Multiconfigurational Research Articles

Symmetry-Adapted Perturbation Theory Based on Multiconfigurational Wave Function Description of Monomers.

We present a formulation of the multiconfigurational (MC) wave function symmetry-adapted perturbation theory (SAPT). The method is applicable to noncovalent interactions between monomers which require a multiconfigurational description, in particular when the interacting system is strongly correlated or in an electronically excited state. SAPT(MC) is based on one- and two-particle reduced density matrices of the monomers and assumes the single-exchange approximation for the exchange energy contributions. Second-order terms are expressed through response properties from extended random phase approximation (ERPA). The dispersion components of SAPT(MC) have been introduced in our previous works [HapkaM.J. Chem. Theory Comput.2019, 15, 1016−102730525591; HapkaM.J. Chem. Theory Comput.2019, 15, 6712–672331670950]. SAPT(MC) is applied either with generalized valence bond perfect pairing (GVB) or with complete active space self-consistent field (CASSCF) treatment of the monomers. We discuss two model multireference systems: the H2 ··· H2 dimer in out-of-equilibrium geometries and interaction between the argon atom and excited state of ethylene. Using the C2H4* ··· Ar complex as an example, we examine second-order terms arising from negative transitions in the linear response function of an excited monomer. We demonstrate that the negative-transition terms must be accounted for to ensure qualitative prediction of induction and dispersion energies and develop a procedure allowing for their computation. Factors limiting the accuracy of SAPT(MC) are discussed in comparison with other second-order SAPT schemes on a data set of small single-reference dimers.

Semiclassical Dispersion Corrections Efficiently Improve Multiconfigurational Theory with Short-Range Density-Functional Dynamic Correlation.

Multiconfigurational wave functions are known to describe the electronic structure across a Born-Oppenheimer surface qualitatively correct. However, for quantitative reaction energies, dynamic correlation originating from the many configurations involving excitations out of the restricted orbital space, the active space, must be considered. Standard procedures involve approximations that eventually limit the ultimate accuracy achievable (most prominently, multireference perturbation theory). At the same time, the computational cost increases dramatically due to the necessity to obtain higher-order reduced density matrices. It is this disproportion that leads us here to propose an MC-srDFT-D hybrid approach of semiclassical dispersion (D) corrections to cover long-range dynamic correlation in a multiconfigurational (MC) wave function theory, which includes short-range (sr) dynamic correlation by density functional theory (DFT) without double counting. We demonstrate that the reliability of this approach is very good (at negligible cost), especially when considering that standard second-order multireference perturbation theory usually overestimates dispersion interactions.

Multiconfigurational short-range density-functional theory for open-shell systems.

Many chemical systems cannot be described by quantum chemistry methods based on a single-reference wave function. Accurate predictions of energetic and spectroscopic properties require a delicate balance between describing the most important configurations (static correlation) and obtaining dynamical correlation efficiently. The former is most naturally done through a multiconfigurational (MC) wave function, whereas the latter can be done by, e.g., perturbation theory. We have employed a different strategy, namely, a hybrid between multiconfigurational wave functions and density-functional theory (DFT) based on range separation. The method is denoted by MC short-range DFT (MC-srDFT) and is more efficient than perturbative approaches as it capitalizes on the efficient treatment of the (short-range) dynamical correlation by DFT approximations. In turn, the method also improves DFT with standard approximations through the ability of multiconfigurational wave functions to recover large parts of the static correlation. Until now, our implementation was restricted to closed-shell systems, and to lift this restriction, we present here the generalization of MC-srDFT to open-shell cases. The additional terms required to treat open-shell systems are derived and implemented in the DALTON program. This new method for open-shell systems is illustrated on dioxygen and [Fe(H2O)6]3+.

Measures for the non-Markovianity of a harmonic oscillator coupled to a discrete bath derived from numerically exact references

System-bath problems in physics and chemistry are often described by Markovian master equations. However, the Markov approximation, i.e., neglect of bath memory effects is not always justified, and different measures of non-Markovianity have been suggested in the literature to judge the validity of this approximation. Here we calculate several computable measures of non-Markovianity for the non-trivial problem of a harmonic oscillator coupled to a large number of bath oscillators. The Multi Configurational Time Dependent Hartree method is used to provide a numerically converged solution of the system-bath Schrodinger equation, from which the appropriate quantities can be calculated. In particular, we consider measures based on trace-distances and quantum discord for a variety of initial states. These quantities have proven useful in the case of two-level and other small model systems typically encountered in quantum optics, but are less straightforward to interpret for the more complex model systems that are relevant for chemical physics.

Multireference Ab Initio Study of Ligand Field d–d Transitions in Octahedral Transition-Metal Oxide Clusters

We have used multiconfigurational (MC) and multireference (MR) methods (CASSCF, CASPT2, and MRCI) to study d–d transitions and other optical excitations for octahedral [M(H2O)6]n+ clusters (M = Ti, V, Mn, Cr, Fe, Co, Ni, Cu) as models of hematite and other transition-metal oxides of interest in solar fuels. For [Fe(H2O)6]3+, all calculations substantially overestimate the d–d transition energies (∼3.0 versus ∼1.5 eV) compared to what has been experimentally assigned. This problem occurs even though theory accurately describes (1) the lowest d–d transition energy in the atomic ion Fe3+ (∼4.4 eV), (2) the t2g–eg splitting (∼1.4 eV) in [Fe(H2O)6]3+, and (3) the ligand-to-metal charge transfer (LMCT) energy in [Fe(H2O)6]3+. Indeed, the results for Fe3+ and the t2g–eg splitting suggest that the lowest d–d excitation energy in the hexa-aqua complex should be ∼3 eV (or slightly below because of Jahn–Teller stabilization), as we find. Possible origins for the d–d discrepancy are examined, including Fe2+ and low-s...

Lessons on O2 and NO bonding to heme from ab initio multireference/multiconfiguration and DFT calculations

This commentary focuses on the conceptual interpretation of the bonding of O(2) and NO to heme in oxyheme and nitrosylheme complexes, using high-level ab initio complete active space self-consistent field (CASSCF)/molecular mechanical (MM), gas-phase CASSCF, and CASSCF followed by second-order perturbation theory (CASPT2) calculations as well as density functional theory (DFT) calculations. The commentary shows that expanding the complex multiconfigurational (MC) wave functions into valence bond (VB)-type configurations based on localized orbitals provides significant insight into bonding and precise definitions of oxidation states. Furthermore, the VB "reading" of the wave function unifies the descriptions of DFT and MC theories, reconciling controversies and surprises. In so doing, we demonstrate the impact of the protein bulk polarity and its hydrogen-bonding capability on the bonding. The insight provided by "reading" the VB content of the MC wave functions highlights the potential of this approach as a general paradigm in future computational bioinorganic chemistry. A great deal of insight lies in this road.

Evolution of orbital spaces along potential curves for diatomic molecules. A comparative study of Hartree–Fock, Kohn–Sham, Brueckner and multi-configurational orbital spaces

For three isoelectronic systems: CO molecule, CN - and NO + ions we have carried out comparative studies of Kohn–Sham orbitals (KSO) generated by the most typical local, gradient-corrected and hybrid exchange–correlation (xc) potentials, Brueckner orbitals (BO) and Hartree–Fock orbitals (HFO). The relations of orbital subspaces of well defined symmetry properties have been analyzed in terms of numerical proximity indices, treated as some kind of orbital space distances. Geometric interpretation and graphical representation of these proximity relations have been proposed. The orbital space similarity relations have been studied at different interatomic distances, i.e. as a function of orbital quasidegeneracy, or a multi-configurational (MC) state character. For this reason also the orbitals spanning the CASSCF reference determinant (MC-orbitals, MCO) have been included in the analysis. Our investigations indicate, that at equilibrium bond distances KSOs are more close to BOs and MCOs than to HFOs, with the orbital similarity increasing in the sequence: local, gradient-corrected, to hybrid xc-potentials. However in many cases the distance of KSO spaces to BO or MCO ones are larger than, or comparable to the corresponding distance of HFO spaces to BO or MCO ones. Together with increasing internuclear distances the KSOs become relatively more close to BOs than to HFOs; only for the hybrid xc-potential with a large contribution of the Hartree–Fock exchange this relation is inversed. The evolution of KSO–BO proximity relations does not depend qualitatively on the total charge of the system. Our results indicate that the Kohn–Sham calculations are able to describe the MC state character in a limited region around the equilibrium bond distance.

Combining two-body density correlation functionals with multiconfigurational wave functions using natural orbitals and occupation numbers

We propose a procedure that combines multiconfigurational (MC) wave functions with two-body density correlation functionals by transforming the latter into functionals of the MC natural orbitals and occupation numbers. The method is tested with the spectroscopic constants of a set of 11 diatomics, the diradical-involved automerization barrier of cyclobutadiene, the energy difference between triplet and open-shell singlet states in He and the methylene molecule, and the magnetic coupling constants of several systems, such as NiO, KNiF(3), K(2)NiF(4), La(2)CuO(4), alpha-4-dehydrotoluene, 1,1('),5,5(')-tetramethyl-6,6(')-dioxo-3,3(')-biverdazyl, [Cu(2)Cl(6)](-2), copper(II) acetate monohidrate and H-He-H. The procedure is applied to the Colle-Salvetti [Theor. Chim. Acta 37, 329 (1975); 53, 55 (1979)], functional and to a size-consistent functional depending on the on-top pair density (F1-5-N(eff)). On average, the best results are provided by the transformed F1-5-N(eff) [J. Chem. Phys. 114, 2022 (2001)] functional.

Theoretical Study of the Photochemical Isomerization of Colchicine.

The photochemical reaction of colchicine to β- and γ-lumicolchicine, through a mechanism involving a disrotatory cyclization, is studied by theoretical methods. The energetics of the reaction, including one or two methanol solvent molecules, are studied at the DFT-B3LYP and multireference perturbation levels of theory using the 6-31G(d) basis set. The results show that, in agreement with experimental results, the first excited state of colchicine at ∼3.6 eV can lead to both β- and γ-lumicolchicine, whose energy is about 15 kcal mol(-)(1) above the colchicine energy. Owing to the high steric tension of the condensed four- and five-atom rings arising from cyclization, the two trans-lumicolchicines are higher in energy (>60 kcal mol(-)(1)), and their formation appears much less probable. A partial inclusion of the solvent effects through the addition of two solvent molecules does not alter the general conclusions based on the free energy in the gas phase. The photochemical reaction path is studied by choosing the distance between the two carbon atoms which form the new σ bond as the leading coordinate of the minimum-energy path of both the ground and the first singlet excited states. The energies are computed by Multi Configurational self-consistent-field calculations on a model molecule, retaining those atoms that presumably play an active role in the reaction. A reasonable mechanism starting from colchicine in the first singlet excited state and leading to γ-lumicolchicine is proposed. On the contrary, a high-energy transition state is found for trans-lumicolchicines, whose formation, although not strictly forbidden for energetic reasons, appears to be rather improbable.

Time-resolved gas-phase kinetic and quantum chemical studies of the reaction of silylene with oxygen

Time-resolved kinetic studies of the reaction of silylene, SiH2, generated by laser flash photolysis of phenylsilane, have been carried out to obtain rate constants for its bimolecular reaction with O(2). The reaction was studied in the gas phase over the pressure range 1-100 Torr in SF(6) bath gas, at five temperatures in the range 297-600 K. The second order rate constants at 10 Torr were fitted to the Arrhenius equation: [see text] The decrease in rate constant values with increasing temperature, although systematic is very small. The rate constants showed slight increases in value with pressure at each temperature, but this was scarcely beyond experimental uncertainty. From estimates of Lennard-Jones collision rates, this reaction is occurring at ca. 1 in 20 collisions, almost independent of pressure and temperature. Ab initio calculations at the G3 level backed further by multi-configurational (MC) SCF calculations, augmented by second order perturbation theory (MRMP2), support a mechanism in which the initial adduct, H(2)SiOO, formed in the triplet state (T), undergoes intersystem crossing to the more stable singlet state (S) prior to further low energy isomerisation processes leading, via a sequence of steps, ultimately to dissociation products of which the lowest energy pair are H2O+SiO. The decomposition of the intermediate cyclo-siladioxirane, via O-O bond fission, plays an important role in the overall process. The bottleneck for the overall process appears to be the T-->S process in H2SiOO. This process has a small spin-orbit coupling matrix element, consistent with an estimate of its rate constant of 1x10(9) s-1 obtained with the aid of RRKM theory. This interpretation preserves the idea that, as in its reactions in general, SiH2 initially reacts at the encounter rate with O2. The low values for the secondary reaction barriers on the potential energy surface account for the lack of an observed pressure dependence. Some comparisons are drawn with the reactions of CH2+O2 and SiCl2+O2.

Application and development of multiconfigurational localized perturbation theory

Generalization of localized perturbation theory, which results with a method able to span the spin space correctly, is presented. This generalization is achieved by using a multiconfigurational (MC) wave function as the reference. This is the most comprehensive expansion used within MC–LMP2 approach to date, with, however, low computational cost [computational scaling with system size (N) of the new method is O(N3)]. Recently, we have reported the successful Jaguar2 (J2) model for calculating atomization energies. Within the MC–LMP2 framework, the J2 model for calculating heats of formation is based on the generalized valence bond–perfect pairing (GVB–PP) wave function. The J2 model was applied only to closed shell cases because of the perfect pairing (PP) restriction in the reference function. In order to describe other systems, the PP restriction needs to be lifted. This work describes efforts in that direction. The PP restriction can be lifted by a restricted configuration interaction (RCI) procedure applied to the GVB–PP wave function. In this paper, the equations describing the application of LMP2 theory to self-consistent RCI wave function are derived and explained. The RCI wave function is a “true” MC expansion as opposed to the GVB–PP, which uses only a single spin eigenfunction (SEF). We also present the self-consistent (SC) optimization of the RCI wave function. The SC–RCI–LMP2 is the first MC–LMP2 method where the spin space is spanned in the reference. This is important for describing the nondynamical correlation (near degeneracy) effects associated, for example, with bond breaking processes. The SC–RCI–LMP2 is an efficient method applicable to large systems; it is shown to reproduce the potential energy surfaces calculated by the complete active space–second order perturbation (CAS–SCF–PT2) method. This is demonstrated, for the first time, on some widely used test cases.

Mono and multiconfigurational wave functions with DFT correlation energy: the case of fluorine

It is known that the Hartree–Fock–Kohn–Sham method does not constitute an improvement over the Hartree–Fock method when the latter gives a poor description of the system, as is the case for the fluorine molecule. On the contrary, it is known that a multiconfigurational (MC) wave function gives a rather good description of this molecule. We show that adding the correlation energy computed from a variety of correlation energy functionals to the MC energy improves the potential energy curve and the spectroscopic constants significantly. We consider two different ways of including the correlation energy, either in a self-consistent (SCF) fashion or through a post-SCF procedure.

Improvement of Multiconfigurational Wave Functions and Energies by Correlation Energy Functionals

Several aspects of hybrid methods resulting from combining multiconfigurational (MC) wave functions with correlation energy functionals are discussed, in particular for a family of functionals recently proposed.1,2 It is found that adding the correlation at the end of a MC calculation instead of including it in the self-consistent procedure constitutes an excellent approximation. MC potential energy curves and spectroscopic constants show a significant improvement after being corrected with correlation energies from this family of functionals.

Symmetry adaptation of configuration basis in MCSCF method

A novel approach of space symmetry adaptation is developed for multiconfigurational (MC) functions in fully optimized reaction space and complete active space SCF calculations. The bonded tableau and two box symmetric tableau are basic representations (rep) of configuration functions; the group symmetric localized orbitals are used as one-electron orbitals. The method is proposed for generating a complete and orthonormal set of MC single excited functions. The redundant variable in MCSCF can be eliminated by symmetry adaptation.

A potentially size-consistent multiconfiguration based coupled electron pair approximation

A coupled electron pair approximation is derived and illustrative calculations are presented. The present approximation, which we refer to as the unitary coupled electron pair approximation (UCEPA), provides multiconfigurational (MC) reference capability and is as computationally tractable as multireference configuration interaction (MRCI). The method is capable of yielding size-consistent energies if the MC reference function is of the complete active space (CAS) variety. The coefficient matrix of the resultant set of simultaneous linear equations is evaluated using internal/external orbital space partitioning within a unitary group approach (UGA) treatment of the state space. We demonstrate the accuracy of the method on several small benchmark molecules for which full CI results are known, and on nontrivial studies on the singlet–triplet splitting in methylene and the electron affinity of the oxygen atom.

Potential energy surface for collinear Mg + FH → MgF + H reaction

The potential energy surface for collinear Mg + FH → MgF + H has been studied at both the Restricted Hartree-Fock (RHF) and Multiconfigurational (MC) direct minimization levels of “ab initio” electronic structure theory. About one hundred points calculated on the surface were obtained from a two configuration MC direct minimization function by using a DZ GTO basis set. The final surface obtained by fitting with splines is employed to discuss qualitatively the dynamics of the reaction.

Pseudopotential, MC-SCF and limited CI calculations on nickel-bis-dithiolene

The pseudopotential method is used to perform multiconfigurational (MC) SCF and full valence configuration interaction (CI) calculations on the dianion of nickel-bis-dithiolene, [Ni(S 2C 2H 2) 2] 2−. Both a planar (D 2h) and a near tetrahedral (D 2d) conformation are considered. Charge distributions are calculated, yielding a charge on Ni of 0.22 in the planar and 0.57 in the tetrahedral case. The experimental ground state, the planar 1Ag state, approaches a 3d 9 configuration on Ni (the d-population equals 8.87), while the nickel 3d-population of all other calculated states ranges from 8.35 to 8.69. The single d—d excitation spectrum for the planar complex is discussed, and the first transition is computed to be 1Ag → 1B 1g. Intramolecular rotation, i.e., one ligand ring rotating with respect to the other, is also discussed. The rotation is proposed to occur via a tetrahedral triplet intermediate since the relevant singlet tetrahedral state is estimated to lie more than 1 eV higher than the triplet intermediate. The discussion of the intramolecular rotation is made difficult by an error in the calculated singlet—triplet splittings. Possible causes of this error are pointed out.