The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.     

Extending spin-symmetry projected coupled-cluster to large model spaces using an iterative null-space projection technique

Recently, we introduced an orbital-invariant approximate coupled-cluster (CC) method in the spin-projection manifold. The multi-determinantal property of spin-projection means that the parametrization in the spin-extended CC (ECC) ansatz is nonorthogonal and overcomplete. Therefore, the linear dependencies must be removed by an orthogonalization procedure to obtain meaningful solutions. Multi-reference methods often achieve this by diagonalizing a metric of the equation system, but this is not feasible with ECC because of the enormous size of the metric, a consequence of the incomplete active space of the spin-projected Hartree–Fock reference. As a result, the applicability of ECC has been limited to small benchmark systems, for which the ansatz was shown to be superior to the configuration interaction and linearized approximations. In this article, we provide a solution to this problem that completely avoids the metric diagonalization by iteratively projecting out its null-space from the working equations. As the additional computational cost required for this iterative projection is only marginal, it greatly expands the application range of ECC. We demonstrate the potential of approximate ECC by studying the complete basis set limit of F₂ and transition metal complexes such as NiO, Mn₂, and [Cu₂O₂]²⁺ , which have all been hindered by the prohibitively large metric size. We also identify the potential inadequacy of the molecular orbitals given by spin-projected Hartree–Fock in some cases, and propose possible solutions. © 2018 Wiley Periodicals, Inc.     

Accurate Interaction Energies of CO2 with the 20 Naturally Occurring Amino Acids

We have performed a series of highly accurate calculations between CO₂ and the 20 naturally occurring amino acids for the investigation of the attractive noncovalent interactions. Different nucleophilic groups present in the amino acid structures were considered (α-NH₂, COOH, side groups), and the stronger binding sites were identified. A database of accurate reference interactions energies was compiled as computed by explicitly-correlated coupled-cluster singles-and-doubles, together with perturbative triples extrapolated to the complete-basis-set limit. The CCSD(F12)(T)/CBS reference values were used for comparing a variety of popular density functionals with different basis sets. Our results show that most density functionals with the triple-zeta basis set def2-TZVPP align with the CCSD(F12)(T)/CBS reference values, but errors range from 0.1 kcal/mol up to 1.0 kcal/mol.     

New massively parallel linear-response coupled-cluster module in ACES III: application to static polarisabilities of closed-shell molecules and oligomers and of open-shell radicals

ABSTRACT

Stemming from our implementation of parallel coupled-cluster (CC) capabilities for electron spin resonance properties [J. Chem. Phys. 139, 174103 (2013)], we present a new massively parallel linear response CC module within ACES III. Unlike alternative parallel CC modules, this general purpose module evaluates any type of first- and second-order CC properties of both closed- and open-shell molecules employing restricted, unrestricted and restricted-open-shell Hartree–Fock (HF) references. We demonstrate the accuracy and usefulness of this module through the calculation of static polarisabilities of large molecules. Closed-shell calculations are performed at the following levels: second-order many-body perturbation theory [MBPT(2)], CC with single- and double-excitations (CCSD), coupled-perturbed HF and density functional theory (DFT), and open-shell calculations at the unrestricted CCSD (UCSSD) one. Applications involve eight closed-shell organic-chemistry molecules (Set I), the first four members of the closed-shell thiophene oligomer series (Set II), and five open-shell radicals (Set III). In Set I, all calculated average polarisabilities agree reasonably well with experimental data. In Set II, all calculated average polarisabilities vs. the number of monomers show comparable values and saturation patterns and demonstrate that experimental polarisabilities may be inaccurate. In Set III, UCCSD perpendicular polarisabilities show a reasonable agreement with previous UCCSD(T) and restricted-open-shell-MBPT(2) values.     

Model study of the impact of orbital choice on the accuracy of coupled-cluster energies. II. Valence-universal coupled-cluster method

The impact of the choice of molecular orbital sets on the results of the valence-universal coupled cluster method involving up to three-body amplitudes (VU-CCSDT) was studied for the H4 model. This model offers a straightforward way of representing all possible symmetry-adapted orbitals. Moreover, the degree of quasi-degeneracy of its lowest ¹A₁ states can be varied over a wide range by changing its geometry. Calculations were performed both for 13 sets of standard quantum chemical orbitals and for a vast variety of nonstandard orbital sets defined by nodes of a two-dimensional orbital grid. The performance of various standard orbital sets in VU-CCSDT calculations is compared. It is also documented that for every quasi-degeneracy region there exist nonstandard orbital sets which allow one to obtain more accurate VU-CCSDT energies than the standard orbital sets. In an attempt to provide a general interpretation for some of the alternative orbital sets, we defined a set of orbitals which maximize the proximity of the model and target spaces—maximum proximity orbitals (MPO). It is demonstrated that outside the strong quasi-degeneracy region the energies obtained for the VU-CCSDT approach based on the MPOs are more accurate than for the standard orbital sets. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 221–237, 1998     

钫原子磁偶极超精细结构常数及其同位素的磁偶极矩的理论计算

应用基于B样条基组的相对论耦合簇理论方法,计算了~(212)Fr原子的n S (n=7—12), n P (n=7—12)和n D (n=6—11)态的磁偶极超精细结构常数.与精确实验值的比较说明这套理论方法能精确计算出磁偶极超精细结构常数,其中7P态的磁偶极超精细常数的理论值与实验值之间的差异小于1%.在忽略场移效应对Fr原子7P态超精细结构常数的影响下,通过结合实验值进一步定出了~(207-213,220-228)Fr核磁偶极矩μ,这些值与已有的测量值具有非常好的一致性.本文报道了12S, n P (n=9—12)和n D (n=10—11)态的磁偶极超精细结构常数.     

A coupled cluster study of the magnetisability,rotational g-tensor and quadrupole moment of NF3, PF3 and AsF3

ABSTRACT

A coupled-cluster investigation of magnetic and electric properties of NF₃, PF₃ and AsF₃ provides for a comparison with recent experimental data. For PF₃, achieving reliable values for the magnetisability and rotational g-tensor of PF₃ has been particularly challenging. We report the most accurate calculations to date for PF₃; for the vibrationally corrected anisotropic magnetisability, our extrapolated CCSD(T)/CBS value of ?0.290 a.u is within the uncertainty limits of the most recent experimental value of ?0.286 ± 0.042 a.u. For the rotational g-tensor of PF₃, agreement between theory and experiment for the g_⊥ component is excellent (deviation of less than 0.0006 a.u.). However, the g_|| component remains problematic even though our vibrationally corrected CCSD(T)/CBS value of ?0.0387 a.u is in closer agreement with the recently revised experimental value of ?0.0470 ± 0.0020 a.u. than the original value of ?0.0815 ± 0.0020 a.u. The origin of the remaining discrepancy remains unclear. Dipole and quadrupole moments have also been investigated.     

Determination of consistent semiempirical one-centre integrals based on coupled-cluster theory

ABSTRACT

We examine the one-centre integrals used in semiempirical molecular orbital theory, for the elements H–Ne. The currently used parameters do not provide good estimates for the relative energies of ionised states of atoms. Directly calculating the one-electron integrals U _ss and U _pp with coupled-cluster theory and fitting the two-electron repulsion integrals G _ss and G _pp to accurate coupled-cluster ionisation curves improves this behaviour. Since all the remaining parameters can be derived from these, the number of fitted variables is reduced from seven to two. The two-parameter model provides qualitative agreement with coupled-cluster theory for all ionisation potentials (IPs) and the principal electron affinity (EA). To obtain quantitative agreement for the principal IP and EA, U _ss and U _pp are included as variables in a four-parameter model. We discuss the new parameters and implications for the development of new, consistent semiempirical Hamiltonians.