A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initioMO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initioSCF–MO calculations.

Optimization of equilibrium geometries and transition structures

A method is proposed for carrying out molecular dynamics simulations of processes that involve electronic transitions. The time dependent electronic Schrodinger equation is solved self‐consistently with the classical mechanical equations of motion of the atoms. At each integration time step a decision is made whether to switch electronic states, according to probabilistic ‘‘fewest switches’’ algorithm. If a switch occurs, the component of velocity in the direction of the nonadiabatic coupling vector is adjusted to conserve energy. The procedure allows electronic transitions to occur anywhere among any number of coupled states, governed by the quantum mechanical probabilities. The method is tested against accurate quantal calculations for three one‐dimensional, two‐state models, two of which have been specifically designed to challenge any such mixed classical–quantal dynamical theory. Although there are some discrepancies, initial indications are encouraging. The model should be applicable to a wide variety of gas‐phase and condensed‐phase phenomena occurring even down to thermal energies.

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Molecular dynamics with electronic transitions

6. Rehybridization of the Acceptor (RICT) 3908 7. Planarization of the Molecule (PICT) 3909 III. Fluorescence Spectroscopy 3909 A. Solvent Effects and the Model Compounds 3909 1. Solvent Effects on the Spectra 3909 2. Steric Effects and Model Compounds 3911 3. Bandwidths 3913 4. Isoemissive Points 3914 B. Dipole Moments 3915 C. Radiative Rates and Transition Moments 3916 1. Quantum Yields and Radiationless Deactivation Processes 3916

Structural Changes Accompanying Intramolecular Electron Transfer: Focus on Twisted Intramolecular Charge-Transfer States and Structures

The equations usually given relating fluorescence lifetime to absorption intensity are strictly applicable only to atomic systems, whose transitions are sharp lines. This paper gives the derivation of a modified formula 1/τ0=2.880×10−9n2〈νf−3〉Av−1(gl/gu) ∫ edlnν, which should be valid for broad molecular bands when the transition is strongly allowed.Lifetimes calculated by this formula have been compared with measured lifetimes for a number of organic molecules in solution. In most cases the values agree within experimental error, indicating that the formula is valid for such systems. The limitations of the formula and the results expected for weak or forbidden transitions are also discussed.

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Relationship between Absorption Intensity and Fluorescence Lifetime of Molecules

Cation-pi interactions in protein structures are identified and evaluated by using an energy-based criterion for selecting significant sidechain pairs. Cation-pi interactions are found to be common among structures in the Protein Data Bank, and it is clearly demonstrated that, when a cationic sidechain (Lys or Arg) is near an aromatic sidechain (Phe, Tyr, or Trp), the geometry is biased toward one that would experience a favorable cation-pi interaction. The sidechain of Arg is more likely than that of Lys to be in a cation-pi interaction. Among the aromatics, a strong bias toward Trp is clear, such that over one-fourth of all tryptophans in the data bank experience an energetically significant cation-pi interaction.

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Cation-π interactions in structural biology

A series of papers describing advances made in recent years in obtaining explicit functions for the potential energy surfaces of small polyatomic molecules. Employs techniques drawn from quantum chemistry, molecular dynamics, gas kinetics, and vibrational spectroscopy in developing the functions required for studying the rates of simple chemical reactions, molecular beam scattering cross sections, and vibrational and rotational spectroscopy. Emphasizes the general features that potential energy functions should possess rather than the particular functions.

Molecular Potential Energy Functions

The theory of the electronic spectra of organic molecules

Previous work has indicated that in calculating rates using activated-complex theory one should omit symmetry numbers from the rotational partition functions and multiply by a statistical factor; this factor is the number of equivalent forms of the activated complex that can arise from the reactants. This procedure can lead to error if the activated complex is chosen to have such a high degree of symmetry that it can form more than one form of the reactants or products. It is shown from a consideration of multi-dimensional potential-energy surfaces that a single activated complex cannot exist at the intersection of two valleys; three valleys, for example, will give rise to three separate activated states, each of which represents the lowest pass between a pair of valleys. The implications of this conclusion are considered with reference to several reactions, including that between hydrogen and iodine.

Symmetries of activated complexes

An analytic potential for stable states of diatomic molecules is proposed V=–De(1 +a1r+a2r2+a3r3)e–a1r. The constants a1, a2 and a3 are obtained from the harmonic, cubic and quartic force constants. The potential is shown to be superior to the Hulburt–Hirschfelder potential when tested on a least-squares basis against the spectroscopic RKR potential.

New analytic form for the potential energy curves of stable diatomic states

The intensities of the vibrational components of the A ← X and B ← X band systems of NH3 have been calculated from vibrational functions appropriate to the best one-dimensional potential curves for the inversion mode. For the A ← X system the experimental data is fitted by an expression: The Franck-Condon approximation in which the Q 2 matrix element is neglected gives a poor fit to the data. We deduce that the electronic transition moment increases as the planar molecule is bent. For the B ← X system, which is a forbidden electronic transition, the experimental data are fitted well by the matrix element 2 which arises from the leading term in the Taylor expansion of the electronic transition moment. Using the same potential functions for ND3, and a reduced mass based on a pure bending mode, gives good agreement for the intensities of the corresponding ND3 bands. New experimental values for the extinction coefficients of the A ← X system have been obtained.

John N. Murrell

Papers

Molecular Potential Energy Functions

The theory of the electronic spectra of organic molecules

Symmetries of activated complexes

New analytic form for the potential energy curves of stable diatomic states

Calculation of the intensities of the vibrational components of the ammonia ultra-violet absorption bands