Question

I've been trying to learn quantum chemistry at an introductionary level. While reading I've found out that the Born-Oppenheimer approximation seems to be the reason for the basic and crucial model of molecules for chemists; molecular structure in space as nuclei occupying some positions with chemical bonds between them. Thus, I've tried to learn more about this approximation. However, I've gotten stuck at the reasoning for wanting to try to separate the wavefunction of the molecule into a product of two functions, one depending on the positions of the nuclei and one depending on the positions of the electrons as well as depending parametrically on the positions of the nuclei. All resources state that the reasoning is because nuclei move considerably slower than electrons, but I have not found one that could elaborate why this is the case.

Most books and other resources simply states that Because the nuclei is over thousand times heavier than an electron, in the timescale of the movements of the electron the nuclei can be considered to be standing still., or Because of the great difference in mass, the electrons can respond almost instantaneously to the movements of the nuclei, and then quickly moves on. However, in this answer on Born–Oppenheimer adiabaticity it was said that nuclei at average are moving considerably slower than electrons because of the equipartition theorem.

In the book Introduction to Statistical Physics by S. R. A. Salinas, the theory of equipartition of energy is formulated by:

Consider a classical system of n degrees of freedom given by the Hamiltonian
H=H0+ϕp2j where (i) H0 and ϕ are functions that do not depend on the particular coordinate pj; (ii) the function ϕ is always positive; and (iii) the coordinate pj is defined from −[infinity] to +[infinity]. It then shows how the expectation value of ϕp2j equals 12kBT, where kB is the Boltzmann constant and T temperature. Thus, we can conclude that for any system (in thermal equilibrium) that has energy contributions that are quadratic in such a variable, the average energy is 12kBT for each of those degrees of freedom. Since kinetic energy is always quadratic in the momentum, it will have an average energy of 12kBT in thermal equilibrium in each degree of freedom if the rest of the Hamiltonian does not depend on the momentum (which is the case for the Hamiltonian of a molecule). This elaborates the reasoning in the linked answer. However, this seems to only apply for macroscopic systems of molecules which also have to be in thermal equilibrium. I am not completely satisfied with this, especially considering when this is only about the average kinetic energy. I would assume that the distribution of kinetic energy for the different degrees of freedom in a general system at room temperature is such that a decent portion are thousands of times higher than others. I could not find any answer that rectified this.

One book I read mentioned something different than others; that the nuclei move considerably slower than electrons because of conservation of momentum. However, it gave no elaboration. One text I remember reading somewhere on a website said something along the lines of: The electrostatic forces on nuclei and electrons from their pairwise interaction are the same, and so the changes in their momenta from these forces must be the same. Therefore, one might assume that the actual momenta of the nuclei and electrons are of similar magnitude. This is the explanation for the reasoning that makes the most sense to me, but I have not seen it anywhere else. Since it came from a source that wasn't necessarily trustworthy it made me unsure, until I saw the statement of conservation of momentum which felt related.

Does the statement of equipartition make sense as the explanation for the reasoning? If yes, why? Does the statement of changes in momenta make sense as the explanation for the reasoning? Or maybe there is something different that is the sensible explanation? Please do help me understand more clearly, I've been at this for weeks and I cannot seem to get any answer from books and other texts.

Answer 1

The Born-Oppenheimer approximation can be justified based on the mass difference between nuclei and electrons in combination with the equipartition theorem and conservation of momentum, which together explain why nuclei move significantly slower than electrons.

Step-by-step explanation:

The reasoning behind the Born-Oppenheimer approximation is grounded in the mass disparity between nuclei and electrons, as well as fundamental physics principles such as the equipartition theorem and conservation of momentum. The equipartition theorem suggests that all degrees of freedom in a system at thermal equilibrium have the same average energy, specifically ½kBT, were kB is the Boltzmann constant and T is temperature. Heavy particles like nuclei, in comparison to the much lighter electrons, move slower, meaning the average kinetic energy results in much slower motion for nuclei because kinetic energy depends on mass.

Moreover, because of conservation of momentum, when considering the electrostatic forces acting on both electrons and nuclei, we find that for the same force, a much larger mass (the nucleus) will have a much smaller acceleration compared to a smaller mass (the electron) due to Newton's second law (F = ma). This contributes to the reason why we can treat the nuclei as 'almost' stationary in comparison to the rapidly moving electrons in the Born-Oppenheimer approximation.

Overall, these principles collaboratively justify why the wavefunction of a molecule can be separated into nuclear and electronic parts - the motions of each are on vastly different timescales due to their mass difference. This separation simplifies the mathematical calculation of molecular structure and dynamics, which is crucial within quantum chemistry.