MCSCF coupling coefficients

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. 

The number of variational parameters in internally contracted MCSCF-SCEP wavefunctions rarely exceeds 10 even if large basis sets and complex reference wavefunctions are employed. In contrast to the number of variational parameters, the number of coupling coefficients depends on the number of reference configurations, and can become very large if CASSCF references are used. The main problem is, therefore, the calculation and storage of the coupling coefficients. It would be very helpful if at least part of them could be recalculated each time they are needed. As will be discussed in Section 111.I, this would also allow one to relax the contraction coefficients in each direct Cl iteration, thereby improving the quality of the wavefunction. 

In the following we divide the internal orbital space into three subspaces the core orbitals, which are doubly occupied and not correlated the closed-shell orbitals, which are correlated and the active orbitals, which are only partially filled in the MCSCF wavefunction. We will show that it is sufficient to evaluate coupling coefficients only for the active subspace. For the special case that the reference wavefunction is a single closed-shell determinant, the algorithm then reduces to the closed-shell SCEP method described by Meyer, and no coupling coefficients have to be calculated explicitly. ... 

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 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. 

Grayson and Sauer computed the coefficients in a Karplus-type equation of the spin-spin coupling constants for a series of rotated ethane geometries. The coupling constants were calculated at the SCF, SOPPA, and SOPPA-CCSD levels and compared with results of previous calculations and experimental data. It was found that the coefficients in the Karplus equation calculated at the SOPPA-CCSD level are in good agreement with coefficients derived from experimental coupling constant data or results of MCSCF calculations. 

Hence, if we are to make bE = 0, we can avoid these terms. But to do so we have to have E optimum with respect to the location of the atomic basis functions, t (R) the MO coefficients, c(R) and the Cl coefficients, C(R). The first cannot be satisfied unless the atomic orbital basis set is floated off the atomic centers to an optimum location [105], while the second requires optimum MO coefficients, and the third optimum Cl coefficients. In practice, we will introduce atomic orbital derivatives explicitly, so the AOs can follow their atoms. Now focusing only on the MO and Cl coefficients, in SCF we have optimum MOs and no Cl term. In MCSCF, both terms would vanish, whUe in Cl, the MO derivatives would remain, but the Cl coefficients contribution would vanish. In the non-variational coupled-cluster theory, neither will vanish and this means that CC theory forces us into some new considerations for analytical forces. 

Linear response theory expression Alternatively, the spin-spin coupling constant can be expressed using the linear response theory formalism. Let us write the electronic energy of the system perturbed by the nuclear magnetic dipole moments M/f in the form E = E(Mjf, A), where A are the variational parameters of the wave function. A may represent orbital rotation parameters for the SCF wave function, or orbital rotation parameters and coefficients of the configuration interaction expansion for the MCSCF... 

In this way, the Cl coefficients finally enter the SCF equations, which is necessary to couple spinor and Cl coefficient optimizations in MCSCF calculations. 

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