# First-variational and second-variational eigenvectors

1. Mar 12, 2014

### Douasing

Dear all,
Recenty,I am reading the source code of the first-principle software.I meet some words that I haven't found in those DFT books.For example,it mentions the first-variational and second-variational eigenvectors. Similarly,the first-variational and second-variational eigenvalues are mentioned.In my opinion,the first-vatiational eigenvectors and eigenvalues should be the wave functions and eigen energies of Kohn-sham equation.But what is the meaning of the second-variational eigenvectors and eigenvalues ?
Regards.

2. Mar 12, 2014

### UltrafastPED

Usually it is an alternative expression of the variational principle using an alternative coordinate scheme. Thus you have two ways of expressing the eigenvectors. This is often useful, but not always.

3. Mar 12, 2014

### Douasing

Thank you for your explaining.So,actually,the first-variational eigenvectors and eigenvalues are equal to the second, am I right for understanding that ?
Could you give me a simple example to illustrate that using the following Kohn-sham equation,i.e.,
$$[\frac{1}{2}∇^{2}+V(r)+U(r)+V_{xc}(r)]\phi_{i}(r)=E_{i}\phi_{i}(r)$$
Maybe I don't clear why it is often useful but not always.

4. Mar 12, 2014

### bsmile

If I understand UltrafastPED right, then you can choose different basis sets to expand the Hamiltonian, and solve the ks equation. As the two basis sets are related by unitary transformation (hope so), their eigenvalues are the same, but eigenvectors wouldn't. But I doubt this is the first/second variational principles.

5. Mar 14, 2014

### Douasing

Yes,bsmile,your analysis is very reasonable.Maybe UltrafastPED is not right.So-called the second-variational eigenvectors or eigenvalues,actually,it means that in the second-variational step, the magnetic fields, spin-orbit coupling and A field are added using the first-variational step as a basis.