Orbital minimization

The localization of such hybrids comes from the combination of the CASSCF orbitals minimizing the energy for a wavefunction written in terms of structures 1-4 the degree of polarization can thus be easily studied for all geometries through these kinds of plots. 

The Hartree-Fock equation is obtained by requiring that the orbitals minimize the expectation value of the energy. Those orbitals satisfy... 

Such diffuse orbitals are found on atoms of small x and small 77. Diffuse orbitals also overlap at larger atomic distances. This, in turn, permits an increase in the coordination number. Electron-electron repulsion is greatest when both electrons are on the same atom in this case diffuse orbitals minimize the repulsion energy. ... 

Equation (73) implies that the search for the exact ground-state wavefunction must be carried out by a combined intra-orbit and inter-orbit minimization [7]. The former reflects the charge consistency variational principle, whereas the latter the... 

The LCAO method can be used to obtain molecular orbitals for any diatomic molecule from the appropriate atomic orbitals. Minimizing the energy leads to an equation known as the secular determinant, from which the best energies for the molecular orbitals can be calculated. 

The orbital model explains the facts about double bonds listed in Table 3.1. Rotation about a double bond is restricted because, for rotation to occur, we would have to break the pi bond, as seen in Figure 3.6. For ethylene, it takes about 62 kcal/mol (259 kj/mol) to break the pi bond, much more thermal energy than is available at room temperature. With the pi bond intact, the sp orbitals on each carbon lie in a single plane. The 120° angle between those orbitals minimizes repulsion between the electrons in them. Finally, the carbon-carbon double bond is shorter than the carbon-carbon single bond because the two shared electron pairs draw the nuclei closer together than a single pair does. 

We do not yet know if the spin orbitals that are the solution to Equation 2.30 are also the looked-for spin orbitals, minimizing the energy expectation value in Equation 2.27. The proof will be carried out in steps. First, we have to construct a many-electron operator that corresponds to the Fock one-electron operator. This is easy to do if we subtract a constant term to compensate for the fact that electron repulsion is counted twice. We obtain the many-electron Hamiltonian as... 

It is possible to choose spin orbitals where the coefficients of the singly substituted determinant are equal to zero, even if doubly and higher substitutions are introduced. In a Cl type expansion using these orbitals, the Slater determinant with the latter spin orbitals has the maximum possible overlap with the true wave function. The new spin orbitals are therefore called best overlap orbitals. In a Slater determinant O, these orbitals minimize... 

These orbitals minimize the total energy E = . In a finite basis set representation, the molecular orbitals (MOs) are expanded in terms of known basis orbitals as ... 

Student Annotation Because electrons are negatively charged, they repel one another. Maximizing parallel spins in separate orbitals minimizes the electron-electron repulsions in a subshell. 

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... 

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