Exact decoupling of the Dirac Hamiltonian. I. General theory

An efficient implementation of two-component relativistic exact-decoupling methods for large molecules

We present an efficient algorithm for one- and two-component relativistic exact-decoupling calculations. Spin-orbit coupling is thus taken into account for the evaluation of relativistically transformed (one-electron) Hamiltonian. As the relativistic decoupling transformation has to be evaluated with primitive functions, the construction of the relativistic one-electron Hamiltonian becomes the bottleneck of the whole calculation for large molecules. For the established exact-decoupling protocols, a minimal matrix operation count is established and discussed in detail. Furthermore, we apply our recently developed local DLU scheme [D. Peng and M. Reiher, J. Chem. Phys. 136, 244108 (2012)] to accelerate this step. With our new implementation two-component relativistic density functional calculations can be performed invoking the resolution-of-identity density-fitting approximation and (Abelian as well as non-Abelian) point group symmetry to accelerate both the exact-decoupling and the two-electron part. The capability of our implementation is illustrated at the example of silver clusters with up to 309 atoms, for which the cohesive energy is calculated and extrapolated to the bulk.

Spin-orbit coupling splitting in the X(2)Π, A(2)Δ, B(2)Σ(-), C(2)Σ(+), D(2)Σ(+), F(2)Π and a(4)Σ(-) Λ-S states of SiH radical

The potential energy curves (PECs) of eleven Ω states generated from the seven Λ-S bound states of SiH radical are studied in detail using an ab initio quantum chemical method. The PECs are calculated for internuclear separations from about 0.10 to 1.00nm by the CASSCF method, which is followed by the internally contracted MRCI approach with the Davidson modification. The spin-orbit coupling is accounted for with the Breit-Pauli Hamiltonian. Core-valence correlation corrections are included with an aug-cc-pCVTZ basis set. Scalar relativistic correction calculations are made by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. To discuss the effect of core-electron correlations on the spectroscopic parameters, in particular on the Te, the all-electron basis set, aug-cc-pCVTZ with and without core-electron correlations, is employed to calculate the spin-orbit coupling splitting in the X(2)Π Λ-S state. The all-electron aug-cc-pCVTZ basis set with core-electron correlations is chosen for the present spin-orbit coupling studies. The convergent behavior of present calculations is discussed with respect to the basis set and level of theory. All the PECs are extrapolated to the complete basis set limit. The spectroscopic parameters of seven Λ-S and eleven Ω bound states are evaluated. The splitting energy of X(2)Π Λ-S state is determined as 141.12cm(-1), which agrees well with the recent measurements of 142.896cm(-1). Moreover, other spectroscopic parameters are also in excellent agreement with available measurements. It demonstrates that the spectroscopic parameters of eleven Ω bound states reported here can be expected to be reliable predicted ones.

Accurate theoretical study on 18 Λ-S and 50 Ω states of CS in the gas phase: potential energy curves, spectroscopic parameters, and spin-orbit coupling

The potential energy curves (PECs) of 50 Ω states generated from the 18 Λ-S states are studied for the first time using the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification. All the 18 Λ-S states correlate to the first dissociation limit, C((3)Pg) + S((3)Pg), of CS molecule, of which only the 2(5)Π is repulsive and the A(1)Π, A(') (1)Σ(+), and 2(3)Σ(+) possess double wells. The spin-orbit (SO) coupling is accounted for by the state interaction approach with the Breit-Pauli Hamiltonian. Core-valence correlation correction is taken into account with an all-electron cc-pCV5Z basis set. Scalar relativistic correction calculations are made by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pVTZ basis set. All the PECs are extrapolated to the complete basis set limit. The a(') (3)Σ(+), e(3)Σ(-), 1(5)Σ(+), 1(5)Π, and d(3)Δ are found to be the inverted Λ-S states with the SO coupling included. The spectroscopic parameters of 17 Λ-S and 41 Ω bound states are evaluated. The comparisons between the present results and available measurements are performed, and excellent agreement has been found. It shows that the spectroscopic results reported here can be expected to be reliable predicted ones.

Theoretical study on ten Λ-S states of Si2(-) anion: potential energy curves, spectroscopy and spin-orbit couplings

The potential energy curves (PECs) of seventeen Ω states generated from the ten Λ-S states of the Si2(-) anion are studied in detail using an ab initio quantum chemical method for the first time. The PECs are calculated for internuclear separations from 0.10 to 1.20 nm by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification. The spin-orbit coupling is accounted for by the Breit-Pauli Hamiltonian. Core-valence correlation and scalar relativistic corrections are considered. Core-valence correlation corrections are included using an aug-cc-pCVTZ basis set. Scalar relativistic correction calculations are made with the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. Obvious effect of core-valence correlation corrections on the PECs is observed, in particular for the two lowest (2)Πu and (2)Σg(+) Λ-S states. All the PECs are extrapolated to the complete basis set limit. The lowest (2)Πu Λ-S state is found to be the ground state of Si2(-) anion. The convergence observations of present calculations are made and the convergent behavior is discussed with respect to the basis set and level of theory. The effects of core-electron correlations on the energy splitting are studied by the all-electron aug-cc-pCVTZ basis set. Using these PECs, the spectroscopic parameters of Λ-S and Ω states involved are determined. The vibrational manifolds are evaluated for each Λ-S and Ω state of non-rotation Si2(-) anion. It shows that the spectroscopic parameters and molecular constants of ten Λ-S and seventeen Ω states reported here can be expected to be reliable predicted ones.

Is near-"spectroscopic accuracy" possible for heavy atoms and coupled cluster theory? An investigation of the first ionization potentials of the atoms Ga-Kr

Recent developments in ab initio coupled cluster (CC) theory and correlation consistent basis sets have ushered in an era of unprecedented accuracy when studying the spectroscopy and thermodynamics of molecules containing main group elements. These same developments have recently seen application to heavier inorganic or transition metal-containing species. The present work benchmarks conventional single reference coupled cluster theory (up to full configuration interaction for valence electron correlation and coupled cluster with up to full pentuple excitations (CCSDTQP) for core-valence correlation) and explicitly correlated coupled cluster methods [CC with single, double, and perturbative triple substitutions (CCSD(T)-F12)] for the atomic ionization potentials of the six 4p elements (Ga-Kr), a property with experimental error bars no greater than a few cm(-1). When second-order spin orbit coupling effects are included, a composite methodology based on CCSD(T) calculations yielded a mean signed error of just -0.039 kcal mol(-1) and a mean unsigned error of 0.043 kcal mol(-1). Inclusion of post-CCSD(T) correlation corrections reduced both of these values to -0.008 kcal mol(-1) and 0.025 kcal mol(-1), respectively, with the latter corresponding to an average error of just 9 cm(-1). The maximum signed error in the latter scheme was just -0.043 kcal mol(-1) (15 cm(-1)).

Accurate ab initio study on the A2Π, 1(4)Σ+, 1(4)Π, 2(4)Π and 1(6)Σ+ electronic states of AlO radical including spin-orbit coupling

The potential energy curves (PECs) of 15 Ω states generated from five Λ-S states (A2Π, 1(4)Σ+, 1(4)Π, 2(4)Π and 1(6)Σ+) of AlO radical are studied in detail using high level ab initio quantum chemical method for the first time. All the PEC calculations are made by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification (MRCI+Q). The spin-orbit coupling effect is included by the Breit-Pauli Hamiltonian with the aug-cc-pCVTZ basis set. Convergent behavior is discussed and excellent convergence has been observed with respect to the basis sets and level of theory. To improve the quality of PECs, core-valence correlation and scalar relativistic corrections are taken into account. Core-valence correlation corrections are included employing a cc-pCVQZ basis set. Scalar relativistic corrections are calculated by the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. All the PECs are extrapolated to the complete basis set limit by the total-energy extrapolation scheme. With these PECs including all the corrections used here, on the one hand, the spectroscopic parameters of all the Λ-S and Ω states are calculated, which are in reasonable agreement with the experimental and other theoretical results; on the other hand, the vibrational levels and inertial rotation constants of X2Σ+, A2Π, B2Σ+ Λ-S states as well as A2Π3/2 and A2Π1/2 Ω states are determined, which also agree well with the measurements. The vibrational levels and inertial rotation constants of A(2)Π3/2 and A2Π1/2 Ω states as well as the spectroscopic parameters of four Λ-S states (1(4)Σ+, 1(4)Π, 2(4)Π and 1(6)Σ+) and their corresponding 13 Ω states can be expected to be reliable predicted ones.

Spectroscopic and thermochemical properties of the c-C6H7 radical: a high-level theoretical study

The electronic ground state (X(2)B(1)) of the cyclohexadienyl radical (c-C(6)H(7)) has been studied by explicitly correlated coupled cluster theory at the RCCSD(T)-F12x (x = a, b) level, partly in combination with the double-hybrid density functional method B2PLYP. An accurate equilibrium structure has been established and the ground-state rotational constants are predicted to be A(0) = 5347.3 MHz, B(0) = 5249.7 MHz, and C(0) = 2692.5 MHz. The calculated vibrational wavenumbers agree well with the recent p-H(2) matrix IR data [M. Bahou, Y.-J. Wu, and Y.-P. Lee, J. Chem. Phys. 136, 154304 (2012)] and several predictions have been made. A low value of 6.803 ± 0.005 eV is predicted for the adiabatic ionization energy of c-C(6)H(7). Owing to a moderately large change in the equilibrium structure upon ionization, the first band of the photoelectron spectrum is dominated by the adiabatic peak (100%) and only the peaks corresponding to excitation of the two lowest totally symmetric vibrations (ν(12) and ν(11)) by one vibrational quantum have relative intensities of more than 15%. The C(6)H(6)-H dissociation energy is calculated to be D(0) = 85.7 kJ mol(-1), with an estimated error of ~2 kJ mol(-1).

MRCI study on electronic spectrum of 13 electronic states of SiP molecule

The potential energy curves (PECs) of the X(2)Π, A(2)Σ(+), a(4)Σ(+), B(2)Π, c(4)Δ, C(2)Σ(+), d(4)Σ(-), D(2)Φ, E(2)Σ(-), G(2)Δ, H(2)Π, I(2)Σ(+) and f(4)Δ electronic states of the SiP molecule are calculated employing an ab initio quantum chemical method. The PEC calculations are performed for internuclear separations from 0.10 to 1.10nm using the complete active space self-consistent field (CASSCF) method, which is followed by the internally contracted multireference configuration interaction (MRCI) approach in combination with a correlation-consistent aug-cc-pV6Z basis set. To improve the quality of the PECs, core-valence correlation and scalar relativistic corrections are included. Scalar relativistic correction calculations are carried out using the third-order Douglas-Kroll Hamiltonian approximation at the level of a cc-pV5Z basis set. Core-valence correlation corrections are included using a cc-pCVQZ basis set. The PECs obtained by the MRCI calculations are corrected for size-extensivity errors by means of the Davidson modification. The PECs are extrapolated to the complete basis set limit. The spectroscopic parameters are obtained by fitting the vibrational levels, which are calculated by solving the ro-vibrational Schrödinger equation. The spectroscopic results are compared in detail with those reported in previous literature. Excellent agreement is found between the present spectroscopic results and the experimental ones. Using the Breit-Pauli operator, the spin-orbit (SO) coupling effect on the spectroscopic parameters is included in the X(2)Π, D(2)Φ and H(2)Π electronic states at the level of a cc-pCVTZ basis set. The energy separation of the X(2)Π and A(2)Σ(+) electronic states is accurately determined by including the Davidson modification, SO coupling and core-valence correlation and scalar relativistic corrections. Using the PECs determined by the MRCI+Q/CV+DK+56 calculations, the G(υ), B(υ) and D(υ) are calculated for each vibrational state of each electronic state, and those of the first 20 vibrational states are reported for each electronic state of the non-rotation (29)Si(31)P molecule. Comparison with the measurements demonstrates that the present results are accurate. The spectroscopic parameters of the a(4)Σ(+), B(2)Π, c(4)Δ, d(4)Σ(-), D(2)Φ, E(2)Σ(-), G(2)Δ, H(2)Π, I(2)Σ(+) and f(4)Δ electronic states and the G(υ), B(υ) and D(υ) of all the electronic states obtained here are expected to be reliable predicted results.

A theoretical revisit to the lowest ionization potentials in metal-metal bonds

The ties that bind: The electrostatic potential of quadruply bonded W(2)(hpp)(4) and its cation support the idea that the very low ionization energies are due to excitations from the metal-metal bonding core. Theoretical MRCI computations based on CASSCF orbitals show that the relativistic effect influences the lowest four ionization energies, leading to distinct IE sequences: IE(δ) < IE(π) < IE(σ) in W(2)(hpp)(4), and IE(δ) < IE(σ) < IE(π) in Mo(2)(hpp)(4).

Investigation of spectroscopic properties and spin-orbit splitting in the XΠ² and AΠ² electronic states of the SO ⁺cation

The potential energy curves (PECs) of the X(2)Π and A(2)Π electronic states of the SO(+) ion are calculated using the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction (MRCI) approach for internuclear separations from 0.08 to 1.06 nm. The spin-orbit coupling effect on the spectroscopic parameters is included using the Breit-Pauli operator. To improve the quality of PECs and spin-orbit coupling constant (A(0)), core-valence correlation and scalar relativistic corrections are included. To obtain more reliable results, the PECs obtained by the MRCI calculations are corrected for size-extensivity errors by means of the Davidson modification (MRCI+Q). At the MRCI+Q/aug-cc-pV5Z+CV+DK level, the A(0) values of the SO(+)(X(2)Π(1/2, 3/2)) and SO(+)(A(2)Π(1/2, 3/2)) are 362.13 and 58.16 cm(-1) when the aug-cc-pCVTZ basis set is used to calculate the spin-orbit coupling splitting, and the A(0) of the SO(+)(X(2)Π(1/2, 3/2)) and SO(+)(A(2)Π(1/2, 3/2)) are 344.36 and 52.90 cm(-1) when the aug-cc-pVTZ basis set is used to calculate the spin-orbit coupling splitting. The conclusion is drawn that the core-valence correlations correction makes the A(0) slightly larger. The spectroscopic results are obtained and compared with those reported in the literature. Excellent agreement exists between the present results and the measurements. The vibrational manifolds are calculated, and those of the first 30 vibrational states are reported for the J = 0 case. Comparison with the measurements shows that the present vibrational manifolds are both reliable and accurate.

Electronic spectrum of 17 electronic states of BN molecule: a theoretical study

The potential energy curves (PECs) of the X(3)Π, a(1)Σ(+), b(1)Π, A(3)Σ(+), B(3)Σ(-), c(1)Δ, D(3)Π, 1(5)Π, 3(1)Σ(+), 3(3)Π, 2(1)Π, 2(3)Σ(+), 1(3)Δ, 1(5)Σ(+), 4(3)Π, 2(3)Σ(-) and 1(5)Σ(-) electronic states of the BN molecule are calculated using an ab initio quantum chemical method. The PEC calculations have been made for internuclear separations from 0.06 to 1.20 nm using the complete active space self-consistent field (CASSCF) method, which is followed by the valence internally contracted multireference configuration interaction (MRCI) approach in combination with a correlation-consistent aug-cc-pV5Z basis set. To improve the quality of PECs, core-valence correlation and relativistic corrections are included. Relativistic correction calculations are carried out using the third-order Douglas-Kroll Hamiltonian (DKH3) approximation. Core-valence correlation corrections are included using a cc-pCVQZ basis set. Relativistic corrections are calculated at the level of a cc-pVQZ basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are corrected for size-extensivity errors by means of the Davidson modification (MRCI+Q). These PECs are extrapolated to the complete basis set limit by the total-energy extrapolation scheme. The spectroscopic parameters are determined by fitting the vibrational levels, which are calculated in a direct forward manner from the analytic potential by solving the ro-vibrational Schrödinger equation using Numerov's method. The spectroscopic results have been compared in detail with those reported in the literature. Excellent agreement has been found between the present spectroscopic results and the experimental ones. Using the Breit-Pauli operator, the spin-orbit coupling effect on the spectroscopic parameters is included in the X(3)Π and D(3)Π electronic states. The vibrational level, inertial rotation and centrifugal distortion constants are calculated for each vibrational state of each electronic state. And those of the first 20 vibrational states of each electronic state are reported when the rotational quantum number equals zero. Comparison with the measurements shows that the present results are accurate. The spectroscopic parameters of the c(1)Δ, 1(5)Π, 3(1)Σ(+), 3(3)Π, 2(1)Π, 2(3)Σ(+), 1(3)Δ, 1(5)Σ(+), 4(3)Π and 1(5)Σ(-) electronic states and the vibrational manifolds of all the electronic states obtained here are expected to be reliable predicted results.

A 4D wave packet study of the CH3I photodissociation in the A-band. Comparison with femtosecond velocity map imaging experiments

The time-resolved photodissociation dynamics of CH(3)I in the A-band has been studied theoretically using a wave packet model including four degrees of freedom, namely the C-I dissociation coordinate, the I-CH(3) bending mode, the CH(3) umbrella mode, and the C-H symmetric stretch mode. Clocking times and final product state distributions of the different dissociation (nonadiabatic) channels yielding spin-orbit ground and excited states of the I fragment and vibrationless and vibrationally excited (symmetric stretch ν(1) and umbrella ν(2) modes) CH(3) fragments have been obtained and compared with the results of femtosecond velocity map imaging experiments. The wave packet calculations are able to reproduce with very good agreement the experimental reaction times for the CH(3)(ν(1), ν(2))+I*((2)P(1/2)) dissociation channels with ν(1) = 0 and ν(2) = 0,1,2, and also for the channel CH(3)(ν(1) = 0, ν(2) = 0)+I((2)P(3/2)). However, the model fails to predict the experimental clocking times for the CH(3)(ν(1), ν(2))+I((2)P(3/2)) channels with (ν(1), ν(2)) = (0, 1), (0, 2), and (1, 0), that is, when the CH(3) fragment produced along with spin-orbit ground state I atoms is vibrationally excited. These results are similar to those previously obtained with a three-dimensional wave packet model, whose validity is discussed in the light of the results of the four-dimensional treatment. Possible explanations for the disagreements found between theory and experiment are also discussed.

MRCI study on the spectroscopic parameters and molecular constants of the X1Σ+, a3Σ+, A1Π and C1Σ- electronic states of the SiO molecule

The potential energy curves (PECs) of the X(1)Σ(+), a(3)Σ(+), A(1)Π and C(1)Σ(-) electronic states of the SiO molecule are studied using an ab initio quantum chemical method. The calculations have been made employing the complete active space self-consistent field (CASSCF) method, which is followed by the valence internally contracted multireference configuration interaction (MRCI) approach in combination with several correlation-consistent basis sets. The effect on the PECs by the core-valence correlation and relativistic corrections is included. The way to consider the relativistic correction is to use the third-order Douglas-Kroll Hamiltonian approximation. The core-valence correlation correction is carried out with the cc-pCVQZ basis set, and the relativistic correction is performed at the level of the cc-pVQZ basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are also corrected for size-extensivity errors by means of the Davidson modification (MRCI+Q). The PECs of these electronic states are extrapolated to the complete basis set limit by the total-energy extrapolation scheme. Employing these PECs, the spectroscopic parameters are calculated and compared with those reported in the literature. With these PECs determined by the MRCI+Q/CV+DK+56 calculations, by solving the radial Schrödinger equation of nuclear motion, 110 vibrational states for the X(1)Σ(+), 69 for the a(3)Σ(+), 54 for the A(1)Π and 67 for the C(1)Σ(-) electronic state are predicted when the rotational quantum number J equals zero. The vibrational manifolds of the first 20 vibrational states are reported and compared with the available RKR data for each electronic state. On the whole, as expected, the most accurate spectroscopic parameters and molecular constants of the SiO molecule are obtained by the MRCI+Q/CV+DK+56 calculations. And the present molecular constants of the a(3)Σ(+), C(1)Σ(-) and A(1)Π electronic states determined by the MRCI+Q/CV+DK+56 calculations should be good prediction for future laboratory experiment.

Benchmarking Experimental and Computational Thermochemical Data: A Case Study of the Butane Conformers

Due to its crucial importance, numerous studies have been conducted to determine the enthalpy difference between the conformers of butane. However, it is shown here that the most reliable experimental values are biased due to the statistical model utilized during the evaluation of the raw experimental data. In this study, using the appropriate statistical model, both the experimental expectation values and the associated uncertainties are revised. For the 133-196 and 223-297 K temperature ranges, 668 ± 20 and 653 ± 125 cal mol(-1), respectively, are recommended as reference values. Furthermore, to show that present-day quantum chemistry is a favorable alternative to experimental techniques in the determination of enthalpy differences of conformers, a focal-point analysis, based on coupled-cluster electronic structure computations, has been performed that included contributions of up to perturbative quadruple excitations as well as small correction terms beyond the Born-Oppenheimer and nonrelativistic approximations. For the 133-196 and 223-297 K temperature ranges, in exceptional agreement with the corresponding revised experimental data, our computations yielded 668 ± 3 and 650 ± 6 cal mol(-1), respectively. The most reliable enthalpy difference values for 0 and 298.15 K are also provided by the computational approach, 680.9 ± 2.5 and 647.4 ± 7.0 cal mol(-1), respectively.