Matrix equation of motion

Equation (3.21) is the matrix equation of motion and it can be into the coupled equations ... 

Spin-spin relaxation is handled by second order perturbation theory [ Abr 1 ] of the density matrix equation of motion (2.2.62) in the rotating frame (RCF) [Abrl],... 

The general matrix equation of motion for a damped linear structure is given by... 

The matrix equation of motion Eq. 37 can be recast into the following first-order matrix equation ... 

Evidently, the vector s and its modal components s, are independent of the absolute magnitudes of naturally emerge when performing the modal analysis steps indicated in Eqs. 22c and 22d. Namely, when substituting for the load form adopted in Eq. 47 and writing the i modal matrix equation of motion, one gets... 

The basic equation [8] is tlie equation of motion for the density matrix, p, given in equation (B2.4.25), in which H is the Hamiltonian. 

This Liouville-space equation of motion is exactly the time-domain Bloch equations approach used in equation (B2.4.13). The magnetizations are arrayed in a vector, and anything that happens to them is represented by a matrix. In frequency units (1i/2ti = 1), the fomial solution to equation (B2.4.26) is given by equation (B2.4.27) (compare equation (B2.4.14H. 

Relaxation or chemical exchange can be easily added in Liouville space, by including a Redfield matrix, R, for relaxation, or a kinetic matrix, K, to describe exchange. The equation of motion for a general spin system becomes equation (B2.4.28). 

Projecting the nuclear solutions Xt( ) oti the Hilbert space of the electronic states (r, R) and working in the projected Hilbert space of the nuclear coordinates R. The equation of motion (the nuclear Schrddinger equation) is shown in Eq. (91) and the Lagrangean in Eq. (96). In either expression, the terms with represent couplings between the nuclear wave functions X (K) and X (R). that is, (virtual) transitions (or admixtures) between the nuclear states. (These may represent transitions also for the electronic states, which would get expressed in finite electionic lifetimes.) The expression for the transition matrix is not elementaiy, since the coupling terms are of a derivative type. 

The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

A key feature of the Car-Parrinello proposal was the use of molecular dynamics a simulated annealing to search for the values of the basis set coefficients that minimise I electronic energy. In this sense, their approach provides an alternative to the traditioi matrix diagonalisation methods. In the Car-Parrinello scheme, equations of motion ... 

Write the equations of motion in matrix form. Take mi and m2 to be unit masses (say 1.000 kg each). 

We have found the principal axes from the equation of motion in an arbitrary coordinate system by means of a similarity transformation S KS (Chapter 2) on the coefficient matrix for the quadratic containing the mixed terms... 

There are various ways to obtain the solutions to this problem. The most straightforward method is to solve the full problem by first computing the Lagrange multipliers from the time-differentiated constraint equations and then using the values obtained to solve the equations of motion [7,8,37]. This method, however, is not computationally cheap because it requires a matrix inversion at every iteration. In practice, therefore, the problem is solved by a simple iterative scheme to satisfy the constraints. This scheme is called SHAKE [6,14] (see Section V.B). Note that the computational advantage has to be balanced against the additional work required to solve the constraint equations. This approach allows a modest increase in speed by a factor of 2 or 3 if all bonds are constrained. 

Compare this with Eq. (7-59) and observe the order of the factors on the right Equation (7-83) is the equation of motion of the statistical matrix in the Schrodinger representation p is constant, of course, in the Heisenberg representation. 

Here, po is time independent density matrix and can be defined for initial state I. The excitation of electrons caused by absorption of a single photon is regarded as a polarization of the electron density, which is measured by the linear polarizability = Tr p uj)6). The equation of motion for the... 

U 2 - JI density matrix is second-order and the initial state I) can evolve after the second scattering caused in both bra and ket states caused by the applied field. Further, three-photon absorptions with frequencies uj, 0J2, and can be described by p t) = ///dwidw2dw3/2(a i,u 2,W3)e= ( i+ 2+" ), which obeys a third-order equation of motion,... 

Note that for general parameterizations this metric matrix is neither skew diagonal nor constant-, see below. The equations of motion expressed in Eq. (2.6) are obtained by using the Principle of Stationary Action, 5A = 0, with Lagrangian... 

INTRODUCTION DENSITY MATRIX TREATMENT Equation of motion for the density operator Variational method for the density amplitudes THE EIKONAL REPRESENTATION The eikonal representation for nuclear motions... 

Perhaps the best starting point in a review of the nonequilibrium field, and certainly the work that most directly influenced the present theory, is Onsager s celebrated 1931 paper on the reciprocal relations [10]. This showed that the symmetry of the linear hydrodynamic transport matrix was a consequence of the time reversibility of Hamilton s equations of motion. This is an early example of the overlap between macroscopic thermodynamics and microscopic statistical mechanics. The consequences of time reversibility play an essential role in the present nonequilibrium theory, and in various fluctuation and work theorems to be discussed shortly. 

From the time-reversible nature of the equations of motion, Eq. (12), it is readily shown that the matrix is block-asymmetric ... 

In the first part to follow, the equations of motion of a soft solid are written in the harmonic approximation. The matrices that describe the potential, and hence the structure, of the material are then considered in a general way, and their properties under a normal mode transformation are discussed. The same treatment is given to the dissipation terms. The long wavelength end of the spectral density is of interest, and here it seems that detailed matrix calculations can be replaced by simple scaling arguments. This shows how the inertial term, usually absent in molecular problems, is magnified to become important in the continuum limit. 

The equation of motion is further generalized by imposition of a 3xn external force matrix o, so that we finally have... 

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