Diagonal mass matrix

For ease of presentation, we consider the case of just one quantum degree of freedom with spatial coordinate x and mass m and N classical particles with coordinates q e and diagonal mass matrix M e tj Wxsjv Upon... 

The emphasis in other chapters of this book is on initial value solution of classical equations of motion (e.g. the Newton s equations). The Newton s equations are second order differential equations - MX = —dU/dX, where X [X e is the coordinate vector. Throughout this chapter X is assumed to be a Cartesian vector, M is a 3N x 3N (diagonal) mass matrix, and U is the potential energy. A widely used algorithm that employs the coordinates and the velocities (V) at a specific time and integrates the equations of motion in small time steps is the Verlet algorithm [1] ... 

In this book, we shall frequently use a compact, vectorial notation, where q and q represent vectors of the positions and velocities, and Af is a diagonal mass matrix, so the equations of motion (1.2) become... 

Mr = F(r) = -V,V(r) where M is a diagonal mass matrix with diagonal... 

As the constrained system is described redundantly, the resulting ODEs are not minimal. A considerable reduction of the computational cost of the right hand side of the ODEs results from describing the free system with a constant and diagonal mass-matrix. In addition, the linear algebra computations have to be well organised and the sparsity of the system-matrices exploited. Since the sparsity structure is independent of time, the reduction obtained by sparse matrix techniques is considerable even for relatively small systems. This is discussed in section 3. 

For every multibody system there is a description with a constant diagonal mass matrix M such that each body is modelled separately, as detailed in the next section. This description leads to large systems of equations when compared to alternative descriptions (e.g. describing a spanning tree of the system in minimal coordinates), but has other advantages (see next section). We describe here how to use the special structure of the equations to reduce significantly the computational cost of the linear algebra calculations. 

For simplicity, take the specific case where ki = k2 = k. Write the matrix of force constants analogous to matrix (5-29). Diagonalize this matrix. What are the roots Discuss the motion of the double pendulum in contrast to two coupled, tethered masses (Fig. 5-1). 

The mass matrix M enters the Hamiltonian for convenience of expression and is an n X n matrix with on the diagonal elements and 2 on all of the off-diagonal elements the M notation for any matrix will mean a Kronecker product with the 3 X 3 identity matrix, M = M h. 

In analogous manner, residue curve maps of the reactive membrane separation process can be predicted. First, a diagonal [/e]-matrix is considered with xcc = 5 and xbb = 1 - that is, the undesired byproduct C permeates preferentially through the membrane, while A and B are assumed to have the same mass transfer coefficients. Figure 4.28(a) illustrates the effect of the membrane at nonreactive conditions. The trajectories move from pure C to pure A, while in nonreactive distillation (Fig. 4.27(a)) they move from pure B to pure A. Thus, by application of a C-selective membrane, the C vertex becomes an unstable node, while the B vertex becomes a saddle point This is due to the fact that the membrane changes the effective volatilities (i.e., the products xn a/a) of the reaction system such that xcc a. ca > xbbO-ba-... 

In terms of the co-ordinates Rt, 8am, or Q, the kinetic energy is represented by an effective mass matrix whose elements are constants. In a momentum representation this effective mass matrix is the G matrix, a diagonal matrix of the reciprocal atomic masses wm and a unit matrix 1, for the co-ordinates Ri, 8txm, and Qr, respectively. 

Sometimes the effect of off-diagonal elements of the Hessian is significant. This occurs, for example, when pairs of floating spherical Gaussians are used to represent p-orbitals [33]. In this case, in-phase and out-of-phase motion of parameters associated with each lobe of the p—orbital have very different frequencies. When the effect of the full Hessian matrix must be incorporated to decrease the width of the electronic parameter frequency spectrum, the parameter kinetic energy can be generalized to include a mass matrix [33]. 

The coordinates are stored in the vector X X denotes a transposed vector) and throughout this manuscript we use Cartesian coordinates only. A dot denotes a time derivative. The mass matrix M is diagonal, T is the kinetic energy, U is the potential energy, and L is the Lagrangian. We seek trajectories such that the total time, t, and the end points of the trajectories, X (0) and X (t), are fixed, and the action is stationary with respect to path variations. With the above conditions the Newton s equations of motion are obtained by a standard variation of the classical path [4]. Let r] T) be an arbitrary displacement vector from a path, X (t). The stationary condition of the action is obtained from the expression below... 

It is apparent why the separation of translation is problematic for the identification of electrons and nuclei. In the translation-free Hamiltonian the inverse effective mass matrix and the form of the potential functions fij depend intimately on the choice of V and the choice of this is essentially arbitrary. In particular it should be observed that because there are only N—1 translation-free variables they cannot, except in the most conventional of senses, be thought of as particle coordinates and that the non-diagonal nature of fi 1 and the peculiar form of the fij also militate against any simple particle interpretation of the translation-free Hamiltonian. It is thus not an entirely straightforward matter to identify electrons and nuclei once this separation has been made. 

H(r,Q) is the Hamiltonian of the interacting electrons and nuclei. The configuration variables of ne electrons, r=(r1,...,rnc), and nN nuclei, Q=(Qi,...,QnN) are defined with respect to a coordinate frame fixed in the laboratory. The nuclei are identified by their nuclear charges, Z=(eZ,...,eZ N), where e is the electron charge, and the mass vector (or block diagonal 3nNx3nN matrix) M=(m,.m ). 

For simplicity, we will assume the mass matrix is diagonal with ith diagonal element m . Now we write a formula for the time rate of change of the wedge product... 

Paraskevopoulos, E.A. Talaslidis, D.G. 2004. A rational approach to mass matrix diagonal-ization in two-dimensional elastodynamics. International Journal for Numerical Methods in Engineering 61 2639-2659. 

The sum over L corresponds to a triple summation over all values of h, k, and / in the crystal. The second derivatives of potential energy with respect to the components of q are obtained from the converged Hessian matrix at the potential energy minimum. The roots of Eq. (8) yield the frequencies ft), k) for the 37V independent 1-dimensional oscillators, or normal modes of vibration. Associated with each frequency is a polarization vector, k) which is a linear combination of the original mass-weighted atomic displacements u. The polarization vectors are orthonormal, such that = ij (here, the superscript denotes the complex conjugate). These are found by diagonalizing the matrix D(k) ... 

The mass matrix is constant and block-diagonal. This makes it easy to convert Eq. (1.3.11b) into an explicit equation. 

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