Triples Model CC3 Research Articles

Noniterative Doubles Corrections to the Random Phase and Higher Random Phase Approximations: Singlet and Triplet Excitation Energies.

The second‐order noniterative doubles‐corrected random phase approximation (RPA) method has been extended to triplet excitation energies and the doubles‐corrected higher RPA method as well as a shifted version for calculating singlet and triplet excitation energies are presented here for the first time. A benchmark set consisting of 20 molecules with a total of 117 singlet and 71 triplet excited states has been used to test the performance of the new methods by comparison with previous results obtained with the second‐order polarization propagator approximation (SOPPA) and the third order approximate coupled cluster singles, doubles and triples model CC3. In general, the second‐order doubles corrections to RPA and HRPA significantly reduce both the mean deviation as well as the standard deviation of the errors compared to the CC3 results. The accuracy of the new methods approaches the accuracy of the SOPPA method while using only 10–60% of the calculation time. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.

Frequency-dependent hyperpolarizabilities of the Ne, Ar, and Kr atoms using the approximate coupled cluster triples model CC3

The frequency-dependent electric field-induced second harmonic generation (ESHG) second hyperpolarizabilities gamma of neon, argon, and krypton are calculated using the approximate coupled cluster triples model CC3. Systematic basis set investigations are carried out to establish basis set limits, and scalar relativistic effects are accounted for by direct perturbation theory. To estimate higher-order correlation effects, full configuration-interaction results are used to benchmark the accuracy of CC3. The best theoretical estimates obtained thereby for the static second hyperpolarizabilities gamma(0) are 107.4, 1159, and 2589 a.u. for neon, argon, and krypton, respectively. These values as well as the results for the dispersion curve of the parallel component gamma( parallel) agree well with the latest experimental values from electric field-induced second harmonic generation. In addition, the dispersion of the perpendicular component gamma( perpendicular) and the hyperpolarizability ratios gamma( parallel)gamma( perpendicular) has been studied for the first time on a consistently correlated ab initio level. The analysis of the results indicates that, in particular for neon and krypton, the presently available experimental values are flawed.

The second hyperpolarizability of the N 2 molecule calculated using the approximate coupled cluster triples model CC3

The static and frequency-dependent hyperpolarizabilities of the N 2 molecule have been calculated using the coupled cluster approximate triples model CC3. The CC3 basis-set limit of the hyperpolarizability has been estimated. Considering the zero-point vibrational correction and neglecting the relativistic contributions (which are found to be small) the best estimate of the static hyperpolarizability is 912.9 ± 4.3 a.u. (the uncertainty of 4.3 a.u. does not account for correlation effects beyond CC3). The effect of triples is important and increases with frequency. The hyperpolarizabilities calculated at the CC3 level agree well with experiment, the discrepancies being of 0.1–2.2%. The largest CC3 calculations employed 326 basis functions.

The hyperpolarizability of the Ne atom in the approximate coupled cluster triples model CC3

The static and frequency-dependent hyperpolarizability of Ne is calculated using the coupled cluster approximate triples model CC3 and a large correlation-consistent basis set. Relativisitic contributions are taken into account through first-order. Analyzing in details the one- and N-electron errors, the uncertainty of the results is estimated to be about one atomic unit, an accuracy which for the first time is obtained for a many electron system in ab initio calculations of frequency-dependent second hyperpolarizabilities. For the static hyperpolarizability of Ne we obtain a best theoretical estimate of 107.6 ± 1.0 a.u. The frequency-dependent hyperpolarizability of Ne is also calculated at the lower levels of the coupled cluster hierarchy – CCS, CC2, and CCSD – and compared to CC3 to illustrate the importance of the inclusion of the triple excitations in coupled cluster wavefunctions for an accurate determination of hyperpolarizabilities. The results at the CC3 level agree well with the latest experimental results.

The hyperpolarizability of the Ne atom in the approximate coupled cluster triples model CC3

The static and frequency-dependent hyperpolarizability of Ne is calculated using the coupled cluster approximate triples model CC3 and a large correlation-consistent basis set. Relativisitic contributions are taken into account through first-order. Analyzing in details the one- and N-electron errors, the uncertainty of the results is estimated to be about one atomic unit, an accuracy which for the first time is obtained for a many electron system in ab initio calculations of frequency-dependent second hyperpolarizabilities. For the static hyperpolarizability of Ne we obtain a best theoretical estimate of 107.6 ± 1.0 a.u. The frequency-dependent hyperpolarizability of Ne is also calculated at the lower levels of the coupled cluster hierarchy – CCS, CC2, and CCSD – and compared to CC3 to illustrate the importance of the inclusion of the triple excitations in coupled cluster wavefunctions for an accurate determination of hyperpolarizabilities. The results at the CC3 level agree well with the latest experimental results.

Gauge invariance of oscillator strengths in the approximate coupled cluster triples model CC3

The gauge invariance of the oscillator strength is examined for the lowest dipole transitions in Ne, N 2 and H 2O using the coupled cluster model hierarchy, CCS, CC2, CCSD, and CC3, and a variety of correlation consistent basis sets. The deviations between the oscillator strengths in the different gauges decrease from CCSD to CC3 (where connected triples are included) by about the same factor as they decrease from CCS to CC2 (where connected doubles are included). The deviations are very similar in CC2 and CCSD. The quality of the oscillator strengths is improved significantly at each level in coupled cluster hierarchy, also from CC2 to CCSD, where the equivalence between the oscillator strengths in the different gauges is very similar.

Calculation of frequency-dependent polarizabilities using the approximate coupled-cluster triples model CC3

The calculation of frequency-dependent polarizabilities is discussed for the iterative approximate coupled-cluster singles, doubles and triples model CC3. A new implementation of the linear response functions is reported, which has the same computational O(N7) scaling as CC3 ground state calculations and uses an explicitly spin-coupled excitation space. Sample calculations are presented for the static and frequency-dependent polarizabilities of Ne and ethylene, as well as for the static polarizabilities of HF. The largest calculation employs the t-aug-cc-pVTZ basis set for ethylene giving a total of 328 basis functions. The results obtained agree well with the experimental data.

Calculation of ground and excited state potential energy curves of the MgAr complex using the coupled cluster approximate triples model CC3

Potential curves of the ground and six excited states of MgAr are analysed using the coupled cluster approximate triples approach CC3. The ground state curve is obtained from conventional closed shell energy calculations, and excitation energies to the electronically excited states are obtained solving the CC3 response eigenvalue equation, yielding the potential curves for the excited states. The computed spectroscopic parameters ( R e, ω e, D 0 and T 0) are in good agreement with experimental data when sufficiently large basis sets are used and the counterpoise correction is applied.

Calculation of first-order one-electron properties using the coupled-cluster approximate triples model CC3

The calculation of first-order one-electron molecular properties is discussed for the iterative approximate coupled-cluster triples model CC3. The implementation which has the computational scaling N7 uses an explicitly spin coupled excitation space. Sample calculations are presented for the quadrupole moments of ethylene where the largest calculation uses an aug-cc-pVQZ basis which contains 344 basis functions.

CC3 triplet excitation energies using an explicit spin coupled excitation space

Triplet excitation energies are derived in the approximate triples model CC3 using an explicit spin coupled triplet excitation space. The explicit spin coupled excitation space gives considerable computational savings compared to the spin–orbital approach. Sample calculations are performed on the Be2 and CH2 systems and the performance of the CC3 results are evaluated from a comparison with full configuration interaction (FCI) results.

The CC3 model: An iterative coupled cluster approach including connected triples

An alternative derivation of many-body perturbation theory (MBPT) has been given, where a coupled cluster parametrization is used for the wave function and the method of undetermined Lagrange multipliers is applied to set up a variational coupled cluster energy expression. In this variational formulation, the nth-order amplitudes determine the energy to order 2n+1 and the nth-order multipliers determine the energy to order 2n+2. We have developed an iterative approximate coupled cluster singles, doubles, and triples model CC3, where the triples amplitudes are correct through second order and the singles amplitudes are treated without approximations due to the unique role of singles as approximate orbital relaxation parameters. The compact energy expressions obtained from the variational formulation exhibit in a simple way the relationship between CC3, CCSDT-1a [Lee et al., J. Chem. Phys. 81, 5906 (1984)] CCSDT-1b models [Urban et al., J. Chem. Phys. 83, 4041 (1985)], and the CCSD(T) model [Raghavachari et al., Chem. Phys. Lett. 157, 479 (1989)]. Sample calculations of total energies are presented for the molecules H2O, C2, CO, and C2H4. Comparisons are made with full CCSDT, CCSDT-1a, CCSDT-1b, CCSD(T), and full configuration interaction (FCI) results. These calculations demonstrate that CC3 and CCSD(T) give total energies of a similar quality. If results obtained by CC3 and CCSD(T) differ significantly, neither method can be trusted. In contrast to CCSD(T), time-dependent response functions can be obtained for CC3.

Response functions in the CC3 iterative triple excitation model

The derivation of response functions for coupled cluster models is discussed in a context where approximations can be introduced in the coupled cluster equations. The linear response function is derived for the approximate coupled cluster singles, doubles, and triples model CC3. The linear response functions for the approximate triples models, CCSDT-1a and CCSDT-1b, are obtained as simplifications to the CC3 linear response function. The consequences of these simplifications are discussed for the evaluation of molecular properties, in particular, for excitation energies. Excitation energies obtained from the linear response eigenvalue equation are analyzed in orders of the fluctuation potential. Double replacement dominated excitations are correct through second order in all the triples models mentioned, whereas they are only correct to first order in the coupled cluster singles and doubles model (CCSD). Single replacement dominated excitation energies are correct through third order in CC3, while in CCSDT-1a, CCSDT-1b, and CCSD they are only correct through second order. Calculations of excitation energies are reported for CH+, N2, and C2H4 to illustrate the accuracy that can be obtained in the various triples models. The CH+ results are compared to full configuration interaction results, the C2H4 results are compared with complete active space second order perturbation theory (CASPT2) and experiment, and the N2 results are compared to experiment. Double replacement dominated excitations are improved significantly relative to CCSD in all the triples models mentioned, and is of the same quality in CC3 and CCSDT-1a. The single replacement dominated excitation are close to full configuration interaction results for the CC3 model and significantly improved relative to CCSD. The CCSDT-1 results for the single replacement dominated excitations are not improved compared to CCSD.