Coupled Cluster Wave Functions Research Articles

Combining Complex Conjugation, Time-Reversal, and Spin-Flip Symmetry Projection of Coupled Cluster Wave Functions.

Complex conjugation symmetry breaking and restoration generates two nonorthogonal configurations at the Hartree-Fock level that can capture static correlation naturally. In conjunction with broken spin-symmetry coupled cluster theory, the symmetry-projected wave function shows good agreement with full configuration interaction in beryllium hydride insertion, lithium fluoride dissociation, and symmetric stretching of tetrahedral H4. By adding spin flip projection, we can also recover time reversal symmetry in the same coupled cluster framework. We also show results including point group symmetry projection.

Reasons Why Most Single-Reference Coupled Cluster Methods Fail to Provide the Correct Adiabatic Potentials of a Diatomic Carbon Molecule: UCCSDecCCSD Potential Study.

Many single-reference coupled cluster (CC) methods offer adiabatically incorrect potentials when calculating the diatomic carbon molecule, so this problem has been studied extensively. Analysis of the full configuration interaction (FCI) wave function indicates that the main cause of the adiabatic collapse of potentials calculated by the CC method with singles, doubles, and triples (CCSDT) is the strongly increasing bonding character of the T4FCI cluster contribution. In turn, comparative analysis of the CCSDTQ adiabats X1Σg+ and B'1Σg+ demonstrates that the gap between them near the avoided crossing geometry is significantly reduced by quantitatively differentiating the character of the T4 and cluster contributions. These observations clearly indicate the need to take into account the T4 cluster contribution in the standard CC wave function to obtain the correct adiabatic potential. Further analysis of this issue shows that the T4 contribution must be additionally bonding to ensure the adiabatic correctness of the potential. What also seems very interesting is that when the UCCSDecCCSD method [Toboła, Chem. Phys. Lett. 2014, 614, 82-88] is used for the potential calculation, the shape of the potential is entirely determined by a subset of unrestricted Hartree-Fock (UHF) configurations, which are structurally identical to the ground-state configuration in the UHF-based CCSD wave function.

Aufbau Suppressed Coupled Cluster Theory for Electronically Excited States.

We introduce an approach to improve single-reference coupled cluster theory in settings where the Aufbau determinant is absent from or plays only a small role in the true wave function. Using a de-excitation operator that can be efficiently hidden within a similarity transform, we create a coupled cluster wave function in which de-excitations work to suppress the Aufbau determinant and produce wave functions dominated by other determinants. Thanks to an invertible and fully exponential form, the approach is systematically improvable, size consistent, size extensive, and, interestingly, size intensive in a granular way that should make the adoption of some ground state techniques, such as local correlation, relatively straightforward. In this initial study, we apply the general formalism to create a state-specific method for orbital-relaxed, singly excited states. We find that this approach matches the accuracy of similar-cost equation-of-motion methods in valence excitations while offering improved accuracy for charge transfer states. We also find the approach to be more accurate than excited-state-specific perturbation theory in both types of states.

Coupled-cluster theory for trapped bosonic mixtures.

We develop a coupled-cluster theory for bosonic mixtures of binary species in external traps, providing a promising theoretical approach to demonstrate highly accurately the many-body physics of mixtures of Bose-Einstein condensates. The coupled-cluster wavefunction for the binary species is obtained when an exponential cluster operator eT, where T = T(1) + T(2) + T(12) and T(1) accounts for excitations in species-1, T(2) for excitations in species-2, and T(12) for combined excitations in both species, acts on the ground state configuration prepared by accumulating all bosons in a single orbital for each species. We have explicitly derived the working equations for bosonic mixtures by truncating the cluster operator up to the single and double excitations and using arbitrary sets of orthonormal orbitals for each of the species. Furthermore, the comparatively simplified version of the working equations are formulated using the Fock-like operators. Finally, using an exactly solvable many-body model for bosonic mixtures that exists in the literature allows us to implement and test the performance and accuracy of the coupled-cluster theory for situations with balanced as well as imbalanced boson numbers and for weak to moderately strong intra- and interspecies interaction strengths. The comparison between our computed results using coupled-cluster theory with the respective analytical exact results displays remarkable agreement exhibiting excellent success of the coupled-cluster theory for bosonic mixtures. All in all, the correlation exhaustive coupled-cluster theory shows encouraging results and could be a promising approach in paving the way for high-accuracy modeling of various bosonic mixture systems.

High Accuracy Ab Initio Potential Energy Curves and Dipole Moment Functions for the X1Σ+ and a3Π Spin States of the CF+ Diatomic Molecule.

Potential energy curves and dipole moment functions constructed using high-accuracy ab initio methods allow for an in-depth examination of the electronic structure of diatomic molecules. Ab initio computations serve as a valuable complement to experimental data, offering insights into the nature of short-lived molecules such as those encountered within the interstellar medium (ISM). While laboratory experiments provide critical groundwork, the ISM's conditions often permit longer lifetimes for lower stability molecules, enabling unique observations. The CF+ diatomic molecule is one such molecule that has been observed spectroscopically in the ISM. Previous experimental and theoretical work have examined different spectroscopic aspects of the CF+ molecule, but the development of newer, more complete potential energy curves and dipole moment functions allows for even greater insight. We constructed both potential energy curves and dipole moment functions for the ground X1Σ+ and first excited a3Π states of CF+ for both the 12C and 13C isotopologues. The potential energy curves were constructed using coupled cluster with single, double, and perturbative triple excitations (CCSD(T)) at the complete basis set limit with corrections from full triple, quadruple, quintuple, and hextuple excitations within a finite-basis coupled cluster wave function as well as corrections from full configuration interaction and relativistic effects. Rovibrational wave functions were calculated using a vibrational Hamiltonian matrix, which moves beyond the harmonic oscillator approximation. The equilibrium bond length, vibrational constant, and rotational constant were reproduced to within 0.00013 Å, 0.28 cm-1, and 0.00045 cm-1, respectively, of experimental values. Experimental transition energies from rovibrational spectra were reproduced with an error of no larger than 0.63 cm-1. The triplet excited state (a3Π) was found to have a longer equilibrium bond length at 1.21069 Å, a vibrational constant of 1611.29 cm-1, and a rotational constant of 1.56376 cm-1. Rovibrational line lists for the 12C and 13C isotopologues for both the X1Σ+ and the excited a3Π states were generated.

Excited-State-Specific Pseudoprojected Coupled-Cluster Theory.

We present an excited-state-specific coupled-cluster approach in which both the molecular orbitals and cluster amplitudes are optimized for an individual excited state. The theory is formulated via a pseudoprojection of the traditional coupled-cluster wavefunction that allows correlation effects to be introduced atop an excited-state mean field starting point. The approach shares much in common with ground-state CCSD, including size extensivity and an N6 cost scaling. Preliminary numerical tests show that, when augmented with N5 cost perturbative corrections for key terms, the method can improve over excited-state-specific second-order perturbation theory in valence, charge transfer, and Rydberg states.

Toward an efficient implementation of internally contracted coupled-cluster methods.

A new implementation of the internally contracted multireference coupled-cluster with singles and doubles (icMRCCSD) method is presented. The new code employs an efficient tensor contraction kernel and can also avoid full four-external integral transformations, which significantly extends the scope of the applicability of icMRCCSD. The new implementation is currently restricted to the simple case of two active electrons in two orbitals and also supports the computation of spin-adapted doublet and triplet coupled-cluster wavefunctions. This contribution describes the basic approach for the automated derivation of working equations and benchmarks the current code against efficient implementations of standard methods, such as single-reference coupled-cluster singles and doubles (CCSD) and internally contracted multireference configuration interaction (icMRCI). Run times for linearized variants of icMRCCSD are only twice as long as comparable CCSD runs and similar to those of the icMRCI implementation, while non-linear terms of more complete variants of icMRCCSD lead to an order of magnitude longer computation times. Nevertheless, the new code allows for computations at larger scales than it was possible previously, with less demands on memory and disk-space resources. This is exemplified by numerical structure optimizations and harmonic force field determinations of NC2H5 isomers and the singlet and triplet states of m-benzyne. In addition, the exchange coupling of a dinuclear copper complex is determined. This work also defines a new commutator approximation for icMRCCSD, which includes all terms that are also present in the single-reference CCSD method, thus yielding a consistent pair of single-reference and multireference coupled-cluster methods.

How the Exchange Energy Can Affect the Power Laws Used to Extrapolate the Coupled Cluster Correlation Energy to the Thermodynamic Limit.

Finite size error is commonly removed from coupled cluster theory calculations by N-1 extrapolations over correlation energy calculations of different system sizes (N), where the N-1 scaling comes from the total energy rather than the correlation energy. However, previous studies in the quantum Monte Carlo community suggest an exchange-energy-like power law of N-2/3 should also be present in the correlation energy when using the conventional Coulomb interaction. The rationale for this is that the total energy goes as N-1 and the exchange energy goes as N-2/3; thus, the correlation energy should be a combination of these two power laws. Further, in coupled cluster theory, these power laws are related to the low G scaling of the transition structure factor, S(G), which is a property of the coupled cluster wave function calculated from the amplitudes. We show here that data from coupled cluster doubles calculations on the uniform electron gas fit a function with a low G behavior of S(G) ∼ G. The prefactor for this linear term is derived from the exchange energy to be consistent with an N-2/3 power law at large N. Incorporating the exchange structure factor into the transition structure factor results in a combined structure factor of S(G) ∼ G2, consistent with an N-1 scaling of the exchange-correlation energy. We then look for the presence of an N-2/3 power law in the energy. To do so, we first develop a plane-wave cutoff scheme with less noise than the traditional basis set used for the uniform electron gas. Then, we collect data from a wide range of electron numbers and densities to systematically test five methods using N-1 scaling, N-2/3 scaling, or combinations of both scaling behaviors. We find that power laws that incorporate both N-1 and N-2/3 scaling perform better than either alone, especially when the prefactor for N-2/3 scaling can be found from exchange energy calculations.

Mapping renormalized coupled cluster methods to quantum computers through a compact unitary representation of nonunitary operators

Nonunitary theories are commonly seen in the classical simulations of quantum systems. Among these theories, the method of moments of coupled cluster equations (MMCCs) and the ensuing classes of the renormalized coupled cluster (CC) approaches have evolved into one of the most accurate approaches to describe correlation effects in various quantum systems. The MMCC formalism provides an effective way for correcting the energies of approximate CC formulations (parent theories) using moments, or CC equations, that are not used to determine approximate cluster amplitudes. In this paper, we propose a quantum algorithm for computing MMCC ground-state energies that provides two main advantages over classical computing or other quantum algorithms: (i) the possibility of forming superpositions of CC moments of arbitrary ranks in the entire Hilbert space and using an arbitrary form of the parent cluster operator for MMCC expansion, and (ii) significant reduction in the number of measurements in quantum simulation through a compact unitary representation for a generally nonunitary operator. We illustrate the robustness of our approach over a broad class of test cases, including $\ensuremath{\sim}40$ molecular systems with varying basis sets encoded using 4--40 qubits, and we exhibit the detailed MMCC analysis for the 8-qubit half-filled, four-site, single impurity Anderson model and the 12-qubit hydrogen fluoride molecular system from the corresponding noise-free and noisy MMCC quantum computations. We also outline the extension of the MMCC formalism to the case of a unitary CC wave-function Ansatz.

Geometric interpretation for coupled-cluster theory. A comparison of accuracy with the corresponding configuration interaction model.

Although coupled-cluster theory is well-known for its accuracy, the geometry associated with the manifold of wave functions reached by the coupled-cluster Ansatz has not been deeply explored. In this article, we look for an interpretation for the high accuracy of coupled-cluster theory based on how the manifold of coupled-cluster wave functions is embedded within the space of n-electron wave functions. We define the coupled-cluster and configuration interaction manifolds and measure the distances from the full-configuration interaction (FCI) wave function to these manifolds. We clearly observe that the FCI wave function is closer to the coupled-cluster manifold that is curved than to the configuration interaction manifold that is flat for the selected systems studied in this work. Furthermore, the decomposition of the distances among these manifolds and wave functions into excitation ranks gives insights into the failure of the coupled-cluster approach for multireference systems. The present results show a new interpretation for the quality of the coupled-cluster method, as contrasted to the truncated configuration interaction approach, besides the well-established argument based on size extensivity. Furthermore, we show how a geometric description of wave function methods can be used in electronic structure theory.

Comprehensive Theoretical View of the [Cu2 O2 ] Side-on-Peroxo-/Bis-μ-Oxo Equilibria.

Coupled binuclear copper (CBC) sites are employed by many metalloenzymes to catalyze a broad set of biochemical transformations. Typically, the CBC catalytic sites are activated by the O2 molecule to form various [Cu2 O2 ] reactive species. This has also inspired synthesis and development of various biomimetic inorganic complexes featuring the CBC core. From theoretical perspective, the [Cu2 O2 ] reactivity often hinges on the side-on-peroxo-dicopper(II) (P) vs. bis-μ-oxo-dicopper(III) (O) isomerism - an equilibrium that has become almost iconic in theoretical bioinorganic chemistry. Herein, we present a comprehensive calibration and evaluation of the performance of various composite computational protocols available in contemporary computational chemistry, involving coupled-cluster and multireference (relativistic) wave function methods, popular density functionals and solvation models. Starting with the well-studied reference [Cu2 O2 (NH3 )6 ]2+ system, we compared the performance of electronic structure methods and discussed the relativistic effects. This allowed us to select several 'calibrated' DFT functionals that can be conveniently employed to study ten experimentally well-characterized [Cu2 O2 ] inorganic systems. We mostly predicted the lowest-energy structures (P vs. O) of the studied systems correctly. In addition, we present calibration of the used electronic structure methods for prediction of the spectroscopic features of the [Cu2 O2 ] core, mostly provided by the resonance Raman (rR) spectroscopy.

Molecular Auger decay rates from complex-variable coupled-cluster theory.

The emission of an Auger electron is the predominant relaxation mechanism of core-vacant states in molecules composed of light nuclei. In this non-radiative decay process, one valence electron fills the core vacancy, while a second valence electron is emitted into the ionization continuum. Because of this coupling to the continuum, core-vacant states represent electronic resonances that can be tackled with standard quantum-chemical methods only if they are approximated as bound states, meaning that Auger decay is neglected. Here, we present an approach to compute Auger decay rates of core-vacant states from coupled-cluster and equation-of-motion coupled-cluster wave functions combined with complex scaling of the Hamiltonian or, alternatively, complex-scaled basis functions. Through energy decomposition analysis, we illustrate how complex-scaled methods are capable of describing the coupling to the ionization continuum without the need to model the wave function of the Auger electron explicitly. In addition, we introduce in this work several approaches for the determination of partial decay widths and Auger branching ratios from complex-scaled coupled-cluster wave functions. We demonstrate the capabilities of our new approach by computations on core-ionized states of neon, water, dinitrogen, and benzene. Coupled-cluster and equation-of-motion coupled-cluster theory in the singles and doubles approximation both deliver excellent results for total decay widths, whereas we find partial widths more straightforward to evaluate with the former method.

Symmetry projection to coupled-cluster singles plus doubles wave function through the Monte Carlo method

A new method for calculating the symmetry-projected energy of coupled-cluster singles and doubles (CCSD) wave function through the Monte Carlo method is proposed. We present benchmark calculations in considering the three-level Lipkin model which is a simple and minimal model with two phases: spherical and deformed ones. It is demonstrated that this new method gives good ground state energy and low-lying spectra.

Feshbach-Fano approach for calculation of Auger decay rates using equation-of-motion coupled-cluster wave functions. II. Numerical examples and benchmarks.

X-ray photon absorption leads to the creation of highly excited species, which often decay through the Auger process. The theoretical treatment of Auger decay is challenging because of the resonance nature of the initial core-excited or core-ionized states and the continuous nature of the ejected electron. In Paper I [W. Skomorowski and A. I. Krylov, J. Chem. Phys. 154, 084124 (2021)], we have introduced a theoretical framework for computing Auger rates based on the Feshbach-Fano approach and the equation-of-motion coupled-cluster ansätze augmented with core-valence separation. The outgoing Auger electron is described with a continuum orbital. We considered two approximate descriptions-a plane wave and a Coulomb wave with an effective charge. Here, we use the developed methodology to calculate Auger transition rates in core-ionized and core-excited benchmark systems (Ne, H2O, CH4, and CO2). Comparison with the available experimental spectra shows that the proposed computational scheme provides reliable ab initio predictions of the Auger spectra. The reliability, cost efficiency, and robust computational setup of this methodology offer advantages in applications to a large variety of systems.

Feshbach–Fano approach for calculation of Auger decay rates using equation-of-motion coupled-cluster wave functions. I. Theory and implementation

X-ray absorption creates electron vacancies in the core shell. These highly excited states often relax by Auger decay-an autoionization process in which one valence electron fills the core hole and another valence electron is ejected into the ionization continuum. Despite the important role of Auger processes in many experimental settings, their first-principles modeling is challenging, even for small systems. The difficulty stems from the need to describe many-electron continuum (unbound) states, which cannot be tackled with standard quantum-chemistry methods. We present a novel approach to calculate Auger decay rates by combining Feshbach-Fano resonance theory with the equation-of-motion coupled-cluster single double (EOM-CCSD) framework. We use the core-valence separation scheme to define projectors into the bound (square-integrable) and unbound (continuum) subspaces of the full function space. The continuum many-body decay states are represented by products of an appropriate EOM-CCSD state and a free-electron state, described by a continuum orbital. The Auger rates are expressed in terms of reduced quantities, two-body Dyson amplitudes (objects analogous to the two-particle transition density matrix), contracted with two-electron bound-continuum integrals. Here, we consider two approximate treatments of the free electron: a plane wave and a Coulomb wave with an effective charge, which allow us to evaluate all requisite integrals analytically; however, the theory can be extended to incorporate a more sophisticated description of the continuum orbital.

Orbital entanglement and correlation from pCCD-tailored coupled cluster wave functions.

Wave functions based on electron-pair states provide inexpensive and reliable models to describe quantum many-body problems containing strongly correlated electrons, given that broken-pair states have been appropriately accounted for by, for instance, a posteriori corrections. In this article, we analyze the performance of electron-pair methods in predicting orbital-based correlation spectra. We focus on the (orbital-optimized) pair-coupled cluster doubles (pCCD) ansatz with a linearized coupled-cluster (LCC) correction. Specifically, we scrutinize how orbital-based entanglement and correlation measures can be determined from a pCCD-tailored CC wave function. Furthermore, we employ the single-orbital entropy, the orbital-pair mutual information, and the eigenvalue spectra of the two-orbital reduced density matrices to benchmark the performance of the LCC correction for the one-dimensional Hubbard model with the periodic boundary condition as well as the N2 and F2 molecules against density matrix renormalization group reference calculations. Our study indicates that pCCD-LCC accurately reproduces the orbital-pair correlation patterns in the weak correlation limit and for molecules close to their equilibrium structure. Hence, we can conclude that pCCD-LCC predicts reliable wave functions in this regime.

Structural changes upon electronic excitation in 1,3-dimethoxybenzene from Franck-Condon/rotational constants fits of the fluorescence emission spectra[formula omitted

The geometry changes of two rotamers of 1,3-dimethoxybenzene were determined from a combined fit of line intensities in fluorescence emission spectra, obtained via pumping different vibronic transition and the changes of rotational constants upon electronic excitation. The so determined changes are compared to the results of ab initio approximate coupled cluster singles and doubles (CC2) calculations. The bond length variations in the aromatic ring follow the calculated electron density differences from the coupled cluster wavefunctions and are in agreement with excitation to an 1Lb-state for the C2v symmetric B-conformer.

Ab initioproperties of the NaLi molecule in thea3Σ+electronic state

Ultracold polar and magnetic ${}^{23}$Na${}^6$Li molecules in the rovibrational ground state of the lowest triplet $a^3\Sigma^+$ electronic state have been recently produced. Here, we calculate the electronic and rovibrational structure of these 14-electron molecules with spectroscopic accuracy ($<0.5\,$cm$^{-1}$) using state-of-the-art ab initio methods of quantum chemistry. We employ the hierarchy of the coupled-cluster wave functions and Gaussian basis sets extrapolated to the complete basis set limit. We show that the inclusion of higher-level excitations, core-electron correlation, relativistic, QED, and adiabatic corrections is necessary to reproduce accurately scattering and spectroscopic properties of alkali-metal systems. We obtain the well depth, $D_e=229.9(5)\,$cm$^{-1}$, the dissociation energy, $D_0=208.2(5)\,$cm$^{-1}$, and the scattering length, $a_s=-84^{+25}_{-41}\,$bohr, in good agreement with recent experimental measurements. We predict the permanent electric dipole moment in the rovibrational ground state, $d_0=$0.167(1)$\,$debye. These values are obtained without any adjustment to experimental data, showing that quantum chemistry methods are capable of predicting scattering properties of many-electron systems, provided relatively weak interaction and small reduced mass of the system.

Effective Hamiltonians derived from equation-of-motion coupled-cluster wave functions: Theory and application to the Hubbard and Heisenberg Hamiltonians.

Effective Hamiltonians, which are commonly used for fitting experimental observables, provide a coarse-grained representation of exact many-electron states obtained in quantum chemistry calculations; however, the mapping between the two is not trivial. In this contribution, we apply Bloch's formalism to equation-of-motion coupled-cluster wave functions to rigorously derive effective Hamiltonians in Bloch's and des Cloizeaux's forms. We report the key equations and illustrate the theory by application to systems with two or three unpaired electrons, which give rise to electronic states of covalent and ionic characters. We show that Hubbard's and Heisenberg's Hamiltonians can be extracted directly from the so-obtained effective Hamiltonians. By establishing a quantitative connection between many-body states and simple models, the approach facilitates the analysis of the correlated wave functions. We propose a simple diagnostic for assessing the validity of the model space choice based on the overlaps between the target- and model-space states. Artifacts affecting the quality of electronic structure calculations such as spin contamination are also discussed.

Energy of fermionic ground states with low-entanglement single-reference expansions, and tensor-based strong-coupling extensions of the coupled-cluster method

We consider a fermionic system for which there exist a single-reference configuration-interaction (CI) expansion of the ground state wave function that converges, albeit not necessarily rapidly, with respect to excitation number. We show that, if the coefficients of Slater determinants (SD) with $l\leq k$ excitations can be defined with a number of free parameters $N_{\leq k}$ bounded polynomially in $k$, the ground state energy $E$ only depends on a small fraction of all the wave function parameters, and is the solution of equations of the coupled-cluster (CC) form. This generalizes the standard CC method, for which $N_{\leq k}$ is bounded by a constant. Based on that result and low-rank tensor decompositions (LRTD), we discuss two possible extensions of the CC approach for wave functions with general polynomial bound for $N_{\leq k}$. First, one can use LRTD to represent the amplitudes of the CC cluster operator $T$ which, unlike in the CC case, is not truncated with respect to excitation number, and the energy and tensor parameters are given by a LRTD-adapted version of standard CC equations. Second, the LRTD can also be used to directly parametrize the CI coefficients, which involves different equations of the CC form. We derive those equations for up to quadruple-excitation coefficients, using a different type of excitation operator in the CC wave function ansatz, and a Hamiltonian representation in terms of excited particle and hole operators. We complete the proposed CC extensions by constructing compact tensor representations of coefficients, or $T$-amplitudes, using superpositions of tree tensor networks which take into account different possible types of entanglement between excited particles and holes. Finally, we discuss why the proposed extensions are theoretically applicable at larger coupling strengths than those treatable by the standard CC method.