Minimum Energy Conical Intersections Research Articles

Dense and Acidic Organelle-Targeted Visualization in Living Cells: Application of Viscosity-Responsive Fluorescence Utilizing Restricted Access to Minimum Energy Conical Intersection

Cell-imaging methods with functional fluorescent probes are an indispensable technique to evaluate physical parameters in cellular microenvironments. In particular, molecular rotors, which take advantage of the twisted intramolecular charge transfer (TICT) process, have helped evaluate microviscosity. However, the involvement of charge-separated species in the fluorescence process potentially limits the quantitative evaluation of viscosity. Herein, we developed viscosity-responsive fluorescent probes for cell imaging that are not dependent on the TICT process. We synthesized AnP2-H and AnP2-OEG, both of which contain 9,10-di(piperazinyl)anthracene, based on 9,10-bis(N,N-dialkylamino)anthracene that adopts a nonflat geometry at minimum energy conical intersection. AnP2-H and AnP2-OEG exhibited enhanced fluorescence as the viscosity increased, with sensitivities comparable to those of conventional molecular rotors. In living cell systems, AnP2-OEG showed low cytotoxicity and, reflecting its viscosity-responsive property, allowed specific visualization of dense and acidic organelles such as lysosomes, secretory granules, and melanosomes under washout-free conditions. These results provide a new direction for developing functional fluorescent probes targeting dense organelles.

A Nitrogen Out-of-Plane (NOOP) Mechanism for Imine-Based Light-Driven Molecular Motors

Light-driven molecular motors have generated considerable interest due to their potential applications in material and biological systems. Recently, Greb and Lehn reported a new class of molecular motors, chiral N-alkyl imines, which undergo unidirectional rotation induced by light and heat. The mechanism of unidirectional motion in molecular motors containing a C═N group has been assumed to consist of photoinduced torsion about the double bond. In this work, we present a computational study of the photoisomerization dynamics of a chiral N-alkyl imine motor. We find that the location and energetics of minimal energy conical intersections (MECIs) alone are insufficient to understand the mechanism of the motor. Furthermore, a key part of the mechanism consists of out-of-plane distortions of the N atom (followed by isomerization about the double bond). Dynamic effects and out-of-plane distortions are critical to understand the observed (rather low) quantum yield for photoisomerization. Our results provide hints as to how the photoisomerization quantum yield might be increased, improving the efficiency of this class of molecular motors.

Modeling Light-Induced Chromophore Hydration in the Reversibly Photoswitchable Fluorescent Protein Dreiklang.

We report the results of a computational study of the mechanism of the light-induced chemical reaction of chromophore hydration in the fluorescent protein Dreiklang, responsible for its switching from the fluorescent ON-state to the dark OFF-state. We explore the relief of the charge-transfer excited-state potential energy surface in the ON-state to locate minimum energy conical intersection points with the ground-state energy surface. Simulations of the further evolution of model systems allow us to characterize the ground-state reaction intermediate tentatively suggested in the femtosecond studies of the light-induced dynamics in Dreiklang and finally to arrive at the reaction product. The obtained results clarify the details of the photoswitching mechanism in Dreiklang, which is governed by the chemical modification of its chromophore.

Substituent‐controlled aggregate luminescence: Computational unraveling of S1/S0 surface crossing

AbstractWe computationally investigated the molecular aggregation effects on the excited state deactivation processes by considering both the direct vibrational relaxation and the S0/S1 surface crossing, that is, the minimum energy conical intersection (MECI). Taking classical AIEgens bis(piperidyl)anthracenes (BPAs) isomers and the substituted silole derivatives as examples, we show that the deformation of MECI always occurs at the atom with greater hole/electron overlap. Besides, the energetic and structural changes of MECI caused by substituent has been investigated. We find that effective substituent such as the addition of the electron‐donating groups, which can polarize the distribution of hole/electron density of molecules, will lead to the pyramidalization deformation of MECI occurring at the substituent position and simultaneously reduce the required energy to reach MECI. And MECI is sterically restricted by the surrounding molecules in solid phase, which remarkably hinders the non‐radiative decay through surface crossing. Through quantitative computational assessments of the fluorescence quantum efficiency for both solution and solid phases, we elucidate the role of MECI and its dependence on the substitutions through pyramidalization deformation, which give rise to the aggregation‐induced emission (AIE) phenomenon for 9,10‐BPA, to aggregation‐enhance emission (AEE) behavior for 1,4‐BPA, and to conventional aggregation‐caused quenching (ACQ) for 1,5‐BPA. We further verify such mechanism for siloles, for which we found that the substitutions do not change the AIE behavior. Our findings render a general molecular design approach to manipulating the aggregation effect for optical emission.

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.

Nonadiabatic Dynamics of 1,3-Cyclohexadiene by Curvature-Driven Coherent Switching with Decay of Mixing.

The photoinduced ring-opening reaction of 1,3-cyclohexadiene to produce 1,3,5-hexatriene is a classic electrocyclic reaction and is also a prototype for many reactions of biological and synthetic importance. Here, we simulate the ultrafast nonadiabatic dynamics of the reaction in the manifold of the three lowest valence electronic states by using extended multistate complete-active-space second-order perturbation theory (XMS-CASPT2) combined with the curvature-driven coherent switching with decay of mixing (κCSDM) dynamical method. We obtain an excited-state lifetime of 79 fs, and a product quantum yield of 40% from the 500 trajectories initiated in the S1 excited state. The obtained lifetime and quantum yield values are very close to previously reported experimental and computed values, showing the capability of performing a reasonable nonadiabatic ring-opening dynamics with the κCSDM method that does not require nonadiabatic coupling vectors, time derivatives, or diabatization. In addition, we study the ring-opening reaction by initiating the trajectories in the dark state S2. We also optimize the S0/S1 and S1/S2 minimum-energy conical intersections (MECIs) by XMS-CASPT2; for S1/S2, we optimized both an inner and an outer local-minimum-energy conical intersections (LMECIs). We provide the potential energy profile along the ring-opening coordinate by joining selected critical points via linear synchronous transit paths. We find the inner S1/S2 LMECI to be more crucial than the outer one.

Fast Screening of Minimum Energy Crossing Points with Semiempirical Tight-Binding Methods.

The investigation of photochemical processes is a highly active field in computational chemistry. One research direction is the automated exploration and identification of minimum energy conical intersection (MECI) geometries. However, due to the immense technical effort required to calculate nonadiabatic potential energy landscapes, the routine application of such computational protocols is severely limited. In this study, we will discuss the prospect of combining adiabatic potential energy surfaces from semiempirical quantum mechanical calculations with specialized confinement potential and metadynamics simulations to identify S0/T1 minimum energy crossing point (MECP) geometries. It is shown that MECPs calculated at the GFN2-xTB level can provide suitable approximations to high-level S0/S1ab initio conical intersection geometries at a fraction of the computational cost. Reference MECIs of benzene are studied to illustrate the basic concept. An example application of the presented protocol is demonstrated for a set of photoswitch molecules.

The competitive mechanism between photoisomerization and excited state intramolecular proton transfer process of 2′-Hydroxychalcone system

The photoisomerization and excited state intramolecular proton transfer (ESIPT) are the most common processes in molecular photoswitches. Their coexistence and competing relationship in molecules usually make the deactivation process of the molecules become complex. In the current work, we study the underlying deactivation mechanism of trans-form 2′-hydroxychalcone (2′HC), which possesses both photoisomerization and ESIPT in the excited state, by combining high-level electronic structure calculations and on-the-fly surface hopping dynamics simulations. Four minimum energy conical intersections (MECIs) are found to be involved in the whole deactivation process. Following excitation to the S3 state of the most stable trans-enol tautomer, a strong coupling from the ππ* onto the nπ* state facilitates the ultrafast S3/S1 internal conversion process through an intricate conical intersection seam among S1, S2 and S3 state. The subsequent S1 to S0 state relaxation pathway is divided into two branches. The first process is initiated by twisting motion of C8─C9 bridging bond, which decays directly to the ground state via enol MECI. The other one is triggered by an ultrafast ESIPT process firstly and followed by twisting motion of the C1─C7 bond, leading the system funnel to the ground state by a keto type MECI, and finally generates the precursor of coloured flavanone. The proposed S3→(S2) S1 → S0 two step decay pattern is consistent with previous experimental observations. At the molecular microscopic level, the CO bond alternation motion plays a vital role in the whole relaxation process. Our results indicate that further practical application of 2′HC as molecular photoswitches can be achieved by modifying the molecular structure to enhance the ESIPT process.

Analytic First-Order Derivatives of (X)MS, XDW, and RMS Variants of the CASPT2 and RASPT2 Methods.

Crossings between states involve complex electronic structures, making the accurate characterization of the crossing point difficult. In this study, the analytic derivatives of three complete active space second-order perturbation theory (CASPT2) variants as well as an extension of the restricted active space (RASPT2) are developed. These variants are applied to locating minimum energy conical intersections. Our results demonstrate that the three CASPT2 variants predict qualitatively similar results, but a recently developed variant, the rotated multistate CASPT2 (RMS-CASPT2), is least sensitive to the number of states considered in the calculation. We demonstrate that CASPT2 and the reference self-consistent field calculations predict qualitatively different energetics and bond lengths.

InteraChem: Exploring Excited States in Virtual Reality with Ab Initio Interactive Molecular Dynamics.

InteraChem is an ab initio interactive molecular dynamics (AI-IMD) visualizer that leverages recent advances in virtual reality hardware and software, as well as the graphical processing unit (GPU)-accelerated TeraChem electronic structure package, in order to render quantum chemistry in real time. We introduce the exploration of electronically excited states via AI-IMD using the floating occupation molecular orbital-complete active space configuration interaction method. The optimization tools in InteraChem enable identification of excited state minima as well as minimum energy conical intersections for further characterization of excited state chemistry in small- to medium-sized systems. We demonstrate that finite-temperature Hartree-Fock theory is an efficient method to perform ground state AI-IMD. InteraChem allows users to track electronic properties such as molecular orbitals and bond order in real time, resulting in an interactive visualization tool that aids in the interpretation of excited state chemistry data and makes quantum chemistry more accessible for both research and educational purposes.

Analytical Gradient Theory for Spin-Free State-Averaged Second-Order Driven Similarity Renormalization Group Perturbation Theory (SA-DSRG-MRPT2) and Its Applications for Conical Intersection Optimizations.

Second-order multireference-driven similarity renormalization group perturbation theory (DSRG-MRPT2) provides an efficient means of correcting the dynamical correlation with the multiconfiguration reference function. The state-averaged DSRG-MRPT2 (SA-DSRG-MRPT2) method is the simplest means of treating the excited states with DSRG-MRPT2. In this method, the Hamiltonian dressed with dynamical correlation is diagonalized in the CASCI state subspace (SA-DSRG-MRPT2c) or the configuration subspace (SA-DSRG-MRPT2). This work develops analytical gradient theory for spin-free SA-DSRG-MRPT2(c) with the density-fitting approximation. We check the accuracy of the analytical gradients against the numerical gradients. We present applications for optimizing minimum energy conical intersections (MECI) of ethylene and retinal model chromophores (PSB3 and RPSB6). We investigate the dependence of the optimized geometries and energies on the flow parameters and reference relaxations. The smoothness of the SA-DSRG-MRPT2(c) potential energy surfaces near the reference (complete active space self-consistent field) MECI is comparable to the XMCQDPT2 one. These results render SA-DSRG-MRPT2(c) theory a promising approach for studies of conical intersections.

Machine-Learning Photodynamics Simulations Uncover the Role of Substituent Effects on the Photochemical Formation of Cubanes.

Photochemical [2 + 2]-cycloadditions store solar energy in chemical bonds and efficiently access strained organic molecular architectures. Functionalized [3]-ladderdienes undergo [2 + 2]-photocycloadditions to afford cubanes, a class of strained organic molecules. The substituents (e.g., methyl, trifluoromethyl, and cyclopropyl) affect the overall reactivities of these cubane precursors; the yields range from 1 to 48%. However, the origin of these substituent effects on the reactivities and chemoselectivities is not understood. We now integrate single and multireference calculations and machine-learning-accelerated nonadiabatic molecular dynamics (ML-NAMD) to understand how substituents affect the ultrafast dynamics and mechanism of [2 + 2]-photocycloadditions. Steric clashes between substituent groups destabilize the 4π-electrocyclic ring-opening pathway and minimum energy conical intersections by 0.72-1.15 eV and reaction energies by 0.68-2.34 eV. Noncovalent dispersive interactions stabilize the [2 + 2]-photocycloaddition pathway; the conical intersection energies are lower by 0.31-0.85 eV, and the reaction energies are lower by 0.03-0.82 eV. The 2 ps ML-NAMD trajectories reveal that closed-shell repulsions block a 6π-conrotatory electrocyclic ring-opening pathway with increasing steric bulk. Thirty-eight percent of the methyl-substituted [3]-ladderdiene trajectories proceed through the 6π-conrotatory electrocyclic ring-opening, whereas the trifluoromethyl- and cyclopropyl-substituted [3]-ladderdienes prefer the [2 + 2]-photocycloaddition pathways. The predicted cubane yields (H: 0.4% < CH3: 1% < CF3: 14% < cPr: 15%) match the experimental trend; these substituents predistort the reactants to resemble the conical intersection geometries leading to cubanes.

Comparative study of the multistate ESIPT in dihydroxyanthraquinones

We examine the spatio-energetic and dynamics aspects of the S1-S2 conical intersection in positional isomers of dihydroxyanthraquinone – alizarin, anthrarufin, chrysazin and quinizarin – by employing the linear vibronic coupling approach. The S0→ S1 and S0→ S2 transitions are dipole-allowed (bright) and dipole-forbidden (dark), respectively, in alizarin, anthrarufin and quinizarin, while for chrysazin, the reverse holds. Geometries associated with the minimum energy conical intersection and the Franck–Condon point are near-identical in these molecules, wherein slight displacements of O-H vibrations are sufficient for the molecule to reach the conical intersection immediately after photoexcitation. The wavepacket initially evolving on the “bright” state bifurcates at the conical intersection; consequently, a portion of the wavepacket moves to the “dark” state of the molecule. Our findings suggest two intramolecular proton transfer pathways in these molecules, one via the “bright” state and the other via the “dark” state. We also discuss the broad emission features originating from excited-state single and double proton-transferred tautomers of these molecules.

Analytical Gradient Theory for Resolvent-Fitted Second-Order Extended Multiconfiguration Perturbation Theory (XMCQDPT2).

We present the formulation and implementation of an analytical gradient algorithm for extended multiconfiguration quasidegenerate perturbation theory (XMCQDPT2) with the resolvent-fitting approximation by Granovsky. This algorithm is powerful when optimizing molecular configurations with a moderate-sized active space and many electronic states. First, we present the powerfulness and accuracy of resolvent-fitting approximations compared to canonical XMCQDPT2 theory. Then, we demonstrate the utility of the current algorithm in frequency analyses, optimizing the minimum energy conical intersection geometries of the retinal chromophore model RPSB6 and evaluating nuclear gradients when there are many electronic states. Furthermore, we parallelize the algorithm using the OpenMP/MPI hybrid approach. Additionally, we report the computational cost and parallel efficiency of the program.

Switching the relaxation pathway by steric effects in conjugated dienes

Modification of a molecular structure at the minimum-energy conical intersection (MECI) on the relaxation pathway of the smallest conjugated polyene, trans-1,3-butadiene (BD), was achieved by substituting the hydrogen atoms at the end of the molecular chain with methyl groups. At the MECI between S1 and S0, BD is known to have a pyramidalized structure; therefore, the substituted methyl groups are expected to hinder this pyramidalization and change the relaxation pathway. Here, the relaxation dynamics of three conjugated dienes: BD, trans-1,3-pentadiene (PD), and 2,5-dimethyl-2,4-hexadiene (HD), were investigated by time-resolved photoelectron spectroscopy (TRPES) using single-order high harmonic pulses. Ultrafast relaxation to the ground state with decay times of less than 100 fs was observed in all three molecules. The potential energy curves from the theoretical calculation show the common features among these dienes, which explains the similar relaxation dynamics. However, HD was found theoretically to have a transoid structure at MECI between S1 and S0, while BD and PD both had pyramidalized structures. The relaxation pathway switching was corroborated experimentally by analysis of the photoelectron spectra specifically appearing in HD after a few hundred femtoseconds upon photoexcitation. The larger twist in the transoid structure stimulates molecular vibrations that modulated the photoionization probabilities.

Unraveling the Mechanism for Tuning the Fluorescence of Fluorescein Derivatives: The Role of the Conical Intersection and nπ* State.

Although a large number of fluorescein derivatives have been developed and applied in many different fields, the general mechanisms for tuning the fluorescence of fluorescein derivatives still remain uncovered. Herein, we found that the fluorescence quenching of neutral form of fluorescein derivatives in acidic medium resulted from a dark nπ* state, whereas the fluorescence of the anionic form of fluorescein derivatives in the gas phase and alkaline solutions was tuned by minimal energy conical intersection (MECI). The formation of MECI involved significant rotation of benzene ring and flip-flop motion of xanthene moiety, which would be restricted by intermolecular hydrogen bonding and lowering temperature. The energy barrier for reaching MECI depended on the substituents in the benzene moiety in accordance with experimentally observed substituent effects. These unprecedented mechanisms would lead to a recognition of fluorescein derivatives and could provide a correct and instructive design strategy for further developing new fluorescein derivatives.

Photo-dissociation mechanism of trifluoroacetyl chloride in the gas phase: AIMS dynamic simulations.

In this article, the structures and energies of CF3COCl in the low-lying electronic states have been determined by SA-2-CAS(8,7)/6-31G* and SA-2-MSPT2(8,7)/6-31G* calculations, which include equilibrium geometries, transition states, and three minimum-energy conical intersections (CI-1, CI-2, and CI-3) between S0 and S1 states. The AIMS method was used to carry out non-adiabatic dynamic simulations with the ab initio calculation performed at the SA-2-CAS(8,7)/6-31G* level. Upon irradiation to the S1 state, CF3COCl first relaxes to S1 minimum and then overcomes the ∼10 kcal/mol (TSS1_CCl) or ∼30 kcal/mol (TSS1_CO) barrier to the conical intersection region CI-1 or CI-3 (minor), with the S1 → S0 transition probability of 63:1. After non-adiabatic transition to the S0 state through CI-1, trajectories mainly distribute to three different reaction pathways, with one going back to S0 minimum through shortening of the C-Cl bond, the other forming CF3CO and Cl radicals by continuous elongation of the C-Cl distance, and another dissociating into CF3 + CO + Cl and running into the CI-3 region through elongation of C-C and C-Cl distances. Moreover, we found that the trajectories would recross to the S1 state with the recrossing probability of 13.9% through the CI-3 region due to the extremely sloped topographic character of CI-3. On the basis of time evolution of wavefunctions simulated here, the product ratio of CF3 + CO + Cl and CF3CO + Cl is 53.5%:18.4%, which is consistent with the experimental value of 3:1. We further explain the photo-dissociation wavelength dependence of CF3COCl, and the product ratio of CF3 + CO + Cl increases with the increase in total energy.

Exploring the effects of quantum decoherence on the excited-state dynamics of molecular systems

Fewest-switches surface hopping (FSSH) is employed in order to investigate the nonadiabatic excited-state dynamics of thiophene and related compounds and hence to establish a connection between the electronic system, the critical points in configuration space and the deactivation dynamics. The potential-energy surfaces of the studied molecules were calculated with complete active space self-consistent field and time-dependent density-functional theory. They are analyzed thoroughly to locate and optimize minimum-energy conical intersections, which are essential to the dynamics of the system. The influence of decoherence on the dynamics is examined by employing different decoherence schemes. We find that irrespective of the employed decoherence algorithm, the population dynamics of thiophene give results which are sound with the expectations grounded on the analysis of the potential-energy surface. A more detailed look at single trajectories as well as on the excited-state lifetimes, however, reveals a substantial dependence on how decoherence is accounted for. In order to connect these findings, we describe how ensemble averaging cures some of the overcoherence problems of uncorrected FSSH. Eventually, we identify carbon–sulfur bond cleavage as a common feature accompanying electronic transitions between different states in the simulations of all thiophene-related compounds studied in this work, which is of interest due to their relevance in organic photovoltaics.

Strong static and dynamic Jahn-Teller and pseudo-Jahn-Teller effects in niobium tetrafluoride.

We present a first-principles study of the static and dynamic aspects of the strong Jahn-Teller (JT) and pseudo-JT (PJT) effects in niobium tetrafluoride, NbF4, in the manifold of its electronic ground state, 2E, and its first excited state, 2T2. The complex topography of the full-dimensional multi-sheeted adiabatic JT/PJT surfaces is analyzed computationally at the complete-active-space self-consistent-field (CASSCF) and multireference second-order perturbation levels of electronic structure theory, providing a detailed characterization of minima, saddle points, and minimum-energy conical intersection points. The calculations reveal that the tetrahedral (Td) configuration of NbF4 undergoes strong JT distortions along the bending mode of e symmetry, yielding tetragonal molecular structures of D2d symmetry with Td → D2d stabilization energies of about 2000 cm-1 in the X̃2E state and about 6400 cm-1 in the Ã2T2 state. In addition, there exists strong X̃2E-Ã2T2 PJT coupling via the bending mode of t2 symmetry, which becomes important near the crossing seam of the X̃2E and Ã2T2 potential energy surfaces. A five-state five-mode JT/PJT vibronic-coupling Hamiltonian is constructed in terms of symmetry-invariant polynomial expansions of the X̃2E and Ã2T2 diabatic potential energy surfaces in the e and t2 bending coordinates. The parameters of the Hamiltonian are determined by a least-squares fit of its eigenvalues to the CASSCF ab initio data. The vibronic spectra and the time evolution of adiabatic electronic population probabilities are computed with the multi-configuration time-dependent Hartree method. The complexity of the spectra reflects the effects of the exceptionally strong E × e and T2 × e JT couplings and (E + T2) × (e + t2) PJT coupling. The time evolution of the populations of the adiabatic electronic states after the initial preparation of the Ã2T2 state reveals the femtosecond nonadiabatic dynamics through a multidimensional seam of conical intersection. These results represent the first study of the static and dynamical JT/PJT effects in the X̃2E and Ã2T2 electronic states of NbF4.

Optimization of Three State Conical Intersections by Adaptive Penalty Function Algorithm in Connection with the Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory Method (MRSF-TDDFT).

A new adaptive algorithm for penalty function optimization for minimum-energy three-states conical intersections (ME3CI) is suggested. The new algorithm differs from the original penalty function algorithm by (a) removing the redundancy in the target function, (b) using an adaptive increment for the penalty function weighting factor, and (c) using tighter convergence criteria for the energy gap. The latter was introduced to guarantee convergence to a true conical intersection rather than to a narrowly avoided crossing geometry. The new algorithm was tested in the optimization of the ME3CI geometries in butadiene and malonaldehyde, where all of the previously found true ME3CI geometries were recovered. The previously found butadiene's CI3/2/1 turned out to be a narrowly avoided crossing. For butadiene, seven new ME3CI geometries have been located. Because of the removal of the redundancy and the use of the adaptive weighting factor, the convergence rate of the new algorithm is noticeably improved as compared to that of the previously proposed penalty function algorithm. The application to malonaldehyde and butadiene demonstrates that the three-state conical intersections may be more abundant and hence more involved in the photochemistry than previously thought. The recently developed mixed-reference spin flip (MRSF)-TDDFT method yields ME3CI geometries and relative energies quantitatively consistent with the previously reported calculations at a much reduced computational cost.