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Jeffrey R Reimers

Publications and source records attributed to Jeffrey R Reimers.

18 recordsLinked to original sources

The symmetry of single-molecule conduction.

We introduce the conductance point group which defines the symmetry of single-molecule conduction within the nonequilibrium Green's function formalism. It is shown, either rigorously or to within a very good approximation, to correspond to a molecular-conductance point group defined purely in terms of the properties of the conducting molecule. This enables single-molecule conductivity to be described in terms of key qualitative chemical descriptors that are independent of the nature of the molecule-conductor interfaces. We apply this to demonstrate how symmetry controls the conduction through 1,4-benzenedithiol chemisorbed to gold electrodes as an example system, listing also the molecular-conductance point groups for a range of molecules commonly used in molecular electronics research.

Electric Conductivity↗

Charge delocalization in the special-pair radical cation of mutant reaction centers of Rhodobacter sphaeroides from Stark spectra and nonadiabatic spectral simulations.

Stark and absorption spectra for the hole-transfer band of the bacteriochlorophyll special pair in the wild-type and L131LH, M160LH, and L131LH/M160LH mutants of the bacterial reaction center of Rhodobacter sphaeroides are presented, along with extensive analyses based on nonadiabatic spectral simulations. Dramatic changes in the Stark spectra are induced by the mutations, changes that are readily interpreted in terms of the redox-energy asymmetry and degree of charge localization in the special-pair radical cation. The effect of mutagenesis on key properties such as the electronic coupling within the special pair and the reorganization energy associated with intervalence hole transfer are determined for the first time. Results for the L131LH and M160LH/L131LH mutants indicate that these species can be considered as influencing the special pair primarily through modulation of the redox asymmetry, as is usually conceptualized, but M160LH is shown to develop a wide range of effects that can be interpreted in terms of significant mutation-induced structural changes in and around the special pair. The nonadiabatic spectra simulations are performed using both a simple two-state 1-mode and an extensive four-state 70-mode model, which includes the descriptions of additional electronic states and explicitly treats the major vibrational modes involved. Excellent agreement between the two simulation approaches is obtained. The simple model is shown to reproduce key features of the Stark effect of the main intervalence transition, while the extensive model quantitatively reproduces most features of the observed spectra for both the electronic and the phase-phonon regions, thus giving a more comprehensive description of the effect of the mutations on the properties of the special-pair radical cation. These results for a series of closely related mixed-valence complexes show that the Stark spectra provide a sensitive indicator for the properties of the mixed-valence complexes and should serve as an instructive example on the application of nonadiabatic simulations to the study of mixed-valence complexes in general as well as other chemical systems akin to the photosynthetic special pair.

Absorption↗

Density functional theory for charge transfer: the nature of the N-bands of porphyrins and chlorophylls revealed through CAM-B3LYP, CASPT2, and SAC-CI calculations.

While density functional theory (DFT) has been proven to be extremely useful for the prediction of thermodynamic and spectroscopic properties of molecules, to date most functionals used in common implementations of DFT display a systematic failure to predict the properties of charge-transfer processes. While this is explicitly manifest in Rydberg transitions of atoms and molecules and in molecular charge-transfer spectroscopy, it also becomes critical for systems containing extended conjugation such as polyenes and other conducting polymers, porphyrins, chlorophylls, etc. A new density functional, a Coulomb-attenuated hybrid exchange-correlation functional (CAM-B3LYP), has recently been developed specifically to overcome these limitations, and it has been shown to properly predict molecular charge-transfer spectra. Here, we demonstrate that it predicts qualitatively reasonable spectra for porphyrin, some oligoporphyrins, and chlorophyll. However, alternate density functionals developed to overcome the same limitations such as current-density functional theory are shown, in their present implementation, to remain inadequate. The CAM-B3LYP results are shown to be in excellent agreement with complete-active-space plus second-order Møller-Plesset perturbation theory and symmetry-adapted cluster configuration interaction calculations: These depict the N and higher bands of porphyrins and chlorophylls as being charge-transfer bands associated with localization of molecular orbitals on individual pyrrole rings. The validity of the basic Gouterman model for the spectra of porphyrins and chlorophylls is confirmed, rejecting modern suggestions that non-Gouterman transitions lie close in energy to the Q-bands of chlorophylls. As porphyrins and chlorophylls provide useful paradigms for problems involving extended conjugation, the results obtained suggest that many significant areas of nanotechnology and biotechnology may now be realistically treated by cost-effective density-functional-based computational methods.

Journal Article↗

Understanding the inelastic electron-tunneling spectra of alkanedithiols on gold.

We present results for a simulated inelastic electron-tunneling spectra (IETS) from calculations using the "gDFTB" code. The geometric and electronic structure is obtained from calculations using a local-basis density-functional scheme, and a nonequilibrium Green's function formalism is employed to deal with the transport aspects of the problem. The calculated spectrum of octanedithiol on gold(111) shows good agreement with experimental results and suggests further details in the assignment of such spectra. We show that some low-energy peaks, unassigned in the experimental spectrum, occur in a region where a number of molecular modes are predicted to be active, suggesting that these modes are the cause of the peaks rather than a matrix signal, as previously postulated. The simulations also reveal the qualitative nature of the processes dominating IETS. It is highly sensitive only to the vibrational motions that occur in the regions of the molecule where there is electron density in the low-voltage conduction channel. This result is illustrated with an examination of the predicted variation of IETS with binding site and alkane chain length.

Journal Article↗

Density-functional geometry optimization of the 150,000-atom photosystem-I trimer.

We present a linear-scaling method based on the use of density-functional theory (DFT) for the system-wide optimization of x-ray structural coordinates and apply it to optimize the 150,000 atoms of the photosystem-I (PS-I) trimer. The method is based on repetitive applications of a multilevel ONIOM procedure using the PW916-31G(d) DFT calculations for the high level and PM3 for the lower level; this method treats all atoms in the structure equivalently, a structure in which the majority of the atoms can be considered as part of some internal "active site." To obtain a realistic single structure, some changes to the original protein model were necessary but these are kept to a minimum in order that the optimized structure most closely resembles the original x-ray one. Optimization has profound effects on the perceived electronic properties of the cofactors, with, e.g., optimization lowering the internal energy of the chlorophylls by on average 53 kcal mol(-1) and eliminates the enormous 115 kcal mol(-1) energy spread depicted by the original x-ray heavy-atom coordinates. A highly precise structure for PS-I results that is suitable for analysis of device function. Significant qualitative features of the structure are also improved such as correction of an error in the stereochemistry of one of the chlorophylls in the "special pair" of the reaction center, as well as the replacement of a water molecule with a metal cation in a critical region on the C3 axis. The method also reveals other unusual features of the structure, leading both to suggestions concerning device functionality and possible mutations between gene sequencing and x-ray structure determination. The optimization scheme is thus shown to augment the molecular modeling schemes that are currently used to add medium-resolution structural information to the raw scattering data in order to obtain atomically resolved structures. System-wide optimization is now a feasible process and its use within protein x-ray data refinement should be considered.

Binding Sites↗

Molecular origins of conduction channels observed in shot-noise measurements.

Measurements of shot noise from single molecules have indicated the presence of various conduction channels. We present three descriptions of these channels in molecular terms showing that the number of conduction channels is limited by bottlenecks in the molecule and that the channels can be linked to transmission through different junction states. We introduce molecular-conductance orbitals, which allow the transmission to be separated into contributions from individual orbitals and contributions from interference between pairs of orbitals.

Electric Conductivity↗

Coexistence of multiple conformations in cysteamine monolayers on Au(111).

The structural organization, catalytic function, and electronic properties of cysteamine monolayers on Au(111) have been addressed comprehensively by voltammetry, in situ scanning tunneling microscopy (STM) in anaerobic environment, and a priori molecular dynamics (MD) simulation and STM image simulation. Two sets of voltammetric signals are observed. One peak at -(0.65-0.70) V (SCE) is caused by reductive desorption of cysteamine. The other signal, at -(0.25-0.40) V consists of a peak doublet. The pH dependence of the latter suggests that the origin is catalytic dihydrogen evolution. The doublet feature is indicative of two distinct cysteamine configurations. Cysteamine monolayer formation from initial nucleation to a highly ordered phase has been successfully observed in real time using oxygen-free in situ STM. Random cellular patterns, disordered adlayer formation accompanied by high step edge mobility, and ultimately a highly ordered (square root 3 x 4) R30 degrees lattice are observed sequentially. Pits are formed due to enclosure of the mobile edges during the adsorption process. In the highly ordered cysteamine layer, each unit has two spots with apparent 0.6 A height difference in STM images. The coverage 5.7 +/- 0.1 x 10(-10) mol cm(-2) determined by voltammetry supports that the spots represent two individual cysteamine molecules. A priori MD and density functional simulations hold other clues to the image interpretation and indicate that the NH(3)(+) groups dominate the tunneling contrast. A wide range of interface structures, showing variations in the sulfur binding site and orientation, gauche and trans conformers, and especially hydrogen-bonding interactions, are examined, from which it is concluded that the adsorbate structure is controlled by interactions with the solvent rather than with the substrate.

Computer Simulation↗

Overcoming computational uncertainties to reveal chemical sensitivity in single molecule conduction calculations.

In the calculation of conduction through single molecule's approximations about the geometry and electronic structure of the system are usually made in order to simplify the problem. Previously [G. C. Solomon, J. R. Reimers, and N. S. Hush, J. Chem. Phys. 121, 6615 (2004)], we have shown that, in calculations employing cluster models for the electrodes, proper treatment of the open-shell nature of the clusters is the most important computational feature required to make the results sensitive to variations in the structural and chemical features of the system. Here, we expand this and establish a general hierarchy of requirements involving treatment of geometrical approximations. These approximations are categorized into two classes: those associated with finite-dimensional methods for representing the semi-infinite electrodes, and those associated with the chemisorption topology. We show that ca. 100 unique atoms are required in order to properly characterize each electrode: using fewer atoms leads to nonsystematic variations in conductivity that can overwhelm the subtler changes. The choice of binding site is shown to be the next most important feature, while some effects that are difficult to control experimentally concerning the orientations at each binding site are actually shown to be insignificant. Verification of this result provides a general test for the precision of computational procedures for molecular conductivity. Predictions concerning the dependence of conduction on substituent and other effects on the central molecule are found to be meaningful only when they exceed the uncertainties of the effects associated with binding-site variation.

Journal Article↗

First singlet (n,pi*) excited state of hydrogen-bonded complexes between water and pyrimidine.

Hydrogen bonds from water to excited-state formaldehyde and from water to excited-state pyridine have been shown to display novel motifs to traditional hydrogen bonds involving ground states, with, in particular for H2O:pyridine, strong interactions involving the electron-rich pi cloud dominating the (n,pi) excited state. We investigate H2O:pyrimidine and various dihydrated species and reveal another motif, one in which the hydrogen bonding can dramatically alter the electronic structure of the excited state. Such effects are rare for ground-state interactions for which hydrogen bonding usually acts to merely perturb the electronic structure of the participating molecules. It arises as the (n,pi*) excitation of isolated pyrimidine is delocalized over both nitrogens but asymmetric hydrogen bonding causes it to localize on just the noninteracting atom. As a result, the excited-state hydrogen bond in H2O:pyrimidine is suprisingly very similar to the ground-state structure. These results lead to an improved understanding of the spectroscopy of pyrimidine in liquid water, and to the prediction that stable excited-state hydrogen bonds in H2O:pyrimidine should be observable, despite failure of experiments to actually do so. They also provide a simple model for the intricate control over primary charge separation in photosynthesis exerted by hydrogen bonding, and for solvent-induced electron localization in symmetric mixed-valence complexes. All conclusions are based on strong parallels found between the results of calculations performed using density-functional theory (DFT) and time-dependent DFT (TDDFT), complete-active-space self-consistent-field (CASSCF) with second-order perturbation-theory correction (CASPT2) theory, and equation-of-motion coupled cluster (EOM-CCSD) theory, calculations that are verified through detailed comparison of computed properties with experimental data for both the isolated molecules and the ground-state hydrogen bond.

Journal Article↗

The structure, energetics, and nature of the chemical bonding of phenylthiol adsorbed on the Au(111) surface: implications for density-functional calculations of molecular-electronic conduction.

The adsorption of phenylthiol on the Au(111) surface is modeled using Perdew and Wang density-functional calculations. Both direct molecular physisorption and dissociative chemisorption via S-H bond cleavage are considered as well as dimerization to form disulfides. For the major observed product, the chemisorbed thiol, an extensive potential-energy surface is produced as a function of both the azimuthal orientation of the adsorbate and the linear translation of the adsorbate through the key fcc, hcp, bridge, and top binding sites. Key structures are characterized, the lowest-energy one being a broad minimum of tilted orientation ranging from the bridge structure halfway towards the fcc one. The vertically oriented threefold binding sites, often assumed to dominate molecular electronics measurements, are identified as transition states at low coverage but become favored in dense monolayers. A similar surface is also produced for chemisorption of phenylthiol on Ag(111); this displays significant qualitative differences, consistent with the qualitatively different observed structures for thiol chemisorption on Ag and Au. Full contours of the minimum potential energy as a function of sulfur translation over the crystal face are described, from which the barrier to diffusion is deduced to be 5.8 kcal mol(-1), indicating that the potential-energy surface has low corrugation. The calculated bond lengths, adsorbate charge and spin density, and the density of electronic states all indicate that, at all sulfur locations, the adsorbate can be regarded as a thiyl species that forms a net single covalent bond to the surface of strength 31 kcal mol(-1). No detectable thiolate character is predicted, however, contrary to experimental results for alkyl thiols that indicate up to 20%-30% thiolate involvement. This effect is attributed to the asymptotic-potential error of all modern density functionals that becomes manifest through a 3-4 eV error in the lineup of the adsorbate and substrate bands. Significant implications are described for density-functional calculations of through-molecule electron transport in molecular electronics.

Journal Article↗

Wavepacket propagation using time-sliced semiclassical initial value methods.

A new semiclassical initial value representation (SC-IVR) propagator and a SC-IVR propagator originally introduced by Kay [J. Chem. Phys. 100, 4432 (1994)], are investigated for use in the split-operator method for solving the time-dependent Schrodinger equation. It is shown that the SC-IVR propagators can be derived from a procedure involving modified Filinov filtering of the Van Vleck expression for the semiclassical propagator. The two SC-IVR propagators have been selected for investigation because they avoid the need to perform a coherent state basis set expansion that is necessary in other time-slicing propagation schemes. An efficient scheme for solving the propagators is introduced and can be considered to be a semiclassical form of the effective propagators of Makri [Chem. Phys. Lett. 159, 489 (1989)]. Results from applications to a one-dimensional, two-dimensional, and three-dimensional Hamiltonian for a double-well potential are presented.

Journal Article↗

Single molecule conductivity: the role of junction-orbital degeneracy in the artificially high currents predicted by ab initio approaches.

A priori evaluations, using Hartree-Fock self-consistent-field (SCF) theory or density-functional theory (DFT), of the current passing between two electrodes through a single bridging molecule result in predicted conductivities that may be up to one to two orders of magnitude larger than observed ones. We demonstrate that this is, in part, often due to the improper application of the computational methods. Conductivity is shown to arise from tunneling between junction states of the electrodes through the molecule; these states are inherently either quasi two-fold or four-fold degenerate and always comprise the (highest occupied molecular orbital) HOMO band at the Fermi energy of the system. Frequently, in previous cluster based molecular conduction calculations, closed-shell SCF or Kohn-Sham DFT methods have been applied to systems that we demonstrate to be intrinsically open shell in nature. Such calculations are shown to induce artificial HOMO-LUMO (LUMO-lowest unoccupied molecular orbital) band splittings that Landauer-based formalisms for steady-state conduction interpret as arising from extremely rapid through-molecule tunneling at the Fermi energy, hence, overestimating the low-voltage conductivity. It is demonstrated that these shortcomings can be eliminated, dramatically reducing calculated current magnitudes, through the alternate use of electronic-structure calculations based on the spin-restricted open-shell formalism and related multiconfigurational SCF of DFT approaches. Further, we demonstrate that most anomalies arising in DFT implementations arise through the use of hybrid density functionals such as B3LYP. While the enhanced band-gap properties of these functionals have made them the defacto standard in molecular conductivity calculations, we demonstrate that it also makes them particularly susceptible to open-shell anomalies.

Journal Article↗

A unified description of the electrochemical, charge distribution, and spectroscopic properties of the special-pair radical cation in bacterial photosynthesis.

We apply our four-state 70-vibration vibronic-coupling model for the properties of the photosynthetic special-pair radical cation to: (1) interpret the observed correlations between the midpoint potential and the distribution of spin density between the two bacteriochlorophylls for 30 mutants of Rhodobacter sphaeroides, (2) interpret the observed average intervalence hole-transfer absorption energies as a function of spin density for six mutants, and (3) simulate the recently obtained intervalence electroabsorption Stark spectrum of the wild-type reaction center. While three new parameters describing the location of the sites of mutation with respect to the special pair are required to describe the midpoint-potential data, a priori predictions are made for the transition energies and the Stark spectrum. In general, excellent predictions are made of the observed quantities, with deviations being typically of the order of twice the experimental uncertainties. A unified description of many chemical and spectroscopic properties of the bacterial reaction center is thus provided. Central to the analysis is the assumption that the perturbations made to the reaction center, either via mutations of protein residues or by application of an external electric field, act only to independently modify the oxidation potentials of the two halves of the special pair and hence the redox asymmetry E0. While this appears to be a good approximation, clear evidence is presented that effects of mutation can be more extensive than what is allowed for. A thorough set of analytical equations describing the observed properties is obtained using the Born-Oppenheimer adiabatic approximation. These equations are generally appropriate for intervalence charge-transfer problems and include, for the first time, full treatment of both symmetric and antisymmetric vibrational motions. The limits of validity of the adiabatic approach to the full nonadiabatic problem are obtained.

Journal Article↗

Synthesis and physical properties of biquinoxalinyl bridged bis-porphyrins: models for aspects of photosynthetic reaction centres.

The synthesis of biquinoxalinyl-bridged bis-porphyrin 4 and metallated derivatives 5-11 was achieved in high yields. UV-visible spectroscopy and electrochemical experiments indicated weak orbital coupling of the two quinoxalinyl units but minimal orbital coupling between the two porphyrins. The weak electronic communication is attributed to non-planarity, on average, of the molecule because of rotation about the inter-quinoxalinyl connection. This, combined with poor coupling across the fused junctions between the porphyrin and quinoxalinyl units, results in minimal inter-porphyrin communication. As a result, the biquinoxalinyl linkage is appropriate for inclusion in more elaborate synthetic compounds, such as the tris-porphyrin 1, that are designed to model the charge-separation apparatus of Photosynthetic Reaction Centres.

Journal Article↗

The appropriateness of density-functional theory for the calculation of molecular electronics properties.

As molecular electronics advances, efficient and reliable computation procedures are required for the simulation of the atomic structures of actual devices, as well as for the prediction of their electronic properties. Density-functional theory (DFT) has had widespread success throughout chemistry and solid-state physics, and it offers the possibility of fulfilling these roles. In its modern form it is an empirically parameterized approach that cannot be extended toward exact solutions in a prescribed way, ab initio. Thus, it is essential that the weaknesses of the method be identified and likely shortcomings anticipated in advance. We consider four known systematic failures of modern DFT: dispersion, charge transfer, extended pi conjugation, and bond cleavage. Their ramifications for molecular electronics applications are outlined and we suggest that great care is required when using modern DFT to partition charge flow across electrode-molecule junctions, screen applied electric fields, position molecular orbitals with respect to electrode Fermi energies, and in evaluating the distance dependence of through-molecule conductivity. The causes of these difficulties are traced to errors inherent in the types of density functionals in common use, associated with their inability to treat very long-range electron correlation effects. Heuristic enhancements of modern DFT designed to eliminate individual problems are outlined, as are three new schemes that each represent significant departures from modern DFT implementations designed to provide a priori improvements in at least one and possible all problem areas. Finally, fully semiempirical schemes based on both Hartree-Fock and Kohn-Sham theory are described that, in the short term, offer the means to avoid the inherent problems of modern DFT and, in the long term, offer competitive accuracy at dramatically reduced computational costs.

Computer Simulation↗

Examination of the photophysical processes of chlorophyll d leading to a clarification of proposed uphill energy transfer processes in cells of Acaryochloris marinas.

A comprehensive study of the photophysical properties of chlorophyll (Chl) d in 1:40 acetonitrile-methanol solution is performed over the temperature range 170-295 K. From comparison of absorption and emission spectra, time-dependent density-functional calculations and homologies with those of Chl a, we assign the key features of the absorption and fluorescence spectra. Possible photophysical energy relaxation mechanisms are summarized, and thermal equilibration processes are studied in detail by monitoring the observed emission profiles and quantum yields as a function of excitation energy. In particular, we concentrate on emission subsequent to excitation in the extreme far-red tail of the Qy absorption spectrum, with this emission partitioned into contributions from hot-band absorptions as well as uphill energy transfer processes that occur subsequent to absorption. No unusual photophysical processes are detected for Chl d; it appears that all intramolecular relaxation processes reach thermal equilibration on shorter timescales than the fluorescence lifetime even at 170 K. The results from these studies are used to reinterpret a previous study of photochemical processes observed in intact cells and their acetone extracts of the photosynthetic system of Acaryochloris marina. In the study of Mimuro et al., light absorbed by Chl d at 736 nm is found to give rise to emission by another species, believed to also be Chl d, at 703 nm; this uphill energy transfer process is easily rationalized in terms of the thermal equilibration processes that we deduced for Chl d. However, no evidence is found in the experimental results of Mimuro et al. to support claims that (nonequilibrium) uphill energy transfer is additionally observed to Chl a species that emit at 670-680 nm. This finding is relevant to broader issues concerning the nature of the special pair in photosystem II of A. marina because suggestions that it is comprised of Chl a can only be correct if nonthermal uphill energy transfer processes from Chl d are operative.

Chlorophyll↗

Switchable electronic coupling in model oligoporphyrin molecular wires examined through the measurement and assignment of electronic absorption spectra.

The use of a quinone functionality in the linkage unit of laterally bridged oligoporphyrins as a switch for controlling electronic coupling between the termini is examined. The quinone-bridged bisporphyrin P(2)TA-O(2) was synthesized by condensation of 2 equiv of the dione 2,3-dioxo-5,10,15,20-tetrakis(3,5-di-tert-butylphenyl)chlorin with 2,3,5,6-tetraamino-1,4-benzoquinone. The electronic absorption spectra of P(2)TA-O(2) and its conjugated benzenoid analogue P(2)TA are measured and assigned, in conjunction with the spectra of the fragment monomers and porphyrin-bridge compounds. Band homologies and CASPT2 calculations are used to make the assignments. Chemically, the dimer in one case is bridged by a through-conjugated, pi-delocalized 1,4,5,8-tetraazaanthracene molecule. This is shown to display significant inter-porphyrin coupling, with an observed difference in the exciton couplings of the B(x) and B(y) bands being ca. 0.18 eV. However, the other dimer is bridged using a derivative in which the central ring is converted to a cross-conjugated, pi-localized quinonoid form; this molecule displays no observable inter-porphyrin coupling. This scenario provides a paradigm for the use of molecular electronic devices in sensing, control, and high-capacity relatively low-speed data storage applications.

Journal Article↗

An atomistic approach to conduction between nanoelectrodes through a single molecule.

Capacitance and other properties of nanoelectrodes, finite-size metal clusters envisaged for use in complex molecular-electronic devices, are discussed. The applicability of classical electrostatics (Coulomb's and Gauss' law, Poisson's equation, etc.) to atomistic systems is investigated and the self-energy necessary to store a finite charge on an atom is found to be of central importance. In particular, the neglect of electron exchange is found to introduce severe limitations, with quantum calculations predicting fundamentally different electronic structures. Also, the well-known poor representation of the atomic self-energy inherent to modern DFT is discussed, along with its implications for molecular electronics calculations. An INDO/S method is introduced with new parameters for gold. This is the simplest approximate computational scheme that correctly includes quantum electrostatic, resonance, and spin effects, and is capable of describing arbitrary excited electronic states. Encouraging results are obtained for some trial problems. In particular, voltage differential between the electrodes in electrode-molecule-electrode conduction is obtained, not through an a priori prescription but rather by moving whole electrons between the electrodes and analyzing the response. The voltage drops across the molecule-electrode junctions and the central molecular region are then deduced. This alternative to the current Landauer-based 1-particle transmission equations for electrode-molecule-electrode conduction is discussed in terms of the use of the electronic states of the system. It provides a proper description not only of conduction via electrode-to-molecule charge or hole transfer but also of conduction via simultaneous charge and hole transfer via low-lying excited molecular electronic states, including the ability to account for electroluminescence and other chemical effects. In addition, various aspects of our research on the quantitative prediction of the I(V) curves for electrode-molecule-electrode conduction are reviewed, including demonstration of the equivalence of the formalisms generated by the Datta and the Mujica-Ratner groups, and the development of analytically solvable paradigms, including the conduction through a linear-chain Hückel wire.

Electric Capacitance↗