# Physical Mechanism of Nonlinear Spectra in Triangene

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## Abstract

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## 1. Introduction

## 2. Results and Discussion

#### 2.1. OPA Spectrum

_{73}, S

_{66}, S

_{62}, S

_{62}, S

_{91}, S

_{94}, and S

_{98}, and the secondary absorption peaks were caused by S

_{38}, S

_{37}, S

_{36}, S

_{35}, S

_{55}, S

_{53}, and S

_{55}, respectively. Since N = 3TSCs also has an excited state with a similar vibronic intensity near S

_{66}, when the intensity of an oscillator is broadened into the curve of the absorption spectrum, it is subjected to the intensity of that oscillator and forms an even stronger absorption peak. Thus, although S

_{37}has a higher oscillator intensity, it has a smaller molar absorptivity. With the addition of a triangulene unit, the two main absorption peaks of N = 4-1TSCs were red-shifted. The main absorption peak was shifted to 261.1 nm, caused by S

_{62}, and the secondary absorption peak was caused by S

_{36}near 297 nm (Figure S1c). Figure S1d shows the OPA spectrum of 4-2TSCs with a main absorption peak of ${\lambda}_{max}$ = 261.4 nm, caused by S

_{62}. The secondary absorption peak was caused by S

_{35}at 297.1 nm, which is spectroscopically very close to N = 4-1TSCs. This small gap may have occurred because triangulene forms two completely different conformations due to different attachment sites, resulting in different excitation energy between the two configurations at the same wavelength. This difference leads to an increase with increasing size, indicating that the change in conformation has a significant effect on the optical absorption properties of triangulene. When the TSCs form a closed six-membered ring, ${\lambda}_{max}$ = 257.6 nm of the main absorption peak of the N = 6c TSCs is caused by the excited state S

_{98}, and the secondary absorption peak is caused by S55 near 294.5 nm (Figure S1g). The distinctive feature of the absorption spectrum of N = 6cTSCs is greatly enhanced absorbance. The main absorption peaks show a color enhancement effect, mainly due to the large vibrational intensity of the system.

_{50}; ${\lambda}_{max}$ = 573 nm for N = 4-1TSCs, where the absorption peak is mainly caused by S

_{40}; and ${\lambda}_{max}$ = 573.6 nm for N = 4-2TSCs, with absorption peaks mainly caused by S

_{40}and reduced two-photon cross sections compared to N = 4-1TSCs. After changing the connection point of triangulene, N = 4-2TSCs and ${\lambda}_{max}$ = 573.6 nm. Here, the absorption peak is mainly caused by S

_{40}, and the two-photon cross-section decreases compared to N = 4-1TSCs; N = 6cTSCs and ${\lambda}_{max}$ = 573.8 nm are mainly caused by S

_{60}. Figure S2 presents a composite diagram of the two-step transition summation terms of the four configurations of the TPA spectra. It can be seen that the molar absorption intensity of TPA with N = 3TSCs is the largest. In addition, the TPA peak of N = 6cTSCs changes from two to one, mainly because the excited state of the two-photon cross-section is concentrated around 550 nm.

_{38}. TDM shows that there are significant green and yellow areas on the main diagonal. There are also some transition density distributions on both sides of the main diagonal, which are caused by the charge transfer. The density of the hole-electron pairs shows that the holes are mainly distributed at the junctions of two triangulene sheets, and the electrons are distributed on both sides. Therefore, the electronic transition of S

_{38}has obvious charge transfer characteristics. After adding a triangle, TDM shows that the transition density is almost completely distributed on the main diagonal, corresponding to atoms 25–40, namely, the triangle unit in the middle of the triangulene dimer. The blue isosurface on the hole-electron pair represents the area where the electron decreases during the electron transition, and the red isosurface represents the area where the electron increases. In addition to a small number of hole-electron pairs on the triangulene units on both sides, the main isosurface is distributed on the triangulene in the middle, which is very consistent with the distribution of transition density, as shown in Figure 3b. In addition to the relatively high transition density of the intermediate triangulene, there are two areas with weak brightness at the upper-left and upper-right sides of the TDM, which represent the transition density on the left and right sides of the triangulene trimer, respectively. Because these transition densities are mainly distributed on the diagonal, the TDM reflects the local excitation characteristics between adjacent carbon atoms. After adding a triangulene unit, the main diagonal of TDM shows two connected areas with similar brightness, which can be speculated to represent the local excitation of the middle two pieces of triangulene in N = 4-1TSCs. The hole-electron pair density diagram on the right also indicates that the hole-electron is mainly distributed at the connection point of the middle two pieces of triangulene, as shown in Figure 3c. After changing the connection point of triangulene, the distribution of bright areas on TDM is similar, belonging to the local excitation of the middle two pieces of triangulene, as shown in Figure 3d. The ring triangulene spin chain composed of five triangulene N = 5cTSCs also belongs to local excitation in S

_{55}, and the hole-electrons are mainly distributed at the junction of triangulene units on both sides (Figure 3e). After triangulene forms a closed six-member ring, TDM shows that the transition density is also divided into two bright regions, but the transition density of this system is the smallest of the four configurations. From the perspective of the density of hole-electron pairs, the distribution of the hole-electron-equivalent surface is central symmetry. At the same isosurface value (iso value = 0.001), the density of the hole-electron pair is the smallest (Figure 3f), which shows that the electron transition is weakened in this stable ring structure.

_{73}belong to local excitation and the hole-electrons overlap at the junction of two triangulene units, as shown in Figure 4a. The excitation characteristics of N = 3TSCs in S

_{66}are opposite to the secondary absorption peak. From the TDM, the bright yellow and green areas are mainly located at the lower left and upper right areas of the main diagonal. Compared to S

_{37}, the distribution range of the bright regions is greatly enlarged, and their atomic numbers correspond to triangulene on both sides of the system. From the perspective of hole-electron pair density, the hole-electron isosurface is very significant and almost completely distributed on both sides of triangulene, belonging to local excitation between adjacent carbon atoms of triangulene on both sides, as shown in Figure 4b. The electron excitation of N = 4-1TSCs in S

_{62}was transferred to the middle two triangulene units. TDM shows that compared to S

_{36}, the degree of electronic excitation is enhanced. Based on the density of the hole-electron pairs, there is also electron distribution on the middle two triangulene junction points, as shown in Figure 4c. After changing the connection point, the electronic excitation characteristics of N = 4-2TSCs in S

_{62}also belong to local excitation, as shown in Figure 4d. The electronic transition of N = 5cTSCs in S

_{91}comes from the triangulene unit in the middle part, which belongs to local excitation, as shown in Figure 4e. The degree of electronic excitation in the closed six-member ring in S

_{98}is still the smallest, belonging to weak local excitation. Compared to the secondary absorption peak, the hole-electron pairs are distributed on the triangulene unit adjacent to the two directions, as shown in Figure 4f. Figure 5 shows the TDM and hole-electron pair density of the main absorption peak of the N = 6, 8-1, 8-2, and 16cTSCs, where the N = 8-1, 8-2, and 16cTSCs have only one absorption peak. Compared to the previous structure, the transition density of TDM is only distributed on the main diagonal as the system increases. Here, the isosurface of the hole-electron is significantly reduced, indicating that the degree of electron transition is weakened (Figure 5a–c). When the quantum dot is composed of 16 triangulene units, the distribution of the hole-electron disappears, as shown in Figure 5d.

#### 2.2. Analysis of Transition Index of Excited State

_{37}being the largest among the four configurations. The D index is the distance between the hole and the electron centroid, which is derived from the Cartesian coordinates of the hole and the electron centroid and obtained from the following formula:

_{66}reached 0.173 Å, which is the largest among all one-photon excited states. This result was mainly due to the location of the hole-electron exchange on both sides of the triangulene. Additionally, on a triangulene, the holes are mainly distributed at the edge of the triangulene, while the electrons are mainly distributed in the interior. Therefore, compared to other excited states, the electron migration path in S

_{66}is longer. Corresponding to the distance between the centroid of the hole-electron is the separation degree t of the electron and the hole, which is defined as

#### 2.3. Molecular van der Waals Surface Electrostatic Potential

#### 2.4. Two-Photon Transition

_{50}. Figure 7a,b show the TDM and hole-electron pair density from the ground state to the excited state (${\mathrm{S}}_{0}\to {\mathrm{S}}_{2}$). The TDM shows that the transition density is mainly distributed in the middle of the diagonal, and the hole-electron density value is concentrated on the intermediate triangulene. Therefore, the first step of transition belongs to strong local excitation between adjacent carbon atoms on the intermediate triangulene. The second transition comes from the second part of Formula (1), that is, the electronic transition from the excited state to the final state (${\mathrm{S}}_{2}\to {\mathrm{S}}_{50}$). TDM shows that in addition to the transition density on the main diagonal, there are weak bright areas on the non-diagonal, as shown in Figure 7c. Combined with the density of hole-electron pairs, it can be seen that the electrons transfer from the left triangulene of N = 3TSCs to the middle triangulene, as shown in Figure 7d. Therefore, the second transition can be observed as local charge transfer excitation.

_{40}contribution, and its two-photon transition consists of two channels, where the intermediate state of channel I is S

_{36}. From the perspective of TDM and hole-electron pair density, the first step of transition ${\mathrm{S}}_{0}\to {\mathrm{S}}_{36}$ belongs to local excitation between the two adjacent atoms of the middle triangle alkene, as shown in Figure 8a,b. The transition density of transition ${\mathrm{S}}_{36}\to {\mathrm{S}}_{40}$ in the second step is significantly lower than that in the first step, and there is almost no hole-electron pair density (Figure 8c,d). When the intermediate state is S

_{36}, the TPA is mainly caused by the first step. The intermediate state of the other channel in the two-photon transition of S

_{40}is S

_{39}. The TDM of ${\mathrm{S}}_{0}\to {\mathrm{S}}_{39}$ shows that the first step of the transition still belongs to local excitation. Compared to channel I, the degree of local excitation is enhanced. Based on the distribution of the hole-electron pair density, the hole-electron distribution has a very obvious distribution on the four triangles (Figure 8e,f), and the transition characteristics of the second step of transition ${\mathrm{S}}_{39}\to {\mathrm{S}}_{40}$ are similar to those of channel I. The two-photon transition of N = 4-1TSCs shows that the TPA of the triangulene spin chain is mainly caused by the first transition after the system increases.

_{40}and consists of two channels. The dipole moment distance of the first step is smaller than that of the second step. The intermediate state of channel I is S

_{35}, and the TDM and hole-electron pair density of the two-step transition is very similar to those of N = 4-1TSCs, as shown in Figure 9a–d. The other channel, the intermediate state, is S

_{38}. The TDM of ${\mathrm{S}}_{0}\to {\mathrm{S}}_{38}$ shows that in addition to the relatively large transition density on the diagonal, there is also significant transition density distribution in the non-diagonal region, as shown in Figure 9e. Based on the density of hole-electron pairs, the electrons are mainly concentrated on the two triangulene units in the upper half, as shown in Figure 9f. This result indicates that the electron transfer direction extends from the two triangulene units in the lower part to the adjacent triangulene units in the upper part, presenting the characteristics of charge transfer. The second transition ${\mathrm{S}}_{38}\to {\mathrm{S}}_{40}$ belongs to weak charge transfer excitation, as shown in Figure 9g,h. Therefore, a comparison of the two configurations shows that the charge transfer ratio of the two-step transition can be significantly increased by changing the triangulene junction point, and the electronic transition ability can be enhanced.

_{61}. The two-photon transition process includes two channels. The intermediate state of channel I is S

_{56}. From the perspective of TDM, the first step of transition ${\mathrm{S}}_{0}\to {\mathrm{S}}_{56}$ belongs to local excitation. As shown in Figure 10a, the density of the hole-electron pairs shows that the hole-electrons are distributed on the adjacent triangulene units on the upper and lower sides (Figure 10b). Combined with the second transition characteristics of the triangulene spin chain in the previous three configurations, we found that with a further increase in size, the electronic transition degree of the second transition of N = 6cTSCs further decreased, and the distribution of hole-electrons almost disappeared (Figure 10c,d). The intermediate state of channel II is S

_{55}, and the first step of transition ${\mathrm{S}}_{0}\to {\mathrm{S}}_{55}$ also belongs to local excitation. The second step of transition ${\mathrm{S}}_{55}\to {\mathrm{S}}_{61}$ has the same excitation characteristics as channel I. However, unlike channel I, the hole-electron distribution is exactly the opposite, with hole-electrons mainly distributed on left and right single triangulene units.

#### 2.5. Electron Circular Dichroism

_{30}at 301.8 nm, and the negative absorption peak ${\lambda}_{max}$ is caused by S

_{26}at 321.4 nm. After adding a triangulene unit, the circular dichroism of N = 4-1TSCs almost disappears from the perspective of the absorption intensity of the ordinate, which is due to the formation of a triangulene spin chain with a symmetrical structure after adding a triangulene unit. Compared to N = 3TSCs, the molar absorption intensity of N = 4-1TSCs is positive, and the main and secondary absorption peaks near the same wavelength are, respectively, caused by S

_{36}at 297.0 nm and S

_{21}at 322.3 nm, as shown in Figure 11b. After changing the linkage sites, the changing trend of ECD of N = 4-2TSCs is similar to that of N = 4-1TSCs, but the intensity of the molar absorption peak is greatly enhanced. The ${\lambda}_{max}$ of the main absorption peak and the secondary absorption peak are, respectively, caused by S

_{35}at 288.7 nm and S

_{21}at 322.4 nm, as shown in Figure 11c. When triangulene forms a closed six-member ring, the intensity of the absorption peak decreases significantly. Compared to N = 3TSCs, the ECD absorption peak of the same wavelength N = 6cTSCs is reversed, and the ${\lambda}_{max}$ of the negative absorption peak is caused by S

_{33}at 322.3 nm, as shown in Figure 11d.

#### 2.6. Chiral Physical Mechanism

_{30}. Compared to S

_{26}, the transition electric dipole moment density and transition magnetic dipole moment density of S

_{30}in the three components are similar, but the distribution trend is very different. The yellow isosurface and purple isosurface of S

_{26}on the Y and Z components are relatively independent, so the positive and negative values of the transition magnetic dipole moment are highly separated, as shown in Figure 13a–c. The positive and negative values of the transition magnetic dipole moment density of S

_{30}on the Y and Z components are evenly distributed, as shown in Figure 12d–f, which is why circular dichroism changes from negative to positive. After adding a triangulene unit, the transition electric dipole of S

_{36}is mainly caused by the Y component, as shown in Figure 13h. The transition magnetic dipole is mainly caused by the Z component, and the X component also contributes a portion of the transition magnetic dipole moment (as shown in Figure 13j–l). Compared to N = 3TSCs, the transition dipole moment of these components is mainly caused by a single square quantity. However, the density of the transition magnetic coupling of S

_{30}is larger in the Y and Z components, while the transition magnetic coupling of S

_{36}is only larger in the Z component, so the overall tensor product is smaller than S

_{30}, and its rotor strength is also relatively small. Unlike N = 3TSCs, the absorption peak of S

_{36}still shows positive circular dichroism because the isosurface size of the transition electric dipole moment density and the transition magnetic dipole moment density on the three components of S

_{36}are similar to those of S

_{21}, and the distribution trends are also similar. In both, the positive and negative values of the transition electric dipole moment are uniformly distributed on the Y component, and the positive and negative values of the transition magnetic dipole moment are separated on the Z component (as shown in Figure 12g–l and Figure 13g–l. In addition, since the transition dipole moment of S

_{36}on the Y component is significantly greater than that of S

_{21}(Figure 12h and Figure 13h), it has strong circular dichroism.

_{23}. The upper part of Figure 14 shows the density of the transition electric dipole moment and transition magnetic dipole moment of S

_{23}in Cartesian coordinates. Here, the transition dipole moment is mainly caused by the X component, and there are also some values on the Y component. The transition dipole moment density on the Z component completely disappears, as shown in Figure 14a,b. The transition magnetic dipole moment is mainly caused by the Z component, with only a small distribution on the X and Y components, as shown in Figure 14d–f. Compared to N = 4-1TSCs, the transition magnetic dipole moments of the two systems are similar, and the difference between the system and the photon electromagnetic interaction mainly comes from the transition electric dipole moment. When the connection mode of the added triangulene sheet unit and triangulene trimer is in periodic superposition, the distribution of the transition dipole moment on the three components becomes relatively uniform, and the value is relatively small. When triangulene changes the linkage site and triangulene trimer connection, a large transition dipole moment is generated in the X component. Therefore, this semi-closed triangulene spin chain has a larger tensor product of electricity and magnetism and will present greater rotor strength and stronger circular dichroism.

_{33}. Due to its unique ring structure, the complementary characteristics of the transition electric dipole moment and the transition magnetic dipole moment are more significant than the chain structure, and the distribution trend is similar to that of N = 4-2TSCs (Figure 14a–f,g–l.) Therefore, the circular dichroism of the two structures at this wavelength is positive.

_{35}, whose circular dichroism is significantly enhanced compared to that of S

_{23}. For the transition dipole moment, the density values on the X and Y components are larger than those of S

_{23}, as shown in Figure 15a,b. This result mainly shows that the two triangulenes in the upper half of the semi-closed triangulene spin chain also produce a small amount of transition dipole moment density. The change in the transition magnetic dipole moment in the X and Y components is the same. The Z component has an obvious transition magnetic dipole moment density in the whole semi-closed triangulene spin chain, as shown in Figure 15d–f. The large transition electric dipole moment response and transition magnetic dipole moment response on each component make the rotor strength of S

_{35}much greater than that of S

_{23}. The circular dichroism of the closed triangulene six-membered ring N = 6cTSCs at 322.6 nm is negative, and greater chiral inversion occurs compared to N = 4-2TSCs at the same wavelength. The distribution of the transition electric dipole moment density and the transition magnetic dipole moment density in the X and Y components of the two configurations is approximate (Figure 15a–f,g–l). However, there are great differences in the Z component. The transition magnetic coupling of N = 4-2TSCs in the Z component is mainly caused by the yellow isosurface representing a negative value. However, the transition dipole moment density on the closed six-membered ring triangulene spin chain is composed of yellow and purple isosurfaces, and positive and negative values have the same contribution, as shown in Figure 15f–l. This contribution is due to the uneven distribution of transition magnetic dipole moments caused by changes in the triangulene spin chain configuration, resulting in chiral inversion.

## 3. Methods

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Triangulene spin chains trimers (N = 2, 3, 4-1, 4-2, 6, 8-1, 8-2TSCs) and triangulene cyclic spin chains (N = 5c, 6c, 16cTSCs).

**Figure 3.**TDM and hole-electron pair density of one-photon excited states of secondary absorption peaks of triangulene spin chain N = 2 (

**a**), 3 (

**b**), 4-1 (

**c**), 4-2 (

**d**), 5c (

**e**), 6c (

**f**) TSCs.

**Figure 4.**TDM and hole-electron pair density of one-photon excited states of triangulene spin chain N = 2 (

**a**), 3 (

**b**), 4-1 (

**c**), 4-2 (

**d**), 5c (

**e**), 6c (

**f**) TSCs at the main absorption peak.

**Figure 5.**TDM and hole-electron pair density of one-photon excited states of triangulene spin chain N = 6 (

**a**), 8-1 (

**b**), 8-2 (

**c**), 16c (

**d**) TSCs at the main absorption peak.

**Figure 6.**The van der Waals surface electrostatic potential of triangulene spin chain N = 3TSCs (

**a**), N = 4-1TSCs (

**b**), N = 4-2TSCs (

**c**) and N = 6cTSCs (

**d**). The red isosurface represents the positive distribution area of ESPO, and the blue isosurface represents the negative distribution area of ESPO, in kcal/mol. The surface local maximum and minimum of ESP are represented by red and blue spheres respectively. Global minimum and maximum values are marked with italic font.

**Figure 7.**TDM and hole−electron pair density diagram in the first step (

**a**,

**b**) and second step (

**c**,

**d**) of TPA at 544.5 nm (S

_{50}) for N = 3TSCs.

**Figure 8.**TDM and hole-electron pair density of the first step (

**a**,

**b**) and second step (

**c**,

**d**) of TPA when N = 4-1TSCs pass through channel I (intermediate state is S

_{36}) at 573 nm (S

_{40}) and the first step (

**e**,

**f**) and second step (

**g**,

**h**) of TPA when N = 4-1TSCs pass through channel II (intermediate state is S

_{39}).

**Figure 9.**TDM and hole-electron pair density of the first step (

**a**,

**b**) and second step (

**c**,

**d**) of TPA when N = 4-2T

_{S}Cs pass through channel I (intermediate state is S

_{35}) at 574 nm (S

_{40}) and the first step (

**e**,

**f**) and second step (

**g**,

**h**) of TPA when N = 4-2TSCs pass through channel II (intermediate state is S

_{38}).

**Figure 10.**TDM and hole−electron pair density of the first step (

**a**,

**b**) and second step (

**c**,

**d**) of TPA when N = 6cTSCs pass through channel I (intermediate state S

_{56}) at 574 nm (S

_{61}) and the first step (

**e**,

**f**) and second step (

**g**,

**h**) of TPA when N = 6cTSCs pass through channel II (intermediate state S

_{55}).

**Figure 11.**ECD spectra of the spin chain trimer of triangulene ((

**a**), N = 3TSCs), the tetramer of triangulene with different connection modes ((

**b**), N = 4-1TSCs); ((

**c**), N = 4-2TSCs); ((

**d**) N = 6cTSCs) and six closed loops of the hexamer of triangulene.

**Figure 12.**The Cartesian components of TEDM (

**a**–

**c**) and TMDM (

**d**–

**f**) of N = 3TSCs in S

_{26}; N = the Cartesian components of TEDM (

**g**–

**i**) and TMDM (

**j**–

**l**) of 4-1TSCs in S

_{21}(the blue isosurface in TEDM is positive and the yellow isosurface is negative; the purple isosurface in TMDM is positive and the yellow isosurface is negative).

**Figure 13.**The Cartesian components of TEDM (

**a**–

**c**) and TMDM (

**d**–

**f**) of N = 3T

_{S}Cs in S

_{30}; N = the Cartesian components of TEDM (

**g**–

**i**) and TMDM (

**j**–

**l**) of 4-1TSCs in S

_{36}(the blue isosurface in TEDM is positive and the yellow isosurface is negative; the purple isosurface in TMDM is positive and the yellow isosurface is negative).

**Figure 14.**The Cartesian components of TEDM (

**a**–

**c**) and TMDM (

**d**–

**f**) of N = 4-2TSCs in S

_{23}; N = Cartesian components of TEDM (

**g**–

**i**) and TMDM (

**j**–

**l**) of 6cTSCs in S

_{33}.

**Figure 15.**The Cartesian components of TEDM (

**a**–

**c**) and TMDM (

**d**–

**f**) of N = 4-2TSCs in S

_{35}; N = Cartesian components of TEDM (

**g**–

**i**) and TMDM (

**j**–

**l**) of 6cTSCs in S

_{55}.

Excited States | Oscillator Strength | Excited Energy (eV) | H (Å) | D (Å) | t (Å) | S_{r} | |
---|---|---|---|---|---|---|---|

N = 3TSCs | S_{0}→S_{37} | 1.3805 | 4.324 | 7.322 | 0.042 | −3.535 | 0.96.64 |

N = 4-1TSCs | S_{0}→S_{36} | 2.2673 | 4.175 | 8.368 | 0.000 | −4.753 | 0.98551 |

N = 4-2TSCs | S_{0}→S_{35} | 1.2938 | 4.173 | 8.053 | 0.031 | −4.1972 | 0.98536 |

N = 6cTSCs | S_{0}→S_{55} | 2.3274 | 4.210 | 10.222 | 0.001 | −0.975 | 0.98753 |

Excited States | Oscillator Strength | Excited Energy (eV) | H (Å) | D (Å) | t (Å) | S_{r} | |
---|---|---|---|---|---|---|---|

N = 3TSCs | S_{0}→S_{66} | 1.1894 | 5.037 | 9.239 | 0.173 | −2.160 | 0.98102 |

N = 4-1TSCs | S_{0}→S_{62} | 4.5128 | 4.748 | 8.273 | 0.000 | −3.091 | 0.95292 |

N = 4-2TSCs | S_{0}→S_{62} | 2.8618 | 4.744 | 7.95 | 0.053 | −3.979 | 0.95280 |

N = 6cTSCs | S_{0}→S_{98} | 7.0326 | 4.813 | 10.111 | 0.005 | −0.960 | 0.96190 |

Molecule | State | Path | Process | Integral Value (Debye) | The Largest TPA Cross-Section |
---|---|---|---|---|---|

N = 3TSCs | S_{50} | Ⅰ | $\langle {\varphi}_{s0}\left|\mu \right|{\varphi}_{2}\rangle \times \langle {\varphi}_{s2}\left|\mu \right|{\varphi}_{50}\rangle $ | 3.96 × 3.96 | 5.35 × 10^{3} |

N = 4-1TSCs | S_{40} | Ⅰ | $\langle {\varphi}_{s0}\left|\mu \right|{\varphi}_{36}\rangle \times \langle {\varphi}_{s36}\left|\mu \right|{\varphi}_{40}\rangle $ | 22.39 × 20.59 | 2.92 × 10^{3} |

Ⅱ | $\langle {\varphi}_{s0}\left|\mu \right|{\varphi}_{39}\rangle \times \langle {\varphi}_{s39}\left|\mu \right|{\varphi}_{40}\rangle $ | 3.25 × 26.27 | |||

N = 4-2TSCs | S_{40} | Ⅰ | $\langle {\varphi}_{s0}\left|\mu \right|{\varphi}_{35}\rangle \times \langle {\varphi}_{s35}\left|\mu \right|{\varphi}_{40}\rangle $ | 12.81 × 15.48 | 2.19 × 10^{3} |

Ⅱ | $\langle {\varphi}_{s0}\left|\mu \right|{\varphi}_{38}\rangle \times \langle {\varphi}_{s38}\left|\mu \right|{\varphi}_{40}\rangle $ | 10.29 × 17.25 | |||

N = 6cTSCs | S_{61} | Ⅰ | $\langle {\varphi}_{s0}\left|\mu \right|{\varphi}_{56}\rangle \times \langle {\varphi}_{s56}\left|\mu \right|{\varphi}_{61}\rangle $ | 22.32 × 21.15 | 4.61 × 10^{3} |

Ⅱ | $\langle {\varphi}_{s0}\left|\mu \right|{\varphi}_{55}\rangle \times \langle {\varphi}_{s55}\left|\mu \right|{\varphi}_{61}\rangle $ | 22.98 × 14.18 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, N.; Feng, W.; Wen, H.; Feng, N.; Sheng, H.; Huang, Z.; Wang, J.
Physical Mechanism of Nonlinear Spectra in Triangene. *Molecules* **2023**, *28*, 3744.
https://doi.org/10.3390/molecules28093744

**AMA Style**

Zhang N, Feng W, Wen H, Feng N, Sheng H, Huang Z, Wang J.
Physical Mechanism of Nonlinear Spectra in Triangene. *Molecules*. 2023; 28(9):3744.
https://doi.org/10.3390/molecules28093744

**Chicago/Turabian Style**

Zhang, Na, Weijian Feng, Hanbo Wen, Naixing Feng, Hao Sheng, Zhixiang Huang, and Jingang Wang.
2023. "Physical Mechanism of Nonlinear Spectra in Triangene" *Molecules* 28, no. 9: 3744.
https://doi.org/10.3390/molecules28093744