Mass-weighting

The mass weighted position of a single nucleus v in the center-of-mass frame of a molecule with N atomic nuclei at time points t is obtained from an END trajectory and can be expressed as... 

A different approach comes from the idea, first suggested by Flelgaker et al. [77], of approximating the PES at each point by a harmonic model. Integration within an area where this model is appropriate, termed the trust radius, is then trivial. Normal coordinates, Q, are defined by diagonalization of the mass-weighted Flessian (second-derivative) matrix, so if... 

The important underlying components of protein motion during a simulation can be extracted by a Principal Component Analysis (PGA). It stands for a diagonalization of the variance-covariance matrix R of the mass-weighted internal displacements during a molecular dynamics simulation. 

X is a matrix whose elements Xu give the mass-weighted internal displacements of each atomic coordinate i from its average position at a given time step t. N is the total number of integration steps. 

The standard analytic procedure involves calculating the orthogonal transformation matrix T that diagonalizes the mass weighted Hessian approximation H = M 2HM 2, namely... 

To determine the vibrational motions of the system, the eigenvalues and eigenvectors of a mass-weighted matrix of the second derivatives of potential function has to be calculated. Using the standard normal mode procedure, the secular equation... 

Defining mass-weighted Cartesian displacement coordinates r/ . 

Tran sl orm in g to mass-weighted coordinates, equation (210) can be rewritten into a set oI 3N sim u Itan eon s lin ear differen tial equation s... 

The eigenvalues and eigenvectors of the mass-weighted force matrix can be obtained by diagonalizing equation (21.5). Then each eigenvalue corresponds to its normal coordinates, Qj,... 

Mass weighting the generalized displacement coordinates qi and qj retains the form of Eq. (5-37) even when the actual masses are not unit masses. 

These elassieal equations ean more eompaetly be expressed in terms of the time evolution of a set of so-ealled mass weighted Cartesian eoordinates defined as ... 

To illustrate, again consider the H2O molecule in the coordinate system described above. The 3N = 9 mass weighted Cartesian displacement coordinates (Xl, Yl, Zl, Xq, Yq, Zq, Xr, Yr, Zr) can be symmetry adapted by applying the following four projection operators ... 

These nine Qr,j are expressed as unitary transformations of the original mass weighted Cartessian eoordinates ... 

These transformation eoeffieients Crj,k can be used to earry out a unitary transformation of the 9x9 mass-weighted Hessian matrix. In so doing, we need only form bloeks... 

As a result, the 9x9 mass-weighted Hessian eigenvalue problem ean be sub divided into two 3x3 matrix problems ( of ai and b2 symmetry), one 2x2 matrix of bi symmetry... 

Regardless of whether symmetry is used to bloek diagonalize the mass-weighted Hessian, six (for non-linear moleeules) or five (for linear speeies) of the eigenvalues will equal zero. The eigenveetors belonging to these zero eigenvalues deseribe the 3 translations and 2 or 3 rotations of the moleeule. For example. 

So, of the 9 eartesian displaeements, 3 are of ai symmetry, 3 of b2,2 of bi, and 1 of a2- Of these, there are three translations (ai, b2, and b i) and three rotations (b2, b i, and a2). This leaves two vibrations of ai and one of b2 symmetry. For the H2O example treated here, the three non zero eigenvalues of the mass-weighted Hessian are therefore of ai b2, and ai symmetry. They deseribe the symmetrie and asymmetrie streteh vibrations and the bending mode, respeetively as illustrated below. 

The method of vibrational analysis presented here ean work for any polyatomie moleeule. One knows the mass-weighted Hessian and then eomputes the non-zero eigenvalues whieh then provide the squares of the normal mode vibrational frequeneies. Point group symmetry ean be used to bloek diagonalize this Hessian and to label the vibrational modes aeeording to symmetry. 

The eigenvalues (coa of the mass weighted Hessian matrix (see below) are used to compute, for each of the 3N-7 vibrations with real and positive cOa values, a vibrational partition function that is combined to produce a transition-state vibrational partition function ... 

The squares of the desired harmonie vibrational frequeneies co are thus given as eigenvalues of the mass-weighted Hessian H ... 

The MEP is defined as the path of steepest descent in mass-weighted Cartesian coordinates. This is also called intrinsic reaction coordinate (IRC). In reality, we know that many other paths close to the IRC path would also lead to a reaction and the percentage of the time each path is taken could be described by the Boltzmann distribution. 

The potential energy of vibration is a function of the coordinates, xj,. .., z hence it is a function of the mass-weighted coordinates, qj,. .., q3N. For a molecule, the vibrational potential energy, U, is given by the sum of the electronic energy and the nuclear repulsion energy ... 

A mass (weight) density function, given the symbol m and having dimensions mass/(volume)(length), can be defined analogously to population density letting AM be the mass of crystals per unit system volume in the size range E to L + AL,... 

Equation (8) shows that it is the fluctuations of the lowest frequency modes that contribute most to the overall fluctuation of the molecule. For example, in the case of lysozyme, the lowest frequency nonnal mode (out of a total of 6057) accounts for 13% of the total mass-weighted MSF. It is for this reason that it is common to analyze just the lowest frequency modes for the large-scale functional motions. 

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