Explicit expressions for electron correlation at the second-order many-body perturbation-theory [MBPT(2)] level are presented and implemented for the total energy per unit cell and for the band structure of extended systems. In the latter case, a formula is presented for a direct evaluation of the band gap rather than obtaining it as a difference of two large numbers. Application is made to alternating trans-polyacetylene. We assess the convergence of MBPT(2) with the number of unit cells (N) included in the lattice summations, the number of k-points (K) taken for the integrals over k in the first Brillouin zone, and the cutoff threshold (10−C) for the two-electron integrals. The MBPT(2) correlation correction to the band structure converges very slowly with N and demands a large K while the MBPT(2) correction to the total energy per unit cell converges much faster with N and needs a much smaller K. Neither MBPT(2) correction is sensitive to the cutoff of the two-electron atomic orbital integrals, 10−C, when C≥5. For polyacetylene, the MBPT(2) band gap is much improved from the SCF result, but does not agree with previous numerical data. Analysis shows that the previous MBPT(2) results were obtained either with too few unit cells such that the convergence with N had not been reached, or that the zeroth-order Hartree–Fock results were inadequately converged. MBPT(2) with a DZP basis improves the Hartree–Fock band gap from 5.57 to 3.22 eV at the experimentally estimated geometry, compared to the measured ∼2 eV peak in the absorption spectrum of the system. We verify our results with three independent programs. We also study the band gap as a function of geometry.
Explicit expressions for electron correlation at the second-order many-body perturbation-theory [MBPT(2)] level are presented and implemented for the total energy per unit cell and for the band structure of extended systems. In the latter case, a formula is presented for a direct evaluation of the band gap rather than obtaining it as a difference of two large numbers. Application is made to alternating trans-polyacetylene. We assess the convergence of MBPT(2) with the number of unit cells (N) included in the lattice summations, the number of k-points (K) taken for the integrals over k in the first Brillouin zone, and the cutoff threshold (10−C) for the two-electron integrals. The MBPT(2) correlation correction to the band structure converges very slowly with N and demands a large K while the MBPT(2) correction to the total energy per unit cell converges much faster with N and needs a much smaller K. Neither MBPT(2) correction is sensitive to the cutoff of the two-electron atomic orbital integrals, 10−C, when C≥5. For polyacetylene, the MBPT(2) band gap is much improved from the SCF result, but does not agree with previous numerical data. Analysis shows that the previous MBPT(2) results were obtained either with too few unit cells such that the convergence with N had not been reached, or that the zeroth-order Hartree–Fock results were inadequately converged. MBPT(2) with a DZP basis improves the Hartree–Fock band gap from 5.57 to 3.22 eV at the experimentally estimated geometry, compared to the measured ∼2 eV peak in the absorption spectrum of the system. We verify our results with three independent programs. We also study the band gap as a function of geometry.
The convergence behavior of many-body perturbation theory (MBPT) with lattice summations directly reflects the spatial property of electron correlation in extended systems. A knowledge of such properties is essential for both theory and numerical calculations. It is shown solely from theory that the second-order MBPT contribution to the total energy per unit cell in polymers converges with lattice size N as 1/N3 while the second-order MBPT contribution to the band energy converges as 1/N2. These convergence behaviors are demonstrated numerically for polyacetylene.
An application of second-order many-body perturbation theory to energies and energy bands of polymers is often hindered by the steep polynomial dependence of its computational cost on the number of wave vector sampling points (K) in the Brillouin zone (BZ). It is shown that a Hartree-Fock (HF) calculation with a large value of K (120 in the first BZ) followed by a second-order many-body perturbation calculation with a much smaller value (K=6) can lead to reliable, interpolated correlated energy bands and density of states of a polymer at less than 1% of the computational cost of the conventional approach. Quantitative simulations on photoelectron spectra of trans- and cis-polyacetylenes and polyethylene show that the correlated energy bands and densities of states thus obtained agree quantitatively with the observed and are significant (sometimes qualitative) improvements over the HF results. The energy bands and photoelectron spectra of polydiacetylene are predicted by this method to assist in the interpretation of future high-resolution measurements.
We propose progressive downsampling of wave vectors in the Brillouin zone integrations occurring in the second-order many-body or Møller-Plesset perturbation (MP2) method for extended systems of one-dimensional periodicity. Higher-lying unoccupied and lower-lying occupied Bloch orbitals are subject to downsampling by an exponentially increasing factor (with base n), making the total number of Bloch orbitals included in the MP2 lattice sums grow only logarithmically with respect to the number of basis functions per unit cell. Unlike the mod n downsampling scheme proposed earlier, this log n scheme reduces the scaling of the computational cost and thus achieves a greater speedup as the unit cell size increases. Correct band indexing is essential for accuracy. Two-electron integrals entering the MP2 energy and quasiparticle energy expressions must be multiplied by quadrature weights that are a function of the energy bands involved, and an algorithm to compute the weights is proposed. A combined use of the log n and mod n schemes can speedup the MP2/6-31G** calculation of polyacetylene typically by a factor of 20 with an error in the correlation energy within a few percent relative to the conventional calculation. Similar combinations can reproduce the MP2 quasiparticle energy bands accurately at a fraction of the usual computational cost.
Recently the possible divergence of many-body perturbation theory or many-body green function (propagator) methods with lattice summations in extended systems has been raised. The convergence of these methods with lattice summations is not only the key to establishing their correct scaling properties in inhomogeneous systems, but is also a necessity if numerical calculations are to be meaningful. In this report, it is rigorously shown that many-body perturbation theory (MBPT), coupled cluster theory (CC), and many-body green function (MBGF) methods all converge uniformly with lattice summations, although the integrand for the integration over the reciprocal lattice vector, k, could become infinite at special k values. Our proof is given not only for infinite polymers but also for crystals. In our proof, only the continuity of the zeroth-order band structure and Bloch orbitals with k is used. We show that MBPT, CC, and MBGF methods converge with the radius of the lattice summation range R at least as fast as 1/R1/n where n is the dimensionality of the system. It could be much faster if the zeroth-order band structure and Bloch orbitals have continuous first- or even higher-order derivatives with k. In practical numerical calculations, one should not pursue the convergence of the integrand, but the convergence of the correction considered with lattice summations.
Coupled-cluster theory with connected single and double excitation operators (CCSD) and related approximations, such as linearized CCSD, quadratic configuration interaction with single and double excitation operators, coupled-cluster with connected double excitation operator (CCD), linearized CCD, approximate CCD, and second- and third-order many-body perturbation theories, are formulated and implemented for infinitely extended one-dimensional systems (polymers), on the basis of the periodic boundary conditions and distance-based screening of integrals, density matrix elements, and excitation amplitudes. The variation of correlation energies with the truncation radii of short- and long-range lattice sums and with the number of wave vector sampling points in the first Brillouin zone is examined for polyethylene, polyacetylene, and polyyne, and is shown to be a function of the degree of pi-electron conjugation or the fundamental band gaps. The t2 and t1 amplitudes in the atomic orbital (AO) basis are obtained by first computing the t amplitudes in the Bloch-orbital basis and subsequently back-transforming them into the AO basis. The plot of these AO-based t amplitudes as a function of unit cells also indicates that the t2 amplitudes of polyacetylene and polyyne exhibit appreciably slower decay than those of polyethylene, although the asymptotic decay behavior is invariably 1/r3. The AO-based t1 amplitudes appear to correlate strongly with the electronic structure, and they decay seemingly exponentially for polyethylene whereas they stay at a constant magnitude across the seventh nearest neighbors of polyacetylene and polyyne, which attests to far reaching effects of nondynamical electron correlation mediated by orbital rotation. Nonetheless, the unit cell contributions to the correlation energies taper below 10(-6) hartree after 15 A for all three polymers. The basis set dependence of the decay behavior of t2 amplitudes is also examined for linear hydrogen fluoride polymer (HF)infinity and linear beryllium polymer (Be)infinity employing the STO-3G, 6-31G, and 6-31G* basis sets, and proves to be rather small.
A systematically converging series of ab initio, postdensity functional, size-consistent, electron-correlated approximations is desired for predictive computing of electronic band structures of insulating, semiconducting, and metallic solids. A series that meets all of these desiderata (except the applicability to metals) is ab initio many-body Green’s function theory based on Gaussian-type orbital (GTO) basis sets. Here, its leading-order approximation, the second-order Green’s function (GF2) method in the diagonal and frequency-independent approximations with the aug-cc-pVDZ basis set, is applied to the fundamental band gaps of three semiconductors (diamond, silicon, and silicon carbide in the zincblende structure) using cluster models. Corrections are made to the basis set incompleteness errors by the explicit correlation (F12) ansatz (GF2-F12) for the valence band edges. The crystals are modeled as surface-passivated clusters of increasing sizes, whose wave functions are expanded by up to 2709 GTO basis functions. Immense computational costs of these calculations are overcome by the highly scalable stochastic algorithm of the Monte Carlo GF2-F12 method, whose operation cost per state increases only as a cubic power of system size, which has a tiny memory footprint and easily achieves near-perfect parallel efficiency on thousands of CPUs or on hundreds of GPUs. The correlated, F12-corrected highest-occupied and lowest-unoccupied molecular orbital energy (HOMO–LUMO) gap is 5.78 ± 0.07 eV for C 87 H 76 as compared with the experimental value of the fundamental (indirect) band gap of bulk diamond at 5.48 eV. The correlated, F12-corrected HOMO–LUMO gaps for Si 75 H 76 and Si 32 C 43 H 76 are 2.56 ± 0.15 and 3.50 ± 0.12 eV, respectively, which are expected to decrease further with increasing cluster sizes. The experimental fundamental (indirect) band gaps of bulk silicon and silicon carbide are 1.17 and 2.42 eV, respectively.