### Fully analytic energy gradient for the fragment molecular orbital method

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Kurt Brorsen, Iowa State University
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Near exact analytic energy gradients for the fragment molecular orbital (FMO) method were introduced by Kitaura et al. in 2001. This original derivation neglected the response terms due to the external electrostatic potentials, since the response terms would require the solution of the coupled perturbed Hartree-Fock (CPHF) equations. Here, we derive and implement the z-vector equations to solve the CPHF equations for the response terms and the corresponding contribution to the energy gradients in the framework of the FMO scheme. To practically solve the equations for large molecules such as proteins, we introduce a local z-vector procedure, taking advantages of the local nature in FMO. The resulting gradients are shown to be fully analytic by comparing them with the numerical gradients for test molecular systems. We find that the computational times for calculating the response contribution are comparable to or less than that in the self-consistent charge (SCC) calculation in FMO.

**Abstract Author(s): **Kurt R. Brorsen, Takeshi Nagata, Dmitri G. Fedorov, Kazuo Kitauraa, Mark S. Gordon