A new size extensive multireference perturbation theory

A new multireference perturbation series is derived based on the Rayleigh–Schrödinger perturbation theory. It is orbitally invariant. Its computational cost is comparable to the single reference Møller–Plesset perturbation theory. It is demonstrated numerically that the present multireference second‐ and third‐order energies are size extensive by two types of supermolecules composed of H₂ and BH monomers. Spectroscopic constants of as well as the ground state energies of H₂O, NH₂, and CH₂ at three bond lengths have been calculated with the second multireference perturbation theory. The dissociation behaviors of CH₄ and HF have also been investigated. Comparisons with other approximate theoretical models as well as the experimental data have been carried out to show their relative performances. © 2013 Wiley Periodicals, Inc.     

An ab initio benchmark study of the H + CO --> HCO reaction

The H + CO --> HCO reaction has been characterized with correlation consistent basis sets at five levels of theory in order to benchmark the sensitivities of the barrier height and reaction ergicity to the one-electron and n-electron expansions of the electronic wave function. Single and multireference methods are compared and contrasted. The coupled cluster method RCCSD(T) was found to be in very good agreement with Davidson-corrected internally-contracted multireference configuration interaction (MRCI+Q). Second-order Moller-Plesset perturbation theory (MP2) was also employed. The estimated complete basis set (CBS) limits for the barrier height (in kcal/mol) for the five methods, including harmonic zero-point energy corrections, are MP2, 4.66; RCCSD, 4.78; RCCSD(T), 4.15; MRCI, 5.10; and MRCI+Q, 4.07. Similarly, the estimated CBS limits for the ergicity of the reaction are: MP2, -17.99; RCCSD, -13.34; RCCSD(T), -13.79; MRCI, -11.46; and MRCI+Q, -13.70. Additional basis set explorations for the RCCSD(T) method demonstrate that aug-cc-pVTZ sets, even with some functions removed, are sufficient to reproduce the CBS limits to within 0.1-0.3 kcal/mol.     

Diagrammatic formulation of the second‐order many‐body multipartitioning perturbation theory

The second‐order multireference perturbation theory employing multiple partitioning of the many‐electron Hamiltonian into a zero‐order part and a perturbation is formulated in terms of many‐body diagrams. The essential difference from the standard diagrammatic technique of Hose and Kaldor concerns the rules of evaluation of energy denominators which take into account the dependence of the Hamiltonian partitioning on the bra and ket determinantal vectors of a given matrix element, as well as the presence of several two‐particle terms in zero‐order operators. The novel formulation naturally gives rise to a “sum‐over‐orbital” procedure of correlation calculations on molecular electronic states, particularly efficient in treating the problems with large number of correlated electrons and extensive one‐electron bases. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 395–401, 1999     

Intruder state avoidance multireference Møller-Plesset perturbation theory

A new perturbation approach is proposed that enhances the low‐order, perturbative convergence by modifying the zeroth‐order Hamiltonian in a manner that enlarges any small‐energy denominators that may otherwise appear in the perturbative expansion. This intruder state avoidance (ISA) method can be used in conjunction with any perturbative approach, but is most applicable to cases where small energy denominators arise from orthogonal‐space states—so‐called intruder states—that should, under normal circumstances, make a negligible contribution to the target state of interests. This ISA method is used with multireference Møller–Plesset (MRMP) perturbation theory on potential energy curves that are otherwise plagued by singularities when treated with (conventional) MRMP; calculation are performed on the 1³Σ state of O₂; and the 2¹Δ, 3¹Δ, 2³Δ, and 3³Δ states of AgH. This approach is also applied to other calculations where MRMP is influenced by intruder states; calculations are performed on the ³Π_u state of N₂, the ³Π state of CO, and the 2¹A′ state of formamide. A number of calculations are also performed to illustrate that this approach has little or no effect on MRMP when intruder states are not present in perturbative calculations; vertical excitation energies are computed for the low‐lying states of N₂, C₂, CO, formamide, and benzene; the adiabatic ¹A₁–³B₁ energy separation in CH₂, and the spectroscopic parameters of O₂ are also calculated. Vertical excitation energies are also performed on the Q and B bands states of free‐base, chlorin, and zinc–chlorin porphyrin, where somewhat larger couplings exists, and—as anticipated—a larger deviation is found between MRMP and ISA‐MRMP. © 2002 Wiley Periodicals, Inc. J Comput Chem 10: 957–965, 2002     

Excitation Energies of Canonical Nucleobases Computed by Multiconfigurational Perturbation Theories

In this computational work, we assessed the performance of ab initio multireference (MR) methods for the calculation of vertical excitation energies of five nucleobases: adenine, guanine, cytosine, thymine and uracil. In total, we have studied 38 singlet and 30 triplet excited states. Where possible we used the multireference configuration interaction (MRCI) method as a reference for various flavors of multireference perturbation theory to second order. In particular, we have benchmarked CASPT2, NEVPT2 and XMCQDPT2. For CASPT2, we have analyzed the single‐state, multistate (MS) and extended MS variants. In addition, we have assessed the effect of the ionization potential electron affinity (IPEA) shift. For NEVPT2, we have used the partially and the strongly contracted variants. Further, we have tested the commonly used RI‐CC2, RI‐ADC2 and EOM‐CCSD methods. Generally, we observe the following trends for singlet excited states: NEVPT2 is the closest MR method to MRCISD+Q, closely followed by CASPT2 with the default IPEA shift. The same trend is observed for triplet states, although NEVPT2 and CASPT2‐IPEA are getting closer. Interestingly, the n, π* singlet excited states were described more accurately than π, π* excited states, while for triplet states the trend is inverted except for NEVPT2. This work is an important benchmark for future photochemical investigations.     

CCSD(T) and MRCI studies on the ground and excited states of BrOClO and ClOBrO

In this work, three forms (cis, trans and nonplanar) of ClOBrO and BrOClO were optimized at CCSD(T)/cc‐pVTZ level of theory. At the most stable forms (nonplanar form) of ClOBrO and BrOClO, the vertical excitation energies for the lowest six singlet states and two triplet states were calculated at the multireference internally contracted configuration interaction (MRCI) level of theory using cc‐pVDZ, Aug‐cc‐pVDZ, cc‐pVTZ, and Aug‐cc‐pVTZ basis sets. The scalar relativistic effect on the excited states of BrOClO and ClOBrO were estimated. In addition, the potential energy curves of the lowest six singlet states and two triplet states of BrOClO and ClOBrO, as well as BrOOCl were calculated at both MCSCF (complete active space self‐consistent field) and MRCI levels of theory using Aug‐cc‐pVDZ basis set on the active space (18e,12o) along the distances of BrO? ClO, ClO? BrO, and BrO? OCl. The results were compared among BrOOCl, ClOBrO, and BrOClO. The first singlet excited state of BrOOCl is 1.12 eV higher than that of BrOClO and 1.36 eV higher than that of ClOBrO at MRCI/cc‐pVTZ level of theory. The first triplet excited state of BrOOCl is 0.77 eV higher than that of BrOClO and 0.86 eV higher than that of ClOBrO at MRCI/cc‐pVTZ level of theory. Most of the excited states of BrOClO studied in this work are unbound states; but most of the ClOBrO and BrOOCl excited states studied in this work are weakly bound states at MRCI level of theory. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Spectroscopic constants and molecular properties of A3Σu+, B3Πg,W3Δu,and B′3Σu− electronic states of the N2 molecule

The potential energy curves (PECs) of A³Σ, B³Π_g, W³Δ_u, and B^′3Σ electronic states of the N₂ molecule have been studied for internuclear separations from 0.05 to 2.0 nm using the full valence complete active space self‐consistent‐field method followed by the highly accurate valence internally contracted multireference configuration interaction (MRCI) approach in conjunction with the correlation‐consistent basis sets. Effects on the PECs by the core–valence correlation and relativistic corrections are taken into account. The way to consider the relativistic correction is to use the second‐order Douglas‐Kroll Hamiltonian approximation. The core–valence correlation correction is made with the cc‐pCV5Z basis set. And the relativistic correction is performed at the level of cc‐pV5Z basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are also corrected for size‐extensivity errors by the Davidson modification (MRCI+Q). These PECs are extrapolated to the complete basis set limit. The spectroscopic parameters of ¹⁴N₂, ¹⁴N¹⁵N, and ¹⁵N₂ isotopologs have been evaluated and compared with those reported in the literature. Excellent agreement has been found between the present results and the Rydberg‐Klein‐Rees (RKR) data. With the PECs obtained by the MRCI+Q/CV+DK+56 calculations, the first 30 vibrational states for three species are computed for each electronic state. And for each electronic state of each species, the vibrational level G(ν), inertial rotation constant B_ν, and centrifugal distortion constant D_ν have been determined, which agree well with the RKR data. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Prediction of a Neutral Noble Gas Compound in the Triplet State

Discovery of the HArF molecule associated with H?Ar covalent bonding [Nature, 2000 , 406, 874–876] has revolutionized the field of noble gas chemistry. In general, this class of noble gas compound involving conventional chemical bonds exists as closed‐shell species in a singlet electronic state. For the first time, in a bid to predict neutral noble gas chemical compounds in their triplet electronic state, we have carried out a systematic investigation of xenon inserted FN and FP species by using quantum chemical calculations with density functional theory and various post‐Hartree–Fock‐based correlated methods, including the multireference configuration interaction technique. The FXeP and FXeN species are predicted to be stable by all the computational methods employed in the present work, such as density functional theory (DFT), second‐order Møller–Plesset perturbation theory (MP2), coupled‐cluster theory (CCSD(T)), and multireference configuration interaction (MRCI). For the purpose of comparison we have also included the Kr‐inserted compounds of FN and FP species. Geometrical parameters, dissociation energies, transition‐state barrier heights, atomic charge distributions, vibrational frequency data, and atoms‐in‐molecules properties clearly indicate that it is possible to experimentally realize the most stable state of FXeP and FXeN molecules, which is triplet in nature, through the matrix isolation technique under cryogenic conditions.     

Quadratic Padé approximants and the intruder state problem of multireference perturbation methods

Simple and quadratic Padé resummation methods are applied to high‐order series from multireference many‐body perturbation theory (MR‐MBPT) calculations using various partitioning schemes (Møller–Plesset, Epstein–Nesbet, and forced degeneracy) to determine their efficacy in resumming slowly convergent or divergent series. The calculations are performed for the ground and low‐lying excited states of (i) CH₂, (ii) BeH₂ at three geometries, and (iii) Be, for which full configuration interaction (CI) calculations are available for comparison. The 49 perturbation series that are analyzed include those with oscillatory and monotonic divergence and convergence, including divergences that arise from either frontdoor or backdoor intruder states. Both the simple and quadratic Padé approximations are found to speed the convergence of slowly convergent or divergent series. However, the quadratic Padé method generally outperforms the simple Padé resummation. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Spin-orbit configuration interaction study of the electronic structure of the 5f (2) manifold of U(4+) and the 5f manifold of U(5+)

The energy levels of the 5f configuration of U(5+) and 5f(2) configuration of U(4+) have been calculated in a dressed effective Hamiltonian relativistic spin-orbit configuration interaction framework. Electron correlation is treated in the scalar relativistic scheme with either the multistate multireference second-order multiconfigurational perturbation theory (MS-CASPT2) or with the multireference single and double configuration interaction (MRCI) and its size-extensive Davidson corrected variant. The CASPT2 method yields relative energies which are lower than those obtained with the MRCI method, the differences being the largest for the highest state (1)S(0) of the 5f(2) manifold. Both valence correlation effects and spin-orbit polarization of the outer-core orbitals are shown to be important. The satisfactory agreement of the results with experiments and four-component correlated calculations illustrates the relevance of dressed spin-orbit configuration interaction methods for spectroscopy studies of heavy elements.     

Multiconfiguration perturbation theory: size consistency at second order

A modified version of a previously elaborated multiconfiguration perturbation theory (MCPT) [Rolik et al. J. Chem. Phys. 119, 1922 (2003)] is presented. In the modified formulation size consistency is ensured at second order in energy, by omitting projectors from the zero order Hamiltonian operator. This MCPT formulation is abbreviated as SC2-MCPT (size consistent at second order). To ensure proper separability, we also require that energy denominators are constructed as differences of some one-particle energies. A similar choice for energy denominators also renders the well-known multireference Moller-Plesset (MRMP) energy size consistent at second order. The same thing applies to the related multireference perturbation theory by Witek, Nakano, and Hirao.     

Full CI benchmark calculations for molecular properties

Full CI calculations of first- and second-order properties are presented to provide benchmark results for comparisons with other methods, such as multireference CI(MRCI). The full CI(FCI) polarizability of F^– is computed using a double zeta plus polarization plus diffuse basis set. These FCI results are compared to those obtained at other levels of theory; the CASSCF/MRCI with Davidson correction results are in excellent agreement with the FCI. Differences between the polarizability results computed as a (numerical) second derivative of the energy or as an induced dipole moment are also discussed. FCI calculations are presented for the dipole moment and polarizability of HF, CH₂ and SiH₂ using a DZP basis set. Again, the CASSCF/MRCI values are in excellent agreement with the FCI results, whereas SDCI values, whether computed as an expectation value or as an energy derivative, are much worse. The results obtained using the CPF approach are in considerably better agreement with the FCI results than SDCI, and are similar in quality to the SDCI energy derivative results with the inclusion of Davidson's correction.     

Reciprocal adjustment of approximate coupled cluster and configuration interaction approaches

The linear version of the externally corrected coupled cluster method with singles and doubles (ecLCCSD), the recently proposed coupled cluster corrections to the multireference configuration interaction (ccMRCI) energies, and the so‐called self‐consistent, size‐consistent [(SC)²] approaches, which are designed to correct for the dynamic correlation effects and the size inconsistency of the MRCI energies, are analyzed and compared using several illustrative examples, including the dissociation of a triple‐zeta (TZ) model of the N₂ molecule. It is emphasized that the exponential cluster ansatz for the wave function is the basis of all these approaches, and appropriate cluster analysis of the MRCI wave function is the key step for both ecLCCSD and ccMRCI. The contributions from the orthogonal complement of the MRCI space, which can be generated by relying on such a cluster analysis, are responsible for a substantial part of the missing correlation energy. The ecLCCSD approach seems to represent a particularly attractive alternative to other highly accurate methods for the calculation of the ground‐state energy in the presence of quasidegeneracy, both due to its efficiency and affordability. It may in fact be regarded as a simple alternative to the iterative reduced multireference (RMR) CCSD method. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 693–703, 2000     

Improved third-order Møller-Plesset perturbation theory

Based on a partitioning of the total correlation energy into contributions from parallel‐ and antiparallel‐spin pairs of electrons, a modified third‐order Møller–Plesset (MP) perturbation theory is developed. The method, termed SCS–MP3 (SCS for spin‐component‐scaled) continues previous work on an improved version of MP2 (S. Grimme, J Chem Phys 2003, 118, 9095). A benchmark set of 32 isogyric reaction energies, 11 atomization energies, and 11 stretched geometries is used to assess to performance of the model in comparison to the standard quantum chemical approaches MP2, MP3, and QCISD(T). It is found, that the new method performs significantly better than usual MP2/MP3 and even outperforms the more costly QCISD method. Opposite to the usual MP series, the SCS third‐order correction uniformly improves the results. Dramatic enhancements are especially observed for the more difficult atomization energies, some of the stretched geometries, and reaction and ionization energies involving transition metal compounds where the method seems to be competitive or even superior to the widely used density functional approaches. Further tests performed for other complex systems (biradicals, C₂₀ isomers, transition states) demonstrate that the SCS–MP3 model yields often results of QCISD(T) accuracy. The uniformity with which the new approach improves for very different correlation problems indicates significant robustness, and suggests it as a valuable quantum chemical method of general use. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1529–1537, 2003     

Electronic spectra of nitroethylene

A systematic study of the electronic excited states of nitroethylene (C₂H₃NO₂) was carried out using the approximate coupled‐cluster singles‐and‐doubles approach with the resolution of the identity (RI‐CC2), the time dependent density functional theory with the CAMB3LYP functional (TDDFT/CAMB3LYP) and the DFT multireference configuration interaction (DFT/MRCI) method. Vertical transition energies and optical oscillator strengths were computed for a maximum of 20 singlet transitions. Semiclassical simulations of the ultraviolet (UV) spectra were performed at the RI‐CC2 and DFT/MRCI levels. The main features in the UV spectrum were assigned to a weak n‐π* transition, and two higher energy π_CC+O‐π* bands. These characteristics are common to molecules containing NO₂ groups. Simulated spectra are in good agreement with the experimental spectrum. The energy of the bands in the DFT/MRCI simulation agrees quite well with the experiment, although it overestimates the band intensities. RI‐CC2 produced intensities comparable to the experiment, but the bands were blue shifted. A strong π_CC+O‐π* band, not previously measured, was found in the 8–9 eV range. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Spectroscopic constants and molecular properties of the X2Π and a4Σ− electronic states of the SiF radical

The potential energy curves (PECs) of the X²Π and a⁴Σ^? electronic states of the SiF radical have been studied by an ab initio quantum chemical method. The calculations have been made using the complete active space self‐consistent field (CASSCF) method, which is followed by the valence internally contracted multireference configuration interaction (MRCI) approach in combination with several correlation‐consistent basis sets. The effects on the PECs by the core‐valence correlation and relativistic corrections are included. The way to consider the relativistic correction is to use the third‐order Douglas–Kroll Hamiltonian approximation. The relativistic corrections are made at the level of cc‐pV5Z basis set. The core‐valence correlation corrections are performed using the cc‐pCV5Z basis set. To obtain more reliable results, the PECs determined by the MRCI calculations are also corrected for size‐extensivity errors by means of the Davidson modification (MRCI+Q). These PECs are extrapolated to the complete basis set limit by the total‐energy extrapolation scheme. Using these PECs, the spectroscopic parameters are determined and compared with those reported in the literature. With these PECs obtained by the MRCI+Q/CV+DK+56 calculations, the vibrational levels, inertial rotation, and centrifugal distortion constants of the first 20 vibrational state of each electronic state are calculated when the rotational quantum number J equals zero. Comparison with the Rydberg‐Klein‐Rees (RKR) data shows that the present results are reliable and accurate. The molecular constants of the X²Π and a⁴Σ^? electronic states determined by the MRCI+Q/CV+DK+56 calculations should be good prediction for future laboratory experiment. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Initial excited‐state relaxation of locked retinal protonated schiff base chromophore. An insight from coupled cluster and multireference perturbation theory calculations

The initial S₁ excited‐state relaxation of retinal protonated Schiff base (RPSB) analog with central C11C12 double bond locked by eight‐membered ring (locked‐11.8) was investigated by means of multireference perturbation theory methods (XMCQDPT2, XMS‐CASPT2, MS‐CASPT2) as well as single‐reference coupled‐cluster CC2 method. The analysis of XMCQDPT2‐based geometries reveals rather weak coupling between in‐plane and out‐of‐plane structural evolution and minor energetical relaxation of three locked‐11.8 conformers. Therefore, a strong coupling between bonds length inversion and backbone out‐of‐plane deformation resulting in a very steep S₁ energy profile predicted by CASSCF/CASPT2 calculations is in clear contradiction with the reference XMCQDPT2 results. Even though CC2 method predicts good quality ground‐state structures, the excited‐state structures display more advanced torsional deformation leading to ca. 0.2 eV exaggerated energy relaxation and significantly red shifted (0.4–0.7 eV) emission maxima. According to our findings, the initial photoisomerization process in locked‐11.8, and possibly in other RPSB analogs, studied fully (both geometries and energies) by multireference perturbation theory may be somewhat slower than predicted by CASSCF/CASPT2 or CC2 methods. © 2018 Wiley Periodicals, Inc.     

Global X2A′ potential energy surface of Li2H and quantum dynamics of H + Li2 (X1Σg+) → Li + LiH (X1Σ+) reaction

A global potential energy surface (PES) for the electronic ground state of Li₂H system is constructed over a large configuration space. About 30 000 ab initio energy points have been calculated by MRCI‐F12 method with aug‐cc‐pVTZ basis set. The neural network method is applied to fit the PES and the root mean square error of the current PES is only 1.296 meV. The reaction dynamics of the title reaction has been carried out by employing time‐dependent wave packet approach with second order split operator on the new PES. The reaction probability, integral cross section and thermal rate constant are obtained from the dynamics calculation. In most of the collision energy regions, the integral cross sections are in well agreement with the results reported by Gao et al. The rate constant calculated from the new PES increases in the temperature range of present investigation.     

Density functional and multiconfigurational ab initio study of the ground and excited states of Os2

Multiconfigurational ab initio methods predict that the ⁵Π_u state as the ground state instead of the ⁷Δ_u state. Although multiconfigurational perturbation theory correctly predicts the ground state, they overestimate the bond dissociation energy (BDE). Only multireference configuration interaction method can reasonably calculate the BDE. The spin‐orbit effect on the spectroscopic constants is not significant. The results calculated by density functional theory (DFT) vary significantly depending on the selection of a DFT functional. No DFT functional gives the same energy ordering as calculated by the second‐order multiconfigurational perturbation theory (CASPT2). The old generalized gradient approximations functionals are well suited for predicting the ground state and calculating the bond length and the vibrational frequency of Os₂. According to the CASPT2 calculation, the ground state of Os₂ has a quadruple bond. © 2014 Wiley Periodicals, Inc.