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# Linear Scaling Solution of the Time-Dependent Self-Consistent-Field Equations 

Matt Challacombe ∗ 

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 

A new approach to solving the Time-Dependent Self-Consistent-Field equations is developed based on the double quotient formulation of Tsiper [J. Phys. B, 34 L401 (2001)]. Dual channel, quasi-independent non-linear optimization of these quotients is found to yield convergence rates approaching those of the best case (single channel) Tamm-Dancoff approximation. This formulation is variational with respect to matrix truncation, admitting linear scaling solution of the matrix-eigenvalue problem, which is demonstrated for bulk excitons in the polyphenylene vinylene oligomer and the (4,3) carbon nanotube segment. 

The Time-Dependent Self-Consistent-Field equations together with models that include some portion of the Hartree-Fock (HF) exchange admit control over the range of self-interaction in the optical response [1, 7, 9, 16], and are related to new models of electron correlation based on the Random Phase Approximation (RPA) [5, 6, 8, 18]. Solving the TD-SCF equations is challenging due to an unconventional J -symmetric structure of the naive molecular orbital (MO) representation, 

where A and B are Hermitian blocks corresponding to 4 th order tensors spanning transitions between occupied and virtual sub-spaces, ω is the real excitation energy and ⃗ v = ( ⃗ X ⃗ Y ) is the corresponding transition density. By construction, the MO representation allows strict sepa- ration between the dyadic particle-hole ( ph ) and hole- particle ( hp ) solutions, ⃗ X and ⃗ Y , for which specialized algorithms exist. Nevertheless, convergence of the naive J -symmetric problem is typically much slower than the corresponding Hermitian Tamm-Dancoff approximation (TDA), $_{A}$⃗ X = ω ⃗ X , which is of reduced dimensionality in the MO representation. 

Several TD-SCF eigensolvers are based on the oscil- lator picture ( 0 K T 0 ) ( ⃗ p ⃗ q ) = ω ( ⃗ p ⃗ q ) , with $^{K}$= $^{A}$+ B and $^{T}$= A$_{−}$ B the Hermitian potential and kinetic matrices, and the dual { ⃗ p, ⃗ q } = { ⃗ X − ⃗ Y , ⃗ X + ⃗ Y } corresponding to position and momentum. This picture avoids the im- balance ∥ ⃗ X ∥ ≫ ∥ ∥$^{⃗}$ Y ∥ whilst admitting conventional solu- tions based on the Hermitian matrix G = K · T , as shown by Tamara and Udagawa [17] and extended by Narita and Shibuya with second order optimization of the quo- tient ω $^{2}$[ ⃗ p, ⃗ q ] = ⃗ q · G · ⃗ p/ | ⃗ p · ⃗ q | [10]. More recently, Tsiper considered the quotients 

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and developed a corresponding dual channel Lanczos solver. Subspace solvers in this dual representation have recently been surveyed by Tretiak, Isborne, Niklasson and Challacombe (TINC) [19], with comparative results for semi-empirical models. 

Another challenge is dimensionality and scaling. Writ- ing Eq. (1) in the general form L · ⃗ v = ω⃗ v , admitting arbitrary representation, the superoperator matrix L is a ∼ N 2 × N 2 tetradic, with N the number of basis func- tions, assumed proportional to system size. In prac- tice the action of L onto ⃗ v is carried out implicitly as L [ v ] = [ F , v ] + [ G [ v ] , P ] , using an existing framework for construction of the effective Hamiltonian (Fockian) F , where P is the one-particle reduced density matrix, G is a screening operator involving Coulomb, exchange and/or exchange-correlation terms and the correspon- dence between superoperator and functional notation is given by a tensorial mapping between diadic and matrix, ⃗ v$_{N}$ 2$_{×}$ 1 ⇔ v$_{N}$$_{×}$$_{N}$ . 

Recent efforts have focused on addressing the prob- lem of dimensionality by employing linear scaling meth- ods that reduce the cost of L [ · ] within Density Func- tional Theory (DFT) to O ( N ) . However, this remains an open problem for the Hartree-Fock (HF) exchange, an ingredient in models that account for charge transfer in the dynamic and static response, including the Ran- dom Phase Approximation (RPA) at the pure HF level of theory. Likewise, scaling of the TD-SCF eigenprob- lem remains formidable due to associated costs of lin- ear algebra, even when using powerful Krylov subspace methods. Underscoring this challenge, one of the most successful approaches to linear scaling TD-DFT avoids the matrix eigenproblem entirely through explicit time- evolution [21, 22]. 

Linear scaling matrix methods exploit quantum local- ity, manifest in approximate exponential decay of matrix elements expressed in a well posed, local basis; with the dropping of small elements below a threshold, τ$_{mtx}$ , this decay leads to sparse matrices and O ( N ) complexity at the forfeit of full precision [3, 4, 11]. Likewise, linear scaling methods for computing the HF exchange employ an advanced form of direct SCF, exploiting this decay 

