Frequency-Dependent Polarization Propagator Calculation for Quantum Dots Using Optimized Inverse Krylov Subspace and Folded-Spectrum Method

TL;DR

This work tackles the challenge of predicting frequency-dependent optical responses in quantum dots by formulating the polarization propagator as a resolvent of a Hamiltonian superoperator and incorporating MP2-based correlation up to second order. It introduces an inverted Krylov subspace combined with the folded-spectrum method to selectively target interior eigenvalues near a chosen frequency, enabling matrix-free calculations and reducing the cost of large-scale excitonic problems. Diagrammatic evaluation and Monte Carlo integration of two-electron integrals further streamline the computation, allowing application to PbS and CdS quantum dots with sizes spanning from sub-nanometer to several nanometers. The results show that first-order correlations red-shift spectral peaks and enhance intensities, while second-order corrections provide refined adjustments; the approach achieves substantial computational savings while maintaining accuracy, and it connects to the Riesz projector framework for interior-spectrum analysis.

Abstract

Accurate prediction of the frequency response of quantum dots under electromagnetic radiation is essential for investigating absorption spectra, excitonic effects, and nonlinear optical behavior in quantum dots and semiconductor nanoparticles. The polarization propagator provides a rigorous framework for evaluating these properties, but its construction is computationally demanding. Challenges arise from the level of electron correlation, the size of the excitonic basis, and the cost of evaluating two-electron integrals. This work addresses these difficulties by developing first- and second-order frequency-dependent polarization propagator calculations for PbS and CdS quantum dots. The propagator is formulated using the electron propagator approach and expressed as the resolvent of the Hamiltonian superoperator. Light-matter interaction is treated using the dipole approximation and represented in a particle-hole excitation operator basis. The correlated ground state is treated at the MP2 level, and all response-matrix terms up to second order in the fluctuating potential are included. A frequency-dependent inverse Krylov subspace method is derived and combined with the folded-spectrum technique to isolate excitation energies within a chosen frequency window. This strategy avoids full diagonalization of the response matrix and significantly reduces computational cost for large systems. The method is implemented in a matrix-free manner in which no explicit response matrix is assembled, and all operations rely on matrix-vector products. UV-VIS excitation spectra of PbS and CdS quantum dots were computed, demonstrating that the inverse Krylov subspace projection approach provides an efficient and accurate approximation for excitation spectra when full diagonalization is computationally prohibitive.

Figures (15)

• Figure 1: The contributing terms of the $\mathbf{M}$ matrix in terms of Hugenholz diagrams. Each diagram is labeled with the piece of the $\mathbf{M}$ matrix that it comes from ($\mathbf{A}$, $\mathbf{B}$, or $\mathbf{S}$). These represent the expectation values required to calculate the $\mathbf{M}$ operator up to $2^{\mathrm{nd}}$ order.
• Figure 2: Absorption spectra of CdS QDs from the zeroth order HF approximation of the M-operator
• Figure 3: Absorption spectra of PbS QDs from the zeroth order HF approximation of the M-operator
• Figure 4: Absorption spectra of PbS QDs from the $1^{\mathrm{st}}$ order correction to the approximation of the M-operator
• Figure 5: Absorption spectra of CdS QDs from the $1^{\mathrm{st}}$ order correction to the approximation of the M-operator