Second-order Perturbation Correction Research Articles

A Decapeptide Hydrated by Two Waters: Conformers Determined by Theory and Validated by Cold Ion Spectroscopy.

The intrinsic structures of biomolecules in the gas phase may not reflect their native solution geometries. Microsolvation of the molecules bridges the two environments, enabling a tracking of molecular structural changes upon hydration at the atomistic level. We employ density functional calculations to compute a large pool of structures and vibrational spectra for a gas-phase complex, in which a doubly protonated decapeptide, gramicidin S, is solvated by two water molecules. Though most vibrations of this large complex are treated in a harmonic approximation, the water molecules and the vibrations of the host ion coupled to them are locally described by a quantum mechanical vibrational self-consistent field theory with second-order perturbation correction (VSCF-PT2). Guided and validated by the available cold ion spectroscopy data, the computational analysis identifies structures of the three experimentally observed conformers of the complex. They, mainly, differ by the hydration sites, of which the one at the Orn side chain is the most important for reshaping the peptide toward its native structure. The study demonstrates the ability of a quantum chemistry approach that intelligently combines the semiempirical and ab initio computations to disentangle a complex interplay of intra- and intermolecular hydrogen bonds in large molecular systems.

Optimal information transfer in enzymatic networks: A field theoretic formulation.

Signaling in enzymatic networks is typically triggered by environmental fluctuations, resulting in a series of stochastic chemical reactions, leading to corruption of the signal by noise. For example, information flow is initiated by binding of extracellular ligands to receptors, which is transmitted through a cascade involving kinase-phosphatase stochastic chemical reactions. For a class of such networks, we develop a general field-theoretic approach to calculate the error in signal transmission as a function of an appropriate control variable. Application of the theory to a simple push-pull network, a module in the kinase-phosphatase cascade, recovers the exact results for error in signal transmission previously obtained using umbral calculus [Hinczewski and Thirumalai, Phys. Rev. X 4, 041017 (2014)2160-330810.1103/PhysRevX.4.041017]. We illustrate the generality of the theory by studying the minimal errors in noise reduction in a reaction cascade with two connected push-pull modules. Such a cascade behaves as an effective three-species network with a pseudointermediate. In this case, optimal information transfer, resulting in the smallest square of the error between the input and output, occurs with a time delay, which is given by the inverse of the decay rate of the pseudointermediate. Surprisingly, in these examples the minimum error computed using simulations that take nonlinearities and discrete nature of molecules into account coincides with the predictions of a linear theory. In contrast, there are substantial deviations between simulations and predictions of the linear theory in error in signal propagation in an enzymatic push-pull network for a certain range of parameters. Inclusion of second-order perturbative corrections shows that differences between simulations and theoretical predictions are minimized. Our study establishes that a field theoretic formulation of stochastic biological signaling offers a systematic way to understand error propagation in networks of arbitrary complexity.

A framework for second-order parton showers

A framework is presented for including second-order perturbative corrections to the radiation patterns of parton showers. The formalism allows to combine O(αs2)-corrected iterated 2→3 kernels for “ordered” gluon emissions with tree-level 2→4 kernels for “unordered” ones. The combined Sudakov evolution kernel is thus accurate to O(αs2). As a first step towards a full-fledged implementation of these ideas, we develop an explicit implementation of 2→4 shower branchings in this letter.

Pathways for Excited-State Nonradiative Decay of 5,6-Dihydroxyindole, a Building Block of Eumelanin

The photophysics of 5,6-dihydroxyindole (DHI) following excitation to its lowest two optically bright states was investigated using the complete active space self-consistent field method with second-order perturbative energy corrections. There is a barrierless pathway for the molecule to relax from the second-lowest bright state (21ππ*) to the lowest bright state (11ππ*). The 11ππ* state has a conical intersection with the optically dark 11πσ* state, which further intersects with the ground state along the NH and OH stretching coordinates. Moreover, the 11ππ* has out-of-plane conical intersections with the ground state. For accessing the conical intersections with the ground state, there are energy barriers, which are higher than the available energy following vertical excitation to the lowest bright state. The nature of the calculated deactivation pathways helps interpret the experimentally estimated lifetimes of the lowest two bright states of DHI. The relatively long lifetime of the lowest excited state suggests that isolated DHI in monomeric form cannot rationalize the ultrafast deactivation property of eumelanin.

Correlations in the low-density Fermi gas: Fermi-liquid state, dimerization, and Bardeen-Cooper-Schrieffer pairing

We present ground state calculations for low-density Fermi gases described by two model interactions, an attractive square-well potential and a Lennard-Jones potential, of varying strength. We use the optimized Fermi-Hypernetted Chain integral equation method which has been proved to provide, in the density regimes of interest here, an accuracy better than one percent. We first examine the low-density expansion of the energy and compare with the exact answer by Huang and Yang (H. Huang and C. N. Yang, {\em Phys. Rev.\/} {\bf 105}, 767 (1957)). It is shown that a locally correlated wave function of the Jastrow-Feenberg type does not recover the quadratic term in the expansion of the energy in powers of $\a0\KF$, where $\a0$ is the vacuum $s$-wave scattering length and $\KF$ the Fermi wave number. The problem is cured by adding second-order perturbation corrections in a correlated basis. Going to higher densities and/or more strongly coupled systems, we encounter an instability of the normal state of the system which is characterized by a divergence of the {\em in-medium\/} scattering length. We interpret this divergence as a phonon-exchange driven dimerization of the system, similar to what one has at zero density when the vacuum scattering length $\a0$ diverges. We then study, in the stable regime, the superfluid gap and its dependence on the density and the interaction strength. We identify two different corrections to low-density expansions: One is medium corrections to the pairing interaction, and the other one finite-range corrections. We show that the most important finite-range corrections are a direct manifestation of the many-body nature of the system.

Approximate first-principles anharmonic calculations of polyatomic spectra using MP2 and B3LYP potentials: comparisons with experiment.

Anharmonic vibrational spectroscopy calculations using MP2 and B3LYP computed potential surfaces are carried out for a series of molecules, and frequencies and intensities are compared with those from experiment. The vibrational self-consistent field with second-order perturbation correction (VSCF-PT2) is used in computing the spectra. The test calculations have been performed for the molecules HNO3, C2H4, C2H4O, H2SO4, CH3COOH, glycine, and alanine. Both MP2 and B3LYP give results in good accord with experimental frequencies, though, on the whole, MP2 gives very slightly better agreement. A statistical analysis of deviations in frequencies from experiment is carried out that gives interesting insights. The most probable percentage deviation from experimental frequencies is about -2% (to the red of the experiment) for B3LYP and +2% (to the blue of the experiment) for MP2. There is a higher probability for relatively large percentage deviations when B3LYP is used. The calculated intensities are also found to be in good accord with experiment, but the percentage deviations are much larger than those for frequencies. The results show that both MP2 and B3LYP potentials, used in VSCF-PT2 calculations, account well for anharmonic effects in the spectroscopy of molecules of the types considered.

Second-order perturbative corrections to the restricted active space configuration interaction with the hole and particle approach.

Second-order corrections to the restricted active space configuration interaction (RASCI) with the hole and particle truncation of the excitation operator are developed. Theoretically, the computational cost of the implemented perturbative approach, abbreviated as RASCI(2), grows like its single reference counterpart in MP2. Two different forms of RASCI(2) have been explored, that is the generalized Davidson-Kapuy and the Epstein-Nesbet partitions of the Hamiltonian. The preliminary results indicate that the use of energy level shift of a few tenths of a Hartree might systematically improve the accuracy of the RASCI(2) energies. The method has been tested in the computation of the ground state energy profiles along the dissociation of the hydrogen fluoride and N2 molecules, the computation of correlation energy in the G2/97 molecular test set, and in the computation of excitation energies to low-lying states in small organic molecules.

A second-order multi-reference perturbation method for molecular vibrations

We present a general multi-reference framework for treating strong correlation in vibrational structure theory, which we denote the vibrational active space self-consistent field (VASSCF) approach. Active configurations can be selected according to excitation level or the degrees of freedom involved, or both. We introduce a novel state-specific second-order multi-configurational perturbation correction that accounts for the remaining weak correlation between the vibrational modes. The resulting VASPT2 method is capable of accurately and efficiently treating strong correlation in the form of large anharmonic couplings, at the same time as correctly resolving resonances between states. These methods have been implemented in our new dynamics package DYNAMOL, which can currently treat up to four-body Hamiltonian coupling terms. We present a pilot application of the VASPT2 method to the trans isomer of formic acid. We have constructed a new analytic potential that reproduces frozen core CCSD(T)(F12*)/cc-pVDZ-F12 energies to within 0.25% RMSD over the energy range 0-15 000 cm(-1). The computed VASPT2 fundamental transition energies are accurate to within 9 cm(-1) RMSD from experimental values, which is close to the accuracy one can expect from a CCSD(T) potential energy surface.

Assessment of the second-order perturbative corrections to PNOF5

In a recent paper (J. Chem. Phys. 139, 064111, 2013), an antisymmetrised product of strongly orthogonal geminals with the expansion coefficients explicitly expressed by means of the occupation numbers was used to generate the Piris natural orbital functional 5 (PNOF5). This functional describes most of the non-dynamical effects, but also an important part of the intrapair (intrageminal) electron correlation. Second-order corrections to the generating PNOF5 wave function were derived using the multiconfigurational perturbation theory size consistent at the second order (SC2-MCPT) to include the missing interpair (intergeminal) electron correlation. A modified version of the SC2-MCPT involving double excitations only from different geminals was introduced and denoted as PNOF5-PT2. In this paper, the ground-state energies of 36 closed-shell species belonging to the G2/97 test set of molecules are studied by the PNOF5-PT2 and PNOF5-SC2-MCPT methods. The numerical performance of both methods on eight dimers, with different strength of hydrogen bonds, and 13 isogyric reactions is also assessed. The results are in reasonable agreement with those obtained using the complete active space second-order perturbation theory and coupled-cluster method with singles, doubles, and noniterative triples corrections.

Convergence Acceleration of Parallel Monte Carlo Second-Order Many-Body Perturbation Calculations Using Redundant Walkers.

A Monte Carlo (MC) integration of the second-order many-body perturbation (MP2) corrections to energies and self-energies eliminates the usual computational bottleneck of the MP2 algorithm (i.e., the basis transformation of two-electron integrals), thereby achieving near-linear size dependence of its operation cost, a negligible core and disk memory cost, and a naturally parallel computational kernel. In this method, the correlation correction expressions are recast into high-dimensional integrals, which are approximated as the sums of integrands evaluated at coordinates of four electron random walkers guided by a Metropolis algorithm for importance sampling. The gravest drawback of this method, however, is the inevitable statistical uncertainties in its results, which decay slowly as the inverse square-root of the number of MC steps. We propose an algorithm that can increase the number of MC sampling points in each MC step by many orders of magnitude by having 2m electron walkers (m > 2) and then using m(m - 1)/2 permutations of their coordinates in evaluating the integrands. Hence, this algorithm brings an O(m(2))-fold increase in the number of MC sampling points at a mere O(m) additional cost of propagating redundant walkers, which is a net O(m)-fold enhancement in sampling efficiency. We have demonstrated a large performance increase in the Monte Carlo MP2 calculations for the correlation energies of benzene and benzene dimer as well as for the correlation corrections to the energy, ionization potential, and electron affinity of C60. The calculation for C60 has been performed with a parallel implementation of this method running on up to 400 processors of a supercomputer, yielding an accurate prediction of its large electron affinity, which makes its derivative useful as an electron acceptor in bulk heterojunction organic solar cells.

Interpair electron correlation by second-order perturbative corrections to PNOF5

An antisymmetrized product of strongly orthogonal geminals with the expansion coefficients explicitly expressed by the occupation numbers is used to generate the Piris natural orbital functional 5 (PNOF5). Second-order corrections to PNOF5 are derived and implemented using the multiconfigurational perturbation theory size consistent at second-order (SC2-MCPT). A modified version of the SC2-MCPT including only doubly excited determinants from different geminals is proposed to describe the dispersion interaction in the helium dimer, and to avoid the breakdown of curves in homolytic dissociations of FH, CO, and N2. Comparisons of calculated properties to experimental data are included to verify the accuracy of the formulation.

The angular dependence of spin-state energy splittings in the core

Spin-state energy splittings are highly relevant for catalysis, molecular magnetism, and materials science, yet continue to pose a challenge for electronic structure methods. For a Fe2O2+ 2 core, we evaluate the bridging angle dependence of energy splittings between ferromagnetically and antiferromagnetically coupled states for different exchange–correlation functionals, and compare with complete active space self-consistent field (CASSCF) values, also including second-order perturbative corrections (CASPT2). CASSCF and CASPT2 yield strong antiferromagnetic coupling, with the smallest coupling at 100°, and a smooth dependence on the angle for Fe−O−Fe angles of 70° to 120°. Interestingly, this is qualitatively the same behaviour as often found for stable dinuclear transition metal complexes. While all functionals show the same angular dependence as CASPT2, they favour the antiferromagnetic state less strongly. Pure functionals such as BP86, BLYP, SSB-D, and TPSS come closer to the CASPT2 results (with energy splittings by about 60 kJ/mol smaller than the CASPT2 ones) than hybrid functionals. The hybrid functionals B3LYP, B3LYP⋆, and PBE0 favour the antiferromagnetic state even less strongly, resulting in ferromagnetic coupling for angles around 100°. The good qualitative agreement between CASPT2 and CASSCF on the one hand and CASPT2 and density functional theory on the other hand for angles between 70° and 110° suggests that the chosen active space of 18 electrons in 14 orbitals may be adequate for spin-state energy splitting of Fe2O2+ 2 in that region (possibly due to error cancellation), while angles of 60° or 120° may require larger active spaces. This study is complemented by an analysis of local spins, local charges, and CASSCF natural orbitals.

Stochastic evaluation of second-order Dyson self-energies

A stochastic method is proposed that evaluates the second-order perturbation corrections to the Dyson self-energies of a molecule (i.e., quasiparticle energies or correlated ionization potentials and electron affinities) directly and not as small differences between two large, noisy quantities. With the aid of a Laplace transform, the usual sum-of-integral expressions of the second-order self-energy in many-body Green's function theory are rewritten into a sum of just four 13-dimensional integrals, 12-dimensional parts of which are evaluated by Monte Carlo integration. Efficient importance sampling is achieved with the Metropolis algorithm and a 12-dimensional weight function that is analytically integrable, is positive everywhere, and cancels all the singularities in the integrands exactly and analytically. The quasiparticle energies of small molecules have been reproduced within a few mEh of the correct values with 10(8) Monte Carlo steps. Linear-to-quadratic scaling of the size dependence of computational cost is demonstrated even for these small molecules.

Stochastic evaluation of second-order many-body perturbation energies

With the aid of the Laplace transform, the canonical expression of the second-order many-body perturbation correction to an electronic energy is converted into the sum of two 13-dimensional integrals, the 12-dimensional parts of which are evaluated by Monte Carlo integration. Weight functions are identified that are analytically normalizable, are finite and non-negative everywhere, and share the same singularities as the integrands. They thus generate appropriate distributions of four-electron walkers via the Metropolis algorithm, yielding correlation energies of small molecules within a few mE(h) of the correct values after 10(8) Monte Carlo steps. This algorithm does away with the integral transformation as the hotspot of the usual algorithms, has a far superior size dependence of cost, does not suffer from the sign problem of some quantum Monte Carlo methods, and potentially easily parallelizable and extensible to other more complex electron-correlation theories.

Avoided crossings, conical intersections, and low-lying excited states with a single reference method: The restricted active space spin-flip configuration interaction approach

The restricted active space spin-flip CI (RASCI-SF) performance is tested in the electronic structure computation of the ground and the lowest electronically excited states in the presence of near-degeneracies. The feasibility of the method is demonstrated by analyzing the avoided crossing between the ionic and neutral singlet states of LiF along the molecular dissociation. The two potential energy surfaces (PESs) are explored by means of the energies of computed adiabatic and approximated diabatic states, dipole moments, and natural orbital electronic occupancies of both states. The RASCI-SF methodology is also used to study the ground and first excited singlet surface crossing involved in the double bond isomerization of ethylene, as a model case. The two-dimensional PESs of the ground (S(0)) and excited (S(1)) states are calculated for the complete configuration space of torsion and pyramidalization molecular distortions. The parameters that define the state energetics in the vicinity of the S(0)/S(1) conical intersection region are compared to complete active space self-consistent field (CASSCF) results. These examples show that it is possible to describe strongly correlated electronic states using a single reference methodology without the need to expand the wavefunction to high levels of collective excitations. Finally, RASCI is also examined in the electronic structure characterization of the ground and 2(1)A(g)(-), 1(1)B(u)(+), 1(1)B(u)(-), and 1(3)B(u)(-) states of all-trans polyenes with two to seven double bonds and beyond. Transition energies are compared to configuration interaction singles, time-dependent density functional theory (TDDFT), CASSCF, and its second-order perturbation correction calculations, and to experimental data. The capability of RASCI-SF to describe the nature and properties of each electronic state is discussed in detail. This example is also used to expose the properties of different truncations of the RASCI wavefunction and to show the possibility to use an excitation operator with any number of α-to-β electronic promotions.

Extracting Stochastic Airwake Models from a Database for Engineering Analysis and Simulation

A framework is presented for extracting interpretive models of airwake autocorrelation and autospectrum as well as crosscorrelation and cross-spectrum from a database. These models have a simple analytical structure that aids routine simulation and application as a predictive tool. Airwake refers to turbulence shed from the ship superstructure, and the database, to a set of spectral (autospectral and cross-spectral) points of flow velocity data from experimental and computational fluid dynamics–based investigations. The framework is developed from first principles: It is based on perturbation theory; it addresses all three velocity components, and it is tested against a comprehensive database under different superstructure and wind-over-deck conditions. For each velocity component, the autocorrelation and cross-correlation are represented by separate perturbation series in which the first terms have a form of the von Karman longitudinal or lateral correlation function. These series are then transformed into equivalent perturbation series of autospectra and cross-spectra. The perturbation coefficients are evaluated by satisfying the algorithmic constraints and fitting a curve on a set of selected spectral data points in the low-frequency bandwidth (0≤f(Hz)≤1.6); the emphasis is on extracting spectral models for this bandwidth. Generally, no more than a second-order perturbation correction (a three-term perturbation series) is necessary, and the extracted models lend themselves well to construction of shaping filters driven by white noise. The framework's strengths and weaknesses are discussed as well.

Perturbative correction for the basis set incompleteness error of complete-active-space self-consistent field

To reduce the basis set incompleteness of the complete-active-space self-consistent field (CASSCF) wave function and energy we develop a second-order perturbation correction due to single excitations to complete set of unoccupied states. Other than the one- and two-electron integrals, only one- and two-particle reduced density matrices are required to compute the correction, denoted as [2](S). Benchmark calculations on prototypical ground-state bond-breaking problems show that only the aug-cc-pVXZ basis is needed with the [2](S) correction to match the accuracy of CASSCF energies of the aug-cc-pV(X+1)Z quality.

Computational study of the equilibrium geometry and anharmonic vibrational spectra of PbX2 ··· NO and PbX2 ··· ON (X = F, Cl, Br, I) complexes

Equilibrium geometry parameters of the open shell PbX2 ··· NO and PbX2 ··· ON (X = F, Cl, Br, I) complexes have been computed by second-order Z-averaged perturbation theory (ZAPT2) with Stevens–Basch–Krauss–Jasien–Cundari (SBKJC) scalar-relativistic effective core potentials (RECP) and basis sets on all atoms. Equilibrium geometries of both PbX2 ··· NO and PbX2 ··· ON bonding isomers conform to Cs symmetry structure with end-on ligand coordination, and are characterized by Pb-N bond length within 266.6–271.7 pm range, Pb–O distance of 267.8–275.8 pm, Pb–N–O angle within 109.2–120.7° range, and Pb–O–N angle of 117.1–127.8°. Anharmonic vibrational spectra of the PbX2 ··· NO and PbX2 ··· ON complexes have been calculated by direct correlation-corrected vibrational self-consistent field (CC-VSCF) method enhanced with second-order perturbative correction using potential energy surfaces (PESs) determined at ZAPT2/SBKJC+(d) level in curvilinear (internal) coordinates. Fundamental ν(Pb–N) stretching mode has been computed at 232.8 to 209.0 cm−1 within PbX2 ··· NO series. whereas ν(Pb–O) stretching mode fundamental evaluation in PbX2 ··· ON series afforded wavenumbers within 183.2–150.7 cm−1 range. Blue shift of the ν(N=O) stretching mode wavenumber upon PbX2 ··· NO complex formation, computed in anharmonic approximation, 15.8–14.6 cm−1, correctly reproduces the effect observed in the low-temperature Ar matrix spectra of PbX2 ··· NO compounds. Influence of complex formation on the νs(Pb–X) and νas(Pb–X) fundamentals of PbX2 halides has also been discussed. Two-dimensional mapping of the (Qi , Qj ) mode-mode coupling potential has been used to rationalize the origin of mode coupling related anharmonic corrections.

Computational Study of Promising Organic Dyes for High-Performance Sensitized Solar Cells

The energy transition from the ground state to the first excited singlet of four organic dye candidates to be used as sensitizers in solar cells, D5, D7, D9, and D11, has been computationally explored and compared to experimental results with TDDFT (B3LYP, ωB97, and ωB97X functionals) and the CIS(D) and SOS-CIS(D) wave function based methods. The second-order perturbation correction to CI singles’ excitation energies are superior to any TDDFT functional employed here. The performance of SOS-CIS(D) is especially interesting, being in good agreement with absorption spectra and having important computational savings. The best TDDFT results are obtained by the ωB97X functional. Solvation effects on the excitation energies have been studied with three different models, i.e., the Onsager reaction field model, SS(V)PE, and SM8.

Generalizing the bra state in the symmetry-adapted cluster singles and doubles method and the second-order perturbation correction

We present a procedure to overcome the quasi-degeneracy problem in the CCSD and SACSD methods. A multi-configuration bra state in the SACSD equations (MRbra-SACSD equation) together with a second-order perturbation correction gives quantitatively correct potential curves for bond dissociations. The benchmark results for HF, F 2, and H 2O show that the perturbative triple correction reduces the deviations from Full-CI energies to within 3 mHartree (1.9 kcal/mol). Within the benchmark calculations, the results of the present method are comparable with the recent post-CCSD corrections, the CR-CCSD(T) and CCSD(2) T methods, and are slightly less accurate than the CR-CC(2,3) method.