Introduction to Relativistic Quantum Chemistry
Abstract
This book provides an introduction to the essentials of relativistic effects in quantum chemistry, and a reference work that collects all the major developments in this field. It is designed for the graduate student and the computational chemist with a good background in nonrelativistic theory. In addition to explaining the necessary theory in detail, at a level that the non-expert and the student should readily be able to follow, the book discusses the implementation of the theory and practicalities of its use in calculations. After a brief introduction to classical relativity and electromagnetism, the Dirac equation is presented, and its symmetry, atomic solutions, and interpretation are explored. Four-component molecular methods are then developed: self-consistent field theory and the use of basis sets, double-group and time-reversal symmetry, correlation methods, molecular properties, and an overview of relativistic density functional theory. The emphases in this section are on the basics of relativistic theory and how relativistic theory differs from nonrelativistic theory. Approximate methods are treated next, starting with spin separation in the Dirac equation, and proceeding to the Foldy-Wouthuysen, Douglas-Kroll, and related transformations, Breit-Pauli and direct perturbation theory, regular approximations, matrix approximations, and pseudopotential and model potential methods. For each of these approximations, one-electron operators and many-electron methods are developed, spin-free and spin-orbit operators are presented, and the calculation of electric and magnetic properties is discussed. The treatment of spin-orbit effects with correlation rounds off the presentation of approximate methods. The book concludes with a discussion of the qualitative changes in the picture of structure and bonding that arise from the inclusion of relativity.
... The third term is the contribution from SOMO. The Kramers pair orbitals are related by the time reversal operator (τ ) [24]: ...
... The details of the basis sets are given in Table I (uncontracted [14], kinetically balanced [24] Gaussian Type Orbitals (GTOs), cc-pV DZ and TZ for F, Cl and Br [28], Dyall's basis for I [29], and Dyall's c2v and c3v basis sets for Hg [29]). We did not freeze any of our occupied orbitals. ...
Mercury monohalides are promising candidates for electron electric dipole moment searches. This is due to their extremely large values of effective electric fields, besides other attractive experimental features. We have elucidated the theoretical reasons of our previous work. We have also presented a detailed analysis of our calculations, by including the most important of the correlation effects' contributions. We have also analyzed the major contributions to the effective electric field, at the Dirac- Fock level, and identified those atomic orbitals' mixings that contribute significantly to it.
... where α and β = β − 1 4×4 are 4×4 Dirac matrices in their standard representation, c is the speed of light. [54] The Zeeman operator,ĥ B , and the hyperfine operators,ĥ m K , are defined as:ĥ ...
... [53] One-electron basis The 4c molecular spinors, eigenfunctions of the DC Hamiltonian, are expanded in scalar finite basis sets. [54] The small component basis functions (χ S ) are generated from the large component functions (χ L ) by the restricted kinetic balance (RKB) or the restricted magnetic balance (RMB) prescription, which properly describes the relation between large and small components in the presence of external magnetic fields. The question of magnetic balance in 4c calculations was initially addressed by Aucar and coworkers [46] and Kutzelnigg, [55,56] though later investigations have shown these yielded mixed results. ...
We report an implementation of the nuclear magnetic resonance (NMR) shielding (), isotope-independent indirect spin-spin coupling (K) and the magnetizability () tensors in the frozen density embedding (FDE) scheme using the four-component (4c) relativistic Dirac--Coulomb (DC) Hamiltonian and the non-collinear spin density functional theory (SDFT). The formalism takes into account the magnetic balance between the large and the small components of molecular spinors and assures the gauge-origin independence of NMR shielding and magnetizability results. This implementation has been applied to hydrogen-bonded HXHOH complexes (X = Se, Te, Po) and compared with the supermolecular calculations and with the approach based on the integration of the magnetically induced current density vector. A comparison with the approximate Zeroth-Order Regular Approximation (ZORA) Hamiltonian indicates non-negligible differences in and K in the HPoHOH complex, and calls for a thourough comparison of ZORA and DC in the description of environment effects on NMR parameters for molecular systems with heavy elements.
... However, the TGNOF results (orange dots) lie below their FCI counterparts in the x ∈ It is known that the non-relativistic limit can be approached by setting the value of the speed of light c to a large value in the 4c-DHF. 52,111,113,114 Then, as discussed in Sec. II, in this limit the KR-4c-DHF solution approaches the THF solution (instead of the RHF one) when the RHF and THF solutions differ because (i) the THF solution can be lower in energy, (ii) the THF and the KR-4c-DHF methods work with the extra flexibility provided by complex orbitals, and (iii) both methods are built to preserve time-reversal symmetry. ...
... which is widely used in quadratic convergent methods and similar algorithms by updating the parameters κ pq with the Newton-Raphson step κ = −G −1 · g.3,14,52,[73][74][75][76][77] At the stationary point, the gradient vector vanishes (g = 0) and the diagonalization of the Hessian matrix G provides valuable information about the type of stationary point one has reached: it is a minimum when all the eigenvalues are positive, a kth-order saddle point when there are k negative eigenvalues, or a maximum when all eigenvalues are negative. ...
Reduced density matrix functional theory (RDMFT) and coupled cluster theory restricted to paired double excitations (pCCD) are emerging as efficient methodologies for accounting for the so-called non-dynamic electronic correlation effects. Up to now, molecular calculations have been performed with real-valued orbitals. However, before extending the applicability of these methodologies to extended systems, where Bloch states are employed, the subtleties of working with complex-valued orbitals and the consequences of imposing time-reversal symmetry must be carefully addressed. In this work, we describe the theoretical and practical implications of adopting time-reversal symmetry in RDMFT and pCCD when allowing for complex-valued orbital coefficients. The theoretical considerations primarily affect the optimization algorithms, while the practical implications raise fundamental questions about the stability of solutions. Specifically, we find that complex solutions lower the energy when non-dynamic electronic correlation effects are pronounced. We present numerical examples to illustrate and discuss these instabilities and possible problems introduced by N-representability violations.
... where c nq,i is the linear expansion coefficient, A [16] is the anti-symmetrization operator for the two electrons [35], and P ζ[16] G is a projector corresponding to the ζ irreducible representation (irrep) of the G point group [38,51]. ...
... In orbital-based approaches, the construction of the one-particle projectors is straightforward [51,56,57]. So, it may first sound like a practical idea to apply an orbital-based approach only for the construction of the positive-energy projector ('non-interacting space corresponding to positive energies') and use it in the full computation (also including the electron-electron interaction) with an explicitly correlated basis set [35,58,59]. ...
This work is concerned with two-spin-1/2-fermion relativistic quantum mechanics, and it is about the construction of one-particle projectors using an inherently two(many)-particle, `explicitly correlated' basis representation, necessary for good numerical convergence of the interaction energy. It is demonstrated that a faithful representation of the one-particle operators, which appear in intermediate but essential computational steps, can be constructed over a many-particle basis set by accounting for the full Hilbert space beyond the physically relevant anti-symmetric subspace. Applications of this development can be foreseen for the computation of quantum-electrodynamics corrections for a correlated relativistic reference state and high-precision relativistic computations of medium-to-high-Z helium-like systems, for which other two-particle projection techniques are unreliable.
... In relativistic theory, basis sets such as the Dyall family 60 are optimized to describe the relativistic electronic structure of a nuclear charge distribution. [61][62][63][64] However, as the nuclear position is still static, even these electronic basis sets are not suitable for describing the quantum nuclei in multicomponent calculations. Although the Dyall basis sets may offer an improved description of predominantly spherically symmetric ground state protonic densities, they do not correctly describe asymmetric ground state protonic densities or the excited vibronic states, where the proton vibrational wavefunctions have nodes and often are even more asymmetric. ...
The nuclear-electronic orbital (NEO) approach incorporates nuclear quantum effects into quantum chemistry calculations by treating specified nuclei quantum mechanically, equivalently to the electrons. Within the NEO framework, excited states are vibronic states representing electronic and nuclear vibrational excitations. The NEO multireference configuration interaction (MRCI) method presented herein provides accurate ground and excited vibronic states. The electronic and nuclear orbitals are optimized with a NEO multiconfigurational self-consistent field (NEO-MCSCF) procedure, thereby including both static and dynamic correlation and allowing the description of double and higher excitations. The accuracy of the NEO-MRCI method is illustrated by computing the ground state protonic densities and excitation energies of the vibronic states for four molecular systems with the hydrogen nucleus treated quantum mechanically. In addition, revised conventional electronic basis sets adapted for quantized nuclei are developed and shown to be essential for achieving this level of accuracy. The NEO-MRCI approach, as well as the strategy for revising electronic basis sets, will play a critical role in multicomponent quantum chemistry.
... To alleviate the high computational cost associated with four-component methods, various two-component theories [8][9][10][11][12][13][14] have been introduced. Among these, the exact two-component (X2C) theory 10,13,[15][16][17] has gained recognition as a particularly prominent approach, which can reduce the computational cost of the integral transformation step. The literature covers a diverse array of X2C-CC methods, each providing various approaches and enhancements suited to different applications and computational requirements. ...
We present an efficient and cost-effective implementation for the exact two-component atomic mean field (X2CAMF) based coupled cluster (CC) method, which integrates frozen natural spinors (FNS) and the Cholesky decomposition (CD) technique. The use of CD approximation greatly reduces the storage requirement of the calculation without any significant reduction in accuracy. Compared to four-component methods, the FNS and CD-based X2CAMF-CC approach gives similar accuracy as that of the canonical four-component relativistic coupled cluster method at a fraction of the cost. The efficiency of the method is demonstrated by the calculation of a medium-sized uranium complex involving the correlation of over 1000 virtual spinors.
... Gaussian charge distribution is considered as nuclear model where the nuclear parameters 29 are taken as default values of DIRAC10 program package. The basis functions are represented in scalar basis and restricted kinetic balance (RKB) 30 condition is applied to generate the small component basis functions from the large component basis functions. The unphysical solutions are removed by diagonalizing the free particle Hamiltonian. ...
The Z-vector method in the relativistic coupled-cluster framework is employed to calculate the parallel and perpendicular components of the magnetic hyperfine structure constant of a few small alkaline earth hydrides (BeH, MgH, and CaH) and fluorides (MgF and CaF). We have compared our Z-vector results with the values calculated by the extended coupled-cluster (ECC) method reported in Phys. Rev. A 91 022512 (2015). All these results are compared with the available experimental values. The Z-vector results are found to be in better agreement with the experimental values than those of the ECC values.
... For light element compounds such an approximation indeed does not lead to noticeable changes in molecular parameters (see e.g. [23]). ...
Recently a number of diatomic and polyatomics molecules has been identified as a prospective systems for Doppler/Sisyphus cooling. Doppler/Sisyphus cooling allows to decrease the kinetic energy of molecules down to microkelvin temperatures with high efficiency and then capture them to molecular traps, including magneto-optical trap. Trapped molecules can be used for creation of molecular fountains and/or performing controlled chemical reactions, high-precision spectra measurements and a multitude of other applications. Polyatomic molecules with heavy nuclei present considerable interest for the search for "new physics" outside of Standard Model and other applications including cold chemistry, photochemistry, quantum informatics etc. Herein we would like to attract attention to radium monohydroxide molecule (RaOH) which is on the one hand an amenable object for laser cooling and on the other hand provides extensive possibilities for searching for P-odd and P,T-odd effects. At the moment RaOH is the heaviest polyatomic molecule proposed for direct cooling with lasers.
... Finite size of nucleus with Gaussian charge distribution is considered as the nuclear model where the nuclear parameters [38] are taken as default values of DIRAC10. Small component basis functions are generated from the large component by applying restricted kinetic balance (RKB) [39] condition. The basis functions are represented in scalar basis and unphysical solutions are removed by means of the diagonalization of free particle Hamiltonian. ...
We have employed both Z-vector method and the expectation value approach in the relativistic coupled-cluster framework to calculate the scalar-pseudoscalar (S-PS) P, T -odd interaction constant (W_s) and the effective electric field (Eeff) experienced by the unpaired electron in the ground electronic state of RaF. Further, the magnetic hyperfine structure constants of ^{223}Ra in RaF and ^{223}Ra+ are also calculated and compared with the experimental values wherever available to judge the extent of accuracy obtained in the employed methods. The outcome of our study reveals that the Z-vector method is superior than the expectation value approach in terms of accuracy obtained for the calculation of ground state property. The Z-vector calculation shows that RaF has a high E_eff (52.5 GV/cm) and W_s (141.2 kHz) which makes it a potential candidate for the eEDM experiment.
... While gradients of diagonal states including SOC are already described for semi-empirical wavefunctions, 176 no such gradients are available up to now for ab initio electronic structure methods and neither are the corresponding non-adiabatic couplings. Furthermore, quantum chemistry in the diagonal basis including SOCs (see references 171,172,177 and references therein) is in many cases significantly more expensive than quantum chemistry in the MCH basis. The higher cost is related to the fact that the spin-orbit Hamiltonian does not commute withŜ 2 orŜ z and hence it is not block-diagonal. ...
Intersystem crossing is a radiationless process that can take place in a molecule irradiated by UV-Vis light, thereby playing an important role in many environmental, biological and technological processes. This paper reviews different methods to describe intersystem crossing dynamics, paying attention to semiclassical trajectory theories, which are especially interesting because they can be applied to large systems with many degrees of freedom. In particular, a general trajectory surface hopping methodology recently developed by the authors, which is able to include non-adiabatic and spin-orbit couplings in excited-state dynamics simulations, is explained in detail. This method, termed SHARC, can in principle include any arbitrary coupling, what makes it generally applicable to photophysical and photochemical problems, also those including explicit laser fields. A step-by-step derivation of the main equations of motion employed in surface hopping based on the fewest-switches method of Tully, adapted for the inclusion of spin-orbit interactions, is provided. Special emphasis is put on describing the different possible choices of the electronic bases in which spin-orbit can be included in surface hopping, highlighting the advantages and inconsistencies of the different approaches.
... The property integrals are constructed by using a locally modified version of DIRAC10 program package. Large and small components basis are linked through restricted kinetic balance condition [32]. The basis functions are expressed in scalar basis and all unphysical solutions are removed by means of the diagonalization of free particle Hamiltonian. ...
The scalar-pseudoscalar interaction constant of PbF in its ground state electronic configuration is calculated using the Z-vector method in the relativistic coupled-cluster framework. The precise calculated value is very important to set upper bound limit on P,T-odd scalar-pseudoscalar interaction constant, k_s, from the experimentally observed P,T-odd frequency shift. Further, the ratio of the effective electric field to the scalar-pseudoscalar interaction constant is also calculated which is required to get an independent upper bound limit of electric dipole moment of electron, d_e, and k_s and how these (d_e and k_s) are interrelated is also presented here.
... Gaussian charge distribution is considered for the finite size of the nucleus where the the nuclear parameters are taken from Ref. [40]. Restricted kinetic balance [41] condition is used to link small and large component basis function. No virtual pair approximation (NVPA) is used to solve DHF equation. ...
The Z-vector method in the relativistic coupled-cluster framework is used to calculate magnetic hyperfine structure constant () of alkali metals and singly charged alkaline earth metals in their ground state electronic configuration. The Z-vector results are in very good agreement with the experiment. The values of Li, Na, K, Rb, Cs, Be, Mg, Ca, and Sr obtained in the Z-vector method are compared with the extended coupled-cluster results taken from Phys. Rev. A 91, 022512 (2015). The same basis and cutoff are used for the comparison purpose. The comparison shows that Z-vector method with the singles and double approximation can produce more precise wavefunction in the nuclear region than the ECC method.
... [59] and [60]. This transformation is extensively used across various fields, including optic [61], condensed matter physics [62], nuclear physics [63], quantum chemistry [64], quantum field theory [65], electrodynamics [66], and gravitation [67]. For a broader perspective on its impact and applications, see Refs. ...
In this paper, we investigate the non-relativistic limit of the Dirac equation for relativistic spin-1/2 particles within the framework of the conformable fractional derivative (CFD) using the Foldy–Wouthuysen (FW) transformation. This approach leads to the derivation of a conformable fractional Schrödinger–Pauli equation. We propose and employ a conformable fractional version of the FW transformation, thoroughly examining its efficacy and behavior in the non-relativistic limit. Additionally, based on perturbation theory, we compute the energy shifts within the context of CFD and derive a conformable fractional fine structure of the hydrogen spectrum.
... Additionally, it supports Bohr's model of the hydrogen atom, which was previously regarded as empirical [22,23]. Accurate solutions to the Pauli equation are crucial for predictive quantum chemistry, enabling precise predictions of multi-electron atoms and molecules [24][25][26][27]. ...
This study explores the time-dependent Dunkl-Pauli oscillator in two dimensions. We constructed the Dunkl-Pauli Hamiltonian, which incorporates a time-varying magnetic field and a harmonic oscillator characterized by time-dependent mass and frequency, initially in Cartesian coordinates. Subsequently, we reformulated the Hamiltonian in polar coordinates and analyzed the eigenvalues and eigenfunctions of the Dunkl angular operator, deriving exact solutions using the Lewis-Riesenfeld invariant method. Our findings regarding the total quantum phase factor and wave functions reveal the significant impact of Dunkl operators on quantum systems, providing precise expressions for wave functions and energy eigenvalues. This work enhances the understanding of quantum systems with deformed symmetries and suggests avenues for future research in quantum mechanics and mathematical physics.
... However, if one electron is removed from the negative energy states, it would leave a hole corresponding to the creation of a positively charged particle, the positron. This reinterpretation becomes clearer if we split the sum over all state into a sum over the positive energy states and another over the negative energy ones.In this picture, the negative energy electron creation (annihilation) operators are reinterpreted as positive energy positron annihilation (creation) operators (see Ref.s [56,[60][61][62]). In this framework, the field operator can be decomposed in this way: ...
We present a new ab-initio approach to study molecules containing heavy atoms strongly interacting with quantum fields in optical devices. The relativistic quantum electrodynamics (QED) theory has been rewritten with a formalism close to relativistic quantum chemistry. This general framework represents the ideal starting point to extend the main quantum chemistry methods to relativistic QED. The Polaritonic Dirac Hartree Fock (Pol-DHF) approach is the first method we propose based on this theory. Pol-DHF allows for the simulation of field induced effects on the ground and excited state properties of heavy transition metals molecular complexes. The method is able to include not only the effects of the photons but can be easily extended also to include explicit interactions with positrons. Application of Pol-DHF to three metal hydrides revealed the importance of including radiative QED corrections to the treatment in strong coupling conditions. Due to an accurate description of spin-orbit coupling, the method is able to reproduce polaritonic effects happening at the crossing between singlet and triplet potential energy surfaces.
... The state-of-the-art post-HF four-component relativistic calculations employ the no-pair approximation. 101 This approximation is well-substantiated for applications of chemical scale. Formally, it yields a Hamiltonian analogous to the non-relativistic case ...
Scientific groups are struggling to adapt their codes to quickly-developing GPU-based HPC platforms. The domain of distributed coupled cluster (CC) calculations is not an exception. Moreover, our applications to tiny QED effects require higher-order CC which include thousands of tensor contractions, which makes automatic treatment imperative. The challenge is to allow efficient implementation by capturing key symmetries of the problem, while retaining the abstraction from the hardware. We present the tensor programming framework tenpi, which seeks to find this balance. It features a python library user interface, global optimization of intermediates, a visualization module and Fortran code generator that bridges the DIRAC package for relativistic molecular calculations to tensor contraction libraries. tenpi brings higher-order CC functionality to the massively parallel module of DIRAC. The architecture and design decision schemes are accompanied by benchmarks and by first production calculations on Summit, Frontier and LUMI along with state-of-the-art of tensor contraction software.
... As noted before, 123 both the (AA|AA) and (AB|BA) types of ERIs appear in the main diagonal blocks, whereas the (AA|AB) and (AB|AB) types of ERIs enter the first and second off-diagonal blocks, respectively, of the block pentadiagonal Hamiltonian matrix over SDs that are ordered in a descent order of their M K values. It will be shown in Sec. ...
In parallel to the unified construction of relativistic Hamiltonians based solely on physical arguments [J. Chem. Phys. 160, 084111 (2024)], a unified implementation of relativistic wave function methods is achieved here via programming techniques (e.g., template metaprogramming and polymorphism in C++). That is, once the code for constructing the Hamiltonian matrix is made ready, all the rest can be generated automatically from existing templates used for the nonrelativistic counterparts. This is facilitated by breaking a second-quantized relativistic Hamiltonian down to diagrams that are topologically the same as those required for computing the basic coupling coefficients between spin-free configuration state functions (CSF). Moreover, both time reversal and binary double point group symmetries can readily be incorporated into molecular integrals and Hamiltonian matrix elements. The latter can first be evaluated in the space of (randomly selected) spin-dependent determinants and then transformed to that of spin-dependent CSFs, thanks to simple relations in between. As a showcase, we consider here the no-pair four-component relativistic iterative configuration interaction with selection and perturbation correction (4C-iCIPT2), which is a natural extension of the spin-free iCIPT2 [J. Chem. Theory Comput. 17, 949 (2021)], and can provide near-exact numerical results within the manifold of positive energy states (PES), as demonstrated by numerical examples.
... The usual starting point of relativistic quantum-chemical calculations is the no-pair Dirac-Coulomb (DC) Hamiltonian, 9,10 which consists of the fully relativistic one-electron Dirac Hamiltonian and of the non-relativistic Coulomb potential for the two-electron part. ...
We present an implementation for the use of Cholesky decomposition (CD) of two-electron integrals within the spin-free Dirac-Coulomb (SFDC) scheme that enables to perform high-accuracy coupled-cluster (CC) calculations at costs almost comparable to those of their non-relativistic counterparts. While for non-relativistic CC calculations atomic-orbital (AO) based algorithms, due to their significantly reduced disk-space requirements, are the key to efficient large-scale computations, such algorithms are less advantageous in the SFDC case due to their increased computational cost on that case. Here, molecular-orbital (MO) based algorithms exploiting the CD of the two-electron integrals allow to reduce disk-space requirements, and lead to computational cost in the CC step that are more or less the same as in the non-relativistic case. The only remaining overhead in a CD-SFDC-CC calculation are due to the need to compute additional two-electron integrals, the somewhat higher cost of the Hartree-Fock calculation in the SFDC case, and additional cost in the transformation of the Cholesky vectors from the AO to the MO representation. However, these additional costs typically amount to less than 5-15 % of the total wall time and are thus acceptable. We illustrate the efficiency of our CD scheme for SFDC-CC calculations on a series of illustrative calculations for the X(CO) molecules with X = Ni, Pd, Pt.
... We utilize a 4-component wavefunction, which is expanded using distinct basis sets for both large and small components. The kinetic balance condition is applied to smaller components to prevent the wavefunction from experiencing a variational collapse into the negative energy continuum [33]. The DHF Coulomb Hamiltonian is employed with the approximation proposed by Visscher [34], wherein the contribution from the (SS|SS) integrals is replaced by an inter-atomic correction. ...
In this article, the molecular permanent electric dipole moments and components of static dipole polarizabilities for the electronic ground state of singly charged aluminum monohalides are reported. The coupled-cluster method by considering single and double excitations (CCSD) together with relativistic Dyall basis sets have been used to carry out these molecular property calculations. The contribution from triple excitations are incorporated through perturbative triples (CCSD(T)). The results from a series of progressively larger basis sets are extrapolated to the complete basis set limit. Further, the role of correlation and relativistic effects, and also the effect of augmentation over the considered basis sets on the valence molecular properties are studied. Our results are compared with those available in the literature.
... Furthermore, different relativistic methods [22][23][24][25][26][27] have been developed to calculate some physical quantities linked to diatomic molecules such as the Dirac-Hartree-Fock-Breit method (program BERTHA) [28], coupled cluster-CCSDT [29], relativistic configuration-interaction valence-bond method (with direct accounting for the spin-orbit interaction) [30], relativistic many-body perturbation theory formalism and empirical approaches [31]. ...
In this paper, the Dirac equation with the q-deformed Scarf potential for spin symmetry is solved for an arbitrary spin-orbit quantum number κ, in the presence of Coulomb-like potential tensor. Using the Feynman path integral formalism and the Pekeris approximation of the centrifugal term, we obtain the bound state energy eigenvalues and the associated spinor of the Dirac particle. Furthermore, this method is used to determine the spectrum of two diatomic molecules Li 2(6¹Π u ) and KRb(B ⁻¹Π). The obtained results are compared to the experimental and numerical ones.
... The finite size nuclei modelled by the Gaussian charge distribution and the default values of the nuclear parameters [72] in DIRAC10 are considered in the present study. Small component basis are generated from the large component basis using the restricted kinetic balance [73] condition. All our calculations are performed using the Dirac-Coulomb Hamiltonian unless otherwise specified. ...
We employ the four-component relativistic extended–coupled–cluster (ECC) method, a variational coupled–cluster (CC) approach, to compute the permanent electric dipole moment (PDM) of open-shell diatomic molecules (CaH, CaF, SrH and SrF) in their ground electronic state. The ECC results are compared with the PDM values estimated by the experiments as well as other single-reference CC-based approaches (the Z-vector technique, the expectation value method and the finite field approach) within the four-component relativistic framework to test the efficacy of the employed method. Our study reveals that the relativistic ECC method can yield reliable results for the PDMs of the considered molecular systems. We also observe that the computed results of the dipole moment improve upon the augmentation of diffused functions to the basis set.
Single-molecule magnets (SMMs) are molecular entities with strongly anisotropic magnetic moment. As a result, SMMs display slow relaxation of magnetization at the macroscopic scale. Up to date all experimentally characterized...
The chemical bond lengths and angles of group 16 dihydrides were investigated. The relativistic effects are essential for heavy elements molecules calculations. Here we implement two relativistic effects, that is, scalar relativistic effects and spin‐orbit coupled zero‐order regular approximation. Concerning molecular symmetry, scalar relativistic effects and spin‐orbit relativistic effects show different descriptions. They are single group and double group, respectively. In addition, non‐relativistic effects were used for very weak relativistic effects on molecules and for comparing with and without relativistic effects for heavy element molecules. From H 2 O to PoH 2 , the bonding lengths and angles are due to sp hybridization orbitals, while LvH is mainly due to p‐orbital bonding, resulting in a different configuration of bond lengths and angles than other group 16 dihydrides. The chemical bonding of group 16 dihydrides was analyzed from a single group point of view by operating the double group results to a single group.
The formation of [CuSn5Sb3]24- serves as a template for heterometallic species to evaluate the resulting aromatic properties. Our results indicate the spherical aromatic characteristics of the [CuSn5Sb3]2- building unit remain...
We have developed a new basis set family, denoted aug-cc-pVnZ-F12 (or aVnZ-F12 for short), for explicitly correlated calculations. The sets included in this family were constructed by supplementing the corresponding cc-pVnZ-F12 sets with additional diffuse functions on the higher angular momenta (i.e., additional d-h functions on non-hydrogen atoms, and p-g on hydrogen), optimized for the MP2-F12 energy of the relevant atomic anions. The new basis sets have been benchmarked against electron affinities of the first- and second-row atoms, the W4-17 dataset of total atomization energies, the S66 dataset of noncovalent interactions, the BEGDB water clusters subset, and the WATER23 subset of the GMTKN24 and GMTKN30 benchmark suites. The aVnZ-F12 basis sets displayed excellent performance, not just for electron affinities but also for noncovalent interaction energies of neutral and anionic species. Appropriate CABS (complementary auxiliary basis sets) were explored for the S66 noncovalent interactions benchmark: between similar-sized basis sets, CABS were found to be more transferable than generally assumed.
Heavy atom compounds represent a challenge for computational chemistry due to the need for simultaneous treatment of relativistic and correlation effects. Often such systems also exhibit strong correlation, which hampers the application of perturbation theory or single-reference coupled cluster (CC) methods. As a viable alternative, we have proposed externally correcting the CC method using the density matrix renormalization group (DMRG) wave functions, yielding the DMRG-tailored CC method. In a previous paper [J. Chem. Phys.2020, 152, 174107], we reported a first implementation of this method in the relativistic context, which was restricted to molecules with real double group symmetry. In this work, we present a fully general implementation of the method, covering complex and quaternion double groups as well. The 4c-TCC method thus becomes applicable to polyatomic molecules, including heavy atoms. For the assessment of the method, we performed calculations of the chiral uranium compound NUHFI, which was previously studied in the context of the enhancement of parity violation effects. In particular, we performed calculations of a cut of the potential energy surface of this molecule along the stretching of the N–U bond, where the system exhibits strong multireference character. Since there are no experimental data for NUHFI, we have performed also an analogous study of the (more symmetric) NUF3 molecule, where the vibrational frequency of the N–U bond can be compared with spectroscopic data.
The structures and some vertical excitation energies of third-row transition metal hexafluorides (MF6, M = Re, Os, Ir, Pt, Au, Hg) were calculated using the generalized-active-space configuration interaction (GASCI) theory based on the exact two-component (X2C) Hamiltonian. The spin–orbit coupling (SOC) was included at the Hartree–Fock level, enabling us to analyze the SOC at the orbital level (spinor-representation). The excitation spectra were assigned based on the double group, a relativistic group theory applicable to states with the SOC. This study provides a fundamental understanding of the ligand field splitting, including the SOC, that is useful for the photochemistry and spin chemistry involving heavy elements.
Low-valent palladium nanoparticles are efficient species promoting catalytic activity and selectivity in a number of chemical reactions. Recently, an atom-centered cuboctahedral Pd13 motif has been characterized as a ligand-protected [Pd13(Tr)6]2+...
Full-dimensional (12D) vibrational states of the methanol molecule (CH3OH) have been computed using the GENIUSH-Smolyak approach and the potential energy surface from Qu and Bowman (2013). All vibrational energies are converged better than 0.5 cm–1 with respect to the basis and grid size up to the first overtone of the CO stretch, ca. 2000 cm–1 beyond the zero-point vibrational energy. About 70 torsion-vibration states are reported and assigned. The computed vibrational energies agree with the available experimental data within less than a few cm–1 in most cases, which confirms the good accuracy of the potential energy surface. The computations are carried out using curvilinear normal coordinates with the option of path-following coefficients, which minimize the coupling of the small- and large-amplitude motions. It is important to ensure tight numerical fulfillment of the C3v(M) molecular symmetry for every geometry and coefficient set used to define the curvilinear normal coordinates along the torsional coordinate to obtain a faithful description of degeneracy in this floppy system. The reported values may provide a computational reference for fundamental spectroscopy, astrochemistry, and for the search of the proton-to-electron mass ratio variation using the methanol molecule.
In this article, in the context of the Magueijo–Smolin (MS) model and by using the Foldy-Wouthuysen transformation, we investigate the non-relativistic limit (NRL) of the Duffin–Kemmer–Petiau (DKP) equation for both zero and unity spin particles within the doubly special relativity. Our analysis yields deformed Schrödinger and Schrödinger–Pauli equations, allowing us to assess the influence of the MS model on the NRL of the DKP equation. Additionally, we examine the efficacy and behavior of the Foldy–Wouthuysen transformation in deriving the NRL in the presence of MS model.
Atomically precise gold superatoms are of interest owing to their suitable use as building blocks for cluster-assembled materials favoring ordered structures with advanced properties. In this sense, expanding their versatility...
Most nonrelativistic electron correlation methods can be adapted to account for relativistic effects, as long as the relativistic molecular spinor integrals are available, from either a four-, two-, or one-component mean-field calculation. However, relativistic multireference correlation methods remain a relatively unexplored area, with mixed evidence regarding the improvements brought by perturbative treatments. We report, for the first time, the implementation of state-averaged four-component relativistic multireference perturbation theories to second and third order based on the driven similarity renormalization group (DSRG). With our methods, named 4c-SA-DSRG-MRPT2 and 3, we find that the dynamical correlation included on top of 4c-CASSCF references can significantly improve the spin–orbit splittings in p-block elements and potential energy surfaces when compared to 4c-CASSCF and 4c-CASPT2 results. We further show that 4c-DSRG-MRPT2 and 3 are applicable to these systems over a wide range of the flow parameter, with systematic improvement from second to third order in terms of both improved error statistics and reduced sensitivity with respect to the flow parameter.
The aggregation of halide atoms into gold clusters offers an interesting scenario for the development of novel metal-based cavities for anion recognition and sensing applications. Thus, further understanding of the...
Precision physics aims to use atoms and molecules to test and develop the fundamental theory of matter, possibly beyond the Standard Model. Most of the atomic and molecular phenomena are described by the QED (quantum electrodynamics) sector of the Standard Model. Do we have the computational tools, algorithms, and practical equations for the most possible complete computation of atoms and molecules within the QED sector? What is the fundamental equation to start with? Is it still Schr\"odinger's wave equation for molecular matter, or is there anything beyond that? This paper provides a concise overview of the relativistic QED framework and recent numerical developments targeting precision physics and spectroscopy applications with common features with the robust and successful relativistic quantum chemistry methodology.
We investigate actinide covalency effects in two [AnCptt3] (An = Th, U) complexes recently studied with pulsed electron paramagnetic resonance spectroscopy, using the Hyperion package to obtain relativistic hyperfine coupling constants from relativistic multiconfigurational wave functions. ¹H and ¹³C HYSCORE simulations using the computed parameters show excellent agreement with the experimental data, highlighting the accuracy of modern relativistic ab initio methods. The extent of covalency indicated from the calculations on [ThCptt3] is in agreement with the original report based on traditional spectral fitting methods, while the covalency in [UCptt3] is found to be previously overestimated. The latter is due to the paramagnetic spin–orbit effect that arises naturally in a relativistic theory of hyperfine coupling and yet was not accounted for in the original study, thus highlighting the necessity of relativistic approaches for the interpretation of magnetic resonance data pertaining to actinides.
The implementation of an efficient self-consistent field (SCF) method including both scalar-relativistic effects and spin-orbit interaction in density functional theory (DFT) is presented. We make use of Gaussian-type orbitals and all integrals are evaluated in real space. Our implementation supports density functional approximations up to the level of meta-generalized gradient approximations for SCF energies and gradients. The latter can be used to compute the stress tensor and consequently allow us to optimize the cell structure. Considering spin-orbit interaction requires the extension of the standard procedures to a two-component formalism and a noncollinear approach for open-shell systems. Here, we implemented both the canonical and the Scalmani-Frisch noncollinear DFT formalisms, with hybrid and range-separated hybrid functionals being presently restricted to SCF energies. We demonstrate both efficiency and relevance of spin-orbit effects for the electronic structure of discrete systems and systems periodic in one to three dimensions.
In this paper, in the context of the Magueijo–Smolin (MS) model and using the Foldy–Wouthuysen transformation for relativistic spin-1/2 particles, we study the nonrelativistic limit (NRL) of the Dirac equation within the doubly special relativity. This leads to obtaining a deformed Schrödinger–Pauli equation; there, we test the influence of the MS model on the NRL of the Dirac equation. However, we also examine the efficacy and behavior of the Foldy–Wouthuysen transformation in deriving the NRL in the presence of MS model. Besides, we compute that the energies shift in the context of MS model and check if the deformed Schrödinger–Pauli equation form still shows explicitly the g-factor 2.
In this paper, we study the relationship between the Dirac–Fock model and the electron-positron Hartree–Fock model. We justify the Dirac–Fock model as a variational approximation of QED when the vacuum polarization is neglected and when the fine structure constant is small and the velocity of light c is large. As a byproduct, we also prove, when is small or c is large, the no-unfilled shells theory in the Dirac–Fock theory for atoms and molecules. The proof is based on some new properties of the Dirac–Fock model.
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