We may write the nonrelativistic Hamiltonian for a molecule as a sum of five terms:

(2) |

(3) |

Unfortunately, the
term prevents us
from separating into nuclear and electronic parts, which
would allow us to write the molecular wavefunction as a product of
nuclear and electronic terms,
. We thus introduce the Born-Oppenheimer
approximation, by which we conclude that this nuclear and electronic
separation is *approximately* correct. The term
is large and cannot be neglected; however, we
can make the dependence parametric, so that the total
wavefunction is given as
. The
Born-Oppenheimer approximation rests on the fact that the nuclei are
much more massive than the electrons, which allows us to say that the
nuclei are nearly fixed with respect to electron motion. We can fix
, the nuclear configuration, at some value ,
and solve for the electronic wavefunction
,
which depends only parametrically on . If we do this for a range
of , we obtain the potential energy curve along which the
nuclei move.

We now show the mathematical details. Initially,
can be neglected since is smaller than
by a factor of , where is the
reduced mass of an electron. Thus for a *fixed* nuclear
configuration, we have

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

Consider again the original Hamiltonian (1). An exact
solution can be obtained by using an (infinite) expansion of the form

(11) |

Expressions for the nuclear wavefunctions
can be
obtained by inserting the expansion (10) into the
total Schrödinger equation yields

(12) |

(13) |

The last term can be expanded using the chain rule to yield

At this point, a more compact notation is very helpful. Following
Tully [1], we introduce the following quantities:

(16) |

(17) |

(18) |

(19) |

(20) |

(21) |

(22) |

(23) |

or

This is equation (14) of Tully's article [1]. Tully simplifies this equation by one more step, removing the term . By taking the derivative of the overlap of it is easy to show that this term must be zero when the electronic wavefunction can be made real. If we use electronic wavefunctions which diagonalize the electronic Hamiltonian, then the electronic basis is called

In most cases, the couplings on the right-hand side of the preceeding
equation are small. If they can be safely neglected, and
assuming that the wavefunction is real, we obtain the following
equation for the motion of the nuclei on a given Born-Oppenheimer
potential energy surface:

**The Born-Oppenheimer Diagonal Correction** In a perturbation
theory analysis of the Born-Oppenheimer approximation, the first-order
correction to the Born-Oppenheimer electronic energy due to the nuclear
motion is the Born-Oppenheimer diagonal correction (BODC),

(26) |

For many years, it was only possible
to compute the BODC for Hartree-Fock or multiconfigurational
self-consistent-field wavefunctions. However, in 2003 the evaluation
of the BODC using general configuration interaction wavefunctions
was implemented [6] and its convergence toward the
*ab initio* limit was investigated for H, BH, and HO.
This study found that the absolute value of the BODC is difficult
to converge, but errors in estimates of the BODC largely cancel each
other so that even BODC's computed using Hartree-Fock theory
capture most of the effect of the adiabatic correction on relative
energies or geometries. Table 1 displays
the effect of the BODC on the barrier to linearity in the water
molecule and the convergence of this quantity with respect to
basis set and level of electron correlation. Although the absolute
values of the BODC's of bent and linear water change significantly
with respect to basis set and level of electron correlation, their
difference does not change much as long as a basis of at least
cc-pVDZ quality is used. For the cc-pVDZ basis, electron correlation
changes the
differential BODC correction by about 1 cm.
Table 2
displays the effect of the BODC on the equilibrium bond lengths and
harmonic vibrational frequencies of the BH, CH, and NH molecules
[7] and demonstrates somewhat larger changes
to the spectroscopic constants than one might have expected, particularly
for BH.

Basis | Method | |||

DZ | RHF | 613.66 | 587.69 | -25.97 |

DZ | CISD | 622.40 | 596.43 | -25.97 |

DZ | CISDT | 623.62 | 597.56 | -26.06 |

DZ | CISDTQ | 624.56 | 598.28 | -26.28 |

DZ | CISDTQP | 624.61 | 598.32 | -26.29 |

DZP | RHF | 597.88 | 581.32 | -16.56 |

cc-pVDZ | RHF | 600.28 | 585.20 | -15.08 |

cc-pVDZ | CISD | 615.03 | 599.15 | -15.88 |

cc-pVDZ | CISDT | 616.82 | 600.62 | -16.20 |

cc-pVTZ | RHF | 596.53 | 581.43 | -15.10 |

cc-pVTZ | CISD | 611.89 | 596.73 | -15.16 |

cc-pVQZ | RHF | 595.57 | 580.72 | -14.85 |

Data from Valeev and Sherrill [6]. | ||||

The difference between the adiabatic correction | ||||

for the and structures. |

BH | CH | NH | |

0.00066 | 0.00063 | 0.00027 | |

-2.25 | -2.81 | -1.38 | |

Data from Temelso, Valeev, and | |||

Sherrill [7]. |