Fock Reference Research Articles

Relative energies without electronic perturbations via alchemical integral transform.

We show that the energy of a perturbed system can be fully recovered from the unperturbed system's electron density. We derive an alchemical integral transform by parametrizing space in terms of transmutations, the chain rule, and integration by parts. Within the radius of convergence, the zeroth order yields the energy expansion at all orders, restricting the textbook statement by Wigner that the p-th order wave function derivative is necessary to describe the (2p + 1)-th energy derivative. Without the need for derivatives of the electron density, this allows us to cover entire chemical neighborhoods from just one quantum calculation instead of single systems one by one. Numerical evidence presented indicates that predictive accuracy is achieved in the range of mHa for the harmonic oscillator or the Morse potential and in the range of machine accuracy for hydrogen-like atoms. Considering isoelectronic nuclear charge variations by one proton in all multi-electron atoms from He to Ne, alchemical integral transform based estimates of the relative energy deviate by only few mHa from corresponding Hartree-Fock reference numbers.

Symmetry Breaking Slows Convergence of the ADAPT Variational Quantum Eigensolver.

Because quantum simulation of molecular systems is expected to provide the strongest advantage over classical computing methods for systems exhibiting strong electron correlation, it is critical that the performance of VQEs be assessed for strongly correlated systems. For classical simulation, strong correlation often results in symmetry breaking of the Hartree-Fock reference, leading to Löwdin's well-known "symmetry dilemma", whereby accuracy in the energy can be increased by breaking spin or spatial symmetries. Here, we explore the impact of symmetry breaking on the performance of ADAPT-VQE using two strongly correlated systems: (i) the "fermionized" anisotropic Heisenberg model, where the anisotropy parameter controls the correlation in the system, and (ii) symmetrically stretched linear H4, where correlation increases with increasing H-H separation. In both of these cases, increasing the level of correlation of the system leads to spontaneous symmetry breaking (parity and , respectively) of the mean-field solutions. We analyze the role that symmetry breaking in the reference states and orbital mappings of the fermionic Hamiltonians have in the compactness and performance of ADAPT-VQE. We observe that improving the energy of the reference states by breaking symmetry has a deleterious effect on ADAPT-VQE by increasing the length of the ansatz necessary for energy convergence and exacerbating the problem of "gradient troughs".

Potential energy curves for electronic states of the sodium dimer with multireference coupled cluster calculations

Accurate potential energy curves (PECs) were obtained for the 26 lowest lying electronic states of the sodium dimer. The current approach is based on the first principle size-extensive multireference coupled cluster method formulated in the (2,0) sector of the Fock space (FS). The (2,0) sector of FS provides the description of the states obtained by the attachment of two electrons to the reference system. This makes it possible to adopt the doubly ionised sodium dimer as a reference with its very concrete advantage in the calculations of the PECs, i.e. it dissociates into closed shell fragments (i.e. sodium cations) hence the restricted Hartree–Fock reference can be used in the whole range of interatomic distances. This results in the correct description of the interatomic potentials for all bond lengths. The theoretical PECs stay very close to the experimental curves wherever the latter are available and the computed spectroscopic parameters reproduce the experiment with very good accuracy. Relativistic corrections included at the second-order Douglass–Kroll level have a non-negligible effect on the accuracy of computed excitation and dissociation energies.

Quantifying and reducing spin contamination in algebraic diagrammatic construction theory of charged excitations.

Algebraic diagrammatic construction (ADC) theory is a computationally efficient and accurate approach for simulating electronic excitations in chemical systems. However, for the simulations of excited states in molecules with unpaired electrons, the performance of ADC methods can be affected by the spin contamination in unrestricted Hartree-Fock (UHF) reference wavefunctions. In this work, we benchmark the accuracy of ADC methods for electron attachment and ionization of open-shell molecules with the UHF reference orbitals (EA/IP-ADC/UHF) and develop an approach to quantify the spin contamination in charged excited states. Following this assessment, we demonstrate that the spin contamination can be reduced by combining EA/IP-ADC with the reference orbitals from restricted open-shell Hartree-Fock (ROHF) or orbital-optimized Møller-Plesset perturbation (OMP) theories. Our numerical results demonstrate that for open-shell systems with strong spin contamination in the UHF reference, the third-order EA/IP-ADC methods with the ROHF or OMP reference orbitals are similar in accuracy to equation-of-motion coupled cluster theory with single and double excitations.

UREMP, RO-REMP, and OO-REMP: Hybrid perturbation theories for open-shell electronic structure calculations.

An accurate description of the electron correlation energy in closed- and open-shell molecules is shown to be obtained by a second-order perturbation theory (PT) termed REMP. REMP is a hybrid of the Retaining the Excitation degree (RE) and the Møller-Plesset (MP) PTs. It performs particularly encouragingly in an orbital-optimized variant (OO-REMP) where the reference wavefunction is given by an unrestricted Slater determinant whose spin orbitals are varied such that the total energy becomes a minimum. While the approach generally behaves less satisfactorily with unrestricted Hartree-Fock references, reasonable performance is observed for restricted Hartree-Fock and restricted open-shell Hartree-Fock references. Inclusion of single excitations to OO-REMP is investigated and found-as in similar investigations-to be dissatisfying as it deteriorates performance. For the non-multireference subset of the accurate W4-11 benchmark set of Karton et al. [Chem. Phys. Lett. 510, 165-178 (2011)], OO-REMP predicts most atomization and reaction energies with chemical accuracy (1 kcalmol-1) if complete-basis-set extrapolation with augmented and core-polarized basis sets is used. For the W4-11 related test-sets, the error estimates obtained with the OO-REMP method approach those of coupled-cluster with singles, doubles and perturbative triples [CCSD(T)] within 20%-35%. The best performance of OO-REMP is found for a mixing ratio of 20%:80% MP:RE, which is essentially independent of whether radical stabilization energies, barrier heights, or reaction energies are investigated. Orbital optimization is shown to improve the REMP approach for both closed and open shell cases and outperforms coupled-cluster theory with singles and doubles (CCSD), spin-component scaled Møller-Plesset theory at second order (SCS-MP2), and density functionals, including double hybrids in all the cases considered.

A black-box, general purpose quadratic self-consistent field code with and without Cholesky decomposition of the two-electron integrals

We present the implementation of a quadratically convergent self-consistent field (QCSCF) algorithm based on an adaptive trust-radius optimisation scheme for restricted open-shell Hartree–Fock (ROHF), restricted Hartree–Fock (RHF), and unrestricted Hartree–Fock (UHF) references. The algorithm can exploit Cholesky decomposition (CD) of the two-electron integrals to allow calculations on larger systems. The most important feature of the QCSCF code lies in its black-box nature – probably the most important quality desired by a generic user. As shown for pilot applications, it does not require one to tune the self-consistent field (SCF) parameters (damping, Pulay's DIIS, and other similar techniques) in difficult-to-converge molecules. Also, it can be used to obtain a very tight convergence with extended basis sets – a situation often needed when computing high-order molecular properties – where the standard SCF algorithm starts to oscillate. Nevertheless, trouble may appear even with a QCSCF solver. In this respect, we discuss what can go wrong, focusing on the multiple UHF solutions of ortho-benzyne.

Spin contamination in MP2 and CC2, a surprising issue

When calculating the spin multiplicity at either the second-order Møller-Plesset (MP2) or the iterative second-order approximate coupled-cluster singles and doubles (CC2) levels of theory using the same strategy for the calculation of the expectation value as in regular CC theory together with the usual definitions of the MP2 and CC2 density matrices, artificial spin contamination occurs in closed-shell molecules. Non-intuitively, for open-shell systems, results at the MP2 or CC2 levels of theory based on this procedure even suggest stronger contamination at the correlated level than for the Hartree-Fock reference, although treatment of electron correlation should lower spin contamination. In this Communication, the reasons behind this inconsistency are investigated and a solution is proposed, which removes spin contamination for closed-shell molecules and leads to physically meaningful results for open-shell cases. Additionally, we show that CC2 significantly outperforms MP2 in describing systems with a strongly spin-contaminated reference with a performance similar to that of full coupled-cluster with singles and doubles substitutions (CCSD).

Intermediate state representation approach to physical properties of molecular electron-attached states: Theory, implementation, and benchmarking.

Computational schemes for comprehensive studies of molecular electron-attached states and the calculation of electron affinities (EAs) are formulated and implemented employing the intermediate state representation (ISR) formalism and the algebraic-diagrammatic construction approximation for the electron propagator (EA-ADC). These EA-ADC(n)/ISR(m) schemes allow for a consistent treatment of not only electron affinities and pole strengths up to third-order of perturbation theory (n = 3) but also one-electron properties of electron-attached states up to second order (m = 2). The EA-ADC/ISR equations were implemented in the Q-Chem program for Ŝz-adapted intermediate states, allowing also open-shell systems to be studied using unrestricted Hartree-Fock references. For benchmarking of the EA-(U)ADC/ISR schemes, EAs and dipole moments of various electron-attached states of small closed- and open-shell molecules were computed and compared to full configuration interaction data. As an illustrative example, EA-ADC(3)/ISR(2) has been applied to the thymine-thymine (6-4) DNA photolesion.

Static dipole polarizabilities of atoms and ions from Z = 1 to 20 calculated within a single theoretical scheme

Static dipole polarizabilities for the first 20 atoms and ions of the Periodic Table are calculated within a single theoretical scheme: the coupled-cluster method with a Hartree–Fock reference wavefunction using an augmented-polarization-consistent and polarization-consistent basis set. The values of the atomic polarizabilities of neutral atoms obtained here agree extremely well with the experimentally measured values. For the ions, all the calculations are consistent with experimental values with the highest resolution so far, except for the monovalent calcium ion, probably due to strong relativistic effects. To the best of our knowledge, this is the first time, applying a single theoretical scheme successfully predicts static polarizabilities of all 20 consecutive elements across 8 columns and more than 3 full periods being consistent with the experimentally measured values both for neutrals and ions. The results clearly signal the universality of this theoretical scheme for the atomic polarizability calculation, and should be easy to extend to other elements with weak relativistic effects. We adopt the CCSD(T) method with an augmented polarization-consistent and polarization-consistent basis set to calculate the atomic and ionic polarizabilities for the first 20 atoms and ions of the Periodic Table. Our calculation results are in good agreement with the experimental data with the highest accuracy so far and the reported high accuracy calculation results. Semi-logarithm plot of calculated static polarizabilities of neutral atoms (filled circles, black line) and ions (open circles). The colored lines join isoelectronic species.

Analytic gradients and derivative couplings for configuration interaction with all single excitations and one double excitation-En route to nonadiabatic dynamics.

We present analytic gradients and derivative couplings for the simplest possible multireference configuration interaction method, CIS-1D, an electronic structure Ansatz that includes all single excitations and one lone double excitation on top of a Hartree-Fock reference state. We show that the resulting equations are numerically stable and require the evaluation of a similar number of integrals as compared to standard CIS theory; one can easily differentiate the required frontier orbitals (h and l) with minimal cost. The resulting algorithm has been implemented within the Q-Chem electronic structure package and should be immediately useful for understanding photochemistry with S0-S1 crossings.

Adcc: A versatile toolkit for rapid development of algebraic‐diagrammatic construction methods

AbstractADC‐connect (adcc) is a hybrid python/C++ module for performing excited state calculations based on the algebraic‐diagrammatic construction scheme for the polarization propagator (ADC). Key design goal is to restrict adcc to this single purpose and facilitate connection to external packages, for example, for obtaining the Hartree–Fock references, plotting spectra, or modeling solvents. Interfaces to four self‐consistent field codes have already been implemented, namely pyscf, psi4, molsturm, and veloxchem. The computational workflow, including the numerical solvers, is implemented in python, whereas the working equations and other expensive expressions are done in C++. This equips adcc with adequate speed, making it a flexible toolkit for both rapid development of ADC‐based computational spectroscopy methods as well as unusual computational workflows. This is demonstrated by three examples. Presently, ADC methods up to third order in perturbation theory are available in adcc, including the respective core‐valence separation and spin‐flip variants. Both restricted or unrestricted Hartree–Fock references can be employed.This article is categorized under: Software > Simulation Methods Electronic Structure Theory > Ab Initio Electronic Structure Methods Theoretical and Physical Chemistry > Spectroscopy Software > Quantum Chemistry

Main-group test set for materials science and engineering with user-friendly graphical tools for error analysis: systematic benchmark of the numerical and intrinsic errors in state-of-the-art electronic-structure approximations

Understanding the applicability and limitations of electronic-structure methods needs careful and efficient comparison with accurate reference data. Knowledge of the quality and errors of electronic-structure calculations is crucial to advanced method development, high-throughput computations and data analyses. In this paper, we present a main-group test set for computational materials science and engineering (MSE), that provides accurate and easily accessible crystal properties for a hierarchy of exchange-correlation approximations, ranging from the well-established mean-field approximations to the state-of-the-art methods of many-body perturbation theory. We consider cohesive energy, lattice constant and bulk modulus of a set of materials that representatives for the first- and second-row elements and their binaries with cubic crystal structures and various bonding characters. A strong effort is made to achieve high numerical accuracy for cohesive properties as calculated using the local-density approximation (LDA), several generalized gradient approximations (GGAs), meta-GGAs and hybrids in all-electron resolution, and the second-order Møller–Plesset perturbation theory (MP2) and the random-phase approximation (RPA) both with frozen-core approximation based on all-electron Hartree–Fock, PBE and/or PBE0 references. This results in over 10 000 calculations, which record a comprehensive convergence test with respect to numerical parameters for a wide range of electronic-structure methods within the numerical atom-centered orbital framework. As an indispensable part of the MSE test set, a web site is established http://mse.fhi-berlin.mpg.de. This not only allows for easy access to all reference data but also provides user-friendly graphical tools for post-processing error analysis.

Fock space coupled cluster study of the 11Πg state of the Li2 molecule

ABSTRACTRecently reported (M. Musiał, J. Chem. Phys., 136, 134111 (2012)) multireference coupled cluster method formulated in the (2,0) sector of the Fock space proved to be an efficient scheme in the treatment of potential energy curves (PECs) for alkali metal diatomics Me2. The (2,0) sector provides description of states obtained by an attachment of two electrons to the reference which for the Me2 is a doubly ionised Me+ 22 system. The latter has a very concrete advantage in the calculations of the PECs since it dissociates into closed shell fragments (Me+ 22 → Me++Me+); hence, the restricted Hartree–Fock reference can be used in the whole range of interatomic distances. In this work, an accurate PEC and vibrational energy levels are obtained for the state of the lithium dimer. The average deviation of the vibrational term values from experiment is 0.79 wavenumber and is significantly lower than that provided by other theoretical works.

Explicitly correlated coupled-cluster theory with Brueckner orbitals.

Brueckner orbitals are the optimal orbitals for use in F12 explicitly correlated coupled-cluster (CC) treatments. A novel approach, Brueckner coupled-cluster doubles with perturbative triples BCCD(T)(F12*) is presented that avoids the expensive re-evaluation of F12 integrals throughout the orbital optimisation and includes a newly derived basis set correction to the Brueckner reference energy. The generalisation of F12 theory to arbitrary non-Hartree-Fock references and to Fock operators that include scalar relativistic effects is also presented. The performance of the new Brueckner F12 method is assessed for a test set of 50 open- and closed-shell reactions and for the ionisation potentials and electron affinities (EAs) of the first-row transition metal atoms. Benchmark basis set limit coupled-cluster singles, doubles and perturbative triples (CCSD(T)) and BCCD(T) values are reported for all energies in the test sets. BCCD(T)(F12*) performs systematically better than CCSD(T)(F12*) for electron affinities where orbital relaxation effects are significant.

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

ABSTRACTStemming 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.

On the dipole polarisability and dipole sum rules of ozone

Ab initio calculations of the dipole polarisability and other Cauchy moments of the dipole oscillator strength distribution (DOSD) of ozone are reported to help resolve discrepancies between theory and experiment. A number of coupled-cluster methods based on a Hartree–Fock reference function, multiconfiguration-reference configuration interaction methods, and perturbatively corrected, complete-active-space self-consistent field methods are used. The C DOSD of Kumar and Thakkar is probably preferable to their B1 distribution. Our best estimate of the mean polarisability is atomic units.

Giant resonances based on unitarily transformed two-nucleon plus phenomenological three-nucleon interactions

We investigate giant resonances of spherical nuclei on the basis of the Argonne V18 potential after unitary transformation within the similarity renormalization group or the unitary correlation operator method supplemented by a phenomenological three-body contact interaction. Such Hamiltonians can provide a good description of ground-state energies and radii within Hartree–Fock plus low-order many-body perturbation theory. The standard random phase approximation is applied here to calculate the isoscalar monopole, isovector dipole, and isoscalar quadrupole excitation modes of the , , and nuclei. Thanks to the inclusion of the three-nucleon interaction and despite the minimal optimization effort, a reasonable agreement with experimental centroid energies of all three modes has been achieved. The role and scope of the Hartree–Fock reference state in RPA methods are discussed.

Relativistic Small-Core Pseudopotentials for Actinium, Thorium, and Protactinium

Small-core pseudopotentials for actinium, thorium, and protactinium have been energy-adjusted to multiconfiguration Dirac-Hartree-Fock reference data based on the Dirac-Coulomb-Breit Hamiltonian and the Fermi nucleus model. Corresponding optimized valence basis sets of polarized valence quadruple-ζ quality are presented. Atomic test calculations for the first four ionization potentials show satisfactory results at both the Hartree-Fock and the multireference averaged coupled-pair functional level. Highly correlated Fock-space coupled cluster calculations demonstrate that the new pseudopotentials yield ionization potentials, which are in excellent agreement with corresponding all-electron results and experimental data. The pseudopotentials and basis sets supplement a similar set previously published for uranium.

The coupled cluster singles, doubles, and a hybrid treatment of connected triples based on the split virtual orbitals

We have proposed a simple strategy for splitting the virtual orbitals with a large basis set into two subgroups (active and inactive) by taking a smaller basis set as an auxiliary basis set. With the split virtual orbitals (SVOs), triple or higher excitations can be partitioned into active and inactive subgroups (according to the number of active virtual orbitals involved), which can be treated with different electron correlation methods. In this work, the coupled cluster (CC) singles, doubles, and a hybrid treatment of connected triples based on the SVO [denoted as SVO-CCSD(T)-h], has been implemented. The present approach has been applied to study the bond breaking potential energy surfaces in three molecules (HF, F(2), and N(2)), and the equilibrium properties in a number of open-shell diatomic molecules. For all systems under study, the SVO-CCSD(T)-h method based on the unrestricted Hartree-Fock (UHF) reference is an excellent approximation to the corresponding CCSDT (CC singles, doubles, and triples), and much better than the UHF-based CCSD(T) (CC singles, doubles, and perturbative triples). On the other hand, the SVO-CCSD(T)-h method based on the restricted HF (RHF) reference can also provide considerable improvement over the RHF-based CCSD(T).

The coupled cluster approach with a hybrid treatment of connected triple excitations: Spectroscopic constants in open-shell diatomic molecules, and bond-breaking or twisting potential energy surfaces

A coupled cluster singles, doubles, and a hybrid treatment of triples [denoted as CCSD(T)-h] has been applied to investigate the equilibrium geometries and harmonic frequencies of five diatomic open-shell molecules, bond breaking potential energy surfaces in C2 and F2+, and the twisting potential energy surface of ethylene. In the present work, CCSD(T)-h calculations are based on the restricted or restricted open-shell Hartree–Fock (RHF or ROHF) reference. A general procedure for constructing the active RHF or ROHF orbitals is proposed. A comparison of CCSD(T)-h with other CC methods show that for all systems CCSD(T)-h is an excellent approximation to CCSDT, being much better than CCSD(T) especially when a molecule exhibits strong multireference character.