Self-consistent field method iterative minimization

The self-consistent-field algorithm is an iterative method for finding the coefficients c which minimize E(c) subject to these constraints. 

In complete active space self-consistent field (CASSCF) calculations with long configuration expansions the most expensive part is often the optimization of the Cl coefficients. It is, therefore, particularly important to minimize the number of Cl iterations. In conventional direct second-order MCSCF procedures , the Cl coefficients are updated together with the orbital parameters in each micro-iteration. Since the optimization requires typically 100-150 micro-iterations, such calculations with many configurations can be rather expensive. A possible remedy to this problem is to decouple the orbital and Cl optimizations , but this causes the loss of quadratic convergence. The following method allows one to update the Cl coefficients much fewer times than the orbital parameters. This saves considerable time without loss of the quadratic convergence behaviour. 

Quantum Mechanics (QM). The objective of QM is to describe the spatial positions of electrons and nuclei. The most commonly implemented QM method is the molecular orbital (MO) theory, in which electrons are allowed to flow around fixed nuclei (the Born-Oppenheimer approximation) until the electrons reach a self-consistent field (SCF). The nuclei are then moved, iteratively, until the energy of the system can go no lower. This energy minimization process is called geometry optimization. 

Referred to as the DIIS, this method has been developed by Pulay in the context of ah initio molecular orbital calculations [173, 174, 175], On the minimization, further improvement is possible only after updating the iterative subspace of (rt) by new vectors introducing new dimensions that cannot be reduced to a linear combination of the previous ones. The basis vectors are then being updated by several conventional self-consistent field iterative loops [173] or through the approximate Fock matrix at the minimum point (4.A.29) [174, 175]. 

Let us give a few examples of how conformational transition states may be determined in an automated manner. The calculations have been carried out with HERMIT-SIRIUS-ABACUS [31, 32, 33] at the self-consistent field level using a minimal basis set. A simple example is provided by PH5, whose equilibrium structure is a trigonal bipyramid of D h symmetry. Starting at equilibrium and applying the TRIM method without symmetry constraints, we arrive after nine iterations at a square pyramidal transition state of C4V symmetry. This structure represents the barrier to a Berry pseudo-rotation connecting two equivalent Dsh equilibrium structures, as confirmed by walking down the other side of the barrier. 

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