Spin-free Hamiltonian Research Articles

A systematic improvement to UGA-SSMRCCSD equations and its implication for potential energy curves.

The Unitary Group Adaptation (UGA) offers a very compact and efficient spin adaptation strategy for any spin-free Hamiltonian in a many body framework. Our use of UGA in the context of state-specific (SS) Jeziorski-Monkhorst Ansatz based multireference coupled cluster (MRCC) theory obviates the non-commutativity between the spin-free cluster operators via a normal ordered exponential parametrization in the wave operator. A previous formulation of UGA-SSMRCC by us [R. Maitra, D. Sinha, and D. Mukherjee, J. Chem. Phys. 137, 024105 (2012)], using the same ansatz, employed certain sufficiency conditions to reach the final working equations, which cannot be improved systematically. In this article, we will present a more rigorous formulation that follows from an exact factorization of the unlinked terms of the Bloch equation, resulting in equations on which a hierarchy of approximations can be systematically performed on the emergent additional terms. This derivation was shown in our recent article [D. Chakravarti, S. Sen, and D. Mukherjee, Mol. Phys. 119, e1979676 (2021)] in the context of a single open shell CC formalism and was applied to spectroscopic energy differences where the contribution of the new terms was found to be of the order of ∼0.001eV for ionization potential, electron affinity, and excitation energy. In the current work, we will present a comparison between the earlier and current formulations via both a theoretical analysis and a numerical demonstration of the dramatic effect of the additional terms brought in by the factorization on potential energy curves. The contribution of such terms was found to gain importance with an increase in the number of singly occupied active orbitals in the model space functions.

Mechanism of L2,3-edge x-ray magnetic circular dichroism intensity from quantum chemical calculations and experiment-A case study on V(IV)/V(III) complexes.

In this work, we present a combined experimental and theoretical study on the V L2,3-edge x-ray absorption (XAS) and x-ray magnetic circular dichroism (XMCD) spectra of VIVO(acac)2 and VIII(acac)3 prototype complexes. The recorded V L2,3-edge XAS and XMCD spectra are richly featured in both V L3 and L2 spectral regions. In an effort to predict and interpret the nature of the experimentally observed spectral features, a first-principles approach for the simultaneous prediction of XAS and XMCD spectra in the framework of wavefunction based ab initio methods is presented. The theory used here has previously been formulated for predicting optical absorption and MCD spectra. In the present context, it is applied to the prediction of the V L2,3-edge XAS and XMCD spectra of the VIVO(acac)2 and VIII(acac)3 complexes. In this approach, the spin-free Hamiltonian is computed on the basis of the complete active space configuration interaction (CASCI) in conjunction with second order N-electron valence state perturbation theory (NEVPT2) as well as the density functional theory (DFT)/restricted open configuration interaction with singles configuration state functions based on a ground state Kohn-Sham determinant (ROCIS/DFT). Quasi-degenerate perturbation theory is then used to treat the spin-orbit coupling (SOC) operator variationally at the many particle level. The XAS and XMCD transitions are computed between the relativistic many particle states, considering their respective Boltzmann populations. These states are obtained from the diagonalization of the SOC operator along with the spin and orbital Zeeman operators. Upon averaging over all possible magnetic field orientations, the XAS and XMCD spectra of randomly oriented samples are obtained. This approach does not rely on the validity of low-order perturbation theory and provides simultaneous access to the calculation of XMCD A, B, and C terms. The ability of the method to predict the XMCD C-term signs and provide access to the XMCD intensity mechanism is demonstrated on the basis of a generalized state coupling mechanism based on the type of the excitations dominating the relativistically corrected states. In the second step, the performance of CASCI, CASCI/NEVPT2, and ROCIS/DFT is evaluated. The very good agreement between theory and experiment has allowed us to unravel the complicated XMCD C-term mechanism on the basis of the SOC interaction between the various multiplets with spin S' = S, S ± 1. In the last step, it is shown that the commonly used spin and orbital sum rules are inadequate in interpreting the intensity mechanism of the XAS and XMCD spectra of the VIVO(acac)2 and VIII(acac)3 complexes as they breakdown when they are employed to predict their magneto-optical properties. This conclusion is expected to hold more generally.

Second-Order Perturbation Theory with Spin-Symmetry-Projected Hartree-Fock.

We propose two different schemes for second-order perturbation theory with spin-projected Hartree-Fock. Both schemes employ the same ansatz for the first-order wave function, which is a linear combination of spin-projected configurations. The first scheme is based on the normal-ordered projected Hamiltonian, which is partitioned into the Fock-like component and the remaining two-particle-like contribution. In the second scheme, the generalized Fock operator is used to construct a spin-free zeroth-order Hamiltonian. To avoid the intruder state problem, we adopt the level-shift techniques frequently used in other multireference perturbation theories. We describe both real and imaginary shift schemes and compare their performances on small systems. Our results clearly demonstrate the superiority of the second perturbation scheme with an imaginary shift over other proposed approaches in various aspects, giving accurate potential energy curves, spectroscopic constants, and singlet-triplet splitting energies. We also apply these methods to the calculation of spin gaps of transition-metal complexes as well as the potential energy curve of the chromium dimer.

Simulating Magnetic Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory in Gauge Including Atomic Orbitals.

Magnetic circular dichroism (MCD) spectra are able to provide insights into the geometric, electronic, and magnetic properties of chemical systems. However, they can be challenging to understand and simulate given the need to simultaneously treat both the finite magnetic and optical fields. Thus, efficient simulations are desired to understand the spectra and resolve the molecular electronic states. Real-time dynamics are used widely in the simulation of electronic spectroscopies such as absorption as well as electronic circular dichroism, but simulating MCD with real-time dynamics is technically and theoretically challenging. In this work, we introduce a real-time dynamics-based ab initio method with a nonperturbative treatment of a static magnetic field with London orbitals for simulating the MCD spectra of closed shell systems. Effects of a magnetic field are included variationally in the spin-free nonrelativistic Hamiltonian. Real-time time-dependent density functional theory dynamics are then performed, from which we compute the response function in the presence of the external magnetic field, giving the MCD spectrum. The method developed in this paper is applied to simulate the MCD spectra for pyrimidine, pyrazine, and 1,4-naphthoquinone. Results are discussed and compared to the experiment.

An ab Initio Linear Response Method for Computing Magnetic Circular Dichroism Spectra with Nonperturbative Treatment of Magnetic Field.

Magnetic circular dichroism (MCD) experiments provide a sensitive tool for exploring geometric, magnetic, and electronic properties of chemical complexes and condensed matter systems. They are also challenging to simulate because of the need to simultaneously treat the perturbations of a finite magnetic field as well as an optical field. In this work, we introduce an ab initio approach that treats the external magnetic field nonperturbatively with London orbitals for simulating the MCD spectra of closed-shell systems. Effects of a magnetic field are included variationally in the spin-free nonrelativistic Hamiltonian, followed by a linear response formalism to directly calculate the difference in absorption between the left and right circularly polarized light. In addition to the presentation of underlying mathematical formalism and implementation, the method developed in this paper has been applied to simulations of MCD spectra of the sodium anion, 2,2,6,6-tetramethylcyclohexanone, and 3-methyl-2-hexanone. Results are discussed and compared to experiments.

Equation-of-motion coupled-cluster theory based on the 4-component Dirac-Coulomb(-Gaunt) Hamiltonian. Energies for single electron detachment, attachment, and electronically excited states.

We report in this paper an implementation of a 4-component relativistic Hamiltonian based Equation-of-Motion Coupled-Cluster with singles and doubles (EOM-CCSD) theory for the calculation of ionization potential, electron affinity, and excitation energy. In this work, we utilize the previously developed double group symmetry-based generalized tensor contraction scheme and also extend it in order to carry out tensor contractions involving non-totally symmetric and odd-ranked tensors. Several approximated spin-free and two-component Hamiltonians can also be accessed in this implementation. We have applied this method to the halogen monoxide (XO, X = Cl, Br, I, At, Ts) species, in order to assess the quality of a few other recent EOM-CCSD implementations, where spin-orbit coupling contribution has been approximated in different degrees. Besides, we have also studied various excited states of CH2IBr, CH2I2, and (as well as single electron attachment and detachment electronic states of the same species) where comparison has been made with a closely related multi-reference coupled-cluster method, namely, Intermediate Hamiltonian Fock Space Coupled-Cluster singles and doubles theory.

Inclusion of orbital relaxation and correlation through the unitary group adapted open shell coupled cluster theory using non-relativistic and scalar relativistic Hamiltonians to study the core ionization potential of molecules containing light to medium-heavy elements.

The orbital relaxation attendant on ionization is particularly important for the core electron ionization potential (core IP) of molecules. The Unitary Group Adapted State Universal Coupled Cluster (UGA-SUMRCC) theory, recently formulated and implemented by Sen et al. [J. Chem. Phys. 137, 074104 (2012)], is very effective in capturing orbital relaxation accompanying ionization or excitation of both the core and the valence electrons [S. Sen et al., Mol. Phys. 111, 2625 (2013); A. Shee et al., J. Chem. Theory Comput. 9, 2573 (2013)] while preserving the spin-symmetry of the target states and using the neutral closed-shell spatial orbitals of the ground state. Our Ansatz invokes a normal-ordered exponential representation of spin-free cluster-operators. The orbital relaxation induced by a specific set of cluster operators in our Ansatz is good enough to eliminate the need for different sets of orbitals for the ground and the core-ionized states. We call the single configuration state function (CSF) limit of this theory the Unitary Group Adapted Open-Shell Coupled Cluster (UGA-OSCC) theory. The aim of this paper is to comprehensively explore the efficacy of our Ansatz to describe orbital relaxation, using both theoretical analysis and numerical performance. Whenever warranted, we also make appropriate comparisons with other coupled-cluster theories. A physically motivated truncation of the chains of spin-free T-operators is also made possible by the normal-ordering, and the operational resemblance to single reference coupled-cluster theory allows easy implementation. Our test case is the prediction of the 1s core IP of molecules containing a single light- to medium-heavy nucleus and thus, in addition to demonstrating the orbital relaxation, we have addressed the scalar relativistic effects on the accuracy of the IPs by using a hierarchy of spin-free Hamiltonians in conjunction with our theory. Additionally, the contribution of the spin-free component of the two-electron Gaunt term, not usually taken into consideration, has been estimated at the Self-Consistent Field (ΔSCF) level and is found to become increasingly important and eventually quite prominent for molecules with third period atoms and below. The accuracies of the IPs computed using UGA-OSCC are found to be of the same order as the Coupled Cluster Singles Doubles (ΔCCSD) values while being free from spin contamination. Since the UGA-OSCC uses a common set of orbitals for the ground state and the ion, it obviates the need of two N5 AO to MO transformation in contrast to the ΔCCSD method.

Internally Contracted Multireference Coupled Cluster Calculations with a Spin-Free Dirac-Coulomb Hamiltonian: Application to the Monoxides of Titanium, Zirconium, and Hafnium.

We combine internally contracted multireference coupled cluster theory with a four-component treatment of scalar-relativistic effects based on the spin-free Dirac-Coulomb Hamiltonian. This strategy allows for a rigorous treatment of static and dynamic correlation as well as scalar-relativistic effects, which makes it viable to describe molecules containing heavy transition elements. The use of a spin-free formalism limits the impact of the four-component treatment on the computational cost to the non-rate-determining steps of the calculations. We apply the newly developed method to the lowest singlet and triplet states of the monoxides of titanium, zirconium, and hafnium and show how the interplay between electronic correlation and relativistic effects explains the electronic structure of such molecules.

Heavy Element Effects in the Diagonal Born-Oppenheimer Correction within a Relativistic Spin-Free Hamiltonian.

Methodologies beyond the Born-Oppenheimer (BO) approximation are nowadays important to explain high precision spectroscopic measurements. Most previous evaluations of the BO correction are, however, focused on light-element molecules and based on a nonrelativistic Hamiltonian, so no information about the BO approximation (BOA) breakdown in heavy-element molecules is available. The present work is the first to investigate the BOA breakdown for the entire periodic table, by considering scalar relativistic effects in the Diagonal BO correction (DBOC). In closed shell atoms, the relativistic EDBOC scales as Z(1.25) and the nonrelativistic EDBOC scales as Z(1.17), where Z is the atomic number. Hence, we found that EDBOC becomes larger in heavy element atoms and molecules, and the relativistic EDBOC increases faster than nonrelativistic EDBOC. We have further investigated the DBOC effects on properties such as potential energy curves, spectroscopic parameters, and various energetic properties. The DBOC effects for these properties are mostly affected by the lightest atom in the molecule. Hence, in X2 or XAt molecule (X = H, Li, Na, K, Rb, and Cs) the effect of DBOC systematically decreases when X becomes heavier but in HX molecules, the effect of DBOC seems relatively similar among all the molecules.

The (E + A) × (e + a) Jahn-Teller and pseudo-Jahn-Teller Hamiltonian including spin-orbit coupling for trigonal systems.

The Hamiltonian describing E × e Jahn-Teller (JT) coupling and (E + A) × (e + a) pseudo-JT (PJT) coupling is developed beyond the standard JT theory for the example of XY3 systems, taking the bending modes of a and e symmetry into account. For the electrostatic (spin-free) Hamiltonian, the conventional Taylor expansion up to second order in symmetry-adapted displacements is replaced by an expansion in invariant polynomials up to arbitrarily high orders. The relevance of a systematic high-order expansion in the three large-amplitude bending modes is illustrated by the construction of an eighth-order three-sheeted three-dimensional ab initio potential-energy surface for PH3+. The theory of spin-orbit coupling in trigonal JT/PJT systems is extended beyond the standard model of JT theory by an expansion of the microscopic Breit-Pauli operator up to second order in symmetry-adapted vibrational coordinates. It is shown that a linear E × e JT effect of relativistic origin exists in C(3v) systems which vanishes at the planar (D(3h)) geometry. The linear relativistic 2E – 2A PJT coupling, on the other hand, persists at the planar geometry

A relativistic time-dependent density functional study of the excited states of the mercury dimer

In previous works on Zn2 and Cd2 dimers we found that the long-range corrected CAMB3LYP gives better results than other density functional approximations for the excited states, especially in the asymptotic region. In this paper, we use it to present a time-dependent density functional (TDDFT) study for the ground-state as well as the excited states corresponding to the (6s(2) + 6s6p), (6s(2) + 6s7s), and (6s(2) + 6s7p) atomic asymptotes for the mercury dimer Hg2. We analyze its spectrum obtained from all-electron calculations performed with the relativistic Dirac-Coulomb and relativistic spinfree Hamiltonian as implemented in DIRAC-PACKAGE. A comparison with the literature is given as far as available. Our result is excellent for the most of the lower excited states and very encouraging for the higher excited states, it shows generally good agreements with experimental results and outperforms other theoretical results. This enables us to give a detailed analysis of the spectrum of the Hg2 including a comparative analysis with the lighter dimers of the group 12, Cd2, and Zn2, especially for the relativistic effects, the spin-orbit interaction, and the performance of CAMB3LYP and is enlightened for similar systems. The result shows, as expected, that spinfree Hamiltonian is less efficient than Dirac-Coulomb Hamiltonian for systems containing heavy elements such as Hg2.

Jahn-Teller theory beyond the standard model

In the standard model of Jahn-Teller (JT) theory, the spin-free electronic Hamiltonian is expanded up to second order in normal-mode displacements, while the spin-orbit (SO) coupling operator is approximated in zeroth order. In this article, the systematic extension of JT theory beyond the standard model is outlined. For the classic E × e and T2 × t2 JT problems in C3v and Td symmetry, respectively, it is shown how the conventional Taylor expansion of the spin-free Hamiltonian can be replaced by an expansion in invariant polynomials up to arbitrarily high orders. The theory of SO coupling in JT systems is extended beyond the standard model by an expansion of the Breit-Pauli SO-coupling operator up to first order in normal-mode displacements. The spectroscopic relevance of the higher-order electrostatic and the linear relativistic JT coupling terms is illustrated for the vibronic structures of the photoelectron spectra of tetrahedral P4 and Sb4.

Relativistic diffusion Monte Carlo method: Zeroth-order regular approximation-diffusion Monte Carlo method in a spin-free formalism

A diffusion Monte Carlo (DMC) method for the relativistic zeroth-order regular approximation (ZORA) is proposed. In this scheme, a novel approximate Green's function is derived for the spin-free ZORA Hamiltonian. Several numerical tests on atoms and small molecules showed that by combining with the relativistic cusp-correction scheme, the present approach can include both relativistic and electron-correlation effects simultaneously. The correlation energies recovered by the ZORA-DMC method are comparable with the nonrelativistic DMC results and superior to the coupled cluster singles and doubles with perturbative triples correction results when the correlation-consistent polarized valence triple-zeta Douglas-Kroll basis set is used. For the heavier CuH molecule, the ZORA-DMC estimation of its dissociation energy agrees with the experimental value within the error bar.

Relativistic Time-Dependent Density Functional Theory and Excited States Calculations for the Zinc Dimer

I present a time-dependent density functional study of the 20 low-lying excited states as well the ground states of the zinc dimer Zn2, analyze its spectrum obtained from all electrons calculations performed using time-depended density functional with a relativistic 4-component and relativistic spin-free Hamiltonian as implemented in Dirac-Package, and show a comparison of the results obtained from different well-known and newly developed density functional approximations, a comparison with the literature and experimental values as far as available. The results are very encouraging, especially for the lowest excited states of this dimer. However, the results show that long-range corrected functionals such as CAMB3LYP gives the correct asymptotic behavior for the higher states, and for which the best result is obtained. A comparable result is obtained from PBE0 functional. Spin-free Hamiltonian is shown to be very efficient for relativistic systems such as Zn2.

Electrically polarized valence basis sets for the SBKJC effective core potential developed for calculations of dynamic polarizabilities and Raman intensities

Sadlej’s electric polarization method of Gaussian basis functions was applied to the double-zeta effective core potential basis sets of Stevens, Basch, Krauss, Jasien and Cundari to generate a new augmented polarized valence double-zeta set, named as pSBKJC, which is appropriate for the calculation of dynamic polarizabilities and Raman intensities. The pSBKJC basis set was developed for the atoms of families 14–17 (from C to F, Si to Cl, Ge to Br and Sn to I). In order to assess the performance of this new basis set, these properties were compared to those evaluated using Sadlej’s set, available in the EMSL online library under the name of Sadlej-pVTZ. In these tests, Hartree-Fock/pSBKJC calculations have proved to be less demanding of the computer than the Hartree-Fock/Sadlej-pVTZ ones but give results in excellent agreement with those from the Sadlej-pVTZ basis set. Since the Stevens et al. pseudopotential can represent the scalar relativistic effects, the results obtained at the Hartree-Fock/pSBKJC level show a better agreement with the results of Dirac-Hartree-Fock/Sadlej-pVTZ relativistic calculations using Dyall’s spin-free Hamiltonian. When comparing Hartree-Fock/pSBKJC data of Raman scattering activities, at the excitation wavelength of 488 nm, with those of spin-free Dirac-Hartree-Fock/Sadlej-pVTZ calculations, a very good agreement is observed, where the RMS error is 8.5 A4a.m.u.−1 and the averaged percentage error is 3.4%. In terms of computer savings in calculations of dynamic Raman intensities, a 20% reduction in the CPU time in the coupled cluster singles and doubles intensities of C6H6 and about 40% reduction in the time-dependent Hartree-Fock intensities for C6F6 molecules were attained.

Analytical evaluation of first-order electrical properties based on the spin-free Dirac-Coulomb Hamiltonian

We report an analytical scheme for the calculation of first-order electrical properties using the spin-free Dirac-Coulomb (SFDC) Hamiltonian, thereby exploiting the well-developed density-matrix formulations in nonrelativistic coupled-cluster (CC) derivative theory. Orbital relaxation effects are fully accounted for by including the relaxation of the correlated orbitals with respect to orbitals of all types, viz., frozen-core, occupied, virtual, and negative energy state orbitals. To demonstrate the applicability of the presented scheme, we report benchmark calculations for first-order electrical properties of the hydrogen halides, HX with X = F, Cl, Br, I, At, and a first application to the iodo(fluoro)methanes, CH(n)F(3 - n)I, n = 0-3. The results obtained from the SFDC calculations are compared to those from nonrelativistic calculations, those obtained via leading-order direct perturbation theory as well as those from full Dirac-Coulomb calculations. It is shown that the full inclusion of spin-free (SF) relativistic effects is necessary to obtain accurate first-order electrical properties in the presence of fifth-row elements. The SFDC scheme is also recommended for applications to systems containing lighter elements because it introduces no extra cost in the rate-determining steps of a CC calculation in comparison to the nonrelativistic case. On the other hand, spin-orbit contributions are generally small for first-order electrical properties of closed-shell molecules and may be handled efficiently by means of perturbation theory.

Gradients for configuration interaction energies with spin‐orbit coupling in a semiempirical framework

We present a method for the analytical evaluation of the molecular energy gradient for a semiempirical configuration interaction (CI) wavefunction, taking into account the spin-orbit coupling. We show how to proceed in the simplest case where all the wavefunctions belonging to the CI subspace considered are mixed by the spin-orbit interaction, and we develop an original procedure for the more complex case where only a limited number of CI eigenvectors of the spin-free Hamiltonian are mixed.

Magnetism and spin-free quantum chemistry

It is shown that for a system whose fine and hyperfine structure can be neglected, its magnetic properties are determined solely by the spectrum of its spin-free Hamiltonian.

First-principles calculations of magnetic circular dichroism spectra

An elaborate approach for the prediction of magnetic circular dichroism (MCD) spectra in the framework of highly correlated multiconfigurational ab initio methods is presented. The MCD transitions are computed by the explicit treatment of spin-orbit coupled (SOC) and spin-spin coupled (SSC) N-electron states. These states are obtained from the diagonalization of the SOC and SSC operators along with the spin and orbital Zeeman operators in the basis of a preselected number of roots of the spin-free Hamiltonian. Therefore, zero-field splittings due to the SOC and SSC interactions along with the magnetic field splittings are explicitly accounted for in the ground as well as the excited states. This makes it possible to calculate simultaneously all MCD A, B, and C terms even beyond the linear response limit. The SOC is computed using a multicenter mean-field approximation to the Breit-Pauli Hamiltonian. Two-electron SSC terms are included in the treatment without further approximations. The MCD transition intensities are subjected to numerical orientational averaging in order to treat the most commonly encountered case of randomly oriented molecules. The simulated MCD spectra for the OH, NH, and CH radicals as well as for [Fe(CN)(6)](3-) are in good agreement with the experimental spectra. In the former case, the significant effects of the inert gas matrices in which the experimental spectra were obtained were modeled in a phenomenological way.

A Diabatic Representation Including Both Valence Nonadiabatic Interactions and Spin−Orbit Effects for Reaction Dynamics

A diabatic representation is convenient in the study of electronically nonadiabatic chemical reactions because the diabatic energies and couplings are smooth functions of the nuclear coordinates and the couplings are scalar quantities. A method called the fourfold way was devised in our group to generate diabatic representations for spin-free electronic states. One drawback of diabatic states computed from the spin-free Hamiltonian, called a valence diabatic representation, for systems in which spin-orbit coupling cannot be ignored is that the couplings between the states are not zero in asymptotic regions, leading to difficulties in the calculation of reaction probabilities and other properties by semiclassical dynamics methods. Here we report an extension of the fourfold way to construct diabatic representations suitable for spin-coupled systems. In this article we formulate the method for the case of even-electron systems that yield pairs of fragments with doublet spin multiplicity. For this type of system, we introduce the further simplification of calculating the triplet diabatic energies in terms of the singlet diabatic energies via Slater's rules and assuming constant ratios of Coulomb to exchange integrals. Furthermore, the valence diabatic couplings in the triplet manifold are taken equal to the singlet ones. An important feature of the method is the introduction of scaling functions, as they allow one to deal with multibond reactions without having to include high-energy diabatic states. The global transformation matrix to the new diabatic representation, called the spin-valence diabatic representation, is constructed as the product of channel-specific transformation matrices, each one taken as the product of an asymptotic transformation matrix and a scaling function that depends on ratios of the spin-orbit splitting and the valence splittings. Thus the underlying basis functions are recoupled into suitable diabatic basis functions in a manner that provides a multibond generalization of the switch between Hund's cases in diatomic spectroscopy. The spin-orbit matrix elements in this representation are taken equal to their atomic values times a scaling function that depends on the internuclear distances. The spin-valence diabatic potential energy matrix is suitable for semiclassical dynamics simulations. Diagonalization of this matrix produces the spin-coupled adiabatic energies. For the sake of illustration, diabatic potential energy matrices are constructed along bond-fission coordinates for the HBr and the BrCH(2)Cl molecules. Comparison of the spin-coupled adiabatic energies obtained from the spin-valence diabatics with those obtained by ab initio calculations with geometry-dependent spin-orbit matrix elements shows that the new method is sufficiently accurate for practical purposes. The method formulated here should be most useful for systems with a large number of atoms, especially heavy atoms, and/or a large number of spin-coupled electronic states.