Gradients and Force Constants

The evaluation of all the diagrams in Fig. 5.1 would thus give the desired second-order property consistent through first order in electronic interaction. 

The determination of minima and saddle points on the potential-energy surface of a molecule plays an important role (Schaefer and Miller, 1977, Chapter 4) in describing the electronic structure and chemical reactivity of molecules. In this section, we show how such stationary points on a molecule s potential energy surface may be found by using an approach similar to that employed in Section 5.B. We first consider how the electronic Hamiltonian changes when the nuclear positions are changed from an initial set of positions, to R, i.e., R - R + u. The electron-nuclear interaction is the only term in the Hamiltonian that depends explicitly on the nuclear position. Performing a Taylor expansion of this potential about the point R, we obtain 

We may thus identify the changes in the electronic Hamiltonian through second order in the nuclear displacements (u ) as 

Fj clearly represents the forces on the electrons due to the nuclear displacement, whereas F2 describes electric-field gradient terms induced by movement of the nuclei. A stationary point on the potential energy surface occurs when the average value of the first-order term in zero  

The total energy that contains all terms through second order in the nuclear displacement may thus be expressed as 

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The limits of accuracy were probed in a study which focused on the gradient and force constants in diatomics like FIF. As more basis functions are added to one of 6-31G type, changes of the order of 1 % occur in the force constant, at both correlated and SCF levels. The superposition contribution to the gradient is fairly large, and can account for variation of as much as 0.004 A in the bond length. 

In this subsection we discuss several practical considerations that arise when constructing RPH s for polyatomic systems from a set of ab initio calculations of the energy, gradient, and force constant matrix at a series of points along a reaction coordinate. To illustrate some of these considerations, we will use the CH3 + H2 CH H reaction and the inversion "reaction of NH3 as examples. For the CH3 + H2 reaction. 

Now, the construction of the gradient and force constant matrices in internal coordinates is possible ... 

Famulari, A., Gianinetti, E., Raimondi, M. and Sironi, M. (1998) Implementation of gradient optimisation algorithms and force constant computations in BSSE free direct and conventional SCF... 

As regards SCF and SCF-MI calculations, the GAMESS-US program was employed, in which the SCF-MI algorithm including evaluation of analytic gradient, geometry optimisation and force constant matrices computation is available [18,41,42]. 

A. Famulari, E. Gianinetti, M. Raimondi, M. Sironi, Int. J. Quant. Chem. 69, 151 (1998). Implementation of Gradient-Optimization Algorithms and Force Constant Computations in BSSE-Free Direct and Conventional SCF Approaches. 

TABLE 5. Calculated bond distances Re(A), total bond dissociation energies Do (kcalmol-1) which include ZPE corrections and force constants of the totally symmetric mode ke (Ncm-1) of the molecules EH4 and ECI4, using relativistic gradient-corrected DFT ... 

Here go, Fo, and Go are, respectively, the first (gradient), second (force constants), and third energy derivatives evaluated at xQ. The square brackets... 

Gianinetti, E., Vandoni, 1., Famulari, A. and Raimondi, M (1998) Extension of the SCF-MI method to the case of K fragments one of which is an open-shell system, Adv. Quantum Chem., 31, 251-266. Famulari, A., Gianinetti. E.. Raimondi. M. and Sironi. M. (1998) Implementation of gradient op timisation algorithms and force constant computations in BSSE free direct and conventional SCF... 

Here gg, Fg, and Gg are, respectively, the first (gradient), second (force constants), and third energy derivatives evaluated at Xg. The square brackets indicate that the three-dimensional array of third derivatives is contracted with the vector of coordinate changes to yield a square matrix. If Xg is a stationary point, i.e., a minimum, then the usual theory of small vibrations applies the gradient term vanishes, and truncation after the second-order term leads to separable, harmonic normal modes of vibration. However, on the MEP, the gradient term generally is not zero. The second relevant expansion is the Taylor series representation of the path (of the solution to Eq. [8]) in the arc length parameter, s, about the same point, Xg ... 

For all the other studied five-membered rings, the magnitudes of the gradients, the force constants, and the AI values were obtained by applying the same procedure. The results are presented in Figure 20.4. 

Once one has the gradient of the energy with respect to the nuclear coordinates, one can use it to efficiently determine various characteristics of a potential energy surface, such as equilibrium and saddle point geometries and force constants. 

Pulay P, FogarasI G, Pang F and Boggs J E 1979 Systematic ab initio gradient calculation of molecular geometries, force constants and dipole moment derivatives J. Am. Chem. Soc. 101 2550... 

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

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