Coupled-cluster Approximation Research Articles

Operator Relationship between Conventional Coupled Cluster and Unitary Coupled Cluster

The chemistry community has long sought the exact relationship between the conventional and the unitary coupled cluster ansatz for a single-reference system, especially given the interest in performing quantum chemistry on quantum computers. In this work, we show how one can use the operator manipulations given by the exponential disentangling identity and the Hadamard lemma to relate the factorized form of the unitary coupled-cluster approximation to a factorized form of the conventional coupled cluster approximation (the factorized form is required, because some amplitudes are operator-valued and do not commute with other terms). By employing the Trotter product formula, one can then relate the factorized form to the standard form of the unitary coupled cluster ansatz. The operator dependence of the factorized form of the coupled cluster approximation can also be removed at the expense of requiring even more higher-rank operators, finally yielding the conventional coupled cluster. The algebraic manipulations of this approach are daunting to carry out by hand, but can be automated on a computer for small enough systems.

Flexibility of the factorized form of the unitary coupled cluster Ansatz.

The factorized form of the unitary coupled cluster Ansatz is a popular state preparation Ansatz for electronic structure calculations of molecules on quantum computers. It is often viewed as an approximation (based on the Trotter product formula) for the conventional unitary coupled cluster operator. In this work, we show that the factorized form is quite flexible, allowing one to range from a conventional configuration interaction, to conventional unitary coupled cluster, to efficient approximations that lie in between these two. The variational minimization of the energy often allows simpler factorized unitary coupled cluster approximations to achieve high accuracy, even if they do not accurately approximate the Trotter product formula. This is similar to how quantum approximate optimization algorithms can achieve high accuracy with a small number of levels.

Decomposition of high-rank factorized unitary coupled-cluster operators using ancilla and multiqubit controlled low-rank counterparts

The factorized form of the unitary coupled-cluster approximation is one of the most promising methodologies to prepare trial states for strongly correlated systems within the variational quantum eigensolver framework. The factorized form of the UCC ansatz can be systematically applied to a reference state to generate the desired entanglement. The difficulty associated with such an approach is the requirement of simultaneously entangling a growing number of qubits, which quickly exceeds the hardware limitations of today's quantum machines. In particular, while circuits for singles and double excitations can be performed on current hardware, higher-rank excitations require too many gate operations. In this work, we propose a set of new schemes that trade off using extra qubits for a reduced gate depth to decompose high-rank UCC excitation operators into significantly lower depth circuits. These results will remain useful even when fault-tolerant machines are available to reduce the overall state-preparation circuit depth.

Limitations of perturbative coupled-cluster approximations for highly accurate investigations of Rb2+

We reveal limitations of several standard coupled-cluster (CC) methods with perturbation-theory based noniterative or approximate iterative treatments of triple excitations when applied to the determination of highly accurate potential energy curves (PECs) of ionic dimers, such as the XΣg+2 electronic ground state of Rb2 +. Such computations are of current interest for the understanding of ion-atom interactions in the ultracold regime. We demonstrate that these CC methods lead to an unphysical long-range barrier for the Rb2 + system. The barrier is small but spoils the long-range behavior of the PEC. The effect is also found for other X2 + systems, such as X = Li, Na, and K. Calculations using a flexible framework for obtaining leading perturbative triples corrections derived using an analytic CC singles and doubles energy derivative formulation demonstrate that the origin of this problem lies in the use of T̂3 amplitudes obtained from approximate CC singles, doubles, and triples amplitude equations. It is shown that the unphysical barrier is related to a symmetry instability of the underlying Hartree-Fock mean-field solution, leading to orbitals representing two +0.5-fold charged ions in the limit of separated fragments. This, in turn, leads to a wrong 1/R asymptote of the interaction potential computed by perturbation-based CC approximations. Physically meaningful perturbative corrections in the long-range tail of the PEC may instead be obtained using symmetry-broken reference determinants.

Polarizability of atomic Pt, Pt+, and Pt.

Electrostatic properties are important for understanding and modeling many phenomena, such as the adsorption of a catalytic metal upon an oxide support. The charge transfer between the metal and the support can lead to positive or negative charges on the metal. Here, the static dipole polarizability is computed for atomic platinum in charge states 0, +1, and -1 in several low-lying electronic terms and levels. Core pseudopotentials are used along with coupled-cluster theory. The best results are estimates for the coupled-cluster CCSDTQ/q-aug-cc-pwCV∞Z-PP values for atomic terms, combined with compositional data from spin-orbit configuration interaction. The polarizability of the anion Pt- is especially challenging for the theory with wildly varying results from different coupled-cluster perturbative approximations such as CCSD(T). For atomic mercury (Hg), selected as a nearby experimental value, our polarizability volume is larger than experiment by 0.8 bohrs3 (or 0.12 × 10-30 m3). For the ground level of neutral platinum, Pt(3D3), we find α0 = (41.2 ± 1.1) bohrs3 or (6.10 ± 0.16) × 10-30 m3. A handful of density functional theory methods are tested and found generally within 10% of our best values.

A coupled cluster framework for electrons and phonons.

We describe a coupled cluster framework for coupled systems of electrons and harmonic phonons. Neutral and charged excitations are accessed via the equation-of-motion version of the theory. Benchmarks on the Hubbard-Holstein model allow us to assess the strengths and weaknesses of different coupled cluster approximations, which generally perform well for weak to moderate coupling. Finally, we report progress toward an implementation for ab initio calculations on solids and present some preliminary results on finite-size models of diamond with a linear electron-phonon coupling. We also report the implementation of electron-phonon coupling matrix elements from crystalline Gaussian type orbitals within the PySCF program package.

Analytic Gradients of Approximate Coupled Cluster Methods with Quadruple Excitations.

The analytic gradient theory for both iterative and noniterative coupled-cluster approximations that include connected quadruple excitations is presented. These methods include, in particular, CCSDT(Q), which is an analog of the well-known CCSD(T) method which starts from the full CCSDT method rather than CCSD. The resulting methods are implemented in the CFOUR program suite, and pilot applications are presented for the equilibrium geometries and harmonic vibrational frequencies of the simplest Criegee intermediate, CH2OO, as well as to the isomerization pathway between dimethylcarbene and propene. While all methods are seen to approximate the full CCSDTQ results well for "well-behaved" systems, the more difficult case of the Criegee intermediate shows that CCSDT(Q), as well as certain iterative approximations, display problematic behavior.

Toward the efficient local tailored coupled cluster approximation and the peculiar case of oxo-Mn(Salen).

We introduce a new implementation of the coupled cluster method with single and double excitations tailored by the matrix product state wave functions (DMRG-TCCSD), which employs the local pair natural orbital (LPNO) approach. By exploiting locality in the coupled cluster stage of the calculation, we were able to remove some of the limitations that hindered the application of the canonical version of the method to larger systems and/or with larger basis sets. We assessed the accuracy of the approximation using two systems: tetramethyleneethane (TME) and oxo-Mn(Salen). Using the default cut-off parameters, we were able to recover over 99.7% and 99.8% of the canonical correlation energy for the triplet and singlet state of TME, respectively. In the case of oxo-Mn(Salen), we found that the amount of retrieved canonical correlation energy depends on the size of the complete active space (CAS)-we retrieved over 99.6% for the larger 27 orbital CAS and over 99.8% for the smaller 22 orbital CAS. The use of LPNO-TCCSD allowed us to perform these calculations up to quadruple-ζ basis set, amounting to 1178 basis functions. Moreover, we examined dependence of the ground state of oxo-Mn(Salen) on the CAS composition. We found that the inclusion of 4dxy orbital plays an important role in stabilizing the singlet state at the DMRG-CASSCF level via double-shell effect. However, by including dynamic correlation, the ground state was found to be triplet regardless of the size of the basis set or the composition of CAS, which is in agreement with previous findings by canonical DMRG-TCCSD in smaller basis.

Range-separated double-hybrid density-functional theory with coupled-cluster and random-phase approximations.

We construct range-separated double-hybrid (RSDH) schemes which combine coupled-cluster or random-phase approximations (RPAs) with a density functional based on a two-parameter Coulomb-attenuating-method-like decomposition of the electron-electron interaction. We find that the addition of a fraction of short-range electron-electron interaction in the wave-function part of the calculation is globally beneficial for the RSDH scheme involving a variant of the RPA with exchange terms. Even though the latter scheme is globally as accurate as the corresponding scheme employing only second-order Møller-Plesset perturbation theory for atomization energies, reaction barrier heights, and weak intermolecular interactions of small molecules, it is more accurate for the more complicated case of the benzene dimer in the stacked configuration. The present RSDH scheme employing a RPA thus represents a new member in the family of double hybrids with minimal empiricism which could be useful for general chemical applications.

Linear and quadratic internally contracted multireference coupled-cluster approximations.

Linear and quadratic approximations to the internally contracted multireference coupled-cluster (icMRCC) method are implemented and analyzed by using the linked and unlinked coupled-cluster formalisms. This includes methods based on perturbation theory as well as the coupled-electron pair approximation, CEPA(0). The similarities and differences between all the approximations serve to highlight and provoke discussion about methodological peculiarities of the icMRCC ansatz. When calculating potential energy curves (PECs), discontinuities are observed for the linear icMRCC energies. Using a diagrammatic representation, the terms that cause but also reduce these discontinuities are identified. For benchmarking test cases such as calculating PECs, singlet-triplet splittings, and barrier heights, the multireference CEPA(0) approximation performs well; however, it suffers from a lack of size consistency and so cannot represent a step forward to the goal of developing a computationally cheap and accurate icMRCC method.

Computing the dipole polarizability of Ca48 with increased precision

We compute the electric dipole polarizability of 48Ca with an increased precision by including more correlations than in previous studies. Employing the coupled-cluster method we go beyond singles and doubles excitations and include leading-order three-particle-three-hole (3p-3h) excitations for the ground state, excited states, and the similarity transformed operator. We study electromagnetic sum rules, such as the bremsstrahlung sum rule m_0 and the polarizability sum rule alpha_D using interactions from chiral effective field theory. To gauge the quality of our coupled-cluster approximations we perform several benchmarks with the effective interaction hyperspherical harmonics approach in 4He and with self consistent Green's function in 16O. We compute the dipole polarizability of 48Ca employing the chiral interaction N2LOsat [Ekstroem et al., Phys. Rev. C 91, 051301 (2015)] and the 1.8/2.0 (EM) [Hebeler et al., Phys. Rev. C 83, 031301 (2011)]. We find that the effect of 3p-3h excitations in the ground state is small for 1.8/2.0 (EM) but non-negligible for N2LOsat. The addition of these new correlations allows us to improve the precision of our 48Ca calculations and reconcile the recently reported discrepancy between coupled-cluster results based on these interactions and the experimentally determined alpha_D from proton inelastic scattering in 48Ca [Birkhan et al., Phys. Rev. Lett. 118, 252501 (2017)]. For the computation of electromagnetic and polarizability sum rules, the inclusion of leading-order 3p-3h excitations in the ground state is important, while less so for the excited states and the similarity-transformed dipole operator.

Non-iterative triple excitations in equation-of-motion coupled-cluster theory for electron attachment with applications to bound and temporary anions.

The impact of residual electron correlation beyond the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) approximation on positions and widths of electronic resonances is investigated. To establish a method that accomplishes this task in an economical manner, several approaches proposed for the approximate treatment of triple excitations are reviewed with respect to their performance in the electron attachment (EA) variant of EOM-CC theory. The recently introduced EOM-CCSD(T)(a)* method [D. A. Matthews and J. F. Stanton, J. Chem. Phys. 145, 124102 (2016)], which includes non-iterative corrections to the reference and the target states, reliably reproduces vertical attachment energies from EOM-EA-CC calculations with single, double, and full triple excitations in contrast to schemes in which non-iterative corrections are applied only to the target states. Applications of EOM-EA-CCSD(T)(a)* augmented by a complex absorbing potential (CAP) to several temporary anions illustrate that shape resonances are well described by EOM-EA-CCSD, but that residual electron correlation often makes a non-negligible impact on their positions and widths. The positions of Feshbach resonances, on the other hand, are significantly improved when going from CAP-EOM-EA-CCSD to CAP-EOM-EA-CCSD(T)(a)*, but the correct energetic order of the relevant electronic states is still not achieved.

Spectral functions of the uniform electron gas via coupled-cluster theory and comparison to theGWand related approximations

We use, for the first time, ab initio coupled-cluster theory to compute the spectral function of the uniform electron gas at a Wigner-Seitz radius of $r_\mathrm{s}=4$. The coupled-cluster approximations we employ go significantly beyond the diagrammatic content of state-of-the-art $GW$ theory. We compare our calculations extensively to $GW$ and $GW$-plus-cumulant theory, illustrating the strengths and weaknesses of these methods in capturing the quasiparticle and satellite features of the electron gas. Our accurate calculations further allow us to address the long-standing debate over the occupied bandwidth of metallic sodium. Our findings indicate that the future application of coupled-cluster theory to condensed phase material spectra is highly promising.

Detecting Positron-Atom Bound States through Resonant Annihilation and Scattering

SynopsisA method is proposed for detecting positron-atom bound states by observing enhanced positron annihilation due to electronic Feshbach resonances. Relativistic coupled-cluster single-double approximation with third-order corrections is used to calculate positron binding energies and resonances for all atoms in Periodic Table.

Theoretical study of isomerism of compounds of C-doped aluminum clusters Al13C2, TiAl12C2, and Al42C2

Segments of the potential energy surfaces corresponding to isomers of the Al13C2, TiAl12C2, and Al42C2 clusters with two carbon atoms located in different positions at the surface and inside the Al13, TiAl12, and Al42 aluminum cages have been calculated by the density functional theory method. Their energies and structural and vibrational characteristics have been evaluated. Low-lying isomers with high-coordinated carbon atom C* have been localized. The energy parameters of Al13C2 have been refined in the coupled-cluster approximation. The effect of Ti doping on the relative energies of the isomers have been studied. The results for the small Al13C2 and TiAl12C2 clusters have been compared with the calculations of the related clusters Al13L and TiAl12L, where L = C2H2, CO, CO2, and N2. For the bulkier Al42C2 cluster, it has been shown that five- and six-coordinate configurations are typical of the C* atoms located both at the periphery and in the inner “core” of the Al42 cage.

Analytic evaluation of the dipole Hessian matrix in coupled-cluster theory

The general theory required for the calculation of analytic third energy derivatives at the coupled-cluster level of theory is presented and connected to preceding special formulations for hyperpolarizabilities and polarizability gradients. Based on our theory, we have implemented a scheme for calculating the dipole Hessian matrix in a fully analytical manner within the coupled-cluster singles and doubles approximation. The dipole Hessian matrix is the second geometrical derivative of the dipole moment and thus a third derivative of the energy. It plays a crucial role in IR spectroscopy when taking into account anharmonic effects and is also essential for computing vibrational corrections to dipole moments. The superior accuracy of the analytic evaluation of third energy derivatives as compared to numerical differentiation schemes is demonstrated in some pilot calculations.

Convergence of the Interaction Energies in Noncovalent Complexes in the Coupled-Cluster Methods Up to Full Configuration Interaction.

The CCSD(T) method stands out among various coupled-cluster (CC) approximations as the "golden standard" in computational chemistry and is widely and successfully used in the realm of covalent and noncovalent interactions. The CCSD(T) method provides reliable interaction energies, but their surprising accuracy is believed to arise partially from an error compensation. The convergence of the CC expansion has been investigated up to fully iterative pentuple excitations (CCSDTQP); for the smallest eight electron complexes, the full CI calculations have also been performed. We conclude that the convergence of interaction energy at noncovalent accuracy (0.01 kcal/mol) for the complexes studied is reached already at CCSDTQ or CCSDT(Q) levels. When even higher accuracy (spectroscopic accuracy of 1 cm(-1), or 3 cal/mol) is required, then the noniterative CCSDTQ(P) method could be used.

Relativistic linearized coupled-cluster single-double calculations of positron-atom bound states

Relativistic coupled-cluster single-double approximation is used to calculate positron-atom bound states. The method is tested on closed-shell atoms such as Be, Mg, Ca, Zn, Cd, and Hg where a number of accurate calculations is available. It is then used to calculate positron binding energies for a range of open-shell transition metal atoms from Sc to Cu, from Y to Pd, and from Lu to Pt. These systems possess Feshbach resonances, which can be used to search for positron-atom binding experimentally through resonant annihilation or scattering.

Asymmetric-Lanczos-Chain-Driven Implementation of Electronic Resonance Convergent Coupled-Cluster Linear Response Theory.

We present an implementation of the damped coupled-cluster linear response function based on an asymmetric Lanczos chain algorithm for the hierarchy of coupled-cluster approximations CCS (coupled-cluster singles), CC2 (coupled-cluster singles and approximate doubles), and CCSD (coupled-cluster singles and doubles). Triple corrections to the excitation energies can be included via the CCSDR(3) (coupled-cluster singles and doubles with noniterative-triples-corrected excitation energies) approximation. The performance and some of the potentialities of the approach are investigated in calculations of the visible/ultraviolet absorption spectrum and the dispersion of the real polarizability in near-resonant regions of pyrimidine, the near-edge absorption fine structure (NEXAFS) of ammonia, and the direct determination of the C6 dipole-dipole dispersion coefficient of the benzene dimer.

Accuracy of Calculated Chemical Shifts in Carbon 1s Ionization Energies from Single-Reference ab Initio Methods and Density Functional Theory

A database of 77 adiabatic carbon 1s ionization energies has been prepared, covering linear and cyclic alkanes and alkenes, linear alkynes, and methyl- or fluoro-substituted benzenes. Individual entries are believed to carry uncertainties of less than 30 meV in ionization energies and less than 20 meV for shifts in ionization energies. The database provides an unprecedented opportunity for assessing the accuracy of theoretical schemes for computing inner-shell ionization energies and their corresponding chemical shifts. Chemical shifts in carbon 1s ionization energies have been computed for all molecules in the database using Hartree-Fock, Møller-Plesset (MP) many-body perturbation theory of order 2 and 3 as well as various approximations to full MP4, and the coupled-cluster approximation with single- and double-excitation operators (CCSD) and also including a perturbational estimate of the energy effect of triple-excitation operators (CCSD(T)). Moreover, a wide range of contemporary density functional theory (DFT) methods are also evaluated with respect to computing experimental shifts in C1s ionization energies. Whereas the top ab initio methods reproduce the observed shifts almost to within the experimental uncertainty, even the best-performing DFT approaches meet with twice the root-mean-squared error and thrice the maximum error compared to CCSD(T). However, a number of different density energy functionals still afford sufficient accuracy to become tools in the analysis of complex C1s photoelectron spectra.