General functions Newton-Raphson methods

Generalizing the Newton-Raphson method of optimization (Chapter 1) to a surface in many dimensions, the function to be optimized is expanded about the many-dimensional position vector of a point xq... 

For the special case for which n = 2, it can be shown that the linearization method defined above becomes identical to the Newton-Raphson method. The result may be generalized to apply to any homogeneous function of degree n. 

Computationally the super-CI method is more complicated to work with than the Newton-Raphson approach. The major reason is that the matrix d is more complicated than the Hessian matrix c. Some of the matrix elements of d will contain up to fourth order density matrix elements for a general MCSCF wave function. In the CASSCF case only third order term remain, since rotations between the active orbitals can be excluded. Besides, if an unfolded procedure is used, where the Cl problem is solved to convergence in each iteration, the highest order terms cancel out. In this case up to third order density matrix elements will be present in the matrix elements of d in the general case. Thus super-CI does not represent any simplification compared to the Newton-Raphson method. 

For the general case of n independent equations in n unknowns, the Newton-Raphson method may be formulated in terms of n functions in n unknowns. For the kth trial, the resulting set of Newton-Raphson equations may be represented as follows... 

The intramolecular energy is now expressed in terms the unrotated CRU Cartesian coordinates, v, as foUows. In the Newton-Raphson method the energy is required as a quadratic function of the j ameter displacements. The valence coordinate energy functions are easily exjmnded as quadratics in r, 6, < >, but then transformations to the basis parameters must be providecL Let q, q be generalized valence coordinates (any of R, 6, or ( >) expressing an interaction between a set... 

Moreover, the second-generation MCSCF parametrizes the wave function in a way that enables the simultaneous optimization of spinors and Cl coefficients, in this context then called orbital or spinor rotation parameters and state transfer parameters, respectively. Then, a Newton-Raphson optimization method is employed which also requires the second derivatives of the MCSCF electronic energy with respect to the molecular spinor coefficients (more precisely, to the orbital rotation parameters) and to the Cl coefficients. As we have seen, in Hartree-Fock theory the second derivatives are usually not calculated to confirm that a solution of the SCF procedure has indeed reached a minimum with respect to the large component and not a saddle point. Now, these general MCSCF methods could, in principle, provide such information, although it is often not needed in practice. 

This is a general function that implements a method (such as the Newton-Raphson, Linear Interpolation, Gauss Elimination). This function is portable so that it can be called by other input-output programs and/or from the MATLAB work space (with parameters). 

---