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Rigorous Analysis of Finite-Size Error Scaling in Periodic Coupled Cluster Theory


Core Concepts
The finite-size error in periodic coupled cluster calculations for three-dimensional insulating systems exhibits an inverse volume scaling, even in the absence of any correction schemes. This is reconciled by showing that the Madelung constant correction can effectively reduce the finite-size errors in both orbital energies and electron repulsion integral contractions from the inverse length scaling to the inverse volume scaling.
Abstract
The content discusses the finite-size error scaling in periodic coupled cluster theory for three-dimensional insulating systems. Key points: Finite-size errors can significantly affect the accuracy of quantum chemistry calculations, even for systems with thousands of atoms. Understanding the finite-size scaling and employing correction schemes are crucial. Previous studies have shown that the finite-size error in Hartree-Fock and MP2 calculations exhibits inverse volume scaling, but the scaling in coupled cluster (CC) theory remained unclear. The authors analyze the finite-size error in coupled cluster doubles (CCD) theory, which is the simplest form of CC theory. They decompose the finite-size error into errors in three basic components: energy calculation using exact amplitudes, electron repulsion integral (ERI) contractions using exact amplitudes, and orbital energies. The authors show that the Madelung constant correction can reduce the finite-size errors in both orbital energies and ERI contractions from the inverse length scaling to the inverse volume scaling. When the Madelung constant correction is applied to both orbital energies and ERI contractions, the overall finite-size error in CCD(n) and converged CCD calculations scales as inverse volume. The authors provide rigorous mathematical analysis and numerical validation to support their findings, reconciling the seemingly paradoxical observations about the finite-size error scaling in coupled cluster theory.
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Deeper Inquiries

How can the finite-size error scaling be extended to other quantum chemistry methods beyond coupled cluster theory

The finite-size error scaling observed in coupled cluster theory can be extended to other quantum chemistry methods by considering the underlying principles and mathematical analysis used to derive the scaling behavior. The key lies in understanding the sources of finite-size errors in quantum chemistry calculations for periodic systems and how these errors manifest in different methods. By applying similar error analysis techniques and considering the impact of singularity structures in integrands, one can potentially generalize the inverse volume scaling of finite-size errors to other post-Hartree-Fock methods or wave function-based approaches used in quantum chemistry. This extension would involve adapting the theoretical framework and error correction strategies specific to each method while considering the unique characteristics of the method in question.

What are the implications of the inverse volume scaling of the finite-size error in coupled cluster theory for practical applications

The inverse volume scaling of the finite-size error in coupled cluster theory has significant implications for practical applications in quantum chemistry. This scaling behavior suggests that as the size of the system increases, the finite-size error decreases at a rate inversely proportional to the volume of the system. In practical terms, this means that for larger systems, the relative error in the calculations diminishes more rapidly than the increase in system size. This has profound implications for computational efficiency and accuracy in studying complex periodic systems, as it indicates that with larger system sizes, the accuracy of the calculations can be significantly improved without a proportional increase in computational resources. This understanding of finite-size error scaling can guide researchers in optimizing computational strategies for periodic systems, leading to more accurate and efficient quantum chemistry simulations.

Can the singularity subtraction technique used in this work be applied to reduce finite-size errors in other many-body quantum chemistry methods for periodic systems

The singularity subtraction technique used in the context of coupled cluster theory to reduce finite-size errors can potentially be applied to other many-body quantum chemistry methods for periodic systems. By identifying and addressing the singularities present in the integrands of calculations, it is possible to mitigate the impact of these singularities on the accuracy of the results. The singularity subtraction method involves subtracting the leading singular terms from the integrands in numerical quadratures, effectively reducing the errors associated with these singularities. This approach can be generalized to other methods that encounter similar singularities in their calculations, providing a systematic way to improve the accuracy and reliability of quantum chemistry simulations for periodic systems.
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