Wavefunctions optimization

For a very large number of variables, the question of storing the approximate Hessian or inverse Hessian F becomes important. Wavefunction optimization problems can have a very large number of variables, a million or more. Geometry optimization at the force field level can also have thousands of degrees of freedom. In these cases, the initial inverse Hessian is always taken to be diagonal or sparse, and it is best to store the... 

Trial wavefunction Optimized parameter value(s) IE(calc) (eV) ... 

After the CASSCF calculation with the above choice of orbitals, in order to perform an efficient VB analysis, it is better in this case to resort to an overcomplete non-orthogonal hybrid set. The five active orbitals, in fact, can be split into ten hybrids, in term of which the VB transcription of the wavefunction turns out to be the simplest and the most compact. Such kinds of overcomplete basis sets are commonly used in constructing the so called non-paired spatial orbital structures (NPSO, see for example [35]), but it should be remarked that their use is restricted to gradient methods of wavefunction optimization, such as steepest descent, because other methods, which need to invert the hessian matrix (like Newton--Raphson) clearly have problems with singularities. 

In the wavefunction optimization procedure outlined in the previous sections the coefficients cj can be optimized by any available Cl procedure. 

Section V consists of a detailed discussion of redundant variables. The special case of MCSCF wavefunction optimization for two-electron systems is discussed in some detail. The relation between the configuration expansion space and the orbital variation space is quite straightforward for this case and this simplicity may be used to advantage in understanding the generalization to arbitrary numbers of electrons. There are two aspects of redundant variables that are important in the MCSCF method. First, if redundant variables are allowed to remain in the wavefunction variation space, then the optimization procedure becomes undefined or at least numerically ill-conditioned. Secondly, if the redundant variables are known for a given wavefunction then this flexibility may be used to transform the wavefunction to a form that is qualitatively easier to understand. The qualitative interpretation of MCSCF wavefunctions is one of the assets of the MCSCF method. 

The details of the computer implementation of various MCSCF methods are discussed in Section VI. Modern high-speed computers are biased in their ability to perform certain operations efficiently. MCSCF methods that involve simple vector and matrix operations have a distinct advantage on these types of computers. The reduction of unnecessary I/O (input/output to external storage) is also very important on these computers. This is because the capacity of these machines to perform arithmetic operations outpaces their I/O capacity. These considerations have had a significant impact on the choice of MCSCF wavefunction optimization methods and on the specific details of the implementation of these methods. 

The formalism developed in the preceding sections may now be used to derive the working equations for MCSCF wavefunction optimization. It will be assumed that the wavefunction is expanded in a set of orthonormal CSFs, as described in Section II, that are appropriate eigenfunctions of the operator S. It will also be assumed that the one- and two-electron integrals of the Flamiltonian operator, the CSF expansion coefficients and the orbital transformation coefficients are all real. As discussed previously, this is not a limitation of the formalism, but rather is imposed for reasons of computational efficiency. 

A. Energy Expressions for Multiconfiguration Self-consistent Field Wavefunction Optimization... 

The use of the exponential operator MCSCF formalism, or more specifically the use of optimization methods that require only the density matrix instead of the coupling coefficients over the CSF expansion terms (or even worse, over the single excitation expansion terms), has allowed relatively large CSF expansion lengths to be used in MCSCF wavefunction optimization. These larger expansion lengths allow CSFs to be included based on formal analysis or computational facility with little or no penalty in those cases where some of... 

This increase of CSF expansion length upon transformation to symmetry-adapted orbitals potentially affects any of the expansion forms that attempt to describe electron correlation in terms of localized orbitals and that are not invariant to transformations that mix the different localized orbitals. All of the product and direct product expansion forms (including the RCI, PPMC, PPGVB and SOGVB expansions) are potentially of this type. It often happens that these wavefunctions do have the full molecular symmetry even though they are described in terms of localized orbitals and not symmetry-adapted orbitals. The localized orbital description that results from these wavefunction optimizations is therefore both an asset and a liability it aids the chemical interpretability and results in more compact CSF expansions but the computations must be performed in an orbital basis that does not possess the full molecular symmetry. This is computationally important since many steps of the MCSCF wavefunction optimization can exploit such orbital symmetry when it is present. 

The dipole moment of a molecule is a defined quantum mechanical property and may be obtained exactly, for a given wavefunction, by integration using the appropriate operator r. Comparison of calculated with observed dipole moments gives infoimation about the quality of a wavefunction. Since nearly every reported wavefunction includes the molecular dipole moment (if applicable), there is available an extensive literature of calculated dipole moments. These dipole moments may be calculated either at the observed molecular geometry or at wavefunction-optimized geometry. 

Due to the large number of variables in wavefunction optimization problems, it may appear that full second-order methods are impractical. For example, the storage of the Hessian for a modest closed-shell wavefunction with 500... 

Fischer T H and Almidf J 1992 General methods for geometry and wavefunction optimization J, Phys. Chem. 96 9768... 

Alexander et al. [55] used VMC to compute cross sections for the elastic and inelastic scattering of fast electrons and X rays by H2. Novel trial wavefunction optimization schemes has been proposed by Huang and Cao... 

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