Active Space Self-consistent Field Reference Research Articles

Importance of dynamical electron correlation in diabatic couplings of electron-exchange processes.

We demonstrate the importance of the dynamical electron correlation effect in diabatic couplings of electron-exchange processes in molecular aggregates. To perform a multireference perturbation theory with large active space of molecular aggregates, an efficient low-rank approximation is applied to the complete active space self-consistent field reference functions. It is known that kinetic rates of electron-exchange processes, such as singlet fission, triplet-triplet annihilation, and triplet exciton transfer, are not sufficiently explained by the direct term of the diabatic couplings but efficiently mediated by the low-lying charge transfer states if the two molecules are in close proximity. It is presented in this paper, however, that regardless of the distance of the molecules, the direct term is considerably underestimated by up to three orders of magnitude without the dynamical electron correlation, i.e., the diabatic states expressed in the active space are not adequate to quantitatively reproduce the electron-exchange processes.

Correlating Electronic Structure and Magnetic Anisotropy in Actinide Complexes [An(COT)2], AnIII/IV = U, Np, and Pu.

The electronic structures and magnetic anisotropies for compounds [An(COT)2] (An = UIII/UIV, NpIII/NpIV and PuIII/PuIV, COT = cyclooctatetraene) are characterized using scalar relativistic density functional theory calculations and second-order perturbation theory based on a complete active space self-consistent field reference including spin-orbit coupling. The degree of participation of 5f orbitals in actinide-ligand bonding and the associated metal-ligand covalency is found to trend as U > Np ≥ Pu for both the tetra-positive and tripositive An complexes. A spin-Hamiltonian analysis indicates only weak single-molecule magnet (SMM) characteristics for [U(COT)2]- and [Np(COT)2] complexes and no significant SMM behavior for the other complexes. The weak SMM behavior in [U(COT)2]- and [Np(COT)2] is attributed to a subtle interplay between local symmetry and ligand-field splitting. Such a result suggests that magnetic anisotropy in 5f3 ions can be modulated in general by electrostatic ligand field design. In particular, σ-donor ligands oriented 180 degrees relative to one another will have a maximal influence on the 5f-orbital ligand field splitting, while π donors like cyclopentadiene and COT generate ligand field influences that have more acute angles associated with corresponding atoms on the individual ligands. These observations rationalize the differences in SMM characteristics for [U(BcMe)3] (BcMe- = dihydrobis(methylimidazolyl)borate) and [U(BpMe)3] (BpMe- = dihydrobis(methylpyrazolyl)borate) and indicate strategies to design new actinide-based SMMs with high magnetic relaxation barriers.

Analytic Gradients for Complete Active Space Pair-Density Functional Theory

Analytic gradient routines are a desirable feature for quantum mechanical methods, allowing for efficient determination of equilibrium and transition state structures and several other molecular properties. In this work, we present analytical gradients for multiconfiguration pair-density functional theory (MC-PDFT) when used with a state-specific complete active space self-consistent field reference wave function. Our approach constructs a Lagrangian that is variational in all wave function parameters. We find that MC-PDFT locates equilibrium geometries for several small- to medium-sized organic molecules that are similar to those located by complete active space second-order perturbation theory but that are obtained with decreased computational cost.

Assessing the performance of ab initio classical valence bond methods for hydrogen transfer reactions

Ab initio classical valence bond theory in terms of localized orbitals has several advantages over electronic structure methods based on canonical delocalized Hartree-Fock molecular orbitals, with two key distinctions being greater applicability to inherently multiconfigurational systems (also called multireference systems or strongly correlated systems) and great interpretability in terms of covalent and ionic valence bond configurations (also called valence bond structures). This can be especially advantageous for applications to chemical reactions. However, until now only limited work tested the quantitative accuracy for the energetics of chemical reactions. The present study provides such tests and validations for a representative test set of six barrier heights corresponding to forward and reverse barriers of three hydrogen transfer reactions. In particular, we test the valence bond self-consistent-field theory (VBSCF) and three post-VBSCF methods that use VBSCF as a reference function for adding dynamic correlation, in particular valence bond configuration interaction (VBCI), breathing orbital valence bond (BOVB), and valence bond second-order perturbation theory (VBPT2). The VBSCF method itself is, as expected, not quantitatively accurate, with a mean unsigned error (MUE) for the six barrier heights of ∼17kcal/mol. But the post-VBSCF methods are found to be quite successful. Depending on the basis set, and valence bond structure selection we obtain MUEs (in kcal/mol) as low as 3.7 for VBCI, 4.5 for BOVB, and (using a bigger basis-set) 1.3 for VBPT2. These compare well, on the same data set, with 1.6 for coupled clusters with singles and doubles (CCSD) and 1.5 for multireference second order perturbation theory based on a complete active space self-consistent field reference function (MRPT2). We discuss the results in terms of the pros and cons of each of these methods.

Two-component multi-configurational second-order perturbation theory with Kramers restricted complete active space self-consistent field reference function and spin-orbit relativistic effective core potential.

We report the formulation and implementation of KRCASPT2, a two-component multi-configurational second-order perturbation theory based on Kramers restricted complete active space self-consistent field (KRCASSCF) reference function, in the framework of the spin-orbit relativistic effective core potential. The zeroth-order Hamiltonian is defined as the sum of nondiagonal one-electron operators with generalized two-component Fock matrix elements as scalar factors. The Kramers symmetry within the zeroth-order Hamiltonian is maintained via the use of a state-averaged density, allowing a consistent treatment of degenerate states. The explicit expressions are derived for the matrix elements of the zeroth-order Hamiltonian as well as for the perturbation vector. The use of a fully variational reference function and nondiagonal operators in relativistic multi-configurational perturbation theory is reported for the first time. A series of initial calculations are performed on the ionization potential and excitation energies of the atoms of the 6p-block; the results display a significant improvement over those from KRCASSCF, showing a closer agreement with experimental results. Accurate atomic properties of the superheavy elements of the 7p-block are also presented, and the electronic structures of the low-lying excited states are compared with those of their lighter homologues.

Adiabatic potential energy surfaces for the low-energy collisional dynamics of C(+)((2)P) ions with H2 molecules.

The low-energy electronic states of the CH2(+) molecular ion are investigated with multireference configuration interaction calculations based on complete active space self-consistent field reference wave functions using a large C(6s5p4d3f)/H(8s6p3d1f) basis set. The focus is on the three lowest-lying states describing formation and destruction of the astrophysically relevant methylidine cation CH(+). Both processes are discussed in light of the topology of the relevant potential energy surfaces and their intersections.

Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Studies on Mononuclear First-Row Transition-Metal Systems.

A series of model transition-metal complexes, CrF6, ferrocene, Cr(CO)6, ferrous porphin, cobalt corrole, and FeO/FeO(-), have been studied using second-order perturbation theory based on a restricted active space self-consistent field reference wave function (RASPT2). Several important properties (structures, relative energies of different structural minima, binding energies, spin state energetics, and electronic excitation energies) were investigated. A systematic investigation was performed on the effect of: (a) the size and composition of the global RAS space, (b) different (RAS1/RAS2/RAS3) subpartitions of the global RAS space, and (c) different excitation levels (out of RAS1/into RAS3) within the RAS space. Calculations with active spaces, including up to 35 orbitals, are presented. The results obtained with smaller active spaces (up to 16 orbitals) were compared to previous and current results obtained with a complete active space self-consistent field reference wave function (CASPT2). Higly accurate RASPT2 results were obtained for the heterolytic binding energy of ferrocene and for the electronic spectrum of Cr(CO)6, with errors within chemical accuracy. For ferrous porphyrin the intermediate spin (3)A2g ground state is (for the first time with a wave function-based method) correctly predicted, while its high magnetic moment (4.4 μB) is attributed to spin-orbit coupling with very close-lying (5)A1g and (3)Eg states. The toughest case met in this work is cobalt corrole, for which we studied the relative energy of several low-lying Co(II)-corrole π radical states with respect to the Co(III) ground state. Very large RAS spaces (25-33 orbitals) are required for this system, making compromises on the size of RAS2 and/or the excitation level unavoidable, thus increasing the uncertainty of the RASPT2 results by 0.1-0.2 eV. Still, also for this system, the RASPT2 method is shown to provide distinct improvements over CASPT2, by overcoming the strict limitations in the size of the active space inherent to the latter method.

Quasi-degenerate second-order perturbation theory for occupation restricted multiple active space self-consistent field reference functions

A multi-configuration quasi-degenerate second-order perturbation method based on the occupation restricted multiple active space (ORMAS-PT/ORMAS) reference wavefunction is presented. ORMAS gives one the ability to approximate a complete active space self-consistent field (CASSCF) wavefunction using only a subset of the configurations from the CASSCF space. The essential idea behind ORMAS-PT is to use the multi-reference Møller-Plesset formalism to correct the ORMAS reference energy. A computational scheme employing direct CI methodology is presented. Several tests are presented to demonstrate the performance of the ORMAS-PT method.

An ab initio multireference perturbation theory study on the manganese dimer

The potential energy curves of the ground state and of some excited states of the manganese dimer have been calculated over a wide range of internuclear distances using the second order n-electron valence state perturbation theory applied to a complete active space self-consistent field reference wave function. The ground state of Mn(2), for which also the third order NEVPT has been used, is calculated to be a singlet belonging to the Sigma(g) (+) symmetry, characterized by a large equilibrium internuclear distance R(e) of 3.7-3.8 A, by a low dissociation energy D(e) of 0.07-0.08 eV, and by a small harmonic frequency omega(e) of 43 cm(-1). The experimental evidence that Mn(2) is a van der Waals molecule is thus confirmed. Among the excited states, (11)Pi(u), which is usually indicated as the ground state by density functional theory studies, appears as a low-lying state with R(e)=2.50 A, D(e)=1.35 eV, and omega(e)=246 cm(-1).

Photophysics of intramolecularly hydrogen-bonded aromatic systems: ab initio exploration of the excited-state deactivation mechanisms of salicylic acid

Excited state reaction paths and the corresponding energy profiles of salicylic acid have been determined with the CC2 method, which is a simplified version of singles-and-doubles coupled cluster theory. At crucial points of the potential energy hypersurfaces, single-point energy calculations have been performed with the CASPT2 method (second-order perturbation theory based on the complete active space self-consistent field reference). Hydrogen transfer along the intramolecular hydrogen bond as well as torsion and pyramidization of the carboxy group have been identified as the most relevant photochemical reaction coordinates. The keto-type planar S(1) state reached by barrierless intramolecular hydrogen transfer represents a local minimum of the S(1) energy surface, which is separated by a very low barrier from a reaction path leading to a low-lying S(1)-S(0) conical intersection via torsion and pyramidization of the carboxy group. The S(1)-S(0) conical intersection, which occurs for perpendicular geometry of the carboxy group, is a pure biradical. From the conical intersection, a barrierless reaction path steers the system back to the two known minima of the S(0) potential energy surface (rotamer I, rotamer II). A novel structure, 7-oxa-bicyclo[4.2.0]octa-1(6),2,4-triene-8,8-diol, has been identified as a possible transient intermediate in the photophysics of salicylic acid.

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.

Second-order quasi-degenerate perturbation theory with quasi-complete active space self-consistent field reference functions

A quasi-degenerate perturbation theory (QDPT) is presented that is based on quasi-complete active space self-consistent field (QCAS-SCF) reference functions. The perturbation method shown here is an extension of a previously proposed QDPT with CAS-SCF reference functions (CAS-QDPT) but is a more compact perturbation method that can employ a much smaller reference configuration space with the same number of active electrons and orbitals as the CAS case. A computational scheme to second-order using a diagrammatic approach is described. The second-order effective Hamiltonian consists of the contribution from external excitations, which involve core or/and virtual orbitals, and internal excitations, which involve only active orbitals. The importance of the internal excitation contribution is emphasized. The method is tested on the potential energy curves of the LiF molecule, the Rydberg excitation energies of furan, and the transition state barrier height of the reaction, H2CO→H2+CO. The results are in very good agreement with the corresponding CAS-SCF reference QDPT results and available experimental data. The deviations from the CAS-QDPT values in the energy are less than 0.1 eV on the average for the excitation energies of furan and less than 1 kcal for the barrier height of the reaction, H2CO→H2+CO. The deviation from the experimental values is 0.11 eV at most for the excitation energies, and 1.2 kcal/mol, which is within the twice the experimental uncertainty, for the barrier height.

Theoretical study of the valence π → π * excited states of polyacenes: anthracene and naphthacene

The valence π → π* excited states of anthracene and naphthacene are studied with multireference perturbation theory with complete active space self-consistent field reference functions. The predicted spectra provide a consistent assignment of all one- and two-photon spectra and T-T spectra of low-lying valence π → π* excited states of anthracene and naphthacene. The present theory predicts the valence π → π* excitation energies with an accuracy of 0.15 eV for anthracene and of 0.25 eV or better for naphthacene. The excited states of anthracene and naphthacene are compared with those of benzene and naphthalene studied previously. The present calculations predict that, going from anthracene to naphthacene, there is a symmetry reversal of the two lowest singlet state transitions, but not for the triplet, just as indicated by the experimental data. Some general trends of polyacene excited states are discussed based on the calculated results for benzene to naphthacene. Conclusive results obtained for anthracene and naphthacene can be used as a model for understanding the excited states of larger polyacenes.

Different forms of the zeroth-order Hamiltonian in second-order perturbation theory with a complete active space self-consistent field reference function

A new one-particle zeroth-order Hamiltonian is proposed for perturbation theory with a complete active space self-consistent field (CASSCF) reference function. With the new partitioning of the Hamiltonian, reference functions dominated by a closed-shell configuration, on one hand, and an open-shell configuration, on the other hand, are treated in similar and balanced ways. This leads to a better description of excitation energies and dissociation energies. The new zeroth-order Hamiltonian has been tested on CH2, SiH2, NH2, CH3, N2, NO, and O2, for which full configuration interaction (FCI) results are available. Further, excitation energies and dissociation energies for the N2 molecule have been compared to corresponding multireference (MR) CI results. Finally, the dissociation energies for a large number of benchmark molecules containing first-row atoms (the “G1” test) have been compared to experimental data. The computed excitation energies compare very well with the corresponding FCI and MRCI values. In most cases the errors are well below 1 kcal/mol. The dissociation energies, on the other hand, are in general improved in the new treatment but have a tendency to be overestimated when compared to other more accurate methods.

Different forms of the zeroth-order Hamiltonian in second-order perturbation theory with a complete active space self-consistent field reference function

Different forms of the zeroth-order Hamiltonian in second-order perturbation theory with a complete active space self-consistent field reference function

Second-order perturbation theory with a complete active space self-consistent field reference function

The recently implemented second-order perturbation theory based on a complete active space self-consistent field reference function has been extended by allowing the Fock-type one-electron operator, which defines the zeroth-order Hamiltonian to have nonzero elements also in nondiagonal matrix blocks. The computer implementation is now less straightforward and more computer time will be needed in obtaining the second-order energy. The method is illustrated in a series of calculations on N2, NO, O2, CH3, CH2, and F−.

A b i n i t i o calculation of the X 2Σ+ and A 2Π states of CF++

The potential energy, dipole moment, and electronic transition moment curves of the X 2 Σ+ and A 2 Π states of the doubly charged CF++ ion have been calculated from internally contracted multireference CI wave functions with full valence complete active space self-consistent field reference wave functions and large Gaussian basis sets. Both states are predicted to be metastable with equilibrium geometries Re(X 2 Σ+ )=2.139 Bohr and Re(A 2 Π)=3.073 Bohr. These minima lie above and are separated from their asymptotes by broad barriers with a height of 4.99 eV (X 2 Σ+ ) and 1.04 eV (A 2 Π), respectively. Similar to the isoelectronic NO++, the Π state intersects the barrier of the X 2 Σ+ state. Radiative and nonradiative (tunneling) lifetimes of all vibrational levels have been calculated by standard quantum mechanical scattering techniques.