Adaptive potential energy

Rigid Molecule Group theory will be given in the main part of this paper. For example, synunetry adapted potential energy function for internal molecular large amplitude motions will be deduced. Symmetry eigenvectors which factorize the Hamiltonian matrix in boxes will be derived. In the last section, applications to problems of physical interest will be forwarded. For example, conformational dependencies of molecular parameters as a function of temperature will be determined. Selection rules, as wdl as, torsional far infrared spectrum band structure calculations will be predicted. Finally, the torsional band structures of electronic spectra of flexible molecules will be presented. 

Figure Al.6.13. (a) Potential energy curves for two electronic states. The vibrational wavefunctions of the excited electronic state and for the lowest level of the ground electronic state are shown superimposed, (b) Stick spectrum representing the Franck-Condon factors (the square of overlap integral) between the vibrational wavefiinction of the ground electronic state and the vibrational wavefiinctions of the excited electronic state (adapted from [3]).

Figure A3.12.1. Schematic potential energy profiles for tluee types of iinimolecular reactions, (a) Isomerization, (b) Dissociation where there is an energy barrier for reaction in both the forward and reverse directions, (c) Dissociation where the potential energy rises monotonically as for rotational gronnd-state species, so that there is no barrier to the reverse association reaction. (Adapted from [5].)...

Figure A3.12.4. Plots ofN [E-EQ(qversus q for tln-ee models of the C2Hg 2CH3 potential energy fiinction. represents q and the temi on the abscissa represents N. (Adapted from [23].)...

Figure A3.12.5. A model reaction coordinate potential energy curve for a fluxional molecule. (Adapted from [30].)...

One more quantum number, that relating to the inversion (i) symmetry operator ean be used in atomie eases beeause the total potential energy V is unehanged when all of the eleetrons have their position veetors subjeeted to inversion (i r = -r). This quantum number is straightforward to determine. Beeause eaeh L, S, Ml, Ms, H state diseussed above eonsist of a few (or, in the ease of eonfiguration interaetion several) symmetry adapted eombinations of Slater determinant funetions, the effeet of the inversion operator on sueh a wavefunetion P ean be determined by ... 

If the complete potential energy surface has already been computed, a reaction coordinate can be determined using an adaptation of the IRC algorithm. The IRC computation requires very little computer time, but obtaining the potential energy surface is far more computation-intensive than an ah initio IRC calculation. Thus, this is only done when the potential energy surface is being computed for another reason. 

A graphical representation of the potential energy surface or reaction coordinate. The transition state occurs at the saddle point. ( Adapted from Ref. 18.)... 

Figure 6.35. Potential energy diagrams for adsorption and dissociation of N2on a Ru(0001) surface and on the same surface with a monoatomic step, as calculated with a density functional theory procedure. [Adapted from S. Dahl, A. Logadottir, R. Egberg, J. Larsen, I. Chorkendorff,...

Figure 9.12. Potential energy profile along (adapted from reference 10) near the fulvene conical intersection. The branching space consists of stretching and skeletal deformation of the five-membered ring.

F ure 9.13. Potential energy profile (adapted from reference 10) for fulvene in the space spanned by Xj and the coordinate (torsion). 

In the previous section, the adaptation of the RIS model was based on the distance between next-nearest neighbor beads. This approach is obviously inadequate for CH3-CHX-CH2-CHX-CH3, because it necessarily abandons the ability to attribute different conformational characteristics to the meso and racemo stereoisomers. Therefore a more robust adaption of the RIS model to the 2nnd lattice is necessary if one wants to investigate the influence of stereochemical composition and stereochemical sequence on vinyl polymers [156]. Here we describe a method that has this capability. Of course, this method retains the ability to treat chains such as PE in which the bonds are subject to symmetric torsion potential energy functions. 

Figure 2-9. Reaction scheme for the complete catalytic cycle in glutathione peroxidase (left). Numbers represent calculated reaction barriers using the active-site model. The detailed potential energy diagram for the first elementary reaction, (E-SeH) + H2O2 - (E-SeOH) + H2O, calculated using both the active-site (dashed line) and ONIOM model (grey line) is shown to the right (Adapted from Prabhakar et al. [28, 65], Reprinted with permission. Copyright 2005, 2006 American Chemical Society.)...

Figure 2-11. ONIOM protein model (left) with QM atoms shown as spheres and MM atoms as sticks (substrate MCA atoms are shown as tubes). The graph to the right shows potential energy profiles obtained by relaxed scans along the Co—C5 bond in MCM for different computational models (see text for details) (Adapted from Kwiecien et al. [29]. Reprinted with permission. Copyright 2006 American Chemical Society.)...

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

Fig. 1. Conformational energy diagram for the alanine dipeptide (adapted from Ramachandran et al., 1963). Energy contours are drawn at intervals of 1 kcal mol-1. The potential energy minima for p, ofR, and aL are labeled. The dependence of the sequential d (i, i + 1) distance (in A) on the 0 and 0 dihedral angles (Billeter etal., 1982) is shown as a set of contours labeled according to interproton distance at the right of the figure. The da (i, i + 1) distance depends only on 0 for trans peptide bonds (Wright et al., 1988) and is represented as a series of contours parallel to the 0 axis. Reproduced from Dyson and Wright (1991). Ann. Rev. Biophys. Chem. 20, 519-538, with permission from Annual Reviews.

Fig. 3.3. Typical results from a density-of-states simulation in which one generates the entropy for aliquid at fixed N and V (i.e., fixed density) (adapted from [29]). The dimensionless entropy. r/ In ( is shown as a function of potential energy U for the 110-particle Lennard-Jones fluid at p = 0.88. Given an input temperature, the entropy function can be reweighted to obtain canonical probabilities. The most probable potential energy U for a given temperature is related to the slope of this curve, d// /dU(U ) = l/k T, and this temperature-energy relationship is shown by the dotted line. Energy and temperature are expressed in Lennard-Jones units...

Figure 7. Two-dimensional cut of the ground- and excited-state adiabatic potential energy surfaces of Li + H2 in the vicinity of the conical intersection. The Li-EL distance is fixed at 2.8 bohr, and the ground and excited states correspond to Li(2,v) + H2 and Lit2/j ) + H2, where the p orbital in the latter is aligned parallel to the H2 molecular axis, y is the angle between the H-H intemuclear distance, r, and the Li-to-H2 center-of-mass distance. Note the sloped nature of the intersection as a function of the H-H distance, r, which occurs because the intersection is located on the repulsive wall. (Figure adapted from Ref. 140.)...

Figure 9. Comparison of ab initio (full line) and ab m/rfo/interpolated (dashed line) potential energy surfaces for the first electronically excited state of Li + H2 system restricted to C2v geometry. Contours are labeled in eV. (Figure adapted from Ref. 125.)...

Figure 14. The absolute value of the average disrotatory angle as a function of time in femtoseconds. (The disrotatory angle is defined in the upper left inset.) Lower inset A onedimensional cut of the excited-state potential energy surface along the disrotatory and conrotatory coordinates. All other coordinates are kept at their ground-state equilibrium value, and the full and dashed lines correspond to two levels of electronic structure theory (see text for details). (Figure adapted from Ref. 216.)...

FIGURE 3.4. Reactant and products potential energy curves for the reductive cleavage of the R—X bond. For the product curves, dotted line no interaction between fragments, solid line finite interaction (DP) between fragments. Adapted from Figure 5 of reference 13a, with permission from the American Chemical Society. 

FIGURE 3.1 7. Section of the zero-order ( ) and first-order (-) potential energy surfaces along the reaction coordinate in cases where stretching of the cleaving bond is the dominant factor of nuclei reorganization. Adapted from Figure 1 of reference 24a, with permission from the American Chemical Society. 

FIGURE 3.21. Gas-phase potential energy surfaces for the 4-cyanochlorobenzene anion radical as a function of the C-Cl bond length (r) and the bending angle (0). R, reactant system TS, transition state. Adapted from Figure 4 of reference 32, with permission from the American Chemical Society. 

FIGURE 3.25. Potential energy profiles (from B3LYP/6-13G calculations) for the clevage of 3- and 4-nitrobenzyl chloride anion radicals (a and b, respectively) in the gas phase (top) and in a solvent (middle and bottom) (from COSMO solvation calculations with a dielectric constant of 36.6 and 78.4, respectively). Dotted and solid lines best-fitting Morse and dissociative Morse curves, respectively. Adapted from Figure 3 of reference 43, with permission from the American Chemical Society. 

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