Extrapolation to the complete-basis-set limit and the implications of avoided crossings: The X Σ1g+, B Δ1g, and B′ Σ1g+ states of C2

Rapidly convergent quantum Monte Carlo using a Chebyshev projector

The multireference coupled-cluster Monte Carlo (MR-CCMC) algorithm is a determinant-based quantum Monte Carlo (QMC) algorithm that is conceptually similar to Full Configuration Interaction QMC (FCIQMC). It has been shown to offer a balanced treatment of both static and dynamic correlation while retaining polynomial scaling, although application to large systems with significant strong correlation remained impractical. In this paper, we document recent algorithmic advances that enable rapid convergence and a more black-box approach to the multireference problem. These include a logarithmically scaling metric-tree-based excitation acceptance algorithm to search for determinants connected to the reference space at the desired excitation level and a symmetry-screening procedure for the reference space. We show that, for moderately sized reference spaces, the new search algorithm brings about an approximately 8-fold acceleration of one MR-CCMC iteration, while the symmetry screening procedure reduces the number of active reference space determinants with essentially no loss of accuracy. We also introduce a stochastic implementation of an approximate wall projector, which is the infinite imaginary time limit of the exponential projector, using a truncated expansion of the wall function in Chebyshev polynomials. Notably, this wall-Chebyshev projector can be used to accelerate any projector-based QMC algorithm. We show that it requires significantly fewer applications of the Hamiltonian to achieve the same statistical convergence. We benchmark these acceleration methods on the beryllium and carbon dimers, using initiator FCIQMC and MR-CCMC with basis sets up to cc-pVQZ quality.

Diabatic Valence-Hole States in the C2 Molecule: "Putting Humpty Dumpty Together Again"

Despite the long history of spectroscopic studies of the C₂ molecule, fundamental questions about its chemical bonding are still being hotly debated. The complex electronic structure of C₂ is a consequence of its dense manifold of near-degenerate, low-lying electronic states. A global multi-state diabatic model is proposed here to disentangle the numerous configuration interactions that occur within four symmetry manifolds of excited states of C₂ (¹Π_g, ³Π_g, ¹Σ_u⁺ , and ³Σ_u⁺ ). The key concept of our model is the existence of two "valence-hole" configurations, 2σg22σu11πu33σg2 for ^1,3Π_g states and 2σg22σu11πu43σg1 for ^1,3Σ_u⁺ states, that are derived from 3σ_g ← 2σ_u electron promotion. The lowest-energy state from each of the four C₂ symmetry species is dominated by this type of valence-hole configuration at its equilibrium internuclear separation. As a result of their large binding energy (nominal bond order of 3) and correlation with the 2s²2p² + 2s2p³ separated-atom configurations, the presence of these valence-hole configurations has a profound impact on the global electronic structure and unimolecular dynamics of C₂.

Extrapolation in quantum chemistry: Insights on energetics and reaction dynamics

Since there is no exact solution for problems in physics and chemistry, extrapolation methods may assume a key role in quantitative quantum chemistry. Two topics where it bears considerable impact are addressed, both at the heart of computational quantum chemistry: electronic structure and reaction dynamics. In the first, the problem of extrapolating the energy obtained by solving the electronic Schrödinger equation to the limit of the complete one-electron basis set is addressed. With the uniform-singlet-and-triplet-extrapolation (USTE) scheme at the focal point, the emphasis is on recent updates covering from the energy itself to other molecular properties. The second topic refers to extrapolation of quantum mechanical reactive scattering probabilities from zero total angular momentum to any of the values that it may assume when running quasiclassical trajectories, QCT/QM-[Formula: see text]J. With the extrapolation guided in both cases by physically motivated asymptotic theories, realism is seeked by avoiding unsecure jumps into the unknown. Although, mostly review oriented, a few issues are addressed for the first time here and there. Prospects for future work conclude the overview.

Accurate CHIPR Potential Energy Surface for the Lowest Triplet State of C3

We report the first global ab initio-based potential energy surface (PES) for ground-state triplet C₃(³A') based on accurate energies extrapolated to the complete basis set (CBS) limit, and using the combined-hyperbolic-inverse-power-representation method for the analytical modeling. By relying on a cost-effective CBS(D,T) protocol, we ensure that the final form reproduces all topographical features of the PES, including its cyclic-linear isomerization barrier, with CBS(5,6)-quality. To partially account for the incompleteness of the N-electron basis and other minor effects, the available accurate experimental data on the relevant diatomics were used to obtain direct-fit curves that replace the theoretical ones in the many-body expansion. Besides describing properly long-range interactions at all asymptotic channels and permutational symmetry by built-in construction, the PES reported here reproduces the proper exothermicities at dissociation regions as well as the spectroscopy of the diatomic fragments. Bound vibrational state calculations in both linear and cyclic isomers have also been carried out, unveiling a good match of the available data on C₃(ã ³Π_u), while assisting with IR band positions for C₃(³A₂^') that may serve as a guide for its laboratory and astronomical detection.

Simultaneous Attenuation of Both Self-Interaction Error and Nondynamic Correlation Error in Density Functional Theory: A Spin-Pair Distinctive Adiabatic-Connection Approximation

We present a spin-pair distinctive algorithm in the context of adiabatic-connection fluctuation-dissipation (ACFD) theorem, which enables to quantify the self-interaction error (SIE) and the nondynamic/strong correlation error (NCE) in the direct random-phase approximation (dRPA). Using this knowledge, we propose a spin-component scaled dRPA (scsRPA) correlation model with simultaneous attenuation of both the SIE and the NCE. Along with the exact exchange, scsRPA is shown to present a comprehensive improvement over dRPA, as well as the well-established PBE and PBE0 functionals, for bonding energies of pronounced multireference characters and transition-metal complexes of strongly correlated systems, while consistently providing an accurate description for reaction energies, reaction barriers, and noncovalent bond interactions of weakly correlated systems.

CBS extrapolation of Hartree-Fock energy: Pople and Dunning basis sets hand-to-hand on the endeavour

The Hartree-Fock (HF) energy is shown to be extrapolatable from subminimal, minimal, and extended basis sets. Unprecedentedly, it can be reliably extrapolated to the complete basis set limit (CBS) from as small as Pople's STO-2G up to the largest aug-cc-pVXZ basis sets of Dunning's correlation-consistent type and even more complete variants of such ansatzes. The approach is tested on a panoply of small, medium and large molecular systems. As a case study, extrapolation is shown to be particularly reliable and cost-effective (typically an order of magnitude more economical for similar accuracy) when employing a hybrid (STO-2G,extended) basis set pair. The additional cost relative to a single-point calculation with the extended basis set is of no significance. After training the novel scheme with a meager set of 18 systems (TS-18), four test sets covering from neutrals (TS-105) to neutrals plus cations (TS-26) and anions (TS-17), these involving up to third-row elements, as well as tetrapeptide conformers (TS-10) were utilized for its validation. Although the answer to the question of whether all basis sets converge to the same HF/CBS limit is a clear yes, that of whether they show the same convergence rate is open, an issue also examined in the present work. For systems involving atoms up to the second-row of the periodic table the answer seems to be a clear yes, at least when using correlation-consistent basis sets, but it is shown to be a no when the study is extended to anionic systems containing third-row elements. For the latter, augmented correlation consistent basis sets display a faster, perhaps optimal, convergence rate. The implications of this in practice can hardly be anticipated. For example, it cannot be predicted a priori which of such basis sets will yield the best relative energies for those anions, although a logical expectation suggests it to be the augmented ones which turn out to display the fastest convergence. As a blind-test type illustration, the novel scheme is applied to the conformational analysis of 2 tetrapeptides with 5 conformers each, using from subminimal up to extended basis sets as large as cc-pV5Z and aug-cc-pVQZ. The (STO-2G,extended) results so obtained are predicted to be in very good agreement with the best data reported thus far.

Practical approximation of the non-adiabatic coupling terms for same-symmetry interstate crossings by using adiabatic potential energies only

A very simple equation, F_ij^App=[(∂²(V_i^a-V_j^a)/∂Q²)/(V_i^a-V_j^a)]^1/2/2, giving a reliable magnitude of non-adiabatic coupling terms (NACTs, F_ij's) based on adiabatic potential energies only (V_i^a and V_j^a) was discovered, and its reliability was tested for several prototypes of same-symmetry interstate crossings in LiF, C₂, NH₃Cl, and C₆H₅SH molecules. Our theoretical derivation starts from the analysis of the relationship between the Lorentzian dependence of NACTs along a diabatization coordinate and the well-established linear vibronic coupling scheme. This analysis results in a very simple equation, α=2κ/Δ_c, enabling the evaluation of the Lorentz function α parameter in terms of the coupling constant κ and the energy gap Δ_c (Δ_c=|V_i^a-V_j^a|_Qc ) between adiabatic states at the crossing point Q_C. Subsequently, it was shown that Q_C corresponds to the point where F_ij^App exhibit maximum values if we set the coupling parameter as κ=[(V_i^a-V_j^a)⋅(∂²(V_i^a-V_j^a)/∂Q²)]_Qc^1/2/2. Finally, we conjectured that this relation could give reasonable values of NACTs not only at the crossing point but also at other geometries near Q_C. In this final approximation, the pre-defined crossing point Q_C is not required. The results of our test demonstrate that the approximation works much better than initially expected. The present new method does not depend on the selection of an ab initio method for adiabatic electronic states but is currently limited to local non-adiabatic regions where only two electronic states are dominantly involved within a nuclear degree of freedom.

Multiple conical intersections in small linear parameter Jahn-Teller systems: the DMBE potential energy surface of ground-state C3 revisited

A new single-sheeted DMBE potential energy surface for ground-state C3 is reported. The novel analytical form accurately describes the three symmetry-equivalent C2v disjoint seams, in addition to the symmetry-required D3h one, over the entire configuration space. The present formalism warrants by built-in construction the confluence of the above crossings, and the rotation-in-plane of the C2v seams when the perimeter of the molecule fluctuates. Up to 1050 ab initio energies have been employed in the calibration procedure, of which 421 map the loci of intersection. The calculated energies have been scaled to account for the incompleteness of the basis set and truncation of the MRCI expansion, and fitted analytically with chemical accuracy. The novel form is shown to accurately mimic the region defined by the 4 conical intersections, while exhibiting similar attributes to the previously reported one [J. Chem. Phys., 2015, 143, 074302] at the regions of configuration space away from the crossing seams. Despite being mainly addressed to C3, the present approach should be applicable to adiabatic PESs of any X3 system experiencing similar topological attributes, in particular the small-linear-parameter Jahn-Teller molecules.

Cn (n=2-4): current status

The major aspects of the C₂, C₃ and C₄ elemental carbon clusters are surveyed. For C₂, a brief analysis of its current status is presented. Regarding C₃, the most recent results obtained in our group are reviewed with emphasis on modelling its potential energy surface which is particularly complicated due to the presence of multiple conical intersections. As for C₄, the most stable isomeric forms of both triplet and singlet spin states and their possible interconversion pathways are examined afresh by means of accurate ab initio calculations. The main strategies for modelling the ground triplet C₄ potential are also discussed. Starting from a truncated cluster expansion and a previously reported DMBE form for C₃, an approximate four-body term is calibrated from the ab initio energies. The final six-dimensional global DMBE form so obtained reproduces all known topographical aspects while providing an accurate description of the C₄ linear-rhombic isomerization pathway. It is therefore commended for both spectroscopic and reaction dynamics studies.This article is part of the theme issue 'Modern theoretical chemistry'.

Towards Efficient Orbital-Dependent Density Functionals for Weak and Strong Correlation

We present a new paradigm for the design of exchange-correlation functionals in density-functional theory. Electron pairs are correlated explicitly by means of the recently developed second order Bethe-Goldstone equation (BGE2) approach. Here we propose a screened BGE2 (sBGE2) variant that efficiently regulates the coupling of a given electron pair. sBGE2 correctly dissociates H_{2} and H_{2}^{+}, a problem that has been regarded as a great challenge in density-functional theory for a long time. The sBGE2 functional is then taken as a building block for an orbital-dependent functional, termed ZRPS, which is a natural extension of the PBE0 hybrid functional. While worsening the good performance of sBGE2 in H_{2} and H_{2}^{+}, ZRPS yields a remarkable and consistent improvement over other density functionals across various chemical environments from weak to strong correlation.

The Quadruple Bonding in C2 Reproduces the Properties of the Molecule

Ever since Lewis depicted the triple bond for acetylene, triple bonding has been considered as the highest limit of multiple bonding for main elements. Here we show that C2 is bonded by a quadruple bond that can be distinctly characterized by valence-bond (VB) calculations. We demonstrate that the quadruply-bonded structure determines the key observables of the molecule, and accounts by itself for about 90% of the molecule's bond dissociation energy, and for its bond lengths and its force constant. The quadruply-bonded structure is made of two strong π bonds, one strong σ bond and a weaker fourth σ-type bond, the bond strength of which is estimated as 17-21 kcal mol(-1). Alternative VB structures with double bonds; either two π bonds or one π bond and one σ bond lie at 129.5 and 106.1 kcal mol(-1), respectively, above the quadruply-bonded structure, and they collapse to the latter structure given freedom to improve their double bonding by dative σ bonding. The usefulness of the quadruply-bonded model is underscored by "predicting" the properties of the (3)Σ+u state. C2's very high reactivity is rooted in its fourth weak bond. Thus, carbon and first-row main elements are open to quadruple bonding!

DFT assessment of the spectroscopic constants and absorption spectra of neutral and charged diatomic species of group 11 and 14 elements

The spectroscopic constants and absorption spectra of neutral and charged diatomic molecules of group 11 and 14 elements formulated as [M(2)](+/0/-) (M = Cu, Ag, Au), and [E(2)](+/0/-) (E = C, Si, Ge, Sn, Pb) have been calculated at the PBE0/Def2-QZVPP level of theory. The electronic and bonding properties of the diatomics have been analyzed by natural bond orbital analysis approach and topology analysis by the atoms in molecules method. Particular emphasis was given on the absorption spectra of the diatomic species, which were simulated by time-dependent density functional theory calculations employing the hybrid Coulomb-attenuating CAM-B3LYP density functional. The simulated absorption spectra of the [M(2)](+/0/-) (M = Cu, Ag, Au) and [E(2)](+/0/-) (E = C, Si, Ge, Sn, Pb) species are in close resemblance with the experimentally observed spectra whenever available. The neutral M(2) and E(2) diatomics strongly absorb in the ultraviolet region, given rise to UVC, UVA and in a few cases UVB absorptions. In a few cases, weak absorbion bands also occur in the visible region. The absorption bands have thoroughly been analyzed and assignments of the contributing principal electronic transitions associated to individual excitations have been made.

Accurate ab initio-based adiabatic global potential energy surface for the 2(2)A" state of NH2 by extrapolation to the complete basis set limit

A full three-dimensional global potential energy surface is reported first time for the title system, which is important for the photodissociation processes. It is obtained using double many-body expansion theory and an extensive set of accurate ab initio energies extrapolated to the complete basis set limit. Such a work can be recommended for dynamics studies of the N((2)D) + H2 reaction, a reliable theoretical treatment of the photodissociation dynamics and as building blocks for constructing the double many-body expansion potential energy surface of larger nitrogen/hydrogen containing systems. In turn, a preliminary theoretical study of the reaction N((2)D)+H2(X(1)Σg (+))(ν=0,j=0)→NH(a(1)Δ)+H((2)S) has been carried out with the method of quasi-classical trajectory on the new potential energy surface. Integral cross sections and thermal rate constants have been calculated, providing perhaps the most reliable estimate of the integral cross sections and the rate constants known thus far for such a reaction.

Accurate double many-body expansion potential energy surface by extrapolation to the complete basis set limit and dynamics calculations for ground state of NH2

An accurate single-sheeted double many-body expansion potential energy surface is reported for the title system. A switching function formalism has been used to warrant the correct behavior at the H2(X1Σg+)+N(2D) and NH (X3Σ-)+H(2S) dissociation channels involving nitrogen in the ground N(4S) and first excited N(2D) states. The topographical features of the novel global potential energy surface are examined in detail, and found to be in good agreement with those calculated directly from the raw ab initio energies, as well as previous calculations available in the literature. The novel surface can be using to treat well the Renner-Teller degeneracy of the 12A″ and 12A' states of NH 2. Such a work can both be recommended for dynamics studies of the N(2D)+H2 reaction and as building blocks for constructing the double many-body expansion potential energy surface of larger nitrogen/hydrogen-containing systems. In turn, a test theoretical study of the reaction N(2D)+H2(X1Σg+)(ν=0,j=0)→NH (X3Σ-)+H(2S) has been carried out with the method of quantum wave packet on the new potential energy surface. Reaction probabilities, integral cross sections, and differential cross sections have been calculated. Threshold exists because of the energy barrier (68.5 meV) along the minimum energy path. On the curve of reaction probability for total angular momentum J = 0, there are two sharp peaks just above threshold. The value of integral cross section increases quickly from zero to maximum with the increase of collision energy, and then stays stable with small oscillations. The differential cross section result shows that the reaction is a typical forward and backward scatter in agreement with experimental measurement result.

Quadruple bonding in C2 and analogous eight-valence electron species

Triple bonding is conventionally considered to be the limit for multiply bonded main group elements, despite higher metal-metal bond orders being frequently observed for transition metals and lanthanides/actinides. Here, using high-level theoretical methods, we show that C(2) and its isoelectronic molecules CN(+), BN and CB(-) (each having eight valence electrons) are bound by a quadruple bond. The bonding comprises not only one σ- and two π-bonds, but also one weak 'inverted' bond, which can be characterized by the interaction of electrons in two outwardly pointing sp hybrid orbitals. A simple way of assessing the energy of the fourth bond is proposed and is found to be ~12-17 kcal mol(-1) for the isoelectronic species studied, and thus stronger than a hydrogen bond. In contrast, the analogues of C(2) that contain higher-row elements, such as Si(2) and Ge(2), exhibit only double bonding.