Avoiding the Diagonalization Step—Density Matrix-Based SCF

AVOIDING THE DIAGONALIZATION STEP—DENSITY MATRIX-BASED SCF 

By diagonalizing the Fock matrix, the canonical MO coefficient matrix (C) is obtained (see Eq. [7]). However, we have seen in a previous section that almost all elements in the coefficient matrix are significant, which contrasts with the favorable behavior of the one-particle density matrix (P). The density matrix is conventionally constructed from the coefficient matrix by a matrix product (Eq. [8]). Although the Roothaan-Hall equations are useful for small-to medium-sized molecules, it makes no sense to solve first for a nonlocal quantity (C) and generate from this the local quantity (P) in order to compute the Fock matrix or the energy of a molecule. Therefore, the goal is to solve directly for the one-particle density matrix as a local quantity and avoid entirely the use of the molecular orbital coefficient matrix. 

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Avoiding the Diagonalization Step—Density Matrix-Based SCF 43... 

From this density, a new Fock operator is calculated and this process is repeated until convergence - that is, until the density obtained from (10.7.90) becomes identical to the density from which the Fock matrix was calculated. Since this approach involves the solution of a generalized eigenvalue problem (10.7.89), whose cost scales cubically with the size of the system, it becomes prohibitively expensive for large systems. In the density-based SCF method, we avoid the diagonalization step altogether, replacing it by a direct optimization of the AO density matrix. 

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