4.3 Vibronic Coupling in Diabatic Basis
4.3.2 Analysis of Off-Diagonal Strong-Coupled Modes
The two strong-coupled modes separately observed in Fig. 4.12(b) and Fig. 4.13(b) ex-hibit noticeable large off-diagonal vibronic coupling strength comparing to other normal modes of the molecules. In Chapter 5, we will show that the off-diagonal vibronic coup-ling is crucial to the dynamics and these strong-coupled modes dominate the relaxation process in our model. Before moving onto that stage, we analysis these strong-coupled modes and try to uncover their uniqueness.
As shown in Fig. 4.14 are the strong-coupled mode for Chl a and BChl a visualizing in normal mode vectors, their vibrational frequencies equal to 1755 cm−1 and 1749 cm−1 respectively. The two modes possess C−C single-bond and C=C double-bond stretching character along the conjugation path on the macrocycle. For Qyand Qxstates, given their excitation character of π to π∗ transition on the macrocycle, the vibronic coupling with bond stretching normal modes on the conjugation path of macrocycle is not surprising.
Figure 4.15 are the projection of normal mode vectors on the tangent line of the conjuga-tion circle which schematically illustrate the symmetry of the vibraconjuga-tions. The conjugaconjuga-tion
doi:10.6342/NTU201803678 circle defined for i-th atom on the conjugation path is setted up as the circle using Mg atom
as central point with radius equals to the distance between the i-th atom and the central magnesium. The positive (blue in figure) and negative (red in figure) values represent the opposite directions of vector in polar coordinate and the absolute value denotes the mag-nitude of the vector length. Considering the functional groups in chls as the perturbations of higher symmetry counterpart porphine, the irreducible representation of vibrational symmetry is assigned using the point group of porphine, D2h. If we ignore the magnitude and only focus on the sign of the projection shown in Fig. 4.15, both vibrational modes are in b3g symmetry which is allowed in mediating the coupling between Qy (B2u) and Qx (B1u) states.
4.4 Summary
The properties of coupling between electronic and nuclear motions for Chls are dis-cussed in adiabatic and diabatic basis. In adiabatic basis, we evaluate the reorganization energy between the electronic states of the molecules as a descriptor of coupling strength strength. By the Four-Point Method, we obtained the total reorganization energies which is in good agreement with the experimental results. The reorganization energy for each mode is also evaluated under harmonic approximation for vibrational normal mods. The acquired Huang-Rhys factors are used to simulate the absorption and ∆FLN spectra for Chls and are compared with experimental spectra in different solvents. The effect of solvent on the character of spectra is dissused by evaluating the vibronic coupling with and without considering PCM and axial coordination of Mg.
By means of the diabatization procedures described in Chapter 3, the electronic Hamilto-nian with off-diagonal electronic coupling is estimated without the direct calculation of derivative coupling in adiabatic basis which demand tremendous computational effort.
Scanning along the normal mode coordinates, the potential energies surfaces near the equilibrium geometries of S2 state thoroughly obtained. Using finite difference method, we calculate the diagonal and off-diagonal vibronic coupling constant between Qy and Qx states. The ≈1750 cm−1 bond-length alternation mode along the conjugation path of
doi:10.6342/NTU201803678 Chls is observed to has strongest off-diagonal vibronic coupling strength and is
expec-ted to dominate the relaxation dynamics we concerned. The diagonal and off-diagonal coupling constants estimated here can further be used in dynamics studies in Chapter 5.
doi:10.6342/NTU201803678
0 500 1000 1500 2000
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Vibrational energy (cm
−1)
Coupling constant
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−1)
Coupling constant
(b)
Figure 4.12: (a) Diagonal and (b) Off-diagonal vibronic coupling constants in diabatic basis for Chl a.
doi:10.6342/NTU201803678
0 500 1000 1500 2000
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−0.1 0 0.1 0.2
Vibrational energy (cm
−1)
Coupling constant
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0 500 1000 1500 2000
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Vibrational energy (cm
−1)
Coupling constant
(b)
Figure 4.13: (a) Diagonal and (b) Off-diagonal vibronic coupling constants in diabatic basis for BChl a.
doi:10.6342/NTU201803678
(a) (b)
Figure 4.14: Visualization of off-diagonal vibronic stong-coupled mode for (a) Chl a (ω=1755 cm−1) and (b) BChl a (ω=1749 cm−1).
Mg
y axis
x axis
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
(a)
Mg
y axis
x axis
(b)
Figure 4.15: Scheme of the projection of the stong-coupled mode on the conjugation path for (a) Chl a (ω=1755 cm−1) and (b) BChl a (ω=1749 cm−1).
doi:10.6342/NTU201803678
Chapter 5
Internal Conversion Dynamics
Until now, we have shown that the diabatic states constcucted through enforcement of configuration uniformity hold the smoothness of their electronic structures across nuc-lear coordinate space considered. Applying the diabatization along the vibrational normal modes, we obtain the potential energy surfaces and electronic couplings around the op-timized geometry in diabatic basis. The modulations of Qy/Qx electronic energies and electronic couplings by nuclear motions are quantitatively estimated and the correspond-ing couplcorrespond-ing constants for both Chls are obtained.
In this Chapter, we utilize the vibronic coupling constants parametrized from diabatization approach to construct the effective Hamiltonian in our model. The effective spin-boson Hamiltonian in diabatic basis is used to study the dynamics of Chl Qx → Qy internal conversion. Through inverse transforming the effective Hamiltonian into adiabatic basis, the internal conversion rate constant can be evaluated by applying the prominent Fermi’s Golden rule. Finally, we discuss the role of each parameter in the model to explore the ori-gins of structural and solvent dependences of the relaxation time. The agreement between the relaxation timescale estimated and the experimental counterparts assure the mechan-ism of Qx → Qy internal conversion described in our model.
doi:10.6342/NTU201803678