Approximate Natural Orbitals Research Articles

Quantum Information Driven Ansatz (QIDA): Shallow-Depth Empirical Quantum Circuits from Quantum Chemistry.

Hardware-efficient empirical variational ansätze for Variational Quantum Eigensolver (VQE) simulations of quantum chemistry often lack a direct connection to classical quantum chemistry methods. In this work, we propose a method to bridge this gap by introducing a novel approach to constructing a starting point for variational quantum circuits, leveraging quantum mutual information from classical quantum chemistry states to design simple yet effective heuristic ansätze with a topology reflecting the molecular system's correlations. As a first step, we make use of quantum chemistry calculations, such as Mo̷ller-Plesset (MP2) perturbation theory, to initially provide approximate Natural Orbitals, which have been shown to be the best candidate one-electron basis for developing compact empirical wave functions.1 Second, we evaluate the quantum mutual information matrix, which provides insights about the main correlations between qubits of the quantum circuit, and enables a direct design of entangling blocks for the circuit. The resulting ansatz is then used with a VQE to obtain a short-depth variational ground state of the electronic Hamiltonian. To validate our approach, we perform a comprehensive statistical analysis through simulations of various molecular systems (H2, LiH, H2O) and apply it to the more complex NH3 molecule. The reported results demonstrate that the proposed methodology gives rise to highly effective ansätze, outperforming the standard empirical ladder-entangler ansatz. Overall, our approach can be used as an effective state preparation, providing a promising route for designing efficient variational quantum circuits for large molecular systems.

Adsorption and vibrational spectroscopy of CO on the surface of MgO from periodic local coupled-cluster theory.

The adsorption of CO on the surface of MgO has long been a model problem in surface chemistry. Here, we report periodic Gaussian-based calculations for this problem using second-order perturbation theory (MP2) and coupled-cluster theory with single and double excitations (CCSD) and perturbative triple excitations [CCSD(T)], with the latter two performed using a recently developed extension of the local natural orbital approximation to problems with periodic boundary conditions. The low cost of periodic local correlation calculations allows us to calculate the full CCSD(T) binding curve of CO approaching the surface of MgO (and thus the adsorption energy) and the two-dimensional potential energy surface (PES) as a function of the distance from the surface and the CO stretching coordinate. From the PES, we obtain the fundamental vibrational frequency of CO on MgO, whose shift from the gas phase value is a common experimental probe of surface adsorption. We find that CCSD(T) correctly predicts a positive frequency shift upon adsorption of +14.7 cm-1, in excellent agreement with the experimental shift of +14.3 cm-1. We use our CCSD(T) results to assess the accuracy of MP2, CCSD, and several density functional theory (DFT) approximations, including exchange correlation functionals and dispersion corrections. We find that MP2 and CCSD yield reasonable binding energies and frequency shifts, whereas many DFT calculations overestimate the magnitude of the adsorption energy by 5-15 kJ mol-1 and predict a negative frequency shift of about -20 cm-1, which we attribute to self-interaction-induced delocalization errors that are mildly ameliorated with hybrid functionals. Our findings highlight the accuracy and computational efficiency of the periodic local correlation for the simulation of surface chemistry with accurate wavefunction methods.

Calculation of the inner-shell contribution to the correlation energy through DLPNO-CEPA/1 and scaled same-spin second-order Møller-Plesset perturbation theory.

The use of two low-cost methods for the prediction of the inner-shells contribution to the correlation energy is analyzed. The Spin-Component-Scaled second-order Møller-Plesset perturbation theory (SCS-MP2) was reparameterized for the prediction of such contributions. The best results are found when only the same spin term is considered (SSS-MP2). The Coupled Electron Pair Approximation (CEPA) using the Domain-based Local Pair Natural Orbital approximation (DLPNO) was also studied for the same purpose. The methods were tested for atomization energies on the W4-11 test set using basis sets up to quadruple zeta quality. The SSS-MP2 proved to be a marked improvement upon MP2 decreasing the root-mean-square-error (RMSE) from 0.443 to 0.302 kcal mol-1 . The RMSE of DLPNO-CEPA/1 in the test set is only 0.147 kcal mol-1 and its computational cost is very low considering the intended applications. Furthermore, a linear combination of both methods decreased the RMSE to 0.118 kcal mol-1 .

Extension of frozen natural orbital approximation to open-shell references: Theory, implementation, and application to single-molecule magnets.

Natural orbitals are often used to achieve a more compact representation of correlated wave-functions. Using natural orbitals computed as eigenstates of the virtual-virtual block of the state density matrix instead of the canonical Hartree-Fock orbitals results in smaller errors when the same fraction of virtual space is frozen. This strategy, termed frozen natural orbital (FNO) approach, is effective in reducing the cost of regular coupled-cluster (CC) calculations and some multistate methods, such as EOM-IP-CC (equation-of-motion CC for ionization potentials). This contribution extends the FNO approach to the EOM-SF-CC ansatz (EOM-CC with spin-flip). In contrast to EOM-IP-CCSD, EOM-SF-CCSD relies on high-spin open-shell references. Using FNOs computed for an open-shell reference leads to an erratic behavior of the EOM-SF-CC energies and properties due to an inconsistent truncation of the α and β orbital spaces. A general solution to problems arising in the EOM-CC calculations utilizing open-shell references, termed OSFNO (open-shell FNO), is proposed. By means of singular value decomposition (SVD) of the overlap matrix of the α and β orbitals, the OSFNO algorithm identifies the corresponding orbitals and determines virtual orbitals corresponding to the singly occupied space. This is followed by SVD of the singlet part of the state density matrix in the remaining virtual orbital subspace. The so-computed FNOs preserve the spin purity of the open-shell orbital subspace to the extent allowed by the original reference, thus facilitating a safe truncation of the virtual space. The performance of OSFNO is benchmarked for selected diradicals and triradicals.

RPA natural orbitals and their application to post-Hartree-Fock electronic structure methods.

We present a method to approximate post-Hartree-Fock correlation energies by using approximate natural orbitals obtained by the random phase approximation (RPA). We demonstrate the method by applying it to the helium atom, the hydrogen and fluorine molecule, and to diamond as an example of a periodic system. For these benchmark systems, we show that RPA natural orbitals converge the MP2 correlation energy rapidly. Additionally, we calculated full configuration interaction energies for He and H2, which are in excellent agreement with the literature and experimental values. We conclude that the proposed method may serve as a compromise to reach good approximations to correlation energies at moderate computational cost, and we expect the method to be especially useful for theoretical studies on surface chemistry by providing an efficient basis to correlated wave function based methods.

The PNO–MP2 gradient and its application to molecular geometry optimisations

ABSTRACTA preliminary implementation of a gradient for MP2 based on the pair natural orbital approximation (PNO–MP2) is presented. The accuracy of the PNO approximation for geometries is investigated by comparing bond lengths and bond angles of the PNO–MP2 structures with results from canonical MP2. Special emphasis is put on the optimisation of weakly bound complexes, such as differently sized water clusters. It is found that the error introduced by the PNO approximation is already for moderate PNO thresholds negligible compared to the inherent error of MP2. However, too loose PNO thresholds lead to convergence difficulties in geometry optimisations. This is observed in particular for floppy groups as, for example, methyl groups with low rotational barriers. Compared to optimisations with canonical MP2, the convergence thresholds have to be mainly energy and less coordinate based to comply with the slight roughness of the potential energy surface which is introduced by the PNO selection.

Local reduced-density-matrix-functional theory: Incorporating static correlation effects in Kohn-Sham equations

We propose a novel scheme to bring reduced density matrix functional theory (RDMFT) into the realm of density functional theory (DFT) that preserves the accurate density functional description at equilibrium, while incorporating accurately static and left-right correlation effects in molecules and keeping the good computational performance of DFT-based schemes. The key ingredient is to relax the requirement that the local potential is the functional derivative of the energy with respect to the density. Instead, we propose to restrict the search for the approximate natural orbitals within a domain where these orbitals are eigenfunctions of a single-particle hamiltonian with a local effective potential. In this way, fractional natural occupation numbers are accommodated into Kohn-Sham equations allowing for the description of molecular dissociation without breaking spin symmetry. Additionally, our scheme provides a natural way to connect an energy eigenvalue spectrum to the approximate natural orbitals and this spectrum is found to represent accurately the ionization potentials of atoms and small molecules.

Correlated Ab Initio and Density Functional Studies on H2 Activation by FeO(.).

The reaction FeO(+) + H2 → Fe(+) + H2O is a simple model for hydrogen abstraction processes in biologically important heme systems. The geometries of all relevant stationary points on the lowest sextet and quartet surfaces were optimized using several density functionals as well as the CASSCF method. The corresponding energy profiles were computed at the following levels: density functional theory using gradient-corrected, hybrid, meta, hybrid-meta, and perturbatively corrected double hybrid functionals; single-reference coupled cluster theory including up to single, double, triple, and perturbative quadruple excitations [CCSDT(Q)]; correlated multireference ab initio methods (MRCI, MRAQCC, SORCI, SORCP, MRMP2, NEVPT2, and CASPT2). The calculated energies were corrected for scalar relativistic effects, zero-point vibrational energies, and core-valence correlation effects. MRCI and SORCI energies were corrected for size-consistency errors using an a posteriori Davidson correction (+Q) leading to MRCI+Q and SORCI+Q. Comparison with the available experimental data shows that CCSDT(Q) is most accurate and can thus serve as benchmark method for this electronically challenging reaction. Among the density functionals, B3LYP performs best. In the correlated ab initio calculations with a full-valence active space, SORCI+Q yields the lowest deviations from the CCSDT(Q) reference results, with qualitatively similar energy profiles being obtained from MRCI+Q and MRAQCC. SORCI+Q benefits from the quality of the approximate average natural orbitals used in the final step of the SORCI procedure. Many of the tested methods show surprisingly large errors. The present results validate the common use of B3LYP in computational studies of heme systems and offer guidance on which correlated ab initio methods are most suitable for such studies.

Basis Set Convergence of the Post-CCSD(T) Contribution to Noncovalent Interaction Energies.

We investigated the basis set convergence of high-order coupled-cluster interaction energy contributions for 21 small weakly bound complexes. By performing CCSDT(Q) calculations in at least the aug-cc-pVTZ basis set, and CCSDT calculations in at least aug-cc-pVQZ (aug-cc-pVTZ for one system), we found the convergence to be quite slow. In particular, the 6-31G*(0.25) and 6-31G**(0.25,0.15) bases advocated by Hobza et al. (J. Chem. Theory Comput. 2013, 9, 2151; ibid. 2013, 9, 3420) are unsuitable for the post-CCSD(T) effects, with average errors for the CCSDT(Q)-CCSD(T) interaction energy contribution of about 80% for 6-31G**(0.25,0.15) and 110% for 6-31G*(0.25). Upgrading the basis set to aug-cc-pVDZ reduces the average error to about 35% and extremely demanding CCSDT(Q)/aug-cc-pVTZ calculations are necessary for further improvement in accuracy. An error cancellation between basis set incompleteness effects at the CCSDT-CCSD(T) and CCSDT(Q)-CCSDT levels occurs for most (but not all) complexes, making it unproductive to carry out CCSDT calculations in a larger basis set than the more demanding CCSDT(Q) calculations. We also found that the frozen natural orbital approximation at the CCSDT and CCSDT(Q) levels works well only if the thresholds for discarding least occupied natural orbitals are very tight (significantly tighter than the thresholds recommended for molecular correlation energies in the original work of Rolik and Kállay, J. Chem. Phys. 2011, 134, 124111), making the performance gains quite limited. The interaction energy contributions through CCSDT(Q) are both a necessity and a bottleneck in the construction of top-accuracy interaction potentials and further improvements in the efficiency of high-order coupled-cluster calculations will be of great help.

Molecule-Optimized Basis Sets and Hamiltonians for Accelerated Electronic Structure Calculations of Atoms and Molecules

Molecule-optimized basis sets, based on approximate natural orbitals, are developed for accelerating the convergence of quantum calculations with strongly correlated (multireferenced) electrons. We use a low-cost approximate solution of the anti-Hermitian contracted Schrödinger equation (ACSE) for the one- and two-electron reduced density matrices (RDMs) to generate an approximate set of natural orbitals for strongly correlated quantum systems. The natural-orbital basis set is truncated to generate a molecule-optimized basis set whose rank matches that of a standard correlation-consistent basis set optimized for the atoms. We show that basis-set truncation by approximate natural orbitals can be viewed as a one-electron unitary transformation of the Hamiltonian operator and suggest an extension of approximate natural-orbital truncations through two-electron unitary transformations of the Hamiltonian operator, such as those employed in the solution of the ACSE. The molecule-optimized basis set from the ACSE improves the accuracy of the equivalent standard atom-optimized basis set at little additional computational cost. We illustrate the method with the potential energy curves of hydrogen fluoride and diatomic nitrogen. Relative to the hydrogen fluoride potential energy curve from the ACSE in a polarized triple-ζ basis set, the ACSE curve in a molecule-optimized basis set, equivalent in size to a polarized double-ζ basis, has a nonparallelity error of 0.0154 au, which is significantly better than the nonparallelity error of 0.0252 au from the polarized double-ζ basis set.

State-averaged Monte Carlo configuration interaction applied to electronically excited states

We introduce state-averaging into the method of Monte Carlo configuration interaction (SA-MCCI) to allow the stable and efficient calculation of excited states. We show that excited potential curves for H3, including a crossing with the ground state, can be accurately reproduced using a small fraction of the full configuration interaction (FCI) space. A recently introduced error measure for potential curves [J. P. Coe and M. J. Paterson, J. Chem. Phys. 137, 204108 (2012)] is also shown to be a fair approach when considering potential curves for multiple states. We demonstrate that potential curves for LiF using SA-MCCI agree well with the FCI results and the avoided crossing occurs correctly. The seam of conical intersections for CH2 found by Yarkony [J. Chem. Phys. 104, 2932 (1996)] is used as a test for SA-MCCI and we compare potential curves from SA-MCCI with FCI results for this system for the first three triplet states. We then demonstrate the improvement from using SA-MCCI on the dipole of the 2 (1)A1 state of carbon monoxide. We then look at vertical excitations for small organic molecules up to the size of butadiene where the SA-MCCI energies and oscillator strengths are compared with CASPT2 values [M. Schreiber, M. R. Silva-Junior, S. P. A. Sauer, and W. Thiel, J. Chem. Phys. 128, 134110 (2008)]. We finally see if the SA-MCCI results for these excitation energies can be improved by using MCCIPT2 with approximate natural orbitals when the PT2 space is not onerously large.

Development of Monte Carlo configuration interaction: Natural orbitals and second-order perturbation theory

Approximate natural orbitals are investigated as a way to improve a Monte Carlo configuration interaction (MCCI) calculation. We introduce a way to approximate the natural orbitals in MCCI and test these and approximate natural orbitals from Møller-Plesset perturbation theory and quadratic configuration interaction with single and double substitutions in MCCI calculations of single-point energies. The efficiency and accuracy of approximate natural orbitals in MCCI potential curve calculations for the double hydrogen dissociation of water, the dissociation of carbon monoxide, and the dissociation of the nitrogen molecule are then considered in comparison with standard MCCI when using full configuration interaction as a benchmark. We also use the method to produce a potential curve for water in an aug-cc-pVTZ basis. A new way to quantify the accuracy of a potential curve is put forward that takes into account all of the points and that the curve can be shifted by a constant. We adapt a second-order perturbation scheme to work with MCCI (MCCIPT2) and improve the efficiency of the removal of duplicate states in the method. MCCIPT2 is tested in the calculation of a potential curve for the dissociation of nitrogen using both Slater determinants and configuration state functions.

Accurate and fast treatment of large molecular systems: Assessment of CEPA and pCCSD within the local pair natural orbital approximation

The local pair natural orbital approach, which has been combined with two post-Hartree-Fock methods, CEPA-1 and pCCSD-1a, recently, is assessed for its applicability to large real-world problems without abundant computing resources. Test cases are selected based on being representative for computational chemistry problems and availability of reliable reference data. Both methods show a good performance and can be applied easily to systems of up to 100 atoms when very accurate energies are sought after. A considerable demand for basis sets of good quality has been identified and practical guidelines to satisfy this are mapped out.

A test of composite natural orbitals for benzene

Approximate natural orbitals (NO's) of a larger system can be constructed from the NO's of smaller fragment systems. These orbitals, called composite NO's (CNO's) are expected to be useful in configuration interaction (CI) calculations. The effectiveness of these NO's is shown for the benzene molecule. This molecule is considered a combination of three ethylenes. The CI calculations were carried out for the S1 – S3 and T1 – T3 states. We take into account single and double excitations from σ and π electrons in the CI calculations. The calculated excitation energies are in good agreement with the experimental values Keywords: benzene, π–π* excited state, composite natural orbital, ionic and covalent, SDCI.

Interaction potentials in rare gas-halide ion systems

The interaction potentials in Ne…F −, Ne…Cl −, Ar…F −, and Ar…Cl − are calculated at the SCF and correlated levels of approximation with extended basis sets of approximate natural orbitals. The present results are compared with the predictions of the extended Tang-Toennies (ETT) model and with the experimental values obtained by the inversion of the ion mobility data. For systems involving the fluoride ion the calculated interaction parameters (NeF −: R min = 6.10 au, E min = 0.64 mhartree; ArF −: R min = 5.84 au, E min = 3.35 mhartree) differ to some extent from the ETT data; the differences being attributed to some uncertainties in empirical parameters for the F − ion used in ETT calculations. In the case of the Cl − ion interactions (NeCl −: R min = 7.35 au, E min = 0.42 mhartree; ArCl −: R min = 7.05 au, E min = 2.16 mhartree) good agreement with the corresponding ETT data is obtained. Moreover, certain discrepancies between the present results and experimental interaction parameters are found and interpreted in favor of the theoretical data.

Approximate natural orbitals and the convergence of a second order multireference many-body perturbation theory (CIPSI) algorithm

In this paper we discuss a method to obtain approximate natural orbitals (ANOs) by means of the zero-order and first-order one-particle reduced density matrices. Convergence of the multireference second-order perturbation CI by means of CIPSI method, in its two-class and three-class versions, appears to be much improved when using these ANOs. The method requires only a single additional integral transformation. A comparison with canonical and valence virtual orbitals defined through the PHO procedure is also reported. Finally, the use of ANOs together with the multireference analog of the Davidson’s correction, as well as a suitable extrapolation procedure in the CIPSI method is discussed. Results are compared with available full CI calculations for H2O in a double-ξ and a double-ξ plus polarization basis set.

Composite natural orbitals for configuration interaction method

A new method of constructing approximate natural orbitals (NO's) is proposed. A system can usually be regarded as consisting of fragments. The approximate NO's of the entire system are constructed from the NO's of fragments in their ground states. These orbitals are expected to be useful in configuration interaction (CI) calculations of a large system where orbital truncation is necessary. The usefulness of these approximate NO's, which are called composite NO's (CNO's), is demonstrated on C4H6. This molecule may be regarded as a combination of two fragments i.e., C2H4's. Excitation energies calculated by single and double excitation CI with the truncated CNO's (14σ, 7π correlating orbitals) agree reasonably well with those calculated with a full orbital set (37σ, 12π correlating orbitals), while the truncated SCFMO's of the same size give disastrous excitation energies.

Method of approximate natural orbitals

Method of approximate natural orbitals

Inclusion of some higher order effects with the self-consistent electron pairs method for correlated molecular wavefunctions

An efficient technique is presented for the inclusion of some higher order effects in correlated wavefunctions. Diagonal quadruple substitutions (e.g., i2j2→ a2b2) are variationally added to a self-consistent electron pairs (SCEP) wavefunction that includes single and double substitutions relative to a closed shell reference configuration. Through the use of approximate natural orbitals, these configurations include those which are important for the size-consistency correction at the limit of separated systems. While the number of the diagonal quadruple configurations is of the order of the number of doubles, the computational effort for including the quadruples is less than for the doubles, because of the nature of the required matrix elements and the flexibility of SCEP. Variationally additive quadruple corrections to SCEP pair energies are easily found with this new procedure.

A critical study of basis set effects and the use of approximate natural orbitals in SCF-CI calculations of molecular geometries and heats of reaction

SCF-CI calculations have been performed on a number of chemical reactions between closed shell molecules in order to determine the heats of reaction. Contracted Gaussian type atomic basis sets of three different qualities were used and the CI calculations were performed in a truncated approximate natural orbital space. The conclusions to be drawn from these calculations are rather pessimistic. For heats of reaction, errors up to 6 kcal/mole are obtained on the SCF-level with a double zeta plus polarization atomic basis. A further improvement is only possible if extended basis sets are used. Correlation effects on heats of reaction are of the same size and CI calculations are therefore only meaningful with large atomic basis sets. For the CI calculations a one-electron space of approximate natural orbitals, obtained from second order RS perturbation theory, was used. Different truncations, using the occupation number as criterion, were tested. The general conclusion is that errors in energy differences obtained with a truncated basis set are of the same magnitude as the error in the total correlation energy. In practice this means that not more than 20–30% of the approximate natural orbitals can be deleted if the error is to be kept less than a few kcal/mole. Finally the truncation error in calculations of bond distances was tested for a few cases. Errors of around 10% of the total change due to correlation were found when 30% of the lowest occupied natural orbitals were deleted.