Convergence of many-body wave-function expansions using a plane-wave basis: From homogeneous electron gas to solid state systems

Using the finite simulation-cell homogeneous electron gas (HEG) as a model, we investigate the convergence of the correlation energy to the complete-basis-set (CBS) limit in methods utilizing plane-wave wave-function expansions. Simple analytic and numerical results from second-order Møller-Plesset theory (MP2) suggest a $1/M$ decay of the basis-set incompleteness error where $M$ is the number of plane waves used in the calculation, allowing for straightforward extrapolation to the CBS limit. As we shall show, the choice of basis-set truncation when constructing many-electron wave functions is far from obvious, and here we propose several alternatives based on the momentum transfer vector, which greatly improve the rate of convergence. This is demonstrated for a variety of wave-function methods, from MP2 to coupled-cluster doubles theory and the random-phase approximation plus second-order screened exchange. Finite basis-set energies are presented for these methods and compared with exact benchmarks. A transformation can map the orbitals of a general solid state system onto the HEG plane-wave basis and thereby allow application of these methods to more realistic physical problems. We demonstrate this explicitly for solid and molecular lithium hydride.

Figure 1

MP2 correlation energies for the 14 electron gas at $r_s=5.0$ a.u. retrieved as a function of (a) $E_k^{-\frac{3}{2}}$ and (b) $M^{-1}$ (where $M$ is the number of spin orbitals) tend towards a linear relationship as the complete-basis-set limit is approached. In each plot, the dotted lines refer to CBS limits for each basis-set size, which are obtained by a linear extrapolation of this point and the previous three points (sometimes not visible on the graph). Using these extrapolated estimates, it can be seen that the $M^{-1}$ power law is smoother due to fewer finite-size effects; this is due to $M$ being a more appropriate variable to consider how much correlation energy the basis set retrieves. For this system, $M=1030$ corresponds to a kinetic energy cutoff of 1.3077 a.u. or 35.59 eV, which changes with both $N$ and $r_s$ for the HEG.

Figure 2

Discussion and diagrams of cutoffs using momentum transfer vectors. The white circle represents the Fermi sphere, and the volume excluded from the virtual space by occupation effects is not considered. (a) This is a diagram of the local $E_g$ cutoff. Using a simple momentum transfer cutoff scheme [Eq. (27)] makes the excitation space (set of possible virtual orbitals to be excited into) ${\mathbf{k}}_i$,${\mathbf{k}}_j$ $\to$ $\{{\mathbf{k}}_i+\mathbf{g}\}$,$\{{\mathbf{k}}_j+{\mathbf{g}}^{\prime }\}$ dependent on ${\mathbf{k}}_i$ and ${\mathbf{k}}_j$ when they are not at the $\Gamma$ point. This implies that the sets $\{{\mathbf{k}}_i+\mathbf{g}\}$ and $\{{\mathbf{k}}_j+{\mathbf{g}}^{\prime }\}$ are not the same. (b) This is a diagram of the local $E_g$ cutoff. Comparison between a specific excitation ${\mathbf{k}}_i$,${\mathbf{k}}_j$ $\to$ ${\mathbf{k}}_a$,${\mathbf{k}}_b$ and ${\mathbf{k}}_i$,${\mathbf{k}}_j\to {\mathbf{k}}_b,{\mathbf{k}}_a$. These differ only in the permutation of the hole states (or, equivalently, the electron states). In the case of the local $E_g$-cutoff cutoff scheme [Eq. (27)], one term (solid line) is allowed and the other term (dashed line) is disallowed. (c) This is an illustration of the intersection $E_g$ cutoff. One solution to the problem illustrated in (b) is to only allow excitations to the region of $k$ space formed by the overlap of the two regions in (a). Now, both terms are either disallowed (as shown here) or allowed and permutational symmetry is restored. (d) This is an illustration of the union $E_g$ cutoff. A second and different solution is to extend the allowed space of (a) to anywhere that either $\{{\mathbf{k}}_i+\mathbf{g}\}$ or $\{{\mathbf{k}}_j+{\mathbf{g}}^{\prime }\}$ would be allowed, also restoring the permutational symmetry.

Figure 3

Comparison of correlation energy retrieved as a function of basis-set size for a variety of cutoff schemes.

Figure 4

Graphs comparing (a) union $E_g$ cutoff and (b) $E_k$ cutoff for a range of different densities. As the density is lowered, the extrapolations become more distant from the CBS limit even for MP2 theory, due to a rising contribution from exchange in the Hartree-Fock orbital energies that is not well-behaved with respect to $M$. Furthermore, more pronounced finite-size effects are seen. The extrapolated results shown by the dotted lines are only represented with error bars in two cases ($r_s=0.5$ a.u., $r_s=20.0$ a.u.) for clarity. Complete-basis-set limit energies from which these basis-set incompleteness errors are derived are tabulated in Table .

Figure 5

Comparison of correlation energy for the $N=14$ system for (a) $r_s=1.0$, (b) $r_s=5.0$ retrieved as a function of basis-set size for a variety of quantum chemical methods. Their tendency to display a $1/M$ trend varies from method to method as well as with $r_s$ (see text for further discussion).

Figure 6

Comparison between direct extrapolation and single-point extrapolation (SPE) for RPA+SOSEX on the $N=14$, $r_s=1.0$ a.u. gas. In the conventional direct extrapolation, calculations are performed at a series of basis-set sizes $M$ and then extrapolated using a $1/M$ fit to the high-$M$ limit. In the SPE, a single calculation is performed at an overall basis-set size of $M$, in this case $M=682$, and effective basis-set energies are constructed according to Eq. (35) over the full range of $M^{\prime }$. Some of these points are discarded as $M^{\prime }$ approaches $M$ since not all momentum transfer vectors can be accommodated within the basis set (dashed green line, discussed in the text). The points where $M>M^{\prime }$ have not been included because they are not visibly different on this scale. The extrapolations are shown by dotted lines, and agree in the CBS limit within reasonable extrapolation error estimates ($\sim 2\times {10}^{-3}$ a.u.).

Figure 7

Correlation energies calculated for the 14-electron system using RPA+SOSEX at (a) $r_s=1.0$ and (b) $r_s=5.0$ and using CCD (c) $r_s=1.0$ and (d) $r_s=5.0$. Direct extrapolation and single-point extrapolation (SPE) are compared at a variety of basis-set sizes (SPE curves just show extrapolated results). SPE performs best for RPA at $r_s=1.0$, where it converges faster than direct extrapolation, and worst for CCD at $r_s=5.0$, where it converges slower than direct extrapolation.

Figure 8

The MP2 correlation energy of the LiH 2 $\times$ 2 $\times$ 2 supercell retrieved as a function of the basis-set size for a variety of extrapolation schemes. The SPE curve shows the effective basis-set energies produced from a single calculation with $M=2045$.

Figure 9

The MP2 correlation energy of the LiH molecule retrieved as a function of the basis-set size for a variety of extrapolation schemes. The SPE curve shows the effective basis-set energies produced from a single calculation with $M=1647$.