Quantum mechanical/molecular mechanical/continuum style solvation model: Linear response theory, variational treatment, and nuclear gradients

Abstract

Linear response and variational treatment are formulated for Hartree-Fock ͑HF͒ and Kohn-Sham density functional theory ͑DFT͒ methods and combined discrete-continuum solvation models that incorporate self-consistently induced dipoles and charges. Due to the variational treatment, analytic nuclear gradients can be evaluated efficiently for these discrete and continuum solvation models. The forces and torques on the induced point dipoles and point charges can be evaluated using simple electrostatic formulas as for permanent point dipoles and point charges, in accordance with the electrostatic nature of these methods. Implementation and tests using the effective fragment potential ͑EFP, a polarizable force field͒ method and the conductorlike polarizable continuum model ͑CPCM͒ show that the nuclear gradients are as accurate as those in the gas phase HF and DFT methods. Using B3LYP/EFP/CPCM and time-dependent-B3LYP/EFP/CPCM methods, acetone S 0 → S 1 excitation in aqueous solution is studied. The results are close to those from full B3LYP/CPCM calculations.

Quantum mechanical/molecular mechanical/continuum style solvation model: Second order Møller-Plesset perturbation theory

“…8 The MM electrostatic, repulsion, dispersion potentials, induced dipoles, and the induced surface charges in the continuum model implicitly affect E (2) through the HF orbitals and energies. For RMP2, UMP2, and ZAPT2 methods, the first derivative (gradient) of the second order energy correction E (2) with respect to a coordinate x in the QM/MMpol/C style MP2 method can be written using density matrices and basis function integrals, 45,46 …”

“…[1][2][3][4][5][6][7][8][9][10] In these methods, a large molecule is divided into two regions, a relatively small QM region and a relatively large MM region, and the bulk solvent is simplified as a dielectric continuum. We have established a rigorous variational treatment of induced dipoles [11][12][13][14][15] and induced surface charges for QM/MMpol/C style density functional theory (DFT) and Hartree-Fock (HF) self-consistent field methods, 8 as well as time-dependent density functional theory (TDDFT) methods. 16 It is important and advantageous to incorporate solvation models into HF and DFT methods in a variational manner because of their fundamental importance in quantum chemistry: analytic gradients and some other properties can be evaluated efficiently, and post-HF and post-DFT methods can be formulated.…”

“…, 51 a variant of COSMO52 ) and the effective fragment potential (EFP)8,10,42,43 method. (2) The other is for the FixSol 53 solvation model and general induced dipole polarizable force field methods in the quantum chemistry polarizable force field (QuanPol 54 ) program, which is designed for general molecular mechanics (MM) and combined quantum mechanics and molecular mechanics (QM/MM) calculations.…”

Quantum mechanical/molecular mechanical/continuum style solvation model: Time-dependent density functional theory

“…9 Here, the main equations are summarized for the purpose of establishing the notations for the current paper.…”

“…A rigorous variational treatment of induced dipoles and induced surface charges for QM/MMpol/C style HF and DFT methods were described in a previous work. 9 Time-dependent density functional theory (TDDFT) methods are efficient and relatively accurate QM methods for the study of valence and singly excited states. 16,17 Analytical gradients for QM/MMpol style and QM/continuum style TDDFT methods have been derived and implemented by the authors group.…”