Explicitly correlated wave functions: summary and perspective

Estimating the Complete Basis Set Extrapolation Error through Random Walks

We propose a method for estimating the uncertainty of a result obtained through extrapolation to the complete basis set limit. The method is based on an ensemble of random walks that simulate all possible extrapolation outcomes that could have been obtained if results from larger basis sets had been available. The results assembled from a large collection of random walks can then be analyzed statistically, providing a route for uncertainty prediction at the confidence level required in a particular application. The method is free of empirical parameters and compatible with any extrapolation scheme. The proposed technique is tested in a series of numerical trials by comparing the determined confidence intervals with reliable reference data. We demonstrate that the predicted error bounds are reliable and tight yet conservative at the same time.

Non-iterative Triples for Transcorrelated Coupled Cluster Theory

We present an implementation of a perturbative triples correction for the coupled cluster ansatz including single and double excitations based on the transcorrelated Hamiltonian. Transcorrelation introduces explicit electron correlation in the electronic Hamiltonian through similarity transformation with a correlation factor. Due to this transformation, the transcorrelated Hamiltonian includes up to three-body couplings and becomes non-Hermitian. Since the conventional coupled cluster equations are solved by projection, it is well suited to harbor non-Hermitian Hamiltonians. The arising three-body operator, however, creates a huge memory bottleneck and increases the runtime scaling of the coupled cluster equations. As it has been shown that the three-body operator can be approximated, by expressing the Hamiltonian in the normal-ordered form, we investigate this approximation for the perturbative triples correction. Results are compared with a code-generation based transcorrelated coupled cluster implementation up to quadruple excitations.

Orbital optimisation in xTC transcorrelated methods

We present a combination of the bi-orthogonal orbital optimisation framework with the recently introduced xTC version of transcorrelation. This allows us to implement non-iterative perturbation based methods on top of the transcorrelated Hamiltonian. Additionally, the orbital optimisation influences results of other truncated methods, such as the distinguishable cluster with singles and doubles. The accuracy of these methods in comparison to standard xTC methods is demonstrated, and the advantages and disadvantages of the orbital optimisation are discussed.

Accelerated basis-set convergence of coupled-cluster excitation energies using the density-based basis-set correction method

We present the first application to real molecular systems of the recently proposed linear-response theory for the density-based basis-set correction method [J. Chem. Phys., 158, 234107 (2023)]. We apply this approach to accelerate the basis-set convergence of excitation energies in the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method. We use an approximate linear-response framework that neglects the second-order derivative of the basis-set correction density functional and consists in simply adding to the usual Hamiltonian the one-electron potential generated by the first-order derivative of the functional. This additional basis-set correction potential is evaluated at the Hartree-Fock density, leading to a very computationally cheap basis-set correction. We tested this approach over a set of about 30 excitation energies computed for five small molecular systems and found that the excitation energies from the ground state to Rydberg states are the main source of basis-set error. These excitation energies systematically increase when the size of the basis set is increased, suggesting a biased description in favour of the excited state. Despite the simplicity of the present approach, the results obtained with the basis-set-corrected EOM-CCSD method are encouraging as they yield a mean absolute deviation of 0.02 eV for the aug-cc-pVTZ basis set, while it is 0.04 eV using the standard EOM-CCSD method. This might open the path to an alternative to explicitly correlated approaches to accelerate the basis-set convergence of excitation energies.

Towards efficient quantum computing for quantum chemistry: reducing circuit complexity with transcorrelated and adaptive ansatz techniques

The near-term utility of quantum computers is hindered by hardware constraints in the form of noise. One path to achieving noise resilience in hybrid quantum algorithms is to decrease the required circuit depth - the number of applied gates - to solve a given problem. This work demonstrates how to reduce circuit depth by combining the transcorrelated (TC) approach with adaptive quantum ansätze and their implementations in the context of variational quantum imaginary time evolution (AVQITE). The combined TC-AVQITE method is used to calculate ground state energies across the potential energy surfaces of H4, LiH, and H2O. In particular, H4 is a notoriously difficult case where unitary coupled cluster theory, including singles and doubles excitations, fails to provide accurate results. Adding TC yields energies close to the complete basis set (CBS) limit while reducing the number of necessary operators - and thus circuit depth - in the adaptive ansätze. The reduced circuit depth furthermore makes our algorithm more noise-resilient and accelerates convergence. Our study demonstrates that combining the TC method with adaptive ansätze yields compact, noise-resilient, and easy-to-optimize quantum circuits that yield accurate quantum chemistry results close to the CBS limit.

Compactification of determinant expansions via transcorrelation

Although selected configuration interaction (SCI) algorithms can tackle much larger Hilbert spaces than the conventional full CI method, the scaling of their computational cost with respect to the system size remains inherently exponential. In addition, inaccuracies in describing the correlation hole at small interelectronic distances lead to the slow convergence of the electronic energy relative to the size of the one-electron basis set. To alleviate these effects, we show that the non-Hermitian, transcorrelated (TC) version of SCI significantly compactifies the determinant space, allowing us to reach a given accuracy with a much smaller number of determinants. Furthermore, we note a significant acceleration in the convergence of the TC-SCI energy as the basis set size increases. The extent of this compression and the energy convergence rate are closely linked to the accuracy of the correlation factor used for the similarity transformation of the Coulombic Hamiltonian. Our systematic investigation of small molecular systems in increasingly large basis sets illustrates the magnitude of these effects.

Toward Real Chemical Accuracy on Current Quantum Hardware Through the Transcorrelated Method

Quantum computing is emerging as a new computational paradigm with the potential to transform several research fields including quantum chemistry. However, current hardware limitations (including limited coherence times, gate infidelities, and connectivity) hamper the implementation of most quantum algorithms and call for more noise-resilient solutions. We propose an explicitly correlated Ansatz based on the transcorrelated (TC) approach to target these major roadblocks directly. This method transfers, without any approximation, correlations from the wave function directly into the Hamiltonian, thus reducing the resources needed to achieve accurate results with noisy quantum devices. We show that the TC approach allows for shallower circuits and improves the convergence toward the complete basis set limit, providing energies within chemical accuracy to experiment with smaller basis sets and, thus, fewer qubits. We demonstrate our method by computing bond lengths, dissociation energies, and vibrational frequencies close to experimental results for the hydrogen dimer and lithium hydride using two and four qubits, respectively. To demonstrate our approach's current and near-term potential, we perform hardware experiments, where our results confirm that the TC method paves the way toward accurate quantum chemistry calculations already on today's quantum hardware.

Constraints upon Functionals of the 1-Matrix, Universal Properties of Natural Orbitals, and the Fallacy of the Collins "Conjecture"

Reliability of quantum-chemical calculations based upon the density functional theory and its 1-matrix counterpart hinges upon minimizing the extent of empirical parameterization in the approximate energy expressions of these formalisms while imposing as many rigorous constraints upon them as possible. The recently uncovered universal properties of the natural orbitals facilitate the construction of such constraints for the 1-matrix functionals. The benefits of their employment in the validation of these functionals are vividly demonstrated by a critical review of the three incarnations of the so-called Collins conjecture. Although the incorporation of rigorous definitions of the correlation energy and entropy, and the identification of individual potential energy hypersurfaces as probable domains of its applicability turn the originally published unsubstantiated claim into a proper conjecture, the resulting formalism is found to be merely a conduit for incorporation of static correlation effects in electronic structure calculations that is unlikely to allow attaining chemical accuracy.

Nonunitary projective transcorrelation theory inspired by the F12 ansatz

An alternative nonunitary transcorrelation, inspired by the F12 ansatz, is investigated. In contrast to the Jastrow transcorrelation of Boys-Handy, the effective Hamiltonian of this projective transcorrelation features: 1. a series terminating formally at four-body interactions. 2. no spin-contamination within the non-relativistic framework. 3. simultaneous satisfaction of the singlet and triplet first-order cusp conditions. 4. arbitrary choices of pairs for correlation including frozen-core approximations. We discuss the connection between the projective transcorrelation and F12 theory with applications to small molecules, to show that the cusp conditions play an important role to reduce the uncertainty arising from the nonunitary transformation.

Correcting Models with Long-Range Electron Interaction Using Generalized Cusp Conditions

Sources of energy errors resulting from the replacement of the physical Coulomb interaction by its long-range erfc(μr)/r approximation are explored. It is demonstrated that the results can be dramatically improved and the range of μ giving energies within chemical accuracy limits significantly extended if the generalized cusp conditions are used to represent the wave function at small r. The numerical results for two-electron harmonium are presented and discussed.

Automatic generation of complementary auxiliary basis sets for explicitly correlated methods

Explicitly correlated calculations, aside from the orbital basis set, typically require three auxiliary basis sets: Coulomb-exchange fitting (JK), resolution of the identity MP2 (RI-MP2), and complementary auxiliary basis set (CABS). If unavailable for the orbital basis set and chemical elements of interest, the first two can be auto-generated on the fly using existing algorithms, but not the third. In this paper, we present a quite simple algorithm named autoCABS; a Python implementation under a free software license is offered at Github. For the cc-pVnZ-F12 (n = D,T,Q,5), the W4-08 thermochemical benchmark, and the HFREQ2014 set of harmonic frequencies, we demonstrate that autoCABS-generated CABS basis sets are comparable in quality to purpose-optimized OptRI basis sets from the literature, and that the quality difference becomes entirely negligible as n increases.

A Universal Power Law Governing the Accuracy of Wave Function-Based Electronic Structure Calculations

A universal power law governing the accuracy of wave function-based electronic structure calculations is derived from first principles. The resulting expression ΔE(N,N)N≳19π2gNN, where g is a system-specific factor assuming values between zero and one and ≳ stands for asymptotic inequality at the limit of N→∞, allows facile estimation of the error ΔE(N,N) in the electronic energy of a singlet state of an N-electron system computed with a basis set of N one-electron functions. Several approaches to the estimation of the factor g, which depends on the on-top two-electron density, are presented.

Performance of a one-parameter correlation factor for transcorrelation: Study on a series of second row atomic and molecular systems

In this work, we investigate the performance of a recently proposed transcorrelated (TC) approach based on a single-parameter correlation factor [E. Giner, J. Chem. Phys. 154, 084119 (2021)] for systems involving more than two electrons. The benefit of such an approach relies on its simplicity as efficient numerical-analytical schemes can be set up to compute the two- and three-body integrals occurring in the effective TC Hamiltonian. To obtain accurate ground state energies within a given basis set, the present TC scheme is coupled to the recently proposed TC-full configuration interaction quantum Monte Carlo method [Cohen et al., J. Chem. Phys. 151, 061101 (2019)]. We report ground state total energies on the Li-Ne series, together with their first cations, computed with increasingly large basis sets and compare to more elaborate correlation factors involving electron-electron-nucleus coordinates. Numerical results on the Li-Ne ionization potentials show that the use of the single-parameter correlation factor brings on average only a slightly lower accuracy (1.2 mH) in a triple-zeta quality basis set with respect to a more sophisticated correlation factor. However, already using a quadruple-zeta quality basis set yields results within chemical accuracy to complete basis set limit results when using this novel single-parameter correlation factor. Calculations on the H₂O, CH₂, and FH molecules show that a similar precision can be obtained within a triple-zeta quality basis set for the atomization energies of molecular systems.

Basis-set correction for coupled-cluster estimation of dipole moments

The present work proposes an approach to obtain a basis-set correction based on density-functional theory (DFT) for the computation of molecular properties in wave-function theory (WFT). This approach allows one to accelerate the basis-set convergence of any energy derivative of a non-variational WFT method, generalizing previous works on the DFT-based basis-set correction where either only ground-state energies could be computed with non-variational wave functions [Loos et al., J. Phys. Chem. Lett. 10, 2931 (2019)] or properties could be computed as expectation values over variational wave functions [Giner et al., J. Chem. Phys. 155, 044109 (2021)]. This work focuses on the basis-set correction of dipole moments in coupled-cluster with single, double, and perturbative triple excitations [CCSD(T)], which is numerically tested on a set of 14 molecules with dipole moments covering two orders of magnitude. As the basis-set correction relies only on Hartree–Fock densities, its computational cost is marginal with respect to the one of the CCSD(T) calculations. Statistical analysis of the numerical results shows a clear improvement of the basis convergence of the dipole moment with respect to the usual CCSD(T) calculations.

Accurate scaling functions of the scaled Schrödinger equation

The scaling function g of the scaled Schrödinger equation (SSE) is generalized to obtain accurate solutions of the Schrödinger equation (SE) with the free complement (FC) theory. The electron-nuclear and electron-electron scaling functions, g_iA and g_ij, respectively, are generalized. From the relations between SE and SSE at the inter-particle distances being zero and infinity, the scaling function must satisfy the collisional (or coalescent) condition and the asymptotic condition, respectively. Based on these conditions, general scaling functions are classified into "correct" (satisfying both conditions), "reasonable" (satisfying only collisional condition), and "approximate but still useful" (not satisfying collisional condition) classes. Several analytical scaling functions are listed for each class. Popular functions r_iA and r_ij belong to the reasonable class. The qualities of many electron-electron scaling functions are examined variationally for the helium atom using the FC theory. Although the complement functions of FC theory are produced generally from both the potential and kinetic operators in the Hamiltonian, those produced from the kinetic operator were shown to be less important than those produced from the potential operator. Hence, we used only the complement functions produced from the potential operator and showed that the correct-class g_ij functions gave most accurate results and the reasonable-class functions were less accurate. Among the examined correct and reasonable functions, the conventional function r_ij was worst in accuracy, although it was still very accurate. Thus, we have many potentially accurate "correct" scaling functions for use in FC theory to solve the SEs of atoms and molecules.

Accurate energies of transition metal atoms, ions, and monoxides using selected configuration interaction and density-based basis-set corrections

The semistochastic heat-bath configuration interaction method is a selected configuration interaction plus perturbation theory method that has provided near-full configuration interaction (FCI) levels of accuracy for many systems with both single- and multi-reference character. However, obtaining accurate energies in the complete basis-set limit is hindered by the slow convergence of the FCI energy with respect to basis size. Here, we show that the recently developed basis-set correction method based on range-separated density functional theory can be used to significantly speed up basis-set convergence in SHCI calculations. In particular, we study two such schemes that differ in the functional used and apply them to transition metal atoms and monoxides to obtain total, ionization, and dissociation energies well converged to the complete-basis-set limit within chemical accuracy.

Transcorrelated coupled cluster methods

Transcorrelated coupled cluster and distinguishable cluster methods are presented. The Hamiltonian is similarity transformed with a Jastrow factor in the first quantization, which results in up to three-body integrals. The coupled cluster with singles and doubles equations on this transformed Hamiltonian are formulated and implemented. It is demonstrated that the resulting methods have a superior basis set convergence and accuracy to the corresponding conventional and explicitly correlated methods. Additionally, approximations for three-body integrals are suggested and tested.

Self-consistent density-based basis-set correction: How much do we lower total energies and improve dipole moments?

This work provides a self-consistent extension of the recently proposed density-based basis-set correction method for wave function electronic-structure calculations [E. Giner et al., J. Chem. Phys. 149, 194301 (2018)]. In contrast to the previously used approximation where the basis-set correction density functional was a posteriori added to the energy from a wave-function calculation, here the energy minimization is performed including the basis-set correction. Compared to the non-self-consistent approximation, this allows one to lower the total energy and change the wave function under the effect of the basis-set correction. This work addresses two main questions: (i) What is the change in total energy compared to the non-self-consistent approximation and (ii) can we obtain better properties, namely, dipole moments, with the basis-set corrected wave functions. We implement the present formalism with two different basis-set correction functionals and test it on different molecular systems. The main results of the study are that (i) the total energy lowering obtained by the self-consistent approach is extremely small, which justifies the use of the non-self-consistent approximation, and (ii) the dipole moments obtained from the basis-set corrected wave functions are improved, being already close to their complete basis-set values with triple-zeta basis sets. Thus, the present study further confirms the soundness of the density-based basis-set correction scheme.

Focal-point approach with pair-specific cusp correction for coupled-cluster theory

We present a basis set correction scheme for the coupled-cluster singles and doubles (CCSD) method. The scheme is based on employing frozen natural orbitals (FNOs) and diagrammatically decomposed contributions to the electronic correlation energy, which dominate the basis set incompleteness error (BSIE). As recently discussed in the work of Irmler et al. [Phys. Rev. Lett. 123, 156401 (2019)], the BSIE of the CCSD correlation energy is dominated by the second-order Møller-Plesset (MP2) perturbation energy and the particle-particle ladder term. Here, we derive a simple approximation to the BSIE of the particle-particle ladder term that effectively corresponds to a rescaled pair-specific MP2 BSIE, where the scaling factor depends on the spatially averaged correlation hole depth of the coupled-cluster and first-order pair wavefunctions. The evaluation of the derived expressions is simple to implement in any existing code. We demonstrate the effectiveness of the method for the uniform electron gas. Furthermore, we apply the method to coupled-cluster theory calculations of atoms and molecules using FNOs. Employing the proposed correction and an increasing number of FNOs per occupied orbital, we demonstrate for a test set that rapidly convergent closed and open-shell reaction energies, atomization energies, electron affinities, and ionization potentials can be obtained. Moreover, we show that a similarly excellent trade-off between required virtual orbital basis set size and remaining BSIEs can be achieved for the perturbative triples contribution to the CCSD(T) energy employing FNOs and the (T*) approximation.

A new form of transcorrelated Hamiltonian inspired by range-separated DFT

The present work introduces a new form of explicitly correlated factor in the context of the transcorrelated methods. The new correlation factor is obtained from the r₁₂ ≈ 0 mathematical analysis of the transcorrelated Hamiltonian, and its analytical form is obtained such that the leading order in 1/r₁₂ of the scalar part of the effective two-electron potential reproduces the long-range interaction of the range-separated density functional theory. The resulting correlation factor exactly imposes the cusp and is tuned by a unique parameter μ, which controls both the depth of the coulomb hole and its typical range in r₁₂. The transcorrelated Hamiltonian obtained with such a new correlation factor has a straightforward analytical expression depending on the same parameter μ, and its physical contents continuously change by varying μ: One can change from a non-divergent repulsive Hamiltonian at large μ to a purely attractive one at small μ. We investigate the convergence of the ground state eigenvalues and right eigenvectors of such a new transcorrelated Hamiltonian as a function of the basis set and as a function of μ on a series of two-electron systems. We found that the convergence toward the complete basis set is much faster for quite a wide range of values of μ. We also propose a specific value of μ, which essentially reproduces the results obtained with the frozen Gaussian geminal introduced by Ten-no [Chem. Phys. Lett. 330, 169 (2000)].

Construction of explicitly correlated one-electron reduced density matrices

A general construction of an ensemble N-representable one-electron reduced density matrix Γ1(r₁→^';r→₁) is presented. Unlike the conventional spectral representation, it explicitly incorporates the recently derived discontinuity in the fifth derivative of Γ1(r₁→^';r→₁) with respect to |r₁→^'-r→₁|. Its practical relevance in the context of the density-matrix functional theory is discussed.

Canonical and DLPNO-Based Composite Wavefunction Methods Parametrized against Large and Chemically Diverse Training Sets. 2: Correlation-Consistent Basis Sets, Core-Valence Correlation, and F12 Alternatives

A hierarchy of wavefunction composite methods (cWFT), based on G4-type cWFT methods available for elements H through Rn, was recently reported by the present authors [J. Chem. Theor. Comput.2020, 16, 4238]. We extend this hierarchy by considering the inner-shell correlation energy in the second-order Møller–Plesset correction and replacing the Weigend–Ahlrichs def2-mZVPP(D) basis sets used with complete basis set extrapolation from augmented correlation-consistent core–valence triple-ζ, aug-cc-pwCVTZ(-PP), and quadruple-ζ, aug-cc-pwCVQZ(-PP), basis sets, thus creating cc-G4-type methods. For the large and chemically diverse GMTKN55 benchmark suite, they represent a substantial further improvement and bring WTMAD2 (weighted mean absolute deviation) down below 1 kcal/mol. Intriguingly, the lion’s share of the improvement comes from better capture of valence correlation; the inclusion of core–valence correlation is almost an order of magnitude less important. These robust correlation-consistent cWFT methods approach the CCSD(T) complete basis limit with just one or a few fitted parameters. Particularly, the DLPNO variants such as cc-G4-T-DLPNO are applicable to fairly large molecules at a modest computational cost, as is (for a reduced range of elements) a different variant using MP2-F12/cc-pVTZ-F12 for the MP2 component.

A Density-Based Basis-Set Correction for Wave Function Theory

We report a universal density-based basis-set incompleteness correction that can be applied to any wave function method. This correction, which appropriately vanishes in the complete basis-set (CBS) limit, relies on short-range correlation density functionals (with multideterminant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis-set incompleteness error. Contrary to conventional RS-DFT schemes that require an ad hoc range-separation parameter μ, the key ingredient here is a range-separation function μ(r) that automatically adapts to the spatial nonhomogeneity of the basis-set incompleteness error. As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) atomization and correlation energies near the CBS limit for the G2 set of molecules with compact Gaussian basis sets.

Do CCSD and approximate CCSD-F12 variants converge to the same basis set limits? The case of atomization energies

While the title question is a clear "yes" from purely theoretical arguments, the case is less clear for practical calculations with finite (one-particle) basis sets. To shed further light on this issue, the convergence to the basis set limit of CCSD (coupled cluster theory with all single and double excitations) and of different approximate implementations of CCSD-F12 (explicitly correlated CCSD) has been investigated in detail for the W4-17 thermochemical benchmark. Near the CBS ([1-particle] complete basis set) limit, CCSD and CCSD(F12^*) agree to within their respective uncertainties (about ±0.04 kcal/mol) due to residual basis set incompleteness error, but a nontrivial difference remains between CCSD-F12b and CCSD(F12^*), which is roughly proportional to the degree of static correlation. The observed basis set convergence behavior results from the superposition of a rapidly converging, attractive, CCSD[F12]-CCSD-F12b difference (consisting mostly of third-order terms) and a more slowly converging, repulsive, fourth-order difference between CCSD(F12^*) and CCSD[F12]. For accurate thermochemistry, we recommend CCSD(F12^*) over CCSD-F12b if at all possible. There are some indications that the nZaPa family of basis sets exhibits somewhat smoother convergence than the correlation consistent family.

On the (N, Z) dependence of the second-order Møller-Plesset correlation energies for closed-shell atomic systems

Accurate Møller-Plesset (MP2) correlation energies calculated by means of the variational-perturbation and the finite-element methods are presented for several members of the Cu(+) isoelectronic series (N = 28), which represent closed-shell systems containing for the first time the 3d(10)-electron configuration and, consequently, closed M-shell. Total MP2 energies as well as their inner- and inter-shell components are reported for Cu(+), Zn(2+), Ge(4+), Kr(8+), Sr(10+), and Cd(20+). We found that for these ions the Z-dependence of the total MP2 energies is significantly weaker than for the members of the Ar-like series. The origin of this fact is rationalized by a detailed analysis performed at the levels of the shell- and inter-shell contributions to the MP2 energies. To get, for the first time, more general information about the (N, Z) characteristics of the MP2 energies for closed-shell atomic systems, we compare the Z-dependence of the Cu(+)-like systems with the MP2 energies calculated for other isoelectronic series. The weak Z-dependence is found for the He-, Ne-, and Cu(+)-like series, which consist of atoms having perfectly closed-shell K-, KL-, and KLM-electronic structures, respectively. In turn, for the Be-, Mg-, and Ar-series, the Z-dependence is considerably stronger.

Electron-correlated fragment-molecular-orbital calculations for biomolecular and nano systems

Recent developments in the fragment molecular orbital (FMO) method for theoretical formulation, implementation, and application to nano and biomolecular systems are reviewed. The FMO method has enabled ab initio quantum-mechanical calculations for large molecular systems such as protein-ligand complexes at a reasonable computational cost in a parallelized way. There have been a wealth of application outcomes from the FMO method in the fields of biochemistry, medicinal chemistry and nanotechnology, in which the electron correlation effects play vital roles. With the aid of the advances in high-performance computing, the FMO method promises larger, faster, and more accurate simulations of biomolecular and related systems, including the descriptions of dynamical behaviors in solvent environments. The current status and future prospects of the FMO scheme are addressed in these contexts.