Multi Reference Research Articles

Dipole moments and polarizabilities of some small radicals using constrained variational response to Fock-space multi-reference coupled-cluster theory

In this Letter, we present dipole moments and polarizabilities of some open-shell doublet radicals, obtained analytically using constrained variational response to Fock-space (FS) multi-reference (MR) coupled cluster (CC) theory. These radicals can be treated as electron attached (EA) states of the corresponding closed-shell cations. Additionally, we also report analytical polarizabilities of some doublet radicals, which can be considered as ionized (IP) states of the corresponding anions. We compare our results with finite field FSMRCC response and the available benchmark results.

Quantum spin correction scheme based on spin-correlation functional for Kohn–Sham spin density functional theory

We present a simple quantum correction scheme for ab initio Kohn–Sham spin density functional theory (KS-SDFT). This scheme is based on a mapping from ab initio results to a Heisenberg model Hamiltonian. The effective exchange integral is estimated by using energies and spin correlation functionals calculated by ab initio KS-SDFT. The quantum-corrected spin-correlation functional is open to be designed to cover specific quantum spin fluctuations. In this article, we present a simple correction for dinuclear compounds having multiple bonds. The computational results are discussed in relation to multireference (MR) DFT, by which we treat the quantum many-body effects explicitly.

A truncated version of reduced multireference coupled-cluster method with singles and doubles and noniterative triples: Application to F2 and Ni(CO)n (n=1, 2, and 4)

A perturbatively truncated version of the reduced multireference coupled-cluster method with singles and doubles and noniterative triples RMR CCSD(T) is described. In the standard RMR CCSD method, the effect of all triples and quadruples that are singles or doubles relative to references spanning a chosen multireference (MR) model space is accounted for via the external corrections based on the MR CISD wave function. In the full version of RMR CCSD(T), the remaining triples are then handled via perturbative corrections as in the standard, single-reference (SR) CCSD(T) method. By using a perturbative threshold in the selection of MR CISD configuration space, we arrive at the truncated version of RMR CCSD(T), in which the dimension of the MR CISD problem is significantly reduced, thus leaving more triples to be treated perturbatively. This significantly reduces the computational cost. We illustrate this approach on the F2 molecule, in which case the computational cost of the truncated version of RMR CCSD(T) is only about 10%-20% higher than that of the standard CCSD(T), while still eliminating the failure of CCSD(T) in the bond breaking region of geometries. To demonstrate the capabilities of the method, we have also used it to examine the structure and binding energy of transition metal complexes Ni(CO)n with n=1, 2, and 4. In particular, Ni(CO)2 is shown to be bent rather than linear, as implied by some earlier studies. The RMR CCSD(T) binding energy differs from the SR CCSD(T) one by 1-2 kcal/mol, while the energy barrier separating the linear and bent structures of Ni(CO)2 is smaller than 1 kcal/mol.

Study of the C(3P) + OH(X2Π) → CO(XΣg+) + H(2S) Reaction: A Fully Global ab Initio Potential Energy Surface of the XA‘ State

The C((3)P) + OH(X (2)Pi) --> CO(X (1)Sigma(g)(+)) + H((2)S) reaction has been investigated by ab initio electronic structure calculations of the X(2)A' state based on the multireference (MR) internally contracted single and double configuration interaction (SDCI) method plus Davidson correction (+Q) using Dunning aug-cc-pVQZ basis sets. In particular, the multireference space is taken to be a complete active space (CAS). Improvement over previously proposed potential energy surfaces for HCO/COH is obtained in the sense that present surface describes also the potential part where the CO interatomic distance is large. A large number of geometries (around 2000) have been calculated and analytically fitted using the reproducing kernel Hilbert space (RKHS) method of Ho and Rabitz both for the two-body and three-body terms following the many-body decomposition of the total electronic energies. Results show that the global reaction is highly exothermic ( approximately 6.4 eV) and barrierless (relative to the reactant channel), while five potential barriers are located on this surface. The three minima and five saddle points observed are characterized and found to be in good agreement with previous work. The three minima correspond to the formation of HCO and COH complexes and to the CO + H products, with the COH complex being a metastable minimum relative to the product channel. The five saddle points correspond to potential barriers for both the dissociation/formation of HCO and COH into/from CO + H, to barriers for the isomerization of HCO into COH and to barriers for the inversion of HCO and COH through their respective linear configuration.

Ab initio prediction of the infrared absorption spectrum of the C2Br radical

The first three electronic states of the C2Br radical, correlating at linear geometries with 2Σ+ and 2Π states, have been studied ab initio, using Multi Reference Configuration Interaction techniques. The electronic ground state is found to have a bent equilibrium geometry, RCC=1.2621Å, R CBr=1.7967Å, ∠ CCBr=156.1°, with a very low barrier to linearity. Similarly to the valence isoelectronic radicals C2F and C2Cl, this anomalous behaviour is attributed to a strong three-state non-adiabatic electronic interaction. The Σ ,Π1/2,Π3/2 vibronic energy levels and their absolute infrared absorption intensities at a temperature of 5 K have been calculated for the 12 C12 C79Br isotopomer, to an upper limit of 2000 cm−1, using ab initio diabatic potential energy and dipole moment surfaces and a recently developed variational method.

A convenient decontraction procedure of internally contracted state-specific multireference algorithms

Internally contracted state-specific multireference (MR) algorithms, either perturbative such as CASPT2 or NEVPT2, or nonperturbative such as contracted MR configuration interaction or MR coupled cluster, are computationally efficient but they may suffer from the internal contraction of the wave function in the reference space. The use of a low dimensional multistate model space only offers limited flexibility and is not always practicable. The present paper suggests a convenient state-specific procedure to decontract the reference part of the wave function from a series of state-specific calculations using slightly perturbed zero-order wave functions. The method provides an orthogonal valence bond reading of the ground state and an effective valence Hamiltonian, the excited roots of which are shown to be relevant. The orthogonal valence bond functions can be considered quasidiabatic states and the effective valence Hamiltonian gives therefore the quasidiabatic energies and the electronic coupling among the quasidiabatic states. The efficiency of the method is illustrated in two case problems where the dynamical correlation plays a crucial role, namely, the LiF neutral/ionic avoided crossing and the F(2) ground state wave function.

Constrained variational response to Fock-space multi-reference coupled-cluster theory: Some pilot applications

Fock space (FS) multi-reference (MR) coupled-cluster (CC) has been established to be state-of-the-art method for energies of open-shell radicals, ionized, electron attached states and excited states. Due to the multi-root nature of the method, the linear response of MRCC, however, is non-trivial. Constrained variation approach (CVA) makes it plausible to calculate response of the MRCC method efficiently. However, the approach is used for only one root of the problem at a time. In the present article, we make first implementation of the approach and present test results of dipole moments of small radicals.

General-model-space state-universal coupled-cluster method: excitation energies of water

A recently developed multireference (MR), general-model-space (GMS), state-universal (SU) coupled-cluster (CC) method [J. Chem. Phys. 119, 5320 (2003)] in its extended form that considers singly (S) and doubly (D) excited cluster amplitudes relative to the reference configurations spanning the model space—which can differ by up to and including quadruple excitations—is employed to investigate a number of low-lying excited states of the water molecule. The role played by different atomic orbital basis sets is examined by employing both the aug-cc-pVTZ and cc-pVTZ + diff bases, the latter extending the cc-pVTZ basis by an s diffuse function on hydrogen and two sets of s and p diffuse functions on oxygen. The computed vertical excitation energies are compared with those obtained earlier with a cc-pVDZ basis set, as well as with other—both theoretical and experimental—results, when available. For a number of relevant states the equilibrium geometry, and the implied T 0 excitation energies, are also given. It is shown that two sets of s and p diffuse functions on oxygen are essential for an accurate rendering of the experimental spectra and thus play a much more important role than the higher-angular-momentum diffuse functions (i.e. d and f on oxygen and p and d on hydrogen) characterizing the aug-cc-pVTZ basis set.

Recent Development of Multireference Density Functional Theory

Abstract We review a recently developed ab initio multireference (MR)-density functional theory (DFT) approach based on the partially interacting reference systems. Instead of Kohn–Sham equations, we introduce the auxiliary CI equation with the effective DFT potential field. For practice, we classify electrons of the target system into valence electrons and other electrons. The former plays the quantum resonating feature of electronic structure of molecular species, while the latter affects the electronic properties via the effective correlation potential for the former electrons. The theoretical devices in our MR-DFT are introduced to avoid the double counting problem for any combination of a wavefunction and a correlation functional. Some numerical examples are presented.

Multireference density functional theory with orbital‐dependent correlation corrections

AbstractThe features of ab initio wave function‐driven multireference (MR)‐density functional theory (DFT) approach based on the partially interacting reference system for ground and excited states are discussed from the viewpoint of a simple classification of electron correlation effects. Our previous MR‐DFT approach covers (i) structure‐free, and (ii) resonating‐type correlations, but does not include (iii) orbital‐dependent correlation effects. Indeed, the computational results of excitation energies of atoms imply that the state‐dependent correlations remain if the CASCI wave functions are used for the wave function part of MR‐DFT and semilocal correlation functional is used. Thus, we present an orbital‐dependent residual correlation for complete active space (CAS) DFT to describe type (iii) of correlation effects. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2006

Electronic structure and reactivity of the CNO/NCO/CON isomers

Three-dimensional potential energy surfaces (PESs) have been calculated for the lowest electronic states of NCO, CNO and CON isomers, using internally contracted Multi Reference Configuration Interaction (MRCI) and Coupled-Clusters RCCSD(T) ab initio methods. For the low lying doublet and quartet excited states of the three isomers, the N–CO, O–CN, C–NO and C–ON collinear dissociation paths were mapped by the Complete Active Space SCF (CASSCF) approach and the energy variations with the bending coordinate have been explored. Several regions of conical intersections have been located and the spin–orbit interactions between states of different spin symmetry have been evaluated in the region of intersections of these states. The analysis of the PESs allows one to identify the main interactions governing the reactivity of the lowest electronic states. The NCO and CNO isomers have stable X2Π electronic ground states, for CON the X2Π ground state is separated from the dissociative [CO + N] asymptote by a barrier of 0.11 eV and crosses the dissociative 4Σ - state close to its minimum. At their equilibrium ground state geometries the spin–orbit interactions A SO between the two electronic components of the X2Π states were calculated to be -95.6, -109.6 and -57.1 cm−1 for NCO, CNO and CON, respectively. The predissociation of the vibrational levels of the A2Σ+ and B2Π states of NCO has been explained.

Can a Single-Reference Approach Provide a Balanced Description of Ground and Excited States? A Comparison of the Completely Renormalized Equation-of-Motion Coupled-Cluster Method with Multireference Quasidegenerate Perturbation Theory near a Conical Intersection and along a Photodissociation Coordinate in Ammonia

We calculated the two lowest electronically adiabatic potential energy surfaces of ammonia in the region of the conical intersection and at a sequence of geometries along which one of the N-H bonds is broken. We employed both a multireference (MR) method and a single-reference (SR) method. The MR calculations are based on multiconfiguration quasidegenerate perturbation theory (MC-QDPT) with a 6-311+G(3df,3pd) basis set. The SR calculations, carried out with the same basis, employ the completely renormalized equation-of-motion coupled-cluster method with singles and doubles, and a noniterative treatment of triples, denoted CR-EOMCCSD(T). At 91 geometries used for comparison, including geometries near a conical intersection, the surfaces agree to 7% on average.

The coupled-cluster corrections to multi-reference configuration interaction method: the HF and F2 bond breaking description*

Potential energy curves may be difficult to describe by multi-reference (MR) coupled-cluster or perturbation methods because of the intruder state problem. To avoid intruder states one may have to change the reference space while going from one internuclear distance to another which can make the description inconsistent and cause discontinuities on the potential energy curve. Conversely to this, the MR configuration interaction method with singles and doubles (MR-CISD) does not require the reference space that is specially tailored for particular geometries to avoid intruders. A disadvantage of the MR-CISD scheme is the relatively poor description of the dynamic correlation which does not benefit from the efficiency of the cluster expansion. However, if the coupled-cluster corrections to MR-CISD are a posteriori used to improve that description then a very accurate potential energy curve can be obtained. In this preliminary study the corrections are applied in model calculations to investigate the HF and F2 bond breaking cases. Also the problem of size-inextensivity of the MR-CISD method is addressed. A numerical test shows that the corrections are capable of significant reduction of the MR-CISD inextensivity error.

Ab Initio MRD-CI Study of the Electronic Spectrum of BrNO2 and Photofragmentation

Multi reference configuration interaction, MRD-CI, methods with the cc-pVDZ+sp and cc-pVTZ+sp basis sets were employed to determine the low-lying singlet and triplet electronic states of nitryl bromide, BrNO 2 . The calculations predict two very strong transitions, 3 1 A 1 ← X 1 A 1 and 3 1 B 2 ← X 1 A 1 , at 6.15 and 7.27 eV, respectively. At wavelengths that are atmospherically relevant to the BrNO 2 photolysis, 1 1 B 2 and 1 1 B 1 singlet states with vertical transition energies of 3.43 and 3.85 eV, respectively, are computed. The corresponding triplet states are lower in energy by 0.27 and 0.36 eV, respectively. Both singlet and triplet states are found to be highly repulsive along the Br-N coordinate and they dissociate to the ground-state fragments Br and NO 2 . The comparison between the first singlet excited state of BrNO 2 and ClNO 2 demonstrates that the transition is red-shifted by 80 nm from ClNO 2 upon replacement of chlorine by bromine. Furthermore, the most intense transition 3 1 A 1 ← X 1 A 1 of BrNO 2 has a 0.9 eV lower energy than that of ClNO 2 . These energy differences can be explained by the large halogen character of the orbitals involved in the electron excitations.

Ab initio potential energy surfaces of the ion-molecule reaction: C2H2 + O+.

High level ab initio calculations using complete active space self-consistent field and multi reference single and double excitation configuration interaction methods with cc-pVDZ (correlation consistent polarized valence double zeta) and cc-pVTZ (triple zeta) basis sets have been performed to elucidate the reaction mechanism of the ion-molecule reaction, C2H2(1Sigmag+) + O+(4S), for which collision experiment has been performed by Chiu et al. [J. Chem. Phys. 109, 5300 (1998)]. The minor low-energy process leading to the weak spin-forbidden product C2H2+ (2Piu) + O(1D) has been studied previously and will not be discussed here. The major pathways to form charge-transfer (CT) products, C2H2+ (2Piu) + O(3P) (CT1) and C2H2+ (4A2) + O(3P) (CT2), and the covalently bound intermediates are investigated. The approach of the oxygen atom cation to acetylene goes over an energy barrier TS1 of 29 kcal/mol (relative to the reactant) and adiabatically leads the CT2 product or a weakly bound intermediate Int1 between CT2 products. This transition state TS1 is caused by the avoided crossing between the reactant and CT2 electronic states. As the C-O distance becomes shorter beyond the above intermediate, the C1 reaction pathway is energetically more favorable than the Cs pathway and goes over the second transition state TS2 of a relative energy of 39 kcal/mol. Although this TS connects diabatically to the covalent intermediate Int2, there are many states that interact adiabatically with this diabatic state and these lead to the other charge-transfer product CT1 via either of several nonadiabatic transitions. These findings are consistent with the experiment, in which charge transfer and chemical reaction products are detected above 35 and 39 kcal/mol collision energies, respectively.

A state-specific approach to multireference coupled electron-pair approximation like methods: development and applications.

The traditional multireference (MR) coupled-cluster (CC) methods based on the effective Hamiltonian are often beset by the problem of intruder states, and are not suitable for studying potential energy surface (PES) involving real or avoided curve crossing. State-specific MR-based approaches obviate this limitation. The state-specific MRCC (SS-MRCC) method developed some years ago can handle quasidegeneracy of varying degrees over a wide range of PES, including regions of real or avoided curve-crossing. Motivated by its success, we have suggested and explored in this paper a suite of physically motivated coupled electron-pair approximations (SS-MRCEPA) like methods, which are designed to capture the essential strength of the parent SS-MRCC method without significant sacrificing its accuracy. These SS-MRCEPA theories, like their CC counterparts, are based on complete active space, treat all the reference functions on the same footing and provide a description of potentially uniform precision of PES of states with varying MR character. The combining coefficients of the reference functions are self-consistently determined along with the cluster amplitudes themselves. The newly developed SS-MRCEPA methods are size-extensive, and are also size-consistent with localized orbitals. Among the various versions, there are two which are invariant with respect to the restricted rotations among doubly occupied and active orbitals separately. Similarity of performance of this latter and the noninvariant versions at the crossing points of the degenerate orbitals imply that the all the methods presented are rather robust with respect to the rotations among degenerate orbitals. Illustrative numerical applications are presented for PES of the ground state of a number of difficult test cases such as the model H4, H(8) problems, the insertion of Be into H(2), and Li(2), where intruders exist and for a state of a molecule such as CH(2), with pronounced MR character. Results obtained with SS-MRCEPA methods are found to be comparable in accuracy to the parent SS-MRCC and FCI/large scale CI results throughout the PES, which indicates the efficacy of our SS-MRCEPA methods over a wide range of geometries, despite their neglect of a host of complicated nonlinear terms, even when the traditional MR-based methods based on effective Hamiltonians fail due to intruders.

A systematic theoretical investigation of the lowest valence- and Rydberg-excited singlet states of trans-butadiene. The character of the 1 1 B u (V) state revisited

In the present work systematic procedures for the balanced description of the lowest singlet excited valence and Rydberg states of butadiene, especially for the correct description of the 11B u state, are presented. In the first step of the calculation averaged natural orbitals (ANOs) were computed from the density matrices of the ground state, the 21A g and the 11B u states. For the 11B u state the configuration interaction (CI) wave function used for the computation of the respective density matrix contained all possible single and double excitations from the 1b g (π) orbital into all virtual orbitals and double excitations describing σ–π electron correlation. For the ground and 21A g states a standard multireference (MR) CI with singles and doubles (CISD)/complete-active-space (CAS)(4,4) wave function was used. In the second step, these ANOs were used in extended MR-CISD, MR-CISD with Davidson correction and MR averaged quadratic coupled cluster calculations. This scheme was also extended to state-averaging including the four lowest Rydberg states 11B g (3s), 11A u (3p σ ), 21A u (3p σ ) and 21B u (3p π). Our best value for the vertical excitation energy to the 11B u state is 6.18 eV, close to previous equation-of-motion coupled-cluster with singles and doubles including noniterative triples [EOM - CCSD( )] and complete-active-space perturbation theory to second order (CASPT2) results, but significantly lower than most of the previous MR-CI and MR-CI based results. The computed vertical excitation energy to the 21A g state of 6.55 eV is significantly below previous EOM-CCSD(T) and EOM - CCSD( ) results and demonstrates the deficiencies of these methods in the case of MR situations. On the other hand, this excitation energy is larger than previous CASPT2 results for the 21A g state. The character of the 11B u state is predominantly of valence character, but is more diffuse than the ground state. <x 2>values for the 11B u state range between 25.4 and 26.3a 0 2 in the three-state calculations.

Can We Avoid the Intruder-State Problems in the State-Universal Coupled-Cluster Approaches While Preserving Size Extensivity?

Following the analysis of principal bottlenecks in the extension of the single-reference (SR) coupled-cluster (CC) methodology to the multireference (MR) case, we review and discuss some recent developments that facilitate the use of general model spaces (GMSs) within the state universal (SU) or Hilbert space MR CC formalism. The use of a GMS improves our ability to avoid the intruder state problems. This feature is further enhanced by generalizing the idea of the externally corrected (ec) SR CC formalism to the MR situations. In this latter approach we employ the cluster analysis to extract the most important higher-than-pair cluster amplitudes from a suitable set of known wave functions. Similarly to the SR case, the most convenient external source is represented by wave functions that are obtained via a modest size MR configuration interaction (CI), which employs an N-dimensional reference space. The resulting higher-than-pair cluster amplitudes are subsequently used in the SU CCSD method that is based on an M-dimensional GMS avoiding intruders. We discuss general aspects of these developments from various viewpoints and provide selective illustrations of the key concepts and ideas.

An improved determination of the equilibrium structure and molecular properties of XBS and XCP (X = H, F, Cl) molecules from ab initio calculations

The availability of core-valence basis sets for second-row atoms together with highly accurate coupled-cluster calculations allow us to present an improved determination of the equilibrium structure of the XBS and XCP (X = H, F, Cl) molecules. Core correlation effects and basis set incompleteness have been taken into account in order to obtain best estimates of their equilibrium geometries: results which can be quantitatively compared to experiment are reported. In addition, two important molecular properties, namely the dipole moment and the nuclear quadrupole coupling constants, have been investigated. The molecular dipole moment has been calculated at coupled-cluster level using bases of different quality including diffuse functions and taking into account both core correlation effects and basis set incompleteness. The quadrupole coupling constants, evaluated from the electric field gradient at the quadrupolar nuclei, have been computed at the multi reference configuration interaction, Møller–Plesset perturbation to second order, and coupled-cluster levels of theory employing large core-valence basis sets.

In Pursuit of the Elusive Bond-Stretch Isomers by ab Initio MethodsBenzocyclobutene, Benzo[1,2:4,5]dicyclobutadiene, and Some Related Substituted Systems

The problem of bond-stretch isomerism in benzocyclobutene (1) and benzo[1,2:4,5]- dicyclobutadiene is addressed by single-state (SS-CASSCF and SS-CASPT2) as well by two-state (TWS-CASSCF and TWS-CASPT2) theoretical formalisms. It is shown beyond reasonable doubt that benzocyclobutene 1 does not exhibit bond-stretch isomerism. The situation is more complex and interesting in the case of benzo[1,2:4,5]-dicyclobutadiene. The TWS-CASPT2 ( π ) + σ approach, based on the TWS-CASSCF(10,10) π zeroth-order wave function, yields two bond-stretch isomers 2 and 3. Their total energies, including the ZPVE contributions, indicate that the structure 3 is slightly favored. Because they are separated by an almost vanishing potential energy barrier (0.4-1.0 kcal/mol), it is concluded that 3 prevails and that bond-stretch isomerism probably does not occur in the case of benzo[1,2:4,5]dicyclobutadiene either. However, the final asnwer will be given only by high level multireference (MR) ab initio calculations. It is important to stress that 3 exhibits a pattern of 10π electrons fully delocalized over the CC bonds of the molecular perimeter and thus represents a planar pseudo-[10]annulene system par excellence. Finally, it is argued that strategic substitution of benzo[1,2:4,5]-dicyclobutadiene might lead to a discrimination of the isomers 2 and 3 and to the isolation of some judiciously selected derivatives in the laboratory. In particular, it is shown that a number of substituted pseudo-[10]annulene systems are possible.