2.9 Correction to the Born–Oppenheimer Approximation …
27
Table 2.13 Diagonal
born–oppenheimer
contribution (distances in pm,
angles in degrees)
DBOC
H 2 O, r(OH)
0.003
H 2 O, ∠(HOH)
0.015
H 2
0.021
HF
0.002
N 2
0.001
F 2
0.000
Source Experimental, semi-experimental and ab initio equilibrium
structures, Demaison J. Molecular Physics, 105: 3109–3138, Dec
10, 2007, reprinted by permission of the publisher Taylor&Francis
Ltd, http://www.tandfonline.com
E
DBOC
=
ψ
(e)
0
TN
ψ
(e)
0
(2.27)
It is called adiabatic correction (or diagonal BO correction) and is a good approximation if E
(0)
E
(i) . It is usually small and mass dependent (proportional to 1/M).
Table 2.13 gives a few values, which show that this correction is normally negligible.
This correction slightly differentiates the geometries of the deuterated isotopologues
of ethyne (C 2 H 2 ), by increasing the C≡C, C–H, and C–D bond lengths by 0.002,
0.015, and 0.008 pm, respectively (Liévin et al. 2011).
2.10 Born–Oppenheimer Equilibrium Structure, r BO
e
First, it is important to understand how to determine the geometrical parameters of
the molecule.
The energy is a continuous function of the p internal coordinates defining the
geometry of the molecule. It describes a hypersurface called potential energy surface
(PES). The lowest-energy minimum or global minimum corresponds to the Born–
Oppenheimer equilibrium structure, r
BO
e . It has a positive curvature for distortions in
any direction which is characterized by a positive-definite second-derivative matrix
(Hessian). When the approximate position of the global minimum is not known, it
is useful to first conduct a grid mapping of the parameter space.
There are many different methods to find the exact position of the minimum.
At first sight, the simplest one is to fit the energy with a polynomial function. For
instance, for a triatomic molecule, one may use
V (R 1 , R 2 , R 3 ) =
i jk
C i jk (R 1 )
i
(R 2 )
j
(R 3 )
k
(2.28)
It is then easy to find the minimum as well as the force constants (see Sect. 2.14).
The difficulty is that this method requires some skill.
27
Table 2.13 Diagonal
born–oppenheimer
contribution (distances in pm,
angles in degrees)
DBOC
H 2 O, r(OH)
0.003
H 2 O, ∠(HOH)
0.015
H 2
0.021
HF
0.002
N 2
0.001
F 2
0.000
Source Experimental, semi-experimental and ab initio equilibrium
structures, Demaison J. Molecular Physics, 105: 3109–3138, Dec
10, 2007, reprinted by permission of the publisher Taylor&Francis
Ltd, http://www.tandfonline.com
E
DBOC
=
ψ
(e)
0
TN
ψ
(e)
0
(2.27)
It is called adiabatic correction (or diagonal BO correction) and is a good approximation if E
(0)
E
(i) . It is usually small and mass dependent (proportional to 1/M).
Table 2.13 gives a few values, which show that this correction is normally negligible.
This correction slightly differentiates the geometries of the deuterated isotopologues
of ethyne (C 2 H 2 ), by increasing the C≡C, C–H, and C–D bond lengths by 0.002,
0.015, and 0.008 pm, respectively (Liévin et al. 2011).
2.10 Born–Oppenheimer Equilibrium Structure, r BO
e
First, it is important to understand how to determine the geometrical parameters of
the molecule.
The energy is a continuous function of the p internal coordinates defining the
geometry of the molecule. It describes a hypersurface called potential energy surface
(PES). The lowest-energy minimum or global minimum corresponds to the Born–
Oppenheimer equilibrium structure, r
BO
e . It has a positive curvature for distortions in
any direction which is characterized by a positive-definite second-derivative matrix
(Hessian). When the approximate position of the global minimum is not known, it
is useful to first conduct a grid mapping of the parameter space.
There are many different methods to find the exact position of the minimum.
At first sight, the simplest one is to fit the energy with a polynomial function. For
instance, for a triatomic molecule, one may use
V (R 1 , R 2 , R 3 ) =
i jk
C i jk (R 1 )
i
(R 2 )
j
(R 3 )
k
(2.28)
It is then easy to find the minimum as well as the force constants (see Sect. 2.14).
The difficulty is that this method requires some skill.