Canonical HF equation

The occupied single-particle functions (j) and the virtual single-particle functions ( )a are solutions of the corresponding canonical HF equations... 

The generalized two-particle HF equations are seen to have a structure equivalent to their single-particle counterparts, exhibiting the presence of a direct term, written in terms of the density, and an exchange term. As the canonical HF equations, the present expressions do not contain spurious self-interaction terms. However, unlike the single-particle equations, they allow the determination of fully correlated two-particle states removing to this extent the most basic objection to the HF method. 

For such a function, variational optimization of the spin orbitals to make the expectation value stationary produces [30] the canonical HF equations... 

The immense value of the IPM description of atomic and molecular structure, with orbitals determined from the canonical HF equations rests mainly upon the following results. 

This is a legitimate approximation to only when the spin-orbitals satisfy the canonical HF equations for then E, e, and consequently the difference Et, are all stationary against spin-orbital variations, and the same spin-orbitals therefore serve to describe both the neutral system and its ions. 

In view of the above results, molecular orbital theory, with orbitals and orbital energies determined from the canonical HF equations, provides the natural tool for the interpretation of electronic spectroscopy it was indeed developed and used for this purpose, long before accurate solution of the HF equations became feasible, in the classic papers of R.S. Mulliken during the nineteen twenties and thirties. 

The terms with the eia> dependence merely lead to the complex conjugate of the above equation. Because the HF equation and energy are invariant to rotations among just occupied orbitals or among just virtual orbitals, we only need to consider the occupied-virtual block of d, i.e., d - and dj. Furthermore, if we assume the canonical HF wave function as an initial state, the zeroth-order quantities simplify to... 

Both CIS and TDHF have the correct size dependence and can be applied to large molecules and solids (we will shortly substantiate what is meant by the correct size dependence ) [42-51], It is this property and their relatively low computer cost that render these methods unique significance in the subject area of this book despite their obvious weaknesses as quantitative excited-state theories. They can usually provide an adequate zeroth-order description of excitons in solids [50], Adapting the TDHF or CIS equations (or any methods with correct size dependence, for that matter) to infinitely extended, periodic insulators is rather straightforward. First, we recognize that a canonical HF orbital of a periodic system is characterized by a quantum number k (wave vector), which is proportional to the electron s linear momentum kh. In a one-dimensional extended system, the orbital is... 

The zero-order wave function, i.e., the single determinant HF function, consists of canonical HF spinorbitals satisfying the equations ... 

Upon introducing the LCAO approximation and then minimizing the total energy in Eq. (20) with respect to the MO coefficients (subject to the orthonormality of the MO s), the familiar algebraic equations for the canonical orbitals, analogous to the Pople-Nesbet equations [56] in HF theory, are obtained, with the XC matrix elements given by [57]... 

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