Extension of Frozen Natural Orbital Approximation to Open-Shell References: Theory, Implementation, and Application to Single-Molecule Magnets

Abstract

Natural orbitals are often used in quantum chemistry 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 molecular orbitals results in smaller errors when the same fraction of virtual orbitals is frozen. This strategy, termed frozen natural orbitals (FNO) approach, has been successfully used to reduce the cost of state-specific coupled-cluster (CC) calculations, such as ground-state CC, as well as some multi-state methods, i.e., EOM-IP-CC (equation-of-motion CC method for ionization potentials). This contribution extends the FNO approach to the EOM-SF-CC ansatz (EOM-CC with spin-flip), which has been developed to describe certain multi-configurational wave-functions within the single-reference framework. In contrast to EOM-IP-CCSD, which describes open-shell target states by using a closed-shell reference, EOM-SF-CCSD relies on high-spin open-shell references (triplets, quartets, etc). Consequently, straightforward application of FNOs computed for an open-shell reference leads to an erratic behavior of the EOM-SF-CC energies and properties, which can be attributed 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. The OSFNO algo-rithm first identifies corresponding orbitals by means of singular value decomposition (SVD) of the overlap matrix of the α and β molecular 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 the OSFNO approximation in combination with different choices of reference orbitals is benchmarked for a set of diradicals and triradicals. For a set of di-copper single-molecule magnets, a conservative truncation criterion corresponding to a two-fold reduction of the virtual space in a triple-zeta basis leads to errors of 5–18 cm<sup>-1</sup> in the singlet–triplet gaps and errors of ∼2-3 cm<sup>-1</sup> in the spin–orbit coupling constants.

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.…”

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 [22] with the truncation threshold corresponding to 99% of the total preserved occupation in the virtual space. Core electrons were frozen in all calculations.…”

Equation-of-Motion Coupled-Cluster Protocol for Calculating Magnetic Properties: Theory and Applications to Single-Molecule Magnets

“…The core electrons were frozen in all EOM calculations. To speed up the calculations, we applied: (i) Cholesky decomposition [86] with a threshold of 10 −2 for two-electron integral calculations, (ii) open-shell frozen natural orbital (OSFNO) [87] truncation of the virtual space with the total population threshold of 99%, and (iii) single precision execution. [88] IV.…”