MCSCF convergence

The diagonalisation of the Fock matrix during the iterative process of SCF (or MCSCF) convergence. This step is clearly not applicable to non-SCF models. 

Werner H-J and Meyer W 1981 A quadratically convergent MCSCF method for the simultaneous optimization of several states J. Chem. Phys 74 5794... 

CASSCF is a version of MCSCF theory in which all possible configurations involving the active orbitals are included. This leads to a number of simplifications, and good convergence properties in the optimization steps. It does, however, lead to an explosion in the number of configurations being included, and calculations are usually limited to 14 elections in 14 active orbitals. 

For conhguration interaction calculations of double excitations or higher, it is possible to solve the Cl super-matrix for the 2nd root, 3rd root, 4th root, and so on. This is a very reliable way to obtain a high-quality wave function for the hrst few excited states. For higher excited states, CPU times become very large since more iterations are generally needed to converge the Cl calculation. This can be done also with MCSCF calculations. 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

When natural orbitals are determined from a wave function which only includes a limited amount of electron correlation (i.e. not full Cl), the convergence property is not rigorously guaranteed, but since most practical methods recover 80-90% of the total electron correlation, the occupation numbers provide a good guideline for how important a given orbital is. This is the reason why natural orbitals are often used for evaluating which orbitals should be included in an MCSCF wave function (Section 4.6). 

Within the SCF-CI method a fixed set of molecular orbitals is used. This means that during the calculation (leading to slow convergence) the individual molecular orbitals remain unchanged. A method where the linear expansion coefficients and the LCAO coefficients are optimized simultaneously is the multi-configuration SCF (MCSCF). 

In general, these quartic equations must then be solved in an iterative manner and are susceptible to convergence difficulties that are similar to those that arise in MCSCF-type calculations. In any such iterative process, it is important to start with an approximation (to the t amplitudes, in this case) which is reasonably close to the final converged result. Such an approximation is often achieved, for example, by neglecting all of the terms that are nonlinear in the t amplitudes (because these amplitudes are assumed to be less than unity in magnitude). This leads, for the CC working equations obtained by projecting onto the doubly excited CSFs, to ... 

MCSCF calculations in practice require much more technical expertise than do singleconfiguration HF analogs. One particularly difficult problem is that spurious minima in coefficient space can often be found, instead of the variational minimum. Thus, convergence criteria are met for die self-consistent field, but the wave function is not really optimized. It usually requires a careful inspection of the orbital shapes and, where available, some data on relative energetics between related species or along a reaction coordinate to ascertain if this has happened. 

Similar iterative schemes were used to determine the MO s for multiconfigurational wave functions, in the early implementations. Fock-like operators were constructed and diagonalized iteratively. The convergence problems with these methods are, however, even more severe in the MCSCF case, and modem methods are not based on this approach. The electronic energy is instead considered to be a function of the variational parameters of the wave function - the Cl coefficients and the molecular orbital coefficients. Second order (or approximate second order) iterative methods are then used to find a stationary point on the energy surface. 

We shall in this chapter discuss the methods employed for the optimization of the variational parameters of the MCSCF wave function. Many different methods have been used for this optimization. They are usually divided into two different classes, depending on the rate of convergence first or second order methods. First order methods are based solely on the calculation of the energy and its first derivative (in one form or another) with respect to the variational parameters. Second order methods are based upon an expansion of the energy to second order (first and second derivatives). Third or even higher order methods can be obtained by including more terms in the expansion, but they have been of rather small practical importance. 

The Fock matrix (4 42) is in general not Hermitian for a non-converged MCSCF wave function. With optimized orbitals the gradient is zero. The MCSCF Fock matrix is thus Hermitian at this point on the energy surface. This condition has been used as a basis for optimization schemes in earlier developments of the MCSCF methodology. Convergence of such first order optimization schemes is, however, often poor, and they are not very much used today. 

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