Analytic Energy Gradients for the Driven Similarity Renormalization Group Multireference Second-Order Perturbation Theory

We
derive analytic energy gradients of the driven similarity renormalization
group (DSRG) multireference second-order perturbation theory (MRPT2)
using the method of Lagrange multipliers. In the Lagrangian, we impose
constraints for a complete-active-space self-consistent-field reference
wave function and the semicanonical orthonormal molecular orbitals.
Solving for the associated Lagrange multipliers is found to share
the same asymptotic scaling of a single DSRG-MRPT2 energy computation.
A pilot implementation of the DSRG-MRPT2 analytic gradients is used
to optimize the geometry of the singlet and triplet states of p-benzyne. The equilibrium bond lengths and angles are similar
to those computed via other MRPT2s and Mukherjee’s multireference
coupled cluster theory. An approximate DSRG-MRPT2 method that neglects
the contributions of the three-body density cumulant is found to introduce
negligible errors in the geometry of p-benzyne, lending
itself to a promising low-cost approach for molecular geometry optimizations
using large active spaces.