The generalized active space concept in multiconfigurational self-consistent field methods

A multiconfigurational self-consistent field method based on the concept of generalized active space (GAS) is presented. GAS wave functions are obtained by defining an arbitrary number of active spaces with arbitrary occupation constraints. By a suitable choice of the GAS spaces, numerous ineffective configurations present in a large complete active space (CAS) can be removed, while keeping the important ones in the CI space. As a consequence, the GAS self-consistent field approach retains the accuracy of the CAS self-consistent field (CASSCF) ansatz and, at the same time, can deal with larger active spaces, which would be unaffordable at the CASSCF level. Test calculations on the Gd atom, Gd(2) molecule, and oxoMn(salen) complex are presented. They show that GAS wave functions achieve the same accuracy as CAS wave functions on systems that would be prohibitive at the CAS level.     

A critical study of basis set effects and the use of approximate natural orbitals in SCF-CI calculations of molecular geometries and heats of reaction

SCF-CI calculations have been performed on a number of chemical reactions between closed shell molecules in order to determine the heats of reaction. Contracted Gaussian type atomic basis sets of three different qualities were used and the CI calculations were performed in a truncated approximate natural orbital space. The conclusions to be drawn from these calculations are rather pessimistic. For heats of reaction, errors up to 6 kcal/mole are obtained on the SCF-level with a double zeta plus polarization atomic basis. A further improvement is only possible if extended basis sets are used. Correlation effects on heats of reaction are of the same size and CI calculations are therefore only meaningful with large atomic basis sets.For the CI calculations a one-electron space of approximate natural orbitals, obtained from second order RS perturbation theory, was used. Different truncations, using the occupation number as criterion, were tested. The general conclusion is that errors in energy differences obtained with a truncated basis set are of the same magnitude as the error in the total correlation energy. In practice this means that not more than 20–30% of the approximate natural orbitals can be deleted if the error is to be kept less than a few kcal/mole.Finally the truncation error in calculations of bond distances was tested for a few cases. Errors of around 10% of the total change due to correlation were found when 30% of the lowest occupied natural orbitals were deleted.     

Role of doubly excited states in calculations of optical activity

It is shown that doubly excited states play an important role in calculations of the optical activity of molecules with well-conjugated electron systems, such as the DNA bases. In some significant cases it is necessary to include a large number of excited states in the configuration interaction (CI ) to obtain a reliable, converging result. A new version of the CNDO/OPTIC method, which includes doubly excited states in the CI , is proposed. As an application, the electric transition moments in different pyrimidines are considered. The calculated results agree with experimental data and results obtained from ab initio calculations and INDO calculations using doubly excited states in the CI .     

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 (Li₂, B₂, C₂, carbonyl oxide and the transition state for oxidation of H₂S 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.     

Configuration selection as a route towards efficient vibrational configuration interaction calculations

A configuration selective vibrational configuration interaction (CI) approach is presented that efficiently reduces the variational space and thus leads to significant speedups in comparison to standard vibrational CI implementations. Deviations with respect to reference calculations are well below the accuracy of the underlying electronic structure calculations for the potential and hence are essentially negligible. Parallel implementations of the presented configuration selective vibrational CI approaches lead to further significant time savings. Benchmark calculations based on potential energy surfaces of coupled-cluster quality are presented for the fundamental modes of cis- and trans-difluoroethylene. The size-consistency error within the vibrational configuration interaction calculations of the difluoroethylene dimer has been studied in dependence on the excitation level.     

Ab-initio calculations on large molecules and solids by desirable computational procedures

Desirable computational procedures developed here recently for ab-initio calculations on large molecules are outlined. Effective core model potentials (MODPOT) permit calculations of valence electrons only explicitly, yet accurately; a charge-conserving integral prescreening evaluation to decide whether a block of integrals will be larger than a preset threshold and thus be calculated explicitly is effective for spatially extended systems; an efficient MERGE technique to save and reuse common invariant skeletal integrals is useful for geometry variations and for adding basis fcuntions, substituent groups and molecules; and an effective configuration interaction (CI) Hamiltonian into which are folded the effects of the occupied molecular orbitals from which no excitations are allowed is useful for molecular decompositions and intermolecular reactions. These techniques have been extended for CI calculations on breaking a chemical bond in a molecule in a crystal or solid; atom-class/atomic-class potential functions and dispersion calculations have been added. In a new program, POLY-CRYST, all the integral strategies for large molecules are meshed.     

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.     

A method for the calculation of transition moments between electronic states of molecules using a different one-electron basis set for each state

A state-specific approach to the calculation of transition moments between molecular electronic states requires that the wavefunction for each state is expanded in its optimum one-electron basis and that nonorthonormal basis techniques are used for the calculation of the transition moment integrals. A method has been developed for carrying out such nonorthonormal basis calculations, based on the corresponding orbitals transformation and appropriately defined density matrices, which may be used with configuration interaction (CI ) wavefunctions. Further improvements of the method have resulted in a decrease in the time required for the calculations and thus allow its application with moderately large CI expansions for each state. Nonorthonormal basis calculations on transition moments in H₂O have been carried out using the above method. The results are in agreement with those of large MRD -CI calculations.     

Application of the MC SCF method to the π → π* excitation energies of ethylene

The MC SCF method is employed to calculate the N → T and N → V π → π^* vertical excitation energies of ethylene. To obtain accurate excitation energies it is found to be necessary to utilize an expanded valence space containing two π and two π^* orbitals. Relatively small MC SCF calculations, allowing at most one-electron excitations from the sigma space, are found to yield excitation energies and spatial extents of the excited states in excellent agreement with the predictions of large multi-reference or iterative-natural-orbital CI calculations. These results show that within an MC SCF framework σ-σ correlation is unimportant for describing the π → π^* processes. We also conclude that the neglect of the effects of unlinked cluster terms in some of the CI calculations may have introduced small, but important, errors in the excitation energies and predictions of the spatial extent of the V state.     

Energy continuity in multi-reference CI calculations

Configuration interaction calculations based on changing numbers of reference configurations at different geometries have a theoretical inconsistency which can affect the continuity of a calculated potential surface. As the number of reference configurations is increased for adjacent points on a potential curve (e.g. to describe molecular dissociation), the CI space can increase by large quantum jumps. Using the MRD-CI method of Buenker and Peyerimhoff and coworkers, we give several criteria which help to ensure energy continuity across these changes in CI space, and demonstrate these criteria for the hydrogen fluoride potential curve.     

Low-lying electronic excitations and optical absorption spectra of the black dye sensitizer: a first-principles study

We report on ab-initio calculations of the electronic structure and optical absorption response of the black dye sensitizer in gas phase. We show that, despite the large size of this molecule, the second-order multiconfiguration quasi-degenerate perturbation theory (MC-QDPT) can be used to calculate vertical excitation energies, oscillator strengths and optical absorption spectra. The zeroth-order reference states entering perturbation calculations are complete active space (CAS) configuration interaction (CI) wave functions computed for 12 active electrons distributed in 12 active orbitals. We found that the CI approach is not enough for taking into account the strong dynamical correlation effects in this system. In fact, the excitation energies of the CAS-CI target states are strongly renormalized by the MC-QDPT calculations. In the calculated absorption spectra, the analysis of the perturbed wavefunctions revealed that the stronger absorption bands correspond to metal-to-ligand and ligand-to-ligand charge transfer processes. Comparison with independent time-dependent extension (TDDFT) calculations performed with different functionals shows that corrections to the long-range behavior of the functional is pivotal to achieve agreement with the MC-QDPT results.     

The use of semiempirical energy partitioning terms in the study of through space (homoaromatic) interactions

MNDO and AMI calculations, including configuration interactions, were performed on cycloheptatriene (2), 1,6-methano[10]annulene (3), and the tautomeric elassovalenes (4), (5), and (6). The goal of this study is to examine these systems and assess indicators of the importance of through space (homoaromatic) interactions. It is established that the two-center energy partitioning terms are capable of detecting favorable (negative two-center term) through space interactions. In cases of cyclic conjugation (homoconjugation), it is also shown that the inclusion of CI is necessary.     

Molecular docking using shape descriptors

Molecular docking explores the binding modes of two interacting molecules. The technique is increasingly popular for studying protein-ligand interactions and for drug design. A fundamental problem problem with molecular docking is that orientation space is very large and grows combinatorially with the number of degrees of freedom of the interacting molecules. Here, we describe and evaluate algorithms that improve the efficiency and accuracy of a shape-based docking method. We use molecular organization and sampling techniques to remove the exponential time dependence on molecular size in docking calculations. The new techniques allow us to study systems that were prohibitively large for the original method. The new algorithms are tested in 10 different protein-ligand systems, including 7 systems where the ligand is itself a protein. In all cases, the new algorithms successfully reproduce the experimentally determined configurations of the ligand in the protein.     

Direct INDO/SCI method for excited state calculations

Intermediate neglect of differential overlap (INDO) is the most commonly utilized semiempirical technique for performing excited state calculations on large organic systems such as organic semiconductors and fluorescent dyes. The calculations are typically done at the singles-configuration interaction (SCI) level. Direct methods provide a more efficient means of performing configuration interaction (CI) calculations, and the computational trade offs associated with various approaches to direct-CI theory have been well characterized for ab initio Hamiltonians and high-order CI. However, the INDO and SCI approximations lead to a new set of trade offs. In particular, application of the electron-electron interactions in the atomic basis leads to savings in computational time that scale as the number of atomic orbitals, which for a large organic system can be two to three orders of magnitude. These savings are largest when only a few low-lying excited states are generated and when a full SCI basis, which includes excitations between all filled and empty molecular orbitals, is used. In addition, substantial memory savings are achieved in the direct method by avoiding the evaluation of the two electron integrals in the molecular orbital basis. The method is demonstrated by calculating the absorption spectrum of a poly(paraphenylenevinylene) oligomer containing 16 phenyl rings.     

A regionally contracted multireference configuration interaction method: general theory and results of an incremental version

This work proposes to take benefit of the localizability of both occupied and virtual inactive molecular orbitals (MOs) in the context of complete active space singles and doubles configuration interaction (CAS-SDCI). The doubly occupied MOs are partitioned into blocks, or regions, corresponding to a subset of adjacent bonds and lone pairs. The localized virtual MOs are attributed to these regions from a spatial criterion. Then a series of limited post-CAS-CI calculations is performed, using the same reference space, one for each block, and then one per pair of blocks. From these independent CI calculations contracted external functions are defined for each block or for each pair of blocks, and for each state. A general multistate formalism is proposed, the CI matrix being expressed in the space defined by the CAS and the contracted functions. Preliminary numerical studies, resting on the evaluation of single-block and two-block contributions to the dynamical correlation energy of each state, are presented. Provided that size-consistency corrections are taken into account the results of the procedure are shown to be in excellent agreement with those of the nonpartitioned post-CAS-CI. The computational benefits of this evidently parallelizable procedure are underlined.     

Evaluation of magnetic terms in Cu4O4 cubane-like systems from selected configuration interaction calculations: a case study of polynuclear transition-metal systems

We present the evaluation of magnetic terms in a Cu(4)O(4) cubane-like system from truncated CI calculations, as a case study of polynuclear transition-metal complexes. We employ a new excitation selected configuration interaction (EXSCI) method based on the use of local orbitals. Taking advantage of the locality and then of the fact that the interactions vanish when the distance is large, the dimension of the CI is largely reduced. To the best of our knowledge these CI calculations are the largest one performed for polynuclear transition metal systems so far. The results show the presence of two leading ferromagnetic interactions between bridged Cu ions. Also the interactions between the unbridged Cu ions are ferromagnetic, but very weak, in contrast to the experimental data. The nature and amplitude of all the computed interactions are consistent with the relative orientation of the magnetic orbitals in the molecule, and correctly reproduce the susceptibility versus temperature curve. Our results indicate that it is possible to obtain similar fittings with sets of parameters representing different physical effects and put in evidence the drawbacks of the fitting based on oversimplified magnetic models. In this context, the presented computational strategy can be considered as a useful tool to help in the interpretation of the magnetic data and the validation of the magnetic interaction model in the polynuclear magnetic systems.     

Approaching the full CI limit with MRD CI calculations: The X 1A1 state of water with a double-zeta basis

Results of MRD CI calculations with varying numbers of reference configurations for the water molecule employing a double-zeta basis set are compared with the corresponding full CI results of Harrison and Handy as well as with those of other methods. For the three geometries considered a highly uniform percentage (99.8±0.1%) of the available correlation energy in this AO basis is obtained by solving secular equations in the 13–15000 range, i.e. only 5% of the full CI space. Extrapolation of the full CI energy through the use of various correction formulae is found to be unreliable for large bond distances, although such an approach is successful at the equilibrium geometry.     

Direct CI method in restricted configuration spaces

A configuration interaction (CI) method in restricted CI (RCI) space obtained by imposing inequality-type restrictions on the occupancies of groups of molecular orbitals (MOs) was studied. The direct CI approach in such space was analyzed, and some recommendations concerning practical implementation of the RCI method are given. The corresponding program has been written in FORTRAN 77 for an IBM 486 DX personal computer and has been used for electronic structure calculations on transition metal complexes using a valence MO basis with the INDO approximation. © 1996 by John Wiley & Sons, Inc.     

Generalized doubly symbolic formulation for integral-driven direct configuration interaction method

Summary A direct configuration interaction (CI) scheme using the generalized double symbolism both for the external space and for the internal space is proposed in an integral-driven context. The reason why the double symbolism is used in the present formulation is that the main target is in investigating large molecular systems. The integrals, configuration state functions, and energy expressions are systematically classified in terms of the orbital labels and their mutual relations. Various types of CI wavefunctions can be set up flexibly. The resulting structure of integral processings in the sigma vector construction is complicated. The number of unique loop types for two-electron integrals is 1325. Because the parallel architecture is gaining importance in the recent computational platforms, the parallelism is also addressed.     

Selected configuration interaction with truncation energy error and application to the Ne atom

Selected configuration interaction (SCI) for atomic and molecular electronic structure calculations is reformulated in a general framework encompassing all CI methods. The linked cluster expansion is used as an intermediate device to approximate CI coefficients B(K) of disconnected configurations (those that can be expressed as products of combinations of singly and doubly excited ones) in terms of CI coefficients of lower-excited configurations where each K is a linear combination of configuration-state-functions (CSFs) over all degenerate elements of K. Disconnected configurations up to sextuply excited ones are selected by Brown's energy formula, Delta E(K) = (E-H(KK))B(K)2/(1-B(K)2), with B(K) determined from coefficients of singly and doubly excited configurations. The truncation energy error from disconnected configurations, Delta E(dis), is approximated by the sum of Delta E(K)s of all discarded Ks. The remaining (connected) configurations are selected by thresholds based on natural orbital concepts. Given a model CI space M, a usual upper bound E(S) is computed by CI in a selected space S, and E(M) = E(S) + Delta E(dis) + delta E, where delta E is a residual error which can be calculated by well-defined sensitivity analyses. An SCI calculation on Ne ground state featuring 1077 orbitals is presented. Convergence to within near spectroscopic accuracy (0.5 cm(-1)) is achieved in a model space M of 1.4 x 10(9) CSFs (1.1 x 10(12) determinants) containing up to quadruply excited CSFs. Accurate energy contributions of quintuples and sextuples in a model space of 6.5 x 10(12) CSFs are obtained. The impact of SCI on various orbital methods is discussed. Since Delta E(dis) can readily be calculated for very large basis sets without the need of a CI calculation, it can be used to estimate the orbital basis incompleteness error. A method for precise and efficient evaluation of E(S) is taken up in a companion paper.