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6 Molecular Systems
which corresponds to a splitting of the Hamiltonian into two parts, namely,
H =
T nuclei + H el ,
(6.2.3)
and leads to a separation of the total wavefunction into the product of two factors,
that is,
(r; R) = χ(R)ψ el (r; R) .
(6.2.4)
In the ‘clamped nuclei’ approximation the nuclear motion operator
T nuclei is set
equal to 0, so that E nuclear motions = 0, and the Schrödinger equation reduces to
H el ψ el (r; R) = E el (R)ψ el (r; R) ,
(6.2.5)
which depends only parametrically upon a fixed internuclear separation R. A
potential energy function V (R) is then obtained by carrying out calculations of
E el (R) for a set of fixed separations R.
Once the Born–Oppenheimer approximation has been employed to separate the
electronic energy from the energy associated with the motion of the two nuclei,
the potential energy (PE) curve so generated may be utilized as the PE function
in the Schrödinger equation governing the nuclear motion—in terms of the six
coordinates associated with the two nuclei. We solve this equation by transforming
to centre-of-mass (CM) and relative coordinates, which separates the equation into
two parts, one of which gives the overall translational motion of the diatomic
molecule. This equation is solved trivially. The second equation is concerned with
the relative motion of the two nuclei, and is more difficult to solve because,
for example, the energies are not simply additive. Were we wishing to compute
positions of spectroscopic lines using the Schrödinger equation, we would not be
able to proceed further except by carrying out accurate numerical solutions of the
full Schrödinger equation. However, since we do not require such extreme accuracy
for the calculation of thermodynamic quantities (actually, we require about two
to three orders less accuracy for the calculation of thermodynamic quantities than
we do for the calculation of spectroscopic quantities), we can make two further
simplifying approximations at this stage:
• replace V (R) by a parabola which fits near the minimum;
• separate vibrational and rotational motions from one another by assuming that
as far as rotations are concerned, the molecule has a fixed internuclear distance,
which we shall designate by R e : this approximation is commonly known as the
‘rigid-rotor-SHO approximation’.
We can express the end result of these approximations mathematically by writing
down the energy for the diatomic molecule in the form
= tr + rot + vib + el ,
(6.2.6)
6 Molecular Systems
which corresponds to a splitting of the Hamiltonian into two parts, namely,
H =
T nuclei + H el ,
(6.2.3)
and leads to a separation of the total wavefunction into the product of two factors,
that is,
(r; R) = χ(R)ψ el (r; R) .
(6.2.4)
In the ‘clamped nuclei’ approximation the nuclear motion operator
T nuclei is set
equal to 0, so that E nuclear motions = 0, and the Schrödinger equation reduces to
H el ψ el (r; R) = E el (R)ψ el (r; R) ,
(6.2.5)
which depends only parametrically upon a fixed internuclear separation R. A
potential energy function V (R) is then obtained by carrying out calculations of
E el (R) for a set of fixed separations R.
Once the Born–Oppenheimer approximation has been employed to separate the
electronic energy from the energy associated with the motion of the two nuclei,
the potential energy (PE) curve so generated may be utilized as the PE function
in the Schrödinger equation governing the nuclear motion—in terms of the six
coordinates associated with the two nuclei. We solve this equation by transforming
to centre-of-mass (CM) and relative coordinates, which separates the equation into
two parts, one of which gives the overall translational motion of the diatomic
molecule. This equation is solved trivially. The second equation is concerned with
the relative motion of the two nuclei, and is more difficult to solve because,
for example, the energies are not simply additive. Were we wishing to compute
positions of spectroscopic lines using the Schrödinger equation, we would not be
able to proceed further except by carrying out accurate numerical solutions of the
full Schrödinger equation. However, since we do not require such extreme accuracy
for the calculation of thermodynamic quantities (actually, we require about two
to three orders less accuracy for the calculation of thermodynamic quantities than
we do for the calculation of spectroscopic quantities), we can make two further
simplifying approximations at this stage:
• replace V (R) by a parabola which fits near the minimum;
• separate vibrational and rotational motions from one another by assuming that
as far as rotations are concerned, the molecule has a fixed internuclear distance,
which we shall designate by R e : this approximation is commonly known as the
‘rigid-rotor-SHO approximation’.
We can express the end result of these approximations mathematically by writing
down the energy for the diatomic molecule in the form
= tr + rot + vib + el ,
(6.2.6)