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arXiv:0707.3121 · The Journal of Chemical Physics

Targeted excited state algorithms

Jonathan J. Dorando, Johannes Hachmann, Garnet Kin-Lic Chan

Published Aug 28, 2007• 77 citations
arXivHTMLPDF

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 (1996)], we have implemented harmonic Davidson and state-averaged harmonic Davidson algorithms within the context of the density matrix renormalization group (DMRG). 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.

DOI
10.1063/1.2768360

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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 (1996)], we have implemented harmonic Davidson and state-averaged harmonic Davidson algorithms within the context of the density matrix renormalization group (DMRG). 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.
In this thesis, we will describe several extensions of the Density Matrix Renormalization Group (DMRG) and Canonical Transformation Theory (CT). In the first part we describe a new way to solve for excited states in DMRG. To overcome the limitations of the traditional state-averaging approaches in excited state calculations, where one solves for 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 (1996)), we have implemented Harmonic Davidson and State-Averaged Harmonic Davidson algorithms within the context of DMRG. 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. In the second part, we propose an analytic response theory for DMRG whereby response properties correspond to analytic derivatives of DMRG observables with respect to the applied perturbations. Both static and frequencydependent response theories are formulated and implemented. We evaluate our pilot implementation by calculating static and frequency-dependent polar- isabilities of short oligo-di-acetylenes. The analytic response theory is competitive with dynamical DMRG methods and yields significantly improved accuracies when using a small number of density matrix renormalisation group states. Strengths and weaknesses of the analytic approach are discussed. In the third part, we describe how to calculate density matrices in CT theory. Density matrices are useful not only for computing observables but also for characterizing the nature of individual states. We demonstrate this with a preliminary application of the CT theory to understand the low-lying excited states of oligo-phenylvinylenes. We finish by presenting the theory and equations for calculating the response (e.g. to an external field) in the CT and DMRG-CT theories.
Electronic excited state calculation consists of an important issue in theoretical and computational chemistry. Under the Pariser-Parr-Pople (PPP) model and taking polyacenes and polyenes as examples, we compared the accuracy and efficiency of the state specific method and the state average method for the excited state calculation in the density matrix renormalization group formalism. We found that the state specific method can correctly trace the low-lying excited states and can provide more accurate eigen energies than the state average method. Nevertheless, for higher-lying excited states, the state specific method could converge to wrong eigenstates, which should be highly cautioned.
In previous work we have shown that the density matrix renormalization group (DMRG) enables near-exact calculations in active spaces much larger than are possible with traditional complete active space algorithms. Here, we implement orbital optimization with the DMRG to further allow the self-consistent improvement of the active orbitals, as is done in the complete active space self-consistent field (CASSCF) method. We use our resulting DMRG-CASSCF method to study the low-lying excited states of the all-trans polyenes up to C24H26 as well as beta-carotene, correlating with near-exact accuracy the optimized complete pi-valence space with up to 24 active electrons and orbitals, and analyze our results in the light of the recent discovery from resonance Raman experiments of new optically dark states in the spectrum.
We present the theory of a density matrix renormalization group (DMRG) algorithm which can solve for both the ground and excited states of non-Hermitian transcorrelated Hamiltonians and show applications in molecular systems. Transcorrelation (TC) accelerates the basis set convergence rate by including known physics (such as, but not limited to, the electron-electron cusp) in the Jastrow factor used for the similarity transformation. It also improves the accuracy of approximate methods such as coupled cluster singles and doubles (CCSD) as shown by recent studies. However, the non-Hermiticity of the TC Hamiltonians poses challenges for variational methods like DMRG. Imaginary-time evolution on the matrix product state (MPS) in the DMRG framework has been proposed to circumvent this problem, but this is currently limited to treating the ground state and has lower efficiency than the time-independent DMRG (TI-DMRG) due to the need to eliminate Trotter errors. In this work, we show that with minimal changes to the existing TI-DMRG algorithm, namely, replacing the original Davidson solver with the general Davidson solver to solve the non-Hermitian effective Hamiltonians at each site for a few low-lying right eigenstates, and following the rest of the original DMRG recipe, one can find the ground and excited states with improved efficiency compared to the original DMRG when extrapolating to the infinite bond dimension limit in the same basis set. An accelerated basis set convergence rate is also observed, as expected, within the TC framework.