Frozen Natural Orbital Research Articles

Toward Accurate Quantum Chemical Methods for Molecules of Increasing Dimension: The New Family of Pisa Composite Schemes.

The new versions of the Pisa composite scheme introduced in the present paper are based on the careful selection of different quantum chemical models for energies, geometries, and vibrational frequencies, with the aim of maximizing the accuracy of the overall description while retaining a reasonable cost for all the steps. In particular, the computation of accurate electronic energies has been further improved introducing more reliable complete basis set extrapolations and estimation of core-valence correlation, together with improved basis sets for third-row atoms. Furthermore, the reduced-cost frozen natural orbital (FNO) model has been introduced and validated for large molecules. Accurate molecular structures can be obtained avoiding complete basis set extrapolation and evaluating core-valence correlation at the MP2 level. Unfortunately, analytical gradients are not available for the FNO version of the model. Therefore, for large molecules, an accurate reduced-cost alternative is offered by evaluation of valence contributions with a double-hybrid functional in conjunction with the same MP2 contribution for core-valence correlation or by means of a one-parameter approximation. The same double-hybrid functional and basis set are employed to evaluate zero-point energies and partition functions. After the validation of the new models for small systems, a panel of molecular bricks of life has been used to analyze their performances for problems of current fundamental or technological interest. The fully black-box implementation of the computational workflow paves the way toward the accurate yet not prohibitively expensive study of medium- to large-sized molecules also by experimentally oriented researchers.

Frozen Natural Orbitals for the State-Averaged Driven Similarity Renormalization Group.

We present a reduced-cost implementation of the state-averaged driven similarity renormalization group (SA-DSRG) based on the frozen natural orbital (FNO) approach. The natural orbitals (NOs) are obtained by diagonalizing the one-body reduced density matrix from SA-DSRG second-order perturbation theory (SA-DSRG-PT2). We consider three criteria to truncate the virtual NOs for the subsequent electron correlation treatment beyond SA-DSRG-PT2. An additive second-order correction is applied to the SA-DSRG Hamiltonian to reintroduce correlation effects from the discarded orbitals. The FNO SA-DSRG method is benchmarked on 35 small organic molecules in the QUEST database. When keeping 98-99% of the cumulative occupation numbers, the mean absolute error in the vertical transition energies due to FNO is less than 0.01 eV. Using the same FNO threshold, we observe a speedup of 9 times compared to the conventional SA-DSRG implementation for nickel carbonyl with a quadruple-ζ basis set. The FNO approach enables nonperturbative SA-DSRG computations on chloroiron corrole [FeCl(C19H11N4)] with more than 1000 basis functions, surpassing the current limit of a conventional implementation.

Combining state-of-the-art quantum chemistry and machine learning make gold standard potential energy surfaces accessible for medium-sized molecules

Developing full-dimensional machine-learned potentials with the current “gold-standard” coupled-cluster (CC) level is challenging for medium-sized molecules due to the high computational cost. Consequently, researchers are often bound to use lower-level electronic structure methods such as density functional theory or second-order Møller–Plesset perturbation theory (MP2). Here, we demonstrate on a representative example that gold-standard potentials can now be effectively constructed for molecules of 15 atoms using off-the-shelf hardware. This is achieved by accelerating the CCSD(T) computations via the accurate and cost-effective frozen natural orbital (FNO) approach. The Δ-machine learning (Δ-ML) approach is employed with the use of permutationally invariant polynomials to fit a full-dimensional potential energy surface of the acetylacetone molecule, but any other effective descriptor and ML approach can similarly benefit from the accelerated data generation proposed here. Our benchmarks for the global minima, H-transfer TS, and many high-lying configurations show the excellent agreement of FNO-CCSD(T) results with conventional CCSD(T) while achieving a significant time advantage of about a factor of 30–40. The obtained Δ-ML PES shows high fidelity from multiple perspectives including energetic, structural, and vibrational properties. We obtain the symmetric double well H-transfer barrier of 3.15 kcal/mol in excellent agreement with the direct FNO-CCSD(T) barrier of 3.11 kcal/mol as well as with the benchmark CCSD(F12*)(T+)/CBS value of 3.21 kcal/mol. Furthermore, the tunneling splitting due to H-atom transfer is calculated using a 1D double-well potential, providing improved estimates over previous ones obtained using an MP2-based PES. The methodology introduced here represents a significant advancement in the efficient and precise construction of potentials at the CCSD(T) level for molecules above the current limit of 15 atoms.

Toward Accurate Post-Born-Oppenheimer Molecular Simulations on Quantum Computers: An Adaptive Variational Eigensolver with Nuclear-Electronic Frozen Natural Orbitals.

Nuclear quantum effects such as zero-point energy and hydrogen tunneling play a central role in many biological and chemical processes. The nuclear-electronic orbital (NEO) approach captures these effects by treating selected nuclei quantum mechanically on the same footing as electrons. On classical computers, the resources required for an exact solution of NEO-based models grow exponentially with system size. By contrast, quantum computers offer a means of solving this problem with polynomial scaling. However, due to the limitations of current quantum devices, NEO simulations are confined to the smallest systems described by minimal basis sets, whereas realistic simulations beyond the Born-Oppenheimer approximation require more sophisticated basis sets. For this purpose, we herein extend a hardware-efficient ADAPT-VQE method to the NEO framework in the frozen natural orbital (FNO) basis. We demonstrate on H2 and D2 molecules that the NEO-FNO-ADAPT-VQE method reduces the CNOT count by several orders of magnitude relative to the NEO unitary coupled cluster method with singles and doubles while maintaining the desired accuracy. This extreme reduction in the CNOT gate count is sufficient to permit practical computations employing the NEO method─an important step toward accurate simulations involving nonclassical nuclei and non-Born-Oppenheimer effects on near-term quantum devices. We further show that the method can capture isotope effects, and we demonstrate that inclusion of correlation energy systematically improves the prediction of difference in the zero-point energy (ΔZPE) between isotopes.

Basis Set Limit CCSD(T) Energies for Extended Molecules via a Reduced-Cost Explicitly Correlated Approach.

Several approximations are introduced and tested to reduce the computational expenses of the explicitly correlated coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method for both closed and open-shell species. First, the well-established frozen natural orbital (FNO) technique is adapted to explicitly correlated CC approaches. Second, our natural auxiliary function (NAF) scheme is employed to reduce the size of the auxiliary basis required for the density fitting approximation regularly used in explicitly correlated calculations. Third, a new approach, termed the natural auxiliary basis (NAB) approximation, is proposed to decrease the size of the auxiliary basis needed for the expansion of the explicitly correlated geminals. The performance of the above approximations and that of the combined FNO-NAF-NAB approach are tested for atomization and reaction energies. Our results show that overall speedups of 7-, 5-, and 3-times can be achieved with double-, triple-, and quadruple-ζ basis sets, respectively, without any loss in accuracy. The new method can provide, e.g., reaction energies and barrier heights well within chemical accuracy for molecules with more than 40 atoms within a few days using a few dozen processor cores, and calculations with 50+ atoms are still feasible. These routinely affordable computations considerably extend the reach of explicitly correlated CCSD(T).

Basis set truncation corrections for improved frozen natural orbital CCSD(T) energies

A number of approaches are proposed and assessed to reduce the frozen natural orbital (FNO) truncation error of coupled-cluster singles and doubles with perturbative triples [CCSD(T)] energies. The diagrammatic energy decomposition method of Irmler and Grüneis [J. Chem. Phys. 151, 104107 (2019)] is extended to the FNO truncation correction of the particle-particle ladder (PPL) term in the case of closed- and open-shell molecular systems. The approach is tested for reaction, interaction, and atomization energies, and it is found the most robust for a wider range of FNO truncation thresholds outperforming the commonly employed additive MP2 correction. We also show that the linear extrapolation (LE) of FNO-CCSD(T) energies as a function of second-order Møller–Plesset (MP2) energies provides the best correlation energies and most balanced energy differences with tighter FNO thresholds, but it lacks systematic error compensation that would be required for the better performance with looser FNO thresholds. Further insight is gained from a diagrammatic and spin-component decomposition based analysis. Moreover, orbital (pair) specific energy decompositions are utilised to introduce size-consistent variants of the promising PPL and LE FNO corrections and their analogues for (T), which are also readily applicable in the context of popular local correlation methods.

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.

Accurate Reduced-Cost CCSD(T) Energies: Parallel Implementation, Benchmarks, and Large-Scale Applications.

The accurate and systematically improvable frozen natural orbital (FNO) and natural auxiliary function (NAF) cost-reducing approaches are combined with our recent coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementations. Both of the closed- and open-shell FNO-CCSD(T) codes benefit from OpenMP parallelism, completely or partially integral-direct density-fitting algorithms, checkpointing, and hand-optimized, memory- and operation count effective implementations exploiting all permutational symmetries. The closed-shell CCSD(T) code requires negligible disk I/O and network bandwidth, is MPI/OpenMP parallel, and exhibits outstanding peak performance utilization of 50–70% up to hundreds of cores. Conservative FNO and NAF truncation thresholds benchmarked for challenging reaction, atomization, and ionization energies of both closed- and open-shell species are shown to maintain 1 kJ/mol accuracy against canonical CCSD(T) for systems of 31–43 atoms even with large basis sets. The cost reduction of up to an order of magnitude achieved extends the reach of FNO-CCSD(T) to systems of 50–75 atoms (up to 2124 atomic orbitals) with triple- and quadruple-ζ basis sets, which is unprecedented without local approximations. Consequently, a considerably larger portion of the chemical compound space can now be covered by the practically “gold standard” quality FNO-CCSD(T) method using affordable resources and about a week of wall time. Large-scale applications are presented for organocatalytic and transition-metal reactions as well as noncovalent interactions. Possible applications for benchmarking local CCSD(T) methods, as well as for the accuracy assessment or parametrization of less complete models, for example, density functional approximations or machine learning potentials, are also outlined.

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.

Predictive coupled-cluster isomer orderings for some SinCm (m, n ≤ 12) clusters: A pragmatic comparison between DFT and complete basis limit coupled-cluster benchmarks.

The accurate determination of the preferred Si12C12 isomer is important to guide experimental efforts directed towards synthesizing SiC nano-wires and related polymer structures which are anticipated to be highly efficient exciton materials for the opto-electronic devices. In order to definitively identify preferred isomeric structures for silicon carbon nano-clusters, highly accurate geometries, energies, and harmonic zero point energies have been computed using coupled-cluster theory with systematic extrapolation to the complete basis limit for set of silicon carbon clusters ranging in size from SiC3 to Si12C12. It is found that post-MBPT(2) correlation energy plays a significant role in obtaining converged relative isomer energies, suggesting that predictions using low rung density functional methods will not have adequate accuracy. Utilizing the best composite coupled-cluster energy that is still computationally feasible, entailing a 3-4 SCF and coupled-cluster theory with singles and doubles extrapolation with triple-ζ (T) correlation, the closo Si12C12 isomer is identified to be the preferred isomer in the support of previous calculations [X. F. Duan and L. W. Burggraf, J. Chem. Phys. 142, 034303 (2015)]. Additionally we have investigated more pragmatic approaches to obtaining accurate silicon carbide isomer energies, including the use of frozen natural orbital coupled-cluster theory and several rungs of standard and double-hybrid density functional theory. Frozen natural orbitals as a way to compute post-MBPT(2) correlation energy are found to be an excellent balance between efficiency and accuracy.

Accuracy and Efficiency of Coupled-Cluster Theory Using Density Fitting/Cholesky Decomposition, Frozen Natural Orbitals, and a t1-Transformed Hamiltonian.

We present an algorithm for coupled-cluster through perturbative triples [CCSD(T)] based on a t1-dressed Hamiltonian and the use of density fitting (DF) or Cholesky decomposition (CD) approximations for the construction and contraction of all electron repulsion integrals (ERIs). An efficient implementation of this algorithm is then used to explore whether preoptimized density fitting basis sets [specifically, the (aug-)cc-pVXZ-RI series designed for DF-MP2 computations] are suitable for DF-CCSD(T) computations and how they compare to the CD representation of the integrals. The code is also used to systematically explore the accuracy and efficiency of DF/CD combined with frozen natural orbitals (FNOs) to reduce computational costs. The mean absolute errors due to DF/CD in the CCSD(T)/aug-cc-pVDZ interaction energies of 11 van der Waals dimers are only 0.001 kcal mol(-1) for the preoptimized RI basis set and only 0.002 and 0.001 kcal mol(-1) for CD with cutoffs of 10(-4) and 10(-5), respectively. The very similar performance of the aug-cc-pVDZ-RI auxiliary set is a bit surprising considering that the numbers of CD vectors using these thresholds are, on average, 28% and 73% larger than the dimension of the RI set. When FNOs are coupled with DF/CD, the DF/CD error is roughly an order of magnitude less than the FNO truncation error (at a conservative FNO occupation cutoff of 10(-5)). Utilizing t1-dressed three-index integrals, which remove the explicit dependence of the doubles residual equations on the t1-amplitudes, results in a moderate performance acceleration for the CCSD portion of the algorithm. Moreover, the t1-dressing results in a simpler code which will be more amenable to parallelization. Utilizing both CD and FNO techniques, we observe a speedup of four times for the evaluation of the three-body contribution to the interaction energy for the benzene trimer described by an aug-cc-pVDZ basis set; the error incurred by the CD and FNO approximations in the three-body contribution is only 0.002 kcal mol(-1).

Accurate Noncovalent Interaction Energies Using Truncated Basis Sets Based on Frozen Natural Orbitals.

We assess the accuracy of basis set truncations based on natural orbitals determined by second-order perturbation theory for computing noncovalent interaction energies with coupled cluster through perturbative triples [CCSD(T)]. We consider two methods for truncation: (i) the usual frozen natural orbital approach (FNO) in which the basis set truncation occurs before the iterative CCSD computation [FNO CCSD(T)] and (ii) an approach in which the truncation occurs only for the perturbative triples contribution [CCSD+FNO(T)]. The errors incurred are comparable for both methods and are small enough for the methods to be used for benchmark-quality studies of noncovalent interactions. For the FNO CCSD(T) method with a modest natural orbital occupation tolerance of 10(-5), the mean absolute error in the interaction energies for the S22 data set in an aug-cc-pVDZ basis set is only 0.012 kcal mol(-1) versus canonical CCSD(T) values. The same method exhibits a mean absolute error of 0.020 kcal mol(-1) for the S11 data set in the aug-cc-pVTZ basis set versus canonical CCSD(T) values.

Frozen natural orbitals for ionized states within equation-of-motion coupled-cluster formalism

The frozen natural orbital (FNO) approach, which has been successfully used in ground-state coupled-cluster calculations, is extended to open-shell ionized electronic states within equation-of-motion coupled-cluster (EOM-IP-CC) formalism. FNOs enable truncation of the virtual orbital space significantly reducing the computational cost with a negligible decline in accuracy. Implementation of the MP2-based FNO truncation scheme within EOM-IP-CC is presented and benchmarked using ionized states of beryllium, dihydrogen dimer, water, water dimer, nitrogen, and uracil dimer. The results show that the natural occupation threshold, i.e., percentage of the total natural occupation recovered in the truncated virtual orbital space, provides a more robust truncation criterion as compared to the fixed percentage of virtual orbitals retained. Employing 99%-99.5% natural occupation threshold, which results in the virtual space reduction by 70%-30%, yields errors below 1 kcal/mol. Moreover, the total energies exhibit linear dependence as a function of the percentage of the natural occupation retained allowing for extrapolation to the full virtual space values. The capabilities of the new method are demonstrated by the calculation of the 12 lowest vertical ionization energies (IEs) and the lowest adiabatic IE of guanine. In addition to IE calculations, we present the scans of potential energy surfaces (PESs) for ionized (H(2)O)(2) and (H(2))(2). The scans demonstrate that the FNO truncation does not introduce significant nonparallelity errors and accurately describes the PESs shapes and the corresponding energy differences, e.g., dissociation energies.

Systematic truncation of the virtual space in multiconfigurational perturbation theory

A method is suggested which allows truncation of the virtual space in Cholesky decomposition-based multiconfigurational perturbation theory (CD-CASPT2) calculations with systematic improvability of the results. The method is based on a modified version of the frozen natural orbital (FNO) approach used in coupled cluster theory. The idea is to exploit the near-linear dependence among the eigenvectors of the virtual-virtual block of the second-order Moller-Plesset density matrix. It is shown that FNO-CASPT2 recovers more than 95% of the full CD-CASPT2 correlation energy while requiring only a fraction of the total virtual space, especially when large atomic orbital basis sets are in use. Tests on various properties commonly investigated with CASPT2 demonstrate the reliability of the approach and the associated reduction in computational cost and storage demand of the calculations.

Frozen natural orbital coupled-cluster theory: Forces and application to decomposition of nitroethane

The frozen natural orbital (FNO) coupled-cluster method increases the speed of coupled-cluster (CC) calculations by an order of magnitude with no consequential error along a potential energy surface. This method allows the virtual space of a correlated calculation to be reduced by about half, significantly reducing the time spent performing the coupled-cluster (CC) calculation. This paper reports the derivation and implementation of analytical gradients for FNO-CC, including all orbital relaxation for both noncanonical and semicanonical perturbed orbitals. These derivatives introduce several new orbital relaxation contributions to the CC density matrices. FNO-CCSD(T) and FNO-LambdaCCSD(T) are applied to a test set of equilibrium structures, verifying that these methods are capable of reproducing geometries and vibrational frequencies accurately, as well as energies. Several decomposition pathways of nitroethane are investigated using CCSD(T) and LambdaCCSD(T) with 60% of the FNO virtual orbitals in a cc-pVTZ basis, and find differences on the order of 5 kcalmol with reordering of the transition state energies when compared to B3LYP 6-311 + G(3df, 2p).

Frozen Natural Orbitals: Systematic Basis Set Truncation for Coupled-Cluster Theory

The method of frozen natural orbital (FNO) basis set truncation for coupled-cluster theory is described. Numerical comparisons of the FNO potential energy surfaces of a group of small molecules at the CCSD(T) level in DZP, cc-pVTZ, cc-pVQZ bases show that truncation of up to 50% of the virtual space yields CC correlation energies that are accurate to 90 or 95% when added to the full MBPT(2) basis result. The FNO truncation method is also applied to dimethylnitramine (DMNA): both the equilibrium structure and dimer interactions, yielding results at the CCSD(T) level in both a DZP and cc-pVTZ basis set that agree with literature values. CCSD(T) calculations at two possible equilibrium structures of 1,3,5-trinitrohexahydro-1,3,5-triazine (RDX) in a truncated DZP basis are also reported.

Selection of the reduced virtual space for correlated calculations. An application to the energy and dipole moment of H 2O

Many-body perturbation theory and coupled-cluster calculations for the energy and dipole moment of H 2O are performed using different techniques to reduce the virtual space. The optimized virtual orbital space (OVOS) method is compared with frozen natural orbitals (FNO), defined from a relaxed density matrix, and just SCF orbitals to generate an optimized reduced virtual space. It is shown that OVOS as well as FNO orbitals recover more than 90% of the correlated energy even when the virtual space has been reduced by almost 60%. In the full occupied space, the OVOS method is slightly preferable to FNO results in intermediate sized virtual spaces. By eliminating core correlations effectively perfect agreement is found between the FNO orbitals and OVOS.