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Molecular Hamiltonian





H = T_N + H_{\mathit el}


where


T_N = \sum_{a}{- rac{1}{2M_a}
abla_a^2}


is the kinetic energy Operator corresponding to the molecular dynamics and ''Hel'' is the Electronic Molecular Hamiltonian . ''Ma'' are the masses of the nuclei.
abla_a^2 is the Laplacian with respect to the Cartesian Nuclear Coordinates associated to the Molecular Geometry .

The relativistic molecular hamiltonian differs because it contains terms that depend upon the electron Spin and the nuclear spin. The electron spin appears naturally in the solution of the Dirac Equation , but the nuclear spins are added in phenomenologically as if the nuclei were "heavy" electrons.

The Electronic Molecular Hamiltonian can be replaced within the Born-Oppenheimer Approximation by the Potential Energy Surface s. The Schrödinger Equation becomes then an equation describing the "motion" of the nuclei only. The "motion" of the electrons has been taken into account during the diagonalization of the Electronic Molecular Hamiltonian (see Computational Chemistry for more details).

The discrete Eigenvalue s of the molecular Hamiltonian are called molecular Energy Level s.


REFERENCES


  • Richard Moss, ''Advanced Molecular Quantum Mechanics'', ISBN 412-10490-3.