Kohn-Sham MO theory

However, a detailed reexamination within the framework of Kohn-Sham MO theory leads us to a view that differs in a number of ways.51 In the following, we show that the C—Li bond in CH3Li (25) may very well be envisaged as an electron pair bond (24b), although a rather polar one, of course. But our point is not just the shift of the bonding picture back to the more covalent side of the 24a-24b spectrum. In particular, if we go to the oligomers (e.g., the methyllithium tetramer 27), a fundamentally new phenomenon occurs in the C—Li bonding mechanism. This phenomenon emphasizes the presence of dis-... 

Reactivity with Kohn-Sham MO Theory. The E2—SN-2 Mechanistic Spectrum and Other Concepts. 

In Kohn-Sham DFT based approaches, expressions that are of similar structure as Eqs. (9a) and (9b) are obtained, but in the form of contributions from all occupied Kohn-Sham MOs The excited-state wavefunctions are at the same time formally replaced by the unoccupied MOs, and the many-electron perturbation operators /T(M41, etc. by their one-electron counterparts //(M-41, etc. Orbital energies e and ea formally substitute the total energies of the states (see later). Thus, similar interpretations of NMR parameters can be worked out in which the highest occupied MO-lowest unoccupied MO gap (HLG) plays a highly important role. It must be emphasized, though, that there is no one-to-one correspondence between the excited states of the SOS equations and the unoccupied orbitals which enter the DFT expressions, nor between excitation energies and orbital energy differences, i.e., there are no one-determinantal wavefunctions in Kohn-Sham DFT perturbation theory which approximate the reference and excited states. 

This qualitative model, based on semiempirical MO theory, focuses entirely on the so-called electronic effects, as the A—H bonding orbital interactions are often called. However, steric repulsion (i.e., the destabilizing orbital interactions) between the hydrogen substituents in AH3 is just as important in the interplay of mechanisms that determine whether the molecule adopts a planar or a pyramidal shape. In fact, as will become clear from the following discussion, which is based on a Kohn-Sham DFT study at the BP86/TZ2P level,107 108 steric repulsion turns out to be the decisive factor in determining the pucker of our example.133... 

Strictly speaking, the Kohn-Sham (KS) orbitals are fictitious entities, created by a certain mathematical procedure. Of course, the same can be said about molecular orbitals (MOs) in HF theory. But we know that MOs are given reality by their successful use in many applications. The KS orbitals and the HF MOs are not the same, but there is a one-to-one correspondence, and their orbital energies are similar. It would appear that KS orbitals will one day be used in the same way as MOs are now. 

At this point, the Kohn-Sham Hamiltonian operator can be written and the expectation value determined (compare with the above proceedings for the MO theory) ... 

The quantities F(X) can be obtained straightforwardly if the calculations are carried out within the Density Fimctional Theory (DEI) formalism. Indeed, for ionization of an electron from molecular orbital (MO) Kohn-Sham type calculation in DFT, with an occupation 1 - A = 5 in level number k. In the Generalized Transition State (GTS) method, Williams et al. [16] proposed the use of ... 

As we have already observed above, within the hardness (interaction) representation (see Tables 1, 3) the FF indices provide important weighting factors in combination formulas which express the global CS and potentials in terms of the local properties, relevant for the resolution in question. The FF expressions from the EE equations are invalid in the MO resolution, since no equalization of the orbital potentials can take place, due to obvious constraints on the MO occupations in the Hartree-Fock (HF) theory [61]. Moreover, standard chain-rule transformations of derivatives are not applicable in the MO resolution since some of the derivatives involved are not properly defined. Various approaches to the local FF, f(f), have been proposed e.g., those expressing f(f), in terms of the frontier orbital densities [11, 25], or the spin densities [38]. Also the finite difference estimates of the chemical potential (electronegativity) and hardness have been proposed in the MO and Kohn-Sham theories for various electron configurations [10, 11, 19, 52, 61b, 62, 63]. 

While this endeavor was made in the efforts to discredit the MO approach and the orbital concept in general, we believe that atomic orbitals and their linear combination provide the set of elementary properties of mater on which base the whole chemistry can be rationalized based on a single (i.e., the eigen-value problem) principle, either in Schrodinger, Hartree-Fock/ Roothaan or Kohn-Sham/Density Frmctional Theory (see below) approaches. 

From the beginning of the utilization of density functional theory (DFT), the significance of Kohn-Sham (KS) orbitals has been de-emphasized and they were regarded as only auxiliary mathematical constructs. Until recently, there was a widespread belief that the KS orbitals have no physical significance and should not be used for rationalization of experimental data in the same way as conventional MOs, and that their only connection to the observable properties is that the sum of the densities of the occupied KS orbitals is the exact electron density ... 

ADF contains several analysis options, offering the possibility of gaining detailed understanding of the chemical problem at hand. These methods underline the underlying philosophy that the Kohn-Sham orbitals in DFT can be used for a quantitative MO theory. 

Most molecular quantum-mechanical methods, whether SCF, Cl, perturbation theory (Section 16.3), coupled cluster (Section 16.4), or density functional (Section 16.5), begin the calculation with the choice of a set of basis functions Xn which are used to express the MOs (pi as = IiiCriXr [Eq. (14.33)]. (Density-functional theory uses orbitals called Kohn-Sham orbitals P that are expressed as (pf = 1,iCriXn see Section 16.5.) The use of an adequate basis set is an essential requirement for success of the calculation. 

As typically implemented, DFT calculations are similar to HF calculations. A set of MOs called the Kohn-Sham orbitals are iteratively improved until they converge on self-consistency. The number, shape, and symmetry properties of the Kohn-Sham orbitals are similar to the HF orbitals. However, the orbital energies are generally not in good agreement with those from either HF theory or experiment, and so there is no analogue to Koopmans theorem in DFT. 

The highest occupied Kohn-Sham orbital energy is equal to the exact first ionization energy. This is a property that is very desirable in qualitative MO theory in general and is often simply assumed in such theories. 

Because the Fock matrix depends on the one-particle density matrix P constructed conventionally using the MO coefficient matrix C as the solution of the pseudo-eigenvalue problem (Eq. [7]), the SCF equation needs to be solved iteratively. The same holds for Kohn-Sham density functional theory (KS-DFT) where the exchange part in the Fock matrix (Eq. [9]) is at least partly replaced by a so-called exchange-correlation functional term. For both HF and DFT, Eq. [7] needs to be solved self-consistently, and accordingly, these methods are denoted as SCE methods. 

---