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• Crossing point (MESX and MECI) search by AFIR

# Crossing point (MESX and MECI) search by AFIR

Structures of MESX and MECI can be explored automatically by the SC-AFIR method. In one algorithm termed GP/AFIR, the gradient projection (GP) method, which has been used as a MESX / MECI optimization method, is combined with the SC-AFIR method. The GP method uses a composed gradient gGP,

where vDGV corresponds to the difference gradient vector between the two PESs. The matrix P is so called projection matrix, and P for MESX optimization is defined as follows.

On the other hand, P for MECI optimization is defined as,

where v′ is an arbitrary vector not parallel to vDGV on the branching plane (BP), and the derivative coupling vector or a vector obtained by the BP updating (BPU) method is used. In GRRM17, v′ is obtained by BPU as a unit vector perpendicular to vDGV. In the combined GP/AFIR method, the composed gradient gGP/AFIR is,

where J(Q) is the artificial force term in the AFIR function. This equation can be obtained just by substituting the AFIR function for EX and EY into EX and EY in the expression of gGP, respectively. In the SC-AFIR search, the GP/AFIR path is calculated by inducing structural deformations using gGP/AFIR. A GP/AFIR path leads to an approximate MESX / MECI structure within the seam of crossing / conical intersection hyperspace, where different structures can be obtained depending on a fragment pair in the force term, as in the same way in the SC-AFIR search on a single PES. Therefore, the SC-AFIR algorithm using gGP/AFIR generates various approximate MESX / MECI structures automatically. These approximate structures are further optimized to actual MESX / MECI automatically during the search. Thus, many actual MESX / MECI structures are obtained and appear in xxx_EQ_list.log.

An example of input for an automated exploration of S0/T1 MESX structures with the GP/AFIR approach is:

SC-AFIR/UB3LYP/6-31+G**

0 1
C         -0.450261277786          0.001229457926         -0.553082461345
O          0.054687651811         -0.003728650367          0.666230115529
H          0.197675109173          0.761833114909         -1.107795147264
H          0.197898516798         -0.759333922472         -1.108541506919
Options
Gamma=150.0
END
OptX(Seam)
Second Input
UB3LYP/6-31+G**
0 3
END
MaxStepSize=0.1
Stable=Opt

The OptX(Seam) option is used together with the Second Input … END option. Lines between Second Input and END specify the computation method and [spin charge] of the second state. Use of MaxStepSize=x (x ≤ 0.1) and Stable=Opt options is recommended. When the MO GUESS option is used, two MO files, i.e., mmm.chk and mmm.chk_1, are required for the first and second states, respectively. When OptX(Seam) option is specified, EQOnly is also applied. When the two states belong to the same spin and space symmetry, OptX(Conical) is used instead of OptX(Seam). Unlike ADDF with ModelF or SC-AFIR with ModelF, structures obtained by SC-AFIR with OptX are actual MESX or MECI.

In the other algorithm, SC-AFIR is applied to the following model function F(Q) called seam model function (SMF), which consists of a mean energy term for the two target PESs, EX(Q) and EY(Q), and a penalty function for their energy gap,

where Q is the atomic coordinates {Qi} and α is a constant parameter (in default, α = 30 kJ/mol). Typically, α is set to ∼1/10 of the vertical excitation energy in the target system, and in general results are not very sensitive to the value of α. Minimization of F(Q) gives a geometry in which both the mean energy and the energy gap are small. Hence, local minima on F(Q) can be good guesses of MESXs and MECIs. All obtained local minima on F(Q) are automatically optimized to an actual MESX or MECI, and optimized MESXs or MECIs will appear in xxx_EQ_list.log.

An example of input for an automated exploration of S0/T1 MESX structures with the SMF approach is:

SC-AFIR/UB3LYP/6-31+G**

0 1
C         -0.450261277786          0.001229457926         -0.553082461345
O          0.054687651811         -0.003728650367          0.666230115529
H          0.197675109173          0.761833114909         -1.107795147264
H          0.197898516798         -0.759333922472         -1.108541506919
Options
Gamma=150.0
END
ModelF
Second Input
UB3LYP/6-31+G**
0 3
END
MaxStepSize=0.1
Stable=Opt

The ModelF option is used together with the Second Input … END option. Lines between Second Input and END specify the computation method and [spin charge] of the second state. Use of MaxStepSize=x (x ≤ 0.1) and Stable=Opt options is recommended. When the MO GUESS option is used, two MO files, i.e., mmm.chk and mmm.chk_1, are required for the first and second states, respectively. When ModelF option is specified, EQOnly is also applied. In the xxx_EQ_list.log file of this calculation, many local minimum structures on the model function SMF are obtained. Structures obtained by this calculation are just approximate MESX or MECI structures, and therefore need to be reoptimized with OptX.