Molecular energy quantum mechanics

MSS Molecule surface scattering [159-161] Translational and rotational energy distribution of a scattered molecular beam Quantum mechanics of scattering processes... 

Both molecular and quantum mechanics methods rely on the Born-Oppenheimer approximation. In quantum mechanics, the Schrodinger equation (1) gives the wave functions and energies of a molecule. 

HyperChem can calculate geometry optimizations (minimizations) with either molecular or quantum mechanical methods. Geometry optimizations find the coordinates of a molecular structure that represent a potential energy minimum. 

Classical mechanics does not apply to the atomic scale and does not take the quantized nature of molecular vibration energies into account. Thus, in contrast to ordinary mechanics where vibrators can assume any potential energy, quantum mechanical vibrators can only take on certain discrete energies. Transitions in vibrational energy levels can be brought about by radiation absorption, provided the energy of the radiation exactly matches the difference in energy levels between the vibrational quantum states and provided also that the vibration causes a fluctuation in dipole. 

MOLECULAR POTENTIAL ENERGY QUANTUM MECHANICAL PROBLEM... 

In reality, even if a molecular species is at absolute zero, it will contain vibrational zero-point energy quantum mechanically. How to build this into a classical trajectory calculation is one of the enduring problems in this field. 

Little has been said here about computer graphics, but its importance to the birth of molecular modeling should not be underestimated. Energy calculations, whether molecular mechanics, molecular dynamics, quantum mechanics, or whatever, generate an enormous amount of data. Computer graphics (or visualization, as it has come to be called lately) renders all that data manageable and assimilable. 

Pigments are comparatively large molecules. The relationship of the number of atoms versus the computation time for a crystal structure calculation is approximately linear. However, pigment molecules are fairly rigid and have limited conformational flexibility, which reduces computational expenses if only one or a few conformations must be considered. In many cases, even distinct low energy conformations may be obtained from molecular or quantum mechanical calculations. 

The result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

A quantum mechanical treatment of molecular systems usually starts with the Bom-Oppenlieimer approximation, i.e., the separation of the electronic and nuclear degrees of freedom. This is a very good approximation for well separated electronic states. The expectation value of the total energy in this case is a fiinction of the nuclear coordinates and the parameters in the electronic wavefunction, e.g., orbital coefficients. The wavefiinction parameters are most often detennined by tire variation theorem the electronic energy is made stationary (in the most important ground-state case it is minimized) with respect to them. The... 

Traditionally, for molecular systems, one proceeds by considering the electronic Hamiltonian which consists of the quantum mechanical operators for the kinetic energy of the electrons, their mutual Coulombic repulsions, and... 

Information about critical points on the PES is useful in building up a picture of what is important in a particular reaction. In some cases, usually themially activated processes, it may even be enough to describe the mechanism behind a reaction. However, for many real systems dynamical effects will be important, and the MEP may be misleading. This is particularly true in non-adiabatic systems, where quantum mechanical effects play a large role. For example, the spread of energies in an excited wavepacket may mean that the system finds an intersection away from the minimum energy point, and crosses there. It is for this reason that molecular dynamics is also required for a full characterization of the system of interest. 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

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