Geometry Optimizations in a Subsystem Density Functional Theory Formalism: A Benchmark Study

Development of analytic gradients for the Huzinaga quantum embedding method and its applications to large-scale hybrid and double hybrid DFT forces

The theory of analytic gradients is presented for the projector-based density functional theory (DFT) embedding approach utilizing the Huzinaga-equation. The advantages of the Huzinaga-equation-based formulation are demonstrated. In particular, it is shown that the projector employed does not appear in the Lagrangian, and the potential risk of numerical problems is avoided at the evaluation of the gradients. The efficient implementation of the analytic gradient theory is presented for approaches where hybrid DFT, second-order Møller-Plesset perturbation theory, or double hybrid DFT are embedded in lower-level DFT environments. To demonstrate the applicability of the method and to gain insight into its accuracy, it is applied to equilibrium geometry optimizations, transition state searches, and potential energy surface scans. Our results show that bond lengths and angles converge rapidly with the size of the embedded system. While providing structural parameters close to high-level quality for the embedded atoms, the embedding approach has the potential to relax the coordinates of the environment as well. Our demonstrations on a 171-atom zeolite and a 570-atom protein system show that the Huzinaga-equation-based embedding can accelerate (double) hybrid gradient computations by an order of magnitude with sufficient active regions and enables affordable force evaluations or geometry optimizations for molecules of hundreds of atoms.

Analytic gradients for local density fitting Hartree-Fock and Kohn-Sham methods

We present analytic gradients for local density fitting Hartree-Fock (HF) and hybrid Kohn-Sham (KS) density functional methods. Due to the non-variational nature of the local fitting algorithm, the method of Lagrange multipliers is used to avoid the solution of the coupled perturbed HF and KS equations. We propose efficient algorithms for the solution of the arising Z-vector equations and the gradient calculation that preserve the third-order scaling and low memory requirement of the original local fitting algorithm. In order to demonstrate the speed and accuracy of our implementation, gradient calculations and geometry optimizations are presented for various molecular systems. Our results show that significant speedups can be achieved compared to conventional density fitting calculations without sacrificing accuracy.

Theoretical investigations on P-stabilized boryl cation radicals: from the Aufbau principle to SOMO-HOMO conversion

We report the characterization of a series of novel phosphinidene-stabilized (P-stabilized) boryl cation radicals, in which the phosphinidene and boryl are stabilised by ^iPrNHC (^iPrNHC[:C{N(iPr)C(H)}₂]), and the P-stabilized boryl (P → B) moieties are linked by 1,8-naphthalene (1PB-a), 1,10-biphenyl (1PB-b), 1,2-perylene (1PB-c), and 4,5-perylene (1PB-d), to form a series of 1PB compounds. The 2PB series is designed by adding another P-stabilized boryl (P → B) unit into the 1PB series, in which the two P-stabilized boryl (P → B) moieties for each 2PB compound are linked by 1,4,5,8-naphthalene, (2PB-a), 1,5,6,10-biphenyl (2PB-b), 1,2,7,8-perylene (2PB-c), and 4,5,10,11-perylene (2PB-d), respectively. Theoretical calculations demonstrate that for all the studied molecules, the spin density mainly locates on the B atoms. Interestingly, the series of 2PB(a-d) compounds possess SOMO-HOMO conversion properties, while 1PB(a-d) compounds obey the Aufbau principle, resulting from the difference in the number of the P-stabilized boryl (P → B) moieties and an increase of the π-conjugation bridge that lead to the significantly increased HOMO energy in 2PB(a-d) compounds, which should be responsible for the different structural properties of compounds 1PB(a-d) and 2PB(a-d). The natural bond orbital (NBO) and atoms in molecules (AIM) analysis reveal how the interactions contribute to the covalent bond between P and B atoms. Moreover, the absorption properties show that the spectra of the 2PB(a-d) compounds are red-shifted relative to those of the corresponding 1PB(a-d) compounds in the near infrared region. We hope this work can provide new insights into tuning the electronic structures of the well-defined forms of P-stabilized boryl cation radicals and expand their potential application in organic optoelectronics.

Subsystem density-functional theory: A reliable tool for spin-density based properties

Subsystem density-functional theory compiles a set of features that allow for efficiently calculating properties of very large open-shell radical systems such as organic radical crystals, proteins, or deoxyribonucleic acid stacks. It is computationally less costly than correlated ab initio wave function approaches and can pragmatically avoid the overdelocalization problem of Kohn-Sham density-functional theory without employing hard constraints on the electron-density. Additionally, subsystem density-functional theory calculations commonly start from isolated fragment electron densities, pragmatically preserving a priori specified subsystem spin-patterns throughout the calculation. Methods based on subsystem density-functional theory have seen a rapid development over the past years and have become important tools for describing open-shell properties. In this Perspective, we address open questions and possible developments toward challenging future applications in connection with subsystem density-functional theory for spin-dependent properties.

Environmental Effects with Frozen-Density Embedding in Real-Time Time-Dependent Density Functional Theory Using Localized Basis Functions

Frozen-density embedding (FDE) represents a versatile embedding scheme to describe the environmental effect on electron dynamics in molecular systems. The extension of the general theory of FDE to the real-time time-dependent Kohn-Sham method has previously been presented and implemented in plane waves and periodic boundary conditions [Pavanello, M.; J. Chem. Phys. 2015, 142, 154116]. In the current paper, we extend our recent formulation of the real-time time-dependent Kohn-Sham method based on localized basis set functions and developed within the Psi4NumPy framework to the FDE scheme. The latter has been implemented in its "uncoupled" flavor (in which the time evolution is only carried out for the active subsystem, while the environment subsystems remain at their ground state), using and adapting the FDE implementation already available in the PyEmbed module of the scripting framework PyADF. The implementation was facilitated by the fact that both Psi4NumPy and PyADF, being native Python API, provided an ideal framework of development using the Python advantages in terms of code readability and reusability. We employed this new implementation to investigate the stability of the time-propagation procedure, which is based on an efficient predictor/corrector second-order midpoint Magnus propagator employing an exact diagonalization, in combination with the FDE scheme. We demonstrate that the inclusion of the FDE potential does not introduce any numerical instability in time propagation of the density matrix of the active subsystem, and in the limit of the weak external field, the numerical results for low-lying transition energies are consistent with those obtained using the reference FDE calculations based on the linear-response TDDFT. The method is found to give stable numerical results also in the presence of a strong external field inducing nonlinear effects. Preliminary results are reported for high harmonic generation (HHG) of a water molecule embedded in a small water cluster. The effect of the embedding potential is evident in the HHG spectrum reducing the number of the well-resolved high harmonics at high energy with respect to the free water. This is consistent with a shift toward lower ionization energy passing from an isolated water molecule to a small water cluster. The computational burden for the propagation step increases approximately linearly with the size of the surrounding frozen environment. Furthermore, we have also shown that the updating frequency of the embedding potential may be significantly reduced, much less than one per time step, without jeopardizing the accuracy of the transition energies.