[Paper Review] Formulation and Implementation of Frequency-Dependent Linear Response Properties with Relativistic Coupled Cluster Theory for GPU-accelerated Computer Architectures

Xiang Yuan, Loïc Halbert+5 more|arXiv (Cornell University)|Jul 26, 2023

Molecular spectroscopy and chiralityChemistry137 references3 citations

TL;DR

This paper presents a GPU-accelerated implementation of relativistic coupled cluster linear response theory for calculating frequency-dependent molecular properties, including polarizabilities, spin-spin coupling constants, and optical rotation. The method achieves high accuracy by incorporating both relativistic effects and electron correlation, demonstrating superior performance over DFT and revealing significant discrepancies in core-level properties as relativistic effects increase down the periodic table.

ABSTRACT

We present the development and implementation of the relativistic coupled cluster linear response theory (CC-LR) which allows the determination of molecular properties arising from time-dependent or time-independent electric, magnetic, or mixed electric-magnetic perturbations (within a common gauge origin), and take into account the finite lifetime of excited states via damped response theory. We showcase our implementation, which is capable to offload intensive tensor contractions onto graphical processing units (GPUs), in the calculation of: extit{(a)} frequency-(in)dependent dipole-dipole polarizabilities of IIB atoms and selected diatomic molecules, with a emphasis on the calculation of valence absorption cross-sections for the I$_2$ molecule; extit{(b)} indirect spin-spin coupling constants for benchmark systems such as the hydrogen halides (HX, X = F-I) as well the H$_2$Se-H$_2$O dimer as a prototypical system containing hydrogen bonds; and extit{(c)} optical rotations at the sodium D line for hydrogen peroxide analogues (H$_{2}$Y$_{2}$, Y=O, S, Se, Te). Thanks to this implementation, we are able show the similarities in performance--but often the significant discrepancies--between CC-LR and approximate methods such as density functional theory (DFT). Comparing standard CC response theory with the equation of motion formalism, we find that, for valence properties such as polarizabilities, the two frameworks yield very similar results across the periodic table as found elsewhere in the literature; for properties that probe the core region such as spin-spin couplings, we show a progressive differentiation between the two as relativistic effects become more important. Our results also suggest that as one goes down the periodic table it may become increasingly difficult to measure pure optical rotation at the sodium D line, due to the appearance of absorbing states.

Motivation & Objective

• To develop a relativistic coupled cluster linear response (CC-LR) framework capable of computing frequency-dependent electric, magnetic, and mixed electric-magnetic molecular properties with high accuracy.
• To implement this theory on GPU-accelerated architectures to handle the computationally intensive tensor contractions inherent in coupled cluster methods.
• To validate the implementation against experimental data and established codes for key properties such as dipole polarizabilities, spin-spin coupling constants, and optical rotation.
• To assess the performance and accuracy of CC-LR relative to approximate methods like DFT, particularly in systems with heavy elements where relativistic and correlation effects are significant.
• To explore the limitations of using the sodium D-line for optical rotation in heavy chalcogen-containing molecules due to spectral interference from excited states.

Proposed method

• The formulation employs relativistic coupled cluster theory within the linear response formalism, using analytic derivatives to compute molecular properties under time-dependent or time-independent perturbations.
• The implementation leverages a GPU-accelerated tensor operation library (TAL-SH) to offload computationally intensive contractions, enabling efficient execution on modern HPC architectures.
• A complex algebra framework is used to enable straightforward evaluation of the damped linear response function at complex frequencies, facilitating absorption cross-section calculations.
• The method supports both CC-CI and CC-CC wave-function models, allowing for consistent treatment of excitation energies and response properties.
• The implementation integrates with the DIRAC program and uses a subspace framework to solve eigenvalue and linear system problems arising in the response equations.
• The common gauge origin approach is applied for magnetic properties to ensure gauge invariance in the presence of external magnetic fields.

Figure 1: Frequency dependent polarizability of Zn(top), Cd(middle), and Hg (bottom). The left figures show the real (undamped) polarizability with frequency ranging from 0.0 to 0.30 a.u. The right figures, with insets to zoom in on the peak maxima, show the simulated spectra focusing on the B trans

Experimental results

Research questions

• RQ1 How accurately can relativistic coupled cluster linear response theory compute frequency-dependent polarizabilities for heavy atoms and diatomic molecules, especially in the context of valence absorption cross-sections?
• RQ2 To what extent do relativistic and electron correlation effects influence indirect spin-spin coupling constants in hydrogen halides and H2Se-H2O dimers?
• RQ3 How do CC-LR results for optical rotation at the sodium D-line compare with DFT and experimental data, and what are the spectral limitations for heavy chalcogen analogues?
• RQ4 What are the differences between the standard CC-LR and equation-of-motion CC-LR frameworks in describing core-level versus valence properties?
• RQ5 Can the GPU-accelerated implementation reliably compute damped response functions and simulate absorption spectra using complex-frequency response theory?

Key findings

• The relativistic CC-LR implementation shows significantly improved accuracy over non-relativistic Hartree-Fock linear response for I2 polarizability, bringing results much closer to experimental data due to proper treatment of relativistic and correlation effects.
• For valence properties like polarizabilities, the standard CC-LR and EOM-CC-LR frameworks yield very similar results across the periodic table, consistent with literature findings.
• For core-level properties such as spin-spin coupling constants, the two frameworks begin to diverge as relativistic effects increase, with CC-LR showing stronger sensitivity to relativistic corrections.
• Solvent effects on spin-spin coupling in H2Se-H2O are pronounced, with both correlation and relativistic corrections significantly altering the solvent shift, and CC and DFT yielding different absolute values despite similar magnitudes.
• Optical rotation calculations for H2Y2 (Y=O, S, Se, Te) at the sodium D-line reveal increasing challenges with heavier elements due to the presence of nearby excited states, suggesting spectral interference may limit experimental utility.
• The damped response function computed via complex-frequency CC-LR successfully reproduces the I2 absorption cross-section, validating the method’s capability for simulating frequency-dependent optical properties.

Figure 2: Simulated Spectrum of I 2 . The green lines are non-relativistic HF, the red lines are X2C-HF, the blue lines are X2C-CC, and the black lines are experimental value from work of Tellinghuisen 141 . The dashed lines represent the contribution of transition from ground state X ${}^{1}\Sigma_