Born Oppenheimer Approximation and Product Asnatz

[SpectraCat]

ok, let's use your notation.

The ansatz is \Psi = \sum_k\sum_i\phi_i\chi_{i,k}. Now, using the adiabatic approximation, why is the final solution given by \Psi = \sum_k\phi_j\chi_{j,k}?

Using the adiabatic approximation leads to the Born-Oppenheimer equation (better: many Born Oppenheimer equations, one for each electronic surface). The solution of this equation is given by the \chi_{i,k}. I don't see, why all \chi_{i,k} (except one) (with respect to the index i) should vanish.

Well, you could say, that one uses only one electronic surface. But in this case my question is, why \Psi = \sum_k\phi_j\chi_{j,k} is a good approximation, that is, why are all \chi_{i,k} (except one) small, so that we can neglect them.

I am sorry, but I don't understand where your confusion lies. Your last statement reads to me like, "I understand why a 700 nm laser beam appears red, but I don't understand why my red laser pointer says it emits at 700 nm." Note that is not at all intended to be insulting towards you, it is just an attempt to explain how I am having trouble understanding you.

Perhaps this will help .. the Born-Oppenheimer approximation is conceptually equivalent to the following statement: "All nuclear dymanics happens on a single electronic surface". For many processes, like vibration and rotation of covalently bound molecules, that statement is USUALLY valid. It can also SOMETIMES be valid for certain chemical processes, such as proton transfer. However for other processes, such as chemical reactions where covalent bonds between heavy atoms are formed and broken, it is ALMOST NEVER valid.

 

[Derivator]

That is, there is no special reason why \Psi = \sum_k\phi_j\chi_{j,k} is a good approximation to \Psi = \sum_k\sum_i\phi_i\chi_{i,k}. It is just sometimes valid and sometimes not. But one can't give a criterion, that tells us, in which cases this \Psi = \sum_k\phi_j\chi_{j,k} is a good approximation to \Psi = \sum_k\sum_i\phi_i\chi_{i,k}.In the derivation of the Born - Oppenheimer equation, you can clearly see, in which cases the adiabatic approximation is valid, because you neglect terms like \frac{1}{E_k-E_j}, this is obviously only valid if E_k is not similar to E_j, that is the two electronic surfaces are not allowed to become close to each other (like in photo chemical reactions). My problem is, that i don't see, why and how this adabatic approximation has an effect on the ansatz wave function \Psi = \sum_k\sum_i\phi_i\chi_{i,k} in such a way, that one can approximate this wave function as \Psi = \sum_k\sum_i\phi_i\chi_{i,k}.

 

[SpectraCat]

That is, there is no special reason why \Psi = \sum_k\phi_j\chi_{j,k} is a good approximation to \Psi = \sum_k\sum_i\phi_i\chi_{i,k}. It is just sometimes valid and sometimes not. But one can't give a criterion, that tells us, in which cases this \Psi = \sum_k\phi_j\chi_{j,k} is a good approximation to \Psi = \sum_k\sum_i\phi_i\chi_{i,k}.


In the derivation of the Born - Oppenheimer equation, you can clearly see, in which cases the adiabatic approximation is valid, because you neglect terms like \frac{1}{E_k-E_j}, this is obviously only valid if E_k is not similar to E_j, that is the two electronic surfaces are not allowed to become close to each other (like in photo chemical reactions). My problem is, that i don't see, why and how this adabatic approximation has an effect on the ansatz wave function \Psi = \sum_k\sum_i\phi_i\chi_{i,k} in such a way, that one can approximate this wave function as \Psi = \sum_k\sum_i\phi_i\chi_{i,k}.

It seems like this is all just semantics, so in such cases it is usually helpful to define more precisely what you are talking about. So, what do you mean by the overall wavefunction \Psi? What are you trying to represent? Is \Psi an eigenstate of some Hamiltonian, or is it supposed to be a more general time-dependent solution? I have already tried to explain when the simpler single-sum expression applies (or doesn't apply) for some general classes of problems.

 

[Derivator]

\Psi = \sum_k \sum_i \phi_i \chi_{i,k} is the eigenstate of the total hamiltonian, that is it is the eigenvector of the hamiltonian that consists of nucleonic kinetic energy + electronic hamiltonian.

 

[DrDu]

Sorry, I was out for some days without internet connection.

Yes, the operator H in my definition of the matrix \mathbf{H}_{ij} is the full hamiltonian.
However, once you neglect the non-adiabatic coupling matrix elements, you don't get a set a set of BO -equations which are completely independent as they share the same E^{en}.
As the potential surfaces E_i are different for different i, generically only at most one of the BO equations has a non-trivial solution for a given E^{en}, i.e. the other coefficients \chi_{j} with j not equal i have to vanish.

\Psi = \sum_k \sum_i \phi_i \chi_{i,k} is the eigenstate of the total hamiltonian, that is it is the eigenvector of the hamiltonian that consists of nucleonic kinetic energy + electronic hamiltonian.

In your last posting I would rather write
\Psi = \sum_i \phi_i \chi_{i,k} without summing over k. The chi_i,k arise directly as the elements the vector I defined below. However, you may expand them into a common basis of a priori given functions, e.g. of the harmonic oscillator, if you want so. Only in the latter case a summation over k would be appropriate (and introduction of further coefficients, too).