Addition by subtraction in coupled cluster theory. II. Equation-of-motion coupled cluster method for excited, ionized, and electron-attached states based on the nCC ground state wave function

New iterative double and triple excitation corrections to the equation-of-motion coupled cluster (EOM-CC) based upon the recently developed nCC methods [Bartlett and Musia?, J. Chem. Phys. 125, 204105-1 (2006)] are applied to excitation energies (EEs), ionization potentials (IPs), and electron affinities (EAs). The methods have been tested by the evaluation of the vertical EEs, IPs, and EAs for Ne, BH, CH(2), H(2)O, N(2), C(2), CH(+), CO, and C(2)H(4) compared to full configuration interaction, EOM-CCSD, EOM-CCSDT, and experimental data.     

Multireference coupled-cluster theory: the easy way

The multi-ionization equation-of-motion coupled-cluster (CC) method is developed for multireference (MR) problems. It is operationally single reference, depending upon a formal matrix diagonalization step to define the coefficients in the wavefunction in an unbiased way that allows for important MR character. The method is illustrated for the autoisomerization of cyclobutadiene, which has a very large multireference effect and compared to other MR-CC results. The newly implemented methods are also used to obtain the vertical double ionization (DI) potentials of several small molecules (H(2)O, CO, C(2)H(2), C(2)H(4)). Also, the performance of the new methods is analyzed by plotting the potential energy curve for twisted ethylene as a function of a dihedral angle between two methylenes. Evaluation of the total molecular energy via MR-DI-CC calculations makes it possible to avoid an unphysical cusp.     

Error analysis and improvements of coupled-cluster theory

Summary The error in the energy of the traditional coupled-cluster (TCC) approach and of several variants is analyzed in terms of the error of the cluster operatorS. A key feature of this analysis is that TCC can be based on an energy functional (asymmetric inS andS ) that is made stationary with respect to variation ofS . The error of TCC scales with the particle numbern, but it is not quadratic in . An improved coupled-cluster method (ICC) is presented that is the next step in a hierarchy from TCC to an exact variational theory. An alternative hierarchy is possible that leads to the extended coupled-cluster (ECC) method of Arponen. Variational (VCC) and unitary (UCC) coupled cluster theories and their stationary conditions and errors are analyzed along similar lines and practicable VCC or UCC approaches are presented. An infinite summation of certain terms in the VCC expectation value is shown to lead to a coupled-pair functional of the type proposed by Ahlrichs. The various CC schemes discussed here are compared on the CC-D, CC-SD and CC-SDT levels and beyond this. Special aspects referring to properties are also discussed.     

Analysis of a failure of the CC2 coupled-cluster method for bond lengths of SnO and PbO

CC2 model is found to overestimate bond lengths of SnO and PbO by about 0.25 Å, while both second-order Møller–Plesset perturbation theory and coupled-cluster singles and doubles give reasonable results. Previously, analysis shows that the [[U, T ₁], T ₁] term in the doubles equation of CC2 is the origin of failure for CC2 and some truncated CC models have been suggested to achieve reasonable result for ozone, where CC2 is unable to obtain a stable structure. However, these remedies are unable to afford reasonable bond lengths of SnO and PbO. Based on a term-wise analysis, our results indicate that the [U, T ₁] term results in failure of CC2. CC2 model by removing this term will provide results that agree well with those of MP2. Furthermore, the [[U, T ₂], T ₁] term absent in the CC2 while present in doubles equation of CCSD can balance this [U, T ₁] term and CC2 model augmented with this term is able to afford reasonable results for PbO, SnO and ozone.     

Generalization of coupled-cluster response theory to multireference expansion spaces: application of the coupled-cluster singles and doubles effective Hamiltonian

Employing separate cluster ansatz in time-independent and time-dependent wave-operators, coupled-cluster (CC) response theory is generalized to multireference (MR) expansion spaces. For state energies, this corresponds to the MR secular problem with an arbitrary similarity-transformed effective Hamiltonian, H˜=Ω⁻¹ HΩ. The effective Hamiltonian can be generated via size-extensive CC methods. Thus the states in MR linear response theory (MRLRT) maintain the usual CC core-extensive properties. We have used the Gelfand unitary group basis of the spin-adapted configurations to construct the matrix of H˜ in the MR excitation space. As a preliminary application, the CC singles and doubles effective Hamiltonian is applied to excitation and photoionization energies of the CH⁺ and N₂ molecules, and is compared with experimental results and results from other numerical procedures including conventional CC linear response theory (CC-LRT), MR and full configuration interaction (MRCI and FCI) methods. The numerical results indicate that MRLRT reproduces valence and external excited states quantitatively, combining the best features of CC-LRT and MRCI. Received: 2 July 1998 / Accepted: 28 August 1998 / Published online: 11 November 1998     

Approximate treatment of higher excitations in coupled-cluster theory

The possibilities for the approximate treatment of higher excitations in coupled-cluster (CC) theory are discussed. Potential routes for the generalization of corresponding approximations to lower-level CC methods are analyzed for higher excitations. A general string-based algorithm is presented for the evaluation of the special contractions appearing in the equations specific to those approximate CC models. It is demonstrated that several iterative and noniterative approximations to higher excitations can be efficiently implemented with the aid of our algorithm and that the coding effort is mostly reduced to the generation of the corresponding formulas. The performance of the proposed and implemented methods for total energies is assessed with special regard to quadruple and pentuple excitations. The applicability of our approach is illustrated by benchmark calculations for the butadiene molecule. Our results demonstrate that the proposed algorithm enables us to consider the effect of quadruple excitations for molecular systems consisting of up to 10-12 atoms.     

CC Bond Formation by Addition of Carbenium Ions to Alkenes: Kinetics and Mechanism

The addition of carbenium ions to CC double bonds, a key step in many syntheses in organic and macromolecular chemistry, is analyzed using the Lewis acid promoted reactions of alkyl chlorides with alkenes as an example. Stereochemical and kinetic experiments suggest that the transition state is slightly bridged and product-like. Rearrangements of the carbenium ions that result from the electrophilic attack can be minimized by adding salts with nucleophilic counter ions. The thermodynamics of the addition reactions are analyzed, and the conditions necessary in order to observe the back reaction (i.e. the Grob fragmentation) are discussed. Multiparameter equations that predict rate constants are derived from kinetic studies on the reactivities of carbenium ions and alkenes. Reactivity-selectivity relationships over a reactivity range that covers eight orders of magnitude show that the structure of the transition state is only changed by variation of substituents in the immediate vicinity of the reaction center.     

pCCSD: parameterized coupled-cluster theory with single and double excitations

The primary characteristics of single reference coupled-cluster (CC) theory are size-extensivity and size-consistency, invariance under orbital rotations of the occupied or virtual space, the exactness of CC theory for N electron systems when the cluster operator is truncated to N-tuple excitations, and the relative insensitivity of CC theory to the choice of the reference determinant. In this work, we propose a continuous class of methods which display the desirable features of the coupled-cluster approach with single and double excitations (CCSD). These methods are closely related to the CCSD method itself and are inspired by the coupled electron pair approximation (CEPA). It is demonstrated that one can systematically improve upon CCSD and obtain geometries, harmonic vibrational frequencies, and total energies from a parameterized version of CCSD or pCCSD(α,β) by selecting a specific member from this continuous family of approaches. In particular, one finds that one such approach, the pCCSD(-1,1) method, is a significant improvement over CCSD for the calculation of equilibrium structures and harmonic frequencies. Moreover, this method behaves surprisingly well in the calculation of potential energy surfaces for single bond dissociation. It appears that this methodology has significant promise for chemical applications and may be particularly useful in applications to larger molecules within the framework of a high accuracy local correlation approach.     

Application of the coupled-cluster theory to atomic frequency-dependent polarizabilities

Summary A time-dependent coupled-cluster approach may be employed to describe dynamic processes of many-electron systems. Atomic properties, such as the frequency-dependent polarizability, can be treated as a response of the system described by the coupled-cluster expansion to an external radiation field. The major difficulty in the realization of such a formalism is to deal with dynamic pair functions. The procedure reported here is to simplify the full set of single- and pair-excitation expansion equations to a subset of equations which includes polarization and relaxation effects to all orders and is solved by using a complete set of discrete basis functions. Calculations of excitation energies and frequency-dependent electric dipole polarizabilities for helium are presented. Application of the procedure to calculate photoionization cross sections is discussed.     

A note on the direct calculation of excitation energies by quasi-degenerate MBPT and coupled-cluster theory

For excitation energy calculations using quasi-degenerate MBPT or coupled-cluster (CC) theory, the hitherto chosen strategy has been to have particle-hole (p-h) determinants forming the model space and to use standard formalisms originally developed for complete model spaces. In view of our recent analysis, the p-h determinants constitute an incomplete model space for which intermediate normalization (IN) of the wave-operator Ω is not appropriate if a linked formulation is desired. The theoretical status of earlier applications which had used formulae appropriate to IN, yet ignored disconnected diagrams, is analyzed. It is shown that if only excitation energies, rather than the full Ω, are desired, then no theoretical error is made. The situation is, however, fortuitous in that for mp-mh model spaces with m > 1, a similar procedure using IN would necessarily generate disconnected diagrams.     

Linear-response theory for Mukherjee's multireference coupled-cluster method: Static and dynamic polarizabilities

The formalism of response theory is applied to derive expressions for static and dynamic polarizabilities within the state-specific multireference coupled-cluster theory suggested by Mukherjee and co-workers (Mk-MRCC) [J. Chem. Phys. 110, 6171 (1998)]. We show that the redundancy problem inherent to Mk-MRCC theory gives rise to spurious poles in the Mk-MRCC response functions, which hampers the reliable calculation of dynamic polarizabilities. Furthermore, we demonstrate that in the case of a symmetry-breaking perturbation a working response theory is obtained only if certain internal excitations are included in the responses of the cluster amplitudes. Exemplary calculations within the singles and doubles approximation (Mk-MRCCSD) are carried out on aryne compounds to illustrate the impact of a multireference ansatz on the polarizability.     

Comparison of MBPT and coupled-cluster methods with full CI. Importance of triplet excitation and infinite summations

Results from full fourth-order perturbation theory [SDTQ MBPT(4)], and the coupled-cluster single- and double-excitation model (CCSD). are compared with recent full CI results for BH, HF, NH₃, and H₂O. For H₂O, studies include large symmetric displacements of the OH bonds, which offer a severe test for any single-reference MBPT/CC method. In every case. CCSD plus fourth-order triple-excitation terms provide agreement with the full CI to < 2 kcal/mole. SDTQ MBPT(4) has an error 10 kcal/mole for displaced H₂O.     

Frozen natural orbital coupled-cluster theory: forces and application to decomposition of nitroethane

The frozen natural orbital (FNO) coupled-cluster method increases the speed of coupled-cluster (CC) calculations by an order of magnitude with no consequential error along a potential energy surface. This method allows the virtual space of a correlated calculation to be reduced by about half, significantly reducing the time spent performing the coupled-cluster (CC) calculation. This paper reports the derivation and implementation of analytical gradients for FNO-CC, including all orbital relaxation for both noncanonical and semicanonical perturbed orbitals. These derivatives introduce several new orbital relaxation contributions to the CC density matrices. FNO-CCSD(T) and FNO-LambdaCCSD(T) are applied to a test set of equilibrium structures, verifying that these methods are capable of reproducing geometries and vibrational frequencies accurately, as well as energies. Several decomposition pathways of nitroethane are investigated using CCSD(T) and LambdaCCSD(T) with 60% of the FNO virtual orbitals in a cc-pVTZ basis, and find differences on the order of 5 kcalmol with reordering of the transition state energies when compared to B3LYP 6-311 + G(3df, 2p).     

Analytic calculation of the diagonal Born-Oppenheimer correction within configuration-interaction and coupled-cluster theory

Schemes for the analytic calculation of the diagonal Born-Oppenheimer correction (DBOC) are formulated and implemented for use with general single-reference configuration-interaction and coupled-cluster wave function models. Calculations are reported to demonstrate the convergence of the DBOC with respect to electron-correlation treatment and basis set as well as to investigate the size-consistency error in configuration-interaction calculations of the DBOC. The importance of electron-correlation contributions to the DBOC is illustrated in the computation of the corresponding corrections for the reaction energy and activation barrier of the F + H2 --> FH + H reaction as well as of the atomization energy for trans-butadiene.     

On the single-root approach within the framework of coupled-cluster theory in Fock space

A thorough formulation of Fock Space Brillouin–Wigner Coupled Cluster method is presented following previous developments [N.D.K. Petraco, Ľ. Horný, H.F. Schaefer, I. Hubač, J. Chem. Phys. 117 (2002) 9580]. The new method is designed to avoid the intruder states problem, and introduces the single-root solution feature which has not been considered yet within valence-universal methods. The explicit equations for the (0,1) sector of the Fock space are introduced.     

Communication: spin-orbit splittings in degenerate open-shell states via Mukherjee's multireference coupled-cluster theory: a measure for the coupling contribution

We propose a generally applicable scheme for the computation of spin-orbit (SO) splittings in degenerate open-shell systems using multireference coupled-cluster (MRCC) theory. As a specific method, Mukherjee's version of MRCC (Mk-MRCC) in conjunction with an effective mean-field SO operator is adapted for this purpose. An expression for the SO splittings is derived and implemented using Mk-MRCC analytic derivative techniques. The computed SO splittings are found to be in satisfactory agreement with experimental data. Due to the symmetry properties of the SO operator, SO splittings can be considered a quality measure for the coupling between reference determinants in Jeziorski-Monkhorst based MRCC methods. We thus provide numerical insights into the coupling problem of Mk-MRCC theory.     

Partial fourth order schemes of triples in Fock-space coupled-cluster theory: Ionization potentials of ozone

In this paper, three lowest vertical ionization potentials of ozone molecule are presented using four different approximate triples-corrected methods, in addition to the full singles and doubles level, within the Fock-space coupled-cluster theory. One of them is third-order correction and the other three are fourth-order triples approximations, two of which are partial fourth-order corrections. Ozone represents a major challenge due to the multi-reference nature. The value of the partial fourth-order approximations has been highlighted. In particular, we identify a reliable and computationally cheap partial fourth-order scheme.     

Excited states in the multireference state-specific coupled-cluster theory with the complete active space reference

The recently proposed multireference state-specific coupled-cluster theory with the complete active space reference has been used to study electronically excited states with different spatial and spin symmetries. The algorithm for the method has been obtained using the computerized approach for automatic generation of coupled-cluster diagrams with an arbitrary level of the electronic excitation from a formal reference determinant. The formal reference is also used to generate the genuine reference state in the form of a linear combination of determinants contracted to a configuration with the spin and spatial symmetries of the target state. The natural-orbital expansions of the one-electron configuration inferaction density matrix allowed us to obtain the most compact orbital space for the expansion of the reference function. We applied our approach in the calculations of singlet and triplet states of different spatial symmetries of the water molecule. The comparisons of the results with values obtained using other many-particle methods and with the full configuration interaction results demonstrate good ability of the approach to deal with electronic excited states.