Minimum energy coordinates

Let us first define the external MEC in M, consisting of m atoms. Consider the global equilibrium of M in contact with a hypothetical electron reservoir (r) fi0 = fj1 where fi= fi, the chemical potential of r. Let z = N — N° = d/V denotes the vector of a hypothetical AIM electron-population displacements from their equilibrium values N°. Since d/V = - d/Vr, the assumed equilibrium removes the first-order contribution to the associated change due to z in the energy, = M + , of the combined (closed) system (Mir) moreover, taking into account the infinitely soft character of a macroscopic reservoir, the only contribution to the energy change in the quadratic approximation is  

The explicit expression for these quantities immediately follows from Eq. (87), since the definition of Eq. (89) implies x + k = 0. Thus  

The associated hardness (compliance constant) is defined by a similar minimum energy constraint  

We would like to emphasize that, due to the closure constraint, there are only (m — 1) linearly independent internal MEC. Thus, the m vectors defined by Eq. (93) in reality span the (m — 1 )-dimensional space of internal MEC. In order to remove this linear dependence one could adopt the relative internal approach of Sect. 2.1.3. Namely, one then selects the electron population of one atom in the system as dependent upon populations of all remaining atoms, and discards the MEC associated with that atom. All remaining MEC can also be constructed directly from the corresponding internal relative softness matrix. Although the sets of independent internal MEC for alternative choices of the dependent atom will differ from one another, they must span the same (m — 1 )-dimensional linear space of independent internal MEC. For example, in the two-AIM system of Fig. 4 there is only one independent internal MEC direction along the P-line. 

It has been shown numerically that the above two sets of MEC, external and internal, are almost indistinguishable, exhibiting practically identical compli- 

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Another example is provided by the minimum energy coordinates (MECs) of the compliant approach in CSA (Nalewajski, 1995 Nalewajski and Korchowiec, 1997 Nalewajski and Michalak, 1995,1996,1998 Nalewajski etal., 1996), in the spirit of the related treatment of nuclear vibrations (Decius, 1963 Jones and Ryan, 1970 Swanson, 1976 Swanson and Satija, 1977). They all allow one to diagnose the molecular electronic and geometrical responses to hypothetical electronic or nuclear displacements (perturbations). The thermodynamical Legendre-transformed approach (Nalewajski, 1995, 1999, 2000, 2002b, 2006a,b Nalewajski and Korchowiec, 1997 Nalewajski and Sikora, 2000 Nalewajski et al., 1996, 2008) provides a versatile theoretical framework for describing diverse equilibrium states of molecules in different chemical environments. 

Nalewajski, R. F. and A. Michalak. 1995. Use of charge sensitivity analysis in diagnosing chemisorption clusters Minimum-energy coordinate and Fukui function study of model toluene-[V205] Systems. Int. J. Quantum Chem. 56 603-613. 

Swanson, B. 1.1976. Minimum energy coordinates. A relationship between molecular vibrations and reaction coordinates. J. Am. Chem. Soc. 98 3067-3071. 

Meta- Position in the Substituted Benzene Ring Minimum Energy Coordinates Modified Neglect of Differential Overlap [NDDO (Neglect of Diatomic Differential Overlap) Approximation, Method]... 

Swansrai BI, Satija SK (1977) Molecular vibrations and reaction pathways minimum energy coordinates and compliance craistants for some tetrahedral and octahedral complexes. J Am Chem Soc 99 987-991... 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

Consider the coordinate that transforms the nuclear configuration of H(III) at the minimum energy with the corresponding configuration of TS(I-II). In the foiiner, atoms 1 and 4 are close together, as are atoms 2 and 3. The separation between the two pairs is large. In other words, if Rij is the separation between atoms i and j, we have... 

The potential surfaces of the ground and excited states in the vicinity of the conical intersection were calculated point by point, along the trajectory leading from the antiaromatic transition state to the benzene and H2 products. In this calculation, the HH distance was varied, and all other coordinates were optimized to obtain the minimum energy of the system in the excited electronic state ( Ai). The energy of the ground state was calculated at the geometry optimized for the excited state. In the calculation of the conical intersection... 

We have it on good authority (Ege, 1998) that the gauche minimum on the potential energy coordinate is about 0.9 kcal moI higher in energy than the anti conformation. This establishes a two-state energy system for the stable conformers, gauche and anti (Fig. 4-19). 

With Lammerstma and Simonetta in 1982, we studied the parent six-coordinate diprotonated methane (CH/ ), which has two 2e-3c bonding interactions in its minimum-energy structure (Cid- On the basis of ab initio calculations, with Rasul we more recently found that the seven-coordinate triprotonated methane (CHy ) is also an energy minimum and has three 2e-3c bonding interactions in its minimum-energy structure 3 ). These results indicate the general importance of 2e-3c bonding in protonated alkanes. 

There are many different algorithms for finding the set of coordinates corresponding to the minimum energy. These are called optimization algorithms because they can be used equally well for finding the minimum or maximum of a function. 

In order to define how the nuclei move as a reaction progresses from reactants to transition structure to products, one must choose a definition of how a reaction occurs. There are two such definitions in common use. One definition is the minimum energy path (MEP), which defines a reaction coordinate in which the absolute minimum amount of energy is necessary to reach each point on the coordinate. A second definition is a dynamical description of how molecules undergo intramolecular vibrational redistribution until the vibrational motion occurs in a direction that leads to a reaction. The MEP definition is an intuitive description of the reaction steps. The dynamical description more closely describes the true behavior molecules as seen with femtosecond spectroscopy. 

MEP (IRC, intrinsic reaction coordinate, minimum-energy path) the lowest-energy route from reactants to products in a chemical process MIM (molecules-in-molecules) a semiempirical method used for representing potential energy surfaces... 

Figure 2.7 shows a representation of this situation. The ordinate is an energy axis and the abscissa is called the reaction coordinate and represents the progress of the elementary step. In moving from P to H, the particle simply moves from one equilibrium position to another. In the absence of any external forces, the energy of both the initial and final locations should be the same as shown by the solid line in Fig. 2.7. Between the two minima corresponding to the initial and final positions is the energy barrier arising from the dislodging of the particles neighboring the reaction path from their positions of minimum energy.

Figure 5-3 is the reaction coordinate diagram for Fig. 5-2. Note the region of the maximum potential energy on the reaction coordinate this region assumes great importance in kinetic theory. At this point the reacting system is unstable with respect to motion along the reaction coordinate. However, at this same point the system possesses minimum energy with respect to motion along dashed line cd. This portion of the reaction coordinate is called the transition state of the reaction. (This concept was introduced in Fig. 1-1.)...

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