Hamiltonian operator Kohn-Sham

Note that the Kohn-Sham Hamiltonian hKS [Eq. (4.1)] is a local operator, uniquely determined by electron density15. This is the main difference with respect to the Hartree-Fock equations which contain a nonlocal operator, namely the exchange part of the potential operator. In addition, the KS equations incorporate the correlation effects through Vxc whereas they are lacking in the Hartree-Fock SCF scheme. Nevertheless, though the latter model cannot be considered a special case of the KS equations, there are some similarities between the Hartree-Fock and the Kohn-Sham methods, as both lead to a set of one-electron equations allowing to describe an n-electron system. 

In the Kohn Sham equations (A.116) [324, 325], the core Hamiltonian operator h( 1) has the same definition as in HF theory (equation A.6), as does the Coulomb operator, 7(1), although the latter is usually expressed as... 

Note that because the E of Eq. (8.14) that we are minimizing is exact, the orbitals / must provide die exact density (i.e., the minimum must correspond to reality). Further note that it is these orbitals that form the Slatcr-dctcrminantal eigenfunction for the separable noninteracting Hamiltonian defined as the sum of the Kohn-Sham operators in Eq. (8.18), i.e.. 

QR Method. The first relativistic method is the so-called quasi-relativistic (QR) method. It has been developed by Snijders, Ziegler and co-workers (13). In this approach, a Pauli Hamiltonian is included into the self-consistent solution of the Kohn-Sham equations of DFT. The Pauli operator is in a DFT framework given by... 

Taking the derivatives of E with respect to p,A and p,B or Bext involves first taking the derivatives of the operators that constitute the Hamiltonian H or the Kohn-Sham density functional and that are dependent on p,A and/or p,B... 

Since DFT has essentially the same mean-field formalism as the HF theory and share much the same computational algorithm, it is not surprising that it has the excited-state counterparts corresponding to TDHF and CIS. They—TDDFT [83-88] and Tamm-Dancoff TDDFT [89], respectively - can be derived analogously to Section 2.2.1 with the only differences being in the definitions of the operator (now called the Kohn-Sham or KS Hamiltonian) and its derivative with respect to the density matrix [see Eq. (2-7)]. The latter is... 

Within the DFT framework, the molecular Kohn Sham (KS) operator for a molecular solute becomes a sum of the core Hamiltonian h, a Coulomb and (scaled) exchange term, the exchange-correlation (XC) potential Vxc and the solvent reaction operator VPCM of Eq. (7-1), namely ... 

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 ... 

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) ... 

To motivate the Kohn-Sham method, we return to molecular Hamiltonian [Eq. (2)] and note that, were it not for the electron-electron repulsion terms coupling the electrons, we could write the Hamiltonian operator as a sum of one-electron operators and solve Schrodinger equation by separation of variables. This motivates the idea of replacing the electron-electron repulsion operator by an average local representation thereof, w(r), which we may term the internal potential. The Hamiltonian operator becomes... 

The divide-and-conquer relies upon our ability to write the Kohn-Sham Hamiltonian operator for a subsystem of a larger system. Alternatively, one may try to construct Kohn-Sham orbitals for molecular subsystems directly without recourse to a localized version of the Kohn-Sham equations. The idea, which is rooted in a long tradition of orbital localization transformations, is to write the exact Kohn-Sham density matrix as [40]... 

After constructing the Kohn-Sham potential, one must construct the electron density, p(r ), the Hamiltonian matrix, Eq. (86), and the overlap matrix, Eq. (83). Because the basis functions are localized and the Kohn-Sham Hamiltonian is a local operator [cf. Eq. (91)], most of the matrix elements... 

Only HF and DFT methods provide the effective one-electron Hamiltonian (Fock or Kohn-Sham) operator... 

The Kohn-Sham-Dirac equation (27) introduced in the last section is the basis of most relativistic electronic structure calculations in solid state theory. There are certain aspects which make the numerical solution of this four-component equation more involved than its non-relativistic coimterpart The Hamiltonian of the Kohn-Sham-Dirac equation is, unlike its Schrodinger equivalent and unlike the field-theoretical Hamiltonian (7) with the properly chosen normal order, not bounded below. In the limit of free, non-interacting particles the solutions of the Kohn-Sham-Dirac equation are plane waves with energies e(k) = cVk -I- c, where positive energies correspond to electrons and states with negative energy can be interpreted as positrons. For numerical procedures, which preferably use variational techniques to find electronic solutions, this property of the Dirac operator causes a severe problem, which can be circumvented by certain techniques like the application of a squared Dirac operator or a projection onto the properly chosen electronic states according to their above definition after Eq. (19). 

Alternative approaches to the many-electron problem, working in real space rather than in Hilbert space and with the electron density playing the major role, are provided by Bader s atoms in molecule [11, 12], which partitions the molecular space into basins associated with each atom and density-functional methods [3,13]. These latter are based on a modified Kohn-Sham form of the one-electron effective Hamiltonian, differing from the Hartree-Fock operator for the inclusion of a correlation potential. In these methods, it is possible to mimic correlated natural orbitals, as eigenvectors of the first-order reduced density operator, directly... 

The second Hamiltonian H(k = 0) pertains to the Kohn-Sham fictitious system of the noninteracting electrons (it contains our wonder vq, which we solemnly promise to search for, and the kinetic energy operator and nothing else) ... 

Energy derivatives are represented as perturbation terms. Assume the perturbed Hamiltonian operator of the Kohn-Sham method as... 

With spin-orbit interaction, a one-electron spin-orbit interaction operator derived from relativistic ECP procedure is simply added to the Kohn-Sham Hamiltonian ... 

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