Multireference Coupled-cluster Calculations Research Articles

Three Lagrangians for the complete-active space coupled-cluster method.

Three fully variational formulations of the complete-active space coupled-cluster method are derived. The formulations include the ability to approximate the model vectors by smooth manifolds, thereby opening up the possibility for overcoming the exponential wall of scaling for model spaces of complete-active space type. In particular, model vectors of matrix-product states are considered, and it is argued that the present variational formulation allows not only favorably scaling multireference coupled-cluster calculations but also systematic correction of tailored coupled-cluster calculations and of quantum chemical density-matrix renormalization group methods, which are fast and polynomial scaling but lack the ability to properly resolve dynamical correlation at chemical accuracy. The extension of the variational formulations to the time domain is also discussed, with derivations of abstract evolution equations.

Potential energy curves for electronic states of the sodium dimer with multireference coupled cluster calculations

Accurate potential energy curves (PECs) were obtained for the 26 lowest lying electronic states of the sodium dimer. The current approach is based on the first principle size-extensive multireference coupled cluster method formulated in the (2,0) sector of the Fock space (FS). The (2,0) sector of FS provides the description of the states obtained by the attachment of two electrons to the reference system. This makes it possible to adopt the doubly ionised sodium dimer as a reference with its very concrete advantage in the calculations of the PECs, i.e. it dissociates into closed shell fragments (i.e. sodium cations) hence the restricted Hartree–Fock reference can be used in the whole range of interatomic distances. This results in the correct description of the interatomic potentials for all bond lengths. The theoretical PECs stay very close to the experimental curves wherever the latter are available and the computed spectroscopic parameters reproduce the experiment with very good accuracy. Relativistic corrections included at the second-order Douglass–Kroll level have a non-negligible effect on the accuracy of computed excitation and dissociation energies.

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.

Dissociation of homonuclear diatomic halogens via multireference coupled cluster calculations

We have computed the potential energy surfaces (PESs) of F2, Cl2, Br2, and I2 using the size-extensive state specific multireference coupled cluster (SS-MRCC) method. The MR character of the system considered here at large distances and the presence of low-lying intruder states are known to be the major causes of incorrect or inaccurate predictions of the PES. The SS-MRCC theory is tailored to treat degeneracies of varying extent while bypassing the intruder problem. The quality of the computed PES has been gauged by computing spectroscopic constants. The calculated properties show a good agreement with available experimental data and the errors in the calculated molecular properties compare favourably with the most elaborate current-generation calculations of the literature. The accuracy of the computed PES of F2 is such that it has been proved to calculate the vibrational spectrum of the 22 levels with a minimum and maximum absolute deviation of 2 and 57 cm−1, respectively, from the experimental values. The highly satisfactory performance of the SS-MRCC method, vis-a-vis the other sophisticated methods, in describing the vibrational levels is noticeable for one of the more difficult systems such as F2 clearly indicates that the present method is reliable in studying the vibrational energy levels.

Fock-space multireference coupled-cluster calculations of the hyperfine structure of isoelectronic33S−and35,37Cl

Due to its flexibility and possible systematic improvement, the Fock-space (FS) multireference coupled-cluster (MRCC) method remains a very important tool for the computation of energy differences of spectroscopic interest. In the present work, the FS MRCC method for the electron detachment process has been applied to determine the magnetic hyperfine constant ${A}_{J}$ and nuclear quadrupole moments $Q$ (related to electric hyperfine constant ${B}_{J}$) for the lowest multiplets of ${}^{33}$S${}^{\ensuremath{-}}$, ${}^{35}$Cl, and ${}^{37}$Cl with Dirac-Fock orbitals. In addition, we also report ${}^{2}$${P}_{3/2}$([Ne]$3{s}^{2}3{p}^{5}$) $\ensuremath{\rightarrow}$ ${}^{2}$${P}_{1/2}$([Ne]$3{s}^{2}3{p}^{5}$) magnetic dipole transition matrix element and electron affinity of ${}^{35}\mathrm{Cl}$ (i.e., ionization energy of Cl${}^{\ensuremath{-}}$). Calculated properties are in very good agreement with the available new standard or reference values.

Valence universal multireference coupled cluster calculations of the properties of indium in its ground and excited states

In view of its importance in high precision spectroscopy, the valence universal multireference coupled cluster (VU-MRCC) method with four-component relativistic spinors has been applied to compute ionization potential (IP) and excitation energies (EEs) of the indium atom (In I). The effect of electron correlations on the ground and excited state properties is investigated using different levels of CC approximations and basis sets. This study reveals that for a given basis, the linearized VU-MRCC method tends to underestimate the IP, EEs and other one-electron properties such as magnetic hyperfine constant (A) compared to the full blown VU-MRCC method. Our computed results have been compared with available theoretical and experimental data. The IP, EEs, A and oscillator strengths (f) determined at the VU-MRCC level are in excellent agreement with the experimental results. The properties reported here further demonstrate that a basis set with at least h-type of orbitals is ubiquitous to achieve converged results.

Reinvestigation of the Reaction of Ethylene and Singlet Oxygen by the Approximate Spin Projection Method. Comparison with Multireference Coupled-Cluster Calculations

We quantify a spin contamination error caused by a broken-symmetry (BS) method on the geometry at the stationary points and barrier heights of the [2 + 2] reaction between singlet oxygen and ethylene, which goes through a diradical intermediate. Several hybrid GGA, hybrid meta-GGA, and long-range corrected hybrid functionals, O3LYP, B3LYP, PBE0, MPW1B95, BHandHLYP, and omegaB97X, are examined to elucidate their original nature without the spin contamination error. For that purpose, the geometry of each reaction step for the BS state as well as its total energy is corrected by using an approximate spin projection method. The CCSD and CCSD(T) single-point calculations are also carried out at optimized geometries at the DFT level to confirm the results of the DFT methods. The single-point calculations by means of Mukherjee's multireference coupled cluster with single and double excitations at CASSCF(10e,8o)-optimized geometries are also presented to assess the DFT methods. After the energy and geometry corrections, the barrier height of each functional is consistent with conventional closed-shell-type reactions even in the reaction involving singlet diradical species. We also find that the spin contamination error on the geometric change is not negligible especially at the early stage of the reaction ( approximately 3 kcal/mol), where the triplet state is the ground state.

Multi-reference coupled-cluster study of the potential energy surface of the hydrogen fluoride dissociation including excited states

Multi-reference coupled-cluster calculations for the potential energy surface of the HF molecule including excited states are carried out. Using the multi-reference exponential wavefunction [M. Hanrath, J. Chem. Phys. 123 (2005) 84102] and the single-reference formalism based multi-reference coupled-cluster ansatz [J. Chem. Phys. 94 (1991) 1229, J. Chem. Phys. 99 (1993) 1875] the results are analyzed and discussed with respect to full CI results. The present work completes previous studies which considered calculations with partially restricted residual equations only. A certain attention is paid to spatial degeneracy issues.

Fock space multireference coupled cluster calculations based on an underlying bivariational self-consistent field on Auger and shape resonances.

The Fock space multireference coupled cluster based on an underlying bivariational self-consistent field is applied to the problem of computing complex energy associated with Auger and shape resonances in e-atom scattering. It is concluded that the Fock space multireference coupled cluster based on a bivariational self-consistent field provides a useful and practical approach to calculation of resonance parameters. Numerical results are presented for the 2P shape resonance of Mg and Auger 1 s(-1) hole of Be.

Multireference coupled-cluster calculations on the energy of activation in the automerization of cyclobutadiene: Assessment of the state-specific multireference Brillouin–Wigner theory

Hilbert-space state-universal multireference coupled-cluster (MR CC) data on cyclobutadiene [A. Balková and R. J. Bartlett, J. Chem. Phys. 101, 8972 (1994)] were used as a benchmark for testing our recently developed state-specific (single-root) multireference Brillouin-Wigner coupled-cluster (MR BWCC) theory. For the energy of activation in the automerization of cyclobutadiene (i.e., the energy difference between the square and rectangular structures) at the CCSD/[3s2p1d/2s] level of theory, our MR BWCCSD method gives the value of 6.2 kcal/mol, compared to 6.5 kcal/mol given by MR CCSD. With the cc-pVDZ and cc-pVTZ/cc-pVDZ basis sets, the MR BWCCSD activation barrier is 6.4 and 7.0 kcal/mol, respectively. The effect of the triple excitations [in MR CCSD(T)] and of the frozen core approximation were estimated previously to be below 0.1 kcal/mol and in the opposite direction. This shows the way of how to arrive at a more accurate automerization barrier in future calculations: extension of the basis set seems to be more important than going beyond the CCSD(T) or MB BWCCSD level of the theory.

State-selective multi-reference coupled-cluster theory employing the single-reference formalism: Application to an excited state of H8

The state-selective (SS) multi-reference (MR) coupled-cluster (CC) method exploiting the single-reference (SR) formalism [P. Piecuch, N. Oliphant, and L. Adamowicz, J. Chem. Phys. 99, 1875 (1993)] is applied to the first excited totally symmetric singlet state of a prototype molecular system composed of eight hydrogen atoms. Minimum basis set is employed and various geometries are considered. The SS CC energies are compared with the results of the state-universal (SU) MR CC calculations involving single and double excitations (CCSD) as well as with the exact energies obtained using the full configuration interaction method. Comparison is also made with the results of the standard SR CCSD calculations. In both nondegenerate and quasidegenerate regions, our SS CC theory truncated at double excitations [SS CCSD(TQ) method] provides much better description of the first excited state than the genuine multi-determinantal SU CCSD formalism.

Multireference coupled cluster calculations on CH2+

In this paper we examine the potential energy surface of CH2+ to reinvestigate the controversy surrounding the stability of this dication using the Fock space version of multireference coupled cluster theory.

Comparison of complete model space quasidegenerate many-body perturbation theory for LiH with multireference coupled cluster method

The relative efficacy of using low order trucations with large reference space vs high order methods with small reference space is tested by comparing quasidegenerate many-body perturbation theory (QDMBPT) calculations of potential curves for the five lowest electronic states of LiH with the multireference coupled cluster calculations of Ben-Shlomo and Kaldor [J. Chem. Phys. 89, 956 (1988)]. The infinite order coupled cluster calculations use two configurational reference spaces involving the 2σ, 3σ, and 1π orbitals, while the QDMBPT computations are truncated at either second or third orders and employ the full active reference space formed either from the 2σ, 3σ, and 1π or from the 2σ, 3σ, 4σ, and 1π orbitals. This gives us the opportunity of testing the dependence of QDMBPT computations on the size of reference space, the available freedom in choosing valence orbitals and orbital energies, and the order of truncation. Second order, four valence orbital space QDMBPT calculations provide good agreement with the repulsive portion of the coupled cluster potentials, but yield a separated atom limit that is too high and that therefore distorts the remainder of the potential. Third order improves the separated atom limit considerably, providing good agreement with the coupled cluster calculations. The ‘‘full chemical’’ five orbital reference space, on the other hand, yields very good agreement with coupled cluster potentials when using only the simpler second order QDMBPT calculations, and third order corrections in this case are very small but generally improve agreement with coupled cluster potentials. The five orbital reference space calculations are quite insensitive to a wide range of different choices of valence orbitals and orbital energies, demonstrating a robustness to the QDMBPT formalism used.