LVCM (Linear vibronic coupling models)

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The following figures and data are our benchmarking results (for detailed discussion please see ref¹).

Figure 1. Results of the 24-mode LVCM for pyrazine. Panels (a1)-(a4): Population of the state 2. Panels (b1)-(b4): The average dimensionless coordinate $⟨\overline{R}(t)⟩$$\Braket{\bar{R}(t)}$ of the nuclear normal mode v6a. Panels (c1)-(c4): The average dimensionless momentum $⟨\overline{P}(t)⟩$$\Braket{\bar{P}(t)}$ of the nuclear normal mode ${\nu }_{6a}$$\nu_{6a}$. In panels (a1), (b1) and (c1), the green, blue and cyan solid lines represent the results of NaF-cc, NaF-cx and NaF-TW, respectively. In panels (a2), (b2) and (c2), the cyan, pink, orange and magenta solid lines represent the results of NaF-TW, NaF-TW2, NaF-HW and NaF-HW2, respectively. In panels (a3), (b3) and (c3), the orange dashed line, purple dashed line, brown dashed line and cyan solid line denote the results of SH-1, SH-2, SH-3 and NaF-TW, respectively. In panels (a4), (b4) and (c4), the blue dashed line, green dashed line and cyan solid line denote the results of CMM ($\gamma$$\gamma$ = 0.366), CMM ($\gamma$$\gamma$ = 0.5) and NaF-TW, respectively. The numerically exact results produced by MCTDH² are demonstrated by black solid lines with black points in each panel. data

Figure 2. Results of the 3-mode LVCM for pyrazine. Panels (a1)-(a4): Population of the state 2. Panels (b1)-(b4): The average dimensionless coordinate $⟨\overline{R}(t)⟩$$\Braket{\bar{R}(t)}$ of the nuclear normal mode v6a. Panels (c1)-(c4): The average dimensionless momentum $⟨\overline{P}(t)⟩$$\Braket{\bar{P}(t)}$ of the nuclear normal mode ${\nu }_{6a}$$\nu_{6a}$. In panels (a1), (b1) and (c1), the green, blue and cyan solid lines represent the results of NaF-cc, NaF-cx and NaF-TW, respectively. In panels (a2), (b2) and (c2), the cyan, pink, orange and magenta solid lines represent the results of NaF-TW, NaF-TW2, NaF-HW and NaF-HW2, respectively. In panels (a3), (b3) and (c3), the orange dashed line, purple dashed line, brown dashed line and cyan solid line denote the results of SH-1, SH-2, SH-3 and NaF-TW, respectively. In panels (a4), (b4) and (c4), the blue dashed line, green dashed line and cyan solid line denote the results of CMM ($\gamma$$\gamma$ = 0.366), CMM ($\gamma$$\gamma$ = 0.5) and NaF-TW, respectively. The numerically exact results produced by MCTDH² are demonstrated by black solid lines with black points in each panel. data

Figure 3. Results of the 2-mode LVCM for the $\mathrm{Cr}(\mathrm{CO}{)}_5$$\mathrm{Cr(CO)_5}$ molecule. Panel (a1)-(a10): Population dynamics, where the black, red and blue solid lines represent the population of states 1−3, respectively, and the numerically exact results (taken from ref ³) are plotted by dashed lines with corresponding colors. Panel (a1): NaF-cc. Panel (a2): NaF-cx. Panel (a3): NaF-TW. Panel (a4): NaF-TW2. Panel (a5): NaF-HW. Panel (a6): NaF-HW2. Panel (a7): SH-1. Panel (a8): SH-2. Panel (a9): CMM ($\gamma$$\gamma$ = 0.333). Panel (a10): CMM ($\gamma$$\gamma$ = 0.5). Panels (b1)-(b4): The average dimensionless coordinate $⟨\overline{R}(t)⟩$$\Braket{\bar{R}(t)}$ of the second nuclear normal mode. In panel (b1), the green, blue and cyan solid lines represent the results of NaF-cc, NaF-cx and NaF-TW, respectively. In panel (b2), the cyan, pink, orange and magenta solid lines represent the results of NaF-TW, NaFTW2, NaF-HW and NaF-HW2, respectively. In panel (b3), the orange dashed line, purple dashed line and cyan solid line denote the results of SH-1, SH-2 and NaF-TW, respectively. In panel (b4), the blue dashed line, green dashed line and cyan solid line denote the results of CMM ($\gamma$$\gamma$ = 0.333), CMM ($\gamma$$\gamma$ = 0.5) and NaF-TW, respectively. Note that SH-3 is not applicable for this 3-state model. The numerically exact results produced by MCTDH (taken from ref ³) are demonstrated by black solid lines with black points in panels (b1)-(b4). data

Figure 4. Results of the 39-mode LVCM for Thymine parametrized from the CAM-B3LYP calculations. In each panel, population dynamics of 3 states are demonstrated, where the blue, green and orange solid lines represent the population of the $n_O{\pi }^{\ast }1$$n_O\pi^*1$ state, the $\pi {\pi }^{\ast }1$$\pi\pi^*1$ state and the $\pi {\pi }^{\ast }2$$\pi\pi^*2$ state, respectively. Panel (a): NaF-cc. Panel (b): NaF-cx. Panel (c): NaF-TW. Panel (d): NaF-TW2. Panel (e): NaF-HW. Panel (f): NaF-HW2. Panel (g): SH-1. Panel (h): SH-2. Panel (i): CMM ($\gamma$$\gamma$ = 0.261). Panel (j): CMM ($\gamma$$\gamma$ = 0.5). Note that SH-3 is not applicable for this 7-state model. The numerically exact results produced by ML-MCTDH (taken from ref ⁴) are demonstrated by dashed lines with corresponding colors in each panel. data