Multireference Approaches Research Articles

Theoretical Evaluation of Spin-Dependent Auger De-Excitation in Mn2+-Doped Semiconductor Nanocrystals

A quantitative method for analysis of fast Mn2+-centered Auger processes in Mn2+-doped semiconductors is introduced. Analytical expressions have been derived that describe the Coulomb (″direct″) and exchange (″sp–d″) contributions to Mn2+-electron Auger de-excitation rate constants. All of the quantities in these expressions can be calculated using existing electronic structure methods without the computational expense of multireference approaches. Using time-dependent density functional theory to generate input parameters, these expressions have been applied to calculate Auger de-excitation rate constants in Mn2+-doped CdS nanocrystals. The results of these calculations agree well with experiment. Analysis reveals that the large rate constants for Auger de-excitation in Mn2+-doped semiconductors are primarily attributable to effective s–d exchange coupling.

Building a Model of the Blue Cone Pigment Based on the Wild Type Rhodopsin Structure with QM/MM Methods

Understanding the mechanism of color tuning of the retinal chromophore by its host protein became one of the key issues in the research on rhodopsins. While early mutation studies addressed its genetic origin, recent studies advanced to investigate its structural origin, based on X-ray crystallographic structures. For the human cone pigments, no crystal structures have been produced, and homology models were employed to elucidate the origin of its blue-shifted absorption. In this theoretical study, we take a different route to establish a structural model for human blue. Starting from the well-resolved structure of bovine rhodopsin, we derive multiple mutant models by stepwise mutation and equilibration using molecular dynamics simulations in a hybrid quantum mechanics/molecular mechanics framework. Our 30fold mutant reproduces the experimental UV-vis absorption shift of 0.45 eV and provides new insights about both structural and genetic factors that affect the excitation energy. Electrostatic effects of individual amino acids and collaborative structural effects are analyzed using semiempirical (OM2/MRCI) and ab initio (SORCI) multireference approaches.

Biorthogonal moment expansions in coupled-cluster theory: Review of key concepts and merging the renormalized and active-space coupled-cluster methods

After reviewing recent progress in the area of the development of coupled-cluster (CC) methods for quasi-degenerate electronic states that are characterized by stronger non-dynamical correlation effects, including new generations of single- and multi-reference approaches that can handle bond breaking and excited states dominated by many-electron transitions, and after discussing the key elements of the left-eigenstate completely renormalized (CR) CC and equation-of-motion (EOM) CC methods, and the underlying biorthogonal method of moments of CC (MMCC) equations [P. Piecuch, M. Włoch, J. Chem. Phys. 123 (2005) 224105; P. Piecuch, M. Włoch, J.R. Gour, A. Kinal, Chem. Phys. Lett. 418 (2006) 467; M. Włoch, M.D. Lodriguito, P. Piecuch, J.R. Gour, Mol. Phys. 104 (2006) 2149], it is argued that it is beneficial to merge the CR-CC/EOMCC and active-space CC/EOMCC [P. Piecuch, Mol. Phys. 108 (2010) 2987, and references therein] theories into a single formalism. In order to accomplish this goal, the biorthogonal MMCC theory, which provides compact many-body expansions for the differences between the full configuration interaction and CC or, in the case of excited states, EOMCC energies, obtained using conventional truncation schemes in the cluster operator T and excitation operator Rμ, is generalized, so that one can correct the CC/EOMCC energies obtained with arbitrary truncations in T and Rμ for the selected many-electron correlation effects of interest. The resulting moment expansions, defining the new, Flexible MMCC (Flex-MMCC) formalism, and the ensuing CC(P;Q) hierarchy, proposed in the present work, enable one to correct energies obtained in the active-space CC and EOMCC calculations, in which one selects higher many-body components of T and Rμ via active orbitals and which recover much of the relevant non-dynamical and some dynamical electron correlation effects in applications involving potential energy surfaces (PESs) along bond breaking coordinates, for the effects of higher-order, primarily dynamical, correlations missing in the active-space CC/EOMCC considerations. The Flex-MMCC corrections to the active-space CC/EOMCC energies are mathematically similar to the non-iterative energy corrections defining the existing left-eigenstate CR-CC and CR-EOMCC methods, such as CR-CC(2,3) and CR-EOMCC(2,3). The potential advantages of the Flex-MMCC and CC(P;Q) formalisms are illustrated by describing the initial implementation and numerical tests of the novel CC hybrid scheme, abbreviated as CC(t;3), in which one corrects the results of the CC calculations with singles, doubles, and active-space triples, termed CCSDt, for the remaining effects due to connected triple excitations that are missing in the CCSDt considerations, but are present in the MMCC-based CR-CC(2,3) approach. By examining bond breaking in the HF, F2, and F2+ molecules, it is demonstrated that the CC(t;3) method improves the CCSDt and CR-CC(2,3) results, providing PESs that agree with those obtained with the full CC theory with singles, doubles, and triples (CCSDT) to within small fractions of a millihartree, at the fraction of the computer costs of the CCSDT calculations. Different strategies for defining active-space triples within the CC(t;3) scheme and the underlying CCSDt method are discussed. When limited to the ground-state problem, the CC(t;3) approach can be regarded as an improved and rigorously derived extension of the recently proposed CCSD(T)-h method [J. Shen, E. Xu, Z. Kou, S. Li, J. Chem. Phys. 132 (2010) 114115], in which triples corrections of the CCSD(T) type are replaced by their more robust CR-CC(2,3)-style analogs.

Wavefunction quality of single- and multi-reference coupled-cluster and configuration interaction methods: The P4 and H8 model systems

Projective and expectation value energies, variance and full CI overlap calculations for single- and multi-reference coupled cluster and configuration interaction wavefunctions are carried out. The variance proves to give an accurate measure for the quality of the wavefunction and the error of the correlation energy with respect to full CI. Special attention to the consistency case (two non-interacting H2 and four non-interacting H2 systems) is drawn. Due to the lack of valence extensivity [M. Hanrath, Theor. Chem. Acc. 121 (2008) 187] the multi-reference exponential wavefunction approach [M. Hanrath, J. Chem. Phys. 123 (2005) 84102] lacks exact separability for delocalized orbitals. The single reference formalism based multi-reference coupled cluster approach [O. Oliphant, L. Adamowicz, J. Chem. Phys. 94 (1991) 1229; P. Piecuch, O. Oliphant, L. Adamowicz, J. Chem. Phys. 99 (1993) 1875] proves to be size consistent for localized and delocalized orbitals but does not treat all references on the same footing (lack of Fermi vacuum invariance) causing symmetry issues and discontinuities when calculating potential energy surfaces. Externally restricted exponential approaches are shown to remove extensivity and consistency properties of the coupled cluster approaches.

Application of an efficient multireference approach to free-base porphin and metalloporphyrins: Ground, excited, and positive ion states

The improved virtual orbital-complete active space configuration interaction (IVO-CASCI) method is applied to determine the geometries of the ground state of free-base porphin and its metal derivatives, magnesium and zinc porphyrins. The vertical excitation energies and ionization potentials are computed at these optimized geometries using an IVO-based version of multireference Möller-Plesset (IVO-MRMP) perturbation theory. The geometries and excitation energies obtained from the IVO-CASCI and IVO-MRMP methods agree well with experiment and with other correlated many-body methods. We also provide the ground state vibrational frequencies for free-base porphin and Mg-porphyrin. All frequencies are real in contrast to self-consistent field treatments which yield an imaginary frequency. Ground state normal mode frequencies (scaled) of free-base porphin and magnesium porphyrin from IVO-CASCI and complete active space self-consistent field methods are quite similar and are consistent with Becke-Slater-Hartree-Fock exchange and Lee-Yang-Parr correlation density functional theory calculations and with experiment. In addition, geometries are determined for low-lying excited state triplets and for positive ion states of the molecules. To our knowledge, no prior experimental and theoretical data are available for these excited state geometries of magnesium and zinc porphyrins. Given that the IVO-CASCI and IVO-MRMP computed geometries and excitation energies agree favorably with experiment and with available theoretical data, our predicted excited state geometries should be equally accurate.

Vibrational Spectrum and Structure of CoO6: A Model Compound for Molecular Oxygen Reversible Binding on Cobalt Oxides and Salts; A Combined IR Matrix Isolation and Theoretical Study

The formation and structure of a novel species, a disuperoxo-cobalt dioxide complex (CoO(6)), has been investigated using matrix isolation in solid neon and argon, coupled to infrared spectroscopy and by quantum chemical methods. It is found that CoO(6) can be formed by successive complexation of cobalt dioxide by molecular oxygen without activation energy by diffusion of ground state O(2) molecules at 9K in the dark. The IR data on one combination and seven fundamentals, isotopic effects, and quantum chemical calculations are both consistent with an asymmetrical structure with two slightly nonequivalent oxygen ligands complexing a cobalt dioxide subunit. Evidence for other, metastable states is also presented, but the data are not complete. The electronic structure and formation pathway of this unique, formally +VI oxidation state, complex has been investigated using several functionals of current DFT within the broken-symmetry unrestricted formalism. It has been shown that the M06L pure local functional well reproduce the experimental observations. The ground electronic state is predicted to be an open shell (2)A'' doublet with the quartet states above by more than 9 kcal/mol and the sextet lying even higher in energy. The ground state has a strong and complex multireference character that hinders the use of more precise multireference approaches and requires caution in the methodology to be used. The geometrical, energetic, and vibrational properties have been computed.

Multi-reference state-universal coupled-cluster approaches to electronically excited states

The multi-reference (MR), general model space (GMS), state-universal (SU), coupled-cluster (CC) method with singles and doubles (GMS-SU-CCSD), as well as its triple-corrected versions GMS-SU-CCSD(T), are employed to assess their ability to describe low-lying excited states of various molecules, with an emphasis on a simultaneous handling of several states of the same symmetry species. A special attention is given to the role of the so-called C-conditions that account for non-vanishing internal cluster amplitudes when relying on an incomplete GMS, as well as to the choice of suitable model spaces and a perturbative account of secondary triples. The ambiguities arising when using large basis sets are also pointed out. To achieve a general assessment of the potential of the GMS-type SU-CC approaches, the vertical excitation energies of several species, including the challenging BN diatomic as well as larger systems, namely formaldehyde, trans-butadiene, formamide, and benzene are considered. These results are compared with those provided by the equation-of-motion EOM-CCSD method and, whenever available, the density functional theory results and experimental data. These comparisons clearly demonstrate the usefulness of GMS-type MR-CC approaches.

Two‐ and three‐state conical intersections in the electron capture dissociation of disulfides: The importance of multireference calculations

AbstractThe SS bond cleavage produced upon electron attachment to disulfides was generally assumed to be an adiabatic process because the added electron occupies the σ*(SS) antibonding orbital. This is clearly the case in the parent HSSH compound, but not necessarily in XSSX′ derivatives, where the substituents X and X′ are different. Through the use of MS‐CASPT2 calculations, we have shown that the dissociation of the SS two‐center‐three‐electron bond in these asymmetric XSSX′ compounds requires the interaction of at least two states, in order to localize the extra electron in one of the fragments upon dissociation. This is actually the case for the CH3SSNH2 derivative, where the most favorable dissociation process locates the extra electron at the SCH3 moiety, the SCH3· + SNH dissociation limit being 66 kJ mol−1 higher in energy. The situation is still more complex when one of the substituents is an OH group, because, in this particular case, the most favorable process is the dissociation of the SO rather than the SS bond. Besides, the two dissociation limits CH3SS· + OH− and CH3SS− + OH· are accidentally degenerate, so the SO bond fission involves a three‐state conical intersection. This constitutes the first example of a three‐state crossing computed by accurate ab initio calculations involved in a nonphotochemical reaction. These findings highlight the necessity of using multireference approaches to appropriately describe the electron capture dissociation of disulfide bridges. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem 111: 3316–3323, 2011

Multi-reference many-body perturbation theory and coupled cluster developments

Multi-reference methods for the description of quasi-degenerate ground states and excited states of many-electron systems have been one of the main objectives of scientific activity at the Quantum Theory Project (QTP) University of Florida in Gainesville. The pioneering works by Dr. Per-Olov Löwdin on the partitioning technique, and important contributions to the development of multi-reference methods made by Dr. Monkhorst and Dr. Bartlett must be mentioned here. Dr. Bartlett, his group and its visitors have been especially active in this field. In this paper we describe some of the multi-reference developments, problems that have been faced and solved as well as those which still require solving. The focus is on multi-reference many-body perturbation theory and coupled-cluster approaches. We present recent advances and future perspectives in the development of these methods. Special attention is paid to two multi-reference approaches that seem very promising and have been either initiated or developed at QTP.

Formation and Structure of Co2O4: A Combined IR Matrix Isolation and Theoretical Study

The formation and structure of dicobalt tetroxide (Co₂O₄) has been investigated using matrix isolation in solid neon and argon coupled to infrared spectroscopy and quantum chemical methods. It is found that Co₂O₄ can be formed by dimerization of cobalt dioxide without activation energy by diffusion of ground state CoO₂ molecules at 9 K in the dark. The IR data on eight fundamentals, isotopic effects and quantum chemical calculations are both consistent with a centro-symmetrical structure with two pairs of equivalent oxygen atoms, engaged in a stronger terminal Co-O bond and in a weaker bridging Co-O-Co position. Evidence for other, metastable states is also presented, but the data are not conclusive. The electronic structure and formation pathway has been investigated using the Tao-Perdew-Staroverov-Scuseria/triple-zeta valence polarived basis set (TPSS/TZVP) and broken symmetry unrestricted density functional theory (BS-UDFT) approach and the ground electronic state is predicted to be an open shell ¹Ag singlet with the quintet, triplet, septet, and nonet states above by 3.3, 4.9, 9.3, and 27.7 kcal/mol, respectively, but certainly has a complex multireference character that hinders the use of more precise multireference approaches. Different formation pathways have been considered, and the 2(O═Co═O) → Co₂O₄ dimerization reaction is found to be the only barrierless channel and to be strongly exothermic. Comparisons with another transition metal (TM) oxide system (V₂O₄) suggests that the difference in predicted ground state geometries in TM₂O₄ systems might be due in HOMO-LUMO shapes of the isolated dioxide subunits and optimal overlap configurations.

Theoretical Study of the HOCH2OO• + HO2 • Reaction: Detailed Molecular Mechanisms of the Three Reaction Channels

Abstract The HO2 • + HOCH2OO• reaction was theoretically investigated, using various high-level, single-reference Complete Basis Set methods including CBS-QB3, CBS-QCI/APNO and CBS-Q(MPW1B95) and a new multi-reference CI-PT2 approach. Three major product channels under atmospheric conditions were identified and their molecular mechanisms elucidated in great detail by Intrinsic Reaction Coordinate Analyses (IRC) at the B3LYP/6–311G(d,p) level: (i) Direct head-to-tail H-atom abstraction from the hydroperoxy radical by the alkylperoxy, occurring on the triplet Potential Energy Surface (PES) leading to HOCH2OOH + O2; (ii) A two-step rearrangement of the initial singlet HOCH2OOOOH tetroxide complex to form HC(O)OH + •OH + HO2 •; (iii) A multi-step rearrangement of the initial HOCH2OOOOH tetroxide to yield HC(O)OH + O2(1Δ) + H2O, about twice as fast as the former channel on the singlet-surface. The findings provide an explanation for the observed hydroxyl radical formation in this reaction (Jenkin et al., Phys. Chem. Chem. Phys. 9 (2007) 3149) and rationalize the high overall rate and its pronounced negative temperature dependence (Veyret et al., J. Phys. Chem. 93 (1989) 2368).

Communication: Energetics of reaction pathways for reactions of ethenol with the hydroxyl radical: The importance of internal hydrogen bonding at the transition state

We find high multireference character for abstraction of H from the OH group of ethenol (also called vinyl alcohol); therefore we adopt a multireference approach to calculate barrier heights for the various possible reaction channels of OH+C(2)H(3)OH. The relative barrier heights of ten possible saddle points for reaction of OH with ethenol are predicted by multireference Møller-Plesset perturbation theory with active spaces based on correlated participating orbitals (CPOs) and CPO plus a correlated pi orbital (CPO+pi). Six barrier heights for abstracting H from a C-H bond range from 3.1 to 7.7 kcal/mol, two barrier heights for abstracting H from an O-H bond are both 6.0 kcal/mol, and two barrier heights for OH addition to the double bond are -1.8 and -2.8 kcal/mol. Thus we expect abstraction at high-temperature and addition at low temperature. The factor that determines which H is most favorable to abstract is an internal hydrogen bond that constitutes part of a six-membered ring at one of the abstraction saddle points; the hydrogen bond contributes about 3 kcal/mol stabilization.

Ab initio investigation of the ground X2A′ [ X2A1] and low-lying excited electronic states of C 2B

Ab initio single- and multi-reference approaches, i.e. restricted open-shell Hartree–Fock (ROHF), coupled-cluster singles-and-doubles with approximate triples correction [CCSD(T)], complete active space self-consistent field (CASSCF), 2nd-order CAS perturbation theory (CASPT2), multi-reference singles-and-doubles configuration-interaction (MRSDCI), and density-functional B3LYP – in the cc-pVQZ basis set – have been employed to investigate the structure of the ground X 2 A′ [ X 2 A 1] state of the triatomic C 2B. We found that the ground state of the cyclic C 2B is prone to symmetry breaking along the asymmetric stretching coordinate. Single-reference highly correlated CCSD(T)/aug-cc-pVQZ method (as well as ‘static’-correlated CASSCF, and B3LYP) predicts two equivalent global minima of the adiabatic potential energy function, with the barrier between the minima of about 20 cm −1 at C 2v geometry. Wavefunctions almost free from SB are obtained using the MRSDCI approach based on the CASSCF and state-averaged (SA) CASSCF wavefunctions. Highly correlated multi-reference full-valence (FV) CASPT2 model exactly predicts a C 2v equilibrium geometry; however, the highest level FV-CAS-MRSDCI/aug-cc-pVQZ method predicts lower-symmetry structure. We found that this effect is not artifactual as stated in Leonard et al. (2000) [5]. SB arises from a pseudo-Jahn–Teller effect (PJTE), the non-adiabatic interaction with the first excited 1 2 B 2 state, which has the same symmetry as the asymmetric stretching coordinate in the ground state. Several low-lying excited electronic states have been optimized using SA-CAS-MRSDCI/aug-cc-pVTZ procedure. The first excited doublet is 1 2 B 2 (0.95 eV), and the lowest-lying quartet state 1 4 B 2 (1.74 eV). The vertical excitation spectra and transition matrix elements for six low-lying doublet and four quartet states were calculated using full valence SA-CAS-MRSDCI/aug-cc-pVTZ procedure.

Magnetic Interactions in Phenyl-Bridged Nitroxide Diradicals: Conformational Effects by Multireference and Broken Symmetry DFT Approaches

In the present paper we report the key results of a comprehensive computational study aimed at investigating the dependence of the singlet-triplet energy gap in phenyl-bridged bis-nitroxide diradicals, on the basis set and on soft structural parameters like torsion and pyramidalization. We have compared the BS-DFT technique with the post-Hartree-Fock DDCI2 multireference approach. With this latter method we have also studied the different role that sigma and pi core and virtual orbitals have in the resulting singlet-triplet energy gap.The results obtained represent one step forward in the definition of a protocol for an efficient and reliable computation of spin-spin coupling in diradical systems.

Electronic wave packet motion in water dimer cation: A many electron description

The time evolution of a non-stationary electronic wave packet created by a sudden ionization is computed using a multi-reference time-dependent approach at a frozen geometry of the nuclei. The methodology is illustrated for the water dimer cation. The electron density as well as the dipole moment are computed as a function of time and used as a probe for the charge migration following upon the sudden removal of a valence electron. It is shown that there is significant purely electronic dynamics on a few femtosecond time scale and that the characteristics of the charge motion depend of the degree of localization of the initial hole formed in the impulsive ionization process.

Symmetry breaking in the ground state of BNB: A high level multireference study

A series of multireference approaches based on the SA-CASSCF wave function, i.e., CASPT2, MRCI, MRCI + Q, and MRAQCC with single- or multireference states, have been employed to investigate the symmetry breaking effect in the ground state X (2)Sigma(u)(+) of the triatomic BNB radical. We found that the mixing of the reference states contributes significantly to the dynamical correlation energy, which strongly affects the geometry of the ground state. Our results show that BNB in its ground state has a linear noncentrosymmetric structure with two equivalent global minima of the adiabatic potential energy surface and, respectively, two oppositely directed dipole moments of about 2 D. The barrier between the minima is about 20 cm(-1). The origin of the double-minimum potential in the ground state of BNB is explained as due to the pseudo-Jahn-Teller effect involving vibronic interaction with the first excited state A (2)Sigma(g)(+) via the asymmetric stretching vibrations.

State specific equation of motion coupled cluster method in general active space

The state specific equation of motion coupled cluster (SS-EOMCC) method is an internally contracted multireference approach, applicable to both ground and excited states. Attractive features of the method are as follows: (1) the SS-EOMCC wave function is qualitatively correct and rigorously spin adapted, (2) both orbitals and dynamical correlation are optimized for the target state, (3) nondynamical correlation and differential orbital relaxation effects are taken care of by a diagonalization of the transformed Hamiltonian in the multireference configuration-interaction singles space, (4) only one- and two-particle density matrices of a complete-active-space self-consistent-field reference state are needed to define equations for the cluster amplitudes, and (5) the method is invariant with respect to orbital rotations in core, active, and virtual subspaces. Prior applications focused on biradical-like systems, in which only one extra orbital is needed to construct the active space, and similarly, single bond breaking processes. In this paper, the applicability of the method is extended to systems of general active spaces. Studies on F(2), H(2)O, CO, and N(2) are carried out to gauge its accuracy. The convergence strategy is discussed in detail.

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.

Time-reversal symmetry in general coupled cluster theory

In this paper, the symmetry property of time reversal, which is a consequence of the fundamental quantum-mechanical microreversibility principle, is analyzed in the context of relativistic open-shell and multireference coupled cluster theory. The fact is discussed that in projective theories, in contrast to variational theories such as configuration interaction models, the implementation of time-reversal symmetry is far less straightforward and even depends on the projective model theory in question. An operator basis of time reversal even $n$-body operators is introduced which allows for the formulation of a fully time-reversal-adapted and practical general coupled cluster theory, i.e., covering open-shell models and various types of multireference approaches. The resulting model theory is noncommuting but truncating, albeit at a higher order of commutator nesting.

Configuration interaction based on constrained density functional theory: A multireference method

Existing density functional theory (DFT) methods are typically very effective in capturing dynamic correlation, but run into difficulty treating near-degenerate systems where static correlation becomes important. In this work, we propose a configuration interaction (CI) method that allows one to use a multireference approach to treat static correlation but incorporates DFT's efficacy for the dynamic part as well. The new technique uses localized charge or spin states built by a constrained DFT approach to construct an active space in which the effective Hamiltonian matrix is built. These local configurations have significantly less static correlation compared to their delocalized counterparts and possess an essentially constant amount of self-interaction error. Thus their energies can be reliably calculated by DFT with existing functionals. Using a small number of local configurations as different references in the active space, a simple CI step is then able to recover the static correlation missing from the localized states. Practical issues of choosing configurations and adjusting constraint values are discussed, employing as examples the ground state dissociation curves of H(2) (+), H(2), and LiF. Excellent results are obtained for these curves at all interatomic distances, which is a strong indication that this method can be used to accurately describe bond breaking and forming processes.