Accurate and Efficient Spin–Phonon Coupling and Spin Dynamics Calculations for Molecular Solids

Molecular materials are poised to play a significant role in the development of future optoelectronic and quantum technologies. A crucial aspect of these areas is the role of spin–phonon coupling and how it facilitates energy transfer processes such as intersystem crossing, quantum decoherence, and magnetic relaxation. Thus, it is of significant interest to be able to accurately calculate the molecular spin–phonon coupling and spin dynamics in the condensed phase. Here, we demonstrate the maturity of ab initio methods for calculating spin–phonon coupling by performing a case study on a single-molecule magnet and showing quantitative agreement with the experiment, allowing us to explore the underlying origins of its spin dynamics. This feat is achieved by leveraging our recent developments in analytic spin–phonon coupling calculations in conjunction with a new method for including the infinite electrostatic potential in the calculations. Furthermore, we make the first ab initio determination of phonon lifetimes and line widths for a molecular magnet to prove that the commonplace Born–Markov assumption for the spin dynamics is valid, but such “exact” phonon line widths are not essential to obtain accurate magnetic relaxation rates. Calculations using this approach are facilitated by the open-source packages we have developed, enabling cost-effective and accurate spin–phonon coupling calculations on molecular solids.


Figure S1
. Convergence of the equilibrium CF energy levels of 1 using various "spherical" cut-off radii when embedded in a conductor reaction field (upper), compared to the same "spherical" clusters of unit cells without the conductor reaction field (lower).Table S1.Measured and DFT-optimised primitive unit-cell parameters for 1.
Parameter Experimental
. Calculated phonon linewidths for 1 as a function of temperature from 10-300 K for the 203 modes below 100 cm -1 at the 6 unique q-points on a 2×2×2 q-point grid.Points are connected by lines.

Figure S5
. Low-energy DoS generated using the DFT-calculated mode energies and linewidths at 10 K obtained on 2×2×2 and 5×5×5 q-point grids, using an anti-Lorentzian lineshape.The y-axis shown on the same scale as Figure 3b in the main text.

Figure S6
. Low-energy DoS generated using the DFT-calculated mode energies using a 2×2×2 q-point grid and using fixed linewidths of 0.1 (red), 1 (green) and 10 cm -1 (blue), using an anti-Lorentzian lineshape.The y-axis shown on the same scale as Figure 3b in the main text.

Figure S7
. Experimental (black circles) and calculated magnetic relaxation rates for 1.
Calculations are performed using the gas-phase ansatz considering single-phonon transitions only, with fixed linewidths of Γ = 0.1 (red), 1 (green) and 10 (blue) cm -1 .The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates. 2
The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates. 2

Figure S12
. Experimental (black circles) and calculated magnetic relaxation rates for 1.
Calculations using solid-state phonon modes on 2×2×2, 3×3×3, 4×4×4 and 5×5×5 q-point grids, including the infinite crystalline electrostatic potential, and considering single-phonon transitions only.Linewidths given by Equation 1 in the main text (red), mode-dependent ab initio linewidths at 300 K (green), fixed Γ = 10 cm -1 (blue), and the ab initio (mode-and temperature-dependent) linewidths (pink points and dashed lines).The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates. 2

Figure S13
. Experimental (black circles) and calculated magnetic relaxation rates for 1.
Calculations using solid-state phonon modes on 2×2×2, 3×3×3, 4×4×4 and 5×5×5 q-point grids, including the infinite crystalline electrostatic potential, and considering two-phonon transitions only.Linewidths given by Equation 1 in the main text (red), mode-dependent ab initio linewidths at 300 K (green), fixed Γ = 10 cm -1 (blue), and the ab initio (mode-and temperature-dependent) linewidths (pink points and dashed lines).The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates.Calculations using solid-state phonon modes on a 5×5×5 q-point grid, including the infinite crystalline electrostatic potential, considering both single-phonon and two-phonon transitions, with fixed linewidths of Γ = 0.1 (red), 1 (green) and 10 (blue) cm -1 .The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates. 2 Table S2.Contributions to Raman rates for 1 at 10 K, calculated using a 2×2×2 q-point grid with Γ = 1 cm -1 .Only relative contributions > 0.5 shown.S3.Contributions to Raman rates for 1 at 40 K, calculated using a 2×2×2 q-point grid with Γ = 1 cm -1 .Only relative contributions > 0.5 shown.and their symmetry equivalents removed.Column C: all modes listed in Table S3 and their symmetry equivalents removed.Column D: all modes listed in Table S3 and their symmetry equivalents, as well as all modes between 25 and 60 cm -1 removed.

1 )Figure S9 .
Figure S9.Experimental (black circles) and calculated magnetic relaxation rates for 1. Calculations are performed using solid-state phonon modes on 2×2×2 (red), 3×3×3 (green), 4×4×4 (blue) and 5×5×5 (pink) q-point grids, for the same finite supercell expansions, considering single-phonon and two-phonon transitions, with a fixed linewidth of Γ = 1 cm -1 .These calculations were performed without accounting for the infinite crystalline electrostatic potential.The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates.2

2
Figure S9.Experimental (black circles) and calculated magnetic relaxation rates for 1. Calculations are performed using solid-state phonon modes on 2×2×2 (red), 3×3×3 (green), 4×4×4 (blue) and 5×5×5 (pink) q-point grids, for the same finite supercell expansions, considering single-phonon and two-phonon transitions, with a fixed linewidth of Γ = 1 cm -1 .These calculations were performed without accounting for the infinite crystalline electrostatic potential.The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates.2

2
Figure S13.Experimental (black circles) and calculated magnetic relaxation rates for 1. Calculations using solid-state phonon modes on 2×2×2, 3×3×3, 4×4×4 and 5×5×5 q-point grids, including the infinite crystalline electrostatic potential, and considering two-phonon transitions only.Linewidths given by Equation 1 in the main text (red), mode-dependent ab initio linewidths at 300 K (green), fixed Γ = 10 cm -1 (blue), and the ab initio (mode-and temperature-dependent) linewidths (pink points and dashed lines).The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates.2

Figure S14 .
Figure S14.Experimental (black circles) and calculated magnetic relaxation rates for 1. Calculations using solid-state phonon modes on a 5×5×5 q-point grid, including the infinite crystalline electrostatic potential, considering both single-phonon and two-phonon transitions, with fixed linewidths of Γ = 0.1 (red), 1 (green) and 10 (blue) cm -1 .The bars on the experimental data points denote one estimated standard deviation of the distribution of relaxation rates.2