A systematic multireference perturbation-theory study of the low-lying states of SiC3

The three known lowest-energy isomers of SiC(3), two cyclic singlets (2s and 3s) and a linear triplet (1t), have been reinvestigated using multireference second-order perturbation theory (MRPT2). The dependence of the relative energies of the isomers upon the quality of the basis sets and the sizes of the reference active spaces is explored. When using a complete-active-space self-consistent-field reference wave function with 12 electrons in 11 orbitals [CASSCF (12, 11)] together with basis sets that increase in size up to the correlation-consistent polarized core-valence quadruple zeta basis set (cc-pCVQZ), the MRPT2 method consistently predicts the linear triplet to be the most stable isomer. A new parallel direct determinant MRPT2 code has been used to systematically explore reference spaces that vary in size from CASSCF (8,8) to full optimized reaction space [FORS or CASSCF (16,16)] with the cc-pCVQZ basis. It is found that the relative energies of the isomers change substantially as the active space is increased. At the best level of theory, MRPT2 with a full valence FORS reference, the 2s isomer is predicted to be more stable than 3s and 1t by 4.7 and 2.2 kcal/mol, respectively.     

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.     

Fourteen-electron ring model and the anomalous magnetic circular dichroism of meso-triarylsubporphyrins

The MCD spectra of meso-triarylsubporphyrins show a sign anomaly which is correlated with the acceptor properties of the aryl substituent. From the spectra, magnetic moments of the excited states are determined. In the context of a simplified orbital model, the sign change is attributed to the quenching of the magnetic moment of the LUMO by acceptor orbitals of the substituent. The actual calculation of this moment presents a major challenge to computational methods. It is shown that wave function techniques based on CASSCF underestimate the covalency effects that are responsible for the quenching. In contrast, a CI method based on DFT orbitals yields excellent results, which fully support the orbital model.     

Theoretical studies for excited-state tautomerization in the 7-azaindole-(CH3OH)n (n = 1 and 2) complexes in the gas phase

The excited-state tautomerization of 7-azaindole (7AI) complexes bonded with either one or two methanol molecule(s) was studied by systematic quantum mechanical calculations in the gas phases. Electronic structures and energies for the reactant, transition state (TS), and product were computed at the complete active space self-consistent field (CASSCF) levels with the second-order multireference perturbation theory (MRPT2) to consider the dynamic electron correlation. The time-dependent density functional theory (TDDFT) was also used for comparison. The excited-state double proton transfer (ESDPT) in 7AI-CH(3)OH occurs in a concerted but asynchronous mechanism. Similarly, such paths are also found in the two transition states during the excited-state triple proton transfer (ESTPT) of the 7AI-(CH(3)OH)(2) complex. In the first TS, the pyrrole ring proton first migrated to methanol, while in the second the methanol proton moved first to the pyridine ring. The CASSCF level with the MRPT2 correction showed that the former path was much preferable to the latter, and the ESDPT is much slower than the ESTPT. Additionally, the vibrational-mode enhanced tautomerization in the 7AI-(CH(3)OH)(2) complex was also studied. We found that the excitation of the low-frequency mode shortens the reaction path to increase the tautomerization rate. Overall, most TDDFT methods used in this study predicted different TS structures and barriers from the CASSCF methods with MRPT2 corrections.     

Theoretical modeling of the symmetric (C2v) electrophilic attachment of chlorine to ethylene in aqueous solution

The electrophilic attachment of chlorine to ethylene in aqueous solution is studied using the complete active space self‐consistent field (CASSCF) method combined with the polarizable continuum model in a version which includes electrostatic, repulsion, and dispersion solute–solvent interactions. The C_2v symmetry is maintained for all the geometries considered, and the active space is generated distributing six electrons in five orbitals. After the CASSCF calculation a valence bond (VB) analysis has been performed along an approximate reaction coordinate by projecting the wave function onto a set of four classical structures; a reliable explanatory model of the rearrangement of the electronic structure for this process is then derived. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 74: 59–67, 1999     

Photostability via a sloped conical intersection: a CASSCF and RASSCF study of pyracylene

An extensive ab initio study of the ground- and excited-state potential energy surfaces of pyracylene is presented in this work. CASSCF calculations show that there is an accessible sloped S0/S1 conical intersection, which leads to ultrafast internal conversion and explains the observed photostability. RASSCF calculations (using a well-defined subset of the CASSCF configurations) are shown to be able to reproduce CASSCF results satisfactorily and will therefore be useful for larger systems where CASSCF is currently too expensive. MRCI and MRPT2 energy corrections are computed to assess the ionic character of the excited states. Finally, MMVB calculations are also benchmarked against CASSCF, to assess the reliability of this parametrized method for excited states of large conjugated polycyclic aromatic hydrocarbons.     

Spin-coupled theory for 'N electrons in M orbitals' active spaces

Spin-coupled (SC) theory, an ab initio valence bond (VB) approach which uses a compact and an easy-to-interpret single-orbital product wave function comparable in quality to a ‘N in N’ complete-active-space self-consistent field [CASSCF(N,N)] construction, is extended to ‘N in M’ (N ≠ M) active spaces. The SC(N,M) wave function retains the essential features of the original SC model: It involves just the products of nonorthogonal orbitals covering all distributions of N electrons between M orbitals in which as few orbitals as possible, |N – M|, are doubly occupied (for N > M) or missing (for N < M) and all other orbitals are singly occupied; each of these products is combined with a flexible spin function which allows any mode of coupling of the spins of the orbitals within the product. The SC(N,M) wave function remains much more compact than a CASSCF(N,M) construction; for example, the SC(6,7) wave function includes 35 configuration state functions (CSFs) as opposed to the 490 CSFs in the CASSCF case. The essential features of the SC(N,M) method are illustrated through a SC(6,5) calculation on the cyclopentadienyl anion, C5H5(–), and a SC(6,7) calculation on the tropylium cation, C7H7(+). The SC(6,5) and SC(6,7) wave functions for C5H5(–) and C7H7(+) are shown to provide remarkably clear modern VB models for the electronic structures of these aromatic cyclic ions which closely resemble the well-known SC model of benzene and yet recover almost all of the correlation energy included in the corresponding CASSCF(6,5) and CASSCF(6,7) wave functions: over 97% in the case of C5H5(–) and over 95% in the case of C7H7(+).     

Localization of molecular orbitals on fragments

A non‐iterative algorithm for the localization of molecular orbitals (MOs) from complete active space self consistent field (CASSCF) and for single‐determinantal wave functions on predefined moieties is given. The localized fragment orbitals can be used to analyze chemical reactions between fragments and also the binding of fragments in the product molecule with a fragments‐in‐molecules approach by using a valence bond expansion of the CASSCF wave function. The algorithm is an example of the orthogonal Procrustes problem, which is a matrix optimization problem using the singular value decomposition. It is based on the similarity of the set of MOs for the moieties to the localized MOs of the molecule; the similarity is expressed by overlap matrices between the original fragment MOs and the localized MOs. For CASSCF wave functions, localization is done independently in the space of occupied orbitals and active orbitals, whereas, the space of virtual orbitals is mostly uninteresting. Localization of Hartree–Fock or Kohn–Sham density functional theory orbitals is not straightforward; rather, it needs careful consideration, because in this case some virtual orbitals are needed but the space of virtual orbitals depends on the basis sets used and causes considerable problems due to the diffuse character of most virtual orbitals. © 2012 Wiley Periodicals, Inc.     

Orbital optimization in the density matrix renormalization group, with applications to polyenes and beta-carotene

In previous work we have shown that the density matrix renormalization group (DMRG) enables near-exact calculations in active spaces much larger than are possible with traditional complete active space algorithms. Here, we implement orbital optimization with the DMRG to further allow the self-consistent improvement of the active orbitals, as is done in the complete active space self-consistent field (CASSCF) method. We use our resulting DMRG-CASSCF method to study the low-lying excited states of the all-trans polyenes up to C24H26 as well as beta-carotene, correlating with near-exact accuracy the optimized complete pi-valence space with up to 24 active electrons and orbitals, and analyze our results in the light of the recent discovery from resonance Raman experiments of new optically dark states in the spectrum.     

Absorption and magnetic circular dichroism (MCD) studies of 1,4,5,8-naphthalenetetracarboxy diimides in terms of CASSCF method and FC theory

The electronic and geometrical structures of the low-energy states of 1,4,5,8-naphthalenetetracarboxylic dianhydride parent diimide (1) are studied in terms of the complete active space self-consistent field (CASSCF) method employed at different level with respect to the size and the quality of the active space. In the framework of the vibronic model based on the Franck–Condon (FC) effect the absorption and magnetic circular dichroism (MCD) spectra are studied in the excitation region corresponding to two low-energy 1¹A_g → 1¹B_2u and 1¹A_g → 1¹B_3u electronic transitions in diimides. In that (visible) excitation region the CASSCF computations with the 5π[4n]5π active space (i.e., the naphthalene-like π orbitals enriched by the four lone pair orbitals of the oxygen atoms) were found to reproduce very well the empirical absorption and the MCD spectra measured for the dicyclohexyl-N,N^′-substituted diimide (2). At the same CASSCF/5π[4n]5π level, the electronic absorption of diimides in the near UV excitation region were attributed to the 1¹A_g → 2¹B_1u, 1¹A_g → 2¹B_3u and 1¹A_g → 2¹B_2u electronic transitions; the latter two are mostly localized on the “diimide chromophore”. For these transitions the calculated magneto-optical characteristics, such as sign pattern and intensity distribution in the MCD spectrum, were found to be consistent with that experimentally observed for the diimide 2 compound.     

On the stability of Be3: a benchmark complete active space self-consistent field + averaged quadratic coupled cluster study

The optimized geometries and binding energies for the linear and triangular isomers of the beryllium trimer have been obtained through benchmark multireference averaged quadratic coupled cluster (AQCC) calculations using very large complete active space SCF (CASSCF) references (12 active electrons in 13 and 14 orbitals). Geometries were optimized with the cc-pV5Z basis, while the binding energies (including counterpoise correction) were obtained with the significantly larger aug-cc-pV5Z basis set. The binding energies (27.3 and 16.3 kcal/mol for the equilateral and linear isomers, respectively) are larger than the previous full CI benchmark values, while the corresponding Be-Be equilibrium distances of 4.101 and 4.088 a.u. are smaller. In view of the near-size consistency character of the CASSCF + AQCC method, the fact that all 12 electrons are fully correlated, the active reference space includes 14 orbitals, and the very large basis set used here, we propose to consider these results as reference data for Be(3). Using the electron pair localization function obtained at the CASSCF(12,15) level, it is clearly illustrated that the 2p orbitals lying in the molecular plane play a dominant role in the bonding pattern for the equilateral isomer.     

A study of the correlation effects upon the modelization of the double exchange phenomenon

A previous work by the authors has shown that the usual spin Hamiltonian used to model the magnetic spectra of mixed valence compounds was not sufficient to reproduce the magnetic spectrum of the molecule [Fe(2)(OH)(3)(NH(3))(6)](2+). In the present paper, the spin Hamiltonian is confronted to correlated ab initio calculations. The discrepancy between this Hamiltonian and the calculations is investigated and explained. It is pointed out that the multiconfigurational nature of the transition metal is responsible for this discrepancy. However, we show that this effect can easily be treated at the complete active space self-consistent field (CASSCF) level and that no further correlation treatment is needed. The spin Hamiltonian, which reproduces very well the minimal CASSCF results, could not be improved to recover the multireference effects.     

State‐specific multireference perturbation theory with improved virtual orbitals: Taming the ground state of F2, Be2, and N2

Adaptation of improved virtual orbitals (IVOs) in state‐specific multireference perturbation theory using Møller–Plesset multipartitioning of the Hamiltonian (IVO‐SSMRPT) is examined in which the IVO‐complete active space configuration interaction (CASCI) is used as an inexpensive alternative to the more involved CAS‐self‐consistent field (CASSCF) orbitals. Unlike the CASSCF approach, IVO‐CASCI does not bear tedious and costly iterations beyond those in the initial SCF calculation. The IVO‐SSMRPT is intruder‐free, and explicitly size‐extensive. In the present preliminary study, the IVO‐SSMRPT method which relies on a small reference space is applied to study potential energy surfaces (PES) of the ground state of challenging, multiconfigurational F₂, Be₂, and N₂ molecules. These systems provide a serious challenge to any ab initio methodology due to the presence of an intricate interplay of nondynamical and dynamical correlations to the entire PES. The quality of the computed PES has been judged by extracting spectroscopic parameters and vibrational levels. The reported results illustrate that the IVO‐SSMRPT method has a potential to yield accuracies as good as the CASSCF‐SSMRPT one with reduced computational labor. Even with small reference spaces, our estimates demonstrate a good agreement with the available experimental values, and some benchmark computations. The blend of accuracy and low computational cost of IVO‐SSMRPT should deserve future attention for the accurate treatment of electronic states of small to large molecular systems for which the wavefunction is characterized by various configurations. © 2015 Wiley Periodicals, Inc.     

A simple computational scheme for obtaining localized bonding schemes and their weights from a CASSCF wave function

We devise and apply a simple computational scheme for obtaining localized bonding schemes and their weights from a CASSCF wave function. These bonding schemes are close to resonance structures drawn by chemists. Firstly, a CASSCF wave function is computed. Secondly, the CASSCF computation is repeated but now the delocalized complete active space MOs are substituted by Weinhold's localized natural atomic orbitals. In this way the original CASSCF wave function is represented by a sequence of Slater determinants composed of localized natural atomic orbitals. Thus, a standard CASSCF wave function can be reinterpreted in terms of a local picture. To test the method we obtain localized bonding schemes and their weights for the ground and the pi-pi* excited state of ethylene. Moreover, bonding schemes and their weights are derived for the ground, the 1(1)B(u), and the 2(1)Ag pi-pi* excited states of trans-butadiene. The large weight bonding schemes are shown to be a qualitative indicator for the known photochemistry of butadiene. The remarkable stability of the Arduengo carbene is discussed by obtaining bonding schemes that indicate a stabilizing delocalization of the pi electrons. We illustrate that the large weight bonding schemes are in line with the observed reactivity of the Arduengo carbene.     

Balancing single- and multi-reference correlation in the chemiluminescent reaction of dioxetanone using the anti-Hermitian contracted Schrödinger equation

Direct computation of energies and two-electron reduced density matrices (2-RDMs) from the anti-Hermitian contracted Schro?dinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)], it is shown, recovers both single- and multi-reference electron correlation in the chemiluminescent reaction of dioxetanone especially in the vicinity of the conical intersection where strong correlation is important. Dioxetanone, the light-producing moiety of firefly luciferin, efficiently converts chemical energy into light by accessing its excited-state surface via a conical intersection. Our previous active-space 2-RDM study of dioxetanone [L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010)] concluded that correlating 16 electrons in 13 (active) orbitals is required for realistic surfaces without correlating the remaining (inactive) orbitals. In this paper we pursue two complementary goals: (i) to correlate the inactive orbitals in 2-RDMs along dioxetanone's reaction coordinate and compare these results with those from multireference second-order perturbation theory (MRPT2) and (ii) to assess the size of the active space-the number of correlated electrons and orbitals-required by both MRPT2 and ACSE for accurate energies and surfaces. While MRPT2 recovers very different amounts of correlation with (4,4) and (16,13) active spaces, the ACSE obtains a similar amount of correlation energy with either active space. Nevertheless, subtle differences in excitation energies near the conical intersection suggest that the (16,13) active space is necessary to determine both energetic details and properties. Strong electron correlation is further assessed through several RDM-based metrics including (i) total and relative energies, (ii) the von Neumann entropy based on the 1-electron RDM, as well as the (iii) infinity and (iv) squared Frobenius norms based on the cumulant 2-RDM.     

Characterization of the excited states of ethylene by MRCI

The excited states of ethylene are systematically analyzed and characterized according to the natural orbitals (NOs) resulting from multireference configuration interaction singles and doubles (MRCISD) calculations. By comparing the shapes and nodal structures of the NOs with those of hydrogen atomic orbitals, the Rydberg series can be classified. Two or three different types of Rydberg series appear within five excited states for each symmetry of D_2h. For example, in the ¹A_g symmetry there are three series having np and two nf hydrogen‐like atomic orbitals. Electronic correlation effects for the (π→π*) V state are also discussed on the basis of a complete active space self‐consistent field (CASSCF) calculation, showing that electron correlation effects merely within the valence space cannot explain contraction of the V state. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Low-lying excited states of C120 and C151: a multireference perturbation theory study

Excited states of two 7-aminocoumarin derivatives, coumarin 120 (7-amino-4-methylcoumarin) and coumarin 151 (7-amino-4-trifluoromethylcoumarin), were investigated using generalized multiconfigurational quasidegenerate perturbation theory (GMC-QDPT), multiconfigurational quasidegenerate perturbation theory (MC-QDPT) and time-dependent density functional theory (TDDFT) with the B3LYP and CAM-B3LYP functionals. The absorption and fluorescence spectra of C120 and C151 were calculated. We elucidated the characters of the low-lying states of C120 and C151. The absorption spectra calculated with GMC-QDPT and TDDFT B3LYP agreed well with the experimental data, while for the fluorescence spectra, the TDDFT calculations overestimated the fluorescence spectra compared to GMC-QDPT calculations. Utilizing active spaces with large numbers of electrons and orbitals for reference functions, GMC-QDPT showed a better performance than MC-QDPT with a complete active space self-consistent field (CASSCF) reference of active space with smaller number of electrons and orbitals. In our gas phase calculation, we found that the optimized structures for the first excited states have a planar amino group with a CN single bond, while the amino group is pyramidal in the ground state.     

A CASSCF-CASPT2 study of the excited-state intramolecular proton transfer reaction in 1-amino-3-propenal using different active spaces

In this work we analyze how the choice of the active space in the CASSCF (the complete-active-space multiconfiguration self-consistent-field method) and CASPT2 (the second-order perturbation theory based on the CASSCF reference wave function) calculations affects the computed potential energy curves (PECs) for the intramolecular proton transfer reaction in the ground state and the two lowest lying singlet excited states of 1-amino-3-propenal. As anticipated, the results revealed that, qualitatively, the proton transfer in the different states can be correctly described even by minimal active spaces, which include the orbitals involved in the electronic excitation of the considered state and the antibonding sigma orbital corresponding to the bond formed by the molecule with the migrating hydrogen atom. However, quantitatively, the relative energies of the two tautomers and the energy barriers computed at the CASSCF level change when the active space is increased, indicating importance of the dynamic electron correlation. Introducing the dynamic correlation effects via CASPT2 makes the calculated energy parameters more uniform among the different active spaces. The analysis suggested certain optimal active spaces for studying proton transfer reactions in systems similar to 1-amino-3-propenal. The PEC calculations for excited states showed that the results are sensitive to the molecular geometries used in the calculations, particularly near the transition point. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1422–1431 (1999)     

A multireference configuration interaction method based on the separated electron pair wave functions

A multireference configurational interaction method based on the separated electron pair (SEP) wave functions, SEP‐CI approach, has been developed as an approximation to the traditional CASSCF method. It differs from the CASSCF method in that active orbitals are obtained from the SEP wave function without further optimization in the subsequent CI calculations, and the active space is automatically constructed according to the occupation coefficients of SEP natural orbitals. These features make the present SEP‐CI method computationally much less demanding than the CASSCF method. The applicability of the SEP‐CI method is illustrated with sample calculations on the insertion reaction of BeH₂ and dissociation energies of LiH, BH, FH, H₂O, and N₂. © 2005 Wiley Periodicals, Inc. J Comput Chem 27: 39–47, 2006