Canonical Kohn-Sham orbitals

The idea of distributed dipole moments has also been transferred to the dynamic domain and we shall discuss recent work from our laboratory in this section in more detail. With the help of maximally localized Wannier functions local dipoles and charges on atoms can be derived. The Wannier functions are obtained by Boys localization scheme [217]. Thus, Wannier orbitals [218] are the condensed phase analogs of localized molecular orbitals known from quantum chemistry. Access to the electronic structure during a CPMD simulation allows the calculation of electronic properties. Through an appropriate unitary transformation U of the canonical Kohn-Sham orbitals maximally localized Wannier functions (MLWFs)... 

Everytime the orbital parameters are changed, one produces a new wavefunction belonging to a different orbit. This follows from the one to one correspondence between iV-representable densities and wavefunctions within a given orbit. As a result, optimization of these transformed orbitals in the kinetic energy expression leads to the minimum value Ts[ corresponding to the parameters The orbitals are, in general, non-canonical Kohn-Sham orbitals, i.e., they... 

The orbitals /i mm) 2=i are not yet the canonical Kohn-Sham orbitals of Eq. (85). The canonical Kohn-Sham orbitals are obtained by rotating the former ones through an angle 0 = Oks Explicitly, the rotated orbitals are ... 

Substituting O Eq. 16,12 in O Eq. 16.5 yields the canonical Kohn-Sham orbital equations ... 

Figure 8 shows one of the orbitals after the isopycnic transformation of the canonical Kohn-Sham orbitals. Having occupation number 1 for each spin channel, it nicely represents the 2c-2e bond between the nearest carbon atoms. 

We may again chose a unitary transfonnation which makes tlie matrix of the Lagrange multiplier diagonal, producing a set of canonical Kohn-Sham (KS) orbitals. The resulting pseudo-eigenvalue equations are known as the Kohn-Sham equations. 

As noted in Sect. 4, a unitary transformation (/> —> ( = leaves both the density n(r) and the total energy invariant. Any unitary transformation of the Kohn-Sham orbitals is thus a valid set of orbitals. Canonical orbitals are a special set of such orbitals which diagonalize the Kohn-Sham Hamiltonian. Localized orbitals on the other hand are obtained by finding the unitary transformation U so as to optimize the expectation value of a two electrons operator Q ... 

The centrality of the FNA has spawned considerable research into improvement of the approach. The strategies for obtaining better nodes are numerous. Canonical HF orbitals, Kohn-Sham orbitals from density functional theory (DFT), and natural orbitals from post-HF methods have been used. The latter do not necessarily yield better nodes than single configuration wave functions [39-41]. More success has been found with alternative wave function forms that include correlation more directly than sums of Slater determinants. These include antisymmetrized geminal power functions [42,43], valence-bond [44,45] and Pfaffian [46] forms as well as... 

The 6ji in O Eq. 16.35 are the undetermined Lagrange multipliers. Because the electronic density is invariant to unitary transformations of the occupied molecular orbitals (MOs) it is possible (and convenient) to choose a set of MOs for which the off-diagonal multipliers are zero. These MOs are called canonical and are the solutions of the canonical Kohn-Sham equations. 

This section summarizes the TDDFT linear response approach to compute optical rotation and circular dichroism. For reasons of brevity, assume a closed shell system, real orbitals, and a complete basis set (see Sect. 2.4 for comments regarding basis set incompleteness issues). From solving the canonical ground state Kohn-Sham (KS) equations,... 

Density functional theory also offers an attractive computational scheme, the Kohn-Sham (KS) theory [2], similar to the Hartree-Fock (HF) approach, which in principle takes into account both the electron exchange and correlation effects. The canonical KS orbitals thus offer certain interpretative advantages over the widely used HF orbitals, especially for describing the bond dissociation and the open system characteristics, when the electrons are added or removed from the system [3,82,126-130]. For this reason, a determined effort has been made to calculate the reactivity indices from the KS DFT calculations [3,82,83,112,118,119,121, 131-136]. 

Kohn-Sham determinant represents an eigenfunction of the fictitious system. Scientists compared the Kohn-Sham otbitals with the canonical Hartree-Fock orbitals with great interest. It turns out that the diffetenees are small. 

The POOs can be expanded in terms of a set of ortho-normalized virtual orbitals (e.g. the set of canonical virtu-als). Although later we eliminate explicit reference to the set of virtual orbitals of the Fock/Kohn-Sham operator (i.e., everything will be written only in terms of the occupied orbitals or equivalently in terms of the corresponding density matrix), with the help of the resolution of identity, we give the explicit form of the coefficient matrix linking the POOs with the virtuals ... 

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