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.     

Calibration of the n-electron valence state perturbation theory approach

Extensive tests have been performed to benchmark and to compare with second-order perturbation theory based on a complete active space self-consistent field reference function (CASPT2), the recently developed n-electron valence state perturbation theory at second order (NEVPT2). Test calculations included the group fifteen diatomic molecules X(2) (X=N, P, As, and Sb) and the (4)S/(2)D and (4)S/(2)P splittings for the corresponding atoms, the (1)A(1)-(3)B(1) splittings for CH(2) and SiH(2), and the absorption spectra of pyrrole and of Cu(Imidazole)(2)(SH)(SH(2))(+), which is a model for plastocyanin. Comparisons with full configuration-interaction calculations and experimental data show that the accuracy of NEVPT2 is in most cases even better than CASPT2. Where intruder states hamper the CASPT2 calculations, NEVPT2 performs significantly better. Care is needed in the choice of active orbitals, for example in the calculation of the (4)S/(2)D and (4)S/(2)P splittings for the group fifteen atoms. This is due to the different treatment of orbitals belonging to the inactive or active spaces, making the NEVPT2 not invariant for the choice of active space, even in cases where the multiconfiguration self-consistent field energy is invariant.     

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.     

A two‐scale approach to electron correlation in multiconfigurational perturbation theory

We present a new approach for the calculation of dynamic electron correlation effects in large molecular systems using multiconfigurational second‐order perturbation theory (CASPT2). The method is restricted to cases where partitioning of the molecular system into an active site and an environment is meaningful. Only dynamic correlation effects derived from orbitals extending over the active site are included at the CASPT2 level of theory, whereas the correlation effects of the environment are retrieved at lower computational costs. For sufficiently large systems, the small errors introduced by this approximation are contrasted by the substantial savings in both storage and computational demands compared to the full CASPT2 calculation. Provided that static correlation effects are correctly taken into account for the whole system, the proposed scheme represent a hierarchical approach to the electron correlation problem, where two molecular scales are treated each by means of the most suitable level of theory. © 2014 Wiley Periodicals, Inc.     

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)     

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.     

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.     

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.     

A theoretical investigation of the excited states of OCLO radical, cation, and anion using the CASSCF/CASPT2 method

Using the complete active space self-consistent field method with a large atomic natural orbital basis set, 10, 13, and 9 electronic states of the OClO radical, OClO(+) cation, and OClO(-) anion were calculated, respectively. Taking the further correlation effects into account, the second-order perturbation (CASPT2) calculations were carried out for the energetic calibration. The photoelectron spectroscopy of the OClO radical and OClO(-) anion were extensively studied in the both case of the adiabatic and vertical ionization energies. The calculated results presented the relatively complete assignment of the photoelectron bands of the experiments for OClO and its anion. Furthermore, the Rydberg states of the OClO radical were investigated by using multiconfigurational CASPT2 (MS-CASPT2) theory under the basis set of large atomic natural orbital functions augmented with an adapted 1s1p1d Rydberg functions that have specially been built for this study. Sixteen Rydberg states were obtained and the results were consistent with the experimental results.     

OClO里德堡态激发能的准确预测及其阴离子低能激发态的从头算研究

采用全活化空间自洽场方法(CASSCF)研究了OClO阴离子7个低能电子态及其自由基的基态. 为了进一步考虑动态电子相关效应, 采用二级多组态微扰理论(CASPT2)获得更加可靠的能量值. 此外, 在ANO-L基组的基础上, 在OClO自由基的电荷中心增加了为研究里德堡态所建立的1s1p1d的波函数, 并应用多组态二级微扰理论(MS-CASPT2)方法获得了里德堡态的准确电子激发能.     

Multireference composite approaches for the accurate study of ground and excited electronic states: C2, N2, and O2

A multireference analog of the correlation consistent composite approach (MR-ccCA) based on complete active space with second-order perturbation theory (CASPT2) has been utilized in an investigation of the ground and valence excited states of C(2), N(2), and O(2). The performance of different second-order multireference perturbation theory methods including second-order n-electron valence state perturbation theory, second-order multireference M?ller-Plesset, and second-order generalized van Vleck perturbation theory has been analyzed as potential alternatives to CASPT2 within MR-ccCA. The MR-ccCA-P predicts spectroscopic constants with overall mean absolute deviations from experimental values of 0.0006 A?, 7.0 cm(-1), and 143 cm(-1) for equilibrium bond length (r(e)), harmonic frequency (ω(e)), and term values (T(e)), respectively, which are comparable to the predictions by more computationally costly multireference configuration interaction-based methods.     

Quasi-degenerate perturbation theory with general multiconfiguration self-consistent field reference functions

The quasi-degenerate perturbation theory (QDPT) with complete active space (CAS) self-consistent field (SCF) reference functions is extended to the general multiconfiguration (MC) SCF references functions case. A computational scheme that utilizes both diagrammatic and sum-over-states approaches is presented. The second-order effective Hamiltonian is computed for the external intermediate configurations (including virtual or/and core orbitals) by the diagrammatic approach and for internal intermediate configurations (including only active orbitals) by the configuration interaction matrix-based sum-over-states approach. The method is tested on the calculations of excitation energies of H(2)O, potential energy curves of LiF, and valence excitation energies of H(2)CO. The results show that the present method yields very close results to the corresponding CAS-SCF reference QDPT results and the available experimental values. The deviations from CAS-SCF reference QDPT values are less than 0.1 eV on the average for the excitation energies of H(2)O and less than 1 kcal/mol for the potential energy curves of LiF. In the calculation of the valence excited energies of H(2)CO, the maximum deviation from available experimental values is 0.28 eV.     

Ab initio calculations of electronically excited states of cyano-substituted polyacetylene cations

The transition energies for the lowest energy pi --> pi* electronic excitations are calculated with the complete active space self-consistent field method (CASSCF) and with the complete active space second-order perturbation theory method (CASPT2) for the linear cyano-substituted polyacetylene cations, H-Cn-CN+, n = 4-11, and NC-Cn-CN+, n = 2-10. These systems are models for an important class of interstellar species. We demonstrate the utility of the theoretical calculations in assigning the experimental spectra.     

Communication: extended multi-state complete active space second-order perturbation theory: energy and nuclear gradients

The extended multireference quasi-degenerate perturbation theory, proposed by Granovsky [J. Chem. Phys. 134, 214113 (2011)], is combined with internally contracted multi-state complete active space second-order perturbation theory (XMS-CASPT2). The first-order wavefunction is expanded in terms of the union of internally contracted basis functions generated from all the reference functions, which guarantees invariance of the theory with respect to unitary rotations of the reference functions. The method yields improved potentials in the vicinity of avoided crossings and conical intersections. The theory for computing nuclear energy gradients for MS-CASPT2 and XMS-CASPT2 is also presented and the first implementation of these gradient methods is reported. A number of illustrative applications of the new methods are presented.     

CASPT2 study of the decomposition of nitrosomethane and its tautomerization reactions in the ground and low-lying excited states

The dissociation reaction of nitrosomethane into methyl and nitric oxide and the tautomerization reactions to formaldehyde oxime, nitrone, and methoxy nitrene have been studied with the second-order multiconfigurational perturbation theory (CASPT2) by the computation of numerical energy gradients. The prevailing reactions in both the ground and the excited states are dissociations. The structures of the ground and excited states are compared with the corresponding complete active space SCF (CAS-SCF) geometries. It is found that changes in the individual bond lengths are rather large (0.01-0.02 A), while the character and energetics of the CASPT2 optimizations remain similar to the CAS-SCF values.     

Ground and excited state CASPT2 geometry optimizations of small organic molecules

A method for computing second-order multiconfigurational perturbation theory (CASPT2) energy gradients numerically has been implemented and applied to a range of elementary organic chromophores, including 1,3 butadiene, acrolein, and two protonated Schiff bases. Geometries of ground and excited states-as well as conical intersections-are compared with the corresponding CASSCF structures, illustrating the effect of including the correction for dynamical electron correlation. It is shown that the differences between the two methods are not readily categorized, but that, while individual changes in bond lengths can be quite large ( approximately 0.01-0.02 A), the natures and CASPT2 energetics of the structures remain similar. Exceptions to this tend to be systems that have a strong ionic character and that are not well described at the CASSCF level. Basis set effects (double- vs. triple-zeta) were examined for a limited number of examples, and found to be quite dramatic at both levels of theory.     

The internal rotational barrier of biphenyl studied with multiconfigurational second-order perturbation theory (CASPT2)

Summary A detailedab initio study of the molecular structure and rotational barriers of biphenyl has been performed. First, non-dynamical correlation effects involving the system are taken into account at the CASSCF level. These wave functions are subsequently employed as reference functions in a multiconfigurational second-order perturbation treatment (CASPT2). The performance single-reference approaches is in addition analysed. The molecular geometries of biphenyl in twisted, coplanar, and perpendicular conformations have been optimized at the CASSCF level. A rotational angle of 44.3° is predicted for the minimum energy conformer in agreement with gas-phase electron diffraction data (44.4±1.2°). The highest level of theory employed yields the values 12.93 (6.0±2.1) and 6.40 (6.5±2.5) kJ/mol for the barrier heights at 0° and 90°, respectively (electron diffraction data within parentheses). In the light of the present findings, the reliability of the available experimental data is discussed.     

Orbital-dependent correlation energy in density-functional theory based on a second-order perturbation approach: success and failure

We have developed a second-order perturbation theory (PT) energy functional within density-functional theory (DFT). Based on PT with the Kohn-Sham (KS) determinant as a reference, this new ab initio exchange-correlation functional includes an exact exchange (EXX) energy in the first order and a correlation energy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlation energy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully self-consistent calculations for any orbital-dependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For two-electron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both second-order Moller-Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital- and eigenvalue-dependent algebraic structure of the correlation functional. The failure emerges in the self-consistent approach but not in perturbative post-EXX calculations, emphasizing the crucial importance of self-consistency in testing new orbital-dependent energy functionals.