Targeted excited state algorithms

Abstract

To overcome the limitations of the traditional state-averaging approaches in excited state calculations, where one solves and represents all states between the ground state and excited state of interest, we have investigated a number of new excited state algorithms. Building on the work of van der Vorst and Sleijpen ͓SIAM J. Matrix Anal. Appl. 17, 401 ͑1996͔͒, we have implemented harmonic Davidson and state-averaged harmonic Davidson algorithms within the context of the density matrix renormalization group ͑DMRG͒. We have assessed their accuracy and stability of convergence in complete-active-space DMRG calculations on the low-lying excited states in the acenes ranging from naphthalene to pentacene. We find that both algorithms offer increased accuracy over the traditional state-averaged Davidson approach, and, in particular, the state-averaged harmonic Davidson algorithm offers an optimal combination of accuracy and stability in convergence.

Linear response theory for the density matrix renormalization group: Efficient algorithms for strongly correlated excited states

“…One way to relieve this drawback of the SA-DMRG algorithm is to use the stateaveraged harmonic Davidson (SA-HD) DMRG algorithm to target higher excited states directly. 20 Recently, excitations have been constructed on top of a reference MPS wavefunction, [21][22][23][24][25][26][27] analogous to the concept of particle-hole excitations on top of a reference Slater determinant. In this post-MPS or post-DMRG theory, the reference MPS wavefunction provides a site-based meanfield ansatz, 22,28 and excitations consist of "local" changes in this mean-field ansatz.…”

An Efficient Variational Principle for the Direct Optimization of Excited States

“…Indeed, the presence of H 2 makes the straightforward evaluation of Ω drastically more expensive than the ground state function E, which is why studies that have worked implicitly with this function in the past [16,17] have, to the best of our knowledge, always approximated this term (see discussion of harmonic Ritz methods below). Here we avoid explicitly squaring H by resolving identities via complete sums over states,…”

“…Note the similarity of this eigenvalue equation to the harmonic Davidson equation that arises in applications [16,17,27,28] of the harmonic Ritz principle [29,30] for targeting interior eigenvalues of a matrix. In fact, some of these approaches [16,17] appear to have been minimizing an approximation to Ω with respect to linear parameters, in which P H 2 P was approximated by P HP HP , where P is the projector into the subspace corresponding to the linear parameters in question.…”

“…In fact, some of these approaches [16,17] appear to have been minimizing an approximation to Ω with respect to linear parameters, in which P H 2 P was approximated by P HP HP , where P is the projector into the subspace corresponding to the linear parameters in question. Except for its controllable statistical uncertainty, the present approach makes no approximation when evaluating Ω and can optimize both linear and nonlinear parameters alike.…”

“…For example, linear response (LR) methods such as time de- * Electronic mail: eneuscamman@berkeley.edu pendent HF and DFT [8], CI singles (CIS) [8], equation of motion CC with singles and doubles (EOM-CCSD) [9], and LR DMRG [10][11][12] are limited by the requirement that all excited states of interest must be found in the ground state's LR space, which for a nonlinear ansatz is typically much less flexible than its full variational space. In many other cases, such as state-averaged complete active space methods [13,14], some VMC approaches [15], and directly targeted DMRG [16], crucial ansatz components (often the one particle basis) are required to be the same for the ground and all excited states. While these methods clearly do not take full advantage of an ansatz's variational freedom, they have been preferred due to the lack of an efficiently optimizable function whose minimum is an excited state.…”