Molecular energy levels
Hierarchy of motions and energies in molecules
The different types of motion in a molecule (electronic, vibrational, rotational,: : :) take
place on different time scales and are associated with different contributions to the total
energy.
This hierarchy of time scales implies the approximate separability of the corresponding
motions that lies behind the so-called Born-Oppenheimer approximation: The nuclei
appear immobile to the fast moving electrons; hardly any rotation takes place during a
vibrational period. The vibrational motion can thus be viewed as taking place in an
\average" electronic cloud, the rotational motion at a nuclear configuration corresponding
to an average over the vibrational motion.
The Born-Oppenheimer Approximation
The exact Hamiltonian H for a molecule, with the electronic coordinates
expressed in the molecule-fixed axis system, is rather difficult to derive.
For details see Bunker (Molecular Symmetry and Spectroscopy)
In standard procedure, the non-relativistic Hamiltonian is approximated by sum of
three operators:
^TN---------the nuclear kinetic energy,
^Te----------the electron kinetic energy
^V-----------The electrostatic potential energy for the nuclei
and electrons: e-e, e-N, N-N
Then the Hamiltonian is expressed in appropriate coordinate system,
as for instance in spherical coordinates, cylindrical, parabolical etc.
The Born-Oppenheimer approximation
The Born-Oppenheimer (BO) approximation enables the separation of electronic
and nuclear degrees of freedom in a molecule. The Hamilton operator of a
molecule has the form
* vib. and rot.
term operat.
and the Schrödinger equation
involves 3k + 3N coordinates.
After transformation to the “internal interaction Hamiltonian” and use the
spherical coordinate system, the approximate separability can be expressed as:
where e, v, r and ns stand for electronic, vibrational, rotational and nuclear spin,
respectively, and the superscripts indicate which motion is averaged out.
The separability enables one to reduce a problem of high dimensionality to several
problems of reduced dimensionality. It also justies the chemist's description of
molecules in terms of “rigid" geometries.
Q and qi are the position vectors of the nuclei (N = 1, 2,…, k) and the electrons
(i = 1, 2,…,N) and θ, φ are the Euler angles describing the orientation of the
molecule axis system in the lab frame. R is used in ϕe instead of Q to
indicate a parametric dependence.
By making the BO approximation one reduces the dimensionality of
the problem and solves an equation of 3N variables for the electronic motion and
an equation of 3k variables for the nuclear motion. Separating the center-ofmass translational motion and the overall rotation of the molecule enables the
reduction of the number of nuclear degrees of freedom by 6 for nonlinear
molecules and by 5 for linear molecules.
In the BO treatment, Ψn(Q; qi) is represented as a product (separability)
**
of a nuclear wavefunction ϕ (n) m (Q) and an electronic wavefunction ϕn(qi; Q).
The treatment exploits the fact that the nuclei hardly move during a period of the
electronic motion. The index m is needed to distinguish different states of the
nuclear motion (e.g. vibrational, rotational) associated with a given electronic
state n.
The BO treatment consists of two steps:
Step 1: The nuclei are frozen at a given geometry Rα : Qα - Rα. As a result Tk = 0
and the first term of Eq.* becomes constant. A purely electronic Schrödinger
equation is then solved (the nuclear coordinates are treated as parameters, not
as variables):
***
Inserting Eqs. ** and *** in Hamiltonian equation :
we obtain
Because neither ^ Te nor ^ V act on the nuclear coordinates Qα ,
ϕ(n) m (Q) cancels out :
The solutions of this equation are sets of electronic wavefunctions ϕn(qi; Rα ) and
energies Un(Rα ), where n = 1, 2,…and n corresponds to a label for the electronic
states. Determining Un(Rα ) for a large number of possible congurations Rα one
obtains the so-called BO potential hypersurface, which represents the dependence
of the electronic energy on the nuclear coordinates.
The dimensionality of the BO hypersurface is 3k - 5 for linear molecules and 3k - 6
for nonlinear molecules and equals the number of internal degrees of freedom f.
Un(Rα) does not depend on the mass of the nuclei and is therefore isotope
independent.
Assuming that at least 10 points must be calculated per internal degree of
freedom, the equation must be solved 10 times for a diatomic molecule (f = 1),
1000 times for a nonlinear triatomic molecule (H2O, f = 3) and 1030 times for C6H6
(f = 30), if one wants to determine the complete hypersurface. Stable molecular
configurations correspond to local minima on the BO hypersurface and adiabatic
isomerization reactions can be viewed as trajectories connecting two local
minima on the Born-Oppenheimer hypersurface.
Example:
Rough estimate of the error in the dissociation energy of H2+ associated with the
BO approximation: H2+, f=1
At R Æ~ , H2+ dissociate in H+ + H. The dissociation energy De of H2 and D2+
are identical within the BO approximation and amounts to De = 22500 cm-1.
The order of magnitude error of the BO approximation on the dissociation energy of H2+
can be estimated to be ~30 cm-1 (i.e. 0.13 %) because the ionization energy of H and D
differs by 30 cm-1: IE/hc (H) = RH = 109677 cm-1 and IE/hc (D) = RD = 109706 cm-1.
Step 2: The Schrödinger equation describing the nuclear motion on the n-th potential
hypersurface is solved. Inserting
with the solution ϕ (n) m (Qα ) and ϕ n(qi; Q) in
leads to an equation describing the nuclear motion
(X)
Then one can rewrite (x):
Term T1 is the direct contribution to the nuclear kinetic energy. Terms T2 and
T3 are indirect contributions to the kinetic energy of the nuclei which originate
from the dependence of the electronic wavefunction on the nuclear coordinate.
In low electronic states for which the BO hypersurfaces are usually well
separated, this variation is smooth and these terms can be neglected to a good
approximation. We thus get, when neglecting T2 and T3:
Elimination of the electron ccordinates
This equation has the usual form for a Schrödinger equation and consists of a kinetic
energy term (first term in the square brackets) and a potential energy term Un(Q). The
equation can be solved numerically and Enm represents the total (rovibronic) energy
and ϕ (n) m the nuclear wavefunction.
To make this procedure less abstract we consider the case of diatomic molecules as
a simple illustration in the next subsection (next lecture). This illustration will enable us
to visualize the nuclear motion as consisting of a vibrational and a rotational motion.