A two-component variant of the Douglas–Kroll relativistic linear combination of Gaussian-type orbitals density-functional method: Spin–orbit effects in atoms and diatomics

Scalable Ab Initio Electronic Structure Methods with Near Chemical Accuracy for Main Group Chemistry

This study evaluates the precision of widely recognized quantum chemical methodologies, CCSD(T), DLPNO-CCSD(T), and localized ph-AFQMC, for determining the thermochemistry of main group elements. DLPNO-CCSD(T) and localized ph-AFQMC, which offer greater scalability compared to canonical CCSD(T), have emerged over the past decade as pivotal in producing precise benchmark chemical data. Our investigation includes closed-shell, neutral molecules, focusing on their heat of formation and atomization energy sourced from four specific small molecule data sets. First, we selected molecules from the G2 and G3 data sets, noted for their reliable experimental heat of formation data. Additionally, we incorporate molecules from the W4-11 and W4-17 sets, which provide high-level theoretical reference values for atomization energy at 0 K. Our findings reveal that both DLPNO-CCSD(T) and ph-AFQMC methods are capable of achieving a root-mean-square deviation of less than 1 kcal/mol across the combined data set, aligning with the threshold for chemical accuracy. Moreover, we make efforts to confine the maximum deviations within 2 kcal/mol, a degree of precision that significantly broadens the applicability of these methods in fields such as biology and materials science.

Multifunctional Photoelectroactive Materials for Optoelectronic Applications Based on Thieno[3,4-b]pyrazines and Thieno[1,2,5]thiadiazoles

In this study, we introduce a novel family of symmetrical thiophene-based small molecules with a Donor-Acceptor-Donor structure. These compounds feature three different acceptor units: benzo[c][1,2,5]thiadiazole (Bz), thieno[3,4-b]pyrazine (Pz), and thieno[1,2,5]thiadiazole (Tz), coupled with electron donor units based on a carbazole-thiophene derivative. Using Density Functional Theory (DFT), we investigate how the molecular geometry and strength of the central acceptor unit impact the redox and spectroscopic properties. Notably, the incorporation of Pz and Tz moieties induces a significant redshift in the absorption and emission spectra, which extend into the near-infrared (NIR) region, simultaneously reducing their energy gaps (~1.4-1.6 eV). This shift is attributed to the increased coplanarity of the oligomeric inner core, both in the ground (S0 ) and excited (S1 ) states, due to the enhanced quinoidal character as supported by bond-length alternation (BLA) analysis. These structural changes promote better π-electron delocalization and facilitate photoinduced charge transfer processes in optoelectronic devices. Notably, we show that Pz- and Tz-containing molecules exhibit NIR electrochromic behavior and present ambivalent character in bulk heterojunction (BHJ) solar cells. Finally, theoretical calculations suggest that these molecules could serve as effective two-photon absorption (2PA) probes, further expanding their potential in optoelectronic applications.

Single-Hydroxide Bridged Dimers of U and Np Actinyls: A Density Functional Study on Their Existence and Structure in Aqueous Solution

With quantum chemical calculations at the density functional theory level, we examined the structure and the stability of diactinyl monohydroxo complexes [(AnO₂)₂(OH)]^3+/+ in aqueous solution for An = U(VI), Np(VI), and Np(V). In particular, this study contributes to understanding the hydrolysis of Np(VI) and Np(V), which is less well characterized than for U(VI). [(UO₂)₂(OH)]³⁺ is a known hydrolysis complex of U(VI) at low pH. Although not yet found in experiments, [(NpO₂)₂(OH)]³⁺ is suggested to exist due to the similarity between Np(VI) and U(VI) complexes, while [(NpO₂)₂(OH)]⁺ is a hypothetical species thus far. Our calculations suggest that the An(VI) complexes favor the parallel orientation of actinyls, whereas for the Np(V) complex a perpendicular arrangement is stabilized by hydrogen bonds between aqua ligands and the actinyl oxygen atoms. The Np(VI) complex [(NpO₂)₂(OH)]³⁺ features a structure and stability similar to its U(VI) analogue. From calculated formation constants for An(VI) diactinyl monohydroxo complexes, we find qualitative agreement with the experiment for U(VI). Both An(VI) complexes are only slightly less stable than the separate mononuclear constituents, the actinyl aqua and the monohydroxo complex. For the Np(V) species [(NpO₂)₂(OH)]⁺, we calculated a considerably lower complexation constant than for its An(VI) analogues, but it is more stable against decay into its constituents. Thus, this complex may exist at about the pH where Np(V) hydrolysis starts at not too low Np(V) concentrations.

Calculation of Metallocene Ionization Potentials via Auxiliary Field Quantum Monte Carlo: Toward Benchmark Quantum Chemistry for Transition Metals

The accurate ab initio prediction of ionization energies is essential to understanding the electrochemistry of transition metal complexes in both materials science and biological applications. However, such predictions have been complicated by the scarcity of gas phase experimental data, the relatively large size of the relevant molecules, and the presence of strong electron correlation effects. In this work, we apply all-electron phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) utilizing multideterminant trial wave functions to six metallocene complexes to compare the computed adiabatic and vertical ionization energies with experimental results. We find that ph-AFQMC yields mean absolute errors (MAEs) of 1.69 ± 1.02 kcal/mol for the adiabatic energies and 2.85 ± 1.13 kcal/mol for the vertical energies. We also carry out density functional theory (DFT) calculations using a variety of functionals, which yields MAEs of 3.62-6.98 kcal/mol and 3.31-9.88 kcal/mol, as well as one variant of localized coupled cluster calculations (DLPNO-CCSD(T₀) with moderate PNO cutoffs), which has MAEs of 4.96 and 6.08 kcal/mol, respectively. We also test the reliability of DLPNO-CCSD(T₀) and DFT on acetylacetonate (acac) complexes for adiabatic energies measured in the same manner experimentally, and we find higher MAEs, ranging from 4.56 to 10.99 kcal/mol (with a different ordering) for DFT and 6.97 kcal/mol for DLPNO-CCSD(T₀). Finally, by utilizing experimental solvation energies, we show that accurate reduction potentials in solution for the metallocene series can be obtained from the AFQMC gas phase results.

Relativistic Four-Component DFT Calculations of Vibrational Frequencies

We investigate the effect of relativity on harmonic vibrational frequencies. Density functional theory (DFT) calculations using the four-component Dirac–Coulomb Hamiltonian have been performed for 15 hydrides (H₂X, X = O, S, Se, Te, Po; XH₃, X = N, P, As, Sb, Bi; and XH₄, X = C, Si, Ge, Sn, Pb) as well as for HC≡CPbH₃. The vibrational frequencies have been calculated using finite differences of the molecular energy with respect to geometrical distortions of the nuclei. The influences of the choice of basis set, exchange–correlation functional, and step length for the numerical differentiation on the calculated harmonic vibrational frequencies have been tested, and the method has been found to be numerically robust. Relativistic effects are noticeable for the heavier congeners H₂Te and H₂Po, SbH₃ and BiH₃, and SnH₄ and PbH₄ and are much more pronounced for the vibrational modes with higher frequencies. Spin–orbit effects constitute a very small fraction of the total relativistic effects, except for H₂Te and H₂Po. For HC≡CPbH₃ we find that only the frequencies of the modes with large contributions from Pb displacements are significantly affected by relativity.

Size-dependent properties of transition metal clusters: from molecules to crystals and surfaces--computational studies with the program ParaGauss

In the so-called scalable regime the size-dependent behavior of the physical and chemical properties of transition metal clusters is described by scaling relationships. For most quantities this scalable regime is reached for cluster sizes between a few tens and a few hundreds of atoms, hence for systems for which an accurate treatment by density functional theory is still feasible. Thus, by invoking scaling relations one is able to obtain properties of very large nanoparticles and even the bulk limit from the results of a series of smaller cluster models. In this invited review we illustrate this strategy by exploiting results from computational studies that mostly were carried out with the density functional theory software ParaGauss. We address mainly the size-dependent behavior of the properties of transition metal clusters. To this end, we first present benchmark studies probing various approximations that are used in such density functional calculations. Subsequently we show how physical insight may be gained by exploring less understood types of systems. These applications range from bare clusters to nanoislands and nanoalloys to adsorption complexes.

A fully variational spin-orbit coupled complete active space self-consistent field approach: application to electron paramagnetic resonance g-tensors

In this work, a relativistic version of the state-averaged complete active space self-consistent field method is developed (spin-orbit coupled state-averaged complete active space self-consistent field; CAS-SOC). The program follows a "one-step strategy" and treats the spin-orbit interaction (SOC) on the same footing as the electron-electron interaction. As opposed to other existing approaches, the program employs an intermediate coupling scheme in which spin and space symmetry adapted configuration space functions are allowed to interact via SOC. This adds to the transparency and computational efficiency of the procedure. The approach requires the utilization of complex-valued configuration interaction coefficients, but the molecular orbital coefficients can be kept real-valued without loss of generality. Hence, expensive arithmetic associated with evaluation of complex-valued transformed molecular integrals is completely avoided. In order to investigate the quality of the calculated wave function, we extended the method to the calculation of electronic g-tensors. As the SOC is already treated to all orders in the SA-CASSCF process, first order perturbation theory with the Zeeman operator is sufficient to accomplish this task. As a test-set, we calculated g-tensors of a set of diatomics, a set of d(1) transition metal complexes MOX4(n-), and a set of 5f(1) actinide complexes AnX6(n-). These calculations reveal that the effect of the wavefunction relaxation due to variation inclusion of SOC is of the same order of magnitude as the effect of inclusion of dynamic correlation and hence cannot be neglected for the accurate prediction of electronic g-tensors.

The Beijing Density Functional (BDF) Program Package: Methodologies and Applications

The Beijing Density Functional (BDF) program package is such a code that can perform nonrelativistic, one-, two-, and four-component relativistic density functional calculations on medium-sized molecular systems with various functionals in most compact and yet sufficient basis set expansions. The mergence of different approaches in a single code facilitates direct and systematic comparisons between different Hamiltonians, since they share all the same numerical and technical issues. In this account, the methodologies adopted in the code will be discussed in great detail and some applications of the code will be briefly presented.

A Quasirelativistic Two-component Density Functional and Hartree-Fock Program

This work describes the implementation of an efficient two-component quasirelativistic density functional and Hartree-Fock program. The fact that the basis functions are real can be exploited if a special internal representation of operators and density matrices is used. This also leads to a considerable reduction of the effort in the closed shell case. While in most applications to open shell molecules, the noncollinear approach to define a relativistic spin density is preferable, the collinear approach finds its application in the calculation of magnetic anisotropy energies. Linear algebra steps in the SCF procedure have a higher relative weight compared to the nonrelativistic case, therefore some care was necessary to make them fast when parallelizing the code. The quasirelativistic Hamiltonians that have been implemented are the ´zeroth-order regular approximation´ (ZORA) Hamiltonian, the Douglas-Kroll-Hess Hamiltonian up to the sixth order, together with an accurate approximation to treat the picture change effect of the electron interaction, and effective core potential (ECP) matrix elements. Geometry gradients are available for the ZORA and ECP methods.

Density functional theory for transition metals and transition metal chemistry

We introduce density functional theory and review recent progress in its application to transition metal chemistry. Topics covered include local, meta, hybrid, hybrid meta, and range-separated functionals, band theory, software, validation tests, and applications to spin states, magnetic exchange coupling, spectra, structure, reactivity, and catalysis, including molecules, clusters, nanoparticles, surfaces, and solids.

Magnetic anisotropy from density functional calculations. Comparison of different approaches: Mn12O12 acetate as a test case

Magnetic anisotropy is the capability of a system in a triplet or higher spin state to store magnetic information. Although the source of the magnetic anisotropy is the zero-field splitting of the ground state of the system, there is a difference between these two quantities that has to be fully rationalized before one makes comparisons. This is especially important for small spins such as triplets, where the magnetic anisotropy energy is only half of the zero-field splitting. Density functional calculations of magnetic anisotropy energies correspond to a high-field limit where the spins are aligned by the external magnetic field. Data are presented for the well-studied molecular magnet Mn(12)O(12) acetate. Both perturbative and self-consistent treatments, different quasirelativistic Hamiltonians (zeroth order regular approximation, Douglas-Kroll, effective core potentials) and exchange-correlation functionals are compared. It is shown that some effects usually considered minor, such as the inclusion of the exchange-correlation potential in the effective one-particle spin-orbit operator, lead to sizable differences when computing magnetic anisotropy energies. Higher-order contributions, that is, the difference between self-consistent and perturbative results, increase the magnetic anisotropy energy somewhat but do not introduce sizeable quartic terms or an in-plane anisotropy. In numerical experiments, on can switch off and on spin-orbit coupling at individual atomic sites. This procedure yields single-site contributions to the overall magnetic anisotropy energy that could be used as parameters in phenomenological spin Hamiltonians. If ferrimagnetic systems are treated with broken symmetry density functional methods where the Kohn-Sham reference function is not a spin eigenfunction, corrections are needed which depend on the size of the exchange couplings in the system and must therefore be evaluated case by case.

Self-consistent treatment of spin-orbit interactions with efficient Hartree-Fock and density functional methods

Efficient self-consistent field (SCF) schemes including both scalar relativistic effects and spin-orbit (SO) interactions at Hartree-Fock (HF) and density functional (DFT) levels are presented. SO interactions require the extension of standard procedures to two-component formalisms. Efficiency is achieved by using effective core potentials (ECPs) and by employing the resolution-of-the-identity approximation for the Coulomb part (RI-J) in pure DFT calculations as well as also for the HF-exchange part (RI-JK) in the case of HF or hybrid-DFT treatments. The procedures were implemented in the program system TURBOMOLE; efficiency is demonstrated for comparably large systems, such as Pb54. Relevance of SO effects for electronic structure and stability is illustrated by treatments of small Pb and Po clusters with and without accounting for SO effects.

Density-functional model cluster studies of EPR g tensors of Fs+ centers on the surface of MgO

We report g tensors of surface color centers, so-called F(s) (+) centers, of MgO calculated with two density-functional approaches using accurately embedded cluster models. In line with recent UHV measurements on single-crystalline MgO film, we determined only small g-tensor anisotropies and negative shifts Deltag identical with g-g(e) for all F(s) (+) sites considered, namely, (001)-terrace, step, edge, and corner sites. The g values are very sensitive to the local structure of the defect: relaxation reverses the sign of Deltag. However, accounting for the spin-orbit interaction either self-consistently or perturbatively yields very similar results. In addition to the values of the tensor components, their direction with respect to the surface was determined. In contrast to edges, significant deviations from ideal C(2v) symmetry were found for F(s) (+) centers at steps. Recent data on single-crystalline thin films are reevaluated in the light of these results.

Importance of direct spin-spin coupling and spin-flip excitations for the zero-field splittings of transition metal complexes: a case study

This work reports the evaluation of several theoretical approaches to the zero-field splitting (ZFS) in transition metal complexes. The experimentally well-known complex [Mn(acac)3] is taken as an example. The direct spin-spin contributions to the ZFS have been calculated on the basis of density functional theory (DFT) or complete active space self-consistent field (CASSCF) wave functions and have been found to be much more important than previously assumed. The contributions of the direct term may exceed approximately 1 cm(-1) in magnitude and therefore cannot be neglected in any treatment that aims at a realistic quantitative modeling of the ZFS. In the DFT framework, two different variants to treat the spin-orbit coupling (SOC) term have been evaluated. The first approach is based on previous work by Pederson, Khanna, and Kortus, and the second is based on a "quasi-restricted" DFT treatment which is rooted in our previous work on ZFS. Both approaches provide very similar results and underestimate the SOC contribution to the ZFS by a factor of 2 or more. The SOC is represented by an accurate multicenter spin-orbit mean-field (SOMF) approximation which is compared to the popular effective DFT potential-derived SOC operator. In addition to the DFT results, direct "infinite order" ab initio calculations of the SOC contribution to the ZFS based on CASSCF wave functions, the spectroscopy-oriented configuration interaction (SORCI), and the difference-dedicated CI (DDCI) approach are reported. In general, the multireference ab initio results provide a more realistic description of the ZFS in [Mn(acac)3]. The conclusions likely carry over to many other systems. This is attributed to the explicit treatment of the multiplet effects which are of dominant importance, since the calculations demonstrate that, even in the high-spin d4 system MnIII, the spin-flip excitations make the largest contribution to the SOC. It is demonstrated that the ab initio methods can be used even for somewhat larger molecules (the present calculations were done with more than 500 basis functions) in a reasonable time frame. Much more economical but still fairly reasonable results have been achieved with the INDO/S treatment based on CASSCF and SOC-CI wave functions.

Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas-Kroll-Hess transformation up to arbitrary order

In order to achieve exact decoupling of the Dirac Hamiltonian within a unitary transformation scheme, we have discussed in part I of this series that either a purely numerical iterative technique (the Barysz-Sadlej-Snijders method) or a stepwise analytic approach (the Douglas-Kroll-Hess method) are possible. For the evaluation of Douglas-Kroll-Hess Hamiltonians up to a pre-defined order it was shown that a symbolic scheme has to be employed. In this work, an algorithm for this analytic derivation of Douglas-Kroll-Hess Hamiltonians up to any arbitrary order in the external potential is presented. We discuss how an estimate for the necessary order for exact decoupling (within machine precision) for a given system can be determined from the convergence behavior of the Douglas-Kroll-Hess expansion prior to a quantum chemical calculation. Once this maximum order has been accomplished, the spectrum of the positive-energy part of the decoupled Hamiltonian, e.g., for electronic bound states, cannot be distinguished from the corresponding part of the spectrum of the Dirac operator. An efficient scalar-relativistic implementation of the symbolic operations for the evaluation of the positive-energy part of the block-diagonal Hamiltonian is presented, and its accuracy is tested for ground-state energies of one-electron ions over the whole periodic table. Furthermore, the first many-electron calculations employing sixth up to fourteenth order DKH Hamiltonians are presented.

Basis-set extensions for two-component spin-orbit treatments of heavy elements

The accuracy of standard basis sets of quadruple-zeta and lower quality for the use in two-component self-consistent field procedures including spin-orbit coupling is investigated for the elements In-I and Au-At. Spin-orbit coupling leads to energetic and spatial splittings of inner shells, which are not described accurately with standard basis sets optimized for scalar relativistic calculations. This results in large errors in total atomic energies and significant errors in atomization energies of compounds containing these atoms. We show how these errors can be corrected by adding just a few steep sets of basis functions and demonstrate the quality of the resulting extended basis sets.

Resolution of identity Dirac-Kohn-Sham method using the large component only: Calculations of g-tensor and hyperfine tensor

A new relativistic two-component density functional approach, based on the Dirac-Kohn-Sham method and an extensive use of the technique of resolution of identity (RI), has been developed and is termed the DKS2-RI method. It has been applied to relativistic calculations of g and hyperfine tensors of coinage-metal atoms and some mercury complexes. The DKS2-RI method solves the Dirac-Kohn-Sham equations in a two-component framework using explicitly a basis for the large component only, but it retains all contributions coming from the small component. The DKS2-RI results converge to those of the four-component Dirac-Kohn-Sham with an increasing basis set since the error associated with the use of RI will approach zero. The RI approximation provides a basis for a very efficient implementation by avoiding problems associated with complicated integrals otherwise arising from the elimination of the small component. The approach has been implemented in an unrestricted noncollinear two-component density functional framework. DKS2-RI is related to Dyall's [J. Chem. Phys. 106, 9618 (1997)] unnormalized elimination of the small component method (which was formulated at the Hartree-Fock level and applied to one-electron systems only), but it takes advantage of the local Kohn-Sham exchange-correlation operators (as, e.g., arising from local or gradient-corrected functionals). The DKS2-RI method provides an attractive alternative to existing approximate two-component methods with transformed Hamiltonians (such as Douglas-Kroll-Hess [Ann. Phys. 82, 89 (1974); Phys. Rev. A 33, 3742 (1986)] method, zero-order regular approximation, or related approaches) for relativistic calculations of the structure and properties of heavy-atom systems. In particular, no picture-change effects arise in the property calculations.

Relativistic calculation of indirect NMR spin-spin couplings using the Douglas-Kroll-Hess approximation

We have employed the Douglas-Kroll-Hess approximation to derive the perturbative Hamiltonians involved in the calculation of NMR spin-spin couplings in molecules containing heavy elements. We have applied this two-component quasirelativistic approach using finite perturbation theory in combination with a generalized Kohn-Sham code that includes the spin-orbit interaction self-consistently and works with Hartree-Fock and both pure and hybrid density functionals. We present numerical results for one-bond spin-spin couplings in the series of tetrahydrides CH(4), SiH(4), GeH(4), and SnH(4). Our two-component Hartree-Fock results are in good agreement with four-component Dirac-Hartree-Fock calculations, although a density-functional treatment better reproduces the available experimental data.

Accurate and efficient treatment of two-electron contributions in quasirelativistic high-order Douglas-Kroll density-functional calculations

Two-component quasirelativistic approaches are in principle capable of reproducing results from fully relativistic calculations based on the four-component Dirac equation (with fixed particle number). For one-electron systems, this also holds in practice, but in many-electron systems one has to transform the two-electron interaction, which is necessary because a picture change occurs when going from the Dirac equation to a two-component method. For one-electron properties, one can take full account of picture change in a manageable way, but for the electron interaction, this would spoil the computational advantages which are the main reason to perform quasirelativistic calculations. Exploiting those picture change effects are largest in the atomic cores, which in molecular applications do not differ too much from the cores of isolated neutral atoms, we propose an elegant, efficient, and accurate approximation to the two-electron picture change problem. The new approach, called the "model potential" approach because it makes use of atomic (four- and two-component) data to estimate picture change effects in molecules, shares with the nuclear-only approach that the Douglas-Kroll operator needs to be constructed only once (not in each self-consistent-field iteration) and that no time-consuming multicenter relativistic two-electron integrals need to be calculated. The new approach correctly describes the screening of both the nearest nucleus and distant nuclei, for the scalar-relativistic as well as the spin-orbit parts of the Hamiltonian. The approach is tested on atomic and molecular-orbital energies as well as spectroscopic constants of the lead dimer.

Efficient treatment of the Hartree interaction in the relativistic Kohn-Sham problem

We elaborate the two-component Douglas-Kroll reduction of the Dirac-Kohn-Sham problem of relativistic density-functional theory as introduced by Matveev and Rosch [J. Chem. Phys. 118, 3997 (2003)]. That method retains corrections to the Coulomb self-interaction (or Hartree) term of the energy functional that are due to the picture change. Using analytic expressions for the matrix elements, one is able to abandon the resolution of the identity approach for a crucial step of the relativistic transformation. Thus, a major source of uncertainties of the method is eliminated because basis sets no longer have to be extended by functions of higher angular momentum, previously required to ensure kinetic balance. This approach also relies on the electron charge-density fitting scheme via an auxiliary basis set. An efficient approximate implementation results if one restricts the relativistic transformation to the spherically symmetric atom-centered auxiliary functions. It provides accurate results while simplifying greatly the expressions for the matrix elements of the relativistically transformed operators and significantly reducing the computational effort. We demonstrate the performance of the method for the fine structure of one-electron levels of the Hg atom, the g-tensor shifts of NO2, and the properties of the diatomic molecules Bi2, Pb2, PbO, and TlH.

Scalar relativistic all-electron density functional calculations on periodic systems

Scalar relativistic effects are included in periodic boundary conditions calculations with Gaussian orbitals. This approach is based on the third-order Douglas-Kroll-Hess approximation, allowing the treatment of all electrons on an equal footing. With this methodology, we are able to perform relativistic all-electron density functional calculations using the traditional local spin-density and generalized gradient approximations (GGA), as well as meta-GGA and hybrid density functionals. We present benchmark results for the bulk metals Pd, Ag, Pt, and Au, and the large band gap semiconductors AgF and AgCl.