Vibrational subsystem analysis: A method for probing free energies and correlations in the harmonic limit

Abstract

A new vibrational subsystem analysis ͑VSA͒ method is presented for coupling global motion to a local subsystem while including the inertial effects of the environment. The premise of the VSA method is a partitioning of a system into a smaller region of interest and a usually larger part referred to as environment. This method allows the investigation of local-global coupling, a more accurate estimation of vibrational free energy contribution for parts of a large system, and the elimination of the "tip effect" in elastic network model calculations. Additionally, the VSA method can be used as a probe of specific degrees of freedom that may contribute to free energy differences. The VSA approach can be employed in many ways, but it will likely be most useful for estimating activation free energies in QM/MM reaction path calculations. Four examples are presented to demonstrate the utility of this method.

Comparative study of various normal mode analysis techniques based on partial Hessians

“…(5) can be written as $$\left(\begin{array}{cc}{H}_{\mathrm{normalss}}& H_{\mathrm{normalse}}\\ H_{\mathrm{normales}}& H_{\mathrm{normalee}}\end{array}\right)\left(\begin{array}{c}{v}_{\mathrm{normals}}\\ v_{\mathrm{normale}}\end{array}\right)={\omega }^2\left(\begin{array}{cc}{M}_{\mathrm{normals}}& 0\\ 0& M_{\mathrm{normale}}\end{array}\right)\left(\begin{array}{c}{v}_{\mathrm{normals}}\\ v_{\mathrm{normale}}\end{array}\right).$$Projection onto the s -subspace yields $$false(H_{\mathrm{ss}}+H_{\mathrm{se}}{false({\omega }^2{M}_{\mathrm{e}}-H_{\mathrm{ee}}false)}^{-1}{H}_{\mathrm{es}}false)v_{\mathrm{s}}={\omega }^2{M}_{\mathrm{s}}{v}_{\mathrm{s}}.$$For the low frequency modes one can expand to first order ${false({\omega }^2{M}_{\mathrm{e}}-H_{\mathrm{ee}}false)}^{-1}\approx -H_{\mathrm{ee}}^{-1}-{\omega }^2{H}_{\mathrm{ee}}^{-1}{M}_{\mathrm{e}}{H}_{\mathrm{ee}}^{-1}$ and derive the VSA normal mode equation as $$false(H_{\mathrm{ss}}-H_{\mathrm{se}}{H}_{\mathrm{ee}}^{-1}{H}_{\mathrm{es}}false)v={\omega }^{2}false(M_{\mathrm{s}}+H_{\mathrm{se}}{H}_{\mathrm{ee}}^{-1}{M}_{\mathrm{e}}{H}_{\mathrm{ee}}^{-1}{H}_{\mathrm{es}}false)v.$$This can be considered an adiabatic approximation, in which the environment follows every movement of the subsystem in order to remain gradient free during the normal mode vibration. 24 The matrix on the left side of eq. (14) is called the effective Hessian of the subsystem and was previously discussed by Hinsen et al 27 in connection with the derivation of an effective pair force constant for protein C α carbons s...…”

Advances in molecular quantum chemistry contained in the Q-Chem 4 program package

“…First, major strides were made in the area of QM/MM normal mode analysis. In 2009, full QM/MM analytic second derivatives were implemented in Q-CHEM (stand-alone and coupled to CHARMM) [296]; both restricted and unrestricted HF and DFT methods are supported. This was closely followed by the parallelisation of these full QM/MM Hessian calculations and extension to the mobile block Hessian formalism, significantly reducing CPU and memory requirements for these intensive calculations [297].…”

Mechanical Properties from Periodic Plane Wave Quantum Mechanical Codes: The Challenge of the Flexible Nanoporous MIL-47(V) Framework

“…53 This expression resembles the general principle of Vibrational Subsystem Analysis (VSA), where the subsystem degrees of freedom relax adiabatically along the normal modes when the environment degrees of freedom are manipulated. [54][55][56] The correction term in Eq. 8 may be regarded as an application of VSA: the unit cell parameters are subsystem coordinates, and the ionic positions are environment coordinates.…”