Finite-size effects in periodic coupled cluster calculations

TL;DR

This work provides the first rigorous finite-size error analysis for periodic coupled cluster theory, focusing on CCD in 3D insulating systems with exact Hartree–Fock orbitals. By recasting finite-size errors as trapezoidal quadrature errors and exploiting the algebraic singularities of Coulomb integrals and CCD amplitudes, the authors prove that CCD(n) energy errors decay as $O(N_{ extbf{k}}^{-1/3})$, with energy errors from exact amplitudes decaying as $O(N_{ extbf{k}}^{-1})$ and MP2 decaying as $O(N_{ extbf{k}}^{-1})$. MP3 energy, being a subset of CCD(2), inherits the $O(N_{ extbf{k}}^{-1/3})$ scaling; under additional assumptions, converged CCD energies share the same rate. The results illuminate the chief source of finite-size error in CCD calculations—the amplitude calculation—and offer a principled framework for finite-size corrections and extrapolation in periodic quantum chemistry methods, while outlining limitations for gapless systems and fully converged CCD procedures.

Abstract

We provide the first rigorous study of the finite-size error in the simplest and representative coupled cluster theory, namely the coupled cluster doubles (CCD) theory, for gapped periodic systems. Assuming that the CCD equations are solved using exact Hartree-Fock orbitals and orbital energies, we prove that the convergence rate of finite-size error scales as $\mathscr{O}(N_\mathbf{k}^{-\frac13})$, where $N_{\mathbf{k}}$ is the number of discretization point in the Brillouin zone and characterizes the system size. Our analysis shows that the dominant error lies in the coupled cluster amplitude calculation, and the convergence of the finite-size error in energy calculations can be boosted to $\mathscr{O}(N_\mathbf{k}^{-1})$ with accurate amplitudes. This also provides the first proof of the scaling of the finite-size error in the third order Møller-Plesset perturbation theory (MP3) for periodic systems.

Figures (2)

• Figure 5.1: Diagrams of all the linear terms in the amplitude calculation. The dashed horizontal line denotes the ERI, and the solid horizontal line denotes the $t$ amplitude. The permuted amplitudes from (a), (b), (c), and (d) are not plotted. A capital letter $P$ refers to an orbital index $(p, \mathbf{k}_p)$. The amplitude calculation in (f) can be formulated as an integral over momentum vector $\mathbf{k}_c$, and in all other subplots as integrals over momentum vector $\mathbf{k}_k$.
• Figure 6.1: Energy and amplitude calculations using exact CCD$(1)$ amplitude. All the amplitudes are evaluated at $\mathbf{k}_i=\mathbf{k}_j= (0,0,0),\mathbf{k}_a=(0,0,\pi)$ and $(i,j,a,b) = (1,1,2,2)$. The power-law extrapolations use the three data points at $N_\mathbf{k} = 5^3,6^3,7^3$ in (a) and $N_\mathbf{k} = 6^3, 8^3, 10^3$ in the remaining subplots. These calculations are estimated theoretically in \ref{['tab:amplitude_error']} to have quadrature errors decay asymptotically in the rate of $N_\mathbf{k}^{-1}$, $N_\mathbf{k}^{-\frac{1}{3}}$, $N_\mathbf{k}^{-1}$, super-algebraically, super-algebraically, and $N_\mathbf{k}^{-1}$, respectively. In subplots (d) and (e), the actual data curve converges faster than the power-law extrapolations using $N_\mathbf{k}^{-1}$ and $N_\mathbf{k}^{-\frac{1}{3}}$. This can be seen as evidence that the quadrature errors decay super-algebraically.

Theorems & Definitions (27)

• Theorem 1: Error of CCD($n$)
• Corollary 2
• Corollary 3: Error of CCD
• Lemma 4: Singularity structure of the amplitude
• Lemma 5: Energy error with exact amplitude
• Lemma 6: Amplitude error in a single iteration
• Lemma 7: Lipschitz continuity of the finite CCD iteration mapping
• Definition 8: Algebraic singularity for univariate functions
• Remark 9
• Definition 10: Algebraic singularity for multivariate functions