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Firstvariational and secondvariational eigenvectors 
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#1
Mar1214, 04:23 AM

P: 33

Dear all,
Recenty,I am reading the source code of the firstprinciple software.I meet some words that I haven't found in those DFT books.For example,it mentions the firstvariational and secondvariational eigenvectors. Similarly,the firstvariational and secondvariational eigenvalues are mentioned.In my opinion,the firstvatiational eigenvectors and eigenvalues should be the wave functions and eigen energies of Kohnsham equation.But what is the meaning of the secondvariational eigenvectors and eigenvalues ? Regards. 


#2
Mar1214, 09:02 AM

Sci Advisor
Thanks
PF Gold
P: 1,908

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
Mar1214, 07:24 PM

P: 33

Thank you for your explaining.So,actually,the firstvariational 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 Kohnsham equation,i.e., [tex][\frac{1}{2}∇^{2}+V(r)+U(r)+V_{xc}(r)]\phi_{i}(r)=E_{i}\phi_{i}(r)[/tex] Maybe I don't clear why it is often useful but not always. 


#4
Mar1214, 07:46 PM

P: 27

Firstvariational and secondvariational eigenvectors
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
Mar1414, 07:58 PM

P: 33

please see:http://www2.mpihalle.mpg.de/theory_...e_development/ But,I am not very clear why it is called the secondvariational step(especially,the word "variational"). 


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