[Pw_forum] Excited State Gradients in TDDFT

dario rocca roccad at gmail.com
Mon Feb 6 20:20:09 CET 2012

Dear Yukihiro

> You mean if we diagonalize the Liouvillean operator by usual method
> instead of using Lanczos chain, and get
> eigenvalue and eigenvectors, we can get excited-state gradient ?

There are not "usual methods" to get eigenvalues and eigenvectors for TDDFT
because the eigenvalue problem is non-Hermitian. I have been lately working
on this type of eigenvalue problems in order to diagonalize the Liouvillian
in the TDDFT formalism without empty states (J. Chem. Phys. 136, 034111
(2012)). This new method seems very promising but at this stage it still
needs an efficient preconditioner to become more practical.

> Are there already formalism to calculate the excited energy gradient
> within occupied state only method ?

I think that a necessary ingredient for the calculation of gradients is the
diagonalization of the Liouvillian. Once this is done I think that the
formalism used for the Casida's equations can be extended to the TDDFT
without empty states.

> Usual Casida's matrix, the dimension of the Matrix is \Omega_{i_j, k_q}
> where, i and j are occupied and unoccupied state (k and q are also occupied
> and unoccupied state pair),
> and the dimension is (2*Nc*Nv) * (2* Nc * Nv) where Nc and Nv is the
> number of the unoccupied and occupied states.
> But your Liouvillean matrix dimension is (2*Nv) * (2 *Nv), and it is very
> small than the usual Casida's one.

The dimension of the Liouvillian is actually (2*Nv*Npw)*(2*Nv*Npw). This
dimension is basically the same as that of Casida's equation when Nc is the
total number of conduction states. The advantage of the Liouvillian
approach is not exactly in the matrix dimension but I would say that some
of the advantages are:
-The matrix is big but never built explicitly and its application to a
vector involve a number of orbitals that is equal to the number of occupied
states (it scales better than Casida's equations, that involve a number of
orbitals equal to Nc)
-The convergence of the spectra with the number of empty states in never an
issue (even at high energy or when there is a strong dependence of the
spectrum at low energy on the states at higher energy)
-The use of plane-waves avoids the convergence problems with respect to the
basis set typical of the localized basis sets

Dario Rocca
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