Analytic gradient methods

The paramount difficulties in the experimental determination of accurate absolute infrared intensities and differential Raman scattering cross sections seem, however, to persist in the theoretical evaluation of these quantities as well. As we will see, predicted Raman intensities appear to be more consistent with experiment as compared to infrared intensities. 

Analytic derivative methods for evaluating polarizability derivatives have been developed simultaneously by the theoretical chemishy groups in Cambridge [340] and Berkley [341]. According to Frisch et al. [341] the analytic evaluation of polarizability derivatives is achieved using the expression 

As can be seen from expression (10.5) no third derivative integrals appear in evaluating polarizability derivatives. No second derivative for the two-electron integrals are also needed. Thus, polarizability derivative calculations do not require much additional time. Second order couple-perturbed Hartree-Fock (CPHF) equations are solved with respect to the six pairs of electric field variables. 

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Analytical gradient energy expressions have been reported for many of the standard models discussed in this book. Analytical second derivatives are also widely available. The main use of analytical gradient methods is to locate stationaiy points on potential energy surfaces. So, for example, in order to find an expression for the gradient of a closed-shell HF-LCAO wavefunction we might start with the electronic energy expression from Chapter 6,... 

Shepard R (1995) The analytic gradient method for configuration interaction wave functions. Yarkony DR (ed) In Modern electronic structure theory part I, World Scientific, Singapore, p 345... 

As a first application of a new analytical gradient method employing UHF reference functions, seven different methods for inclusion of correlation effects were employed to optimize the geometry and calculate the harmonic vibrational frequencies and dipole moments of the lowest open-shell states for three simple hydrides including 3Z i SiH2228. As the degree of correlation correction increased, results approached those from the best multiconfiguration SCF calculation. 

Lengsfield III, B.H., Saxe, P., and Yarkony, D.R. (1984). On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wavefunctions and analytic gradient methods. I, J. Chem. Phys. 81, 4549-4553. 

Cluster, Unitary Coupled-Cluster and MBPT(4) Open-Shell Analytical Gradient Methods. 

Analytic gradient methods became widely used as a result of their implementation for closed-shell self-consistent field (SCF) wavefunctions by Pulay, who has reviewed the development of this topic. Since then, these methods have been extended to deal with all types of SCF wavefunctions, - as well as multi-configuration SCF (MC-SCF), - " configuration-interaction (Cl) wavefunctions, and various non-variational methods such as MoUer-Plesset (MP) perturbation theory - - and coupled-cluster (CC) techniques. - In short, it is possible to obtain analytic energy derivatives for virtually all the standard ab initio approaches. The main use of analytic gradient methods is, and will remain, the location of stationary points on a potential energy siuface, to obtain equilibrium and transition-state geometries. However, there is a specialized use in the calculation of quantities such as dipole derivatives. 

The structures of the reactants, the transition states, and the products are optimized and the intrinsic reaction coordinates (IRC) connecting them are traced by the analytical gradient method at the 6-31G /RHF (for Rl) and MP2 (M011er-Plesset) (for R2) levels. The vibrational frequency analyses [19] along IRC are carried out to obtain the curvature of IRC and the coupling elements among the vibrational modes perpendicular to IRC. 

Scientific, Singapore, 1995, pp. 345—458. The Analytic Gradient Method for Configuration Interaction Wave Functions. 

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