Reference states, diagonal

The disappearance of matrix elements between the HF reference and singly excited states is known as Brillouins theorem. The HF reference state therefore only has nonzero matrix elements with doubly excited determinants, and the full Cl matrix acquires a block diagonal structure. 

For many ionization energies and electron affinities, diagonal selfenergy approximations are inappropriate. Methods with nondiagonal self-energies allow Dyson orbitals to be written as linear combinations of reference-state orbitals. In most of these approximations, combinations of canonical, Hartree-Fock orbitals are used for this purpose, i.e. 

Although P3 procedures perform well for a variety of atomic and molecular species, caution is necessary when applying this method to open-shell reference states. Systems with broken symmetry in unrestricted Hartree-Fock orbitals should be avoided. Systems with high multireference character are unlikely to be described well by the P3 or any other diagonal approximation. In such cases, a renormalized elec-... 

The systematic error caused by the definition of the zeroth order Hamiltonian, as described above, leads to too low relative energies for systems with open shells. A consequence is that dissociation and excitation energies will be too low because the dissociated or excited state has usually more open shell character than the reference state. Is there a way we can remedy this systematic error The diagonal elements of the generalized Fock operator can for an active orbitals be estimated as ... 

The diagonal scale above the saturation curve on the psychrometric chart shows the enthalpy of a unit mass (1 kg or 1 Ibm) of dry air plus the water vapor it contains at saturation. The reference states are liquid water at 1 atm and 0°C (32°F) and dry air at 1 atm and 0°C (Figure 8.4-1) or O F (Figure 8.4-2). To determine the enthalpy from the chart, follow the constant wet-bulb temperature line from the saturation curve at the desired temperature to the enthalpy scale. 

For closed-shell reference states, doublet final states are of primary interest. Here, one may take spin symmetry into account, as well as molecular point groups. Zeroth-order terms appear only in diagonal elements of the 2hp-2hp block, where... 

The elements A22 and V thus contain all matrix elements contained in a configuration interaction calculation within the considered configuration space. When the iterative MCSCF procedure has converged, all elements of V are zero and the interactions between the reference state ()> and the residual states are thus eliminated. The diagonal and off-diagonal matrix elements... 

Since the one-particle density matrix is diagonal for the chosen HF reference state, we have... 

As in general all the y-coefficients do not vanish one has to assume a more general reference state than the single determinant SCF state. This is the rather well-known problem of finding the consistent reference state for the Random Phase Approximation (RPA). It also means that the field operator basis can be enlarged and can for instance include the iV-electron occupation number operators (in this discussion, electron field operators and their adjoints are used referring to a basis of spin orbitals that are the natural spin orbitals of the reference state, as will be discussed below, i.e., the spin orbitals that diagonalize the one-matrix)... 

In electronic structure calculations, the zero point for the energy function has all particles (electrons and nuclei) infinitely removed from each other, and this common reference state allows energies for systems with different numbers of particles to be directly compared. If the same reference is used in force field methods, the energy function becomes an absolute measure of molecular stability. The difference relative to the normal reference state for force field functions is the sum of all bond dissociation energies, at least for a simple diagonal force field. If correction terms are added to the... 

In the previous. section we discussed the reference force field of /-PA (see Table 6.2) derived from the force field of butadiene. In the Og symmetry block, the high frequency C—H stretch is decoupled from the other modes and thus from tt electrons. We are left with three relevant Ug modes their reference and experimental frequencies are reported in Table 6.3 and, as discussed in Section II, fix the matrix and the x cd) curves in Fig. 6.4. The A matrix is written on the basis of the reference normal coordinates Q . It consequently depends on both the G and F matrices and varies with molecular or polymeric structure. The e-ph coupling constants g, thus vary even with isotopic substitution. To define coupling constants independent of mass, we use the symmetry coordinates to solve the GF problem for the reference state. In fact, diagonalization of GF gives both eigenvector matrix L in the S basis. The L matrix is used to transform Jin Eq. (12) back to the S basis ... 

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