Potential energy surface curvature

In addition to optimized geometries, which are minima on the potential energy surface, curvature on the free energy surface is also of interest in many QM/MM studies. Therefore, efficient calculations of free energy are required. One of the most widely used techniques to calculate the free energy in QM/MM methods is free energy perturbation (FEP) theory. Originally developed in MM, this FEP... 

The Newton-Raphson block diagonal method is a second order optimizer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. These derivatives provide information about both the slope and curvature of the potential energy surface. Unlike a full Newton-Raph son method, the block diagonal algorithm calculates the second derivative matrix for one atom at a time, avoiding the second derivatives with respect to two atoms. 

Fig. 4 Free energy reaction coordinate profiles that illustrate a change in the relative kinetic barriers for partitioning of carbocations between nucleophilic addition of solvent and deprotonation resulting from a change in the curvature of the potential energy surface for the nucleophile addition reaction. This would correspond to an increase in the intrinsic barrier for the thermoneutral carbocation-nucleophile addition reaction.

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

The relative magnitude of both second-order terms determines whether the molecule is stable with respect to distortion along the vibrational coordinate Qt. If the relaxation term is of larger magnitude than the distortion term then the geometry is unstable and the potential energy surface has negative curvature for mode Qi. 

Fig. 5. The pseudo-Jahn-Teller effect in ammonia (NH3). (a) CCSD(T) ground state potential energy curve breakdown of energy into expectation value of electronic Hamiltonian (He), and nuclear-nuclear repulsion VNN. (b) CASSCF frequency analysis of pseudo-Jahn-Teller effect showing the effect of including CSFs of B2 symmetry is to couple the ground and 1(ncr ) states to give a negative curvature to the adiabatic ground state potential energy surface for the inversion mode.

A stable nuclear configuration on a potential energy surface is associated with a point for which there is zero slope in any direction and for which there is no direction in which the curvature is negative or zero. Such points are uniquely defined in any system of internal coordinates but we shall see that some other characteristic features of a surface are dependent on the choice of coordinate. 

These discrepancies result (a) from the harmonic approximation used in all calculations [to,- (theory) > v, (exp)], (b) the known deficiencies of minimal and DZ basis sets to describe three-membered rings [polarization functions are needed to describe small CCC bond angles a>,(DZ + P) > w,(DZ) > to,(minimal basis)] and (c) the need of electron correlated wave functions to correctly describe the curvature of the potential energy surface at a minimum energy point [ [Pg.102]

In theory, a properly developed force field should be able to reproduce structures, strain energies, and vibrations with similar accuracies since the three properties are interrelated. However, structures are dependent on the nuclear coordinates (position of the energy minima), relative strain energies depend on the steepness of the overall potential (first derivative), and nuclear vibrations are related to the curvature of the potential energy surface (second derivative). Thus, force fields used successfully for structural predictions might not be satisfactory for conformational analyses or prediction of vibrational spectra, and vice versa. The only way to overcome this problem is to include the appropriate type of data in the parameterization process 501. 

The main advantage of MP2/6-31G optimizations over HF/3-21 ( > or HF/ 6-31G ones is not that the geometries are much better, but rather that for a stationary point, MP2 optimizations followed by frequency calculations are more likely to give the correct curvature of the potential energy surface (Chapter 2) for the species than are HF optimizations/frequencies. In other words, the correlated calculation tells us more reliably whether the species is a relative minimum or merely a transition state (or even a higher-order saddle point see Chapter 2). Thus fluorodiazomethane [91] and several oxirenes [53] are (apparently correctly) predicted by MP2 optimizations to be merely transition states, while HF optimizations... 

By taking as a reference the calculation in vacuo, the presence of the solvent introduces several complications. In fact, besides the direct effect of the solvent on the solute electronic distribution (which implies changes in the solute properties, i.e. dipole moment, polarizability and higher order responses), it should be taken into account that indirect solvent effects exist, i.e. the solvent reaction field perturbs the molecular potential energy surface (PES). This implies that the molecular geometry of the solute (the PES minima) and vibrational frequencies (the PES curvature around minima in the harmonic approximation) are affected by the presence of a solvating environment. Also, the dynamics of the solvent molecules around the solute (the so-called nonequilibrium effect ) has to be... 

The first derivatives of a potential energy function define the gradient of the potential energy surface and the second derivatives describe its curvature (Fig. 4.4). In most molecular mechanics programs the functions used are relatively simple and the derivatives are usually determined analytically. The second derivatives of harmonic oscillators correspond to the force constants. Thus, methods using the entire set of second derivatives result in some direct information on vibrational frequencies. 

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