Low-lying excited states of model proteins: Performances of the CC2 method versus multireference methods.

A benchmark set of relevant geometries of a model protein, the N-acetylphenylalanylamide, is presented to assess the validity of the approximate second-order coupled cluster (CC2) method in studying low-lying excited states of such bio-relevant systems. The studies comprise investigations of basis-set dependence as well as comparison with two multireference methods, the multistate complete active space 2nd order perturbation theory (MS-CASPT2) and the multireference difference dedicated configuration interaction (DDCI) methods. First of all, the applicability and the accuracy of the quasi-linear multireference difference dedicated configuration interaction method have been demonstrated on bio-relevant systems by comparison with the results obtained by the standard MS-CASPT2. Second, both the nature and excitation energy of the first low-lying excited state obtained at the CC2 level are very close to the Davidson corrected CAS+DDCI ones, the mean absolute deviation on the excitation energy being equal to 0.1 eV with a maximum of less than 0.2 eV. Finally, for the following low-lying excited states, if the nature is always well reproduced at the CC2 level, the differences on excitation energies become more important and can depend on the geometry.

Highly efficient perturbative + variational strategy based on orthogonal valence bond theory for the evaluation of magnetic coupling constants. Application to the trinuclear Cu(ii) site of multicopper oxidases.

A new strategy based on orthogonal valence-bond analysis of the wave function combined with intermediate Hamiltonian theory has been applied to the evaluation of the magnetic coupling constants in two AF systems. This approach provides both a quantitative estimate of the J value and a detailed analysis of the main physical mechanisms controlling the coupling, using a combined perturbative + variational scheme. The procedure requires a selection of the dominant excitations to be treated variationally. Two methods have been employed: a brute-force selection, using a logic similar to that of the CIPSI approach, or entanglement measures, which identify the most interacting orbitals in the system. Once a reduced set of excitations (about 300 determinants) is established, the interaction matrix is dressed at the second-order of perturbation by the remaining excitations of the CI space. The diagonalization of the dressed matrix provides J values in good agreement with experimental ones, at a very low-cost. This approach demonstrates the key role of d → d* excitations in the quantitative description of the magnetic coupling, as well as the importance of using an extended active space, including the bridging ligand orbitals, for the binuclear model of the intermediates of multicopper oxidases. The method is a promising tool for dealing with complex systems containing several active centers, as an alternative to both pure variational and DFT approaches.

Magnetic coupling constants of self-assembled Cu(II) [3×3] grids: alternative spin model from theoretical calculations.

This paper reports a theoretical analysis of the electronic structure and magnetic properties of a ferromagnetic Cu(II) [3×3] grid. A two-step strategy, combining calculations on the whole grid and on binuclear fragments, has been employed to evaluate all the magnetic interactions in the grid. The calculations confirm an S = 7/2 ground state, which is in accordance with the magnetisation versus field curve and the thermal dependence of the magnetic moment data. Only the first-neighbour coupling terms present non-negligible amplitudes, all of them in agreement with the structure and arrangement of the Cu 3d magnetic orbitals. The results indicate that the dominant interaction in the system is the antiferromagnetic coupling between the ring and the central Cu sites (J3 = J4 ≈ -31 cm(-1)). In the ring two different interactions can be distinguished, J1 = 4.6 cm(-1) and J2 = -0.1 cm(-1), in contrast to the single J model employed in the magnetic data fit. The calculated J values have been used to determine the energy level distribution of the Heisenberg magnetic states. The effective magnetic moment versus temperature plot resulting from this ab initio energy profile is in good agreement with the experimental curve and the fitting obtained with the simplified spin model, despite the differences between these two spin models. This study underlines the role that the theoretical evaluations of the coupling constants can play on the rationalisation of the magnetic properties of these complex polynuclear systems.

A rational reduction of CI expansions: combining localized molecular orbitals and selected charge excitations.

Based on localized molecular orbitals, the proposed method reduces large configuration interaction (CI) spaces while maintaining agreement with reference values. Our strategy concentrates the numerical effort on physically pertinent CI-contributions and is to be considered as a tool to tackle large systems including numerous open-shells. To show the efficiency of our method we consider two 4-electron parent systems. First, we illustrate our approach by describing the van der Waals interactions in the (H2)2 system. By systematically including local correlation, dispersion and charge transfer mechanisms, we show that 90% of the reference full CI dissociation energy of the H2 dimer is reproduced using only 3% of the full CI space. Second, the conformational cis/trans rotation barrier of the butadiene molecule is remarkably reproduced (97% of the reference value) with less than 1% of the reference space. This work paves the way to numerical strategies which afford the electronic structure determination of large open-shell systems avoiding the exponential limitation. At the same time, a physical analysis of the contents of the wave function is offered.

Do π-π stacking interactions really play a role in the magnetic coupling mechanisms of [Cu2(µ2-CH3COO)2L2(H2O)2]n+ (L = heterocyclic base, n = 0, 2) complexes? An ab initio inspection.

The magnetic properties of two bis-acetate binuclear copper(II) complexes, namely [Cu2(µ2-CH3COO)2(bpydiol-H)2(H2O)2] (bpydiol-H = mono deprotonated 2,2'-bipyridine-3,3'-diol) and [Cu2(µ2-CH3COO)2(phen)2(H2O)2](2+) (phen = 1,10-phenantroline), is revisited using ab initio wave function-based calculations (CASSCF, DDCI). Thanks to an analysis of the magnetic exchange coupling based on localized orbitals, it is shown that, unlike stated in the original work [C. Hou et al. Dalton Trans. 2008, 5970], π-π interactions do not contribute to the overall antiferromagnetism character of these complexes.

Multi-scale multireference configuration interaction calculations for large systems using localized orbitals: partition in zones.

A new multireference configuration interaction method using localised orbitals is proposed, in which a molecular system is divided into regions of unequal importance. The advantage of dealing with local orbitals, i.e., the possibility to neglect long range interaction is enhanced. Indeed, while in the zone of the molecule where the important phenomena occur, the interaction cut off may be as small as necessary to get relevant results, in the most part of the system it can be taken rather large, so that results of good quality may be obtained at a lower cost. The method is tested on several systems. In one of them, the definition of the various regions is not based on topological considerations, but on the nature, σ or π, of the localised orbitals, which puts in evidence the generality of the approach.

Evaluation of magnetic terms in Cu4O4 cubane-like systems from selected configuration interaction calculations: a case study of polynuclear transition-metal systems.

We present the evaluation of magnetic terms in a Cu(4)O(4) cubane-like system from truncated CI calculations, as a case study of polynuclear transition-metal complexes. We employ a new excitation selected configuration interaction (EXSCI) method based on the use of local orbitals. Taking advantage of the locality and then of the fact that the interactions vanish when the distance is large, the dimension of the CI is largely reduced. To the best of our knowledge these CI calculations are the largest one performed for polynuclear transition metal systems so far. The results show the presence of two leading ferromagnetic interactions between bridged Cu ions. Also the interactions between the unbridged Cu ions are ferromagnetic, but very weak, in contrast to the experimental data. The nature and amplitude of all the computed interactions are consistent with the relative orientation of the magnetic orbitals in the molecule, and correctly reproduce the susceptibility versus temperature curve. Our results indicate that it is possible to obtain similar fittings with sets of parameters representing different physical effects and put in evidence the drawbacks of the fitting based on oversimplified magnetic models. In this context, the presented computational strategy can be considered as a useful tool to help in the interpretation of the magnetic data and the validation of the magnetic interaction model in the polynuclear magnetic systems.

Direct selected multireference configuration interaction calculations for large systems using localized orbitals.

A selected multireference configuration interaction (CI) method and the corresponding code are presented. It is based on a procedure of localization that permits to obtain well localized occupied and virtual orbitals. Due to the local character of the electron correlation, using local orbitals allows one to neglect long range interactions. In a first step, three topological matrices are constructed, which determine whether two orbitals must be considered as interacting or not. Two of them concern the truncation of the determinant basis, one for occupied/virtual, the second one for dispersive interactions. The third one concerns the truncation of the list of two electron integrals. This approach permits a fine analysis of each kind of approximation and induces a huge reduction of the CI size and of the computational time. The procedure is tested on linear polyene aldehyde chains, dissociation potential energy curve, and reaction energy of a pesticide-Ca(2+) complex and finally on transition energies of a large iron system presenting a light-induced excited spin-state trapping effect.

A regionally contracted multireference configuration interaction method: general theory and results of an incremental version.

This work proposes to take benefit of the localizability of both occupied and virtual inactive molecular orbitals (MOs) in the context of complete active space singles and doubles configuration interaction (CAS-SDCI). The doubly occupied MOs are partitioned into blocks, or regions, corresponding to a subset of adjacent bonds and lone pairs. The localized virtual MOs are attributed to these regions from a spatial criterion. Then a series of limited post-CAS-CI calculations is performed, using the same reference space, one for each block, and then one per pair of blocks. From these independent CI calculations contracted external functions are defined for each block or for each pair of blocks, and for each state. A general multistate formalism is proposed, the CI matrix being expressed in the space defined by the CAS and the contracted functions. Preliminary numerical studies, resting on the evaluation of single-block and two-block contributions to the dynamical correlation energy of each state, are presented. Provided that size-consistency corrections are taken into account the results of the procedure are shown to be in excellent agreement with those of the nonpartitioned post-CAS-CI. The computational benefits of this evidently parallelizable procedure are underlined.

A study of the fixed-node error in quantum Monte Carlo calculations of electronic transitions: the case of the singlet n-->pi* (CO) transition of the acrolein.

We report fixed-node diffusion Monte Carlo (FN-DMC) calculations of the singlet n-->pi( *) (CO) vertical transition of acrolein. The impact of the fixed-node approximation on the excitation energy is investigated. To do that, trial wave functions corresponding to various nodal patterns are used. They are constructed by using either a minimal complete-active-space self-consistent field (CASSCF) calculation involving an oxygen lone pair n and the pi( *) (CO) molecular orbitals or a more complete set involving all the molecular orbitals expected to play a significant role in the excitation process. Calculations of both states have been performed with molecular orbitals optimized separately for each state via standard "state specific" CASSCF calculations or by using a common set of optimized orbitals ["state averaged" CASSCF calculations] whose effect is to introduce some important correlation between the nodal patterns of the two electronic states. To investigate the role of the basis set three different basis of increasing size have been employed. The comparative study based on the use of all possible combinations of basis sets, active spaces, and type of optimized molecular orbitals shows that the nodal error on the difference of energies is small when chemically relevant active space and state-averaged-type CASSCF wave functions are used, although the fixed-node error on the individual total energies involved can vary substantially. This remarkable result obtained for the acrolein suggests that FN-DMC calculations based on a simple strategy (use of standard ab initio wave functions and no Monte Carlo optimization of molecular orbital parameters) could be a working computational tool for computing electronic transition energies for more general systems.

Ab-initio multireference study of an organic mixed-valence Spiro molecular system.

The electronic structure and some electron transfer properties of a model mixed-valence Spiro molecular cation have been investigated at CAS-SCF, CAS+S, and CAS+SD levels starting from canonical and localized orbitals, using SZ, DZ, and TZP basis sets. The potential energy surfaces of the adiabatic ground and the lowest three excited electronic states have been computed, within a two-state model, and a double-well potential has been obtained for the ground electronic state. We have demonstrated the low coupling interaction between the two redox moieties of this molecular cation by following the charge localization/delocalization in the valence pi system through the reaction coordinate of the intramolecular charge transfer. The effect of dynamical correlation, using either localized or canonical orbitals, was found to be crucial for a quantitative description of the electronic structure and some important electron transfer parameters of this mixed-valence system.

Addressing Through-H Magnetic Interactions: A Comprehensive ab Initio Analysis of This Efficient Coupler.

The exchange coupling in a structuraly characterized Cu(II)2 complex is analyzed to highlight the role of H bonds in the generation of efficient magnetic interactions. The interest for complementary insights which are not accessible through DFT calculations (Desplanches, C. et al. J. Am. Chem. Soc. 2002, 124, 5197) has driven this state-of-the-art ab initio inspection. The wave function expansion based upon localized orbitals allows us to selectively turn on specific mechanisms and quantitatively evaluate their roles in the exchange interactions. Our singlet-triplet splitting calculations demonstrate the enhancement of the magnetic coupling through a concerted oxygen-to-metal charge transfer and electronic redistribution within the OH bond of the OH···O magnetic linker. This mechanism accounts for ∼35% of the total experimentally measured singlet-triplet energy difference. This analysis strongly suggests that H bonds might be particularly useful not only in the establishment of intermolecular contacts but also within the basic units of magnetic materials.

Restoring the size consistency of multireference configuration interactions through class dressings: applications to ground and excited states.

The present paper presents a revised version of a size-consistency correction to the multireference configuration interaction techniques previously proposed by Szalay et al. [J. Phys. Chem. 100, 6288 (1996)]. The method assumes a complete active space reference space and separates the nonreference determinants in several classes according to their number of inactive holes and particles. The correction is formulated as a dressing of the diagonal energies of these determinants, which depends on their class, as originally proposed by Ruttink et al. [J. Chem. Phys. 94, 7212 (1991)]. The exclusion principle violating corrections are evaluated through a simple counting of the various excitation processes which remain possible on each class. The efficiency of the method has been tested on a series of multireference problems for which full configuration interaction results are available (OH(2) bond breaking, Be insertion in H(2), excited states of CH(2)). The dressing of a given state not only provides excellent results for this state but also provides accurate excited roots. The efficiency of state-specific dressings is dramatic. The adaptation of this proposal to difference-dedicated configuration interactions can be extremely fruitful, as illustrated in the calculation of the 1 (1)A(g)-1(1)B(u)(pi->pi(*)) transition energy of the trans-butadiene molecule.

Inspection of the duality of a verdazyl-based radical in transition metal complexes: a pi* donor ligand and a magnetic partner.

The behavior of a verdazyl-based radical bound to open-shell transition metal ions in the structurally and magnetically characterized [M(hfac)2imvd(o)] (M = Mn, Ni; hfac = (1,1,1,5,5,5)hexafluoroacetylacetonate; imvd(o) = 3-(2'-imidazolyl)-1,5-dimethyl-6-oxoverdazyl) complexes is rationalized using ab initio wave-function-based calculations analysis. The calculated exchange coupling constants J (H = -J(s(M) x s(imvd(o)); J(Mn)(calcd) = -63 cm(-1), J(Ni)(calcd) = 205 cm(-1)) are in excellent agreement with the experimental ones (J(Mn)(exp) = -63 cm(-1), J(Ni)(exp) = 193 cm(-1)). Even though both rings are involved through the binding mode of the imvd(o) radical, the spin density remains essentially localized on the nitrogen-rich ring. The singularity stems from its bidentate coordinating character. The analysis of the correlated wave function suggests that the verdazyl-based radical acts as a pi* donor ligand which allows ligand-to-metal charge transfer and excludes metal-to-ligand charge transfer. This reflects the weak covalent character of the M-imvd(o) pi coordination bond. From a magnetic point of view, the through-space exchange governs the ferromagnetic character in the Ni derivative up to 153 cm(-1) as expected from a description limited to the magnetic orbitals. Nevertheless, the CI expansion displays the participation of excited doublet and quartet states (spin polarization) on the verdazyl moiety which leads to a significant additional ferromagnetic contribution (52 cm(-1)). In the [Mn(hfac)2imvd(o)] analogue, the antiferromagnetic contribution arising from kinetic exchange is only one-third of the observed exchange coupling constant. It is necessary to introduce dynamical correlation effects to quantitatively recover the exchange interaction in this compound. Since the pi* donor and spin-polarized characters of the verdazyl moiety dominate over the negligible polarizability of the imidazole part, it is concluded that the noninnocent nature of the imvd(o) radical is held by the verdazyl ring part.

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.

Selected excitation for CAS-SDCI calculations.

A new selected-configuration interaction method is proposed, based on the use of local orbitals. A corresponding code has been written, which is devoted to CI calculations of rather large systems (about 50-100 carbon-like atoms). Taking advantage of the locality, and then of the fact that interactions vanish when the distance is large, the dimension of the CI space is largely reduced. The determinants that would be created by long range excitations are expected to have a small weight in the wave function and are therefore eliminated. This selected excitation CI space is particularly suited for large molecules. It is tested on large polyene chains and on a transition metal complex. For large enough systems, the CPU time saving is important and, what is more noticeable, calculations that were impossible to perform without selection are feasible in this approach.

The Effect of the Basis-Set Superposition Error on the Calculation of Dispersion Interactions: A Test Study on the Neon Dimer.

The dispersion interactions of the Ne2 dimer were studied using both the long-range perturbative and supramolecular approaches: for the long-range approach, full CI or string-truncated CI methods were used, while for the supramolecular treatments, the energy curves were computed by using configuration interaction with single and double excitation (CISD), coupled cluster with single and double excitation, and coupled-cluster with single and double (and perturbative) triple excitations. From the interatomic potential-energy curves obtained by the supramolecular approach, the C6 and C8 dispersion coefficients were computed via an interpolation scheme, and they were compared with the corresponding values obtained within the long-range perturbative treatment. We found that the lack of size consistency of the CISD approach makes this method completely useless to compute dispersion coefficients even when the effect of the basis-set superposition error on the dimer curves is considered. The largest full-CI space we were able to use contains more than 1 billion symmetry-adapted Slater determinants, and it is, to our knowledge, the largest calculation of second-order properties ever done at the full-CI level so far. Finally, a new data format and libraries (Q5Cost) have been used in order to interface different codes used in the present study.

Correlated ab initio study of the excited state of the iron-coordinated-mode noninnocent glyoxalbis(mercaptoanil) ligand.

The intriguing and theoretically unresolved magnetic coupling in the Fe(gma)CN (1) compound [gma = glyoxalbis(mercaptoanil)] has been investigated by means of first-principle correlated ab initio calculations. The low-energy spectrum of the complex has been studied using the difference dedicated configuration interaction method, which is a dynamically correlated multiconfigurational method. In agreement with available spectroscopic information, we found that the ground-state doublet is dominated by the coupling between an iron-centered quartet and the first excited triplet on the gma ligand. The open-shell character of the electronic structure of the ligand clarifies its noninnocent nature. The low-energy spectrum reveals the presence of a first excited quartet of different symmetry lying 200 cm(-1) above. The lowest excitation energy in the ground-state symmetry is found at 4790 cm(-1), thus ruling out the simple description of the system based on a Heisenberg Hamiltonian.

Large systems at ab initio multireference level: a cheap treatment thanks to a division into fragments.

Thanks to the use of localized orbitals and the subsequent possibility of neglecting long-range interactions, the linear-scaling methods have allowed to treat large systems at ab initio level. However, the limitation of the number of active orbitals in a complete active space self consistent-field (CASSCF) calculation remains unchanged. The method presented in this paper suggests to divide the system into fragments containing only a small number of active orbitals. Starting from a guess wave function, each orbital is optimized in its corresponding fragment, in the presence of the other fragments. Once all the fragments have been treated, a new set of orbitals is obtained. The process is iterated until convergence. At the end of the calculation, a set of active orbitals is obtained, which is close to the exact CASSCF solution, and an accurate CASSCF energy can be estimated.

A combined freeze-and-cut strategy for the description of large molecular systems using a localized orbitals approach.

A technique to reduce the computational effort in calculating ab initio energies using a localized orbitals approach is presented. By exploiting freeze strategy at the self-consistent field (SCF) level and a cut of the unneeded atomic orbitals, it is possible to perform a localized complete active space (CAS-SCF) calculation on a reduced system. This will open the possibility to perform ab initio treatments on very large molecular systems, provided that the chemically important phenomena happen in a localized zone of the molecule. Two test cases are discussed, to illustrate the performance of the method: the cis-trans interconversion curves for the (7Z)-13 ammoniotridec-7-enoate, which demonstrates the ability of the method to reproduce the interactions between charged groups; and the cisoid-transoid energy barrier for the aldehydic group in the C13 polyenal molecule.