Second-order Perturbation Correction Research Articles

How do spin-scaled double hybrids designed for excitation energies perform for noncovalent excited-state interactions? An investigation on aromatic excimer models.

Time-dependent double hybrids with spin-component or spin-opposite scaling to their second-order perturbative correlation correction have demonstrated competitive robustness in the computation of electronic excitation energies. Some of the most robust are those recently published by our group (M. Casanova-Páez, L. Goerigk, J. Chem. Theory Comput. 2021, 20, 5165). So far, the implementation of these functionals has not allowed correctly calculating their ground-state total energies. Herein, we define their correct spin-scaled ground-state energy expressions which enables us to test our methods on the noncovalent excited-state interaction energies of four aromatic excimers. A range of 22 double hybrids with and without spin scaling are compared to the reasonably accurate wavefunction reference from our previous work (A. C. Hancock, L. Goerigk, RSC Adv. 2023, 13, 35964). The impact of spin scaling is highly dependent on the underlying functional expression, however, the smallest overall errors belong to spin-scaled functionals with range separation: SCS- and SOS- PBEPP86, and SCS-RSX-QIDH. We additionally determine parameters for DFT-D3(BJ)/D4 ground-state dispersion corrections of these functionals, which reduce errors in most cases. We highlight the necessity of dispersion corrections for even the most robust TD-DFT methods but also point out that ground-state based corrections are insufficient to completely capture dispersion effects for excited-state interaction energies.

Rationale for the extrapolation procedure in selected configuration interaction.

Selected configuration interaction (SCI) methods have emerged as state-of-the-art methodologies for achieving high accuracy and generating benchmark reference data for ground and excited states in small molecular systems. However, their precision relies heavily on extrapolation procedures to produce a final estimate of the exact result. Using the structure of the exact electronic energy landscape, we provide a rationale for the common linear extrapolation of the variational energy as a function of the second-order perturbative correction. In particular, we demonstrate that the energy gap and the coupling between the so-called internal and external spaces are the key factors determining the rate at which the linear regime is reached. Starting from the first principles, we also derive a new non-linear extrapolation formula that improves the post-processing of data generated from SCI methods and can be applied to both ground- and excited-state energies.

Superfluidity of flat band Bose–Einstein condensates revisited

We consider the superfluid weight, speed of sound and excitation fraction of a flat band Bose–Einstein condensate (BEC) within multiband Bogoliubov theory. The superfluid weight is calculated by introducing a phase winding and minimizing the free energy with respect to it. We find that the superfluid weight has a contribution arising from the change in the condensate density and chemical potential upon the phase twist that has been neglected in the previous literature. We also point out that the speed of sound and the excitation fraction are proportional to orbital-position-independent generalizations of the quantum metric and the quantum distance, and reduce to the usual quantum metric (Fubini–Study metric) and the Hilbert–Schmidt quantum distance only in special cases. We derive a second-order perturbation correction to the dependence of the speed of sound on the generalized quantum metric, and show that it compares well with numerical calculations. Our results provide a consistent connection between flat band BEC and quantum geometry, with physical observables being independent of the orbital positions for fixed hopping amplitudes, as they should, and complete formulas for the evaluation of the superfluid weight within the Bogoliubov theory. We discuss the limitations of the Bogoliubov theory in evaluating the superfluid weight.

Perturbing π-clouds with Substituents to Study the Effects on Reaction Dynamics of gauche-1,3-Butadiene to Bicyclobutane Electrocyclization.

The conical intersection (CI) governs the ultra-fast relaxation of excited states in a radiationless manner and are observed mainly in photochemical processes. In the current work, we investigated the effects of substituents on the reaction dynamics for the conversion of gauche-1,3-butadiene to bicyclobutane via photochemical electrocyclization. We incorporated both electron withdrawing (-F) and donating (-CH3 ) groups in the conjugated system. In our study, we optimized the minimum energy conical intersection (MECI) geometries using the multi-configurational state-averaged CASSCF approach, whereas, to study the ground state reaction pathways for the substituted derivatives, dispersion corrected, B3LYP-D3 functional was used. The non-adiabatic surface hopping molecular dynamics simulations were performed to observe the behaviour of electronic states involved throughout the photoconversion process. The results obtained from the multi-reference second-order perturbation correction of energy at the XMS-CASPT2 level of theory, topography analysis, and non-adiabatic dynamics suggest that the -CH3 substituted derivatives can undergo faster thermal conversion to the product in the ground state with a smaller activation energy barrier compared to -F substituted derivative. Our study also reveals that the GBUT to BIBUT conversion follows both conrotatory and disrotatory pathways, whereas, on substitution with -F or -CH3 , the conversion proceeds via the conrotatory pathway.

Stochastic evaluation of four-component relativistic second-order many-body perturbation energies: A potentially quadratic-scaling correlation method.

A second-order many-body perturbation correction to the relativistic Dirac-Hartree-Fock energy is evaluated stochastically by integrating 13-dimensional products of four-component spinors and Coulomb potentials. The integration in the real space of electron coordinates is carried out by the Monte Carlo (MC) method with the Metropolis sampling, whereas the MC integration in the imaginary-time domain is performed by the inverse-cumulative distribution function method. The computational cost to reach a given relative statistical error for spatially compact but heavy molecules is observed to be no worse than cubic and possibly quadratic with the number of electrons or basis functions. This is a vast improvement over the quintic scaling of the conventional, deterministic second-order many-body perturbation method. The algorithm is also easily and efficiently parallelized with 92% strong scalability going from 64 to 4096 processors.

Performance of Vibrational Self-Consistent Field Theory for Accurate Potential Energy Surfaces: Fundamentals, Excited States, and Intensities.

The performance of vibrational structure calculations beyond harmonic approximation in the framework of the vibrational self-consistent field method with second-order perturbation corrections (VSCF-PT2) is investigated in conjunction with very accurate potential energy surfaces (PESs) given by various coupled-cluster electronic structure theories. The quality of anharmonic calculations depends on the accuracy of the underlying multidimensional PES obtained from its functional form, which is given by the level of electronic structure theory. Two such highest levels of typical coupled-cluster electronic structure methods, CCSD and the ″gold standard″ CCSD(T), along with their variants such as CCD, CR-CCL (completely renormalized CR-CC(2,3) approach), and CCSD(TQ) are tested for the construction of accurate anharmonic potentials without any fitting or ad hoc scaling and using cc-pVTZ basis sets. The accuracy of VSCF-PT2 theory in comparison to experimental values is tested for a series of 16 molecules with 135 fundamental bands, 64 overtones, and combination bands and also for 39 intensities. It is found that CCD and CCSD bind the potential tighter than CCSD(T) and the computed VSCF-PT2 transitions are more blue-shifted showing higher deviation from the experiment. In general, VSCF-PT2 results computed at the CCSD(T) potential offer a good cost/accuracy ratio, with the mean absolute deviation and the mean absolute percentage error with the experiment being ∼16 cm-1 and 1.38, respectively, for fundamentals. Additionally, while the CR-CCL and CCSD(TQ) methods offer similar levels of accuracies as compared to CCSD(T), the former offers a better accuracy/cost ratio than the latter and is a suitable alternative to CCSD(T).

Multicomponent heat-bath configuration interaction with the perturbative correction for the calculation of protonic excited states.

In this study, we extend the multicomponent heat-bath configuration interaction (HCI) method to excited states. Previous multicomponent HCI studies have been performed using only the variational stage of the HCI algorithm as they have largely focused on the calculation of protonic densities. Because this study focuses on energetic quantities, a second-order perturbative correction after the variational stage is essential. Therefore, this study implements the second-order Epstein-Nesbet correction to the variational stage of multicomponent HCI for the first time. Additionally, this study introduces a new procedure for calculating reference excitation energies for multicomponent methods using the Fourier-grid Hamiltonian (FGH) method, which should allow the one-particle electronic basis set errors to be better isolated from errors arising from an incomplete description of electron-proton correlation. The excited-state multicomponent HCI method is benchmarked by computing protonic excitations of the HCN and FHF- molecules and is shown to be of similar accuracy to previous excited-state multicomponent methods such as the multicomponent time-dependent density-functional theory and equation-of-motion coupled-cluster theory relative to the new FGH reference values.

Neutral and charged thorium impurity in solid argon

We study the neutral and positively charged thorium impurity (${\mathrm{Th}}^{n+}, n=0$, 1, 2, 3, 4) in solid argon by exploring the nature of chemical bonding in the ThAr diatomic molecule and in clusters ${\mathrm{ThAr}}_{4}, {\mathrm{ThAr}}_{12}, {\mathrm{Th}}^{n+}{\mathrm{Ar}}_{18}$ at the Hartree-Fock level with the second-order perturbation (MP2) correction accounting for the van der Waals forces. The chemical bonding is formed from the valence states of thorium and polarized states of argon in the clusters with ${\mathrm{Th}}^{n+}$ ($n=0$, 1, 2, 3) and solely from polarized states in the clusters with ${\mathrm{Th}}^{4+}$. In all cases with two or more valence electrons of Th, the ground state, influenced by the first Hund rule for the thorium impurity, is the high spin state. Allowing for the cubic to orthorhombic (${D}_{2h}$) symmetry lowering in ${\mathrm{Th}}^{n+}{\mathrm{Ar}}_{18}$, we find that the averaged Th-Ar bond length decreases whereas the binding energy increases with $n$, accounted for by the weakening of the Th-Ar repulsion and the strengthening of polarization. For ${\mathrm{Th}}^{4+}{\mathrm{Ar}}_{18}$, two conformations (cubic and orthorhombic) are found. We conclude that with the Th-Ar bond lengths lying very close to the Ar-Ar bond lengths in a fcc lattice, the solid argon is a material that is well suited for the accommodation of thorium impurities.

Tailoring CIPSI Expansions for QMC Calculations of Electronic Excitations: The Case Study of Thiophene.

The perturbatively selected configuration interaction scheme (CIPSI) is particularly effective in constructing determinantal expansions for quantum Monte Carlo (QMC) simulations with Jastrow–Slater wave functions: fast and smooth convergence of ground-state properties and balanced descriptions of ground and excited states of different symmetries have been reported. In particular, accurate excitation energies have been obtained by the pivotal requirement of using CIPSI expansions with similar second-order perturbation corrections for each state, that is, a similar estimated distance to the full configuration interaction limit. Here, we elaborate on the CIPSI selection criterion for excited states of the same symmetry as the ground state, generating expansions from a common orbital set. Using these expansions in QMC as determinantal components of Jastrow–Slater wave functions, we compute the lowest, bright excited state of thiophene, which is challenging due to its significant multireference character. The resulting vertical excitation energies are within 0.05 eV of the best theoretical estimates, already with expansions of only a few thousand determinants. Furthermore, we relax the ground- and excited-state structures following the corresponding root in variational Monte Carlo and obtain bond lengths that are accurate to better than 0.01 Å. Therefore, while the full treatment at the CIPSI level of this system is quite demanding, in QMC, we can compute high-quality excitation energies and excited-state structural parameters building on affordable CIPSI expansions with relatively few, well-chosen determinants.

Perspective on Simplified Quantum Chemistry Methods for Excited States and Response Properties.

We review recent developments in the framework of simplified quantum chemistry for excited state and optical response properties (sTD-DFT) and present future challenges for new method developments to improve accuracy and extend the range of application. In recent years, the scope of sTD-DFT was extended to molecular response calculations of the polarizability, optical rotation, first hyperpolarizability, two-photon absorption (2PA), and excited-state absorption for large systems with hundreds to thousands of atoms. The recently introduced spin-flip simplified time-dependent density functional theory (SF-sTD-DFT) variant enables an ultrafast treatment for diradicals and related strongly correlated systems. A few drawbacks were also identified, specifically for the computation of 2PA cross sections. We propose solutions to this problem and how to generally improve the accuracy of simplified schemes. New possible simplified schemes are also introduced for strongly correlated systems, e.g., with a second-order perturbative correlation correction. Interpretation tools that can extract chemical structure-property relationships from excited state or response calculations are also discussed. In particular, the recently introduced method-agnostic RespA approach based on natural response orbitals (NROs) as the key concept is employed.

Towards Near-Exact Solutions of Molecular Electronic Structure: Full Coupled-Cluster Reduction with a Second-Order Perturbative Correction.

We introduce a new augmented adaptation of the recently developed full coupled-cluster reduction (FCCR) with a second-order perturbative correction, abbreviated as FCCR(2). FCCR is a selected coupled-cluster expansion aimed at optimally reducing the excitation manifold and commutator expansions for high-rank excitations for obtaining accurate solutions of the electronic Schödinger equation in a size-extensive manner. The present FCCR(2) enables estimating the residual correlation of FCCR by the second-order perturbative correction E(2) from the complementary space of the FCCR projection manifold. The linear relationship between E(2) and the energy of FCCR(2) allows accurate estimates of near-exact energies for a wide variety of molecules with strong electron correlation. The potential of the method is demonstrated using challenging cases, the ground-state electronic energy of the benzene molecule in equilibrium and stretched geometries, and the isomerization energy of the transition metal complex [Cu(NH3)]2O22+.

Multiconfigurational Calculations and Nonadiabatic Molecular Dynamics Explain Tricyclooctadiene Photochemical Chemoselectivity

Sunlight is a renewable energy source that can be stored in chemical bonds using photochemical reactions. The synthesis of exotic and strained molecules is especially attractive with photochemical techniques because of the associated efficient and mild reaction conditions. We have understood the photophysics and subsequent photochemistry of a possible cubane precursor, tricyclo[4,2,0,02,5]octa-3,7-diene with complete active space self-consistent field (CASSCF) calculations with an (8,7) active space and the ANO-S-VDZP basis set to. The CASSCF energies were corrected with a second-order perturbative correction CASPT2(8,7)/ANO-S-VDZP. The S0 → S1 vertical excitation energy of 1 is 6.25 eV, which is a π → π* excitation. The minimum energy path from the S1 Franck-Condon point leads to a 4π-disrotatory electrocyclic ring-opening reaction to afford bicyclo[4,2,0]octa-2,4,7-triene. The 2D potential energy surface scan located a rhomboidal S1/S0 minimum energy crossing point connecting 1 and cubane, suggesting that a cycloaddition is theoretically possible. We used the fewest switches surface hopping to study the photodynamics of this cycloaddition: 85% of 1722 trajectories relaxed to eight products; the major products are bicyclo[4,2,0]octa-2,4,7-triene (30%) and cycloocta-1,3,5,7-tetraene (32%). Only 0.4% of trajectories undergo a [2 + 2] cycloaddition to form cubane.

Finite-temperature many-body perturbation theory in the grand canonical ensemble.

A finite-temperature many-body perturbation theory is presented, which expands in power series the electronic grand potential, chemical potential, internal energy, and entropy on an equal footing. Sum-over-states and sum-over-orbitals analytical formulas for the second-order perturbation corrections to these thermodynamic properties are obtained in a time-independent, nondiagrammatic, algebraic derivation, relying on the sum rules of the Hirschfelder-Certain degenerate perturbation energies in a degenerate subspace as well as nine algebraic identities for the zeroth-order thermal averages of one- through four-indexed quantities and products thereof. They reproduce numerically exactly the benchmark data obtained as the numerical derivatives of the thermal-full-configuration-interaction results for a wide range of temperatures.

Self-Consistent Range-Separated Density-Functional Theory with Second-Order Perturbative Correction via the Optimized-Effective-Potential Method.

We extend the range-separated double-hybrid RSH+MP2 method (Ángyán, J. G.; et al. Phys. Rev. A 2005, 72, 012510), combining long-range HF exchange and MP2 correlation with a short-range density functional to a fully self-consistent version using the optimized-effective-potential technique in which the orbitals are obtained from a local potential including the long-range HF and MP2 contributions. We test this approach, that we name RS-OEP2, on a set of small closed-shell atoms and molecules. For the commonly used value of the range-separation parameter μ = 0.5 bohr-1, we find that self-consistency does not seem to bring any improvement for total energies, ionization potentials, and electronic affinities. However, contrary to the non-self-consistent RSH+MP2 method, the present RS-OEP2 method gives a LUMO energy which physically corresponds to a neutral excitation energy and gives local exchange-correlation potentials which are reasonably good approximations to the corresponding Kohn-Sham quantities. At a finer scale, we find that RS-OEP2 gives largely inaccurate correlation potentials and correlated densities, which points to the need of further improvement of this type of range-separated double hybrids.

Triple Differential Dijet Cross Section at the LHC.

The measurement of the triple-differential dijet production cross section as a function of the average transverse momentum p_{T,avg}, half the rapidity separation y^{*}, and the boost y_{b} of the two leading jets in the event enables a kinematical scan of the underlying parton momentum distributions. We compute for the first time the second-order perturbative QCD corrections to this triple-differential dijet cross section, at leading color in all partonic channels, thereby enabling precision studies with LHC dijet data. A detailed comparison with experimental CMS 8TeV data is performed, demonstrating how the shape of this differential cross section probes the parton densities in different kinematical ranges.

Quantum Package 2.0: An Open-Source Determinant-Driven Suite of Programs.

Quantum chemistry is a discipline which relies heavily on very expensive numerical computations. The scaling of correlated wave function methods lies, in their standard implementation, between and , where N is proportional to the system size. Therefore, performing accurate calculations on chemically meaningful systems requires (i) approximations that can lower the computational scaling and (ii) efficient implementations that take advantage of modern massively parallel architectures. Quantum Package is an open-source programming environment for quantum chemistry specially designed for wave function methods. Its main goal is the development of determinant-driven selected configuration interaction (sCI) methods and multireference second-order perturbation theory (PT2). The determinant-driven framework allows the programmer to include any arbitrary set of determinants in the reference space, hence providing greater methodological freedom. The sCI method implemented in Quantum Package is based on the CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively) algorithm which complements the variational sCI energy with a PT2 correction. Additional external plugins have been recently added to perform calculations with multireference coupled cluster theory and range-separated density-functional theory. All the programs are developed with the IRPF90 code generator, which simplifies collaborative work and the development of new features. Quantum Package strives to allow easy implementation and experimentation of new methods, while making parallel computation as simple and efficient as possible on modern supercomputer architectures. Currently, the code enables, routinely, to realize runs on roughly 2 000 CPU cores, with tens of millions of determinants in the reference space. Moreover, we have been able to push up to 12 288 cores in order to test its parallel efficiency. In the present manuscript, we also introduce some key new developments: (i) a renormalized second-order perturbative correction for efficient extrapolation to the full CI limit and (ii) a stochastic version of the CIPSI selection performed simultaneously to the PT2 calculation at no extra cost.

Tensor-decomposition techniques for ab initio nuclear structure calculations: From chiral nuclear potentials to ground-state energies

The impact of applying state-of-the-art tensor factorization techniques to modern nuclear Hamiltonians derived from chiral effective field theory is investigated. Subsequently, the error induced by the tensor decomposition of the input Hamiltonian on ground-state energies of closed-shell nuclei calculated via second-order many-body perturbation theory is benchmarked. With the aid of the factorized Hamiltonian, the second-order perturbative correction to ground-state energies is decomposed and the scaling properties of the underlying tensor network are discussed. The employed tensor formats are found to lead to an efficient data compression of two-body matrix elements of the nuclear Hamiltonian. In particular, the sophisticated \emph{tensor hypercontraction} (THC) scheme yields low tensor ranks with respect to both harmonic-oscillator and Hartree-Fock single-particle bases. It is found that the tensor rank depends on the two-body total angular momentum $J$ for which one performs the decomposition, which is itself directly related to the sparsity the corresponding tensor. Furthermore, including normal-ordered two-body contributions originating from three-body interactions does not compromise the efficient data compression. Ultimately, the use of factorized matrix elements authorizes controlled approximations of the exact second-order ground-state energy corrections. In particular, a small enough error is obtained from low-rank factorizations in $^{4}$He, $^{16}$O and $^{40}$Ca.

Selected configuration interaction dressed by perturbation.

Selected configuration interaction (sCI) methods including second-order perturbative corrections provide near full CI (FCI) quality energies with only a small fraction of the determinants of the FCI space. Here, we introduce both a state-specific and a multi-state sCI method based on the configuration interaction using a perturbative selection made iteratively (CIPSI) algorithm. The present method revises the reference (internal) space under the effect of its interaction with the outer space via the construction of an effective Hamiltonian, following the shifted-Bk philosophy of Davidson and co-workers. In particular, the multi-state algorithm removes the storage bottleneck of the effective Hamiltonian via a low-rank factorization of the dressing matrix. Illustrative examples are reported for the state-specific and multi-state versions.

Intersystem Crossing Drives Photoisomerization in o-Nitrotoluene, a Model for Photolabile Caged Compounds.

o-Nitrobenzyl (oNB) derivatives are widely used photolabile caged compounds in chemical and biological applications. The primary step in the photoinduced deprotection is an excited state intramolecular hydrogen transfer (ESIHT) leading to tautomerization of the oNB compound and subsequent release of the protecting group. The prototype molecule for studying such ESIHT is o-nitrotoluene (oNT), where hydrogen transfers from the methyl to the nitro group. Using the complete active space self-consistent field (CASSCF) method with second-order perturbative energy corrections (CASPT2), we have comprehensively investigated the photoisomerization and photo decay mechanisms in oNT. We have obtained the minimum energy crossing points (MECPs) between relevant electronic states and identified the singlet and triplet pathways. There is a barrierless path for oNT to relax to the lowest triplet state. In this T1 state, the ESIHT products are more stable than T1 oNT. Hydrogen-transfer occurs on the T1 state followed by relaxation to the ground state to give the isomerized product. A biradical intermediate proposed by previous studies is characterized to be the hydrogen-transferred T1 product. On the singlet pathway, in contrast to the triplet, the ground state tautomer is formed from the S1 oNT through a geometrically distant and energetically higher S1/S0 conical intersection. Although nonadiabatic dynamical studies are essential for determining branching ratios, our study, which considers the accessibility of different MECPs based on geometry and energy, and the magnitude of spin-orbit coupling at singlet-triplet MECPs, suggests that a significant fraction of the isomerization yield is due to the triplet channel.

Adiabatic perturbation theory for atoms and molecules in the low-frequency regime.

There is an increasing interest in the photoinduced dynamics in the low frequency, ω, regime. The multiphoton absorptions by molecules in strong laser fields depend on the polarization of the laser and on the molecular structure. The unique properties of the interaction of atoms and molecules with lasers in the low-frequency regime imply new concepts and directions in strong-field light-matter interactions. Here we represent a perturbational approach for the calculations of the quasi-energy spectrum in the low-frequency regime, which avoids the construction of the Floquet operator with extremely large number of Floquet channels. The zero-order Hamiltonian in our perturbational approach is the adiabatic Hamiltonian where the atoms/molecules are exposed to a dc electric field rather than to ac-field. This is in the spirit of the first step in the Corkum three-step model. The second-order perturbation correction terms are obtained when iℏω∂∂τ serves as a perturbation and τ is a dimensionless variable. The second-order adiabatic perturbation scheme is found to be an excellent approach for calculating the ac-field Floquet solutions in our test case studies of a simple one-dimensional time-periodic model Hamiltonian. It is straightforward to implement the perturbation approach presented here for calculating atomic and molecular energy shifts (positions) due to the interaction with low-frequency ac-fields using high-level electronic structure methods. This is enabled since standard quantum chemistry packages allow the calculations of atomic and molecular energy shifts due to the interaction with dc-fields. In addition to the shift of the energy positions, the energy widths (inverse lifetimes) can be obtained at the same level of theory. These energy shifts are functions of the laser parameters (low frequency, intensity, and polarization).