# First-variational and second-variational eigenvectors

• Douasing
In summary: Could you give me an 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)This is an example of a second-variational equation. The term "variational" means that the equation has a variational component. In this case, the variational component is the addition of the magnetic fields, spin-orbit coupling and A field.
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.

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.

UltrafastPED said:
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.

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.

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.

bsmile said:
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.

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.
please see:http://www2.mpi-halle.mpg.de/theory_department/research/elk_code_development/
But,I am not very clear why it is called the second-variational step(especially,the word "variational").

## 1. What are first-variational and second-variational eigenvectors?

First-variational and second-variational eigenvectors are two types of eigenvectors that are used to understand and analyze complex systems. They are computed from the first and second variations of the system's energy and are used to determine the stability and behavior of the system.

## 2. How are first-variational and second-variational eigenvectors computed?

First-variational and second-variational eigenvectors are computed through the process of variational calculus, which involves taking partial derivatives of the system's energy function with respect to each of its variables. These derivatives are then used to construct a matrix that represents the system's energy and the eigenvectors are computed from this matrix.

## 3. What is the significance of first-variational and second-variational eigenvectors?

First-variational and second-variational eigenvectors provide valuable information about the stability and behavior of a system. They can reveal important insights about the system's critical points, such as local maxima or minima, and can be used to predict how the system will respond to small perturbations.

## 4. How are first-variational and second-variational eigenvectors used in scientific research?

First-variational and second-variational eigenvectors are commonly used in fields such as physics, chemistry, and engineering to study complex systems and phenomena. They are often incorporated into mathematical models and simulations to better understand the behavior of these systems and make predictions about their behavior in different scenarios.

## 5. Are first-variational and second-variational eigenvectors the same as traditional eigenvectors?

No, first-variational and second-variational eigenvectors are different from traditional eigenvectors. Traditional eigenvectors are computed from the eigenvalues of a matrix, while first-variational and second-variational eigenvectors are computed from the first and second variations of a system's energy. However, all types of eigenvectors provide important insights into the behavior of complex systems and can be used together in scientific research.

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