Multireference Perturbation Theory Methods Research Articles

Ab initio multi-reference perturbation theory calculations of the ground and some excited electronic states of the RbYb molecule

The potential energy curves of the ground and twelve low-lying excited electronic states of the RbYb molecule have been calculated using the multi-reference perturbation theory method at the CASSCF/XMCQDPT2 level of theory including the spin-orbit coupling. The electronic term energies, equilibrium internuclear distances, dissociation energies, transition dipole moments, the sequences of vibrational energies, the harmonic vibrational frequencies, and the Franck–Condon factors have been predicted. The potential energy curves at the asymptotic limits are in a good agreement with the experimental data for rubidium and ytterbium atoms. The obtained data would be useful for spectral experiments with RbYb molecules.

Ab initio multi-reference perturbation theory calculations of the ground and low-lying electronic states of the KRb molecule

The potential energy curves of the low-lying electronic states correlating up to the limit K(4p)+Rb(5s) of KRb molecule have been calculated using the multi-reference perturbation theory method at the CASSCF/XMCQDPT2 level of theory without and with spin–orbit coupling. The calculated parameters of the ground X1Σ+ state are in the best agreement among all previously performed ab initio calculations for the KRb molecule. The calculated vibrational intervals of the ground electronic term of the 39K85Rb molecule describe the experiment with the accuracy within ±1cm−1. The calculated intensities of the 21Σ+ (v′=3, J′=26)→X1Σ+ (v″=0…24, J″=25, 27) transitions satisfactory reflect the experimentally observed intensities distribution.

Role of zwitterions in kindling fluorescent protein photochemistry.

Kindling fluorescent protein (KFP), one of the chronologically first members of photoswitchable colored proteins from the green fluorescent protein (GFP) family, increasingly attracts efforts from experimental and theoretical sides. Ambiguous conclusions in solving puzzles of photochemistry of KFP and of its parent natural protein asFP595 are partially explained by lack of reliable theoretical data on chromophore properties in the electronically excited state. We report the results of state-of-the-art quantum chemistry calculations of the structure and energy of the KFP chromophore, 2-acetyl-,4-(p-hydroxybenzylidene)-1-methyl-5-imidazolone (AHBMI), both in the ground and excited states. Ground state equilibrium structures of anionic and zwitterionic protonation states of AHBMI were computed by the conventional MP2 method while excited state structures were characterized by the extended multireference perturbation theory method XMCQDPT2 including optimization of geometry parameters at this level. In particular, the computational results demonstrate that the basicity of the N2 nitrogen atom of the imidazolinone ring should noticeably increase upon electronic excitation, thus affecting excited state proton transfer events in proteins. The results of these simulations as well as of quantum mechanical-molecular mechanical (QM/MM) calculations for model protein systems evidence that KFP conformations with the zwitterionic chromophore are hardly expected to occur in the ground state, but may be populated upon excitation.

Multiconfiguration Pair-Density Functional Theory.

We present a new theoretical framework, called Multiconfiguration Pair-Density Functional Theory (MC-PDFT), which combines multiconfigurational wave functions with a generalization of density functional theory (DFT). A multiconfigurational self-consistent-field (MCSCF) wave function with correct spin and space symmetry is used to compute the total electronic density, its gradient, the on-top pair density, and the kinetic and Coulomb contributions to the total electronic energy. We then use a functional of the total density, its gradient, and the on-top pair density to calculate the remaining part of the energy, which we call the on-top-density-functional energy in contrast to the exchange-correlation energy of Kohn-Sham DFT. Because the on-top pair density is an element of the two-particle density matrix, this goes beyond the Hohenberg-Kohn theorem that refers only to the one-particle density. To illustrate the theory, we obtain first approximations to the required new type of density functionals by translating conventional density functionals of the spin densities using a simple prescription, and we perform post-SCF density functional calculations using the total density, density gradient, and on-top pair density from the MCSCF calculations. Double counting of dynamic correlation or exchange does not occur because the MCSCF energy is not used. The theory is illustrated by applications to the bond energies and potential energy curves of H2, N2, F2, CaO, Cr2, and NiCl and the electronic excitation energies of Be, C, N, N(+), O, O(+), Sc(+), Mn, Co, Mo, Ru, N2, HCHO, C4H6, c-C5H6, and pyrazine. The method presented has a computational cost and scaling similar to MCSCF, but a quantitative accuracy, even with the present first approximations to the new types of density functionals, that is comparable to much more expensive multireference perturbation theory methods.

A theoretical study of the intramolecular charge transfer in 4-(dimethylamino)benzethyne.

We have investigated the non-adiabatic relaxation processes occurring in the singlet manifold of 4-(dimethylamino)benzethyne (DMABE), a molecule isoelectronic with 4-(dimethylamino)benzonitrile (DMABN) but lacking its characteristic dual fluorescence, using multireference perturbation theory methods. The results obtained point out to the existence of a two-fold decay mechanism in which the population of the initially accessed La state bifurcates towards a locally excited (LE) and a πσ* state. Further relaxation to an emitting intramolecular charge transfer (ICT) state is impeded due to the presence of pronounced energy barriers along their associated potential energy surfaces. These results provide further evidence of the role of πσ* states in the non-adiabatic relaxation processes of dialkylaminobenzonitriles.

Quantum Chemical Study of the Initial Step of Ozone Addition to the Double Bond of Ethylene

The mechanisms of the initial step in chemical reaction between ozone and ethylene were studied by multireference perturbation theory methods (MRMP2, CASPT2, NEVPT2, and CIPT2) and density functional theory (OPW91, OPBE, and OTPSS functionals). Two possible reaction channels were considered: concerted addition through the symmetric transition state (Criegee mechanism) and stepwise addition by the biradical mechanism (DeMore mechanism). Predicted structures of intermediates and transition states, the energies of elementary steps, and activation barriers were reported. For the rate-determining steps of both mechanisms, the full geometry optimization of stationary points was performed at the CASPT2/cc-pVDZ theory level, and the potential energy surface profiles were constructed at the MRMP2/cc-pVTZ, NEVPT2/cc-pVDZ, and CIPT2/cc-pVDZ theory levels. The rate constants and their ratio for reaction channels calculated for both mechanisms demonstrate that the Criegee mechanism is predominant for this reaction. These results are also in agreement with the experimental data and previous computational results. The structure of DeMore prereactive complex is reported here for the first time at the CCSD(T)/cc-pVTZ and CASPT2/cc-pVDZ levels. Relative stability of the complexes and activation energies were refined by single-point energy calculations at the CCSD(T)-F12/VTZ-F12 level. The IR shifts of ozone bands due to formation of complexes are presented and discussed.

The role of the magnetic orbitals in the calculation of the magnetic coupling constants from multireference perturbation theory methods

The use of multireference perturbation theory (MRPT) for the calculation of the magnetic coupling in binuclear complexes has shown to give poor results if applied on a minimal active space complete active space self-consistent field (CASSCF) wavefunction. In this work, we identify the origin of this problem in the starting CASSCF orbitals, which are exceedingly localized on the metal atoms. Focusing on the case of antiferromagnetic systems, it is shown that the form of the active orbitals has a dramatic effect on the relative description of the neutral and ionic structures. Finally, a simple and computational inexpensive strategy is proposed for the calculation of a set of magnetic orbitals describing in a more balanced way the neutral and ionic structures. The use of these orbitals, instead the CASSCF ones, in minimal active space MRPT2 calculations leads to a marked improvement of the J values, which become in reasonable agreement with those obtained with the expensive high level difference dedicated configuration interaction approach and with the experimental values.

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 Å with cc-pVQZ compared to the experimental value of 1.68 Å). 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 Å using a cc-pVQZ basis set (experimental value = 3.40 Å) and a binding energy, D(e), of only 0.05 eV (experimental value = 0.1 eV). We obtained R(e) = 3.40 Å 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.

On the role of the triplet state in the cis/trans isomerization of rhodopsin: A CASPT2//CASSCF study of a model chromophore

AbstractThe possibility of population of the lowest‐lying triplet state (T1) in the early events of the photochemical isomerization process of a model chromophore of Rhodopsin (Rh) has been analyzed using multireference perturbation theory (CASPT2//CASSCF) methods. It is shown that the characteristics of the isomerization process namely small S1−T1 gap, presence of hydrogen out of plane active vibrational modes, and existence of a dense manifold of vibrational states, render possible the fulfilment of the conditions needed for the population of T1. The possible consequences for the photochemistry and photophysics of Rh are also discussed. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem 111:3431–3437, 2011

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 Møller-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 Å, 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.

An Ab Initio Study of Substituent Effects on the Excited States of Purine Derivatives

Several excited singlet electronic states of purine nucleobases and related derivatives have been calculated using high-level multireference perturbation theory methods. Purine derivatives with one or two amino or carbonyl groups substituted at positions C(2) and/or C(6) of the purine ring have been included in the study. The effect of the substituents on excited-state energies and wave functions is examined. Some trends have been observed, such as the fact that substitution at the C(2) position decreases the energy of the first pi --> pi* state considerably. Although basic qualitative features of the effects can be explained with the simple frontier molecular orbital theory, ab initio calculations are required to describe the effects quantitatively.

Singlet−Triplet Energy Splitting and Excited States of Phenylnitrene

The vertical and adiabatic singlet-triplet energy splittings (Delta E ST) of phenylnitrene were computed by a variety of multireference configuration interaction and perturbation theory methods employing basis sets of up to quadruple-xi quality and extrapolation to the complete basis set limit. The vertical and adiabatic energy gaps are 18.9 and 15.9 kcal mol (-1), respectively, the latter in reasonable agreement with the revised experimental value of 15.1 +/- 0.2 kcal mol (-1). The energy difference between both states at the geometry of the a (1)A 2 singlet state was also considered and amounts to 13.8 kcal mol (-1). In obtaining accurate state energy splittings, basis set completeness turns out to be a more important issue than the level of dynamical electron correlation treatment. Density functional theory that is frequently employed to investigate phenylnitrenes and their rearrangements yields varying results and, depending on the functional, gives adiabatic energy differences between 9 and 16 kcal mol (-1). The b (1)A 1 state has a similar geometry as the ground state of 1 and is 31 kcal mol (-1) higher in energy. According to best estimates, the next higher singlet states, c (1)A 1 and d (1)B 1, are 57 and 72 kcal mol (-1) above the ground state. In the triplet manifold, vertical excitation energies to the A (3)B 1 and B (3)A 2 states are 71 and 77 kcal mol (-1), respectively.

Can the second order multireference perturbation theory be considered a reliable tool to study mixed-valence compounds?

In this paper, the problem of the calculation of the electronic structure of mixed-valence compounds is addressed in the frame of multireference perturbation theory (MRPT). Using a simple mixed-valence compound (the 5,5(') (4H,4H('))-spirobi[ciclopenta[c]pyrrole] 2,2('),6,6(') tetrahydro cation), and the n-electron valence state perturbation theory (NEVPT2) and CASPT2 approaches, it is shown that the ground state (GS) energy curve presents an unphysical "well" for nuclear coordinates close to the symmetric case, where a maximum is expected. For NEVPT, the correct shape of the energy curve is retrieved by applying the MPRT at the (computationally expensive) third order. This behavior is rationalized using a simple model (the ionized GS of two weakly interacting identical systems, each neutral system being described by two electrons in two orbitals), showing that the unphysical well is due to the canonical orbital energies which at the symmetric (delocalized) conformation lead to a sudden modification of the denominators in the perturbation expansion. In this model, the bias introduced in the second order correction to the energy is almost entirely removed going to the third order. With the results of the model in mind, one can predict that all MRPT methods in which the zero order Hamiltonian is based on canonical orbital energies are prone to present unreasonable energy profiles close to the symmetric situation. However, the model allows a strategy to be devised which can give a correct behavior even at the second order, by simply averaging the orbital energies of the two charge-localized electronic states. Such a strategy is adopted in a NEVPT2 scheme obtaining a good agreement with the third order results based on the canonical orbital energies. The answer to the question reported in the title (is this theoretical approach a reliable tool for a correct description of these systems?) is therefore positive, but care must be exercised, either in defining the orbital energies or by resorting to the third order using for them the standard definition.

Models for dioxygen activation by the CuB site of dopamine β-monooxygenase and peptidylglycine α-hydroxylating monooxygenase

On the basis of spectroscopic and crystallographic data for dopamine beta-monooxygenase and peptidylglycine alpha-hydroxylating monooxygenase (PHM), a variety of ligand sets have been used to model the oxygen-binding Cu site in these enzymes. Calculations which employed a combination of density functional and multireference second-order perturbation theory methods provided insights into the optimal ligand set for supporting eta (1) superoxo coordination as seen in a crystal structure of a precatalytic Cu/O(2) complex for PHM (Prigge et al. in Science 304:864-867, 2004). Anionic ligand sets stabilized eta (2) dioxygen coordination and were found to lead to more peroxo-like Cu-O(2) complexes with relatively exergonic binding free energies, suggesting that these adducts may be unreactive towards substrates. Neutral ligand sets (including a set of two imidazoles and a thioether), on the other hand, energetically favored eta (1) dioxygen coordination and exhibited limited dioxygen reduction. Binding free energies for the 1:1 adducts with Cu supported by the neutral ligand sets were also higher than with their anionic counterparts. Deviations between the geometry and energetics of the most analogous models and the PHM crystal structures suggest that the protein environment influences the coordination geometry at the Cu(B) site and increases the lability of water bound to the preoxygenated reduced form. Another implication is that a neutral ligand set will be critical in biomimetic models in order to stabilize eta (1) dioxygen coordination.

Origin of warping in theE⊗eJahn-Teller problem: Quadratic vibronic coupling versus anharmonicity and application toNaCl:Rh2+and triangular molecules

A general method is suggested which allows one to separate the quadratic, cubic, and pseudo Jahn-Teller contributions to the warping of the adiabatic potential energy surface (APES) of the $E\ensuremath{\bigotimes}e$ problem employing ab initio calculations. Numerical results were obtained for $\mathrm{NaCl}:{\mathrm{Rh}}^{2+}$ in a cluster approximation and triangular molecules $({\mathrm{Na}}_{3},{\mathrm{K}}_{3},{\mathrm{Cu}}_{3},{\mathrm{Ag}}_{3})$ using density functional theory (DFT) and multireference second-order perturbation theory (CASPT2) methods. A largely unexpected result is that the contribution of cubic anharmonicity to the energy barrier between the minima of the lower and upper branches of the APES is dominant in all the systems and amounts for not less than 60% of the total. Another feature is that the three different contributions mentioned above may have different signs, thus either enhancing or diminishing each other, affecting in different ways the lower and upper branches of the APES. Other details of the numerical results are also analyzed.

Effective interaction techniques for the Gamow shell model

We apply a contour deformation technique in momentum space to the newly developed Gamow shell model, using the drip-line nuclei $^{5}\mathrm{He}$, $^{6}\mathrm{He}$, and $^{7}\mathrm{He}$ as test cases. A major problem in Gamow shell-model studies of nuclear many-body systems is the increasing number of many-body configurations due to the large number of resonant and complex-continuum single-particle orbits necessary to reproduce multiparticle bound and resonant states. We address this problem using two different effective operator approaches generalized to the complex momentum plane. These are the Lee-Suzuki similarity transformation method for complex interactions and the multireference perturbation theory method. We show that a combination of these two approaches results in a drastic truncation of the number of relevant configurations compared with direct diagonalization. This offers interesting perspectives for studies of weakly bound systems.

The XΣg+1, BΔg1, and B′Σg+1 states of C2: A comparison of renormalized coupled-cluster and multireference methods with full configuration interaction benchmarks

Unusual bonding and electronic near degeneracies make the lowest-lying singlet states of the C2 molecule particularly challenging for electronic structure theory. Here we compare two alternative approaches to modeling bond-breaking reactions and excited states: sophisticated multireference configuration interaction and multireference perturbation theory methods, and a more "black box," single-reference approach, the completely renormalized coupled-cluster method. These approximate methods are assessed in light of their ability to reproduce the full configuration interaction potential energy curves for the X1Sigmag+, B1Deltag, and B' 1Sigmag+ states of C2, which are numerically exact solutions of the electronic Schrodinger equation within the space spanned by a 6-31G* basis set. Both the multireference methods and the completely renormalized coupled-cluster approach provide dramatic improvements over the standard single-reference methods. The multireference methods are nearly as reliable for this challenging test case as for simpler reactions which break only single bonds. The completely renormalized coupled-cluster approach has difficulty for large internuclear separations R in this case, but over the wide range of R=1.0-2.0 A, it compares favorably with the more complicated multireference methods.

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.

The CIPT2 method: Coupling of multi-reference configuration interaction and multi-reference perturbation theory. Application to the chromium dimer

The potential energy function of is computed using the MRCI+Q (multi-reference configuration interaction with Davidson correction) and CASPT2 (complete active space second-order multi-reference perturbation theory) methods and very large basis sets. In addition, a new hybrid method denoted CIPT2, in which excitations solely from the active space are treated by MRCI and the remaining ones by perturbation theory, is proposed and applied. These methods are used to analyse valence and core correlation effects on the potential energy function. MRCI calculations which fully include correlation of the 3p, 3d, and 4s electrons yield excellent agreement with experimental values for the equilibrium distance Re and the harmonic vibrational constant ω e . Correlation of the (3p) orbitals is found to be very important. At the MRCI+Q level this reduces Re by 0.08 Å and increases ω e by about 200 cm−1. These effects are not correctly reproduced in CASPT2 and CIPT2. It is shown that in CASPT2 the valence correlation effects are significantly overestimated and the distance dependence of the core correlation is incorrect. The basis set limit of the dissociation energies for MRCI+Q and CASPT 2 is estimated to be 1.3 eV and 1.9 eV, respectively. The remaining error (≈ 0.2 − 0.3 eV) of the MRCI+Q dissociation energy is attributed to unlinked cluster and higher excitation effects, which are only approximately accounted for by the Davidson correction.

On the vertical excitation energy of cyclopentadiene.

The vertical excitation energy for the lowest valence pi-->pi(*) transition of cyclopentadiene is investigated. Using a combination of high-level theoretical methods and spectroscopic simulations, the vertical separation at the ground state geometry is estimated to be 5.43+/-0.05 eV. This value is intermediate between those calculated with coupled-cluster and multireference perturbation theory methods and is about 0.13 eV higher than the observed maximum in the absorption profile.