Kohn-Sham Orbitals Research Articles

UMRSF-TDDFT: Unrestricted Mixed-Reference Spin-Flip-TDDFT.

An unrestricted version of Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (UMRSF-TDDFT) was developed based on unrestricted Kohn-Sham orbitals (UKS) with a new molecular orbital (MO) reordering scheme. Additionally, a simple yet accurate method for estimating ⟨S2⟩ expectation values was devised. UMRSF-TDDFT was benchmarked against cases where DFT, TDDFT, and SF-TDDFT traditionally fail to provide accurate descriptions. In an application to the ground and excited states of a Be atom, UMRSF-TDDFT successfully recovers the degenerate states, with its energies slightly reduced compared to its RO counterpart, due to the additional variational flexibility of UKS. A clear difference between UMRSF and U-SF-TDDFT is evident in the bond breaking of the hydrogen fluoride system, as the latter misses an important configuration. In the case of the Jahn-Teller distortion of trimethylenemethane (TMM), the relative singlet energy compared to the triplet is lower by 0.1 and 0.2 eV for UMRSF and U-SF-TDDFT, respectively, than that of MRSF-TDDFT. The reduction in UMRSF energy is attributed to spatial orbital relaxations, whereas the reduction in U-SF-TDDFT energy results from spin contamination. Overall, the additional orbital relaxations afforded by unrestricted Kohn-Sham (UKS) orbitals in UMRSF-TDDFT lead to lower total system energies compared to their restricted open-shell counterparts. This enhancement adds a practical and accurate quantum chemical theory to the existing RO variant for addressing challenging systems where traditional quantum theories suffer.

Kohn-Sham fragment energy decomposition analysis.

We introduce the concept of Kohn-Sham fragment localized molecular orbitals (KS-FLMOs), which are Kohn-Sham molecular orbitals (MOs) localized in specific fragments constituting a generic molecular system. In detail, we minimize the local electronic energies of various fragments, while maximizing the repulsion between them, resulting in the effective localization of the MOs. We use the developed KS-FLMOs to propose a novel energy decomposition analysis, which we name Kohn-Sham fragment energy decomposition analysis, which allows for rationalizing the main non-covalent interactions occurring in interacting systems both in vacuo and in solution, providing physical insights into non-covalent interactions. The method is validated against state-of-the-art energy decomposition analysis techniques and with high-level calculations.

Exciton-Assisted Relaxation of Hot Carriers in Semiconductor Quantum Dots

The work provides computational arguments in support of excitonic approach for the treatment of the photo-induced processes in semiconductor quantum dots. The non-radiative relaxation, non-radiative recombination, and photo-luminescence quantum yield are computed for a range of atomistic models of semiconductor quantum dots (QDs) in the quantum confinement regime. The excitonic (EX) approach is compared to independent orbital approximation (IOA) approach. Both approaches address dissipation of the electronic energy from electronic degrees of freedom to thermal vibrations of the lattice. The difference of two approaches appears in treatment of energies of electronic states and in a way how the electron-phonon interaction is taken into account. IOA approach uses energies of Kohn-Sham orbitals and on the fly non-adiabatic couplings. [1-3] EX approach uses Bethe-Salpeter equation (BSE) for energies.[4-6] The excitonic wavefunctions from BSE is used to construct a linear transformation matrix that transforms IOA-based non-adiabatic couplings into an excitonic basis. Both approaches are compared in application to untrasmall 1 nm diameter Si QD.Results include an evidence that hot excitons relax sooner in the excitonic picture than in the IOA picture. The observed effect is rationalized via smaller subgaps and different available relaxation pathways in the excitonic picture. The most surprising result is found for the simulated emission spectrum. The spectum in the excitonic picture demonstrates intensity in several 5 orders of magnitude higher than in the IOA picture. This observation is related to formation of a bright exciton in the lowest excitation of the ultra-small Si QDs. Obtained evidence favors excitonic approach and promises a reliable interpretation and prediction of time-dependent observables in a range of semiconductor quantum dots of different composition, sizes, and surface environment.[7] Most intriguing results are expected for QDs representing interface between PbSe and CdSe. [8]Support of National Science foundation via NSF CHE-2004197 is gratefully acknowledged.[1] D. S. Kilin and D. A. Micha, “Relaxation of photoexcited electrons at a nanostructured Si(111) surface”, J. Phys. Chem. Lett. 1, 1073-1077 (2010).[2] D. J. Vogel and D. S. Kilin, "First-Principles Treatment of Photoluminescence in Semiconductors" J. Phys. Chem. C 119, 50, 27954–27964 (2015).[3] D. J. Vogel, A. B. Kryjevski, T. M. Inerbaev, and D. S. Kilin, "Photoinduced Single- and Multiple-Electron Dynamics Processes Enhanced by Quantum Confinement in Lead Halide Perovskite Quantum Dots", J. Phys. Chem. Lett. 8, 13, 3032–3039(2017).[4] A. B. Kryjevski and Dmitri Kilin, "Enhanced multiple exciton generation in amorphous silicon nanowires and films", Molec. Phys. 114, 365-379 (2016).[5] M. Rohlfing and S. G. Louie, "Electron-Hole Excitations in Semiconductors and Insulators", Phys. Rev. Lett. 81, 2312-2315 (1998).[6] T. Sander, G. Kresse, "Macroscopic dielectric function within time-dependent density functional theory—Real time evolution versus the Casida approach", J. Chem. Phys. 146, 064110 (2017).[7] S. V. Kilina, P. K. Tamukong, and D. S. Kilin, "Surface Chemistry of Semiconducting Quantum Dots: Theoretical Perspectives", Acc. Chem. Res. 49, 10, 2127–2135 (2016).[8] H. B. Griffin, A. B. Kryjevski, and Dmitri S. Kilin, "Ab initio calculations of through-space and through-bond charge-transfer properties of interacting Janus-like PbSe and CdSe quantum dot heterostructures", Molec. Phys., e2273415 (2023).

A new parameterization of the DFT/CIS method with applications to core-level spectroscopy.

Time-dependent density functional theory (TD-DFT) within a restricted excitation space is an efficient means to compute core-level excitation energies using only a small subset of the occupied orbitals. However, core-to-valence excitation energies are significantly underestimated when standard exchange-correlation functionals are used, which is partly traceable to systemic issues with TD-DFT's description of Rydberg and charge-transfer excited states. To mitigate this, we have implemented an empirically modified combination of configuration interaction with single substitutions (CIS) based on Kohn-Sham orbitals, which is known as "DFT/CIS." This semi-empirical approach is well-suited for simulating x-ray near-edge spectra, as it contains sufficient exact exchange to model charge-transfer excitations yet retains DFT's low-cost description of dynamical electron correlation. Empirical corrections to the matrix elements enable semi-quantitative simulation of near-edge x-ray spectra without the need for significant a posteriori shifts; this should be useful in complex molecules and materials with multiple overlapping x-ray edges. Parameter optimization for use with a specific range-separated hybrid functional makes this a black-box method intended for both core and valence spectroscopy. Results herein demonstrate that realistic K-edge absorption and emission spectra can be obtained for second- and third-row elements and 3d transition metals, with promising results for L-edge spectra as well. DFT/CIS calculations require absolute shifts that are considerably smaller than what is typical in TD-DFT.

Towards adiabatic-connection interpolation model with broader applicability

The adiabatic connection integrand interpolation (ACII) method represents a general path for calculating correlation energies in electronic systems within the density functional theory. ACII functionals include both exact-exchange and the second-order correlation energy, as well as an interpolating function toward the strictly correlated electron (SCE) regime. Several interpolating functions have been proposed in the last years targeting different properties, yet an accurate ACII approach with broad applicability is still missing. Recently, we have proposed an ACII functional that was made accurate for the three-dimensional (3D) uniform electron gas as well as for model metal clusters. In this paper we present an ACII functional (named genISI2), which is very accurate for both three-dimensional (3D) and two-dimensional (2D) uniform electron gases and for the quasi-2D infinite-barrier model, where most of the exchange-correlation functionals fail badly, as well as for strongly correlated two-electrons systems. Using the exact-exchange Kohn-Sham orbitals, we have also assessed the genISI2 for various molecular systems, showing a superior performance with respect to the other ACII methods for total energies, atomization energies, and ionization potentials. The genISI2 functional can thus find application in a broad range of systems and properties. Published by the American Physical Society 2024

Enhanced Twist-Averaging Technique for Magnetic Metals: Applications Using Quantum Monte Carlo.

We propose an improved twist-averaging (TA) scheme for quantum Monte Carlo methods that use converged Kohn-Sham or Hartree-Fock orbitals as the reference. This TA technique is tailored to sample the Brillouin zone of magnetic metals, although it naturally extends to nonmagnetic (NM) conducting systems. The proposed scheme aims to reproduce the reference magnetization and achieves charge neutrality by construction, thus avoiding the large energy fluctuations and the postprocessing needed to correct the energies. It shows the most robust convergence of total energy and magnetism to the thermodynamic limit (TDL) when compared to four other TA schemes. Diffusion Monte Carlo applications are shown on NM Al and ferromagnetic α-Fe. The cohesive energy of Al in the TDL shows an excellent agreement with the experimental result. Furthermore, the magnetic moments in α-Fe exhibit rapid convergence with an increasing number of twists.

Complex-Valued K-Means Clustering of Interpolative Separable Density Fitting Algorithm for Large-Scale Hybrid Functional Enabled Ab Initio Molecular Dynamics Simulations within Plane Waves.

K-means clustering, as a classic unsupervised machine learning algorithm, is the key step to select the interpolation sampling points in interpolative separable density fitting (ISDF) decomposition for hybrid functional electronic structure calculations. Real-valued K-means clustering for accelerating the ISDF decomposition has been demonstrated for large-scale hybrid functional enabled ab initio molecular dynamics (hybrid AIMD) simulations within plane-wave basis sets where the Kohn-Sham orbitals are real-valued. However, it is unclear whether such K-means clustering works for complex-valued Kohn-Sham orbitals. Here, we propose an improved weight function defined as the sum of the square modulus of complex-valued Kohn-Sham orbitals in K-means clustering for hybrid AIMD simulations. Numerical results demonstrate that the K-means algorithm with a new weight function yields smoother and more delocalized interpolation sampling points, resulting in smoother energy potential, smaller energy drift, and longer time steps for hybrid AIMD simulations compared to the previous weight function used in the real-valued K-means algorithm. In particular, we find that this improved algorithm can obtain more accurate oxygen-oxygen radial distribution functions in liquid water molecules and a more accurate power spectrum in crystal silicon dioxide compared to the previous K-means algorithm. Finally, we describe a massively parallel implementation of this ISDF decomposition to accelerate large-scale complex-valued hybrid AIMD simulations containing thousands of atoms (2,744 atoms), which can scale up to 5,504 CPU cores on modern supercomputers.

Avoiding spin contamination and spatial symmetry breaking by exact-exchange-only optimized-effective-potential methods within the symmetrized Kohn-Sham framework.

For open-shell atoms and molecules, Kohn-Sham (KS) methods typically resort to spin-polarized approaches that exhibit spin-contamination and often break spatial symmetries. As a result, the KS Hamiltonian operator and the KS orbitals do not exhibit the space and spin symmetry of the physical electron system. The KS formalism can be symmetrized in a rigorous way only in real space, only in spin space, or both in real and spin space. Within such symmetrized KS frameworks, we present exact-exchange-only optimized-effective-potential (OEP) methods that are free of spin contamination and/or spatial symmetry breaking. The effect of symmetrizations on the total energy and its parts and on the exchange potential is analyzed. The presented exact-exchange-only OEP methods may serve as a starting point for high-level symmetrized KS methods based, e.g., on the adiabatic-connection fluctuation-dissipation theorem.

Orbital-Free Density Functional Theory: An Attractive Electronic Structure Method for Large-Scale First-Principles Simulations.

Kohn-Sham Density Functional Theory (KSDFT) is the most widely used electronic structure method in chemistry, physics, and materials science, with thousands of calculations cited annually. This ubiquity is rooted in the favorable accuracy vs cost balance of KSDFT. Nonetheless, the ambitions and expectations of researchers for use of KSDFT in predictive simulations of large, complicated molecular systems are confronted with an intrinsic computational cost-scaling challenge. Particularly evident in the context of first-principles molecular dynamics, the challenge is the high cost-scaling associated with the computation of the Kohn-Sham orbitals. Orbital-free DFT (OFDFT), as the name suggests, circumvents entirely the explicit use of those orbitals. Without them, the structural and algorithmic complexity of KSDFT simplifies dramatically and near-linear scaling with system size irrespective of system state is achievable. Thus, much larger system sizes and longer simulation time scales (compared to conventional KSDFT) become accessible; hence, new chemical phenomena and new materials can be explored. In this review, we introduce the historical contexts of OFDFT, its theoretical basis, and the challenge of realizing its promise via approximate kinetic energy density functionals (KEDFs). We review recent progress on that challenge for an array of KEDFs, such as one-point, two-point, and machine-learnt, as well as some less explored forms. We emphasize use of exact constraints and the inevitability of design choices. Then, we survey the associated numerical techniques and implemented algorithms specific to OFDFT. We conclude with an illustrative sample of applications to showcase the power of OFDFT in materials science, chemistry, and physics.

What can lattice DFT teach us about real-space DFT?

In this paper we establish a connection between density functional theory (DFT) for lattice models and common real-space DFT. We consider the lattice DFT description of a two-level model subject to generic interactions in Mermin's DFT formulation in the grand canonical ensemble at finite temperature. The case of only density-density and Hund's rule interaction studied in earlier work is shown to be equivalent to an exact-exchange description of DFT in the real-space picture. In addition, we also include the so-called pair-hopping interaction which can be treated analytically and, crucially, leads to non-integer occupations of the Kohn-Sham (KS) levels even in the limit of zero temperature. Treating the hydrogen molecule in a minimal basis is shown to be equivalent to our two-level lattice DFT model. By means of the fractional occupations of the KS orbitals (which, in this case, are identical to the many-body ones) we reproduce the results of full configuration interaction, even in the dissociation limit and without breaking the spin symmetry. Beyond the minimal basis, we embed our HOMO-LUMO model into a standard DFT calculation and, again, obtain results in overall good agreement with exact ones without the need of breaking the spin symmetry.

Machine learning electronic structure methods based on the one-electron reduced density matrix

The theorems of density functional theory (DFT) establish bijective maps between the local external potential of a many-body system and its electron density, wavefunction and, therefore, one-particle reduced density matrix. Building on this foundation, we show that machine learning models based on the one-electron reduced density matrix can be used to generate surrogate electronic structure methods. We generate surrogates of local and hybrid DFT, Hartree-Fock and full configuration interaction theories for systems ranging from small molecules such as water to more complex compounds like benzene and propanol. The surrogate models use the one-electron reduced density matrix as the central quantity to be learned. From the predicted density matrices, we show that either standard quantum chemistry or a second machine-learning model can be used to compute molecular observables, energies, and atomic forces. The surrogate models can generate essentially anything that a standard electronic structure method can, ranging from band gaps and Kohn-Sham orbitals to energy-conserving ab-initio molecular dynamics simulations and infrared spectra, which account for anharmonicity and thermal effects, without the need to employ computationally expensive algorithms such as self-consistent field theory. The algorithms are packaged in an efficient and easy to use Python code, QMLearn, accessible on popular platforms.

Does the Traditional Band Picture Describe the Electronic Structure of Doped Conjugated Polymers? TD-DFT and Natural Transition Orbital Study of Doped P3HT.

Polarons and bipolarons are created when one or two electrons are removed from the π-system of a p-type conjugated polymer, respectively. In the traditional band picture, the creation of a polaron causes two electronic energy levels to move into the band gap. The removal of a second electron to form a bipolaron causes the two intragap states to move further into the gap. Several groups, however, who looked at the energies of the Kohn-Sham orbitals from DFT calculations, have recently argued that the traditional band picture is incorrect for explaining the spectroscopy of doped conjugated polymers. Instead, the DFT calculations suggest that polaron creation causes only one unoccupied state to move into the band gap near the valence band edge while half-filled state in the valence band and the conduction band bend downward in energy. To understand the discrepancy, we performed TD-DFT calculations of polarons and bipolarons on poly(3-hexylthiophene) (P3HT). Not only do the TD-DFT-calculated absorption spectra match the experimental absorption spectra, but an analysis using natural transitional orbitals (NTOs), which provides an approximate one-electron picture from the many-electron TD-DFT results, supports the traditional band picture. Our TD-DFT/NTO analysis indicates that the traditional band picture also works for bipolarons, a system for which DFT calculations were unable to determine the electronic structure.

Potassium complex containing 2,4-dimercaptopyrimidine as ligand: synthesis, x-ray structure and DFT calculations (B3LYP/LanL2DZ)

The reaction of 2,4-dimercaptopyrimidine with potassium borohydride K[BH4] led to the formation of a new polymeric potassium complex. The obtained adduct was characterized by infrared spectroscopy (FT-IR), UV–Vis and single crystal X-ray diffraction, where it was possible to confirm the formation of the product with the following crystallographic parameters: a = 10.4286(16) Å, b = 14.417(2) Å, c = 8.5745(13) Å, α = 90°, β = 108.032(3) ° and γ = 90°. The geometry of the complex, the energy of the Kohn-Sham orbitals (HOMO and LUMO), and the electrostatic potential map (MEP) were calculated using the Density Functional Theory (DFT) at the theoretical level B3LYP/LanL2DZ and the structural parameters are close to those experimentally obtained. Comparison of the formation energies of the title complex and the scorpionate hydrotris{2,4(1H,3H)-pyrimidinedithione}borate, another possible product of the same reaction, indicates that the potassium complex is more favorable by around Hartree.

Dragon: A multi-GPU orbital-free density functional theory molecular dynamics simulation package for modeling of warm dense matter

As progress in electronic structure theoretical methods is made, ab initio molecular dynamics (MD) based on orbital-free density functional theory (OF-DFT) is becoming increasingly more successful at substituting the traditional, very accurate but computationally costly Kohn-Sham (KS) approach for simulations of matter at the challenging warm dense matter (WDM) regime. However, despite the significant cost alleviation of eliminating the dependence on the KS orbitals, OF-DFT MD runs require ∼102 to 103 CPU cores running for days, or even weeks, for simulations of systems comprised of 102 to 103 atoms, depending on thermodynamic conditions. We present Dragon, a multi-GPU OF-DFT MD code for fast and efficient simulations of WDM. With a relatively small allocation of resources (4 to 8 GPU devices) it can provide an order of magnitude speedup for simulations containing O(104) atoms and target systems composed of O(105) atoms at conditions within the WDM regime, which is currently outside the capabilities of CPU codes.

Assessment of DFT functionals for a minimal nitrogenase [Fe(SH)4H]- model employing state-of-the-art abinitio methods.

We have designed a [Fe(SH)4H]- model with the fifth proton binding either to Fe or S. We show that the energy difference between these two isomers (∆E) is hard to estimate with quantum-mechanical (QM) methods. For example, different density functional theory (DFT) methods give ∆E estimates that vary by almost 140 kJ/mol, mainly depending on the amount of exact Hartree-Fock included (0%-54%). The model is so small that it can be treated by many high-level QM methods, including coupled-cluster (CC) and multiconfigurational perturbation theory approaches. With extrapolated CC series (up to fully connected coupled-cluster calculations with singles, doubles, and triples) and semistochastic heat-bath configuration interaction methods, we obtain results that seem to be converged to full configuration interaction results within 5 kJ/mol. Our best result for ∆E is 101 kJ/mol. With this reference, we show that M06 and B3LYP-D3 give the best results among 35 DFT methods tested for this system. Brueckner doubles coupled cluster with perturbaitve triples seems to be the most accurate coupled-cluster approach with approximate triples. CCSD(T) with Kohn-Sham orbitals gives results within 4-11 kJ/mol of the extrapolated CC results, depending on the DFT method. Single-reference CC calculations seem to be reasonably accurate (giving an error of ∼5 kJ/mol compared to multireference methods), even if the D1 diagnostic is quite high (0.25) for one of the two isomers.

Chemical Bonding in Large Systems Using Projected Population Analysis from Real-Space Density Functional Theory Calculations.

We present an efficient and scalable computational approach for conducting projected population analysis from real-space finite-element (FE)-based Kohn-Sham density functional theory calculations (DFT-FE). This work provides an important direction toward extracting chemical bonding information from large-scale DFT calculations on materials systems involving thousands of atoms while accommodating periodic, semiperiodic, or fully nonperiodic boundary conditions. Toward this, we derive the relevant mathematical expressions and develop efficient numerical implementation procedures that are scalable on multinode CPU architectures to compute the projected overlap and Hamilton populations. The population analysis is accomplished by projecting either the self-consistently converged FE discretized Kohn-Sham orbitals or the FE discretized Hamiltonian onto a subspace spanned by a localized atom-centered basis set. The proposed methods are implemented in a unified framework within the DFT-FE code where the ground-state DFT calculations and the population analysis are performed on the same FE grid. We further benchmark the accuracy and performance of this approach on representative material systems involving periodic and nonperiodic DFT calculations with LOBSTER, a widely used projected population analysis code. Finally, we discuss a case study demonstrating the advantages of our scalable approach to extract the quantitative chemical bonding information of hydrogen chemisorbed in large silicon nanoparticles alloyed with carbon, a candidate material for hydrogen storage.

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

Linear-response time-dependent density functional theory approach to warm dense matter with adiabatic exchange-correlation kernels

We present a new methodology for the linear-response time-dependent density functional theory (LR-TDDFT) calculation of the dynamic density response function of warm dense matter in an adiabatic approximation that can be used with any available exchange-correlation (XC) functional across Jacob's Ladder and across temperature regimes. The main novelty of the presented approach is that it can go beyond the adiabatic local density approximation (ALDA) and generalized LDA (AGGA) while preserving the self-consistence between the Kohn-Sham (KS) response function and adiabatic XC kernel for extended systems. The key ingredient for the presented method is the combination of the adiabatic XC kernel from the direct perturbation approach with the macroscopic dynamic KS response from the standard LR-TDDFT method using KS orbitals. We demonstrate the application of the method for the example of warm dense hydrogen, for which we perform a detailed analysis of the KS density response function, the RPA result, the total density response function and of the adiabatic XC kernel. The analysis is performed using LDA, GGA, and meta-GGA level approximations for the XC effects. The presented method is directly applicable to disordered systems such as liquid metals, warm dense matter, and dense plasmas.

High-order harmonic generation from aligned HCN molecules under orthogonally and linearly polarized two-color laser fields

Molecular high-order harmonic generation and molecular orbital ionization probabilities are calculated under orthogonally and linearly polarized two-color laser fields. When a second-harmonic field is applied, the high-order harmonics generated under the linearly polarized two-color laser fields in the antiparallel case are stronger than those generated in the orthogonal polarization case and even stronger than those of the parallel polarization case. The results show that ionization probabilities of various orbitals and harmonic orders are dependent on spatial symmetry of molecular orbitals. It is found that the ionization of low-lying Kohn-Sham molecular orbitals contributes significantly to the ionization and molecular high-order harmonic generation processes. The ionization probability maximum occurs when molecular orbital densities are maximum in the direction of laser field polarization. Furthermore, we show that the degeneracy of π orbitals is broken when the laser-molecule alignment angle deviates from the field axis. Accordingly, we indicated one component of the π orbital is effectively contributed to the ionization and high-order harmonic generation processes. Finally, to confirm the recollision model in the high-order harmonic generation, the quantum time-frequency analysis is used to extract electron paths information on subcycle time scales.

Assessing the Role of the Kohn-Sham Density in the Calculation of the Low-Lying Bethe-Salpeter Excitation Energies.

We adopt the GW many-body perturbation theory in conjunction with the Bethe-Salpeter equation (BSE) to compute 57 excitation energies of a set of 37 molecules. By using the PBEh global hybrid functional and a self-consistent scheme on the eigenvalues in GW, we show a strong dependence of the BSE energy on the starting Kohn-Sham (KS) density functional. This arises from both the quasiparticle energies and the spatial localization of the frozen KS orbitals employed to compute the BSE. In order to address the arbitrariness in the mean field choice, we adopt an orbital-tuning scheme where the amount of Fock exchange, α, is tuned to impose that the KS HOMO matches the GW quasiparticle eigenvalue, thus fulfilling the ionization potential theorem in DFT. The performance of the proposed scheme yields excellent results and it is similar to M06-2X and PBEh with α = 75%, consistent with tuned values of α ranging between 60% and 80%.