Accurate modeling of vibronically driven magnetic relaxation from ab initio calculations is of paramount importance to the design of next-generation single-molecule magnets (SMMs). Previous theoretical studies have been relying on numerical differentiation to obtain spin-phonon couplings in the form of derivatives of spin Hamiltonian parameters. In this work, we introduce a novel approach to obtain these derivatives fully analytically by combining the linear vibronic coupling (LVC) approach with analytic complete active space self-consistent field derivatives and nonadiabatic couplings computed at the equilibrium geometry with a single electronic structure calculation. We apply our analytic approach to the computation of Orbach and Raman relaxation rates for a bis-cyclobutadienyl Dy(III) sandwich complex in the frozen-solution phase, where the solution environment is modeled by electrostatic multipole expansions, and benchmark our findings against results obtained using conventional numerical derivatives and a fully electronic description of the whole system. We demonstrate that our LVC approach exhibits high accuracy over a wide range of coupling strengths and enables significant computational savings due to its analytic, “single-shot” nature. Evidently, this offers great potential for advancing the simulation of a wide range of vibronic coupling phenomena in magnetism and spectroscopy, ultimately aiding the design of high-performance SMMs. Considering different environmental representations, we find that a point charge model shows the best agreement with the reference calculation, including the full electronic environment, but note that the main source of discrepancies observed in the magnetic relaxation rates originates from the approximate equilibrium electronic structure computed using the electrostatic environment models rather than from the couplings.

Here, we report the synthesis of two new sets of dibismuthbridged rare earth molecules. The first series contains a bridging diamagnetic Bi 2 2− anion, (Cp* 2 RE) 2 (μ-η 2 :η 2 -Bi 2 ), 1-RE (where Cp* = pentamethylcyclopentadienyl; RE = Gd (1-Gd), Tb (1-Tb), Dy (1-Dy), Y (1-Y)), while the second series comprises the first Bi 2 3− radical-containing complexes for any d- cryptand), which were obtained from one-electron reduction of 1-RE with KC 8 . The Bi 2 3− radical-bridged terbium and dysprosium congeners, 2-Tb and 2-Dy, are single-molecule magnets with magnetic hysteresis. We investigate the nature of the unprecedented lanthanide−bismuth and bismuth−bismuth bonding and their roles in magnetic communication between paramagnetic metal centers, through single-crystal X-ray diffraction, ultraviolet−visible/ near-infrared (UV−vis/NIR) spectroscopy, SQUID magnetometry, DFT and multiconfigurational ab initio calculations. We find a π z * ground SOMO for Bi 2 3− , which has isotropic spin−spin exchange coupling with neighboring metal ions of ca. −20 cm −1 ; however, the exchange coupling is strongly augmented by orbitally dependent terms in the anisotropic cases of 2-Tb and 2-Dy. As the first examples of p-block radicals beneath the second row bridging any metal ions, these studies have important ramifications for single-molecule magnetism, main group element, rare earth metal, and coordination chemistry at large.

Electron–phonon coupling underlies many physical phenomena, but its microscopic origins are nuanced. This Review derives the spin–phonon interactions in molecules from first principles, and describes an implementation for molecular spin dynamics calculations.

Molecular materials are poised to play a significant role in the development future opto-electronic 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 molecular spin-phonon coupling and spin dynamics in the condensed phase. Here we examine the various approximations inherent in spin-phonon coupling and spin dynamics calculations on molecular solids by performing a case study on a single-molecule magnet. Three key results are: i) finite crystalline slab calculations should be avoided; ii) the phonon spectrum in reciprocal space should be sampled as densely as possible; and iii) phonon linewidths, as calculated by periodic density-functional theory, are likely overestimated at low temperature, but are not essential to obtain accurate magnetic relaxation rates provided point ii is adhered to. Calculations using this approach are facilitated by the open-source packages we have developed, which enable cost-effective and accurate spin-phonon coupling calculations on molecular solids with quantitative accuracy.

The accurate modeling of vibronically driven magnetic relaxation from ab initio calculations is of paramount importance to the design of next generation single-molecule magnets (SMMs). Previous theoretical studies have been relying on numerical differentiation to obtain spin-phonon couplings in the form of derivatives of spin Hamiltonian parameters. In this work, we introduce a novel approach to obtain these derivatives fully analytically by combining the linear vibronic coupling (LVC) approach with analytic CASSCF derivatives and nonadiabatic couplings computed at a single equilibrium geometry and electronic structure. We apply our analytic approach to the computation of Orbach and Raman relaxation rates in a solvated bis-cyclobutadienyl Dy(III) sandwich complex, where the solution environment is modelled by electrostatic multipole expansions, and benchmark our findings against results obtained using conventional numerical derivatives and a fully electronic description of the whole system. Among the ghost, charge and charge+dipole representations, we find that the point charge model shows the best agreement with the reference calculation, but note that the main source of discrepancies observed in the magnetic relaxation rates originates from the approximate equilibrium electronic structure computed using the electrostatic environment models, rather than from the couplings. We demonstrate that our novel LVC approach exhibits high accuracy over a wide range of coupling strengths and enables computational savings due to its analytic, "single-shot'' nature. Evidently, this offers great potential for advancing the simulation of a wide range of vibronic coupling phenomena in magnetism and spectroscopy, ultimately aiding the design of high-performance SMMs.