Inclusion Of Triple Excitations Research Articles

Analytic Gradients for Equation-of-Motion Coupled Cluster with Single, Double, and Perturbative Triple Excitations.

Understanding the process of molecular photoexcitation is crucial in various fields, including drug development, materials science, photovoltaics, and more. The electronic vertical excitation energy is a critical property, for example in determining the singlet-triplet gap of chromophores. However, a full understanding of excited-state processes requires additional explorations of the excited-state potential energy surface and electronic properties, which is greatly aided by the availability of analytic energy gradients. Owing to its robust high accuracy over a wide range of chemical problems, equation-of-motion coupled cluster with single and double excitations (EOM-CCSD) is a powerful method for predicting excited-state properties, and the implementation of analytic gradients of many EOM-CCSD variants (excitation energies, ionization potentials, electron attachment energies, etc.) along with numerous successful applications highlights the flexibility of the method. In specific cases where a higher level of accuracy is needed or in more complex electronic structures, the inclusion of triple excitations becomes essential, for example, in the EOM-CCSD* approach of Saeh and Stanton. In this work, we derive and implement for the first time the analytic gradients of EOMEE-CCSD*, which also provides a template for analytic gradients of related excited-state methods with perturbative triple excitations. The capabilities of analytic EOMEE-CCSD* gradients are illustrated by several representative examples.

Multicomponent MP4 and the inclusion of triple excitations in multicomponent many-body methods.

This study implements the full multicomponent third-order (MP3) and fourth-order (MP4) many-body perturbation theory methods for the first time. Previous multicomponent studies have only implemented a subset of the full contributions, and the present implementation is the first multicomponent many-body method to include any connected triples contribution to the electron-proton correlation energy. The multicomponent MP3 method is shown to be comparable in accuracy to the multicomponent coupled-cluster doubles method for the calculation of proton affinities, while the multicomponent MP4 method is of similar accuracy as the multicomponent coupled-cluster singles and doubles method. From the results in this study, it is hypothesized that the relative accuracy of multicomponent methods is more similar to their single-component counterparts than previously assumed. It is demonstrated that for multicomponent MP4, the fourth-order triple-excitation contributions can be split into electron-electron and electron-proton contributions and the electron-electron contributions ignored with very little loss of accuracy of protonic properties.

The role of high-order electron correlation effects in a model system for non-valence correlation-bound anions.

The diffusion Monte Carlo (DMC), auxiliary field quantum Monte Carlo (AFQMC), and equation-of-motion coupled cluster (EOM-CC) methods are used to calculate the electron binding energy (EBE) of the non-valence anion state of a model (H2O)4 cluster. Two geometries are considered, one at which the anion is unbound and the other at which it is bound in the Hartree-Fock (HF) approximation. It is demonstrated that DMC calculations can recover from the use of a HF trial wave function that has collapsed onto a discretized continuum solution, although larger EBEs are obtained when using a trial wave function for the anion that provides a more realistic description of the charge distribution and, hence, of the nodal surface. For the geometry at which the cluster has a non-valence correlation-bound anion, both the inclusion of triples in the EOM-CC method and the inclusion of supplemental diffuse d functions in the basis set are important. DMC calculations with suitable trial wave functions give EBE values in good agreement with our best estimate EOM-CC result. AFQMC using a trial wave function for the anion with a realistic electron density gives a value of the EBE nearly identical to the EOM-CC result when using the same basis set. For the geometry at which the anion is bound in the HF approximation, the inclusion of triple excitations in the EOM-CC calculations is much less important. The best estimate EOM-CC EBE value is in good agreement with the results of DMC calculations with appropriate trial wave functions.

CCSD[T] Describes Noncovalent Interactions Better than the CCSD(T), CCSD(TQ), and CCSDT Methods.

The CCSD(T) method is often called the "gold standard" of computational chemistry, because it is one of the most accurate methods applicable to reasonably large molecules. It is particularly useful for the description of noncovalent interactions where the inclusion of triple excitations is necessary for achieving a satisfactory accuracy. While it is widely used as a benchmark, the accuracy of CCSD(T) interaction energies has not been reliably quantified yet against more accurate calculations. In this work, we compare the CCSD[T], CCSD(T), and CCSD(TQ) noniterative methods with full CCSDTQ and CCSDT(Q) calculations. We investigate various types of noncovalent complexes [hydrogen-bonded (water dimer, ammonia dimer, water ··· ammonia), dispersion-bound (methane dimer, methane ··· ammonia), and π-π stacked (ethene dimer)] using various coupled-clusters schemes up to CCSDTQ in 6-31G*(0.25), 6-31G**(0.25, 0.15), and aug-cc-pVDZ basis sets. We show that CCSDT(Q) reproduces the CCSDTQ results almost exactly and can thus serve as a benchmark in the cases where CCSDTQ calculations are not feasible. Surprisingly, the CCSD[T] method provides better agreement with the benchmark values than the other noniterative analogs, CCSD(T) and CCSD(TQ), and even than the much more expensive iterative CCSDT scheme. The CCSD[T] interaction energies differ from the benchmark data by less than 5 cal/mol on average (for all complexes and all basis sets), whereas the error of CCSD(T) is 9 cal/mol. In larger systems, the difference between these two methods can grow by as much as 0.15 kcal/mol. While this effect can be explained only as an error compensation, the CCSD[T] method certainly deserves more attention in accurate calculations of noncovalent interactions.

Coupled-cluster response theory for near-edge x-ray-absorption fine structure of atoms and molecules

Based on an asymmetric Lanczos-chain subspace algorithm, damped coupled cluster linear response functions have been implemented for the hierarchy of coupled cluster (CC) models including CC with single excitations (CCS), CC2, CC with single and double excitations (CCSD), and CCSD with noniterative triple corrected excitation energies CCSDR(3). This work is a first step toward the extension of these theoretical electronic structure methods of well-established high accuracy in UV-vis absorption spectroscopies to applications concerned with x-ray radiation. From the imaginary part of the linear response function, the near $K$-edge x-ray absorption spectra of neon, water, and carbon monoxide are determined and compared with experiment. Results at the CCSD level show relative peak intensities in good agreement with experiment with discrepancies in transition energies due to incomplete treatment of electronic relaxation and correlation that amount to 1--2 eV. With inclusion of triple excitations, errors in energetics are less than 0.9 eV and thereby capturing 90$%$, 95$%$, and 98$%$ of the relaxation-correlation energies for C, O, and Ne, respectively.

Pilot applications of internally contracted multireference coupled cluster theory, and how to choose the cluster operator properly

The internally contracted multireference coupled cluster (icMRCC) method allows a highly accurate description of both static and dynamic correlation with a computational scaling similar to single reference coupled cluster theory. The authors show that the method can lose its orbital invariance and size consistency when no special care is taken in the elimination of redundant excitations. Using the BeH(2) model system, four schemes are compared which differ in their treatment of linear dependencies between excitations of different rank (such as between singles and doubles). While the energy curves agree within tens of μE(h) when truncating the cluster operator at double excitations (icMRCCSD), inclusion of triple excitations (icMRCCSDT) leads to significant differences of more than 1 mE(h). One scheme clearly yields the best results, while the others even turn out to be not size consistent. The former procedure uses genuine single and double excitations and discards those linear combinations of (spectator) double and triple excitations which have the same effect on the reference function. With this approach, the equilibrium structure and harmonic vibrational frequencies of ozone obtained with icMRCCSDT are in excellent agreement with CCSDTQ. The authors further apply icMRCC methods to potential energy surfaces of HF, LiF, N(2), and to the singlet-triplet splitting of benzynes. In particular, the latter calculations have been made possible by implementing the method with the proper formal scaling using automated techniques.

Ground state and vertical excitation energies of the diazene isomers with the coupled cluster method

AbstractState‐of‐the‐art coupled cluster methods were used to study ground and excited states of the diazene isomers. Three coupled cluster variants were applied in the ground state calculations: CCSD, CCSDT, and CCSDT(Qf) for basis sets up to aug‐cc‐pV5Z. The ground state equilibrium geometry was found using combination of the analytical and numerical gradients. The computed values compare very well with experimental data if the latter are available. The vertical excitation energies were obtained with the EOM‐CCSD and EOM‐CCSDT approaches for the same set of basis sets. The importance of the connected triple excitations in the accurate evaluation of the excitation energy is discussed. Out of the four lowest lying excited states studied for each isomer at least one belongs to the category for which an inclusion of triple excitations is crucial for its adequate description. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Calculation of frequency-dependent hyperpolarizabilities using general coupled-cluster models

By exploiting the similarities between response theory and analytic derivative theory, we present a scheme for calculating frequency-dependent hyperpolarizabilities at the coupled-cluster level within the framework for analytic third derivatives. This has been implemented for arbitrary levels of coupled-cluster theory up to the full-configuration-interaction limit. An investigation of some small molecules shows that the inclusion of triple excitations is essential for an accurate description of hyperpolarizabilities.

Ab Initio Study of Bonding between Nucleophilic Species and the Nitroso Group

The bonding between anionic nucleophiles and the nitroso group has been studied in the common nitrosating agents nitroso chloride (ONCl), nitroso bromide (ONBr), nitroso thiocyanate (ONSCN), and dinitrogen trioxide (N2O3) in aqueous solution. A variety of theoretical methods were employed, including ab initio, density functional theory (DFT), and composite theoretical techniques, with solvent effects described using the polarizable continuum model (PCM). Experimental nitroso bond heterolytic dissociation free energies were accurately reproduced with a number of composite theoretical methods, the most successful being CBS-Q and G2MP2, with average errors of 3.1 and 3.4 kJ mol(-1), respectively. Using the MP2 and B3LYP methods, calculations were made with correlation consistent basis sets up to quadruple-zeta, extrapolated to the complete basis set (CBS) limit. The MP2/CBS calculations were accurate to around 10 kJ mol(-1), while the B3LYP/CBS calculations routinely overpredicted experimental bond free energies by ca. 40 kJ mol(-1). It is therefore highly recommended that B3LYP energies are not used for nitroso compounds, although other results demonstrate that the B3LYP method provides a good account of nitroso compound geometries, frequencies, and entropies. Single-point CBS energy calculations using MP2/aug-cc-pVQZ geometries and frequencies showed that the MP4(SDTQ) and QCISD(T) methods provide a slight improvement over MP2 at the CBS limit, although the inclusion of triple excitations is necessary to achieve this improvement in accuracy. Enthalpy-entropy compensation was also discovered, with an average isoequilibrium temperature of 825 K. This relatively large isoequilibrium temperature indicates that enthalpic effects dominate over entropic ones.

The structures of m-benzyne and tetrafluoro-m-benzyne

The structures of m-benzyne and its fluorinated derivative, tetrafluoro-m-benzyne, were investigated using coupled cluster methods including triple excitations [CCSD(T) and CCSDT], different reference wave functions (spin-restricted Hartree-Fock, spin-unrestricted Hartree-Fock, and Brueckner), and different basis sets [6-31G(d,p) and correlation-consistent valence triple-zeta (cc-pVTZ)]. The inclusion of triple excitations in conjunction with d- and f-type polarization functions is paramount to correctly describe through-bond delocalization of the monocyclic form. At the highest level of theory, the C1-C3 distance of the minimum energy form of m-benzyne is 2.0 A and the profile of the potential energy surface along the C1-C3 distance is that of an asymmetric, single well, in agreement with previous density-functional theory and coupled cluster studies. In addition, the calculated CCSD(T) fundamental frequencies are in excellent agreement with the measured infrared frequencies, thus confirming the monocyclic form of m-benzyne. For tetrafluoro-m-benzyne, however, the increased eclipsing strain between the ring-external C-X bonds stabilizes the bicyclo[3.1.0]hexatriene form: the C1-C3 distance is calculated at the CCSD(T)/cc-pVTZ level to be approximately 1.75 A, which is in the range of elongated CC bonds. Computed harmonic vibrational frequencies compare reasonably well with the experimental neon-matrix difference spectrum and provide further evidence for the existence of a bicyclic form.

Potential energy of ?H 2 - (2? u + in the bound-state region

An accurate calculation of the lowest negative electronic state of ∞H 2 - (fixed nuclei) is reported using the CCSD(T) method and doubly augmented cc-pv5z basis set. Comparison has been made with the reference data by Senekowitsch et al. [Chem. Phys. Lett. 111 (1984) 211]. Owing to larger size of the basisset and inclusion of triple excitations, no vertical shift in this work is necessary to reproduce the asymptotics of ∞H + ∞H -. In addition, the effect of basis-set truncation is estimated, based on the complete-basis-set extrapolation method. The contribution of correlated electron-proton motion to the electron-energy curve for H2 − dynamics is pointed out.

Electrostatic interactions between molecules from relaxed one-electron density matrices of the coupled cluster singles and doubles model

The influence of electron correlation on the electrostatic interaction between closed shell molecules is studied using the relaxed electron densities of the coupled cluster singles and doubles (CCSD) model. The corresponding CCSD one-electron density matrices are efficiently computed without full four-index transformation by employing the generalized exchange and Coulomb operator technique. Using several representative van der Waals and hydrogen bonded complexes it was found that in most cases the convergence of the Møller-Plesset expansion of the electrostatic energy, restricted to single, double and quadruple excitations, is satisfactory and the fourth-order triple excitation term is more important than the sum of the fifth- and higher-order contributions from CCSD theory. The importance of the CCSD correlation correction to the electrostatic energy was gauged by comparison of the total interaction energy computed by symmetry-adapted perturbation theory (SAPT) and by the super-molecular CCSD(T) approach (coupled cluster singles and doubles model with a non-iterative inclusion of triple excitations). Except for the CO and N2 dimers, very good agreement between the two sets of results is observed. For the difficult case of the CO dimer the difference between the SAPT and CCSD(T) results can be explained by the truncation of the SAPT expansion for the dispersion energy at second order in the intramonomer correlation operator.

Almost variational coupled cluster theory

The energy expectation value of coupled-cluster theory can be formulated as the sum of the energy expression of traditional coupled-cluster (TCC) theory plus a correction term. The latter is simplified if the stationarity conditions of TCC hold. It is then of O(S 4), where S is the coupled-cluster amplitude. The leading error contribution agrees with the leading term of the difference between the TCC energy and the energy expression of extended coupled-cluster (ECC) theory. It is suggested to evaluate this routinely at the end of any TCC calculation as a check of the reliability of the latter. The error of the ECC energy expression with respect to an expectation value is of O(S 6). The error of traditional CCSD with respect to an expectation value is not affected by the inclusion of triple excitations in CCSDT. Approximations to CCSDT are also discussed. A hierarchy of approximations starting from TCC and ending at variational coupled-cluster (VCC) theory, alternative to the previously proposed improved c...

Almost variational coupled cluster theory

The energy expectation value of coupled-cluster theory can be formulated as the sum of the energy expression of traditional coupled-cluster (TCC) theory plus a correction term. The latter is simplified if the stationarity conditions of TCC hold. It is then of O(S 4), where S is the coupled-cluster amplitude. The leading error contribution agrees with the leading term of the difference between the TCC energy and the energy expression of extended coupled-cluster (ECC) theory. It is suggested to evaluate this routinely at the end of any TCC calculation as a check of the reliability of the latter. The error of the ECC energy expression with respect to an expectation value is of O(S 6). The error of traditional CCSD with respect to an expectation value is not affected by the inclusion of triple excitations in CCSDT. Approximations to CCSDT are also discussed. A hierarchy of approximations starting from TCC and ending at variational coupled-cluster (VCC) theory, alternative to the previously proposed improved coupled-cluster (ICC) method is presented.

Adiabatic electron affinities of PF5 and SF6: a coupled-cluster study 5 6

Adiabatic electron affinities (E ea,ad) of the PF5 and SF6 molecules, notoriously difficult for modern density-functional-based methods, are computed at the coupled-cluster method with all singles and doubles and non-iterative inclusion of triple excitations (CCSD(T)) level with rather large basis sets. Our CCSD(T) estimates of 1.02 and 0.92eV for the E ea,ad of PF5 and SF6, respectively, are in good agreement with the corresponding experimental values of 0.75±0.15 and 1.0±0.1 eV. The computed vertical detachment energy of 3.13eV of the SF6 - anion is to be compared to the experimental value of 3.16 eV.

Assessment of Procedures for Calculating Radical Hyperfine Structures

The effects of different basis sets and computational methods on calculated isotropic hyperfine couplings have been investigated for a set of representative small radicals (OH, H2O+, CN, HCN-, FCN-, HCCH-, CH3, CH4+, NH2, NO2, and H2CO+). Particular emphasis has been placed on the performance of the QCISD approach, when used in combination with moderately large basis sets. It is found that the 6-311+G(2df,p) basis set generally gives good results and that the IGLO-III basis set performs nearly as well. The cc-pVXZ and aug-cc-pVXZ basis sets, on the other hand, display large and unpredictable fluctuations in hyperfine couplings even at the cc-pVQZ level. As noted previously, the reason for this erroneous behavior can be traced to the contraction of the s-shell. The error due to the unbalanced nature of the pVXZ basis sets is greatly reduced on going to the core-valence correlated aug-cc-pCVXZ sets. The calculated hyperfine coupling constants are very sensitive to changes in geometry. In turn, the geometries of radical anion systems in particular are sensitive to level of theory. The 6-311+G(2df,p) basis set has also been tested with other spin-unrestricted methods (UHF, UMP2, UQCID, and five DFT functionals), but none of these are found to perform comparably to QCISD. Inclusion of triple excitations (QCISD(T)) leads to hyperfine couplings that generally lie within 2−3 G of the QCISD results.

Oxirene: To Be or Not To Be?

The C{sub 2v}-symmetry structure of oxirene has been examined using ab initio molecular orbital calculations with large basis sets and a variety of methods of including electron correlation. Different qualitative conclusions regarding the nature of oxirene are reached, depending on the choice of basis set and method of electron correlation incorporation. With certain combinations of basis set and theoretical method, the symmetric oxirene structure is found to be a saddle point on the potential energy surface, while for other combinations oxirene is a local minimum. Inclusion of triple excitations in the correlation treatment has a large effect, tending to make the curvature of the surface corresponding to a ring-opening distortion of the C{sub 2v}-symmetry structure less positive (or more negative). This is counterbalanced by a basis set effect, with inclusion of functions making the curvature more positive. At our highest level of theory, CCSD(T) with basis sets of triple-{zeta} quality and including multiple d shells and an f shell on C and O and multiple p shells and a d shell on H, oxirene is a genuine minimum under harmonic vibrational analysis, with a ring-opening frequency of 139-163 cm{sup -1}. 72 refs., 2 tabs.

Linear and cyclic isomers of C4. A theoretical study with coupled-cluster methods and large basis sets

The ground electronic states of linear and rhombic C4 have been studied by high level ab initio quantum chemical techniques. Geometries, harmonic vibrational frequencies, infrared intensities, and other quantities have been determined using 4s3p2d1f correlation consistent basis sets and coupled-cluster methods including triple excitations. The linear–rhombic isomer energy difference has been investigated with a range of basis sets, including a 5s4p3d2f1g correlation consistent set. The linear–rhombic energy difference is influenced significantly by basis set, presence of triple excitations, and the choice of reference function for the open-shell linear isomer. The effect of basis set variation is complex, but once a reasonable quality of basis set has been achieved, further extensions favor the rhombic isomer. The inclusion of triple excitations also favors the rhombic isomer. The use of a restricted Hartree–Fock reference function for the linear isomer yields higher energies at the coupled-cluster level than if an unrestricted Hartree–Fock reference function is used, thereby again favoring the rhombic isomer. The most complete calculations of this study [coupled-cluster singles and doubles with noniterative triples (CCSD(T)) with a 5s4p3d2f1g basis set] indicate that the rhombic isomer is preferred by about 1 kcal mol−1. The coupled-cluster vibrational frequencies of the linear isomer are all real, in agreement with previous work, indicating that this isomer is not bent in the gas phase. The infrared intensities of linear C4 obtained in this work differ significantly from those obtained previously with smaller basis sets and either self-consistent field theory or second-order perturbation theory. The present calculations give a dissociation energy of C4 of 433 kcal mol−1, which is close to a previous value obtained with the aid of an empirical correction, and implies that several experimental estimates of the heat of formation of C4 are unreliable. Electron detachment energies of linear C4− and electron affinities of C4 are computed with larger basis sets than previously and are in very good agreement with recent anion photoelectron data.

Hilbert space multireference coupled-cluster methods. II. A model study on H8

The performance of various coupled-cluster (CC) approaches using both single and multideterminantal references is investigated for the (quasi-)degenerate states of molecular systems, where inclusion of higher excitations (or equivalently nondynamic correlation) proves to be needed. The prototype system H8 represents an adequate model for our study, where we can vary the degree of degeneracy from a completely degenerate situation to a nondegenerate one in a continuous way. To obtain a reliable benchmark for our CC results, the full configuration interaction (FCI) and large-scale complete active space configuration interaction (CAS CI) calculations, respectively, are performed for a variety of geometries and states. The convergence of the approximate single reference CC approaches is found to be extremely sensitive to the level of degeneracies involved. In the nondegenerate case the standard CC method with single and double excitations is found to be quite satisfactory; in the (quasi-)degenerate situations, however, the inclusion of triple excitations and noniterative quadruple excitations is needed to furnish semiquantitative values of correlation energies. The alternative treatment of nondynamic correlation using a multideterminantal Hilbert space coupled-cluster (MRCC) method demonstrates the power of this approach, which provides a balanced description of both dynamic and nondynamic correlation in the degenerate region for all the investigated states of H8. Its convergence for nondegenerate situations, however, is less satisfactory, being affected by an intruder state problem.

Classical and nonclassical forms of the vinyl cation: A coupled cluster study

The energy difference between the classical and nonclassical forms of the vinyl cation is studied using the single and double excitation configuration interaction (CISD) and coupled cluster methods (CCSD and CCSDT-1). The basis sets employed range from double ζ to those including f functions on carbon and d functions on hydrogen. It is shown that the theoretical energy differences are very sensitive to the inclusion of higher-order polarization functions and the inclusion of triple excitations in the correlated wave functions. The results are compared with other theoretical studies in the literature and with the very recent experimental work of Crofton, Jagod, Rehfuss, and Oka. The most reliable single theoretical treatment involves the CCSDT-1 method with basis set C(10s5p2d1f/4s3p2d1f), H(5s2p/3s2p), and predicts 3.3 kcal/mol for the classical/nonclassical energy difference. Basis set extension is expected to increase this separation to about 4.0 kcal/mol, still less than the experimental value, >6 kcal/mol.