Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

ABSTRACT

We describe the implementation of orbital optimisation for the models in the perfect pairing hierarchy. Orbital optimisation, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local minima and σ-π symmetry breaking solutions were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calculations at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimised orbitals are used. PH is found to yield good agreement with previously published density matrix renormalisation group data in the π space, capturing over 95% of the correlation energy. Full-valence calculations made possible by our novel, efficient code reveal that strong correlations are weaker when larger basis sets or active spaces are employed than in previous calculations. The largest full-valence PH calculations presented correspond to a (192e,192o) problem.     

A local framework for calculating coupled cluster singles and doubles excitation energies (LoFEx-CCSD)

ABSTRACT

The recently developed Local Framework for calculating Excitation energies (LoFEx) is extended to the coupled cluster singles and doubles (CCSD) model. In the new scheme, a standard CCSD excitation energy calculation is carried out within a reduced excitation orbital space (XOS), which is composed of localised molecular orbitals and natural transition orbitals determined from time-dependent Hartree–Fock theory. The presented algorithm uses a series of reduced second-order approximate coupled cluster singles and doubles (CC2) calculations to optimise the XOS in a black-box manner. This ensures that the requested CCSD excitation energies have been determined to a predefined accuracy compared to a conventional CCSD calculation. We present numerical LoFEx-CCSD results for a set of medium-sized organic molecules, which illustrate the black-box nature of the approach and the computational savings obtained for transitions that are local compared to the size of the molecule. In fact, for such local transitions, the LoFEx-CCSD scheme can be applied to molecular systems where a conventional CCSD implementation is intractable.     

Iterative diagonalization in augmented plane wave based methods in electronic structure calculations

Due to the increased computer power and advanced algorithms, quantum mechanical calculations based on Density Functional Theory are more and more widely used to solve real materials science problems. In this context large nonlinear generalized eigenvalue problems must be solved repeatedly to calculate the electronic ground state of a solid or molecule. Due to the nonlinear nature of this problem, an iterative solution of the eigenvalue problem can be more efficient provided it does not disturb the convergence of the self-consistent-field problem. The blocked Davidson method is one of the widely used and efficient schemes for that purpose, but its performance depends critically on the preconditioning, i.e. the procedure to improve the search space for an accurate solution. For more diagonally dominated problems, which appear typically for plane wave based pseudopotential calculations, the inverse of the diagonal of (H ? ES) is used. However, for the more efficient “augmented plane wave + local-orbitals” basis set this preconditioning is not sufficient due to large off-diagonal terms caused by the local orbitals. We propose a new preconditioner based on the inverse of (H ? λS) and demonstrate its efficiency for real applications using both, a sequential and a parallel implementation of this algorithm into our WIEN2k code.     

Full-valence density matrix renormalisation group calculations on meta-benzyne based on unrestricted natural orbitals. Revisit of seamless continuation from broken-symmetry to symmetry-adapted models for diradicals

ABSTRACT

In this work, we show that the natural orbitals of unrestricted hybrid density functional theory (UHDFT) can be used as the active space orbitals to perform multireference (MR) calculations, for example, the density matrix renormalisation group (DMRG) and Mukherjee-type (Mk) MR coupled-cluster (CC) method. By including a sufficiently large number of these natural orbitals, full-valence (FV) active space can be identified without recourse of the expensive self-consistent procedures for DMRG-SCF. Several useful chemical indices are derived based on the occupation numbers of the natural orbitals for seamless continuation from broken-symmetry (BS) to symmetry-adapted (SA) methods. These procedures are used on 1,3-didehydrobenzene (meta-benzyne) to calculate its singlet (S)-triplet (T) gap. We compare our results to available experiments and computational results obtained by several other groups. We see our procedures as a seamless bridge between single-reference BS methods, such as UHDFT, and the SA MR methods, such as FV DMRG and MkMRCC.     

An Anderson Impurity Interacting with the Helical Edge States in a Quantum Spin Hall Insulator

Using the natural orbitals renormalization group(NORG)method,we investigate the screening of the local spin of an Anderson impurity interacting with the helical edge states in a quantum spin Hall insulator.It is found that there is a local spin formed at the impurity site and the local spin is completel.y screened by electrons in the quantum spin Hall insulator.Meanwhile,the local spin is screened dominantly by a single active natural orbital.We then show that the Kondo screening mechanism becomes transparent and simple in the framework of the natural orbitals formalism.We project the active natural orbital respectively into real space and momentum space to characterize its structure.We conilrm the spin-momentum locking property of the edge states based on the occupancy of a Bloch state on the edge to which the impurity couples.Furthermore,we study the dynamical property of the active natural orbital represented by the local density of states,from which we observe the Kondo resonance peak.     

UNO DMRG CASCI calculations of effective exchange integrals for m-phenylene-bis-methylene spin clusters

ABSTRACT

Theoretical examinations of the ferromagnetic coupling in the m-phenylene-bis-methylene molecule and its oligomer were carried out. These systems are good candidates for exchange-coupled systems to investigate strong electronic correlations. We studied effective exchange integrals (J), which indicated magnetic coupling between interacting spins in these species. First, theoretical calculations based on a broken-symmetry single-reference procedure, i.e. the UHF, UMP2, UMP4, UCCSD(T) and UB3LYP methods, were carried out with a GAUSSIAN program code under an SR wave function. From these results, the J value by the UHF method was largely positive because of the strong ferromagnetic spin polarisation effect. The J value by the UCCSD(T) and UB3LYP methods improved an overestimation problem by correcting the dynamical electronic correlation. Next, magnetic coupling among these spins was studied using the CAS-based method of the symmetry-adapted multireference methods procedure. Thus, the UNO DMRG CASCI (UNO, unrestricted natural orbital; DMRG, density matrix renormalised group; CASCI, complete active space configuration interaction) method was mainly employed with a combination of ORCA and BLOCK program codes. DMRG CASCI calculations in valence electron counting, which included all orbitals to full valence CI, provided the most reliable result, and support the UB3LYP method for extended systems.     

Projection technique for population analysis of atomic orbitals in crystals

The one-electron density matrix of a crystal in the basis set of localized orbitals is calculated using two variants of the projection technique, namely, the projection of crystal orbitals onto the space of atomic orbitals (technique A) and the projection of atomic functions onto the space of crystal orbitals (technique B). A comparative analysis of the one-electron density matrices thus obtained is carried out, and a simplified version of technique B is proposed to avoid cumbersome calculations with a large number of vacant crystal orbitals. Both techniques are used to calculate the local characteristics of the electronic structure (atomic charges, atomic covalences, bond orders) for a number of crystals (Si, SiC, GaAs, MgO, cubic BN, TiO₂ rutile) in the framework of the density-functional theory within the generalized gradient approximation in the plane wave basis set with the norm-conserving pseudopotentials. It is revealed that both variants of the projection technique lead to close local characteristics of the electronic structure. The local characteristics of the electronic structure of the TiO₂ crystal with a rutile structure are determined by the projection technique and by constructing the Wannier-type atomic functions (WTAF) in the minimal valence basis set in the framework of the variational method with the crystal orbitals calculated in the linear combination of atomic orbitals (LCAO) approximation. It is demonstrated that, although the basis sets used for calculating the crystal orbitals differ significantly (plane waves in the projection technique, LCAO in the WTAF method), the local characteristics of the electronic structure are in good agreement.     

Theoretical study of the photoelectron spectra of gaseous Cu3Cl3

Ab initio calculations have been performed to interpret the photoelectron spectrum of gaseous cuprous chloride, Cu₃Cl³. Density functional calculations revealed Cu₃Cl₃ to be a planar cyclic D_3h molecule. Koopmans' theorem and two-hole/one-particle calculations with canonical Hartree-Fock orbitals were used to interpret the vertical ionization energies. These were compared with similar calculations using B3LYP Kohn-Sham orbitals. The results confirm the claim by Casida that Kohn-Sham orbitals mimic Dyson orbitals.     

Exploring the MRCI method for calculating interaction energies: application to the HeNe potential curve

A multi-reference configuration interaction (MRCI) method is described, which is devised for the calculation of interaction energies of van der Waals complexes and applied to calculating the HeNe potential energy curve. The MRCI calculations make use of a generalized Pople-correction in order to account for the lack of size consistency. The orbital space is partitioned into three subspaces: the first active space (AS1), which contains the strongly occupied orbitals; the second active space (AS2), which contains the main intra-correlating orbitals; and the external space (ES). It is shown that, to keep the error below ± 0.2 K in the excitation scheme and the active orbital space it is sufficient to include only σ-orbitals in AS2 and to use an excitation scheme (labelled Qq-MRCI) that encompasses only up to quadruply excited configurations. The final active orbital space (AS2) turned out to be 2s(He), 2pσ(He), 3s(Ne), 3σ(Ne) and 3dσ(Ne). Other MRCI variants, in which most or all quadruply excited configurations were deleted from the CI expansion (Qt- and Tt-MRCI), were found to be inadequate. Using the Qq-MRCI scheme together with a 197-orbital ‘interaction optimized’ basis set (IO197), the MRCI interaction energy at R = 5.7 _a0 was calculated to be -21.12K. The corresponding values at the MP4 and CCSD(T) levels of theory are -20.06 K and -20.99 K, respectively, indicating that the MP4 method is inappropriate for highly accurate calculations on this system. Fitting the calculated data using a generalized Morse function, including an additional C₆/R⁶ term to account for a correct long-range behaviour of the potential, the MRCI well depth was calculated to be -21.16K at R_eq = 5.73a ₀. The MRCI and CCSD(T) potentials have the same quality and are found to be in good agreement with the Hartree-Fock dispersion (HFD-B) potential of Keil, M., Danielson, L. J., and Dunlop, P. J., 1991, J. Chem. Phys., 94, 296. It is concluded that, for basis IO197, the CCSD(T) method is sufficiently accurate for calculating the HeNe interaction. To recover the small, missing contributions (a few tenths of a Kelvin), MRCI should be used.     

Visualization of deficiencies in approximate molecular wave functions: the orbital amplitude difference function for the matrix Hartree-Fock description of the ground state of the boron fluoride molecule

The orbital amplitude difference function is used to assess the quality of Hartree–Fock orbitals obtained by invoking the algebraic approximation for the BF ground-state. Systematic sequences of even-tempered, spherical-harmonic Gaussian-type basis functions are used to generate orbitals for which the corresponding total Hartree–Fock energy approaches the 1 μE _h level of accuracy. Exact orbitals are obtained from finite difference calculations using a grid based on spheroidal coordinates. The finite basis set approximations for the orbital are discretized. The accuracy of the discretization is assessed. For each occupied orbital a discretized representation of the orbital amplitude difference function is generated and analysed.     

Ab initio calculation of the phase stability,mechanical properties and electronic structure of ZrCr2 Laves phase compounds

Ab initio calculations have been used to investigate the phase stability, mechanical properties and electronic structure of ZrCr₂ Laves phase compounds, based on the method of augmented plane waves plus local orbitals with the generalized gradient approximation. The calculated lattice constants for the C15, C36 and C14 structures are in good agreement with experimental values. The calculation of heats of formation showed that C15 is a ground-state phase, whereas C36 is an intermediate phase and C14 the high-temperature phase. The elastic constants and elastic moduli for the C15 structure were calculated systematically and compared with experiments and previous theoretical calculations. The intrinsic and extrinsic stacking fault energies are found to be 112 and 98?mJ?m^?2, respectively. The equilibrium separations between Schockley are also predicted using the calculated elastic moduli and stacking fault energies. Finally, the calculated electronic structures of these Laves phases are discussed based on these results.     

A CASCI-MRMP method based on Kohn—Sham orbitals

Kohn-Sham orbitals are used in the previously proposed CASCI-MRMP scheme (a multi-reference M?ller-Plesset (MRMP) method with a complete active space configuration interaction (CASCI) reference function). That is, the CASCI wave function was constructed using the Kohn-Sham orbitals and used as a reference function of the MRMP to incorporate the remaining dynamical correlation. The scheme was applied to the potential curves of the ground and low-lying excited states of N₂, the potential curve of the ground state of CO, the barrier height of the H₂CO → H₂ + CO reaction, the valence π-π? and Rydberg excited states of benzene, and the low-lying excited states of ozone. Good agreement between the theory, experiment, and some benchmark calculations was obtained. The various orbitals which are investigated here do not give very different results. Rather, the choice of active space makes a considerable difference, and in particular the perturbation calculation is proved to be very important.     

Large-scale shell model calculations for odd-odd <Superscript>58-62</Superscript>Mn isotopes

Large-scale shell model calculations have been carried out for odd-odd ^58-62Mn isotopes in two different model spaces. The first set of calculations has been carried out in full fp shell valence space with two recently derived fp shell interactions namely GXPF1A and KB3G treating ⁴⁰Ca as core. The second set of calculations has been performed in the fpg _9/2 valence space with the fpg interaction treating ⁴⁸Ca as core and imposing a truncation by allowing up to a total of six particle excitations from the 0f _7/2 orbital to the upper fp orbitals for protons and from the upper fp orbitals to the 0g _9/2 orbital for neutrons. For low-lying states in ⁵⁸Mn , KB3G and GXPF1A both predict good results and for ⁶⁰Mn , KB3G is much better than GXPF1A. For negative-parity and high-spin positive-parity states in both isotopes the fpg interaction is required. Experimental data on ⁶²Mn is sparse and therefore it is not possible to make any definite conclusions. More experimental data on negative-parity states is needed to ascertain the importance of 0g _9/2 and higher orbitals in neutron-rich Mn isotopes.     

Crossing of the πd<Subscript>5/2</Subscript> and πg<Subscript>7/2</Subscript> orbitals in the <Subscript>51</Subscript>Sb isotopes

Self-consistent calculations using the D1S Gogny force have been performed in order to study the mechanism involved in the crossing of the πd _5/2 and πg _7/2 orbitals in the Sb isotopes. This inversion is well predicted by the HFB + blocking calculations with spherical symmetry performed for the odd-A Sb isotopes. In addition, several HFB and HF calculations have been performed for even-even nuclei of the five neighbouring isotopic chains (Z = 46 to 54, from the proton dripline to N = 82). The results obtained for the binding energies of the two proton orbitals indicate that the radii of the systems play an important role in the crossing, even though some particular πν interactions also give a contribution. The spin-orbit interaction, which is known to be concentrated mainly at the nuclear surface, is proposed to be the main responsible of the crossing.     

Simple approximate physical orbitals for GW quasiparticle calculations

Generating unoccupied orbitals within density functional theory (DFT) for use in GW calculations of quasiparticle energies becomes prohibitive for large systems. We show that, without any loss of accuracy, the unoccupied orbitals may be replaced by a set of simple approximate physical orbitals made from appropriately prepared plane waves and localized basis DFT orbitals that represent the continuum and resonant states of the system, respectively. This approach allows for accurate quasiparticle calculations using only a very small number of unoccupied DFT orbitals, resulting in an order of magnitude gain in speed.     

Visualization of deficiencies in approximate molecular wave functions: the local orbital energy function for the matrix Hartree—Fock model

The local orbital energy function is used to assess the quality of approximate Hartree-Fock orbitals obtained by invoking the algebraic approximation and using a finite basis set expansion. Systematic sequences of distributed universal even-tempered basis sets of spherical-harmonic Gaussian-type functions are used to generate orbitals for which the corresponding total Hartree-Fock energy approaches the 1 μE_h level of accuracy. A pilot study of the behaviour of the local energy function is made for the hydrogenic atom described by a sequence of even-tempered Gaussian basis sets. The results of prototype calculations for the Hartree-Fock ground state of the BF molecule at its equilibrium geometry are presented. Sequences of calculations which use atom-centred basis sets are investigated as well as sequences which also include bond centred functions. The effects of the bond centred functions on the local orbital energy function are analysed. The local orbital energy function is seen as a measure of the quality of calculations carried out within the matrix Hartree-Fock approximation which can be employed in cases where the corresponding finite difference Hartree-Fock results are not available.     

能带反折叠方法研究进展

基于密度泛函理论的第一性原理方法已经成为人们研究材料结构、性质以及进行新功能材 料设计的重要手段。对于掺杂和界面体系，人们常常需要使用超胞来描述。超胞的使用导致能带 折叠，从而掩盖能带结构的重要特征，为人们分析掺杂和界面效应对材料能带结构的影响带来困 难。本文概述了超胞导致的能带折叠现象，重点介绍了基于平面波和原子轨道的能带反折叠方法、 声子能带反折叠方法及相关计算工具，给出了该方法在掺杂和界面体系电子、声子能带结构方面 应用的例子，并对该方法进行了展望。     

Convergence acceleration for the multilevel Hartree–Fock model

In the previously introduced multilevel Hartree–Fock (HF) model, the electronic density is optimised in a given region of the molecular system. The approach is based on generating an active occupied and active virtual space by decomposing a start guess density for the entire system. In this work, a diagonalisation based implementation for Roothaan–Hall (RH) with direct inversion in iterative subspace (DIIS) and a quasi-Newton minimisation procedure using the augmented RH (ARH) approach are described for accelerating convergence for the multilevel HF model. The equations are derived to be consistent with convergence acceleration for traditional atomic orbital based HF calculations. The main idea is to formulate all quantities in the molecular orbital basis to exploit that the active molecular orbital basis is significantly smaller than the atomic orbital basis, and thus enable the application of wave function approaches that are well-studied for small molecular systems to large molecular systems. Thus, all equations are formulated such that no atomic orbital density or Fock matrices are needed for the DIIS and ARH algorithms. Results show that the acceleration schemes yield efficient optimisation of the multilevel HF wave function.