Super-CI method

The super-CI method now implies solving the corresponding secular problem and using tpq as the exponential parameters for the orbital rotations. Alternatively we can construct the first order density matrix corresponding to the wave function (4 55), diagonalize it, and use the natural orbitals as the new trial orbitals in I0>. Both methods incorporate the effects of lpq> into I0> to second order in tpq. We can therefore expect tpq to decrease in the next iteration. At convergence all t will vanish, which is equivalent to the condition ... 

The super-CI method thus leads to the stationary point by a direct annihilation of the single excitations (4 54). 

The super-CI method can alternatively be given in a folded form, which includes the coupling between the Cl and orbital rotations. This is done by adding the complementary Cl space, IK>, to the super-CI secular problem. As in the Newton-Raphson approach, it is more efficient to transform the equations back to the original CSF space, and thus work with a super-CI consisting of the Cl basis states plus the SX states. It is left to the reader as an exercise to construct the corresponding secular equation and compare it with the folded one-step Newton-Raphson equations (4 22). 

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. 

Table 1. Convergence in a CASSCF calculation on water, with a DTP basis. The approximate super-CI method was used with and without quasi-Newton update. The active space comprised 8 orbitals (4a12b1, 2b2 in C2v symmetry), yielding 492 CSF s. The Is orbital was inactive.

Calculate the explicit expression, in terms of first and second order density matrices, for the overlap integral (4 57) occurring in the super-CI method. 

This is input for level shift, which is also default. A level shift is used to raise the diagonal elements of the super-CI matrix (if needed), such that the smallest value equals the input value. This is to avoid too large rotations, especially in the beginning of a calculation. The approximate super-CI method used in MOLCAS sometimes diverges, when no level shift is employed. 

The super-CI method can be regarded as an approximation in the augmented Hessian variant of the Newton-Raphson (NR) procedure (see for example Ref 43). However, in its original formulation, it does not eonstitute any simplification when compared with the NR method. The matrix elements between the Brillouin states (41) are actually more difficult to compute than the corresponding Hessian matrix elements, since they involve third-order density matrix elements" . 

The orbital optimization was, in the first implementation of the CASSCF method, performed using an approximate version of the super-CI method, which avoided the calculation of the third-order density matrix s. This was later developed as a procedure entirely based on the average MCSCF Fock operator ... 

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