The Born-Oppenheimer Approximation (BOA) is a central approximation in molecular spectroscopy. It is an assumption that the electronic and nuclear motion can be separated i.e. the wavefunctions can be separated into non-interacting parts. There are no interactions between the discrete adiabatic states within the Born-Oppenheimer approximation. This breaks down at large internuclear separations, where the separation between electronic states is the same order of magnitude as the coupling between them.
The total Hamiltonian can be expressed as:
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(3.4) |
where H
is the non-relativistic electronic Hamiltonian describing the Coulombic interactions between nuclei and electrons. The eigenvalues are the Born-Oppenheimer potentials and the eigenfunctions are the adiabatic wavefunctions,:
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(3.5) |
where m is the mass of an electron (labelled n), V is the electrostatic potential as a function of the electronic co-ordinates
and the nuclear co-ordinate R. H
is the relativistic interaction which is approximated to the spin-orbit interaction here (neglecting spin-spin and hyperfine interactions):
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(3.6) |
This term is responsible for mixing between the and both the
and
states in SiH+ and GeH+.
H
and H
are the radial and rotational components of the nuclear kinetic energy operator T
:
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(3.7) |
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(3.8) |
and
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(3.9) |
where
is the reduced mass. H
furthermore can be separated into a diagonal part (corresponding to the rotational energy of the state):
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(3.10) |
and an off-diagonal part, the second term of which is responsible for rotational electronic (Coriolis) coupling between states of different
but identical
.
This is the origin of the lambda doubling, which is strongly R dependent,:
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(3.11) |
where
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(3.12) | ||
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(3.13) | ||
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(3.14) |