Models

There are several model systems implemented in SHARP Pack, which are summarized in the following table.

Table 1 Models

Keyword	Description
tully1	Tully Model-I: Simple avoided crossing model [3]
tully2	Tully Model-II: Dual avoided crossing model [3]
tully3	Tully Model-III: Extended coupling with reflection model [3]
morse1	Morse Model-I: photo-dissociation model [9]
morse2	Morse Model-II: photo-dissociation model [9]
morse3	Morse Model-III: photo-dissociation model [9]
db2lchain	2-state model coupled with linear chain model [10, 11]
db3lchain	3-state super-exchange model with linear chain model [12]
superexchange	3-state super-exchange model [13]
spinboson	Spin-Boson model coupled to N-harmonic bath [14]
dboc1	Flat Born-Oppenheimer PES model [15, 16]

Tully Models

Diabatic Hamiltonians and parameters for Tully models [3] are:

Model I

\[\begin{split}V_{11}(R) = \begin{cases} A(1 - e^{-B R}) & \text{if } R > 0 \\ -A(1 - e^{B R}) & \text{if } R < 0 \end{cases}\end{split}\]

\[V_{22}(R) = -V_{11}(R)\]

\[V_{12}(R) = V_{21}(R) = C e^{-D R^2}\]

Model II

\[V_{11}(R) = 0\]

\[V_{22}(R) = -A e^{-B R^2} + E_0\]

\[V_{12}(R) = V_{21}(R) = C e^{-D R^2}\]

Model III

\[V_{11}(R) = A, \quad V_{22}(R) = -A\]

\[\begin{split}V_{12}(R) = V_{21}(R) = \begin{cases} B e^{C R}, & \text{if } R < 0 \\ B(2 - e^{-C R}), & \text{if } R > 0 \end{cases}\end{split}\]

Table 2 Tully Model Parameters (in a.u.)

Parameter	Model I	Model II	Model III
A	0.01	0.1	6 \(\times 10^{-4}\)
B	1.6	0.28	0.1
C	0.005	0.015	0.9
D	1.0	0.06	–
E₀	–	0.05	–

Morse Potential Models

Diabatic Hamiltonians and parameters for three Morse models [9] are:

\[V_{ii}(R) = De_i \left(1 - e^{-\beta_i(R - Re_i)} \right)^2 + c_i\]

\[V_{ij}(R) = A_{ij} e^{-a_{ij}(R - R_{ij})^2}, \quad V_{ji}(R) = V_{ij}(R)\]

Table 3 Numerical values of the parameters related to the three Morse potential models (in a.u.)

Model-I	\(De_i\)	\(\beta_i\)	\(Re_i\)	\(c_{i}\)	\(A_{ij}\)	\(R_{ij}\)	\(a_{ij}\)
\(V_{11}\)	0.003	0.65	5.0	0.00
\(V_{22}\)	0.004	0.60	4.0	0.01
\(V_{33}\)	0.003	0.65	6.0	0.006
\(V_{12}\)					0.002	3.40	16.0
\(V_{23}\)					0.002	4.80	16.0
Model-II	\(De_i\)	\(\beta_i\)	\(Re_i\)	\(c_{i}\)	\(A_{ij}\)	\(R_{ij}\)	\(a_{ij}\)
\(V_{11}\)	0.02	0.65	4.5	0.00
\(V_{22}\)	0.01	0.40	4.0	0.01
\(V_{33}\)	0.003	0.65	4.4	0.02
\(V_{12}\)					0.005	3.66	32.0
\(V_{13}\)					0.005	3.34	32.0
Model-III	\(De_i\)	\(\beta_i\)	\(Re_i\)	\(c_{i}\)	\(A_{ij}\)	\(R_{ij}\)	\(a_{ij}\)
\(V_{11}\)	0.02	0.40	4.0	0.02
\(V_{22}\)	0.02	0.65	4.5	0.00
\(V_{33}\)	0.003	0.65	6.0	0.02
\(V_{12}\)					0.005	3.40	32.0
\(V_{13}\)					0.005	4.97	32.0

Linear Chain Model

Linear Chain Model is two-level/three-level system copuled to a signle atom in N-atom linear chain, with the following anharmonic, nearest-neighbor potential energy function:

\[V(\mathbf{R}) = \sum_{i=1}^{N} V_M(R_i - R_{i+1})\]

where

\[V_M(R) = V_0 \left(a^2 R^2 - a^3 R^3 + 0.58 a^4 R^4 \right)\]

and \(R_{N+1}\) is a fixed position and the atom farthest from the quantum systerm (\(R_N\)) is connected to Langevin dynamics with friction constant \(\gamma\). [10, 11, 12]

Table 4 Simulation Parameters for two-level Linear Chain Model

Parameter	Value	Unit
\(N\)	20
\(m\)	12.0	amu
\(V_0\)	175.0	kJ/mol
\(a\)	4.0	\(\angstrom ^{-1}\)
\(\gamma\)	\(\text{10}^{14}\)	\(s^{-1}\)
\(\Delta=\epsilon_2-\epsilon_1\)	8.0	kJ/mol
\(d_{12}\)	-6.0	\(\angstrom ^{-1}\)

Table 5 Simulation Parameters for three-level Superexchange Linear Chain Model

Parameter	Value	Unit
\(N\)	20
\(m\)	12.0	amu
\(V_0\)	175.0	kJ/mol
\(a\)	4.0	\(\angstrom ^{-1}\)
\(\gamma\)	\(\text{10}^{14}\)	\(s^{-1}\)
\(\epsilon_1\)	0	kJ/mol
\(\epsilon_2\)	39.0	kJ/mol
\(\epsilon_3\)	13.0	kJ/mol
\(d_{12}\)	-6.0	\(\angstrom ^{-1}\)
\(d_{23}\)	8.0	\(\angstrom ^{-1}\)
\(d_{13}\)	0	\(\angstrom ^{-1}\)

Super Exchange Model

Diabatic Hamiltonians and parameters for 3-state super exchange model [13] are defined as:

\[V_{ii}(R) = A_i\]

\[V_{ij}(R) = V_{ji}(R) = B_{ij}\, e^{-R^2/2}\]

Table 6 Super Exchange model parameter (in a.u.)

	\(i=1\)	\(i=2\)	\(i=3\)	\(ij=12\)	\(ij=23\)	\(ij=13\)
\(A_i\)	0	0.01	0.005
\(B_{ij}\)				0.001	0.01	0

Spin–Boson Model

The spin-boson model provides a theoretical framework for studying non-adiabatic dynamics in a condensed-phase environment. The corresponding Hamiltonian [14] is given by:

\[H = H_s + H_b + H_{sb}.\]

Two–level system Hamiltonian:

\[\begin{split}H_s = \epsilon \sigma_z + \Delta \sigma_x = \begin{pmatrix} \epsilon & \Delta \\ \Delta & -\epsilon \end{pmatrix},\end{split}\]

with bias \(\epsilon\) and coupling \(\Delta\) is interacting with a bath of harmonic oscillators defined with,

Bath Hamiltonian:

\[H_b = \sum_j \left( \frac{P_j^2}{2 M_j} + \frac{1}{2} M_j \omega_j^2 R_j^2 \right).\]

Two-level system bilaterally interacting with a harmonic bath, defined as System–bath coupling Hamiltonian:

\[\begin{split}H_{sb} = \sigma_z \sum_j c_j R_j = \begin{pmatrix} \sum_j c_j R_j & 0 \\ 0 & -\sum_j c_j R_j \end{pmatrix}.\end{split}\]

Spectral Density

General definition for \(N\) bath modes [17, 18]:

\[J(\omega) = \frac{\pi}{2} \sum_{j=1}^{N} \frac{c_j^2}{M_j \omega_j} \, \delta(\omega - \omega_j),\]

where

• \(M_j\) = mass of \(j\)-th oscillator,

• \(c_j\) = coupling constant between the system and the \(j\),-th oscillator,

• \(\omega_j\) = oscillator frequency.

Debye form

\[J_D(\omega) = \frac{E_r}{2} \, \frac{\omega \, \omega_c}{\omega^2 + \omega_c^2},\]

where

• \(E_r\) = reorganization energy,

• \(\omega_c\) = characteristic bath frequency.

Debye Bath Discretization

Frequencies and couplings are discretizeda as [19, 20]:

\[\omega_j = \omega_c \cdot \tan\left( \frac{j}{N} \tan^{-1}\left( \frac{\omega_{\text{max}}}{\omega_c} \right) \right),\]

\[c_j = \omega_j \sqrt{ \frac{M_j E_r}{\pi N} \, \tan^{-1}\left( \frac{\omega_{\text{max}}}{\omega_c} \right) },\]

with \(\omega_{\text{max}} = 20 \omega_c\), a high-frequency cutoff.

Ohmic form

\[J_O(\omega) = \frac{\pi}{2} \, \alpha \, \omega \, e^{\omega/\omega_c},\]

where

• \(\alpha\) = Kondo parameter that controls the strength of the coupling,

• \(\omega_c\) = characteristic bath frequency.

Ohmic Bath Discretization

Frequencies and couplings are discretizeda as:

\[\omega_j = \omega_c \cdot \log\left(1 - \frac{j}{1+N} \right),\]

\[c_j = \omega_j \sqrt{ \frac{M_j \alpha}{1+N}},\]