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This paper reports the derivation and implementation of the electric dipole-magnetic dipole and electric dipole-electric quadrupole polarizability tensors at the density functional theory level with periodic boundary conditions (DFT-PBC). These tensors are combined to evaluate the Buckingham/Dunn tensor that describes the optical rotation (OR) in oriented chiral systems. We describe several aspects of the derivation of the equations and present test calculations that verify the correctness of the tensor formulation and their implementation. The results show that the full OR tensor is completely origin invariant as for molecules and that PBC calculations match molecular cluster calculations on 1D chains. A preliminary investigation on the choice of density functional, basis set, and gauge indicates a similar dependence as for molecules: the functional is the primary factor that determines the OR magnitude, followed by the basis set and to a much smaller extent the choice of gauge. However, diffuse functions may be problematic for PBC calculations even if they are necessary for the molecular case. A comparison with experimental data of OR for the tartaric acid crystal shows reasonable agreement given the level of theory employed. The development presented in this paper offers the opportunity to simulate the OR of chiral crystalline materials with general-purpose DFT-PBC methods, which, in turn, may help to understand the role of intermolecular interactions on this sensitive electronic property.

 

In this work, we explore the issue of origin dependence in optical rotation (OR) calculations in the length dipole gauge (LG) using standard approximate methods belonging to density functional theory (DFT) and coupled cluster (CC) theory. We use the origin‐invariant LG approach, LG(OI), that we recently proposed as reference for the calculations, and we study whether a proper choice of coordinate origin and molecular orientation can be made such that diagonal elements of the LG‐OR tensor match those of the LG(OI) tensor. Using a numerical search algorithm, we show that multiple spatial orientations can be found where the LG and LG(OI) results match. However, a simple analytical procedure provides a spatial orientation where the origin of the coordinate system is close to the center of mass of the molecule. At the same time, we also show that putting the origin at the center of mass is not an ideal choice for every molecule (relative errors in the OR up to 70% can be obtained in out test set). Finally, we show that the choice of coordinate origin based on the analytical procedure is transferable across different methods and it is superior to putting the origin in the center of mass or center of nuclear charge. This is important because the LG(OI) approach is trivial to implement for DFT, but not necessarily for nonvariational methods in the CC family. Therefore, one can determine an optimal coordinate origin at DFT level and use it for standard LG‐CC response calculations.

 

We present a method for obtaining origin-independent electronic circular dichroism (ECD) in the length-gauge representation LG(OI) without the usage of London atomic orbitals. This approach builds upon the work by Caricato [J. Chem. Phys. 153, 151101 (2020)] and is applied to rotatory strengths and ECD spectra from damped response theory. Numerical results are presented for time-dependent Hartree–Fock and density-functional theory, the second-order algebraic diagrammatic construction method, and linear-response coupled-cluster theory with singles and approximate doubles. We can support the finding that the common choice of placing the gauge origin in the center of mass of a molecule in conventional length-gauge calculations involving chiroptical properties might not be optimal and show that LG(OI) is a valuable alternative for the origin-independent calculation of ECD spectra. We show that, for a limited test set, the convergence of the rotatory strengths calculated with the LG(OI) approach toward the basis-set limit tends to be faster than for the established velocity gauge representation. Relationships between the sum-over-states expression of the optical rotation in the LG(OI) framework and its representation in terms of response functions are analyzed. 

This work presents the first simulations of the full optical rotation (OR) tensor at coupled cluster with single and double excitations (CCSD) level in the modified velocity gauge (MVG) formalism. The CCSD‐MVG OR tensor is origin independent, and each tensor element can in principle be related directly to experimental measurements on oriented systems. We compare the CCSD results with those from two density functionals, B3LYP and CAM‐B3LYP, on a test set of 22 chiral molecules. The results show that the functionals consistently overestimate the CCSD results for the individual tensor components and for the trace (which is related to the isotropic OR), by 10%–20% with CAM‐B3LYP and 20%–30% with B3LYP. The data show that the contribution of the electric dipole–magnetic dipole polarizability tensor to the OR tensor is on average twice as large as that of the electric dipole–electric quadrupole polarizability tensor. The difficult case of (1S,4S)‐(–)‐norbornenone also reveals that the evaluation of the former polarizability tensor is more sensitive than the latter. We attribute the better agreement of CAM‐B3LYP with CCSD to the ability of this functional to better reproduce electron delocalization compared with B3LYP, consistent with previous reports on isotropic OR. The CCSD‐MVG approach allows the computation of reference data of the full OR tensor, which may be used to test more computationally efficient approximate methods that can be employed to study realistic models of optically active materials.