This paper introduces a spin-free formulation of the supporting subspace factorization, enabling a reduction in the computational scaling of extended multi-state complete active space second-order perturbation method (XMS-CASPT2) for arbitrary spins. We present a new derivation of the idea with the purpose of understanding its physical interpretation. The sources that make CASPT2 difficult are seperated into the “same-site interactions” and “inter-site interactions”. We first use the Kronecker sum to remove the same-site interactions in the absence of inter-site interactions, leading to MP2 energy in dressed orbitals. We then recover the inter-site interactions exactly using Lowdin partition, where the supporting subspace concept will naturally arise. 

This work shows that by using supporting subspace method, the CASPT2 analytical gradients can be formulated as MP2 gradients and Fock gradients, thus significantly reduces the computational cost. The key is that the computational bottleneck in gradient calculations, i.e. the "lambda equation", takes the same form as the Bloch equation for solving energy.  This allows the supporting subspace factorization originally proposed for energy to be naturally applied in the gradient calculations. We present results from both ab initio molecular dynamics simulations as well as reaction pathway studies, which demonstrate the potential applications enabled by the new method.  

This work was the first introduction to the supporting subspace method, which enables the reformulation of multi-reference perturbation theory in an exact way, but is much simpler and faster. This work is also the first application of tensor hyper-contraction in the multi-reference context. In the new formulation,  the computational cost of CASPT2 has been reduced from O(N⁶) to O(N⁴).

This work combines automatic differentiation from computer algebra and diagrammatic notation from quantum computing, and showed that analytical gradients for complicated electronic structure methods can be automatically developed. The cost of the gradient calculation is shown to be roughly equal to 3 energy calculations. The tools developed in this work can significantly simplify future developments of analytical gradients and non-adiabatic couplings for new excited state methods. 

This work was the first report of analytical gradients for tensor hyper-contraction. Using MP2 as an example, I showed that THC can also reduce the formal scaling for analytical gradient calculations of correlation methods to O(N⁴), in addition to reducing the scaling of energy calculations as already shown in previous works. 

I also proposed a new variant of THC in this work, called "dual grid THC", which is particularly appealing for gradient calculations due to its simple expressions and fast performance.