An efficient algorithm for calculating non-interacting frequency dependent linear response functions

An O(N) algorithm is proposed for calculating linear response functions of non-interacting electrons. This program is simple and suitable to parallel- and vector- computation. Since it avoids O(N 3) computational effort of matrix diagonalization, it requires only O(N) computational efforts where N is the dimension of the statevector. The use of this O(N) algorithm is very effective since otherwise we have to calculate large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang (L.W. Wang, Phys. Rev. B 49, 10154 (1994);L.W. Wang, Phys. Rev. Lett. 73, 1039 (1994) ) is that our method can calculate linear response functions without any storage of huge statevectors on external storage while their method should store many statevectors. 1An O(N) algorithm is proposed for calculating linear response functions of non-interacting electrons. This program is simple and suitable to parallel- and vector- computation. Since it avoids O(N 3) computational effort of matrix diagonalization, it requires only O(N) computational efforts where N is the dimension of the statevector. The use of this O(N) algorithm is very effective since otherwise we have to calculate large number of eigenstates, i.e., the occupied one-electron states up to the Fermi energy and the unoccupied states with higher energy. The advantage of this method compared to the Chebyshev polynomial method recently developed by Wang (L.W. Wang, Phys. Rev. B 49, 10154 (1994);L.W. Wang, Phys. Rev. Lett. 73, 1039 (1994) ) is that our method can calculate linear response functions without any storage of huge statevectors on external storage while their method should store many statevectors. 1

Jan 1, 19960 citations

The perturbative calculation of linear response functions

Gabriele F. Giuliani, Giovanni Vignale

Cambridge University Press eBooksMar 31, 20051 citation

The Low-Frequency Behaviour of Response Functions

Wilhelm Brenig

Springer eBooksJan 1, 19891 citation

Linear response of independent electrons

Gabriele F. Giuliani, Giovanni Vignale

Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.

Cambridge University Press eBooksMar 31, 20051 citation

An efficient quantum algorithm for ab initio approximations of non-linear response functions

Tyler Kharazi, Torin F. Stetina, Liwen Ko +2 more

Abstract We develop and analyze a fault-tolerant quantum algorithm for non-linear response properties of molecular and condensed phase systems. We consider a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field interacting via the electric dipole approximation. We use the technique of eigenstate filtering, to efficiently resolve excitation energies for dominant dipole transitions. When applied to the electronic structure Hamiltonian with double factorized representation, each significant spectral line can be approximated to a width of ±γ, and to a height of ±ϵ with $O\left({N}^{6}{\eta }^{2}{\gamma }^{-1}{\epsilon }^{-1}\log (1/\epsilon )\right)$ O N 6 η 2 γ − 1 ϵ − 1 log ( 1 / ϵ ) queries to the block encoding of the unperturbed electronic structure Hamiltonian for η electrons over N basis functions. These quantities can be used to compute the nth order response functions for non-linear spectroscopies under limited assumptions using $\widetilde{O}\left({N}^{5n+1}{\eta }^{n+1}/{\gamma }^{n}\epsilon \right)$ O ̃ N 5 n + 1 η n + 1 / γ n ϵ queries to the block encoding of the Hamiltonian.Abstract We develop and analyze a fault-tolerant quantum algorithm for non-linear response properties of molecular and condensed phase systems. We consider a semi-classical description in which the electronic degrees of freedom are treated quantum mechanically and the light is treated as a classical field interacting via the electric dipole approximation. We use the technique of eigenstate filtering, to efficiently resolve excitation energies for dominant dipole transitions. When applied to the electronic structure Hamiltonian with double factorized representation, each significant spectral line can be approximated to a width of ±γ, and to a height of ±ϵ with $$O\left({N}^{6}{\eta }^{2}{\gamma }^{-1}{\epsilon }^{-1}\log (1/\epsilon )\right)$$ O N 6 η 2 γ − 1 ϵ − 1 log ( 1 / ϵ ) queries to the block encoding of the unperturbed electronic structure Hamiltonian for η electrons over N basis functions. These quantities can be used to compute the nth order response functions for non-linear spectroscopies under limited assumptions using $$\widetilde{O}\left({N}^{5n+1}{\eta }^{n+1}/{\gamma }^{n}\epsilon \right)$$ O ̃ N 5 n + 1 η n + 1 / γ n ϵ queries to the block encoding of the Hamiltonian.

npj Quantum InformationJun 9, 20252 citations

O\left({N}^{6}{\eta }^{2}{\gamma }^{-1}{\epsilon }^{-1}\log (1/\epsilon )\right)

O N 6 η 2 γ − 1 ϵ − 1 log ( 1 / ϵ ) queries to the block encoding of the unperturbed electronic structure Hamiltonian for η electrons over N basis functions. These quantities can be used to compute the nth order response functions for non-linear spectroscopies under limited assumptions using

\widetilde{O}\left({N}^{5n+1}{\eta }^{n+1}/{\gamma }^{n}\epsilon \right)

O ̃ N 5 n + 1 η n + 1 / γ n ϵ queries to the block encoding of the Hamiltonian.