Direct selected multireference configuration interaction calculations for large systems using localized orbitals

A selected multireference configuration interaction (CI) method and the corresponding code are presented. It is based on a procedure of localization that permits to obtain well localized occupied and virtual orbitals. Due to the local character of the electron correlation, using local orbitals allows one to neglect long range interactions. In a first step, three topological matrices are constructed, which determine whether two orbitals must be considered as interacting or not. Two of them concern the truncation of the determinant basis, one for occupied/virtual, the second one for dispersive interactions. The third one concerns the truncation of the list of two electron integrals. This approach permits a fine analysis of each kind of approximation and induces a huge reduction of the CI size and of the computational time. The procedure is tested on linear polyene aldehyde chains, dissociation potential energy curve, and reaction energy of a pesticide-Ca(2+) complex and finally on transition energies of a large iron system presenting a light-induced excited spin-state trapping effect.     

The parallel implementation of configuration-selecting multireference configuration interaction method

We report on a scalable implementation of the configuration-selecting multireference configuration interaction method for massively parallel architectures with distributed memory. Based on a residue driven evaluation of the matrix elements, this approach allows the routine treatment of Hilbert spaces of well over 10⁹ determinants, as well as the selective treatment of triple and quadruple excitations with respect to the reference space. We demonstrate the scalability of the method for up to 128 nodes on the IBM-SP2 and for up to 256 nodes on the Cray-T3E. We elaborate on the specific adaptation of the transition residue-based matrix element evaluation scheme that ensures the scalability and load balancing of the method. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1559–1570, 1999     

The direct configuration interaction method for general multireference expansions: Symmetric group approach

A computer implementation of the direct configuration interaction method formulated within the symmetric group approach is discussed. The formulation allows for an open-shell as well as for a multiconfigurational reference state. The number of all necessary formulas, derived by a computer for each integral type rather than for the individual integrals, is lower than in the currently existing techniques, including the unitary group approach. The logical structure of a general program for singly and doubly excited configurations is outlined. The efficiency of the symmetric group approach is demonstrated on a recently developed program, restricted to one reference state only.     

Parallel internally contracted multireference configuration interaction

A parallel implementation of the internally contracted (IC) multireference configuration (MRCI) module of the MOLPRO quantum chemistry program is described. The global array (GA) toolkit has been used in order to map an existing disk-paging small-memory algorithm onto a massively parallel supercomputer, where disk storage is replaced by the combined memory of all processors. This model has enabled a rather complicated code to be ported to the parallel environment without the need for the wholesale redesign of algorithms and data structures. Examples show that the parallel ICMRCI program can deliver results in a fraction of the time needed for equivalent uncontracted MRCI computations. Further examples demonstrate that ICMRCI computations with up to 10⁷ variational parameters, and equivalent to uncontracted MRCI with 10⁹ configurations, are feasible. The largest calculation demonstrates a parallel efficiency of about 80% on 128 nodes of a Cray T3E-300. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1215–1228, 1998     

Explicitly correlated multireference configuration interaction: MRCI-F12

An internally contracted multireference configuration interaction is developed which employs wave functions that explicitly depend on the electron-electron distance (MRCI-F12). This MRCI-F12 method has the same applicability as the MRCI method, while having much improved basis-set convergence with little extra computational cost. The F12b approximation is used to arrive at a computationally efficient implementation. The MRCI-F12 method is applied to the singlet-triplet separation of methylene, the dissociation energy of ozone, properties of diatomic molecules, and the reaction barrier and exothermicity of the F + H(2) reaction. These examples demonstrate that already with basis sets of moderate size the method provides near complete basis set MRCI accuracy, and hence quantitative agreement with the experimental data. As a side product, we have also implemented the explicitly correlated multireference averaged coupled pair functional method (MRACPF-F12).     

Investigation of alkali-metal clusters with pseudopotential multireference double-excitation configuration interaction method

Dimers and tetramers of Li, Na and K are studied utilizing an “ab initio” pseudopotential method followed by extensive multireference Cl. Results for Li and Na are compared with corresponding all-electron calculations. Geometry, optimization of K₄ predicts a comparable stability for singlet rhombic and T-shaped forms and for the triplet square arrangement.     

A non-linear approach to configuration interaction: The low-rank CI method (LR CI)

An alternative to the linear CI expansion is presented, which is based on the spectral resolution of the double excitation CI coefficients arranged as a symmetric matrix. The resulting energy expression is a quartic function of the new variables, where the double excitation operator is given as T̂₂ = ∑_I^Mω_I(d̂_I)², where d̂d_I is a single excitation operator: d̂_I =∑_Ia,d̂^I_iaÊ_ai. Since M is a small number, the number of variables is reduced considerably compared to normal SDCI, with only little loss in accuracy. A program for non-linear CI calculations has been written and is presented with results for H₂O and N₂. The method can easily be extended to include orbital optimization and cluster terms in the wavefunction, still yielding a fully variational approximation to the Schrödinger equation.     

Linear scaling multireference singles and doubles configuration interaction

A linear scaling multireference singles and doubles configuration interaction (MRSDCI) method has been developed. By using localized bases to span the occupied and virtual subspace, local truncation schemes can be applied in tandem with integral screening to reduce the various bottlenecks in a MRSDCI calculation. Among these, the evaluation of electron repulsion integrals and their subsequent transformation, together with the diagonalization of the large CI Hamiltonian matrix, correspond to the most computationally intensive steps in a MRSDCI calculation. We show that linear scaling is possible within each step. The scaling of the method with system size is explored with a system of linear alkane chains and we proceed to demonstrate this method can produce smooth potential energy surfaces via calculating the dissociation of trans-6-dodecene (C(12)H(24)) along the central C[Double Bond]C bond.     

Size extensive modification of local multireference configuration interaction

We recently developed a reduced scaling multireference configuration interaction (MRCI) method based on local correlation in the internal (occupied) and external (virtual) orbital spaces. This technique can be used, e.g., to predict bond dissociation energies in large molecules with reasonable accuracy. However, the inherent lack of size extensivity of truncated CI is a disadvantage that in principle worsens as the system size grows. Here we implement an a priori size-extensive modification of local MRCI known as the averaged coupled pair functional (ACPF) method. We demonstrate that local MR-ACPF recovers more correlation energy than local MRCI, in keeping with trends observed previously for nonlocal ACPF. We test the size extensivity of local ACPF on noninteracting He atoms and a series of hydrocarbons. Basis set and core correlation effects are explored, as well as bond breaking in a variety of organic molecules. The local MR-ACPF method proves to be a useful tool for investigating large molecules and represents a further improvement in predictive accuracy over local MRCI.     

On multireference superdirect configuration interaction in third order

The superdirect configuration interaction (Sup-CI ) method has the usual versatility and stability of the CI methods with computational efficiency typical to that of the many-body methods, such as the many-body perturbation theory (MBPT ). The Hamilton operator is projected into a space of a few trial vectors, such as Krylov, Nesbet, or Møller–Plesset correction vectors. In this space, Hamiltonian matrix elements may be directly computed in the many-body fashion, as weighted sums of integral products over orbital indices. The variation-perturbation method based on the first-order wave function is equivalent to the Sup-CI method with a single correction vector of the Møller–Plesset type. Different points of view on the superdirect CI method are discussed and a version in which third-order contributions are computed for a relatively small (10–100) space of reference and correction vectors is tested. Selection of the best “effective first-order spaces” and size-extensivity corrections in Sup-CI are briefly discussed. Møoller–Plesset, Epstein–Nesbet, and other correction vectors are included in the model calculations on the symmetric stretch of bonds in water, acetylene, and the NH₂ molecule. Errors are almost independent of molecular geometry and the method appears to be superior than the multireference second-order perturbation methods. © 1994 John Wiley & Sons, Inc.     

A determinantal approach to spin-orbit configuration interaction

A general configuration interaction (CI) algorithm incorporating one- and two-electron spin-orbit operators is presented. The algorithm is determinant based and enables the use of highly vectorized non-relativistic algorithms in the most operation-intensive part. Excitations between α and β spin orbitals are avoided in the time consuming parts by performing separate S ₊ and S ₋ operations. The relativistic CI expansions are often very large, so the algorithms require only the presence of segments of vectors in memory. Double-group symmetry is fully accounted for and time-reversal symmetry is exploited for both even and odd numbers of electrons. Received: 12 October 1996 / Accepted: 24 January 1997     

An efficient method for large scale configuration interaction calculations

An efficient method of handling large scale configuration interaction calculations is developed and applied to the H₂O molecule as a test case. The method, which is based upon matrix partitioning, is shown to be capable of calculating the ¹B₁ spectrum of H₂O to an accuracy level of 0.1 eV for each state with very moderate computational effort.     

A multireference configuration interaction method based on the separated electron pair wave functions

A multireference configurational interaction method based on the separated electron pair (SEP) wave functions, SEP‐CI approach, has been developed as an approximation to the traditional CASSCF method. It differs from the CASSCF method in that active orbitals are obtained from the SEP wave function without further optimization in the subsequent CI calculations, and the active space is automatically constructed according to the occupation coefficients of SEP natural orbitals. These features make the present SEP‐CI method computationally much less demanding than the CASSCF method. The applicability of the SEP‐CI method is illustrated with sample calculations on the insertion reaction of BeH₂ and dissociation energies of LiH, BH, FH, H₂O, and N₂. © 2005 Wiley Periodicals, Inc. J Comput Chem 27: 39–47, 2006     

Gradients for configuration interaction energies with spin-orbit coupling in a semiempirical framework

We present a method for the analytical evaluation of the molecular energy gradient for a semiempirical configuration interaction (CI) wavefunction, taking into account the spin-orbit coupling. We show how to proceed in the simplest case where all the wavefunctions belonging to the CI subspace considered are mixed by the spin-orbit interaction, and we develop an original procedure for the more complex case where only a limited number of CI eigenvectors of the spin-free Hamiltonian are mixed.     

Accuracy of energy extrapolation in multireference configuration interaction calculations

The accuracy of extrapolation procedures in conjunction with energy-based configuration selection in CI calculations is examined. The normally high accuracy of such extrapolation can deteriorate in multireference CI calculations when configuration functions of low weight are included in the root (reference) set. This is due to the inadequacy of second-order energy contribution estimates for the very large number of discarded low-contribution functions generated as single and double excitations from the minor members of the root set. The problem may be overcome by increasing the number of configurations included in the zero-order function used for the energy contribution estimation process. Illustrative results are presented for excited states of the H₂O molecule and the H₂O⁺ ion.     

Electron spin-spin coupling from multireference configuration interaction wave functions

We present the implementation of two-electron spin-spin coupling as a quasidegenerate perturbative treatment of the Breit-Pauli spin-spin Hamiltonian. The evaluation is based on a multireference CI treatment and constitutes one of the first efforts in the calculation of this effect within a highly sophisticated consideration of both nondynamical and dynamical correlation. The extension of existing schemes for efficient calculation, in particular, of the spin-coupling elements necessitated some involved derivations, the outline of which is presented herein. Application of the program to calculations of diagonal as well as off-diagonal spin-coupling elements is illustrated with the test cases O(2) and NH.     

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.     

Photochemistry of ethylene: a multireference configuration interaction investigation of the excited-state energy surfaces

Multireference configuration interaction with singles and doubles (MR-CISD) calculations have been performed for the optimization of conical intersections and stationary points on the ethylene excited-state energy surfaces using recently developed methods for the computation of analytic gradients and nonadiabatic coupling terms. Basis set dependence and the effect of various choices of reference spaces for the MR-CISD calculations have been investigated. The crossing seam between the S0 and S1 states has been explored in detail. This seam connects all conical intersections presently known for ethylene. Major emphasis has been laid on the hydrogen-migration path. Starting in the V state of twisted-orthogonal ethylene, a barrierless path to ethylidene was found. The feasibility of ethylidene formation will be important for the explanation of the relative yield of cis and trans H2 elimination.     

On the internally contracted multireference CI method with full contraction

The configuration interaction (CI ) method where the efficiency of the generators of the unitary group is most fully exploited is the internally contracted multireference CI method. In the most recent version of this method the semi-internal configurations were kept uncontracted, which means that the number of configurations can still be quite large. In the present study the necessary formulas are derived for the case where the semi-internal states are also contracted. The highest density matrix that appears in these formulas is of order 5, and the computational treatment of this large matrix is discussed in detail.