Density matrix-based SCF

AVOIDING THE DIAGONALIZATION STEP—DENSITY MATRIX-BASED SCF... 

In the following sections, we will provide an overview of density matrix-based SCF theory that allows one to exploit the naturally local behavior of the one-particle density matrix for molecular systems with a nonvanishing HOMO-LUMO gap. Besides the density matrix-based theories sketched below, " a range of other methods exists, including divide-and-conquer methods, Fermi operator expansions (FOE), °° Fermi operator projection (FOP), ° orbital minimization (OM), ° ° and optimal basis density-matrix minimization (OBDMM). ° ° Although different in detail, many share as a common feature the idea of (imposed or natural) localization regions in order to achieve an overall 0 M) complexity. This notion implies that the density matrix (or the molecule) may be divided into smaller... 

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

In the derivation of density matrix-based SCF theory below, we do not employ the chemical potential introduced by LNV, but instead we follow the derivation of Ochsenfeld and Head-Gordon, because McWeeny s purification automatically preserves the electron number.Therefore, to avoid the diagonalization within the SCF procedure, we minimize the energy functional... 

As discussed, for a formulation of SCF theories suitable for large molecules, it is necessary to avoid the nonlocal MO coefficient matrix, which is conventionally obtained by diagonalizing the Fock matrix. Instead we employ the one-particle density matrix throughout. For achieving such a reformulation of SCF theory in a density matrix-based way, we can start by looking at SCF theory from a slightly different viewpoint. To solve the SCF problem, we need to minimize the energy functional of... 

Density Matrix-Based Quadratically Convergent SCF (D-QCSCF)... 

Density Matrix-Based Coupled Perturbed SCF (D-CPSCF)... 

In the preceding sections, we discussed the optimization of the Hartree-Fock wave function by the iterative SCF procedure, using an approach based on the diagonalization of the Fock matrix as well as an approach based on the optimization of the AO density matrix. The SCF method, in particular when accelerated with the DIIS scheme, is simple to implement and works well in many cases, especially for closed-shell systems close to the equilibrium geometry. Nevertheless, in some cases, conveigence may be difficult to achieve even with the DIIS method. In such cases, it may be better to use one of the standard methods of numerical analysis. 

Step 13 - Comparing the density matrix from the latest c s with the previous density matrix to see if the SCF procedure has converged The density matrix elements based on the c s of C2 are... 

Note that the starting choice for the coefficients of equation 6.2 are based on the assumption there are no two-electron terms, i.e. the input density matrix, cells E 11 to F 12 are set to zero from the link to canonical G 2 to scf G 3. Remember, too, that as in the case of the ground state of helium there is only one doubly-occupied orbital, so the output density matrix is comprised of the coefficients of the 1CT(,+ orbital only. 

A many-body perturbation theory (MBPT) approach has been combined with the polarizable continuum model (PCM) of the electrostatic solvation. The first approximation called by authors the perturbation theory at energy level (PTE) consists of the solution of the PCM problem at the Hartree-Fock level to find the solvent reaction potential and the wavefunction for the calculation of the MBPT correction to the energy. In the second approximation, called the perturbation theory at the density matrix level only (PTD), the calculation of the reaction potential and electrostatic free energy is based on the MBPT corrected wavefunction for the isolated molecule. At the next approximation (perturbation theory at the energy and density matrix level, PTED), both the energy and the wave function are solvent reaction field and MBPT corrected. The self-consistent reaction field model has been also applied within the complete active space self-consistent field (CAS SCF) theory and the eomplete aetive space second-order perturbation theory. ... 

The new scheme is based on the SCF density matrix defined in an atomic orbital (AO) basis as ... 

As we have seen, EHT is a nonself-consistent method but the self-consistency over charge and configuration is included in the MR approximation. The Ab-initio HE SCF method requires the self-consistent calculation of the density matrix (see Chap. 4). This feature of the HE approach is maintained in the semiempirical methods, based on the zero differential overlap (ZDO) approximation. This approximation is used to reduce the number of multicenter integrals appearing in HE LCAO calculations. 

The diagonalization of the one-particle density matrix P on the Cl level yields the natural orbitals (NOs) and the occupation numbers of the molecule. In an RHF-SCF calculation, all occupation numbers are either 0 or 2, which means that an MO is either doubly occupied by 2 electrons or not occupied. If we invoke a Cl calculation, the mixing of configurations results in fractional occupation numbers of the NOs. Still in a normal molecule, such as ethane, water, ammonia, etc., the occupation numbers for the ground state will be close to 0 or 2. The occupation numbers of at least two NOs of excited states usually deviate from the values 0 and 2. In a biradicaloid structure occupation numbers will change and typically two of them will be close to 1. However, this method cannot characterize zwitterions since occupation numbers should be close to 0 and 2, respectively, in this case. The approach by Jug and Poredda yields both zwitterionic and diradical character of species, based on the valence criterion of Gopinathan and Jug. The valence is defined according to equation (5), which can be reformulated as equation (6). 

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