MCSCF density matrix

This is the MCSCF Fock matrix. It is defined in terms of the first and second order reduced density matrices D and P as given by (3 16) and (3 23) or with the present nomenclature ... 

We can simplify the calculation of the matrix (4 50) in the same way as we did for the MCSCF Fock matrix. The same type of integrals are thus needed here. However, the calculation is now much more complicated since we need the transition density matrices. In practical applications one does not work directly with the complementary states IK>. Instead the calculation is performed over the Cl basis states lm>, where the Cl coupling coefficients occur instead of the transition density matrices. 

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. 

These formulae differ from Eqs (39) and (41) only in the use of F " instead of h and the restrictions of the summations. Hence, all second-order density matrix elements involving closed-shell orbitals have been eliminated. However, all operators J and K are still needed in Eq. (88). Since the computational effort for their evaluation depends strongly on the number of optimized orbitals, it would also be useful to eliminate the operators J and K involving any closed-shell orbitals. As shown in the following, this is possible in a direct MCSCF procedure. 

A feature of the template or index-driven approach, which was first proposed by Shavitt for direct Cl algorithms ", is that all contributions to a particular density matrix element, or may be computed together. In the MCSCF procedure, it is also useful to compute quantities of the form <0 erstul ) for a fixed (rstu) and for all possible n>. MCSCF procedures that do not use such a method must instead sort the list of coupling coefficients, which are computed in some arbitrary order, into an order that allows the orderly computation of these transition density matrix elements. The index-driven approach avoids this unnecessary sorting step in those cases where the coupling coefficients are explicitly written to an external file, and it allows the efficient computation of the required coefficients in those cases where they are used as they are computed. The relative merits of the index-driven and CSF-driven approaches are discussed further in Section VI. 

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... 

In an independent second-order MCSCF implementation by Lengsfield" , the density matrix was explicitly constructed and stored. It was correctly assumed that the number of occupied orbitals is sufficiently small for most MCSCF wavefunctions so that the tw o-particle density matrix may be stored in computer memory. This results in a significant improvement in reducing the number of arithmetic operations over the earlier SCI approaches. However, this program still employed a formula file in the construction of the Hessian matrix consisting of the list of integral indices and Hessian matrix indices associated with each density matrix element. 

In these expressions the indices i,j, k and I are used for doubly occupied orbital labels while p and q are used for active orbital labels. Other density matrix element types such as ),p, and dij p are all zero. The above expressions may also be used to simplify the transformation of the density matrices from one orbital basis to another as is required, for example, in the evaluation of MCSCF molecular properties in the AO basis. The use of these identities also eliminates the need for any coupling coefficients involving the doubly occupied orbitals. 

For open-shell and small (e.g. two-configuration) MCSCF wavefunctions, the construction of the AO density matrix according to Eq. (32) is computationally negligible compared to the evaluation of the integral derivatives. The open-shell case is thus computationally identical to the closed-shell case. 

For long MCSCF expansions, construction of the AO density matrix becomes a major computational task, and an alternative method, similar to Cl derivative calculation, must be used. This method, outlined by Meyer (1976), and first used by Brooks et al. (1980), consists of the explicit transformation of the MO two-particle density matrix to AO basis, prior to gradient evaluation. Transformation of the density matrix is obviously superior to the transformation of the 3N gradient components of each integral. 

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 ... 

For the weakest occupied orbitals, it makes sense to interchange particles and holes. The resulting analog to the MCSCF Fock matrix is then 2f- and the hole density matrix is, as before, d = 2 1 - D. 

The general theory outlined in Chapter 14 has been used in the ROHF model. It can also provide the basis for a whole class of multi-determinant models of molecular electronic structure which are constructed from sets o/doubly occupied orbitals. Two such models are outlined in this chapter Paired-Electron MCSCF (PEMCSCF) and the General Valence Bond (GVB). These pair expansion theories are based on the idea of natural orbitals which diagonalise the one-electron density matrix. 

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