Doubly contracted CI method and applications

~~ Doubly contracted CI method and applications~~     

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.     

The prism algorithm for two-electron integrals

A new approach to the evaluation of two-electron repulsion integrals over contracted Gaussian basis functions is developed. The new scheme encompasses 20 distinct, but interrelated, paths from simple shell-quartet parameters to the target integrals, and, for any given integral class, the path requiring the fewest floating-point operations (FLOPS ) is that used. Both theoretical (FLOP counting) and practical (CPU timing) measures indicate that the method represents a substantial improvement over the HGP algorithm.     

A sequential transformation approach to the internally contracted multireference coupled cluster method

The internally contracted multireference coupled cluster (ic-MRCC) approach is formulated using a new wave function ansatz based on a sequential transformation of the reference function (sqic-MRCC). This alternative wave function simplifies the formulation of computationally viable methods while preserving the accuracy of the ic-MRCC approach. The structure of the sqic-MRCC wave function allows folding the effect of the single excitations into a similarity-transformed Hamiltonian whose particle rank is equal to the one of the Hamiltonian. Consequently, we formulate an approximation to the sqic-MRCC method with singles and doubles (included respectively up to fourfold and twofold commutators, sqic-MRCCSD[2]) that contains all terms present in the corresponding single-reference coupled cluster scheme. Computations of the potential energy curves for the dissociation of BeH(2) show that the untruncated sqic-MRCCSD scheme yields results that are almost indistinguishable from the ordinary ic-MRCCSD method. The energy obtained from the computationally less expensive sqic-MRCCSD[2] approximation is found to deviate from the full ic-MRCCSD method by less than 0.2 mE(h) for BeH(2), while, in the case of water, the harmonic vibrational frequencies of ozone, the singlet-triplet splitting of p-benzyne, and the dissociation curve of N(2), sqic-MRCCSD[2] faithfully reproduces the results obtained via the ic-MRCCSD scheme truncated to two commutators. A formal proof is given of the equivalence of the ic-MRCC and sqic-MRCC methods with the internally contracted and full configuration interaction approaches.     

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.     

New concepts in bonded functions theory. I. alternative way of matrix elements evaluation

A new method of evaluation of energy matrix elements has been derived for bonded functions of Boys and a spinless Hamiltonian. In configuration-driven CI algorithm the new approach is two to four times faster than the original Reeves' one, the relative speed depending on the case. In integral-driven CI the new approach is about 10 to 12 times faster than the equivalent approach based on the Reeves' scheme.     

An improved algorithm for the normalized elimination of the small-component method

A new algorithm for the iterative solution of the normalized elimination of the small component (NESC) method is presented that is less costly than previous algorithms and that is based on (1) solving the NESC equations for the uncontracted rather than contracted basis (??First-Diagonalize-then-Contract??), (2) a new iterative procedure for obtaining the NESC Hamiltonian (??iterative TU algorithm??), (3) the renormalization scheme connected to the picture change, and (4) a finite nucleus model with a Gaussian charge distribution. The accuracy of NESC energies, which match those of 4-component Dirac calculations, is demonstrated. Test calculations with CCSD(T), DFT, and large basis sets including high angular momentum basis functions (f,g,h,i) are presented to prove the general applicability of the new NESC algorithm. Comparison with other algorithms of solving the NESC equations are shortly discussed and time savings are presented.     

Relativistic electronic structure theory

The theoretical and technical foundations are presented for the efficient relativistic electronic structure theories to treat heavy-atomic molecular systems. This review contains two surveys of four-component and two-component quasi-relativistic approaches. First, we review our highly efficient computational scheme for four-component relativistic ab initio molecular orbital (MO) methods over generally contracted spherical harmonic Gaussian-type spinors (GTSs). Illustrative calculations, which are performed with a new four-component relativistic ab initio molecular orbital program package REL4D, clearly show the efficiency of our computational scheme by the Dirac-Hartree-Fock (DHF) and Dirac-Hartree-Fock (DKS) methods. Next, in the two-component quasi-relativistic framework, two relativistic Hamiltonians, RESC and higher order Douglas-Kroll (DK) Hamiltonians, are introduced, and several illustrative calculations are shown. Numerical results for several systems show that good accuracy can be obtained with our third-order DK (DK3) Hamiltonian.     

The optimization of MCSCF functions by application of the generalized brillouin theorem: The LiH2 potential energy surface

The generalized Brillouin theorem is used to construct an optimization procedure for MCSCF functions by iterative contracted CI calculations. Special attention is paid to the MO transformation step in each iteration. In this method the MCSCF calculation may easily be augmented by a restricted CI calculation involving a configuration set which is uniquely determined by the trial function. An application to the calculation of the potential energy surface for linear LiH₂ in the reaction LiH + HLi + H₂ leads to the conclusion that this restricted CI is necessary, in order to obtain satisfactory results for the potential energy barrier in this reaction.     

Hard-segment polymorphism in MDI/diol-based polyurethane elastomers

X-ray analyses of methylene-bis(4)phenyl isocynate(MDI)/diol/poly(tetramethylene adipate) polyurethane elastomers prepared using hexanediol (HDO), butanediol (BDO), and propanediol (PDO) point to the development of a second crystalline structure in the hard segments as a result of thermal elongation and annealing. The HDO polymer crystallizes initially in the fully extended conformation, but a second crystal structure, a contracted form, develops with stretching and annealing at 130°C. In contrast, the BDO and PDO polymers crystallize initially in contracted conformation, but fully extended forms develop as a result of elongation and annealing. Only a contracted conformation has been seen so far in hard segments prepared using ethylene glycol (EDO) as the chain extender. These results correlate very well with DSC data for the same polymers. The development of the new crystal structures is accompanied by the appearance of new hard-segment melting peaks in the DSC traces. For the HDO polymer, the new peak for the contracted form appears at a higher temperature than that for the original extended form; for the BDO and PDO polymers the new peaks for the extended forms appear at lower temperatures than those for the contracted forms. Only a single peak is seen for the EDO polymer, for which only one crystal structure has been detected for the hard segments. These results indicate that the development of polymorphic crystal structures must be taken into account in interpreting the multiple melting phenomena seen for the hard domains in polyurethane elastomers.     

Optimizing conical intersections of solvated molecules: the combined spin-flip density functional theory/effective fragment potential method

Solvent effects on a potential energy surface crossing are investigated by optimizing a conical intersection (CI) in solution. To this end, the analytic energy gradient has been derived and implemented for the collinear spin-flip density functional theory (SFDFT) combined with the effective fragment potential (EFP) solvent model. The new method is applied to the azomethane-water cluster and the chromophore of green fluorescent protein in aqueous solution. These applications illustrate not only dramatic changes in the CI geometries but also strong stabilization of the CI in a polar solvent. Furthermore, the CI geometries obtained by the hybrid SFDFT/EFP scheme reproduce those by the full SFDFT, indicating that the SFDFT/EFP method is an efficient and promising approach for understanding nonadiabatic processes in solution.     

The ground-state potential curve for F2

The ground-state potential curve for F₂ has been obtained using large-scale MC SCF and CI methods. MC SCF curves were obtained with the CAS SCF method using a variety of sets of active orbitals. The main conclusion from the CAS SCF calculations is that the 2π_u orbital is important. CI curves were obtained using the contracted CI method. The largest calculations contained 312000 configurations proper spin and space (d_2h) symmetry. The main conclusions from the CI calculations are that the configuration XXX are important, otherwise errors in D_e of 0.3 eV and in r_e of 0.02 Å are found. The remaining errors at the CI level are 0.08 eV for D_e, 0.005 Å for r_e and less than 10 cm^?1 for the lowest vibrational levels.     

Internally contracted multiconfiguration-reference configuration interaction calculations for excited states

Summary The calculation of electronically excited states with the internally contracted multiconfiguration-reference configuration interaction (CMRCI) method is discussed. A straightforward method, in which contracted functions for all states are included in the basis, is shown to be very accurate and stable even in cases of narrow avoided crossings. However, the expense strongly increases with the number of states. A new method is proposed, which employs different contracted basis sets for each state, and in which eigensolutions of the Hamiltonian are found using an approximate projection operator technique. The computational effort for this method scales only linearly with the number of states. The two methods are compared for various applications.Dedicated in honor of Prof. Klaus Ruedenberg     

Opposing influences of ruffling and doming deformation on the 4-N cavity size of porphyrin macrocycles: the role of heme deformations revealed

Ruffle- and dome-type porphyrins were developed as model systems to investigate the role of deformation mode and degree of distortion in heme. Their crystal structures revealed that as the degree of distortion increases, cavity size can be contracted in the ruffle mode and expanded in the dome mode, and the size of cavity can exceed the limit of free metal ions from the fourth period (see scheme).     

Reliability of atomic natural orbital basis sets in calculations involving pseudopotentials

The performance of Atomic Natural Orbital (ANO) basis sets for calculations involving nonempirical core pseudopotentials has been studied by comparing the results for atomic and molecular nitrogen obtained using contracted ANO basis sets with those obtained using both the primitive set and a segmented one. The primitive set has been optimized at the SCF level for atomic N treated as a five-electron pseudo-atom, and consists of 7s and 7p primitive GTOs supplemented by 2d and 1f GTOs optimized at the CI level. From this primitive set three contracted [3s 3p 2d 1f] sets have been obtained. The first one has been derived from the ANOs of the neutral atom, the second has been obtained from an averaged density matrix and the third one is a segmented set. For the atom, the segmented set gives a zero contraction error at the SCF level as it must be in valence-only calculations. The ANO basis sets show some small contraction error at the SCF level but perform better in CI calculations. However, for the diatomic N₂ molecule the ANO basis sets exhibit a rather large contraction error in the calculated SCF energy. A detailed analysis of the origin of this error is reported, which shows that the conventional strategy used to derive ANO basis sets does not work very well when pseudopotentials are involved.     

Core-valence correlation effects in calcium hydride

Accurate multi-reference contracted CI calculations have been performed on CaH to determine the bond distance and the dissociation energy. At the Hartree-Fock limit a bond distance 0.10 au longer than in experiments is found. This value is not improved by a valence CI calculation. Including the core-valence (CV) contribution to the correlation energy makes the bond distance too short by 0.06 au, and agreement with experiments is obtained first when the core-core (CC) contribution from the 3p orbitals is added also. It turns out that the only orbital which needs to be treated of the core is the 3p σ orbital, which shows some bonding character at the CI level. Tight d functions to describe a 3d orbital of the type occupied in the ²D state of Ca^?, and f functions to correlate the 3p orbital are necessary for obtaining a good bond distance. For the dissociation energy a different trend is observed with excellent agreement with experiments at the CV level and no improvement over the SCF level at the CC level. The results are discussed and analyzed in detail in the text, where a section on size consistency is also added.     

A comparison between multireference CI and effective medium theories for diatomic FeN

Summary Internally contracted multireferece CI calculations have been performed for the diatomic molecules FeN, CrN and CrO. For the latter two molecules the calculated dissociation energies are 3.18 eV (3.9±0.2 eV) and 4.09 eV (4.5±0.1 eV) respectively, where the corresponding experimental results are given in parentheses. It is argued that the correct value for CrN must lie in the lower end of the range given by experiment or perhaps slightly below. The best calculated result for FeN, for which no experimental result exists, is 1.69 eV. This value can be compared to an older CI value of 0.9 eV and a recent result of 5.25 eV obtained by an effective medium approach. Based on the results for all the three molecules treated here the correct dissociation energy of FeN can be estimated to be about 2.1 eV. The relevance of the present results for modelling chemical reactions in the regions around on-top positions of transition metal surfaces is discussed.     

Exploiting regularity in systematic sequences of wavefunctions which approach the full CI limit

Summary The ground state total energy and related 1-electron properties are computed for three small molecules (N₂, H₂O, and H₂CN) using several systematic sequences of wavefunctions which approach the full CI. These sequences include multireference CI, averaged coupled pair functional and quasidegenerate variational perturbation theory wavefunctions. It is demonstrated that sufficient regularity exists in the sequence of variationally computed energies to permit extrapolation to the full CI limit using simple analytic expressions. It is furthermore demonstrated that a subset of the original list of configurations employed in the normal singles and doubles CI procedure can be selected using second order perturbation theory without adversely affecting the extrapolation to the full CI limit. This significantly broadens the range of applicability of the method. Along these lines, a scheme is proposed for the extrapolation of the selected CI results to the zero threshold (i.e. unselected) values in cases where the numbers of configurations associated with the latter would render the calculations intractable. Due to the vast reduction in the number of configurations which are handled variationally, the proposed scheme makes it possible to derive estimates of the full CI limit in cases where explicit full CI is either very difficult or currently impossible.Dedicated to Prof. Klaus RuedenbergThe Pacific Northwest Laboratory is operated for the U.S. Department of Energy by Battelle Memorial Institute under contract DE-AC06-76RLO 1830     

Total energies of molecules with the local density functional approximation and gaussian basis sets

Total energies of small molecules were calculated with a local density functional (LDF) approximation within the LCAO MO SCF scheme. The local spin density functional (LSD) of Gunnarsson and Lundqvist was used. The basis sets used are of contracted gaussian type which allow comparison of LSD with Hartree-Fock (HF) results. The program for calculation of the LSD term was incorporated into the standard ab initio package. The LSD binding energies were in better agreement with experiment than those from HF.     

Computed Potential Energy Surfaces and Minimum Energy Pathways for Chemical Reactions

Computed potential energy surfaces are often required for computation of such observables as rate constants as a function of temperature, product branching ratios, and other detailed properties. We have found that computation of the stationary points/reaction pathways using CASSCF/derivative methods, followed by use of the internally contracted CI method with the Dunning correlation consistent basis sets to obtain accurate energetics, gives useful results for a number of chemically important systems. Applications to complex reactions leading to NO_x and soot formation in hydrocarbon combustion are discussed.