Molecular Quantum Mechanics - (162 where and This is known...

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Molecular Quantum Mechanics In this section, we discuss the quantum mechanics of atomic and molecular systems. We begin by writing the Hamiltonian for a collection of nuclei and electrons, and then we introduce the Born-Oppenheimer approximation, which allows us to separate the nuclear and electronic degrees of freedom. The Molecular Hamiltonian We have noted before that the kinetic energy for a system of particles is (160 ) The potential energy for a system of charged particles is (161 ) For a molecule, it is reasonable to split the kinetic energy into two summations--one over electrons, and one over nuclei. Similarly, we can split the potential energy into terms representing interactions between nuclei, between electrons, or between electrons and nuclei. Using and to index electrons, and and to index nuclei, we have (in atomic units)
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Unformatted text preview: (162 ) where , , and . This is known as the ``exact'' nonrelativistic Hamiltonian in field-free space. However, it is important to remember that this Hamiltonian neglects at least two effects. Firstly, although the speed of an electron in a hydrogen atom is less than 1% of the speed of light, relativistic mass corrections can become appreciable for the inner electrons of heavier atoms. Secondly, we have neglected the spin-orbit effects. From the point of view of an electron, it is being orbited by a nucleus which produces a magnetic field (proportional to L); this field interacts with the electron's magnetic moment (proportional to S), giving rise to a spin-orbit interaction (proportional to for a diatomic.) Although spin-orbit effects can be important, they are generally neglected in quantum chemical calculations....
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