Efficient Algorithms for Estimating the Absorption Spectrum within Linear Response TDDFT

Abstract

We present a special symmetric Lanczos algorithm and a kernel polynomial method (KPM) for approximating the absorption spectrum of molecules within the linear response time-dependent density functional theory (TDDFT) framework in the product form. In contrast to existing algorithms, the new algorithms are based on reformulating the original non-Hermitian eigenvalue problem as a product eigenvalue problem and the observation that the product eigenvalue problem is self-adjoint with respect to 1 an appropriately chosen inner product. This allows a simple symmetric Lanczos algorithm to be used to compute the desired absorption spectrum. The use of a symmetric Lanczos algorithm only requires half of the memory compared with the non-symmetric variant of the Lanczos algorithm. The symmetric Lanczos algorithm is also numerically more stable than the non-symmetric version. The KPM algorithm is also presented as a low-memory alternative to the Lanczos approach, but may require more matrix-vector multiplications in practice.We discuss the pros and cons of these methods in terms of their accuracy as well as their computational and storage cost. Applications to a set of small and medium-sized molecules are also presented.

Assessment of time-dependent density functional theory with the restricted excitation space approximation for excited state calculations of large systems

“…Different formalisms of TDDFT can address this problem, for example complex polarisation [14,15], damped response approaches [16] and real-time TDDFT [17,18]. A symmetric Lanczos algorithm and a kernel polynomial method were introduced for low memory determination of absorption spectra within TDDFT [19]. However, within this approach, determining state specific properties is challenging.…”

Serenity: A subsystem quantum chemistry program

“…As iterative eigenvalue solver for the solution of eq. a Davidson solver based on the so‐called K ‐inner product of test vectors is used in S erenity . The Coulomb (and exchange for hybrid and range‐separated hybrid functionals) contributions to the matrix‐vector products for the iterative eigenvalue solvers are most efficiently calculated in a direct fashion using an AO formulation equivalent to CIS or TDHF .…”

“…Arrow representation of the analytical sDFT gradients of a OHradical-water complex at non-equilibrium geometry (PW91, [71] PW91k [91] / def2-TZVPP [72] ). [Color figure can be viewed at wileyonlinelibrary.com] vectors [97,98] is used in SERENITY. The Coulomb (and exchange for hybrid and range-separated hybrid functionals) contributions to the matrix-vector products for the iterative eigenvalue solvers are most efficiently calculated in a direct fashion using an AO formulation equivalent to CIS [99] or TDHF.…”

Accelerating Optical Absorption Spectra and Exciton Energy Computation via Interpolative Separable Density Fitting

“…In Ref. [3,25], we developed a structure preserving Lanczos algorithm for estimating the optical spectrum. The algorithm has been implemented in the BSEPACK [27] library.…”