Minimax principle for the Dirac equation

Toward Accurate Spin-Orbit Splittings from Relativistic Multireference Electronic Structure Theory

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.

Relativistic calculations of magnetic resonance parameters: background and some recent developments

This article outlines some basic concepts of relativistic quantum chemistry and recent developments of relativistic methods for the calculation of the molecular properties that define the basic parameters of magnetic resonance spectroscopic techniques, i.e. nuclear magnetic resonance shielding, indirect nuclear spin-spin coupling and electric field gradients (nuclear quadrupole coupling), as well as with electron paramagnetic resonance g-factors and electron-nucleus hyperfine coupling. Density functional theory (DFT) has been very successful in molecular property calculations, despite a number of problems related to approximations in the functionals. In particular, for heavy-element systems, the large electron count and the need for a relativistic treatment often render the application of correlated wave function ab initio methods impracticable. Selected applications of DFT in relativistic calculation of magnetic resonance parameters are reviewed.

Gaussian-type function set without prolapse for the Dirac-Fock-Roothaan equation (II): 80Hg through 103Lr

We present prolapse-free universal Gaussian-type basis sets for 80Hg through 103Lr. The basis set is determined so that the Dirac-Fock-Roothaan total energy should decrease monotonically toward the numerical Dirac-Fock total energy. The difference between the Dirac-Fock-Roothaan total energy and the numerical Dirac-Fock total energy is less than 3 x 10(-6) hartree for 1H through 102No, and less than 5 x 10(-6) hartree for 103Lr. The exponents of the present sets are determined in an even-tempered manner, aiming to give total energy closer to the numerical Dirac-Fock value as the expansion term increases. The recommended set is expanded by (64, 64, 64, 46, 46, 46, 46) terms for (s+, p-, p+, d-, d+, f-, f+) symmetries, respectively. A practical set with (56, 48, 48, 36, 36, 36, 36) terms is also presented.

Accurate relativistic adapted Gaussian basis sets for Cesium through Radon without variational prolapse and to be used with both uniform sphere and Gaussian nucleus models

Accurate relativistic adapted Gaussian basis sets (RAGBSs) from Cs (Z = 55) through Rn (Z = 86) without variational prolapse were developed by using the polynomial version of the Generator Coordinate Dirac-Fock method. The RAGBSs presented here can be used with any of two popular finite nucleus models, the uniform sphere and the Gaussian models. The largest RAGBS error is 4.5 mHartree for Radon with a size of 30s27p17d11f.

Gaussian-type function set without prolapse 1H through 83Bi for the Dirac-Fock-Roothaan equation

We have developed prolapse free Gaussian basis sets which can be used for 1H to 83Bi, imposing the condition that the Dirac-Fock-Roothaan (DFR) total energy (TE) decreases monotonically toward the numerical DF (NDF) TE as the expansion term increases. An even-tempered basis set was assumed. The resulting sets gave |TE(DFR) - TE(NDF)| < or = 1 x 10(-6) hartree for any atoms less or equal to 83Bi; TE(NDF) = -21 565.638 345, and TE(DFR) = -21,565.638 345 +/- 0.000 001 hartree for Bi when the expansion terms are in the range (58, 58, 58, 36, 36, 36, and 36) and (72, 72, 72, 36, 36, 36, and 36) for (s+, p-, p+, d-, d+, f-, and f+) symmetries, respectively. A practical set with 44, 44, 44, 36, 36, 32, and 32 for the respective symmetries is also proposed where |TE(DFR) - TE(NDF)| < or = 4 x 10(-5).

Gaussian-type function set without prolapse for the Dirac-Fock-Roothaan equation

A Gaussian-type function (GTF) set without a prolapse (variation collapse) is generated for the Dirac-Fock-Roothaan (DFR) equation. The test atom was mercury. The number of primitive GTFs used is between 7 and 62 (abbreviated as 7-62), 6-62, 6-62, 4-36, 4-36, 3-36, and 3-36 for s(+), p(-), p(+), d(-), d(+), f(-), and f(+) symmetries. The respective exponent parameters were determined with even-tempered manner, which requires the minimum and maximum exponents for the respective symmetries. We prepared several sets of these. The total energy (TE) given by the numerical DF (NDF) is -19648.849250 hartree; one of the present sets with largest number of expansion terms gave -19648.849251 hartree. The error (deltaTE) relative to the NDR TE is quite small. We then applied this set to the inert gas atoms Ne (10), Ar (18), Kr (36), Xe (54), Rn (86), and No (102), and also to Es (99) as the representative of the open shell atoms. The absolute values of deltaTE were at most 2.8 x 10(-6) hartree, showing the potential of this set as a universal set.

Four-component relativistic Kohn-Sham theory

A four-component relativistic implementation of Kohn-Sham theory for molecular systems is presented. The implementation is based on a nonredundant exponential parametrization of the Kohn-Sham energy, well suited to studies of molecular static and dynamic properties as well as of total electronic energies. Calculations are presented of the bond lengths and the harmonic and anharmonic vibrational frequencies of Au(2), Hg(2+)(2), HgAu(+), HgPt, and AuH. All calculations are based on the full four-component Dirac-Coulomb Hamiltonian, employing nonrelativistic local, gradient-corrected, and hybrid density functionals. The relevance of the Coulomb and Breit operators for the construction of relativistic functionals is discussed; it is argued that, at the relativistic level of density-functional theory and in the absence of a vector potential, the neglect of current functionals follows from the neglect of the Breit operator.

The Dirac equation in quantum chemistry: strategies to overcome the current computational problems

A perspective on the use of the relativistic Dirac equation in quantum chemistry is given. It is demonstrated that many of the computational problems that plague the current implementations of the different electronic structure methods can be overcome by utilizing the locality of the small component wave function and density. Possible applications of such new and more efficient formulations are discussed.

Minimization methods for the one-particle dirac equation

Taking into account relativistic effects in quantum chemistry is crucial for accurate computations involving heavy atoms. Standard numerical methods can deal with the problem of variational collapse and the appearance of spurious roots only in special cases. The goal of this Letter is to provide a general and robust method to compute particle bound states of the Dirac equation.