Solving the RPA Eigenvalue Equation in Real-Space

Abstract

We present a computational method to solve the RPA eigenvalue equation employing a uniform grid representation in three-dimensional Cartesian coordinates. The conjugate gradient method is used for this purpose as an iterative method for a generalized eigenvalue problem. No construction of unoccupied orbitals is required in the procedure. We expect this method to be useful for systems lacking spatial symmetry to calculate accurate eigenvalues and transition matrix elements of a few low-lying excitations. Some applications are presented to demonstrate the feasibility of the method, considering the simplified mean-field model as an example of a nuclear physics system and the electronic excitations in molecules with time-dependent density functional theory as an example of an electronic system. * ) The precise definition of the RPA differs slightly among various fields. In the present article, we refer to the small amplitude limit of any time-dependent mean-field theory as RPA.

Finite amplitude method for the solution of the random-phase approximation

“…We may control the necessary energy resolution by the smoothing parameter Γ. The numerical application to the BKN functional shows that its efficiency is next to the time-dependent method, better than the other methods including the Green's function method [11] and the diagonalization method [14]. The diagonalization of the RPA matrix is very efficient if we are interested in only a few lowest states, however, it becomes more and more difficult for higher excitation energies.…”

“…In Ref. [14], a numerical method to solve the RPA equation in the coordinate space is proposed, and the similar approaches are used in realistic applications using the Skyrme interaction [8,9]. In those works, one does not need to calculate the particle orbitals, however, the residual interaction must be evaluated in the coordinate-space representation.…”

Real‐time, real‐space implementation of the linear response time‐dependent density‐functional theory

“…(35) can be regarded as an eigenvalue problem in which the eigenvalues ω provide excitation energies [17]. In the 3D grid representation, the dimension of the matrix is N N .…”

Efficient calculation for the quasiparticle random-phase approximation matrix

“…The speed of the calculation depends on the number of iterations required to reach the convergence, which is particularly affected by the precision parameter. In the present case, the number of iterations ranges from about 30 at low energy to about 160 where the strength has a peak (around [15][16][17][18][19][20]. To start the iteration procedure, we need an initial vector x 0 .…”

“…For a given vector (X, Y ), we calculate the densities, (ρ η , κ (±) η ), in Eq. (18). In the i-FAM, since the vector (X, Y ) is updated every iteration, we construct ρ η by the matrix operation as…”