.

MECP_STATE2

Determines the second state for crossing.

TYPE:

INTEGER/INTEGER ARRAY

DEFAULT:

None

OPTIONS:

[,]

find the th excited state with the total spin of ; means the SCF ground state.

RECOMMENDATION:

is ignored for restricted calculations; for unrestricted calculations, can only be 0 or 1.

CIS_S2_THRESH

Determines whether a state is singlet or triplet in unrestricted calculations.

TYPE:

INTEGER

DEFAULT:

120

OPTIONS:

None

RECOMMENDATION:

If set to 120, the states with are treated as triplet states, with other states are treated as singlets.

MECP_PROJ_HESS

Determines whether to project out the coupling vector from the Hessian when using branching plane updating method.

TYPE:

LOGICAL

DEFAULT:

TRUE

OPTIONS:

TRUE

FALSE

RECOMMENDATION:

Use Default.

The MECP between the S and S states of NO is optimized using the direct method:

**Example 9.192** MECP optimization for NO using SOS-CIS(D0)

$MOLECULE -1 1 N1 O2 N1 RNO O3 N1 RNO O2 AONO RNO=1.50 AONO=100 $END $rem jobtype = opt method = soscis(d0) basis = aug-cc-pVDZ aux_basis = rimp2-aug-cc-pVDZ purecart = 1111 cis_n_roots = 4 cis_triplets = false cis_singlets = true mem_static = 900 mem_total = 1950 mecp_opt true mecp_state1 [0,2] mecp_state2 [0,3] mecp_methods mecp_direct $end

The MECP between the S and S states of ethylidene is optimized using the branching-plane update method:

**Example 9.193** MECP optimization for ethylidene using SF-TDDFT

$molecule 0 3 C 0.0446266041 -0.2419241370 0.3571573801 C 0.0089051507 0.6727548956 1.4605006396 H 0.9284257388 -0.1459163900 -0.2720952334 H -0.8310326564 -0.1926895078 -0.2885298629 H -0.0092388670 0.9611331703 2.4799363398 H 0.0683140308 -1.2533580302 0.7788470826 $end $rem jobtype opt mecp_opt true mecp_methods branching_plane MECP_PROJ_HESS true !project out y vector from the hessian GEOM_OPT_COORDS 0 !currently only works for Cartesian coordinate method bhhlyp spin_flip true unrestricted true basis 6-31G(d,p) cis_n_roots 4 mecp_state1 [0,1] mecp_state2 [0,2] CIS_S2_THRESH 120 $end

The MECP between S and S states of twisted-pyramidalized ethylene is optimized using the penalty-constrained method:

**Example 9.194** MECP optimization for ethylene using SF-TDDFT

$molecule 0 3 C -0.0158897609 0.0735325545 -0.0595597308 C 0.0124274563 -0.0024687284 1.3156941918 H 0.8578762360 0.1470146857 -0.7105293671 H -0.9364708648 -0.0116961121 -0.6267613144 H 0.7645577838 0.6633816890 1.7625731128 H 0.7407739370 -0.8697640880 1.3285831079 $end $rem jobtype opt mecp_opt true mecp_methods PENALTY_FUNCTION method bhhlyp spin_flip true unrestricted true basis 6-31G(d,p) cis_n_roots 4 mecp_state1 [0,1] mecp_state2 [0,2] CIS_S2_THRESH 120 $end

The MECP between A and ÃB states of the N ion using the direct method:

**Example 9.195** MECP optimization for N using EOM-EE-CCSD

$MOLECULE 1 1 N1 N2 N1 rNN N3 N2 rNN N1 aNNN rNN=1.54 aNNN=50.0 $END $REM JOBTYPE OPT MECP_OPT TRUE MECP_METHODS mecp_direct METHOD EOM-CCSD BASIS 6-31G ee_singlets [0,2,0,2] XOPT_STATE_1 [0,2,2] XOPT_STATE_2 [0,4,1] ccman2 false GEOM_OPT_TOL_GRADIENT 30 $END

MECP optimization using analytic derivative coupling:

**Example 9.196** MECP between and of ethylidene optimized using BH&HLYP spin-flip TDDFT with analytic derivative coupling

$molecule 0 3 C 0.0446266041 -0.2419241370 0.3571573801 C 0.0089051507 0.6727548956 1.4605006396 H 0.9284257388 -0.1459163900 -0.2720952334 H -0.8310326564 -0.1926895078 -0.2885298629 H -0.0092388670 0.9611331703 2.4799363398 H 0.0683140308 -1.2533580302 0.7788470826 $end $rem jobtype opt mecp_opt true mecp_methods branching_plane MECP_PROJ_HESS true GEOM_OPT_COORDS 0 mecp_state1 [0,1] mecp_state2 [0,2] unrestricted true spin_flip true cis_n_roots 4 cis_der_couple true CIS_DER_NUMSTATE 2 set_iter 50 exchange bhhlyp basis 6-31G(d,p) symmetry_ignore true $end