Multi Reference Research Articles

Study of the Ground State Dissociation of Diatomic Molecular Systems Using State-Specific Multireference Perturbation Theory: A Brillouin-Wigner Scheme.

The size-extensive second-order state-specific (or single root) multireference (MR) perturbation theory (SS-MRPT) in the Brillouin-Wigner (BW) form using Mϕller-Plesset perturbative evaluations of orders up to 2 [termed as SS-MRMPPT(BW)] presents a viable, as well as promising, approach to include both nondynamic and dynamic correlations in the study of the bond-stretching (in multireference/quasidegenerate situations) of molecular species with a manageable cost/accuracy ratio. It combines numerical stability in the presence of an intruder state problem with strict size consistency (when localized orbitals are used). In this paper, the SS-MRMPPT(BW) method has been shown to properly break the bonds (in the ground state) of several diatomic molecules (such as F2, Cl2 and Br2, and BH) that have posed a severe challenge to any many-body theoretical approach due to the presence of quasidegeneracy of varying degrees in the ground state. A comparison of the resulting potentials with the various theoretical results reveals that the method represents a valuable tool that is capable of properly accounting even for very strong quasidegeneracies, while also performing well in nondegenerate situations. In this work, we have also calculated spectroscopic constants (such as equilibrium bond lengths, vibrational frequencies, and dissociation energies) of the ground state of these molecular systems. The SS-MRMPPT spectroscopic constants are compared with the most accurate available ab initio calculations and other theoretical estimates of previous works to calibrate the efficacy of the method. For the sake of completeness, we also compare the computed spectroscopic constants with the experimental observations. The accuracy of computed spectroscopic parameters appears to be rather consistent over a multitude of systems for various basis sets. The SS-MRMPPT enables quantitatively accurate and computationally affordable analysis of potential energy surfaces and spectroscopic constants of various multireference systems in the ground state. It is particularly visible for spectroscopic parameters and nonparallelism error (NPE) calculations. The calculations further reveal that the SS-MRMPPT(BW) method compared to its Rayleigh-Schrödinger counterpart [SS-MRMPPT(RS)] provides a more accurate and consistent solution for the whole dissociation path and spectroscopic constants.

Isotopic effect on the dissociation dynamics of HOD2+ water molecules: An extensive numerical study

We present a detailed study of the potential energy surfaces (PES) of the semi-heavy water dication HOD2+ correlating asymptotically with O(3P) and O(1D). Using multi reference configuration interaction method, we generated a large set of data, which was used to fit the ground state and the first 7 excited PES for bending angles θ ≥ 80o and OH distances ROH ≥ 1.0 a0. The fit is used to investigate the dissociation dynamics along each PES for several initial geometries. We compute the kinetic energy release (KER) distribution and branching ratios of all 3 fragmentation channels and compare them to experimental data, if available. Results show that 2-body dissociation prevails for PES's correlated asymptotically with a deep OH+ potential depth, while the preference for OH rather than OD bond breakage strongly depend on the stiffness of the PES.

Titan’s Ionic Species: Theoretical Treatment of N2H+ and Related Ions

We use different ab initio methods to compute the three-dimensional potential energy surface (3D-PES) of the ground state of N(2)H(+). This includes the standard coupled cluster, the complete active space self-consistent field, the internally contacted multi reference configuration interaction, and the newly developed CCSD(T)-F12 methods. For the description of H and N atoms, several basis sets are tested. Then, we incorporate the 3D-PES analytical representations into variational calculations of the rovibrational spectrum of N(2)H(+)(X(1)Sigma(+)) up to 7200 cm(-1) above the zero point vibrational energy. Our data show that the CCSD(T)-F12/aug-cc-pVTZ approach represents a compromise for good description of the PES and computation cost. This technique is recommended for full dimensional PES generation of atmospheric and astrophysical relevant polyatomic systems. We applied this method to derive the rovibrational spectra of N(2)H(+)(X(1)Sigma(+)) and of N(2)H(++)(X(2)Sigma(+)). Finally, we discuss the existence of the N(2)H(++)(X(2)Sigma(+)) in Titan's atmosphere.

Multireference (MR) configuration interaction (CI) approach for quasidegenerate systems

The multireference (MR) configuration interaction (CI) wavefunctions utilizing generalized Hartree-Fock (GHF) molecular orbitals are discussed in relation to our previous extended HF and recent full-valence MCSCF methods for quasidegenerate systems. The reaction natural orbitals (RNO) for constructing the reference functions are selected on the basis of GHF calculations. The computational procedures for the MR CI are illustrated by considering imino nitroxide (HNNO) as an example. It is found that the present MR CI provides well-balanced wavefunctions between the ground and lower-lying excited states of the species. The correlation corrections coming from higher-order excitations are examined by a Davidson-like formula. The relation of this formula to the coupled-cluster method is stressed.

Energetics of 1,n‐didehydro‐polyene diradicals and performance of reduced multireference coupled‐cluster method

AbstractThe reduced multireference coupled‐cluster method with singles and doubles (RMR CCSD) and its RMR CCSD(T) version are employed to study the energetics of 1,n‐didehydro‐polyenes. The RMR CCSD method accounts for quasi‐degeneracy by exploiting a multireference (MR) CISD wave function as an external source of the most important (primary) triples and quadruples while the subsequent perturbative correction for the secondary triples leads to RMR CCSD(T). The resulting energies are compared with those yielded by the standard single‐reference (SR) CCSD and CCSD(T) approaches. We first determine the optimal geometry for each species considered. Using the CC methods just mentioned we then compute the energy of the lowest‐lying singlet and triplet states, the implied singlet‐triplet splitting, and determine the spin multiplicity of the ground state. We point out the relationship between the degree of the diradical character, the extent of quasi‐degeneracy or the MR nature of the state considered, the distance separating the radical centers and, finally, the size of the largest doubly excited cluster amplitude in the CC wave functions. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Configuration mixing within the energy density functional formalism: Removing spurious contributions from nondiagonal energy kernels

Multi reference calculations along the lines of the generator coordinate method or the restoration of broken symmetries within the nuclear energy density functional (EDF) framework are becoming a standard tool in nuclear structure physics. These calculations rely on the extension of a single-reference energy functional, of the Gogny or the Skyrme types, to nondiagonal energy kernels. There is no rigorous constructive framework for this extension so far. The commonly accepted way proceeds by formal analogy with the expressions obtained when applying the generalized Wick theorem to the nondiagonal matrix element of a Hamilton operator between two product states. It is pointed out that this procedure is ill defined when extended to EDF calculations as the generalized Wick theorem is taken outside of its range of applicability. In particular, such a procedure is responsible for the appearance of spurious divergences and steps in multi reference EDF energies, as was recently observed in calculations restoring particle number or angular momentum. In the present work, we give a formal analysis of the origin of this problem for calculations with and without pairing, i.e., constructing the density matrices from either Slater determinants or quasiparticle vacua. We propose a method to regularize nondiagonal energy kernels such that divergences and steps are removed from multi reference EDF energies. Such a removal is a priori quasiparticle-basis dependent. A special feature of the method we use to proceed to the actual regularization is that it singles out one basis among all possible ones. The regularization method is applicable to calculations based on any symmetry restoration or generator coordinate but is limited to EDFs depending only on integer powers of the normal and anomalous density matrices. Eventually, the method is formally illustrated for particle-number restoration and is specified to configuration mixing calculations based on Slater determinants.

Method of moments for the continuous transition between the Brillouin–Wigner-type and Rayleigh–Schrödinger-type multireference coupled cluster theories†

We apply the method of moments to the multireference (MR) coupled cluster (CC) formalism representing the continuous transition between the Brillouin–Wigner-type and Rayleigh–Schrödinger-type theories based on the Jeziorski–Monkhorst wave function ansatz and derive the formula for the noniterative energy corrections to the corresponding MRCC energies that recover the exact, full configuration interaction energies in the general model space case, including complete and incomplete model spaces. We also extend the relationship between the generalised moments of the state-universal (SU) MRCC equations within the Jeziorski–Monkhorst and Kucharski–Bartlett formulations of the SUMRCC theory to the general model space case. Finally, we argue that in the complete model space case, the relationship between moments of the SUMRCC equations corresponding to the Jeziorski–Monkhorst and Kucharski–Bartlett formulations of the SUMRCC theory, derived in this work, implies an equivalence of these two formulations of the SUMRCC approach, provided that the disconnected linked terms are included in the Kucharski–Bartlett formulation, and verify this statement numerically.

방향 정보를 이용한 H.264/AVC의 고속 인트라 모드 결정

다중 참조 영상을 이용한 최적 부호화 모드를 선택하기 위해 율-왜곡 최적화 기법을 적용한 H.264/AVC 동영상 압축 표준은 기존의 동영상 압축 표준들보다 높은 부호화 효율을 얻었지만 계산 복잡도가 그만큼 증가하게 되었다. 따라서 이런 계산의 복잡도를 줄이기 위한 많은 빠른 모드 결정 방법들이 제안되었다. 그중에 I4MB의 방향성 정보를 이용한 고속 인트라 모드 결정 알고리즘을 제안하였다. 제안한 알고리즘은 JM11.0과 비교하여 무시할만하게 PSNR이 손실되고 비트율의 증가하지만 IPPP 영상들에서 전체 부호화 시간을 평균 15%와 모든 영상을 인트라로 처리한 영상들에서 부호화 시간을 평균 44% 감소시킬 수 있었다. H.264/AVC video coding standard adapting a rate-distortion optimization technique to select the best coding mode with multi reference frames for each macroblock gets a higher coding efficiency than those of previous video coding standards but the computational complexity increases drastically. Therefore, many fast mode decision algorithms are proposed to reduce the computational complexity. Among them, we propose a fast intra mode decision algorithm based on directional information of I4MB. The proposed algorithm achieves consistent time saving about 15% in IPPP sequences and 44% in all I frame sequences with negligible loss in PSNR and small increment of bit rate compared with that of JM11.0.

Multireference ab initio studies of zero-field splitting and magnetic circular dichroism spectra of tetrahedral Co(ii) complexes

A newly developed multireference (MR) ab initio method for the calculation of magnetic circular dichroism (MCD) spectra was calibrated through the calculation of the ground- and excited state properties of seven high-spin (S = 3/2) Co(II) complexes. The MCD spectra were computed by the explicit treatment of spin-orbit coupled (SOC) and spin-spin coupled (SSC) N-electron states. For the complexes studied in this work, we found that the SOC is more important than the SSC for determining the ground state zero field splitting (ZFS). Our computed ZFS parameter D for the [Co(PPh(3))(2)Cl(2)] model complex is -17.6 cm(-1), which is reasonably close to the experimental value of -14.8 cm(-1). Generally, the computed absorption and MCD spectra are in fair agreement with experiment for all investigated complexes. Thus, reliable electronic structure and spectroscopic predictions for medium sized transition metal complexes are feasible on the basis of this methodology. This characterizes the presented method as a promising tool for MCD spectra interpretations of transition metal complexes in a variety of areas of chemistry and biology.

Mechanism of Spectral Tuning Going from Retinal in Vacuo to Bovine Rhodopsin and its Mutants: Multireference ab Initio Quantum Mechanics/Molecular Mechanics Studies

We have investigated photoabsorption spectra of bovine rhodopsin and its mutants (E122Q and E113Q) by hybrid quantum mechanical/molecular mechanical (QM/MM) calculations as well as retinal in vacuo by pure QM calculations, employing multireference (MR) ab initio and TD-B3LYP methods. The sophisticated MR-SORCI+Q and MRCISD+Q methods extrapolated with respect to adopted approximations can reproduce the experimental absorption maxima of retinal very well. The relatively inexpensive MR-DDCI2+Q method gives absorption maxima blue-shifted by ca. 65 nm from experimental values; however, this error is systematic and thus MR-DDCI2+Q can be used to estimate spectral shifts. In MR calculations, the ground-state energy of retinal at B3LYP geometry is significantly lower than that at CASSCF geometry. Therefore, B3LYP geometry is more reliable than CASSCF geometry, which has a blue-shift error as large as 100 nm in the gas phase. The effect of ground-state geometry on the excitation energies is less critical in the polarizing field of protein environments. At the B3LYP geometry, there is no significant charge transfer upon vertical excitation to the S1 excited-state either from Glu1 13 to retinal or from Schiff-base terminal to beta-ionone ring through the polyene chain. All-trans to 11-cis isomerization of retinal in the gas phase has no influence on the calculated S1 absorbing state, in agreement with experiment. The shoulder of the experimental absorption spectrum of retinal in vacuo at the S1 absorbing band appears to be the second electronic transition (S2) in our calculations, contrary to previous tentative assignment to vibrational state of S1 or to the S1 band of a retinal isomer.

The multi-reference retaining the excitation degree perturbation theory: A size-consistent, unitary invariant, and rapidly convergent wavefunction based ab initio approach

The retaining the excitation degree (RE) partitioning [R.F. Fink, Chem. Phys. Lett. 428 (2006) 461(20 September)] is reformulated and applied to multi-reference cases with complete active space (CAS) reference wave functions. The generalised van Vleck perturbation theory is employed to set up the perturbation equations. It is demonstrated that this leads to a consistent and well defined theory which fulfils all important criteria of a generally applicable ab initio method: The theory is proven numerically and analytically to be size-consistent and invariant with respect to unitary orbital transformations within the inactive, active and virtual orbital spaces. In contrast to most previously proposed multi-reference perturbation theories the necessary condition for a proper perturbation theory to fulfil the zeroth order perturbation equation is exactly satisfied with the RE partitioning itself without additional projectors on configurational spaces. The theory is applied to several excited states of the benchmark systems CH2, SiH2, and NH2, as well as to the lowest states of the carbon, nitrogen and oxygen atoms. In all cases comparisons are made with full configuration interaction results. The multi-reference (MR)-RE method is shown to provide very rapidly converging perturbation series. Energy differences between states of similar configurations converge even faster.

An exponential multi-reference wavefunction ansatz: connectivity analysis and application to N2

A connectivity analysis for the exponential multi-reference wavefunction ansatz (MRexpT) (J Chem Phys 123:84102, 2005) is carried out. Assuming a complete model space and separating interactions carrying active labels the cluster operator carrying no active labels is fully connected. The valence (active) part of the MRexpT cluster operator, however remains disconnected. Consequently, the MRexpT correlation energy scales linearly with the number of non-active electrons as single reference coupled cluster does while MRexpT additionally can treat multi reference cases. Therefore, MRexpT should be well suited to be applied to a large number of molecular applications. Its applicability to periodic systems with multi-reference unit cells however seems to be limited. An application to the triple bond breaking of the N2 molecule is presented.

Multireference Model Chemistries for Thermochemical Kinetics.

By combining the generalized valence bond ansatz of correlated participating orbitals (CPO) with the complete-active-space prescription for selecting configurations and with the use of multireference second order perturbation theory (MRMP2) for including dynamical correlation, we define three levels of multireference (MR) theoretical model chemistries for electronic structure calculations of chemical reaction energies and barrier heights. The three levels differ in their choice of which orbitals are considered to be participating; the choices are called nominal (nom-CPO), moderate (mod-CPO), and extended (ext-CPO). Combining any of these three choices with a method for treatment of dynamical correlation energy and a one-electron basis set yields a theoretical model chemistry. Unlike the full-valence choice of active orbitals, the CPO choices lead to active spaces that contain the orbitals needed to include important static correlation effects on chemical reactions but do not increase with the size of the nonparticipating portion of the system, and hence they remain viable computational options even for many large and complex reacting systems. The accuracies of the new levels, combined with the MG3S basis set (a partially augmented, multiply polarized valence triple-ζ basis with appropriately tight d functions for 3p-block elements) and with the fully augmented correlation-consistent aug-cc-pVTZ basis set, are assessed against a previously presented database of barrier heights for diverse reaction types. We find that nom-CPO level captures the bulk of the static correlation energy, and MRMP2/nom-CPO calculations have an average error of only 1.4 kcal/mol in barrier heights, which may be compared to 5.0 kcal/mol for single-reference MP2 theory, 2.5 kcal/mol for CCSD, and 4.1 and 1.0 kcal/mol for the B3LYP and M06-2X density functionals, respectively. The accuracy of MRMP2/CPO for transition structure bond lengths and donor-acceptor distances is excellent, with a mean unsigned error of only 0.007 Å as compared to 0.018 Å for CCSD, 0.019 Å for M06-2X, and 0.039 Å for MP2 and B3LYP. We also introduce a new multireference diagnostic, called the M diagnostic, that allows one to measure the importance of static correlation in a given reagent or transition state.

Partially linearized, fully size-extensive, and reduced multireference coupled-cluster methods. II. Applications and performance

The partially linearized (pl), fully size-extensive multireference (MR) coupled-cluster (CC) method, fully accounting for singles (S) and doubles (D) and approximately for a subset of primary higher than doubles, referred to as plMR CCSD, as well as its plMR CCSD(T) version corrected for secondary triples, as described in Part I of this paper [X. Li and J. Paldus, J. Chem. Phys. 128, 144118 (2008)], are applied to the problem of bond breaking in the HF, F2, H2O, and N2 molecules, as well as to the H4 model, using basis sets of a DZ or a cc-pVDZ quality that enable a comparison with the full configuration interaction (FCI) exact energies for a given ab initio model. A comparison of the performance of the plMR CCSD/CCSD(T) approaches with those of the reduced MR (RMR) CCSD/CCSD(T) methods, as well as with the standard single reference (SR) CCSD and CCSD(T) methods, is made in each case. For the H4 model and N2 we also compare our results with the completely renormalized (CR) CC(2,3) method [P. Piecuch and M. Włoch, J. Chem. Phys. 123, 224105 (2005)]. An important role of a proper choice of the model space for the MR-type methods is also addressed. The advantages and shortcomings of all these methods are pointed out and discussed, as well as their size-extensivity characteristics, in which case we distinguish supersystems involving noninteracting SR and MR subsystems from those involving only MR-type subsystems. Although the plMR-type approaches render fully size-extensive results, while the RMR CCSD may slightly violate this property, the latter method yields invariably superior results to the plMR CCSD ones and is more easy to apply in highly demanding cases, such as the triple-bond breaking in the nitrogen molecule.

Multireference averaged quadratic coupled-cluster (MR-AQCC) method based on the functional of the total energy

In this short paper a new version of the multireference (MR) averaged coupled pair functional (ACPF) and the MR averaged quadratic coupled-cluster (AQCC) methods is presented which corrects the failure of the original formulation in case of close-lying states described with small reference space. This new version is based on a functional which also includes the reference energy, i.e. instead of optimizing the correlation energy, the total energy will be made stationary with respect the wave function parameters. It is demonstrated that this method gives smooth potential energy surfaces in case of avoided crossings.

Molecular ions in ultracold atomic gases: computed electronic interactions for MgH+(X1Σ+) with Rb

The electronic structures of the manifold of potential energy surfaces generated in the lower energy range by the interaction of the MgH+(X1Σ+) cationic molecule with Rb(2S) neutral atom are obtained over a broad range of Jacobi coordinates from strongly correlated ab initio calculations which use a Multireference (MR) wavefunction within a Complete Active Space (CAS) approach. The relative features of the lowest five surfaces are analyzed in terms of possible collisional outcomes when employed to model the ultracold dynamics of ionic molecular partners.

Performance of the Spin-Flip and Multireference Methods for Bond Breaking in Hydrocarbons: A Benchmark Study

Benchmark results for spin-flip (SF) coupled-cluster and multireference (MR) methods for bond-breaking in hydrocarbons are presented. The nonparallelity errors (NPEs), which are defined as an absolute value of the difference between the maximum and minimum values of the errors in the potential energy along bond-breaking curves, are analyzed for (i) the entire range of nuclear distortions from equilibrium to the dissociation limit and (ii) in the intermediate range (2.5-4.5 A), which is the most relevant for kinetics modeling. For methane, the spin-flip and MR results are compared against full configuration interaction (FCI). For the entire potential energy curves, the NPEs for the SF model with single and double substitutions (SF-CCSD) are slightly less than 3 kcal/mol. Inclusion of triple excitations reduces the NPEs to 0.32 kcal/mol. The corresponding NPEs for the MR-CI are less than 1 kcal/mol, while those of multireference perturbation theory are slightly larger (1.2 kcal/mol). The NPEs in the intermediate range are smaller for all of the methods. The largest errors of 0.35 kcal/mol are observed, surprisingly, for a spin-flip approach that includes triple excitations, while MR-CI, CASPT2, and SF-CCSD curves are very close to each other and are within 0.1-0.2 kcal/mol of FCI. For a larger basis set, the difference between MR-CI and CASPT2 is about 0.2 kcal/mol, while SF-CCSD is within 0.4 kcal/mol of MR-CI. For the C-C bond breaking in ethane, the results of the SF-CCSD are within 1 kcal/mol of MR-CI for the entire curve and within 0.4 kcal/mol in the intermediate region. The corresponding NPEs for CASPT2 are 1.8 and 0.4 kcal/mol, respectively. Including the effect of triples by energy-additivity schemes is found to be insignificant for the intermediate region. For the entire range of nuclear separations, sufficiently large basis sets are required to avoid artifacts at small internuclear separations.

Novel Encryption Method Using Multi Reference Patterns in Coaxial Holographic Data Storage

In this paper, we propose an encryption method, in which a multi reference pattern is used in coaxial holographic data storage. The signal data patterns are multiplexed at one spot with multi reference patterns, which are the key code for secure readout. This method is easy to implement in coaxial holographic storage drives, and provides higher security against unauthorized readout and copying of the hologram disk. We experimentally demonstrate the encryption and decryption process with the proper and improper keys. Next, we analyze the degree of security provided by this method, and demonstrate a method of increasing the security further by recording more than 10 holograms at one spot. Finally, we record multiplexed holograms to reach 50 and 100 Gbit/in.2 with this encryption method, and analyze the degradation of the error rate.

CFD MODELING OF GAS-LIQUID HYDRODYNAMICS IN A STIRRED TANK REACTOR

Multiphase impeller stirred tank reactors enhance mixing of reacting species used in a variety of chemical industries. These reactors have been studied based on Computational Fluid Dynamics (CFD) that can be used in the analysis, design and scale up of these reactors. Most of the researches done in this area are limited to single phase reactors, and a few remaining two phase flow investigations have been done based on MRF (Multi Reference Frame) and Snapshot approach. However, the MRF and snapshot approaches cannot be used in rigorous simulation of unsteady state problems. In order to simulate the unsteady state behavior of the multiphase stirred tank reactors we have used sliding mesh technique to solve the problem rigorously. In this work a 3D CFD model is used to investigate hydrodynamics of a fully baffled cylindrical stirred tank containing air-water in which air is sparged. The tank is equipped with a standard Rushton turbine impeller. This work has been done based on two fluid (Eulerian-Eulerian) model and finite volume method, along with standard k-e model to address turbulent behavior of both phases. The results obtained for velocity field show a good agreement with the corresponding data published by researchers in this field for the same case. The effect of gas inlet velocity on the gas holds up distribution has been studied. According to the obtained results, there should be an optimum value for gas inlet velocity in order to achieve appropriate gas distribution in the liquid. The most important parameter affecting the optimum value is impeller rotational speed.

A systematic quantum chemical investigation of the CH bond activation in methane by gas phase vanadium oxide cation VO+

The interaction between a methane molecule and the VO(+) cation in the gas phase has been investigated by means of single reference density functional (B3LYP) and wave function-based multireference (MR) correlation calculations. For the latter, an extrapolation technique is used to evaluate correlation energies at the basis set limit. A comprehensive picture for the C-H activation features a variety of molecular structures corresponding to both minima and transition states. Possible reaction paths are discussed, also taking into account change of the spin multiplicity. Activation of the methane molecule by VO(+) is always an endothermic process. Competing reaction paths might be expected. An evaluation of miscellaneous computational methods is performed using calculated energy differences for various molecular structures. Results obtained from the MR calculations exhibit no systematic convergence with increasing size of the active space used, and for two largest active spaces relative energies still differ by up to 25 kJ/mol. Simple mean difference between the B3LYP results and the best MR values is -50 +/- 19 kJ/mol.