Micro-iterations

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. 

The total number of micro-iterations needed for solving the non-linear equations (53) is often fairly large. It is, therefore, important to make them as efficient as possible. In order to minimize the I/O time, the operators J and K should be kept in high-speed memory whenever possible, since their recovery from disc may be more expensive than their use in the calculation of Y. It is worth while to mention that we often found it advantageous to evaluate the operators G as intermediate quantities. These operators only change if the Cl coefficients are updated, and their calculation is particularly helpful if many micro-iterations are performed between Cl updates. The matrix S is... 

If c is an eigenvector of H, Eqs (167) and (168) produce identical results. The above expression gives the representation of M in the linearly dependent projected basis. Although this projected basis was employed by the author in the direct solution of the optimization equations , it was first employed by Lengsfield and Liu in the micro-iterative solution of these equations where... 

All matrix elements in the Newton-Raphson methods may be constructed from the one- and two-particle density matrices and transition density matrices. The linear equation solutions may be found using either direct methods or iterative methods. For large CSF expansions, such micro-iterative procedures may be used to advantage. If a micro-iterative procedure is chosen that requires only matrix-vector products to be formed, expansion-vector-dependent effective Hamiltonian operators and transition density matrices may be constructed for the efficient computation of these products. Sufficient information is included in the Newton-Raphson optimization procedures, through the gradient and Hessian elements, to ensure second-order convergence in some neighborhood of the final solution. 

The solution of the orbital corrections has usually been uncoupled from the CSF corrections within the micro-iterative procedure with this approach . This usually causes no problems for ground-state calculations, but it is expected to be detrimental for excited states, particularly those with negative eigenvalues of the orbital Hessian matrix at convergence. This could be addressed by using, for example, the PSCI iterative method within a micro-iteration for a fixed operator to solve simultaneously for and Another disadvantage, for large numbers of virtual orbitals, is that the space... 

One promising approach to the problem of effectively reducing the number of direct Cl matrix-vector products is the approximate Hamiltonian operator method of Werner and coworkers - described in Section III. This is an extended micro-iterative method in which the Hamiltonian operator is allowed to be approximated during the solution of the wavefunction correction vector within an MCSCF iteration. 

There are two variations on the above iterative process that have been proposed. In the case of small CSF expansion lengths it is appropriate to perform several c updates within an MCSCF iteration. This is because the expensive step of the MCSCF optimization process in this case is the two-electron integral transformation. The approximate transformations performed in step 3 of the micro-iterative procedure are less expensive than the exact transformations performed in step 1 of the MCSCF iteration. However, in the case of large CSF expansions, these updates should be avoided. This is because the exact transformation becomes a small part of the total iteration effort and it is preferable to perform the expensive CSF coefficient updates only with exact Hamiltonian operators. In this case step 5 and the cycle between steps 5 and 3 is not performed. Continuing this reasoning further, it may even be useful to perform several MCSCF iterations with the same CSF vector c to allow the orbitals to relax further for the exact Hamiltonian operators. 

Werner and Knowles have performed these exact transformations for a fixed CSF vector c but restricted to the active-active block of the orbital transformation matrix only. This transformation is performed using the same subset of integrals as required for the rest of the micro-iterative procedure, namely the integrals that contribute to the Hessian matrix. These additional... 

This is, however, of little importance as long as the MCSCF process converges within a satisfactory number of iterations. It can be estimated that the first-order transformation is 3-5 times as fast as the second-order transformation. Thus a first-order procedure could compete with second-order procedures if the number of iterations needed are not more than three times as many, provided of course that the calculation is not dominated by the Cl step. The most efficient second-order procedure seems to be that of Werner and Knowles. They use three macro-iterations and about 30-50 microiterations to converge to less than 10 a.u. in the energy. A first-order transformation is performed in each micro-iteration and a second-order one in each macro-iteration. It seems that first-order procedures can compete with this performance in many cases. With reasonable starting vectors, convergence to 10 a.u. is often reached in less than 10 iterations. 

Eqs. (57), (59) and (60) are the working equations for the cluster amplitudes. We should note that they are coupled also to the combining coefhcients c s, which are obtained as the elements of the eigenvector from the diagonalization of W dehned in the IMS, in Eq. (50). This is similar to what we had in the SS-MRCC theory for the CMS [38, 39]. Usually, one may get the coefficients from Eq. (50) in a macro-iteration, and get the cluster amplitudes in an inner, micro-iteration. 

Fig. 14 Schematic of the macro-micro iteration scheme Iot QM/MM geometry optimization...

In the previous subsections, we developed Newton s method for the optimization of Cl wave functions and energies. Newton s method requites only a few macro (outer) iterations, but, in each macro iteration (i.e. in each Newton step), a relatively large number of micro (inner) iterations are needed for solving the linear equations (11.5.3). Each micro iteration requires the multiplication of the Hamiltonian matrix by a trial vector (11.5.1). The total number of micro iterations needed for convergence may therefore become quite large with Newton s method. 

---