Active Orbital Space Research Articles

Nuclear quadrupole moment of Au197 from high-accuracy atomic calculations

The electric field gradient (EFG) at the gold nucleus is calculated using a finite field approach, to make the extraction of the nuclear quadrupole moment Q from experimental nuclear quadrupole coupling constants possible. The four-component Dirac-Coulomb Hamiltonian serves as the framework, 51 of the 79 electrons are correlated by the relativistic Fock-space coupled cluster method with single and double excitations, and the contribution of the Gaunt term, the main part of the Breit interaction, is evaluated. Large basis sets (up to 26s22p18d12f8g5h uncontracted Gaussians) are employed. Energy splittings of the 2D5/2 and 2D3/2 levels, rather than level shifts, are used to extract the EFG constants, as the former remain linear with Q up to 10(-5) a.u., whereas the latter display significant nonlinearity even at Q=10(-8) a.u. Larger Q values lead to larger energy changes and better precision. Excellent agreement (0.1%) is obtained between Q values derived from 2D5/2 and 2D3/2 data. Systematic errors connected with neglecting triple and higher excitations, truncating the basis and orbital active space, and approximating the Gaunt contribution are evaluated. The final value of Q(197Au) is 521(7) mb. It is lower than the muonic 547(16) mb and agrees within error bounds with the recent value of 510(15) mb obtained from molecular calculations.

The Spectroscopy of Copper and Silver Monohalides: What Modern Quantum Chemistry Can and Cannot do

A comprehensive analysis on the successes and failures of the theoretical spectroscopy of copper and silver monohalides is presented in light of the recent theoretical versus experimental information for these systems. It is shown that although for copper monohalides the standard quantum chemical multireference SCF, perturbational or CI approaches work well to describe their spectroscopy, for their silver counterparts this is not the case always. While density functional theory is intrinsically unable to deal with the most intense transition (B1Σ+–X1Σ+) observed in the whole CuX and AgX series, the CASSCF+CASPT2 method fails for the AgX family is describing this excitation, even with the largest physically meaningful valence orbital active space. The complexity of silver halides arises from the concatenation of three independent problems: first, the isolated atom spectroscopy is much more complex for Ag than for Cu, since for silver the Rydberg 2P(4d105p1) state lies close but below the valence 2D(4d95s2) one; secondly, the spin-orbit coupling for silver is much larger and thus the fine-structure components of these doublets are intertwined, leading to very large ΛSΣ mixing effects in the molecular case. Finally, rather strong interactions exist between the numerous Ag(4d95s2)X(np5) and Ag(4d105p1)X(np5) neutral configurations with the Ag+(4d95s1)X−(np6) ionic structures, making the accurate description of these mixtures an extremely difficult task, especially for the second 1Σ+ state, where even large CASSCF calculations fail at providing continuous state-specific potential energy curves. This remains a true theoretical impasse related to the still unsolved and complex convergence for the coupled CI-orbital multivariate minimization problem. Therefore, it is only possible to properly describe the electronic structure of the excited states of silver monohalides using the full valence active space, to perform variational treatments of the non-dynamic and dynamic electronic correlation treatments with especially optimized and extended RECP-basis sets; then the spin-orbit effects must imperatively be taken into account to qualitatively explain the nature of the observed AgX spectra. However, non-negligible errors are found for some of the basic spectroscopic quantities such as transition energies and vibrational frequencies for the most spectroscopically active excited states.

Ab Initio Electron Propagator Calculations on the Ionization Energies of Free Base Porphine, Magnesium Porphyrin, and Zinc Porphyrin

Ab initio electron propagator methods are applied to the prediction, assignment, and interpretation of the valence photoelectron spectra of free base porphine and of magnesium and zinc porphyrins. Tests of various approximate self-energies, including the partial third (P3), the outer valence Green's function, and the nondiagonal, renormalized second-order (NR2) methods are performed. Basis set effects and reduced active orbital spaces are examined as well. The P3 method and the one-electron picture of ionization that accompanies it are validated for the first two cationic states and for states with sigma holes that are localized in nitrogen, lone pair regions. In the remaining pi-hole states, there is significant shake-up character and NR2 results provide useful diagnostics of correlation effects.

Implementation of a general multireference configuration interaction procedure with analytic gradients in a semiempirical context using the graphical unitary group approach.

The graphical unitary group approach has been applied in an efficient implementation of a general multireference configuration interaction (MRCI) method for use with small active molecular orbital spaces in a semiempirical framework. Gradients can be computed analytically for molecular orbitals from a closed-shell or a half-electron open-shell Hartree-Fock calculation. CPU times for single point energy and gradient calculations are reported. The code allows MRCI geometry optimizations of large molecules, as illustrated for the singlet ground state and the four lowest triplet states of fullerene C(76).

N‐band Hubbard models II: Cooperative mechanisms of electron–phonon, electron correlation, and many‐band effects toward high‐Tc superconductors

AbstractDiscoveries of superconductivity of MgB2 by Akimitsu et al. and high‐Tc cupurate superconductors by Bednorz and Muller have raised great interest for elucidation of the superconducting mechanism from both experimental and theoretical grounds. The transition temperature (Tc) was found to be 40 K for MgB2, while the very high‐Tc over 130 K was reported for doped copper oxides with layer structures. A crucial role of the electron–phonon (EP) interaction was pointed out as a common mechanism of the superconductivity of both materials. However, such high‐Tc superconductivity may indicate the possibility of cooperative mechanism of the EP interaction with others such as electron correlation (EC) and multiband (MB) effects discussed in part I (Int J Quantum Chem 1990, 37, 167) of this series. Here, as a continuation of part I, theoretical backgrounds are briefly described to select active orbital space for superconductivity and elucidate the nature of EP, EC, and MB effects for cooperative mechanisms. Next, molecular orbital calculations of MgmBn and cage‐type carbon cluster are carried out to elucidate contributions of the EP interaction using McMillan equation. The relative contributions of the EP and EC interactions are also discussed in relation to the screening of the Coulomb repulsion. Pair binding energies are calculated for cage‐type carbon clusters. Finally, the cooperative mechanisms of the EP, EC, and MB effects are discussed to realize the high‐Tc superconductivity in molecule‐based materials such as cage compounds and nanotubes. Several possible candidates are proposed on both experimental and theoretical grounds. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003

N-electron valence state perturbation theory: A spinless formulation and an efficient implementation of the strongly contracted and of the partially contracted variants

The n-electron valence state perturbation theory is reformulated in a spin-free formalism, concentrating on the “strongly contracted” and “partially contracted” variants. The new formulation is based on the introduction of average values in the unperturbed state of excitation operators which bear resemblance with analogous ones occurring in the extended Koopmans’ theorem and in the equations-of-motion technique. Such auxiliary quantities, which allow the second-order perturbation contribution to the energy to be evaluated very efficiently, can be calculated at the outset provided the unperturbed four-particle spinless density matrix in the active orbital space is available. A noticeable inequality concerning second-order energy contributions of the same type between the strongly and partially contracted versions is proven to hold. An example concerning the successful calculation of the potential energy curve for the Cr2 molecule is discussed.

Dipole-allowed excited states of N2: Potential energy curves, vibrational analysis, and absorption intensities

The three lowest adiabatic potential energy curves for each of the two dipole-allowed symmetries, Σu+1 and Πu1, are calculated in the multireference configuration–interaction framework. Diabatic potentials and corresponding coupling elements are obtained by diagonalizing the electronic operator r2 which serves to discriminate Rydberg and valence type states. A large basis set and judiciously chosen active orbital and configuration spaces furnish smooth and reliable potential curves. However, a vibrational analysis of the coupled systems in diabatic representation still shows some disappointing deviations from the experimental interference patterns of overlapping absorption bands that are highly sensitive to potential energy differences. Starting from the calculated curves, a fitting procedure accounting also for empirical information yields potential energy curves and diabatic coupling elements that reproduce all details of the experiment very well. These recommended results also serve to identify residual defects in the ab initio curves mainly as vertical shifts. The performance of other commonly used ab initio methods for the calculation of excited states is briefly discussed.

Effect of correlating core orbitals in calculations of nuclear spin–spin couplings

Electron correlation effects on nuclear spin–spin coupling tensors arising from core-valence interaction were investigated with ab initio calculations based on multiconfigurational self-consistent field (MCSCF) reference states, an adequate basis set, and large restricted active spaces. Calculations of first and second-row hydrides were performed to examine the effect of including the core and semicore orbitals in the active molecular orbital space. The effects of using a multireference wave function instead of a single-reference one, as well as different numbers of electrons in the virtual orbitals, were considered. In second-row hydrides, the inclusion of semicore orbitals is found to be necessary for accurate calculations of spin–spin couplings involving the heavy nucleus. A smaller but nonnegligible effect is observed also for the hydrogen–hydrogen coupling. Comparison is made with nuclear shielding and electric field gradient, for which the effect is considerably smaller. Recommendations for wave functions for calculations of spin–spin couplings are given.

Electronic excitation and ionization spectra of cyclopentadiene: Revisit by the symmetry-adapted cluster–configuration interaction method

Electronic excitation and ionization spectra of cyclopentadiene (CP) were reinvestigated by the symmetry-adapted cluster (SAC) and SAC–configuration interaction (SAC-CI) method with an extended basis set and a wide active orbital space. To give a satisfactory interpretation of the general profile of the observed excitation spectrum, 40 low-lying excited singlet and triplet states (with excitation energies of up to 9.5 eV) were computed. The calculated excitation energies were greatly improved compared to those reported previously. All of the peaks in the experimental spectrum were reassigned theoretically with small deviations. The natures of the low-lying valence and Rydberg-excited states were discussed in detail, and the results were also compared with those of some other recent theoretical studies. The ionization energies calculated by the SAC-CI general-R method agree well with the experimental peaks in the photoelectron spectrum. A number of two-electron shake-up states were calculated below 23 eV.

Density functional theory with fractionally occupied frontier orbitals and the instabilities of the Kohn–Sham solutions for defining diradical transition states: Ring-opening reactions

Density functional theory with fractionally occupied frontier orbitals was combined with an analysis of the instability of the Kohn–Sham solutions and applied to the study of the ring-opening reactions of 1,2-dioxetene, 1,2-dithiete, 1,2-diselenete, and 1,2-ditellurete. These molecules possess transition structures and reactive intermediates for the ring-opening reactions which exhibit strong nondynamical electron correlation. All restricted density functional theory (DFT) solutions for the closed shell transition state structures for all these species are unstable. The solutions with the hybrid Hartree–Fock DFT functionals, B3LYP and B3PW91, are triplet unstable, while for the pure DFT functional BLYP the instability is due to a violation of the Aufbau principle. The same types of instabilities were found for the 1,2-diselenete and 1,2-ditellurete intermediates. Lower energy stable solutions for the diradical transition structures were found with unrestricted DFT methods allowing fractionally occupied orbitals. This DFT approach indicates a decrease in the active orbital space from four fractionally occupied natural orbitals in earlier multireference predictions to two fractionally occupied Kohn–Sham orbitals.

Exploring the MRCI method for calculating interaction energies: application to the HeNe potential curve

A multi-reference configuration interaction (MRCI) method is described, which is devised for the calculation of interaction energies of van der Waals complexes and applied to calculating the HeNe potential energy curve. The MRCI calculations make use of a generalized Pople-correction in order to account for the lack of size consistency. The orbital space is partitioned into three subspaces: the first active space (AS1), which contains the strongly occupied orbitals; the second active space (AS2), which contains the main intra-correlating orbitals; and the external space (ES). It is shown that, to keep the error below ± 0.2 K in the excitation scheme and the active orbital space it is sufficient to include only σ-orbitals in AS2 and to use an excitation scheme (labelled Qq-MRCI) that encompasses only up to quadruply excited configurations. The final active orbital space (AS2) turned out to be 2s(He), 2pσ(He), 3s(Ne), 3σ(Ne) and 3dσ(Ne). Other MRCI variants, in which most or all quadruply excited configurations were deleted from the CI expansion (Qt- and Tt-MRCI), were found to be inadequate. Using the Qq-MRCI scheme together with a 197-orbital ‘interaction optimized’ basis set (IO197), the MRCI interaction energy at R = 5.7 a0 was calculated to be -21.12K. The corresponding values at the MP4 and CCSD(T) levels of theory are -20.06 K and -20.99 K, respectively, indicating that the MP4 method is inappropriate for highly accurate calculations on this system. Fitting the calculated data using a generalized Morse function, including an additional C6/R6 term to account for a correct long-range behaviour of the potential, the MRCI well depth was calculated to be -21.16K at Req = 5.73a 0. The MRCI and CCSD(T) potentials have the same quality and are found to be in good agreement with the Hartree-Fock dispersion (HFD-B) potential of Keil, M., Danielson, L. J., and Dunlop, P. J., 1991, J. Chem. Phys., 94, 296. It is concluded that, for basis IO197, the CCSD(T) method is sufficiently accurate for calculating the HeNe interaction. To recover the small, missing contributions (a few tenths of a Kelvin), MRCI should be used.

Is GVB-CI superior to CASSCF?

Presently the most reliable approach for the study of reaction pathways where chemical bonds are broken and formed is to carry out CASSCF calculations followed by corresponding multireference perturbation or CI treatments. The latter step generally relaxes the “antibonding character” of the CASSCF results. In this study we demonstrate that similar results can be well approximated by using a less optimized MCSCF method and not performing the multireference perturbation or CI step at all. This is accomplished by performing a complete CI calculation within the active orbital space of the generalized valence bond perfect pairing (GVB-PP) model. The local bond/antibond character of the orbital space of the GVB-PP method also allows development of a fast, but robust, Bethe–Goldstone algorithm, which reconstructs the CI energy to an accuracy of a few tenths of a millihartree for most types of bond breaking cases found in chemical reactions. This algorithm executes at a speed proportional to N where Np is the number of localized electron pairs in the active space. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 999–1008, 1999

The unrestricted natural orbital-restricted active space method: methodology and implementation

The full configuration interaction method in the space of fractionally occupied unrestricted natural orbitals (UNO-CAS method) is extended to excited states as well as to strongly correlated and reactive systems with large active spaces. This is accomplished by␣using restricted active space (RAS) wave functions introduced by Olsen et al. [(1988) J Chem Phys 89: 2185] and using the UNOs without the expensive orbital optimization step. In RAS, the space of active orbitals is subdivided into three groups: a group with essentially doubly occupied orbitals (RAS1), the usual CAS space (RAS2), and a space with weakly occupied active orbitals (RAS3). We select these spaces on the basis of the occupation numbers of the UNOs. All possible electron distributions are allowed in the usual CAS space, but the number of vacancies is limited in RAS1 and the number of electrons is limited in RAS3. We discuss an efficient algorithm for generating a RAS wave function. This is based on the Handy-Knowles determinantal expansion with an addressing scheme adopted for the restricted expansion. Results for both ground and excited states of azulene and free base porphyrin are presented.

Group theoretical identification of active localized orbital space in high Tc cuprate superconductors

There is a widely shared consensus that the pair condensate wavefunction in cuprate superconductors has a dominant d x 2− y 2 symmetry. In this paper a group theoretical analysis of the pair condensate wavefunction in real space for cuprate superconductors is presented. The analysis indicates that there is a degenerate pair of active bands in cuprate superconductors which are principally derived from localized e-orbitals and two possible singlet pair states formed from two different orbitals with symmetries a 1, b 1 or a 2, b 2 and all three may contribute to the condensate wavefunction. Hund's rule and the experimental observation that singlet pairing with a short coherence length occurs in cuprate superconductors, indicates that the first option is the most likely candidate in these materials and that hole doping mainly occurs from 1 B 1 pairs of e-orbitals giving rise to singlet electron pairing and the phenomenon of high T c cuprate superconductivity.

On the choice of active space orbitals in MCSCF calculations

We describe a procedure which may be used to aid selection of the active space in multiconfigurational self-consistent field (MCSCF) calculations for general chemical systems. Starting from a restricted Hartree–Fock calculation, we define a hierarchy of interacting virtual orbitals for every occupied orbital. The most strongly interacting orbitals are then taken to constitute the active space in a configuration interaction (CI) calculation. The natural orbital occupation numbers obtained from the CI calculation are then used to choose the active space to be used in a subsequent MCSCF calculation. We illustrate our method on a number of systems (Li 2, B 2, C 2, carbonyl oxide, and the transition state for oxidation of H 2S by dioxirane). In all these cases, `intuitive' active spaces are inadequate, as are active spaces derived from the natural orbitals of unrestricted Hartree–Fock calculations.

Molecular geometry and vibrational frequencies of ozone from compact variational wave functions explicitly including triple and quadruple substitutions

The performance of a multireference CISD method, CISD[TQ], is compared to that of other approaches which include a large degree of electron correlation, including Brueckner methods. The CISD[TQ] method selects as references all single and double substitutions within an active orbital space. Certain triple and quadruple substitutions from the Hartree–Fock reference are included in the CISD[TQ] wave function as singles and doubles from the selected reference set. This wave function has previously been shown in simpler cases to provide results near to those predicted by the configuration interaction wave function, including all single, double, triple, and quadruple substitutions (CISDTQ). For the challenging multireference case of ozone, the CISD[TQ] wave function yields geometries and harmonic vibrational frequencies with an accuracy similar to the full CCSDT method. These promising results suggest that for difficult multireference problems the CISD[TQ] wave function provides an efficient and accurate approach for approximating the complete CISDTQ.

MCSCF calculation of the frequency-dependent hyperpolarizability of the lithium atom

mThe second electric dipole hyperpolarizability of the lithium atom is calculated using a series of multiconfiguration SCF wave functions with an increasing number of active space orbitals. In agreement with other recent calculations, we find a very large correlation correction to the hyperpolarizability. We analyze the frequency dependence of the dc Kerr hyperpolarizabily y K( w) = y( - w; w, 0,O) and observe very significant dispersion effects. 0 1996 John Wiley & Sons, Inc.

Electric properties of the water molecule in 1A1, 1B1, and 3B1 electronic states: Refined CASSCF and CASPT2 calculations

The dipole moments and dipole polarizabilities of the 1A1, 1B1, and 3B1 electronic states of the water molecule have been calculated by using the CASSCF approach followed by the evaluation of the dynamic electron correlation contribution by the second-order perturbation scheme CASPT2. All calculations have been carried out in a specifically extended ANO basis set which accounts for the Rydberg character of the two excited states. In order to estimate the correctness and accuracy of the present data a scan over a variety of different active spaces for the CASSCF wave function has been made. The present results are superior to earlier CASSCF calculations, although their qualitative features remain essentially the same. The dipole moments in 1B1 and 3B1 states are predicted to be about 0.49 a.u. and 0.33 a.u., respectively, and have the opposite orientation with respect to the ground state dipole moment. The dipole polarizability tensors of the excited states are characterized by high anisotropy and are dominated by the in-plane component perpendicular to the symmetry axis. All their components are found to be about an order of magnitude larger than those of the ground state polarizability tensor. The excitation energy dependence on the choice of the active orbital space in the CASSCF reference function is also considered and the analysis of the present data concludes in the concept of what is called the mutually compatible active spaces for the two states involved in excitation. All CASPT2 results are in good agreement with the results of recent calculations carried out in the framework of the open-shell coupled cluster formalism. This agreement confirms the high efficiency of the CASSCF/CASPT2 approach to the treatment of the electron correlation effects.

Generalized valence bond solutions from a constrained coupled cluster method

Abstract The GVB-PP wavefunction is cast into a coupled cluster form with the coupled cluster operator constrained to intrabond double excitations. Following the coupled cluster ansatz, where the trial wave function is assumed to satisfy the Schrodinger equation, projections onto the reference and doubly excited configurations are used to determine the energy and coefficients respectively. A decoupling of these equations results, allowing analytical solutions. The active orbital space is simultaneously optimized to produce the lowest energy. This is carried out efficiently using a procedure previously developed by Head-Gordon and Pople for the direct optimization of a Hartree-Fock wave function. Preliminary calculations for singlet/triplet states and bond dissociations show that although this method is strictly nonvariational, local minima are found which lie within a few tenths of a millihartree of the true GVB-PP energy.

Correlated calculations of indirect nuclear spin-spin coupling constants for XH4 (X = Si, Ge, and Sn)

We report multiconfigurational linear response calculations of the one-bond and geminal indirect nuclear spin-spin couplings constants in SiH4, GeH4, and SnH4. Results of a basis set study on SnH4 using 13 different basis sets on Sn are given, showing the effects on the individual contributions to the coupling constant. In addition to the more conventional choice of active orbital space (one correlating orbital for each strongly occupied valence orbital), calculations employing extended active spaces were carried out. The results are compared with available experimental data showing that correlation effects are very important. For the one-bond couplings good agreement with experiment, within 1–5%, is obtained, whereas the geminai proton-proton couplings show larger deviations. The calculations reported here are, to our knowledge, the first that take into consideration correlation effects in all four contributions to the coupling constants for molecules as large as GeH4 and SnH4. All X-H couplings are dominated by the Fermi contact term while the picture is more diverse for the geminal couplings.