Multireference composite approaches for the accurate study of ground and excited electronic states: C2, N2, and O2

A multireference analog of the correlation consistent composite approach (MR-ccCA) based on complete active space with second-order perturbation theory (CASPT2) has been utilized in an investigation of the ground and valence excited states of C(2), N(2), and O(2). The performance of different second-order multireference perturbation theory methods including second-order n-electron valence state perturbation theory, second-order multireference M?ller-Plesset, and second-order generalized van Vleck perturbation theory has been analyzed as potential alternatives to CASPT2 within MR-ccCA. The MR-ccCA-P predicts spectroscopic constants with overall mean absolute deviations from experimental values of 0.0006 A?, 7.0 cm(-1), and 143 cm(-1) for equilibrium bond length (r(e)), harmonic frequency (ω(e)), and term values (T(e)), respectively, which are comparable to the predictions by more computationally costly multireference configuration interaction-based methods.     

Electronic structure of spherical quantum dots using coupled cluster method

2, 6, 12, and 20 electron quantum dots have been studied using coupled cluster at singles and doubles level and extensive multireference coupled cluster (MRCC) method. A Fock-space version of MRCC (FSMRCC) containing single hole-particle excited determinants has been used to calculate low-lying excited states of the above system. The ionization potential and electron affinity are also calculated. The effect of correlation energy on excitation energy and charge density is shown by calculating them at the high density region (low value of density parameter rs) and at the low density region (high value of density parameter rs).     

The potential energy curves of low-lying electronic states of S2O

Potential energy curves (PECs) of the symmetric and asymmetric bent S(2)O molecules are constructed using the configuration-based multireference second order perturbation theory and multireference configuration interaction with single and double excitations. Based on the PECs, the equilibrium structures of the ground state and several low-lying excited states, as well as the vertical and adiabatic transition energies, are obtained. Furthermore, avoided crossings and intersections displayed on the PECs are studied. The dissociation of states for the asymmetric bent S(2)O, especially the predissociative of the excited (~)C1A' state, is also discussed in detail. According to our calculations, the predissociation limit of (~)C1A' is found to be located in the vicinity of 2(6) or 2(5) (reckoning in the zero-point energy revision) S-S stretching vibration level, which is in good agreement with the available experimental data.     

Kinetics of CH + N2 revisited with multireference methods

The potential energy surface for the CH + N2 reaction was reexamined with multireference ab initio electronic structure methods employing basis sets up to aug-cc-pvqz. Comparisons with related CCSD(T) calculations were also made. The multireference ab initio calculations indicate significant shortcomings in single reference based methods for two key rate-limiting transition states. Transition state theory calculations incorporating the revised best estimates for the transition state properties provide order of magnitude changes in the predicted rate coefficient in the temperature range of importance to the mechanism for prompt NO formation. At higher temperatures, two distinct pathways make a significant contribution to the kinetics. A key part of the transition state analysis involves a variable reaction coordinate transition state theory treatment for the formation of H + NCN from HNCN. The present predictions for the rate coefficients resolve the discrepancy between prior theory and very recent experimental measurements.     

Complex self-consistent field and multireference single- and double-excitation configuration interaction calculations for the 2Pi(g) resonance state of N2-

Self-consistent field and multireference single- and double-excitation configuration interaction calculations employing the complex basis function technique are carried out for the (2)Pi(g) resonance state of the N(2) (-) molecule as well as several other anionic resonance states in the neighboring energy region. The results of calculations employing the same method for the (1)S (2s(2)) state of the He atom and the (1)Sigma(g) (+) (sigma(u) (2)) state of the H(2) molecule are found to be in good agreement with those of earlier work. The present theoretical treatment has succeeded for the first time in satisfying the rigorous criterion of the complex variational principle in computing the N(2) (-) resonance states, namely, a cusp in the plots of real versus imaginary components of the corresponding complex energies has been located at each internuclear distance. On this basis, it is found that the open-shell orbital in the lowest-energy adiabatic N(2) (-) resonance state of (2)Pi(g) symmetry changes its character from quite compact at large internuclear distance to relatively diffuse for r<2.3a(0). This is in contrast to all previous theoretical treatments of this system that have not rigorously satisfied the complex variational principle in their determination of this wave function.     

Molecular dynamics and metadynamics simulations of [2 + 2] photocycloaddition

Triplet state mechanism of [2 + 2] photocycloaddition forming a cyclobutane ring from two ethylenes is investigated in the context of photocatalysis. High‐level ab initio calculations are combined with ab initio adiabatic molecular dynamics and ab initio metadynamics for rare events modeling. In a photocatalytic scheme, a reactant reaches the triplet state either via intersystem crossing (ISC) or triplet sensitization. The model system adopts a biradical structure, which represents energy intersection with the ground state. The system either completes cyclization or undergoes fragmentation into two olefinic units. The potential and free energy surfaces of the cyclobutane/ethylenes system are mapped with multireference approaches describing possible reaction pathways. To obtain a full picture of a double bond photoreactivity, ab initio adiabatic dynamical calculations were used to estimate reaction yields and to model the effects of excess energy. The potential use of density functional theory based approaches for [2 + 2] photocycloaddition was investigated for future simulations and design of realistic photocatalytic systems. Dynamical aspects of [2 + 2] photocycloaddition via a triplet state manifold are investigated by combining ab initio multireference methods and ab initio molecular dynamics and metadynamics. The reaction pathways are studied for a model system of two ethylenes forming a cyclobutane ring to provide a basis for further studies on design of photocatalytic systems.     

The lowest 2A' excited state of the water-hydroxyl complex

Vertical and adiabatic excitation energies of the lowest (2)A(') excited state in the water-hydroxyl complex have been determined using coupled cluster, multireference configuration interaction, multireference perturbation theory, and density-functional methods. A significant redshift of about 0.4 eV in the vertical excitation energy of the complex compared to that of the hydroxyl radical monomer is found with the coupled cluster calculations validating previous results. Electronic excitation leads to a structure with near-equal sharing of the hydroxyl hydrogen by both oxygen atoms and a concomitantly large redshift of the adiabatic excitation energy of approximately 1 eV relative to the vertical excitation energy. The combination of redshifts ensures that the electronic transition in the complex lies well outside the equivalent excitation in the hydroxyl radical monomer. The complex is approximately five times more strongly bound in the excited state than in the ground state.     

Optimizing conical intersections without derivative coupling vectors: application to multistate multireference second-order perturbation theory (MS-CASPT2)

We introduce a new method for optimizing minimal energy conical intersections (MECIs), based on a sequential penalty constrained optimization in conjunction with a smoothing function. The method is applied to optimize MECI geometries using the multistate formulation of second-order multireference perturbation theory (MS-CASPT2). Resulting geometries and energetics for conjugated molecules including ethylene, butadiene, stilbene, and the green fluorescent protein chromophore are compared with state-averaged complete active space self-consistent field (SA-CASSCF) and, where possible, benchmark multireference single- and double-excitation configuration interaction (MRSDCI) optimizations. Finally, we introduce the idea of "minimal distance conical intersections", which are points on the intersection seam that lie closest to some specified geometry such as the Franck-Condon point or a local minimum on the excited state.     

Multireference configuration interaction calculations for the F(2P)+HCl-->HF+Cl(2P) reaction: a correlation scaled ground state (1 2A') potential energy surface

This paper presents a new ground state (1 (2)A(')) electronic potential energy surface for the F((2)P)+HCl-->HF+Cl((2)P) reaction. The ab initio calculations are done at the multireference configuration interaction+Davidson correction (MRCI+Q) level of theory by complete basis set extrapolation of the aug-cc-pVnZ (n=2,3,4) energies. Due to low-lying charge transfer states in the transition state region, the molecular orbitals are obtained by six-state dynamically weighted multichannel self-consistent field methods. Additional perturbative refinement of the energies is achieved by implementing simple one-parameter correlation energy scaling to reproduce the experimental exothermicity (DeltaE=-33.06 kcalmol) for the reaction. Ab initio points are fitted to an analytical function based on sum of two- and three-body contributions, yielding a rms deviation of <0.3 kcalmol for all geometries below 10 kcalmol above the barrier. Of particular relevance to nonadiabatic dynamics, the calculations show significant multireference character in the transition state region, which is located 3.8 kcalmol with respect to F+HCl reactants and features a strongly bent F-H-Cl transition state geometry (theta approximately 123.5 degrees ). Finally, the surface also exhibits two conical intersection seams that are energetically accessible at low collision energies. These seams arise naturally from allowed crossings in the C(infinityv) linear configuration that become avoided in C(s) bent configurations of both the reactant and product, and should be a hallmark of all X-H-Y atom transfer reaction dynamics between ((2)P) halogen atoms.     

DFT,DLPNO-CCSD(T), and NEVPT2 benchmark study of the reaction between ferrocenium and trimethylphosphine

The reaction between ferrocenium and trimethylphosphine was studied using density functional theory (DFT), domain-based local pair natural orbital coupled cluster theory with single-, double-, and perturbative triple excitations (DLPNO-CCSD(T)), and N-electron valence state perturbation theory (NEVPT2). The accuracy of the DFT functionals decreases compared to the DLPNO-CCSD(T) level in the following order: M06-L > TPSS > M06, BLYP > PBE, PBE0, B3LYP > > PWPB95 > > DSD-BLYP. The roles of thermochemical, continuum solvation (SMD), and counterpoise corrections were evaluated. Grimme's D3 empirical dispersion correction is essential for all functionals studied except M06 and M06-L. The reliability of the frequency calculations performed directly within the SMD was confirmed. The systems showed no significant multireference character according to T₁ and T₂ diagnostics and the fractional occupation number (FOD) weighted electron density analysis. The multireference NEVPT2 calculations gave qualitatively valid conclusions about the reaction mechanism. However, a multireference approach is generally not recommended because it requires arbitrary chosen active spaces.     

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.     

Spin-orbit configuration interaction study of the electronic structure of the 5f (2) manifold of U(4+) and the 5f manifold of U(5+)

The energy levels of the 5f configuration of U(5+) and 5f(2) configuration of U(4+) have been calculated in a dressed effective Hamiltonian relativistic spin-orbit configuration interaction framework. Electron correlation is treated in the scalar relativistic scheme with either the multistate multireference second-order multiconfigurational perturbation theory (MS-CASPT2) or with the multireference single and double configuration interaction (MRCI) and its size-extensive Davidson corrected variant. The CASPT2 method yields relative energies which are lower than those obtained with the MRCI method, the differences being the largest for the highest state (1)S(0) of the 5f(2) manifold. Both valence correlation effects and spin-orbit polarization of the outer-core orbitals are shown to be important. The satisfactory agreement of the results with experiments and four-component correlated calculations illustrates the relevance of dressed spin-orbit configuration interaction methods for spectroscopy studies of heavy elements.     

Global potential energy surfaces for the Al+(1S) + H2 system

Global, three-dimensional multireference ab initio potential energy surfaces have been calculated for the AlH2+ system for the two lowest energy singlet states and the lowest energy triplet state. These surfaces were calculated using the multireference configuration interaction level of theory with a large basis set. The accuracy of the surfaces were checked against available experimental data and previous theoretical investigations. The areas of surface crossings between the ground state singlet surface and the lowest energy triplet surface and the first excited singlet surface have been thoroughly investigated in all three dimensions and found to give rise to two regions of surface crossings--an "early" crossing (reduced H2 distance) and a "late" crossing (enlarged H2 distance). It is anticipated that both of these crossings will be important in modeling the dynamics of the system. Each of the global potential energy surfaces were fit by interpolation methodology to obtain analytic representations of the surfaces. A representative classical simulation on the ground state singlet surface was performed and discussed.     

GVVPT2 multireference perturbation theory description of diatomic scandium, chromium, and manganese

With relatively simple model spaces derived from valence bond models, a straightforward zero-order Hamiltonian, and the use of moderate-sized Dunning-type correlation consistent basis sets (cc-pVTZ, aug-cc-pVTZ, and cc-pVQZ), the second order generalized Van Vleck perturbation theory (GVVPT2) method is shown to produce potential energy curves (PECs) and spectroscopic constants close to experimental results for both ground and low-lying excited electronic states of Sc(2), Cr(2) and Mn(2). In spite of multiple quasidegeneracies (particularly for the cases of Sc(2) and Mn(2)), the GVVPT2 PECs are smooth with no discontinuities. Since these molecules have been identified as ones that widely used perturbative methods are inadequate for describing well, due to intruder state problems, unless shift parameters are introduced that can obfuscate the physics, this study suggests that the conclusion about the inadequacy of multireference perturbation theory be re-evaluated. The ground state of Sc(2) is predicted to be X(5)∑(u)(-), and its spectroscopic constants are close to the ones at the MRCISD level. Near equilibrium geometries, the 1(3)∑(u)(-) electronic state of Sc(2) is found to be less stable than the quintet ground state by 0.23 eV. The Cr(2) PEC has several features of the Rydberg-Klein-Rees (RKR) experimental curve (e.g., the pronounced shelf at elongated bond lengths), although the predicted bond length is slightly long (R(e) = 1.80 ? with cc-pVQZ compared to the experimental value of 1.68 ?). The X(1)∑(g)(+) ground state of Mn(2) is predicted to be a van der Waals molecule with a long bond length, R(e), of 3.83 ? using a cc-pVQZ basis set (experimental value = 3.40 ?) and a binding energy, D(e), of only 0.05 eV (experimental value = 0.1 eV). We obtained R(e) = 3.40 ? and D(e) = 0.09 eV at the complete basis set (CBS) limit for ground state Mn(2). Low lying excited state curves have also been characterized for all three cases (Cr(2), Mn(2), and Sc(2)) and show similar mathematical robustness as the ground states. These results suggest that the GVVPT2 multireference perturbation theory method is more broadly applicable than previously documented.     

Modeling the peroxide/superoxide continuum in 1:1 side-on adducts of O2 with Cu

The character of singlet (C(3)N(2)H(5))CuO(2) ranges smoothly between copper(III) peroxide and copper(II) superoxide with variation of the electronic character of the supporting beta-diketiminate ligand. Over the range of the variation, multireference second-order perturbation theory predicts the (1)A(1) singlet state always to be lower in energy than the lowest triplet state ((3)B(1)). The multideterminantal character of the biradical-like superoxide mesomer causes density functional theory sometimes to fail badly in predicting the relative energies of these same states, although its predictions of other properties, such as geometry, are of good quality.     

Multireference second-order perturbation theory: how size consistent is "almost size consistent"

A systematic study of the deviation from size consistency of the multireference second-order Moller-Plesset perturbation theory (MRMP2) method is presented. The size-consistency error is shown to depend on the number of monomers in a supermolecule calculation, size of basis set, number of correlated valence electrons, and size of active space. HF, F(2), and N(2) are used as test cases, with stretched bonds, to include simple, well-defined multireference character. This is essential in ensuring that MRMP2 is being tested as a multireference method. It is concluded that the MRMP2 and other multireference perturbation theory methods can exhibit significant size-consistency errors, and that the size of the error depends on the manner in which the perturbation theory is implemented.     

Reactions of hydrogen atom with hydrogen peroxide

Rate coefficients are calculated using canonical variational transition state theory with multidimensional tunneling (CVT/SCT) for the reactions H + H2O2 --> H2O + OH (1a) and H + H2O2 --> HO2 + H2 (1b). Reaction barrier heights are determined using two theoretical approaches: (i) comparison of parametrized rate coefficient calculations employing CVT/SCT to experiment and (ii) high-level ab initio methods. The evaluated experimental data reveal considerable variations of the barrier height for the first reaction: although the zero-point-exclusive barrier for (1a) derived from the data by Klemm et al. (First Int. Chem. Kinet. Symposium 1975, 61) is 4.6 kcal/mol, other available measurements result in a higher barrier of 6.2 kcal/mol. The empirically derived zero-point-exclusive barrier for (1b) is 10.4 kcal/mol. The electronic structure of the system at transition state geometries in both reactions was found to have "multireference" character; therefore special care was taken when analyzing electronic structure calculations. Transition state geometries are optimized by multireference perturbation theory (MRMP2) with a variety of one-electron basis sets, and by a multireference coupled cluster (MR-AQCCSD) method. A variety of single-reference benchmark-level calculations have also been carried out; included among them are BMC-CCSD, G3SX(MP3), G3SX, G3, G2, MCG3, CBS-APNO, CBS-Q, CBS-QB3, and CCSD(T). Our data obtained at the MRMP2 level are the most complete; the barrier height for (1a) using MRMP2 at the infinite basis set limit is 4.8 kcal/mol. Results are also obtained with midlevel single-reference multicoefficient correlation methods, such as MC3BB, MC3MPW, MC-QCISD/3, and MC-QCISD-MPWB, and with a variety of hybrid density functional methods, which are compared with high-level theory. On the basis of the evaluated experimental values and the benchmark calculations, two possible recommended values are given for the rate coefficients.     

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.     

Interactions of CO2 with cluster models of metal–organic frameworks

The interactions between carbon dioxide and cluster models of coordinatively unsaturated metal–organic frameworks (MOFs) were studied using a variety of ab initio methods. Three metal species and three organic linkers in four structures were considered in these models as a representation of the tunable nature of MOFs and the potential multireference character of such systems. Common single-reference methods, such as MP2 and CCSD(T), were compared with multireference methods based on complete active space self-consistent field theory, going as far as multireference configuration interaction with single and double excitations (MRCISD). Special consideration is taken to avoid issues of size inconsistency in the CI results, where an alternate reference is used in the interaction energy definition. The benchmark values are used to judge the adequacy of a selection of density functionals for the current systems. Symmetry-adapted perturbation theory (SAPT) decomposition was performed to elucidate the important effects that comprise the binding interactions. The systems proved to have very limited multireference character, and MP2 values were closer to the CCSD(T) benchmark than the more difficult MRCISD results. Though the SAPT total energies prove to be relatively poor approximations to the benchmark interaction energies, they reveal (in most cases) the correct trends with respect to the choice of the metal. The SAPT energy decompositions indicate that the CO₂ binding is primarily driven by electrostatics, but induction and dispersion also provide sizable, and quite similar, attractive contributions. Importantly, the small diformate model provides a faithful representation of complexes with large aromatic linkers, both in terms of the total interaction energy and the SAPT decomposition.