Propagator and Equation-of-Motion Methods

Equation (1-1-1) provides a basis for the discussion of the electronic structure and properties of molecules in their stationary states. In the next few chapters we shall deal almost exclusively with stationary states but in later chapters we develop theoretical procedures for calculating the effects of time-dependent perturbations, and we then have to start from (1.1.12)- This latter development will lead to the study of propagator and equation-of-motion methods, which now play an important role in molecular quantum mechanics. 

PROPAGATOR AND EQUATION-OF-MOTION METHODS 13.6 EQUATION-OF-MOTION (EOM) METHODS... 

This edition, a completely rewritten and expanded version of the original, includes second quantization and diagrammatic perturbation theory, symmetric and unitary group methods, new forms of valence bond theory, dynamic properties and response, propagator and equation-of-motion techniques and the theory of intermolecular forces. 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

It is possible to parametarize the time-dependent Schrddinger equation in such a fashion that the equations of motion for the parameters appear as classical equations of motion, however, with a potential that is in principle more general than that used in ordinary Newtonian mechanics. However, it is important that the method is still exact and general even if the trajectories aie propagated by using the ordinary classical mechanical equations of motion. 

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... 

Both the BO dynamics and Gaussian wavepacket methods described above in Section n separate the nuclear and electronic motion at the outset, and use the concept of potential energy surfaces. In what is generally known as the Ehrenfest dynamics method, the picture is still of semiclassical nuclei and quantum mechanical electrons, but in a fundamentally different approach the electronic wave function is propagated at the same time as the pseudoparticles. These are driven by standard classical equations of motion, with the force provided by an instantaneous potential energy function... 

The LFV integration method propagates coordinates and momenta on the basis of the equation of motion (5) by the following relations... 

More recently Equation Of Motion (EOM) methods have been used in connection with other types of wave functions, most notably coupled cluster.Such EOM methods are closely related to propagator methods, and give working equations which are similar to those encountered in propagator theory. 

This e qnession for the propagators is still exact, as long as, the principal sub-manifold h and its complement sub-manifold h arc complete, and the characteristics of the propagator is reflected in the construction of these submanifolds (47,48). It should be noted that a different (asymmetric) metric for the superoperator space, Eq. (2.5), could be invoked so that another decoupling of the equations of motion is obtained (62,63,82-84). Such a metric will not be explored here, but it just shows the versatility of the propagator methods. 

An important advance in making explicit polarizable force fields computationally feasible for MD simulation was the development of the extended Lagrangian methods. This extended dynamics approach was first proposed by Sprik and Klein [91], in the sipirit of the work of Car and Parrinello for ab initio MD dynamics [168], A similar extended system was proposed by van Belle et al. for inducible point dipoles [90, 169], In this approach each dipole is treated as a dynamical variable in the MD simulation and given a mass, Mm, and velocity, p.. The dipoles thus have a kinetic energy, JT (A)2/2, and are propagated using the equations of motion just like the atomic coordinates [90, 91, 170, 171]. The equation of motion for the dipoles is... 

P. O. Lowdin. Some aspects of the hamiltonian and liouvillian formalisms, the special propagator method, and the equation of motion approach. Adv. Quantum Chem., 17 285, 1985. 

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