Quantum Mechanics. The Schrodinger Equation

The quantum mechanical state of a particle like an atom or a molecule, in particular the various contributions to the partition function Q, can be calculated from the solution of the Schrodinger equation for quantum wave mechanics or from approximations thereof. The Schrodinger equation reflects that a particle has both a corpuscular and a wave-like behavior. Classical mechanics, dealing with large objects, only considers the first property but at the atomic scale both characteristics have to be accounted for. 

Consider an atom, a particle consisting of a nucleus and electrons. The former generates a potential energy field, V r), in which the electrons move and contribute with their kinetic energy to the total energy E of the particle. In three dimensional space and in the stationary state the equation for the wave motion of a single particle with mass m can be written 

W i) describes the spatial amplitude of the matter wave as a function of its position in space, defined with respect to the nucleus. The symbol r represents a space vector, h is the Planck constant and E is the energy of the particle, consisting of potential and kinetic contributions. Mathematically (1.7.2-1) is an eigenvalue problem and is the eigenfunction. It can be solved exactly for the hydrogen molecule, with one nucleus and one electron, but problems arise when the wave function of a particle with a number of electrons has to be calculated because of their interaction. The Hartree approach deals with an 7V-electron wave function as the product of N independent single orbital functions and does not explicitly account for the interaction between the electrons [Hartree, 1928 Levine, 1999]. 

The steady state of a molecule corresponds to a minimum energy. The minimization of the energy has to start from an approximation of the molecular structure by means of what is called basis functions. These represent an atomic orbital by a linear combination of Gaussian functions with different exponents. The number of Gaussian functions does not necessarily have to be the same for the inner shell and the valence shell electrons. 

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Before moving deeper into understanding what quantum mechanics means, it is useful to learn how the wavefunctions E are found by applying the basic equation of quantum mechanics, the Schrodinger equation, to a few exactly soluble model problems. Knowing the solutions to these easy yet chemically very relevant models will then facilitate learning more of the details about the structure of quantum mechanics because these model cases can be used as concrete examples. ... 

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. 

Tunnelling solutions may occur when the energy of particles, incident from the left, is less than the height of the potential barrier. Classically none of these would be transmitted, but this is not necessarily so in quantum mechanics. The Schrodinger equations to be solved are ... 

As already mentioned in the Introduction, the exact solution of the main equation of quantum mechanics - the Schrodinger equation - lies beyond the potentialities of modem mathematics and computer technology. But a number of important inferences about the behaviour, structure and properties of a given quantum-mechanical many-particle system can be drawn without solving this equation, just by examining its symmetry properties. 

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