Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12

An explicitly correlated complete active space second-order perturbation (CASPT2-F12) method is presented which strongly accelerates the convergence of CASPT2 energies and properties with respect to the basis set size. A Slater-type geminal function is employed as a correlation factor to represent the electron-electron cusp of the wave function. The explicitly correlated terms in the wave function are internally contracted. The required density matrix elements and coupling coefficients are the same as in conventional CASPT2, and the additional computational effort for the F12 correction is small. The CASPT2-F12 method is applied to the singlet-triplet splitting of methylene, the dissociation energy of ozone, and low-lying excited states of pyrrole.     

Role of the coordination of the azido bridge in the magnetic coupling of copper(II) binuclear complexes

It is well-known that the azido bridge gives rise antiferromagnetic (AF) or ferromagnetic (F) coupling depending on its coordination mode, namely end-to-end or end-on, respectively. The aim of the present work is to analyse the factors contributing to this different magnetic behaviour. The difference dedicated configuration interaction (DDCI) method is applied to several binuclear Cu(II) azido-bridged models with both types of coordination. In end-on complexes, the direct exchange and the spin polarisation contributions are found to be responsible for the ferromagnetic coupling. In end-to-end complexes, both the direct exchange and the spin polarisation are small and the leading term is the antiferromagnetic dynamical polarisation contribution. The most relevant physical effects are included in the DDCI calculations so that good quantitative agreement is reached for the coupling constant as well as the spin densities.     

Heisenberg behavior of some carbon‐beryllium compounds: How well truncated‐CI approaches work

This works tries to establish the performance of truncated CI calculations on the evaluation of magnetic coupling parameters with respect to available FCI estimates on a set of carbon‐beryllium clusters. First‐, second‐ and third‐neighbor magnetic coupling constants have been evaluated and many body effective parameters as the cyclic terms. They result from the fitting of the low‐lying states to the eigenvalues of an extended Heisenberg Hamiltonian, involving not only two‐body isotropic terms but also cyclic terms. SDCI and DDCI calculations have been carried out and their performance compared with FCI ones. The impact of the basis set choice and size‐consistency errors have been explored. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2011     

Ab Initio CI Determination of the Exchange Coupling Constant of Doubly-Bridged Nickel(II) Dimers

Ab initio DDCI2 (difference-dedicated configuration interaction) calculations are performed on the exchange coupling constant of the doubly-bridged Ni(II) complexes [Ni(en)(2)Cl](2)(2+) and [Ni(terpy)(N(3))](2)(2+), which are modeled by substituting the external ligands with ammonia groups. The variational CI space is selected on the grounds of the effective Hamiltonian theory and includes all the second-order contributions to the difference between the lowest quintet, triplet, and singlet states. Both complexes are found to be ferromagnetic, with coupling constants of 1.8 and 21.1 cm(-1), in good agreement with the experiment. A transformation of the molecular orbitals is also proposed for large systems, enabling the molecular orbital set to be significantly truncated-as well as the file of two-electron integrals and the DDCI2 space-with no loss of efficiency.     

Mo2(CN)11]:5- A detailed description of ligand-field spectra and magnetic properties by first-principles calculations

An ab initio multiconfigurational approach has been used to calculate the ligand-field spectrum and magnetic properties of the title cyano-bridged dinuclear molybdenum complex. The rather large magnetic coupling parameter J for a single cyano bridge, as derived experimentally for this complex by susceptibility measurements, is confirmed to a high degree of accuracy by our CASPT2 calculations. Its electronic structure is rationalized in terms of spin-spin coupling between the two constituent hexacyano-monomolybdate complexes. An in-depth analysis on the basis of Anderson's kinetic exchange theory provides a qualitative picture of the calculated CASSCF antiferromagnetic ground-state eigenvector in the Mo dimer. Dynamic electron correlations as incorporated into our first-principles calculations by means of the CASPT2 method are essential to obtain quantitative agreement between theory and experiment.     

Unusual magnetic properties of mixed-valence system: multiconfigurational method theoretical study on Mn2+ cation

The geometry structure, dissociation energy, vibrational frequencies, and low-lying spin-state energy spectrum of Mn2+ are investigated by using ab initio CASSCF/ECP10MDF, complete active space self-consistent field/atomic natural orbital basis sets (CASSCF/ANO-s), CASPT2/ECP10MDF, and second-order perturbation theory with CASSCF reference function/atomic natural orbital basis sets (CASPT2/ANO-s) levels of theory. For the ground state the dissociation energy of 1.397 eV calculated at the CASPT2/ANO-s level supports Jarrlod's experimental value of 1.39 eV. The equilibrium bond length and vibrational frequency are 2.940 A calculated at the CASPT2/ANO-s level of theory and 214.4 cm-1 calculated at the CASSCF/ANO-s level of theory, respectively. On the basis of the mixed-valence model, the Heisenberg exchange constant J(-71.2 cm-1) and the double-exchange constant B(647.7 cm-1) are extracted explicitly from the low-lying energy spectrum calculated at the higher levels of theory. The magnetic competition between the weaker Heisenberg exchange interactions and the stronger double-exchange interactions makes the ground state a 12Sigmag+ state, consistent with electron paramagnetic resonance experimental observation, which explains unusual magnetic properties of Mn2+, quite different from the antiferromagnetic ground state of Mn2 and Cr2. On the other hand, the results calculated at the higher levels of theory show the consistent antiferromagnetic Heisenberg exchange interactions between 3d-3d for Cr2, Mn2+, and Mn2.     

Nuclear magnetic resonance J coupling constant polarizabilities of hydrogen peroxide: A basis set and correlation study

In this article, we present the so far most extended investigation of the calculation of the coupling constant polarizability of a molecule. The components of the coupling constant polarizability are derivatives of the nuclear magnetic resonance (NMR) indirect nuclear spin–spin coupling constant with respect to an external electric field and play an important role for both chiral discrimination and solvation effects on NMR coupling constants. In this study, we illustrate the effects of one‐electron basis sets and electron correlation both at the level of density functional theory as well as second‐order polarization propagator approximation for the small molecule hydrogen peroxide, which allowed us to perform calculations with the largest available basis sets optimized for the calculation of NMR coupling constants. We find a systematic but rather slow convergence with the one‐electron basis set and that augmentation functions are required. We observe also large and nonsystematic correlation effects with significant differences between the density functional and wave function theory methods. © 2012 Wiley Periodicals, Inc.     

Energetics, transition states, and intrinsic reaction coordinates for reactions associated with O(3P) processing of hydrocarbon materials

Electronic structure calculations based on multiconfiguration wave functions are used to investigate a set of archetypal reactions relevant to O(3P) processing of hydrocarbon molecules and surfaces. These include O(3P) reactions with methane and ethane to give OH plus methyl or ethyl radicals, O(3P) + ethane to give CH3O + CH3, and secondary reactions of the OH product radical with ethane and the ethyl radical. Geometry optimization is carried out with CASSCF/cc-pVTZ for all reactions, and with CASPT2/cc-pVTZ for O(3P) + methane/ethane. Single-point energy corrections are applied with CASPT2, CASPT3, and MRCI + Q with the cc-pVTZ and cc-pVQZ basis sets, and the energies extrapolated to the complete basis set limit (CBL). Where comparison of computed barriers and energies of reaction with experiment is possible, the agreement is good to excellent. The best agreement (within experimental error) is found for MRCI + Q/CBL applied to O(3P) + methane. For the other reactions, CASPT2/CBL and MRCI + Q/CBL predictions differ from experiment by 1-5 kcal/mol for 0 K enthalpies of reaction, and are within 1 kcal/mol of the best-estimate experimental range of 0 K barriers for O(3P) + ethane and OH + ethane. The accuracy of MRCI + Q/CBL is limited mainly by the quality of the active space. CASPT2/CBL barriers are consistently lower than MRCI + Q/CBL barriers with identical reference spaces.     

Relative energy of the high-(5T2g) and low-(1A1g) spin states of [Fe(H2O)6]2+, [Fe(NH3)6]2+, and [Fe(bpy)3]2+: CASPT2 versus density functional theory

High-level ab initio calculations using the CASPT2 method and extensive basis sets were performed on the energy differences of the high-[(5)T(2g):t(2g) (4)e(g) (2)] and low-[(1)A(1g):t(2g) (6)] spin states of the pseudo-octahedral Fe(II) complexes [Fe(H(2)O)(6)](2+), [Fe(NH(3))(6)](2+), and [Fe(bpy)(3)](2+). The results are compared to the results obtained from density functional theory calculations with the generalized gradient approximation functional BP86 and two hybrid functionals B3LYP and PBE0, and serve as a calibration for the latter methods. We find that large basis set CASPT2 calculations may provide results for the high-spin/low-spin splitting DeltaE(HL) that are accurate to within 1000 cm(-1), provided they are based on an adequately large CAS[10,12] reference wave function. The latter condition was found to be much more stringent for [Fe(bpy)(3)](2+) than for the other two complexes. Our "best" results for DeltaE(HL) (including a zero-point energy correction) are -17 690 cm(-1) for [Fe(H(2)O)(6)](2+), -8389 cm(-1) for [Fe(NH(3))(6)](2+), and 3820 cm(-1) for [Fe(bpy)(3)](2+).     

A convenient decontraction procedure of internally contracted state-specific multireference algorithms

Internally contracted state-specific multireference (MR) algorithms, either perturbative such as CASPT2 or NEVPT2, or nonperturbative such as contracted MR configuration interaction or MR coupled cluster, are computationally efficient but they may suffer from the internal contraction of the wave function in the reference space. The use of a low dimensional multistate model space only offers limited flexibility and is not always practicable. The present paper suggests a convenient state-specific procedure to decontract the reference part of the wave function from a series of state-specific calculations using slightly perturbed zero-order wave functions. The method provides an orthogonal valence bond reading of the ground state and an effective valence Hamiltonian, the excited roots of which are shown to be relevant. The orthogonal valence bond functions can be considered quasidiabatic states and the effective valence Hamiltonian gives therefore the quasidiabatic energies and the electronic coupling among the quasidiabatic states. The efficiency of the method is illustrated in two case problems where the dynamical correlation plays a crucial role, namely, the LiF neutral/ionic avoided crossing and the F(2) ground state wave function.     

A theoretical study of the kinetics of the benzylperoxy radical isomerization

The rate constant of the benzylperoxy isomerization reaction has been computed using 54 different levels of theory and has been compared to the experimental value reported at 773 K. The aim of this methodology work is to demonstrate that standard theoretical methods are not adequate to obtain quantitative rate constants for the reaction under study. The use of the elaborated CASPT2 method is essential to estimate a quantitative rate constant. Geometry optimizations and vibrational frequency calculations are performed using three different methods (B3LYP, MPW1K, and MP2) and six different basis sets (6-31G(d,p), 6-31+G(d,p), 6-31++G(d,p), 6-311G(d,p), 6-311+G(d,p), and cc-pVDZ). Single-point energy calculations are performed with the highly correlated ab initio coupled cluster method in the space of single, double, and triple (pertubatively) electron excitations CCSD(T) using the 6-31G(d,p) basis set, and with the CASPT2 level of theory with the ANO-L-VDZP basis set. Canonical transition-state theory with a simple Wigner tunneling correction is used to predict the high-pressure limit rate constants as a function of temperature. We recommend the use of the CASPT2/ANO-L-VDZP//B3LYP/cc-pVDZ level of theory to compute the temperature dependence of the rate constant of the four-center isomerization of the benzylperoxy radical. It is given by the following relation: k(600-2000 K) (in s (-1)) = (1.29 x 10 (10)) T (0.79) exp[(-133.1 in kJ mol (-1))/ RT]. These parameters can be used in the thermokinetic models involving aromatic compounds at high pressure. This computational procedure can be extended to predict rate constants for other similar reactions where no available experimental data exist.     

The restricted active space followed by second-order perturbation theory method: theory and application to the study of CuO2 and Cu2O2 systems

A multireference second-order perturbation theory using a restricted active space self-consistent field wave function as reference (RASPT2/RASSCF) is described. This model is particularly effective for cases where a chemical system requires a balanced orbital active space that is too large to be addressed by the complete active space self-consistent field model with or without second-order perturbation theory (CASPT2 or CASSCF, respectively). Rather than permitting all possible electronic configurations of the electrons in the active space to appear in the reference wave function, certain orbitals are sequestered into two subspaces that permit a maximum number of occupations or holes, respectively, in any given configuration, thereby reducing the total number of possible configurations. Subsequent second-order perturbation theory captures additional dynamical correlation effects. Applications of the theory to the electronic structure of complexes involved in the activation of molecular oxygen by mono- and binuclear copper complexes are presented. In the mononuclear case, RASPT2 and CASPT2 provide very similar results. In the binuclear cases, however, only RASPT2 proves quantitatively useful, owing to the very large size of the necessary active space.     

Benchmarking NMR indirect nuclear spin-spin coupling constants: SOPPA, SOPPA(CC2), and SOPPA(CCSD) versus CCSD

Accurate calculations of NMR indirect nuclear spin-spin coupling constants require especially optimized basis sets and correlated wave function methods such as CCSD or SOPPA(CCSD). Both methods scale as N(6), where N is the number of orbitals, which prevents routine applications to molecules with more than 10-15 nonhydrogen atoms. We have therefore developed a modification of the SOPPA(CCSD) method in which the CCSD singles and doubles amplitudes are replaced by CC2 singles and doubles amplitudes. This new method, called SOPPA(CC2), scales only as N(5), like the original SOPPA-method. The performance of the SOPPA(CC2) method for the calculation of indirect nuclear spin-spin coupling constants is compared to SOPPA and SOPPA(CCSD) employing a set of benchmark molecules. We also investigate the basis set dependence by employing three different basis sets optimized for spin-spin coupling constants, namely the HuzIV-su4, ccJ-pVTZ, and ccJ-pVQZ basis sets. The results of the corresponding CCSD calculations are used as a theoretical reference.     

Is the choice of a standard zeroth‐order hamiltonian in CASPT2 ansatz optimal in calculations of excitation energies in protonated and unprotonated schiff bases of retinal?

To account for systematic error of CASPT2 method empirical modification of the zeroth‐order Hamiltonian with Ionization Potential‐Electron Affinity (IPEA) shift was introduced. The optimized IPEA value (0.25 a.u.), called standard IPEA (S‐IPEA), was recommended but due to its unsatisfactory performance in multiple metallic and organic compounds it has been questioned lately as a general parameter working properly for all molecules under CASPT2 study. As we are interested in Schiff bases of retinal, an important question emerging from this conflict of choice, to use or not to use S‐IPEA, is whether the introduction of the modified zeroth‐order Hamiltonian into CASPT2 ansatz does really improve their energetics. To achieve this goal, we assessed an impact of the IPEA shift value, in a range of 0–0.35 a.u., on vertical excitation energies to low‐lying singlet states of two protonated (RPSBs) and two unprotonated (RSBs) Schiff bases of retinal for which experimental data in gas phase are available. In addition, an effect of geometry, basis set, and active space on computed VEEs is also reported. We find, that for these systems, the choice of S‐IPEA significantly overestimates both S₀→S₁ and S₀→S₂ energies and the best theoretical estimate, in reference to the experimental data, is provided with either unmodified zeroth‐order Hamiltonian or small value of the IPEA shift in a range of 0.05–0.15 a.u., depending on active space and basis set size, equilibrium geometry, and character of the excited state. © 2018 Wiley Periodicals, Inc.     

Active space and basis set effects in CASPT2 models of the 1,3‐butadiene‐ethene cycloaddition and the 1,3‐butadiene dimerization

The first step of the asynchronous biradical, stepwise biradical, and concerted mechanisms of the 1,3‐butadiene Diels–Alder reactions with both ethene and itself was studied using CASPT2 to determine the influence of basis set and active space on reaction barriers. CASPT2(6,6) with the cc‐pVDZ, 6‐311+G(3df,2p) and cc‐pVTZ basis sets provided the best results with average errors below 3.1 kJ mol^?1 with respect to the experimental result. Increasing the active space size also had little effect on the calculated reaction barriers. With respect to experimental results, uncontracted multireference averaged quadratic coupled cluster (MRAQCC) produced superior barriers than internally contracted MRAQCC by 16.1–39.3 kJ mol^?1. The inability of CASSCF to locate transition states for some of the cycloadditions across the butadiene‐ethene and butadiene dimerization reactions is also rationalized. CASPT2 suggests a preference for the concerted mechanism of the butadiene‐ethene reaction, however, no basis set yielded a preference for any of the butadiene dimerization pathways. © 2015 Wiley Periodicals, Inc.     

Theoretical studies on the low-lying electronic states of the HSO neutral radical and its cation

Using the complete active space self-consistent field (CASSCF) method with large atomic natural orbital (ANO-L) basis set, four electronic states of the HSO neutral radical are optimized. The vertical transitions of the HSO neutral radical are investigated by using the same method under the basis set of ANO-L functions augmented with a series of adapted 1s1p1d Rydberg functions, through which eight valence states and eight Rydberg states are probed. Ionic states of the HSO neutral radical are extensively studied in both cases of the adiabatic and vertical ionization, from which the relatively complete understanding of ionization energies is given. To include further correlation effects, the second-order perturbation method (CASPT2) is implemented, and the comparison between CASSCF and CASPT2 methods is performed.     

Computation of conical intersections by using perturbation techniques

Multiconfigurational second-order perturbation theory, both in its single-state multiconfigurational second-order perturbation theory (CASPT2) and multistate (MS-CASPT2) formulations, is used to search for minima on the crossing seams between different potential energy hypersurfaces of electronic states in several molecular systems. The performance of the procedures is tested and discussed, focusing on the problem of the nonorthogonality of the single-state perturbative solutions. In different cases the obtained structures and energy differences are compared with available complete active space self-consistent field and multireference configuration interaction solutions. Calculations on different state crossings in LiF, formaldehyde, the ethene dimer, and the penta-2,4-dieniminium cation illustrate the discussions. Practical procedures to validate the CASPT2 solutions in polyatomic systems are explored, while it is shown that the application of the MS-CASPT2 procedure is not straightforward and requires a careful analysis of the stability of the results with the quality of the reference wave functions, that is, the size of the active space.