Gradient calculations

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... 

Caution Geometry optimizations of large molecules may take longer than you expect. The number of computing cycles required for a conjugate gradient calculation is approximately proportional... 

An analytical gradient calculation is invariably faster than a numerical one. To repeat the argument from Chapter 14, with real wavefunction Hamiltonian H (including the field terms) and parameter a (where a is a component of the external electric field)... 

Boggs, J. E., and F. R. Cordell. 1981. Accurate Ab Initio Gradient Calculation of the Structures and Conformations of Some Boric and Fluoroboric Acids. Basis-Set Effects on Angles Around Oxygen. J. Mol. Struct. (Theochem) 76, 329-347. 

McKean, D. C., J. E. Boggs, and L. Schafer. 1984. CH Bond Length Variations Due to the Intramolecular Environment a Comparison of the Results Obtained by the Method of Isolated CH Stretching Frequencies and by Ab Initio Gradient Calculations. J. Mol. Struct. 116, 313-330. 

Pulay, P., G. Fogarasi, F. Pang, and J. E. Boggs. 1979b. Systematic Ab Initio Gradient Calculation of Molecular Geometries, Force Constants, and Dipole Moment Derivatives. J. Am. Chem. Soc. 101, 2550-2560. 

The value of the square root of the two-phase multiplier is read from Figure 7.13, or calculated from equation 7.85 or 7.86, and the two-phase frictional pressure gradient calculated from... 

Figure 7.19 Ratio of the pressure gradient calculated using the generalized Newtonian method to the pressure gradient calculated using the numerical method as a function of barrel diameter...

Spalding, M. A. and Campbell, G.A., The Accuracy of Standard Drag Flow and Pressure Gradient Calculations for Singie-Screw Extruders, SPE ANTEC Tech. Papers, 54, 262 (2008)... 

On the other hand, the optimal control problem with a discretized control profile can be treated as a nonlinear program. The earliest studies come under the heading of control vector parameterization (Rosenbrock and Storey, 1966), with a representation of U t) as a polynomial or piecewise constant function. Here the mode is solved repeatedly in an inner loop while parameters representing V t) are updated on the outside. While hill climbing algorithms were used initially, recent efficient and sophisticated optimization methods require techniques for accurate gradient calculation from the DAE model. 

Gradient calculations for the x variables are obtained from implicit reformulations of the DAE model. Clearly the easiest, but least accurate, way is simply to re-solve the model for each perturbation of the parameters. Sargent and Sullivan (1977, 1979) derived these gradients using an adjoint formulation. In addition, they were able to accelerate the adjoint computations by retaining the information from the model solution (the forward step) for the adjoint solution in the backward step. This approach was later refined for variable stepsize methods by Morison (1984). The adjoint approach to parameterized optimal control was also used by Jang et al. (1987) and Goh and Teo (1988). 

Either approach results in gradient calculations with costs proportional to problem size effort for evaluating gradients with adjoint approaches is... 

All the data reported in the literature seem to confirm that the NH3-HCI complex possesses C3V symmetry (see Figure 2). Table 4 shows the geometrical parameters of the complex reported in the pertinent literature. Bacskay et al [26,27] confirm the C3V symmetry of the complex both at the MP2 and SCF level. A moderate variation from the C3V symmetry, however, results from EFG (Electric Field Gradient) calculations [22]. 

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