Simulation Method

Introduction

The three-dimensional structure of each molecule $D_{obs}$ is expressed by a linear combination of all $m$ vibrational modes of symmetries $\Gamma$ with coefficients $d$. By rearranging the sum, a separation of the vibrational normal modes into out-of-plane ($oop$) and in-plane ($ip$) symmetries can be made.

$$\large D_{obs} = \sum_{\Gamma,m}d_m^{\Gamma} D_m^{\Gamma} = \sum_{\Gamma_{oop},m}d_m^{\Gamma_{oop}} D_m^{\Gamma_{oop}} + \sum_{\Gamma_{ip},m}d_m^{\Gamma_{ip}} D_m^{\Gamma_{ip}} = D_{obs}^{oop} + D_{obs}^{ip}$$

For non-linear molecules there are 3N-6 degrees of freedom which results 66 modes for the 24 framework atoms of the C₂₀N₄ perimeter (Porphyrin, Porphycene, Corrphycene). For the C₁₉N₄ (Corrole) and C₁₈N₄ perimeter (Norcorrole) this results in 63 and 60 modes respectively. Of this 3N-6 modes N-3 modes are out-of-plane distortions (21 for Porphyrin, Porphycene, Corrphycene - 20 for Corrole, 19 for Norcorrole) and 2N-3 modes (45 for Porphyrin, Corrphycene, Porphycene, 43 for Corrole, 41 for Norcorrole) are in-plane distortions.

Simulation

Simulation Procedure

These modes are used as references when simulating the experimental structure (extended basis uses second set of modes). Die displacement vectors of each mode are created by calculating the mean square plane deviation for each atom of the reference structure. These 6 (or 12) vectors form the reference matrix $D_{oop}^{mxn}$. Using the Matrix QR Algorithm the following equation system is solved:

$$\large \hat{D}_{oop} = \hat{d}_{oop} * D_{oop}^{mxn}$$

The solution $\hat{d}_{oop}$ beeing the coefficients of the linear combination which correspond to the mode strengths. With these coefficients a simulated distortion is computed by multiplying the coefficients with the normalized references.

Displacement parameter

One important value is the overall displacement parameter $D_{oop}$ which quantifies the overall out-of-plane distortion by using the euclidean norm of all atom displacements. The value is calculated as follows:

$$\large D_{oop} = \sqrt{\sum_{i=1}^m(\Delta_i^z)^2}$$

For estimating the goodness of the fit the experimental displacement parameter can be compared to the simulated one. This is often called $\delta_{oop}$.

Out-of-Plane Symmetries

Porphyrin (D_4h)

The Porphyrin Macrocycle has 24 perimeter atoms and it's pointgroup is D_4h. Therefore there are 21 out-of-plane distortions. The D_4h point group contains 5 symmetries for each, out-of-plane and in-plane distortions (B_2u, B_1u, A_2u, E_g(x,y), A_1u, and A_1g, A_2g, B_1g, B_2g and E_u(x,y), respectively). These out-of-plane modes are distributed as follows:

$$\large \Gamma_{oop} = 2A_{1u} + 3A_{2u}+3B_{1u}+3B_{2u}+5E_{g}$$

Each out-of-plane symmetry corresponds to a specific mode:

• Doming: $A_{2u}$
• Saddling: $B_{2u}$
• Ruffling: $B_{1u}$
• Waving: $E_{g}$
• Propellering: $A_{1u}$

To see the mode representations, visit this site: Modes.

Corrole (C_2v)

The Corrole Macrocycle has 23 perimeter atoms and it's pointgroup is C_2v. Therefore there are 20 out-of-plane distortions. The C_2v point group contains 2 symmetries for each, out-of-plane and in-plane distortions (B₁, A₂ and A₁, B₂ respectively). These out-of-plane modes are distributed as follows:

$$\large \Gamma_{oop} = 10A_{2} + 10B_{1}$$

Each out-of-plane symmetry corresponds to a specific mode:

• Doming: $B_{1}$
• Saddling: $A_{2}$
• Ruffling: $B_{1}$
• Waving: $B_{1} / A_{2}$
• Propellering: $A_{2}$

To see the mode representations, visit this site: Modes.

Norcorrole (D_2h)

The Norcorrole Macrocycle has 22 perimeter atoms and it's pointgroup is D_2h. Therefore there are 19 out-of-plane distortions. The D_2h point group contains 4 symmetries out-of-plane(B_3u, A_u, B_1g and B_2g).

These out-of-plane modes are distributed as follows:

$$\large \Gamma_{oop} = 5B_{3u} + 4A_u + 4B_{1g} + 5B_{2g}$$

Each out-of-plane symmetry corresponds to a specific mode:

• Doming: $B_{3u}$
• Saddling: $A_{u}$
• Ruffling: $B_{3u}$
• Waving: $B_{1g} / A_{2g}$
• Propellering: $A_{u}$

To see the mode representations, visit this site: Modes.

Porphycene (D_2h)

The Porphycene Macrocycle has 24 perimeter atoms and it's pointgroup is D_2h. Therefore there are 21 out-of-plane distortions. The D_2h point group contains 4 symmetries out-of-plane(B_3u, A_u, B_1g and B_2g).

These out-of-plane modes are distributed as follows:

$$\large \Gamma_{oop} = 5B_{3u} + 6A_u + 5B_{1g} + 5B_{2g}$$

Each out-of-plane symmetry corresponds to a specific mode:

• Doming: $B_{3u}$
• Saddling: $A_{u}$
• Ruffling: $B_{3u}$
• Waving: $B_{1g} / A_{2g}$
• Propellering: $A_{u}$

To see the mode representations, visit this site: Modes.

Corrphycene (C_2v)

The Corrphycene Macrocycle has 24 perimeter atoms and it's pointgroup is C_2v. Therefore there are 21 out-of-plane distortions. The C_2v point group contains 2 symmetries for each, out-of-plane and in-plane distortions (B₁, A₂ and A₁, B₂ respectively). These out-of-plane modes are distributed as follows:

$$\large \Gamma_{oop} = 11A_{2} + 10B_{1}$$

Each out-of-plane symmetry corresponds to a specific mode:

• Doming: $B_{1}$
• Saddling: $A_{2}$
• Ruffling: $B_{1}$
• Waving: $B_{1} / A_{2}$
• Propellering: $A_{2}$

To see the mode representations, visit this site: Modes.

Symmetry Table