Towards the Solution of the Many-Electron Problem in Real Materials: Equation of State of the Hydrogen Chain with State-of-the-Art Many-Body Methods

We present numerical results for the equation of state of an infinite chain of hydrogen atoms. A variety of modern many-body methods are employed, with exhaustive cross-checks and validation. Approaches for reaching the continuous space limit and the thermodynamic limit are investigated, proposed, and tested. The detailed comparisons provide a benchmark for assessing the current state of the art in many-body computation, and for the development of new methods. The ground-state energy per atom in the linear chain is accurately determined versus bond length, with a confidence bound given on all uncertainties.

Popular Summary

One of the grand challenges of modern science is to understand and predict the quantum-mechanical behavior of a large ensemble of interacting electrons. Physical and chemical properties of materials and molecules are often the result of a delicate balance between competing behaviors in a many-electron system. Accurate computations are essential for predicting outcomes, but performing such calculations in a straightforward manner requires unattainable computational expense. Researchers have therefore come up with a variety of theoretical and numerical techniques to work around this hurdle. We present a comprehensive benchmark study of quantum many-body computational methods for addressing this challenge.

Our work focuses on two key aspects in treating real materials: the presence of long-ranged Coulomb interactions (the electrostatic force felt among electrons) and the need to study the continuum and thermodynamic limits. We characterize the relative accuracy and capabilities of 16 methods for performing many-electron calculations, which provides a survey of the state-of-the-art to guide applications. A large amount of data is produced that will be useful in benchmarking other existing and future electronic structure methods. Combining the strengths of complementary methods, we determine the equation of state (energy per atom versus interatomic spacing) of an infinite linear chain of hydrogen atoms in the continuum limit to high accuracy.

Future work can extend these benchmarks to more complex materials. Any progress in addressing the computational challenge of many-electron systems will be fundamental to “materials genome” initiatives, which attempt to discover and design new materials for a range of applications.

Figure 1

Potential energy curve of ${\mathrm{H}}_{10}$ (top) and deviations from FCI (bottom), in the minimal STO-6G basis.

Figure 2

Top: PEC of ${\mathrm{H}}_{10}$ in the continuous space, or complete basis set (CBS), limit. Bottom: Detailed comparison of the final ${\mathrm{H}}_{10}$ PECs, using $\mathrm{MRCI}+Q+\mathrm{F}12$ results at CBS as reference.

Figure 3

Detailed comparison of the ${\mathrm{H}}_{10}$ PEC in each basis set ($\mathrm{cc}\text{−}\mathrm{pV}x\text{Z}$, $x=2$, 3, 4, 5), using $\mathrm{MRCI}+Q$ as reference.

Figure 4

Basis set dependence of the PEC in ${\mathrm{H}}_{10}$ and extrapolation to the CBS limit. $\mathrm{MRCI}+Q$ and SBDMRG results are shown in the main figure for selected basis sets (to avoid cluttering). The inset shows the extrapolation of the correlation energy (from $\mathrm{MRCI}+Q$) for two sequences of basis sets, $\mathrm{cc}\text{−}\mathrm{pV}x\text{Z}$ and with F12, together with LR-DMC, for $R=1.8$.

Figure 5

Top: Computed equation of state in the minimal basis at the thermodynamic limit. Bottom: Detailed comparison using DMRG results as reference.

Figure 6

Top: Computed equation of state of the hydrogen chain in the thermodynamic limit. The inset shows the corresponding correlation energy per particle. Bottom: Detailed comparison using AFQMC results as reference (LR-DMC for the shortest two bond lengths). The empty circles indicate AFQMC results after a correction is applied from the difference with DMRG at the cc-pVDZ level. The pink error bands indicate all statistical uncertainties.

Figure 7

Illustration of the extrapolations to the CBS and TDL limits. Results are shown for $R=1.8a_B$. The left-hand panel shows extrapolation of $E_{\mathrm{CBS}}(N)$ versus $1/N$, while the right-hand panel shows extrapolation of $E_{\mathrm{cc}\text{−}\text{pV}x\mathrm{Z}}(N\to \infty )$ versus $1/x^3$. (The correlation energy is shown on the right, shifted by the CBS RHF energy.) Final results are consistent within statistical errors and independent of the order with which the limits are taken.

Figure 8

Diagrammatic equation of state in the TDL, at STO-6G level. Two $\mathrm{SC}\text{−}GW$ curves underline the ambiguity of formulating the Hamiltonian terms that are identically zero because of the Pauli principle (see text). The SC-$GW(\widehat{H})$ curve corresponds to the protocol usually used for realistic ab initio Hamiltonians, while the SC-$GW({\widehat{H}}^{\prime })$ curve corresponds to the one used in the lattice model Hamiltonian community. By accounting for higher-order vertex corrections, the ${\mathrm{BDMC}}_n$ results are observed to converge towards the estimates from other methods.

Figure 9

Top: EOS in the thermodynamic limit computed with finite basis sets (cc-pVDZ and cc-pVTZ). Middle: Detailed comparison using DMRG and AFQMC as reference. Error bars on the DMRG data indicate estimates of the TDL extrapolation uncertainties based on the results at the STO-6G level in Fig. 15. $\mathrm{AFQMC}+\mathrm{\Delta }{\mathrm{DMRG}}_{\mathrm{DZ}}$ is shown as reference for TZ (empty red circle), where the correction is obtained from the energy difference between DMRG and AFQMC at DZ. Bottom: Correlation energy per particle in the TDL, at cc-pVDZ (left) and cc-pVTZ (right) level.

Figure 10

Equation of state in the thermodynamic limit with select finite basis sets, by DMRG and SBDMRG. AFQMC results at the CBS limit are also shown for reference.

Figure 11

Illustration of the extrapolation to the CBS limit. Results are shown for ${\mathrm{H}}_{10}$ at bond lengths, $R=1.8$ and $2.8a_B$. The unrestricted Hartree-Fock energy is fitted to an exponential function of the index $x=2$, 3, 4, 5 of the cc-$\mathrm{pV}x\text{Z}$ basis, the correlation energy to the power law $\alpha +\beta x^{-3}$.

Figure 12

Effect of the F12 correction on $\mathrm{MRCI}+Q$ data. A cc-$\mathrm{pV}x\text{Z}+\mathrm{F}12$ calculation yields an energy of cc-$\mathrm{pV}(x+1)\text{Z}$ quality. Extrapolations to the CBS agree with each other within uncertainties from the fitting procedure.

Figure 13

Extrapolation to the thermodynamic limit using periodic (PBC) versus open (OBC) boundary conditions. Results are shown from LR-DMC for $R=1.8a_B$. Both sets of calculations converge to the same result for $N\to \infty$, but convergence is more rapid with OBC. The final result for $R=1.8a_B$ from AFQMC (from Fig. 6) is also indicated for reference.

Figure 14

Extrapolation of the EOS to the TDL limit, in the minimal STO-6G basis. The energy per particle is fitted to a second-order polynomial $A_0+A_1{N}^{-1}+A_2{N}^{-2}$ in $N^{-1}$, reflecting the presence of a bulk and a boundary contribution.

Figure 15

Illustration of the extrapolation to TDL. Residual errors of fits from different pairs $(N_1,N_2)$ of finite sizes are shown versus the reference extrapolation from $N=10$, 30, 50. Data are for the minimal STO-6G basis. Large system sizes are investigated in some cases to verify convergence.

Figure 16

Top: Difference with respect to extrapolated energies $E_{\mathrm{TDL}}(R)\equiv E(\infty ,\infty ,R)$ in STO-6G DMET calculations of increasing full system size $N$ (using a fixed fragment of size [22]) for three different geometries. (b) Difference with respect to extrapolated energies $E_{\mathrm{TDL}}(R)\equiv E(\infty ,\infty ,R)$ in STO-6G DMET calculations of increasing fragment size $N_f$ (using a fixed full system size $N_0$: ${\mathrm{H}}_{1200}$ for 1.0 and $1.4a_B$ and ${\mathrm{H}}_{450}$ for $1.8a_B$). The difference between the asymptotic value $E(N_0,\infty ,R)$ and the final extrapolation $E(\infty ,\infty ,R)$ is estimated as $E(\infty ,\infty ,R)-E(N_0,\infty ,R)\simeq E(\infty ,22,R)-E(N_0,22,R)$