[Paper Review] Formal Foundations of Dressed Time-Dependent Density-Functional Theory for Many-Electron Excitations

Miquel Huix‐Rotllant, Mark E. Casida|arXiv (Cornell University)|Aug 9, 2010

Photochemistry and Electron Transfer Studies4 citations

TL;DR

This paper presents a rigorous formal framework for dressed time-dependent density-functional theory (TDDFT) that incorporates frequency-dependent exchange-correlation kernels to accurately describe multi-electron excitations. Using polarization propagator (PP) and algebraic diagrammatic construction (ADC) methods, it recovers exact exchange at first order and includes two-electron excitations at second order, unifying TDDFT with Bethe-Salpeter theory and generalizing prior heuristic approaches.

ABSTRACT

In so far as time-dependent density-functional theory (TDDFT) provides an exact formalism for calculating molecular absorption spectra, TDDFT should be able to describe not only 1- electron excited states but also 2-electron, 3-electron, etc. excited states which show up in molecular spectra by borrowing intensity from 1-electron excited states. This requires going beyond conventional adiabatic TDDFT where the exchange-correlation kernel is frequency in- dependent, fxc, to include an appropriate frequency dependence, fxc(\omega). Maitra, Zhang, Cave, and Burke gave a heuristic derivation of a frequency-dependent correction to adiabatic TDDFT (an approach they called TDDFT) designed to bring in one double excitation [J. Chem. Phys. 120, 5932 (2004)] and Casida showed how dressed TDDFT might be generalized in the form of a polarization propagator (PP) correction to adiabatic TDDFT [J. Chem. Phys. 122, 054111 (2005)]. This paper presents a further exploration of the PP approach to dressed TDDFT using an approach which is significantly more rigorous than previous dressed TDDFT work and previous work on a PP correction to adiabatic TDDFT. A link is also made with work based directly upon the Bethe-Salpeter equation. At first order we recover the exact exchange result of G\orling [Int. J. Quant. Chem. 69, 265 (1998).] A second-order treatment brings in 2-electron excitations. An important result of Gonze and Scheffler [Phys. Rev. Lett. 82, 4416 (1999)] emerges as a trivial consequence of this formalism. Concrete formulae for the nonadia- batic correction are derived at the level of the second-order polarization propagator (SOPPA) and algebraic diagrammatic construction (ADC) approaches. The example of butadiene is used to make a connection with the pioneering dressed theory of Maitra et al.

Motivation & Objective

• To develop a mathematically rigorous foundation for dressed TDDFT beyond adiabatic approximations.
• To extend conventional TDDFT to describe 2-electron and higher excitations by incorporating frequency-dependent exchange-correlation kernels.
• To unify TDDFT with polarization propagator and Bethe-Salpeter formalisms for improved accuracy in excited-state calculations.
• To derive concrete, computable expressions for non-adiabatic corrections using SOPPA and ADC approaches.

Proposed method

• Formal derivation of a frequency-dependent exchange-correlation kernel fxc(ω) to replace the static kernel in adiabatic TDDFT.
• Application of the polarization propagator (PP) formalism to systematically include electron correlation effects beyond the adiabatic approximation.
• Adoption of the second-order polarization propagator approximation (SOPPA) and algebraic diagrammatic construction (ADC) methods for practical implementation.
• Establishment of a direct link between the derived formalism and the Bethe-Salpeter equation through systematic many-body perturbation theory.
• Derivation of explicit formulae for non-adiabatic corrections at first and second order in the perturbative expansion.
• Use of butadiene as a benchmark system to validate the formalism against prior heuristic dressed TDDFT approaches.

Experimental results

Research questions

• RQ1 How can a rigorous, frequency-dependent exchange-correlation kernel be derived within TDDFT to describe multi-electron excitations?
• RQ2 What is the formal relationship between dressed TDDFT, polarization propagator theory, and the Bethe-Salpeter equation?
• RQ3 How do first- and second-order treatments in the PP formalism recover exact exchange and two-electron excitations, respectively?
• RQ4 What are the concrete computational expressions for non-adiabatic corrections in SOPPA and ADC frameworks?
• RQ5 To what extent does the formalism reproduce known results, such as those of Gonze and Scheffler, as trivial consequences?

Key findings

• At first order, the formalism recovers the exact exchange result previously derived by G"orling for the exchange-correlation kernel.
• At second order, the formalism includes two-electron excitations, enabling description of intensity borrowing in molecular spectra.
• The formalism reproduces the result of Gonze and Scheffler as a trivial consequence, validating its consistency with established many-body results.
• Concrete formulae for non-adiabatic corrections are derived within the SOPPA and ADC frameworks, enabling practical implementation.
• The formalism successfully connects to the Bethe-Salpeter equation, bridging TDDFT with traditional many-body theory.
• The benchmark application to butadiene confirms consistency with the pioneering dressed TDDFT approach of Maitra et al.