Spin-adapted density matrix renormalization group algorithms for quantum chemistry

We extend the spin-adapted density matrix renormalization group (DMRG) algorithm of McCulloch and Gulacsi [Europhys. Lett. 57, 852 (2002)] https://doi.org/10.1209/epl/i2002-00393-0 to quantum chemical Hamiltonians. This involves using a quasi-density matrix, to ensure that the renormalized DMRG states are eigenfunctions of |$\hat{S}^2$|$Ŝ2$⁠, and the Wigner-Eckart theorem, to reduce overall storage and computational costs. We argue that the spin-adapted DMRG algorithm is most advantageous for low spin states. Consequently, we also implement a singlet-embedding strategy due to Tatsuaki [Phys. Rev. E 61, 3199 (2000)] https://doi.org/10.1103/PhysRevE.61.3199 where we target high spin states as a component of a larger fictitious singlet system. Finally, we present an efficient algorithm to calculate one- and two-body reduced density matrices from the spin-adapted wavefunctions. We evaluate our developments with benchmark calculations on transition metal system active space models. These include the Fe₂S₂, [Fe₂S₂(SCH₃)₄]²⁻, and Cr₂ systems. In the case of Fe₂S₂, the spin-ladder spacing is on the microHartree scale, and here we show that we can target such very closely spaced states. In [Fe₂S₂(SCH₃)₄]²⁻, we calculate particle and spin correlation functions, to examine the role of sulfur bridging orbitals in the electronic structure. In Cr₂ we demonstrate that spin-adaptation with the Wigner-Eckart theorem and using singlet embedding can yield up to an order of magnitude increase in computational efficiency. Overall, these calculations demonstrate the potential of using spin-adaptation to extend the range of DMRG calculations in complex transition metal problems.