The linear response eigenvalue problem, also known as the random phase approximation eigenvalue problem, arises from computing excitation states (energies) of physical systems. Such an eigenvalue problem is usually of large scale -- the matrix dimensions for a molecule of reasonable size can easily get up to tens of millions. It is considered much more difficult than the symmetric eigenvalue problem because it is non-Hermitian in nature. But it has internal symmetric structures in each of the submatrix blocks, which previously have been largely unexploited. This project involves the development of new theory and advanced computational methods through fully exploiting the internal symmetric structures. A systematic study will be thoroughly conducted to uncover new min/maximization principles. These principles should mirror those for the symmetric eigenvalue problem, and will make it possible and guide the investigator to transform some of existing and successful techniques for the symmetric eigenvalue problem for use in the linear response eigenvalue problem research, It is highly expected that new efficient algorithms that are capable of computing several smallest positive eigenvalues simultaneously will emerge as the result of this project. In addition to advancing research in the linear response eigenvalue problem, the investigator will recruit and train graduate students in computational mathematics and interdisciplinary studies.
The linear response eigenvalue problem is a major tool in computing energy excitation states of electrons and molecules. This project will critically advance current understanding and solution techniques for the eigenvalue problem in the context of mathematical theory, computational methods, and software. With the successful completion of the project, a significant contribution will be made to the state-of-the-art physical excitation energy computations via random phase approximations, a proven technique that is widely used in computational quantum chemistry and physics.
