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

Abstract

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.

Is Solid Copper Oxalate a Spin Chain or a Mixture of Entangled Spin Pairs?

“…We used the open-shell frozen natural orbital truncation of the virtual space with the truncation threshold corresponding to 99% of the total preserved occupation in the virtual space. Core electrons were frozen in all calculations.…”

On the Basis Set Selection for Calculations of Core-Level States: Different Strategies to Balance Cost and Accuracy

“…For example, for molecules with just 4 second row atoms, the total number of basis functions in uC-aug-cc-pV5Z exceeds 500. Aside the obvious choice of using a smaller set, the cost of the calculations can be reduced by using single-precision execution [51] and truncation of virtual space using frozen natural orbitals approach [52,53]. Using single-precision execution limits the convergence thresholds.…”

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

“…Though the OVO and FNO approaches exhibited similar performance in benchmark studies [38,39], thanks to its simplicity, the latter gained wider acceptance. It was extended to larger systems relying on reduced-scaling density matrix construction techniques [40][41][42][43], FNO-CC analytic gradients were implemented [36], and various excited-state extensions are also available [37,[44][45][46][47]. Besides the cost-reduction of conventional CC methods, various types of NOs are widely used to speed up local CC approaches.…”

“…Concerning recent applications of FNOs to CC methods in the past decade, Krylov and co-workers extended the FNO method to the equation-of-motion CCSD treatment of open-shell systems [37,45], DePrince and Sherrill combined FNOs with their DF-CCSD(T) implementation [58], and Aquilante and co-workers explored ideas to extrapolate toward the complete virtual basis result using FNOs [50]. Sorathia and Tew [59], motivated by Helgaker's two-point extrapolation formula [17], proposed an extrapolation approach to reduce the PNO truncation error in PNO-based local correlation methods.…”