Multireference Averaged Quadratic Coupled Cluster Research Articles

Polyradical character assessment using multireference calculations and comparison with density-functional derived fractional occupation number weighted density analysis.

The biradicaloid character of different types of polycyclic aromatic hydrocarbons (PAHs) based on small band gaps is an important descriptor to assess their opto-electronic properties. In this work, the unpaired electron densities and numbers of unpaired electrons (NU values) calculated at the high-level multireference averaged quadratic coupled-cluster (MR-AQCC) method are used to develop a test set to assess the capabilities of different biradical descriptors based on density functional theory. A benchmark collection of 29 different compounds has been selected. The DFT descriptors contain primarily the fractional occupation number weighted electron density (FOD) based on simplified thermally-assisted-occupation density functional theory (TAO-DFT) calculations, but the singlet-triplet energy difference and other descriptors denoted as y0 and nLUNO have been considered as well. After adjustment of the literature-recommended finite temperatures, a very good, detailed agreement between unpaired density and FOD analysis is observed which is also manifested in excellent statistical correlations. The other two descriptors also show good correlations even though the absolute scaling is not satisfactory. A new linear fit of FOD data to the MR-AQCC reference values leads to an improved regression relation for determining the recommended finite temperature value in dependence of the Hartree-Fock exchange. This provides the basis for fast and reliable assessment of the biradical character of many classes of PAHs without the need for performing computationally extended MR calculations.

Active space and basis set effects in CASPT2 models of the 1,3‐butadiene‐ethene cycloaddition and the 1,3‐butadiene dimerization

The first step of the asynchronous biradical, stepwise biradical, and concerted mechanisms of the 1,3‐butadiene Diels–Alder reactions with both ethene and itself was studied using CASPT2 to determine the influence of basis set and active space on reaction barriers. CASPT2(6,6) with the cc‐pVDZ, 6‐311+G(3df,2p) and cc‐pVTZ basis sets provided the best results with average errors below 3.1 kJ mol−1 with respect to the experimental result. Increasing the active space size also had little effect on the calculated reaction barriers. With respect to experimental results, uncontracted multireference averaged quadratic coupled cluster (MRAQCC) produced superior barriers than internally contracted MRAQCC by 16.1–39.3 kJ mol−1. The inability of CASSCF to locate transition states for some of the cycloadditions across the butadiene‐ethene and butadiene dimerization reactions is also rationalized. CASPT2 suggests a preference for the concerted mechanism of the butadiene‐ethene reaction, however, no basis set yielded a preference for any of the butadiene dimerization pathways. © 2015 Wiley Periodicals, Inc.

Studies on the electric dipole moment function and line parameters for high overtone bands of NO

An accurate electric dipole moment function (EDMF) has been obtained by fitting the best available data including the pure rotational band 0–0 for individual ro-vibrational transitions for nitric oxide. Line intensities calculated using the fitted EDMF agree better with the measurements than the current HITRAN database. Moreover, the accurate ab initio potential energy curve (PEC) and EDMF were found using the multi-reference averaged quadratic coupled-cluster (AQCC) approach with the basis set aug-cc-pV6Z (aV6Z). The good agreement of the vibrational transition moments, computed using the Rydberg–Klein–Rees (RKR) potential and the fitted EDMFs, with experiments shows the present fitted EDMF could be more accurate than other fitted ones at the valid range. Our ab initio vibrational transition moments agree better with experimental data than the ones calculated using the previous theoretical PEC and EDMF, especially at the high overtones. We expect that the present study will be helpful for the transitions including high υ and high overtone bands of NO.

Analysis of and remedies for unphysical ground states of the multireference averaged coupled-pair functional

A Multireference Configuration Interaction (MRCI) wavefunction includes both static and dynamic electron correlation. MRCI's well-known flaw, a lack of size extensivity, can be ameliorated with the Multireference Averaged Coupled-Pair Functional (MRACPF). However, the original MRACPF is frequently unstable, sometimes producing unphysical results. The more Multireference Averaged Quadratic Coupled-Cluster and MRACPF2 functionals also occasionally exhibit unphysical behavior. We find that these instabilities are avoided crossings with unphysical solutions to the MRACPF equations. We present two approaches to avoid the undesirable unphysical solutions.

Study of the Diradicaloid Character in a Prototypical Pancake‐Bonded Dimer: The Stacked Tetracyanoethylene (TCNE) Anion Dimer and the Neutral K2TCNE2 Complex

The π-bonded tetracyanoethylene anion dimer (TCNE2(2-)) and the neutral K2TCNE2 system have been investigated to obtain new insights into the unique features of two-electron multicenter (2e-mc) π-pancake bonding. The inter-radical interaction leads to a significant diradicaloid character described by two singly occupied molecular orbitals (SOMOs) of the monomers. A highly correlated approach, the multireference averaged quadratic coupled-cluster (MR-AQCC) method, has been used to achieve a balanced description of the different types of electron correlation that occur in this system. The analysis of the interaction energies for the two systems in the singlet and the lowest triplet states and of the unpaired electron densities demonstrate the importance of diradical π bonding in addition to the conventional van der Waals interactions that occur in intermolecular interactions. In this analysis, the separation of the repulsive Coulomb interaction energies from the remaining terms turned out to be a crucial prerequisite to achieve consistent results. Our calculations also confirm that the driving force behind the energetic stability of the pancake bonds predominantly derives from the overlap of the SOMO-SOMO bonding interaction.

The accurate transition dipole moment and line intensities of NO(X2П) based on averaged quadratic coupled-cluster calculations

For NO(X2П), the vibrational dipole moments calculated using previous accurate ab initio electric dipole moment functions (EDMFs) are not consistent with experiments, especially for high-overtone bands 3–0, 4–0, 21–17, 5–0, 6–0, and 7–0. Moreover, the band intensities of the current HITRAN database agree only roughly with recent experimental results for high-overtone bands 3–0, 4–0, and 5–0. Thus, further theoretical calculation is necessary. We adopt the highly accurate multi-reference averaged quadratic coupled-cluster (MR-AQCC) approach and four different basis sets to investigate the potential-energy curves (PECs) and EDMFs of NO(X2П). The PECs and EDMFs obtained using the aug-cc-pV6Z basis set agree well with Rydberg–Klein–Rees (RKR) potential and experimental EDMFs. They are used to deduce the vibrational dipole moments and Herman–Wallis factors, which are further be utilized to evaluate line intensities and band intensities. Both the calculated transition dipole moments and the calculated band intensities are in remarkably better agreement with the experimental results for high-overtone bands. The Herman–Wallis factors of band 7–0 are also fairly consistent between experiment and our result. We expect that these reliable and accurate theoretical results will be helpful in producing accurate theoretical predictions concerning the spectroscopic characteristics and the analysis of some experimentally measured properties of NO(X2П).

The first metals in Mendeleiev’s table: further arguments to place He above Ne and not above Be

In a recent paper in this Journal, one of us argued against placing He above Be in Mendeleiev’s system of the elements. In it the goal was to dispute the notion that in Mendeleiev’s system of the elements the location of He should in fact lie above Be, which has a very similar electronic configuration, rather than above the noble gas column. That paper was based on rather old, Hartree–Fock limit studies on the strikingly limited non-additive contributions in the He3 and He4 systems in contrast with the much larger non-additivity obtained for the Be3, Be4 and Be5 oligomers. In a recent benchmark multireference Averaged Quadratic Coupled Cluster results on Be2 and Be3 we showed that the delocalized non-additive contribution comprises 94 % of the binding energy of Be3. Here we use this and other pertinent information (drawn from the same paper) to conclude that He may not be associated with Be in Mendeleiev’s Table, despite their quite similar spectroscopic ground states. Furthermore, we use the new results to show that the large non-additivity implies that less than 2 % of the Be3 binding is located in each Be pair contained within the Be trimer. The rest of the interaction energy is necessarily delocalized over all three Be atoms. This might actually announce the bulk properties (i.e. “the electron gas”) that in solid-state physics explain the large electric and heat conduction for the solid Be metal. Thus, in the case of beryllium the metallic characteristics are already evident in Be3, a far cry from the monoatomic helium gas.

Ground and excited state geometries via Mukherjee’s multireference coupled-cluster method

A comprehensive study of molecular equilibrium structures is conducted to benchmark the multireference coupled-cluster (CC) method suggested by Mukherjee and coworkers (Mk-MRCC). We determine equilibrium structures and adiabatic excitation energies by applying the Mk-MRCC method within the singles and doubles (SD) approximation to ground and excited states of various small and medium-sized molecules. The results are compared to those obtained using other multireference or single-reference CC methods. For most molecules with a multireference ground state, it is found that equilibrium structures and excitation energies computed at the Mk-MRCCSD, equation-of-motion CCSD, multireference averaged-quadratic CC, and CCSD(T) levels of theory qualitatively agree, while the single-reference CCSD method often leads to distinctly different results.

New Way of Describing Static and Dynamic Deformations of the Jahn–Teller Type in Ring Molecules

A new method is presented to describe deformations of an N-membered planar ring (N-ring) molecule in terms of deformation vectors that can be expressed by a set of 2N-3 deformation amplitudes and phase angles. The deformation coordinates are directly derived from the normal vibrational modes of the N-ring and referenced to a regular polygon (N-gon) of unit length. They extend the conceptual approach of the Cremer-Pople puckering coordinates (J. Am. Chem. Soc. 1975, 97, 1354) to the planar ring and make it possible to calculate, e.g., a planar ring of special deformation on a Jahn-Teller surface. It is demonstrated that the 2N-3 deformation parameters are perfectly suited to describe the pseudorotation of a bond through the ring as it is found in cyclic Jahn-Teller systems. In general, an N-membered planar ring can undergo N-2 different bond pseudorotations provided the energetics of such a process is feasible. The Jahn-Teller distortions observed in ring compounds correspond either directly to the basic pseudorotation modes or to linear combinations of them. Any deformed ring molecule can be characterized in terms of the new ring deformation coordinates, which help to identify specific electronic effects. The usefulness of the ring deformation coordinates is demonstrated by calculating the Jahn-Teller surfaces for bond pseudorotation in the case of the cyclopropyl radical cation and cyclobutadiene as well as the ring deformation surfaces of disulfur dinitride and its dianion employing multireference averaged quadratic coupled cluster (MR-AQCC) theory, equation-of-motion coupled cluster theory in form of EOMIP-CCSD, and single determinant coupled cluster theory in form of CCSD(T).

Large-Scale Parallel Uncontracted Multireference-Averaged Quadratic Coupled Cluster: The Ground State of the Chromium Dimer Revisited

The accurate prediction of the potential energy function of the X1Sigmag+ state of Cr2 is a remarkable challenge; large differential electron correlation effects, significant scalar relativistic contributions, the need for large flexible basis sets containing g functions, the importance of semicore valence electron correlation, and its multireference nature pose considerable obstacles. So far, the only reasonable successful approaches were based on multireference perturbation theory (MRPT). Recently, there was some controversy in the literature about the role of error compensation and systematic defects of various MRPT implementations that cannot be easily overcome. A detailed basis set study of the potential energy function is presented, adopting a variational method. The method of choice for this electron-rich target with up to 28 correlated electrons is fully uncontracted multireference-averaged quadratic coupled cluster (MR-AQCC), which shares the flexibility of the multireference configuration interaction (MRCI) approach and is, in addition, approximately size-extensive (0.02 eV in error as compared to the MRCI value of 1.37 eV for two noninteracting chromium atoms). The best estimate for De arrives at 1.48 eV and agrees well with the experimental data of 1.47 +/- 0.056 eV. At the estimated CBS limit, the equilibrium bond distance (1.685 A) and vibrational frequency (459 cm-1) are in agreement with experiment (1.679 A, 481 cm-1). Large basis sets and reference configuration spaces invariably result in huge wave function expansions (here, up to 2.8 billion configuration state functions), and efficient parallel implementations of the method are crucial. Hence, relevant details on implementation and general performance of the parallel program code are discussed as well.

CASSCF and multireference CI with singles and doubles study of low‐lying valence and Rydberg states of 2H‐tetrazole

Complete active space self-consistent field (CASSCF) and multireference CI with singles and doubles (MR-CISD) calculations [including extensivity corrections, at MR-CISD+Q and multireference averaged quadratic coupled cluster (MR-AQCC) levels] have been performed to characterize the low-lying valence and the Rydberg states of 2H-tetrazole. The highest level results (MR-AQCC/d'-aug'-cc-pVDZ) indicate the following ordering of the valence singlet excited states: S(1) (n-pi*), 6.06 eV; S(2) (n-pi*), 6.55 eV; S(3) (pi-pi*), 6.55 eV. The MR-CISD+Q/d'-aug'-cc-pVDZ results indicate the same ordering, but at slight higher energies: 6.16, 6.68, and 6.69 eV, respectively. According to our MR-CISD+Q/d'-aug'-cc-pVDZ results, the next two states are Rydberg states, at 7.69 eV (pi-3s) and 7.89 eV (n-3s). The calculated energies of these two states, as well as their proximity, are consistent with the conclusion reached by Palmer and Beveridge (Chem Phys 1987, 111, 249) that the first band of the photoelectron spectrum of 2H-tetrazole is likely to be associated to the first two ionizations processes (of pi and N lone pair electrons), at energies close to 11.3 eV.

An extended multireference study of the electronic states of para-benzyne

A state-averaged, multireference complete active space (CAS) approach was used for the determination of the vertical excitation energies of valence and Rydberg states of para-benzyne. Orbitals were generated with a 10- and 32-state averaged multiconfigurational self-consistent field approach. Electron correlation was included using multireference configuration interaction with singles and doubles, including the Pople correction for size extensivity, multireference averaged quadratic coupled cluster (MR-AQCC), and MR-AQCC based on linear response theory. There is a very high density of electronic states in this diradical system-there are more than 17 states within 7 eV of the ground state including two 3s Rydberg states. All excitations, except 2 (1)A(g), are from the pi system to the sigmasigma(*) system. Of the 32 states characterized, 15 were multiconfigurational, including the ground (1)A(g) state, providing further evidence for the necessity of a multireference approach for p-benzyne. The vertical singlet-triplet splitting was also characterized using a two-state averaged approach. A CAS(2,2) calculation was shown to be inadequate due to interaction with the pi orbitals.

Multireference averaged quadratic coupled-cluster (MR-AQCC) method based on the functional of the total energy

In this short paper a new version of the multireference (MR) averaged coupled pair functional (ACPF) and the MR averaged quadratic coupled-cluster (AQCC) methods is presented which corrects the failure of the original formulation in case of close-lying states described with small reference space. This new version is based on a functional which also includes the reference energy, i.e. instead of optimizing the correlation energy, the total energy will be made stationary with respect the wave function parameters. It is demonstrated that this method gives smooth potential energy surfaces in case of avoided crossings.

A theoretical study of FeCN in the 6Δ electronic ground state

The three-dimensional potential energy and dipole moment surfaces for the electronic ground state 6Δ of FeCN have been computed at the MR-SDCI + Q + E rel/[Roos ANO (Fe), aug-cc-pVQZ (C, N)] level of theory, where MR-SDCI means ‘multi-reference single and double excitation configuration interaction’ and ANO means ‘atomic natural orbital’. Based on these potential energy and dipole moment surfaces, the spectroscopic parameters, rovibronic energies, structural parameters, vibrational transition moments, and the wavenumbers and intensities of selected rotation–vibration transitions have been calculated. The equilibrium structure is linear with r e(Fe–C) = 2.048 Å and r e(C–N) = 1.168 Å, and the zero-point averaged structure is bent with 〈 r(Fe–C)〉 0 = 2.082 Å, 〈 r(C–N)〉 0 = 1.172 Å, and 〈∠(Fe–C–N)〉 0 = 170(5)°. At all the MR-SDCI + Q and the size-extensive multi-reference averaged quadratic coupled-cluster (MR-AQCC) levels of theory, with and without relativistic correction E rel, that were employed in the present work, 6Δ FeCN is predicted to be slightly more stable than 6Δ FeNC. For example, the energy difference between the two isomers is approximately 150 cm −1 at the highest level of theory employed, MR-AQCC + E rel/[Roos ANO (Fe), aug-cc-pVQZ (C, N)] with zero-point energy correction. The electronic structure of 6Δ FeCN has also been compared with that of 6Δ FeNC. At present, no experimental spectroscopic data are available for 6Δ FeCN. It is hoped that the present work will stimulate experimental investigations of this molecule.

Van der Waals interactions and dipole polarizabilities of lanthanides: Tm(F2)–He and Yb(S1)–He potentials

Anisotropic dipole polarizabilities of Tm(2F), Tm+2(2F), and Yb(1S) are calculated using the finite-field multireference averaged quadratic coupled cluster (MR-AQCC) (Tm and Tm+2) and RCCSD(T) (Yb) methods with small-core relativistic pseudopotentials ECP28MWB combined with the augmented ANO basis sets. The lanthanide atoms are strongly polarizable with the scalar part originating from the 6s electrons and the tensorial part from the open 4f shells. The adiabatic interaction potentials 2Sigma+, 2Pi, 2Delta, and 2Phi of Tm(2F)-He and Tm+2(2F)-He were examined by the multireference approaches, multireference configuration interaction and MR-AQCC, using the basis sets designed in the polarizability calculations. A closed-shell lanthanide system Yb(1S)-He was included for comparison. The Tm-He 2Sigma+, 2Pi, 2Delta, and 2Phi interaction potentials are very shallow and nearly degenerate (within 0.01 cm(-1)), with the well depths in the range of 2.35-2.36 cm(-1) at R=6.17 A. The basis-set saturated well depths are expected to be larger by ca. 25%, as estimated using the bond-function augmented basis set. The interactions of lanthanide atoms with He are one order of magnitude less anisotropic than those involving first-row transition metal atoms. The suppression of anisotropy is chiefly attributed to the screening effected by the 6s shell. When these electrons are removed as in the di-cation complex Tm+2(2F)-He, the potentials deepen to a thousand wave number range and their anisotropy is enhanced 500-fold.

Geometry optimization of excited valence states of formaldehyde using analytical multireference configuration interaction singles and doubles and multireference averaged quadratic coupled-cluster gradients, and the conical intersection formed by the 1 1B1(σ-π*) and 2 1A1(π-π*) states

Extended MR-CISD (multireference configuration interaction singles and doubles), MR-CISD+Q (multireference configuration interaction singles, doubles, and quadrupole), and MR-AQCC (multireference averaged quadratic coupled clusters) calculations have been performed on the following valence states of formaldehyde: 1 1A1 (planar ground state), 1 1A2 and 1 1A″ (planar and nonplanar n-π* state), 1 1B1 and 2 1A1 (planar σ-π* and π-π* states) and their nonplanar counterparts 2 1A′ and 3 1A′. Full geometry optimizations have been performed using analytic gradient techniques developed for the MR-CISD and MR-AQCC methods as implemented into the COLUMBUS program system. Basis set extrapolation techniques have been used for the determination of high-accuracy geometries and adiabatic excitation energies. Harmonic vibrational frequencies have been computed also. Agreement between calculated and available experimental data is very good. Especially for the σ-π* and π-π* states experimental information is extremely scarce and our results provide reliable predictions. The major new result of our work is the finding that for the 2 1A1(π-π*) state the structure optimized under planarity constraints is only a saddle point and not a minimum. This fact is the result of a conical intersection between the σ-π* and π-π* states (1 1B1 and 2 1A1). The final result is that neither on the 2 1A′ nor on the 3 1A′ surface a stationary minimum can be assigned to the π-π* state.

Excitation energies and transition moments by the multireference averaged quadratic coupled cluster (MR-AQCC) method

The MR-AQCC (multireference averaged quadratic coupled cluster) and the MR-ACPF (multireference averaged coupled pair functional) methods can be considered as size-extensivity corrected MR-CISD procedures. So far, they have been applied mostly to ground state calculations of a given symmetry. By formulating MR-AQCC and MR-ACPF in terms of a diagonal shift to the Hamiltonian this restriction is removed. In order to compute transition moments, linear response theory (LRT) is used since there is no wavefunction defined within these methods. A LRT-compliant variant of MR-AQCC and MR-ACPF is developed which allows a consistent formulation of transition energies and moments. In calculations on CH2, CH+ and C2 (where full CI results are available) it is shown that MR-AQCC brings about a substantial improvement for excitation energies and transition moments as compared to MR-CISD.

Does N2− exist? A coupled-cluster study

Potential energy curves of the ground-state N2 molecule and its doublet N2− anion are calculated at the coupled-cluster level with single and double excitations and with noniterative triples [CCSD(T)] as well as with the multireference averaged-quadratic coupled-cluster (MR-AQCC) method. The N2− anion is shown to be temporary and decays to its neutral parent plus a free electron at bond lengths shorter than ≈1.4 and larger than ≈2.5 Å. Thus, the N2− anion exists within the 1.4⩽R(N–N)⩽2.5 Å range at the Born–Oppenheimer approximation.

Tunnelling lifetimes of the rovibronic levels in the B electronic state of the CH radical obtained from ab initio data

Rovibronic spectroscopic parameters have been obtained for the X and B states of the CH radical using accurate ab initio potentials. The potentials have been calculated with the multi-reference averaged quadratic coupled cluster (MR-AQCC) method using a large reference space and the cc-pVQZ basis. The one-dimensional Schrödinger equation of the nuclei has been solved exactly to get the vibrational—rotational energy levels. The energies and tunnelling predissociation widths of quasibound levels on these potentials have been calculated, plus the Franck—Condon factors for many of the rovibronic transitions.

High-quality theoretical potential energy surface for Be 2 by using the multireference averaged quadratic coupled-cluster (MR-AQCC) method and large basis sets

Both electron correlation and basis set effects are important in the description of the bond in Be 2. We apply a new version of the multireference averaged quadratic coupled-cluster method which is an approximately size extensive modification of the multireference configuration interaction method and was shown to yield potential energy surfaces parallel to full-CI ones. We show this in the case of Be 2 as well. Since the method is considerably cheaper than full-CI, we were able to account for core correlation and calculate the whole potential energy surface. Equilibrium geometry, vibrational frequencies and dissociation energy are obtained. For the former two quantities the experimental data are confirmed by the calculations, while the results suggest that the dissociation energy might be considerably higher than the one obtained from the lowest vibrational frequencies. Special attention is paid to the basis set superposition error. Our best calculation of the dissociation energy D e is 864 cm −1 which is considered to be a lower bound.