A first-principles linear response theory for open quantum systems and its application to Orbach and direct magnetic relaxation in Ln-based coordination polymers

Technical Deep Dive: A First-Principles Linear Response Theory for Open Quantum Systems and Its Application to Magnetic Relaxation in Ln-Based Coordination Polymers

Overview

This work presents a new first-principles methodology for computing the complex a.c. magnetic susceptibility of lanthanide-based molecular magnets. Unlike previous approaches that infer relaxation rates from quantum master equations, this framework directly connects microscopic spin-phonon coupling to the experimentally measured a.c. susceptibility.

The key innovations are:

• Formulation of the open quantum system response in terms of the reduced response density operator and its Nakajima-Zwanzig equation of motion
• Specialized to linear vibronic spin-phonon coupling with a harmonic phonon bath
• Numerical implementation using Liouville-space superoperators, including ab initio evaluation of spin-phonon coupling
• Ability to capture direction-dependent relaxation channels and their relative weights in powder measurements

The methodology is applied to three cyanido-bridged Ln/Y coordination polymers, reproducing both the direct-process and Orbach regimes in excellent agreement with experiment. Limitations are also identified, motivating future extensions to higher-order spin-phonon coupling.

Methodology

• Developed a first-principles linear response theory for open quantum systems, building on the Nakajima-Zwanzig projection-operator formalism
• Specialized to linear vibronic spin-phonon coupling with a harmonic phonon bath
• Expressed the frequency-dependent magnetic susceptibility χ(ω) in terms of Liouville-space superoperators:
  • Homogeneous projected Liouvillian M_L
  • Fourier-Laplace transformed memory kernel M̂Kr(ω)
  • Inhomogeneous boundary term M̂_ψ(ω)
  • Initial condition MρA(0)
• Evaluated spin-phonon coupling operators Y^q,j directly from ab initio molecular Hamiltonian gradients, without mapping to crystal-field parameters
• Developed a custom Brillouin-zone multigrid integration scheme to efficiently sample the acoustic phonon region

Results

Applied the methodology to three cyanido-bridged Ln/Y coordination polymers:

Compound 1 (Yb-based)

• Reproduced the direct-process regime and its field dependence in very good agreement with experiment
• Demonstrated the importance of asymmetric phonon broadening implied by the ω→0 limit
• Showed that the full powder-averaged susceptibility can be captured, unlike previous rate-based approaches

Compound 2 (Tb-based)

• Correctly captured the high-temperature Orbach regime
• Identified the need to include two-phonon Raman/LMP contributions at lower temperatures
• Phonon density of states analysis revealed the origins of the different relaxation mechanisms

Compound 3 (Dy-based)

• Severely underestimated the experimental relaxation times
• Identified an efficient direct one-phonon excitation channel to the second excited doublet as the cause
• Suggested limitations from the finite cluster approximation and missing hyperfine interactions

Limitations & Uncertainties

• Computationally demanding treatment of the inhomogeneous term M̂ψ and correlated initial condition Mρ_A(0)
• Current second-order formulation insufficient for Raman-dominated relaxation regimes
• Accuracy limited by mismatch between electronic structure and phonon calculations, finite cluster approximation, and missing physical ingredients like hyperfine interactions

Future Work

• Extension to fourth-order spin-phonon coupling to capture Raman processes
• Inclusion of hyperfine and dipolar field effects relevant to QTM
• Systematic refinement of electronic and vibrational models

Despite these limitations, the methodology represents a significant advance, providing both quantitative comparison with experiment and mechanistic insight into relaxation processes in lanthanide-based molecular magnets. It is expected to become a useful tool for in silico screening and rational design of future single-molecule magnets.