A basis-set error correction based on density-functional theory for strongly correlated molecular systems

Exploring the role of mean-field potentials and short-range wave function behavior in the adiabatic connection

In this article, we explore the construction of Hamiltonians with long-range interactions and their corrections using the short-range behavior of the wave function. A key aspect of our investigation is the examination of the one-particle potential, kept constant in our previous work, and the effects of its optimization on the adiabatic connection. Our methodology involves the use of a parameter-dependent potential dependent on a single parameter to facilitate practical computations. We analyze the energy errors and densities in a two-electron system (harmonium) under various conditions, employing different confinement potentials and interaction parameters. The study reveals that while the mean-field potential improves the expectation value of the physical Hamiltonian, it does not necessarily improve the energy of the system within the bounds of chemical accuracy. We also delve into the impact of density variations in adiabatic connections, challenging the common assumption that a mean field improves results. Our findings indicate that as long as energy errors remain within chemical accuracy, the mean field does not significantly outperform a bare potential. This observation is attributed to the effectiveness of corrections based on the short-range behavior of the wave function, a universal characteristic that diminishes the distinction between using a mean field or not.

Long-Range Configuration Interaction with an Ab Initio Short-Range Correction and an Asymptotic Lower Bound†

Short-range corrections to long-range selected configuration interaction calculations are derived from perturbation theory considerations and applied to harmonium (with two to six electrons for some low-lying states). No fitting to reference data is used, and the method is applicable to ground and excited states. The formulas derived are rigorous when the physical interaction is approached. In this regime, the second-order expression provides a lower bound to the long-range full configuration interaction energy. A long-range/short-range separation of the interaction between electrons at a distance of the order of one atomic unit provides total energies within chemical accuracy, and, for the systems studied, provide better results than short-range density functional approximations.

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.

Modified Expression for the Hamiltonian Expectation Value Exploiting the Short-Range Behavior of the Wave Function

The expectation value of the Hamiltonian using a model wave function is widely used to estimate the eigenvalues of electronic Hamiltonians. We explore here a modified formula for models based on a long-range interaction. It scales differently the singlet and triplet components of the repulsion between electrons not present in the model (its short-range part). The scaling factors depend uniquely on the parameter used in defining the model interaction and are constructed using only exact properties. We show results for the ground states and low-lying excited states of Harmonium with two to six electrons. We obtain important improvements for the estimation of the exact energy, not only over the model energy but also over the expectation value of the Hamiltonian.

A density-fitting implementation of the density-based basis-set correction method

This work reports an efficient density-fitting implementation of the density-based basis-set correction (DBBSC) method in the MOLPRO software. This method consists in correcting the energy calculated by a wave-function method with a given basis set by an adapted basis-set correction density functional incorporating the short-range electron correlation effects missing in the basis set, resulting in an accelerated convergence to the complete-basis-set limit. Different basis-set correction density-functional approximations are explored and the complementary-auxiliary-basis-set single-excitation correction is added. The method is tested on a benchmark set of reaction energies at the second-order Møller-Plesset (MP2) level and a comparison with the explicitly correlated MP2-F12 method is provided. The results show that the DBBSC method greatly accelerates the basis convergence of MP2 reaction energies, without reaching the accuracy of the MP2-F12 method but with a lower computational cost.

Basis Set Limit of CCSD(T) Energies: Explicit Correlation Versus Density-Based Basis-Set Correction

A thorough comparison is carried out for explicitly correlated and density-based basis-set correction approaches, which were primarily developed to mitigate the basis-set incompleteness error of wave function methods. An efficient implementation of the density-based scheme is also presented, utilizing the density-fitting approximation. The performance of these approaches is comprehensively tested for the second-order Møller-Plesset (MP2), coupled-cluster singles and doubles (CCSD), and CCSD with perturbative triples [CCSD(T)] methods with respect to the corresponding complete basis set references. It is demonstrated that the density-based correction together with complementary auxiliary basis set (CABS)-corrected Hartree-Fock energies is highly robust and effectively reduces the error of the standard approaches; however, it does not outperform the corresponding explicitly correlated methods. Nevertheless, what still makes the density-corrected CCSD and CCSD(T) methods competitive is that their computational costs are roughly half of those of the corresponding explicitly correlated variants. Additionally, an incremental approach for standard CCSD and CCSD(T) is introduced. In this simple scheme, the total energies are corrected with the CABS correction and explicitly correlated MP2 contributions. As demonstrated, the resulting methods yield surprisingly good results, below 1 kcal/mol for thermochemical properties even with a double-ζ basis, while their computational expenses are practically identical to those of the density-based basis-set correction approaches.

Ground- and Excited-State Dipole Moments and Oscillator Strengths of Full Configuration Interaction Quality

We report ground- and excited-state dipole moments and oscillator strengths (computed in different "gauges" or representations) of full configuration interaction (FCI) quality using the selected configuration interaction method known as Configuration Interaction using a Perturbative Selection made Iteratively (CIPSI). Thanks to a set encompassing 35 ground- and excited-state properties computed in 11 small molecules, the present near-FCI estimates allow us to assess the accuracy of high-order coupled-cluster (CC) calculations including up to quadruple excitations. In particular, we show that incrementing the excitation degree of the CC expansion (from CC with singles and doubles (CCSD) to CC with singles, doubles, and triples (CCSDT) or from CCSDT to CC with singles, doubles, triples, and quadruples (CCSDTQ)) reduces the average error with respect to the near-FCI reference values by approximately 1 order of magnitude.

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.

Explicitly Correlated Electronic Structure Calculations with Transcorrelated Matrix Product Operators

In this work, we present the first implementation of the transcorrelated electronic Hamiltonian in an optimization procedure for matrix product states by the density matrix renormalization group (DMRG) algorithm. In the transcorrelation ansatz, the electronic Hamiltonian is similarity-transformed with a Jastrow factor to describe the cusp in the wave function at electron-electron coalescence. As a result, the wave function is easier to approximate accurately with the conventional expansion in terms of one-particle basis functions and Slater determinants. The transcorrelated Hamiltonian in first quantization comprises up to three-body interactions, which we deal with in the standard way by applying robust density fitting to two- and three-body integrals entering the second-quantized representation of this Hamiltonian. The lack of hermiticity of the transcorrelated Hamiltonian is taken care of along the lines of the first work on transcorrelated DMRG [ J. Chem. Phys. 2020, 153, 164115] by encoding it as a matrix product operator and optimizing the corresponding ground state wave function with imaginary-time time-dependent DMRG. We demonstrate our quantum chemical transcorrelated DMRG approach at the example of several atoms and first-row diatomic molecules. We show that transcorrelation improves the convergence rate to the complete basis set limit in comparison to conventional DMRG. Moreover, we study extensions of our approach that aim at reducing the cost of handling the matrix product operator representation of the transcorrelated Hamiltonian.

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 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.

Variational coupled cluster for ground and excited states

In single-reference coupled-cluster (CC) methods, one has to solve a set of non-linear polynomial equations in order to determine the so-called amplitudes that are then used to compute the energy and other properties. Although it is of common practice to converge to the (lowest-energy) ground-state solution, it is also possible, thanks to tailored algorithms, to access higher-energy roots of these equations that may or may not correspond to genuine excited states. Here, we explore the structure of the energy landscape of variational CC and we compare it with its (projected) traditional version in the case where the excitation operator is restricted to paired double excitations (pCCD). By investigating two model systems (the symmetric stretching of the linear H₄ molecule and the continuous deformation of the square H₄ molecule into a rectangular arrangement) in the presence of weak and strong correlations, the performance of variational pCCD (VpCCD) and traditional pCCD is gauged against their configuration interaction (CI) equivalent, known as doubly occupied CI, for reference Slater determinants made of ground- or excited-state Hartree-Fock orbitals or state-specific orbitals optimized directly at the VpCCD level. The influence of spatial symmetry breaking is also investigated.

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.