Wavefunction classical

To extract infomiation from the wavefimction about properties other than the probability density, additional postulates are needed. All of these rely upon the mathematical concepts of operators, eigenvalues and eigenfiinctions. An extensive discussion of these important elements of the fomialism of quantum mechanics is precluded by space limitations. For fiirther details, the reader is referred to the reading list supplied at the end of this chapter. In quantum mechanics, the classical notions of position, momentum, energy etc are replaced by mathematical operators that act upon the wavefunction to provide infomiation about the system. The third postulate relates to certain properties of these operators ... 

Even expression ( B3.4.31), altiiough numerically preferable, is not the end of the story as it does not fiilly account for the fact diat nearby classical trajectories (those with similar initial conditions) should be averaged over. One simple methodology for that averaging has been tln-ough the division of phase space into parts, each of which is covered by a set of Gaussians [159, 160]. This is done by recasting the initial wavefunction as... 

In molecular dynamics applications there is a growing interest in mixed quantum-classical models various kinds of which have been proposed in the current literature. We will concentrate on two of these models the adiabatic or time-dependent Born-Oppenheimer (BO) model, [8, 13], and the so-called QCMD model. Both models describe most atoms of the molecular system by the means of classical mechanics but an important, small portion of the system by the means of a wavefunction. In the BO model this wavefunction is adiabatically coupled to the classical motion while the QCMD model consists of a singularly perturbed Schrddinger equation nonlinearly coupled to classical Newtonian equations, 2.2. 

Approximation Property We assume that the classical wavefunction 4> is an approximate 5-function, i.e., for all times t G [0, T] the probability density 4> t) = 4> q,t) is concentrated near a location q t) with width, i.e., position uncertainty, 6 t). Then, the quality of the TDSCF approximation can be characterized as follows ... 

The algorithms of the mixed classical-quantum model used in HyperChem are different for semi-empirical and ab mi/io methods. The semi-empirical methods in HyperChem treat boundary atoms (atoms that are used to terminate a subset quantum mechanical region inside a single molecule) as specially parameterized pseudofluorine atoms. However, HyperChem will not carry on mixed model calculations, using ab initio quantum mechanical methods, if there are any boundary atoms in the molecular system. Thus, if you would like to compute a wavefunction for only a portion of a molecular system using ab initio methods, you must select single or multiple isolated molecules as your selected quantum mechanical region, without any boundary atoms. 

The square of the wavefunction is finite beyond the classical turrfing points of the motion, and this is referred to as quantum-mechanical tunnelling. There is a further point worth noticing about the quantum-mechanical solutions. The harmonic oscillator is not allowed to have zero energy. The smallest allowed value of vibrational energy is h/2jt). k /fj. 0 + j) and this is called the zero point energy. Even at a temperature of OK, molecules have this residual energy. 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

In their classic paper Mathematical Problems in the Complete Quantum Predictions of Chemical Phenomena , Boys and Cook (1960) divided the determination of an ab initio electronic wavefunction into distinct logical stages which include the... 

A sum over all final vibration-rotation states f lying below e, for which the geometry Qo is within the classically allowed region of the corresponding vibration-rotation wavefunction Xf (Q) (so that vq is real) of... 

For Q = Q , this density function describes electronic motions for given nuclear positions, while for Q = Q it describes the quantal correlation of nuclear positions at time f, which should be small for classical-like variables. The equation of motion for the density function could be derived from the original LvN equation. Instead, it is more convenient to construct it from the wavefunctions. The phase factor and the preexponential factor are trial functions to be determined from the TDVP. The procedure followed here parallels that in ref. (23). 

The main difficulty in the theoretical study of clusters of heavy atoms is that the number of electrons is large and grows rapidly with cluster size. Consequently, ab initio "brute force" calculations soon meet insuperable computational problems. To simplify the approach, conserving atomic concept as far as possible, it is useful to exploit the classical separation of the electrons into "core" and "valence" electrons and to treat explicitly only the wavefunction of the latter. A convenient way of doing so, without introducing empirical parameters, is provided by the use of generalyzed product function, in which the total electronic wave function is built up as antisymmetrized product of many group functions [2-6]. 

Finally, it is a weU-known result of quantum mechanics" that the wavefunctions of harmonic oscillators extend outside of the bounds dictated by classical energy barriers, as shown schematically in Figure 10.1. Thus, in situations with narrow barriers it can... 

The basis of the scanning tunnelling microscope, illustrated schematically in Figure 3.5, lies in the ability of electronic wavefunctions to penetrate a potential barrier which classically would be forbidden. Instead of ending abruptly at a... 

Changes in the degrees of freedom in a reaction can be classified in two ways (1) classical over the barrier for frequencies o) such that hot) < kBT and (2) quantum mechanical through the barrier for two > kBT. In ETR, only the electron may move by (1) all the rest move by (2). Thus, the activated complex is generated by thermal fluctuations of all subsystems (solvent plus reactants) for which two < kBT. Within the activated complex, the electron may penetrate the barrier with a transmission coefficient determined entirely by the overlap of the wavefunctions of the quantum subsystems, while the activation energy is determined entirely by the motion in the classical subsystem. 

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

This is a recurrent theme of quantum theory. Many quantum systems can be formulated exactly in terms of a wave equation and the behaviour of the system will be described exactly by the wavefunction, the solution to the wave equation. What is not always appreciated is that this is a mathematical description only, which does not ensure understanding of the event in terms of a comprehensible physical model. The problem lies therein that the description is only possible in terms of a wave formalism. Understanding of the physical behaviour however, requires reduction to a particle model. The wave description is no more than a statistically averaged picture of the behaviour of many particles, none of which follows the actual statistically predicted course. The wave description is non-classical, and the particle model is classical. Mechanistic understanding is possible only in terms of the classical approach, and a mathematically precise description only in terms of the wave formalism. The challenge of quantum theory is to reconcile the two points of view. 

The essence of Schrodinger s treatment was to replace the classical orbit of Bohr s semi-classical (particle) model of the H-atom by a corresponding wavelike orbital (single-electron wavefunction) L. Instead of specifying the electron s... 

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