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 **g**^{GP},

where **v**^{DGV} 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 **v**^{DGV} 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 **v**^{DGV}. In the combined GP/AFIR method, the composed gradient **g**^{GP/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 *E*^{X} and *E*^{Y} into *E*^{X} and *E*^{Y} 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 S_{0}/T_{1} MESX structures with the GP/AFIR approach is:

` # SC-AFIR/UB3LYP/6-31+G** 0 1C -0.450261277786 0.001229457926 -0.553082461345O 0.054687651811 -0.003728650367 0.666230115529H 0.197675109173 0.761833114909 -1.107795147264H 0.197898516798 -0.759333922472 -1.108541506919Options Add InteractionGamma=150.0END OptX(Seam)Second InputUB3LYP/6-31+G**0 3ENDMaxStepSize=0.1Stable=Opt `

The ** OptX(Seam)** option is used together with the

**option. Lines between**

`Second Input … END`**and**

`Second Input`**specify the computation method and [spin charge] of the second state. Use of**

`END`**(x ≤ 0.1) and**

`MaxStepSize=`*x***options is recommended. When the**

`Stable=Opt`**option is used, two MO files,**

`MO GUESS`*i.e.*, mmm.chk and mmm.chk_1, are required for the first and second states, respectively. When

**option is specified,**

`OptX(Seam)`**is also applied. When the two states belong to the same spin and space symmetry,**

`EQOnly`**is used instead of**

`OptX(Conical)`**. Unlike ADDF with ModelF or SC-AFIR with**

`OptX(Seam)``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, *E*^{X}(**Q**) and *E*^{Y}(**Q**), and a penalty function for their energy gap,

where **Q** is the atomic coordinates {*Q _{i}*} 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 S_{0}/T_{1} MESX structures with the SMF approach is:

` # SC-AFIR/UB3LYP/6-31+G** 0 1C -0.450261277786 0.001229457926 -0.553082461345O 0.054687651811 -0.003728650367 0.666230115529H 0.197675109173 0.761833114909 -1.107795147264H 0.197898516798 -0.759333922472 -1.108541506919Options Add InteractionGamma=150.0END ModelFSecond InputUB3LYP/6-31+G**0 3ENDMaxStepSize=0.1Stable=Opt `

The ** ModelF** option is used together with the

**option. Lines between**

`Second Input … END`**and**

`Second Input`**specify the computation method and [spin charge] of the second state. Use of**

`END`**(x ≤ 0.1) and**

`MaxStepSize=`*x***options is recommended. When the**

`Stable=Opt`**option is used, two MO files,**

`MO GUESS`*i.e.*, mmm.chk and mmm.chk_1, are required for the first and second states, respectively. When

**option is specified,**

`ModelF`**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**

`EQOnly`**.**

`OptX`##### Frequently used options

See Global reaction route mapping by single-component algorithm (SC-AFIR).