MCSCF method rotations

The final step is the orbital optimization for the truncated SDTQ-CI expansion. We used the Jacobi-rotation-based MCSCF method of Ivanic and Ruedenberg (55) for that purpose. Table 1 contains the results for the FORS 1 and FORS2 wavefiinctions of HNO and NCCN, obtained using cc-pVTZ basis sets (51). In all cases, the configurations were based on split-localized orbitals. For each case, four energies are listed corresponding to (i) whether the full or the truncated SDTQ-CI expansion was used and (ii) whether the split-localized orbitals were those deduced from the SD naturals orbitals or were eventually MCSCF optimized. It is seen that... 

Some types of matrices that appear frequently are Hermitian matrices, for which A = A anti-Hermitian matrices, for which A = — A and unitary matrices, for which = U The MCSCF methods diseussed in most detail in this review will involve only operations of real matrices. In this case these matrix types reduce to symmetric matrices, for which A = A , antisymmetric matrices, for which A = — A, and orthogonal matrices, for which U = U. A particular type of orthogonal matrix is called a rotation matrix and satisfies the relation Det(R)= -I-1, where Det(R) is the usual definition of a determi-... 

The real transformation matrix U is actually a rotation matrix since Det(U) = Det(exp(K)) = exp(Tr(K)) = exp(0)= +1. The orbital phases are not important in the MCSCF method so that this loss of generality, compared to more general orthogonal transformations, is not significant. There are two representations of the K operator that are useful. The first results directly from Eq. (108) and is given as... 

For SCF methods, p represents elements of the one- and two-particle density matrices. In correlated methods, the one-particle part of p is the sum of the aaual reduced density and a contribution that is proportional to the derivative of the energy with respect to orbital rotations. For the MCSCF method, the orbitals are variationally optimum, and this latter term vanishes. Similarly, for FCI there is no orbital contribution. However, it is required for other Cl, CC, and MBPT correlated methods, because the energy in these approaches is not stationary with respect to first-order changes in the molecular orbitals. This... 

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. 

However, until today no systematic comparison of methods based on MpUer-Plesset perturbation (MP) and Coupled Cluster theory, the SOPPA or multiconfigurational linear response theory has been presented. The present study is a first attempt to remedy this situation. Calculations of the rotational g factor of HF, H2O, NH3 and CH4 were carried out at the level of Hartree-Fock (SCF) and multiconfigurational Hartree-Fock (MCSCF) linear response theory, the SOPPA and SOPPA(CCSD) [40], MpUer-Plesset perturbation theory to second (MP2), third (MP3) and fourth order without the triples contributions (MP4SDQ) and finally coupled cluster singles and doubles theory. The same basis sets and geometries were employed in all calculations for a given molecule. The results obtained with the different methods are therefore for the first time direct comparable and consistent conclusions about the performance of the different methods can be made. 

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. 

Previous results show that the EJR method is very promising to solve the convergence problem in MCSCF computations. The only snag is the number of integral transformations to be done in the exact formulation. However, this problem can be obviated if instead of the "exact" EJR algorithm outlined before, some kind of accumulated EJR is used /I/, where the integrals are only transformed after the full set of rotations is done, in the way this problem is treated in the exponential transformations. Preliminary work in this direction shows that the efficiency of the method is kept while the amount of time in the intregral transformations is lowered. 

The subset of operators satisfying Eq. (232) is usually known before the MCSCF optimization process is initiated. In fact, many CSF generation methods, including the graphical unitary group approach, allow the specification of the operators of Eq. (232) or the orbital subsets that define these rotation operators. The corresponding orbital rotation parameters may then be deleted from the MCSCF optimization process for any state and for any geometry for wavefunctions expanded in this CSF space. The direct product... 

As the accuracy of electron correlation methods is Hmited by the choice of the dimension of the active orbital space, the convergence behavior of the spin density with respect to the size of the active space must be studied to ensure that accurate ah initio spin densities are obtained. Although the CASSCF spin densities were quantitatively converged for medium-sized active orbital spaces, larger active spaces (more than 13 electrons correlated in 13 orbitals) were found to be unstable, that is, active space orbitals have been rotated out of the active space during the MCSCF procedure, while the spin densities started to diverge compared to the smaller sized CASSCF results. 

One of the most popular methods of MCSCF class is the complete active space SCF (CASSCF) method by Roos [118] where FCI is performed in active space and op>-timization of MOs is also done. Note that active-active orbital rotations are irrelevant in the framework of this method. 

In this section, we shall ply the second-order Newton method to the optimization of the Hartree-Fock energy, considering both a method that works with the AO density matrix and is applicable to large systems, and a method that carries out rotations among the MOs and is applicable to general single-configuration states [16-18]. Our discussion is here restricted to minimizations - in the discussion of MCSCF theory in Chapter 12, we shall consider also the second-order localization of saddle points. 

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