Kohn-Sham Hamiltonian

H° = is the effective Kohn-Sham Hamiltonian evaluated at the reference... 

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 Hamiltonian, the SVWN exchange-correlation functional was used. Equation 4.12 was applied to calculate the electron density of folate, dihydrofolate, and NADPH (reduced nicotinamide adenine dinucleotide phosphate) bound to the enzyme— dihydrofolate reductase. For each investigated molecule, the electron density was compared with that of the isolated molecule (i.e., with VcKt = 0). A very strong polarizing effect of the enzyme electric field was seen. The largest deformations of the bound molecule s electron density were localized. The calculations for folate and dihydrofolate helped to rationalize the role of some ionizable groups in the catalytic activity of this enzyme. The results are,... 

The SCRF approach became a standard tool167 for estimating solvent effects and was combined with various quantum chemical methods that range from semi-empirical161 to the post-Hartree-Fock ab initio ones. It can also be combined with the Kohn-Sham formalism where the Kohn-Sham Hamiltonian (Eq. 4.2) is used for the gas-phase Hamiltonian in Eq. 4.15. The effective Kohn-Sham Hamiltonian for the system embedded in the dielectric environment takes the following form ... 

Hgas corresponds to the gas-phase Kohn-Sham Hamiltonian of the embedded electron density ... 

The combination of TDDFT with a QM/MM approach is, in principle, straightforward. The surrounding system of point charges modifies the electrostatic potential of the system, which enters the perturbation equations through the Kohn-Sham Hamiltonian Hks. This causes a change in the excitation wavelenghts which reflects the influence of the environment. 

Kohn-Sham Hamiltonian within Keldysh Green s functions. 126... 

The Kohn-Sham Hamiltonian is expanded similarly and we obtain... 

The Fourier transformation of the first-order perturbed Kohn-Sham Hamiltonian is given by... 

In the implementation of the QM/MM approach with the real-space method, the QM cell that contains the real-space grids is embedded in the MM cell. One should take care for the evaluation of the potential upc(r) defined as Eq. (17-20). When a point charge in MM region goes inside the QM cell, it makes a singularity in the effective Kohn-Sham Hamiltonian, which may give rise to a numerical instability. To circumvent the problem, we replace a point charge distribution... 

Finally we also expand the time-dependent Kohn-Sham Hamiltonian as ... 

The Kohn-Sham Hamiltonian (41)for the non-interacting electrons is written in this basis set as a matrix = a Hs (3). Solving the Schrodinger equation in the basis set [Pg.240]

This appears at first sight different from the condition that the one-electron orbitals of the auxiliary non-interacting electron system diagonalize the Kohn-Sham Hamiltonian ... 

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 ... 

In calculating matrix elements of the Kohn-Sham Hamiltonian of Eq. (2), the greatest problem is posed by the electronic Coulomb repulsion. To render this term tractable, it is convenient to cast the electron density p(r) in a model form, so as to calculate the potential by one-dimensional integrations. This is accomplished by approximating p by a multicenter overlapping multipolar expansion pu [37] ... 

In order to tackle large and complex structures, new methods have recently been developed for solving the eleetronie part of the problem. These are mostly applied to the pseudopotential plane wave method, because of the simplicity of the Hamiltonian matrix elements with plane wave basis functions and the ease with which the Hellmann-Feynman forces can he found. Conventional methods of matrix diagonalization for finding the energy eigenvalues and eigenfunctions of the Kohn-Sham Hamiltonian in (9) can tackle matrices only up to about 1000 x 1000. As a basis set of about 100 plane waves per atom is needed, this restricts the size of problem to... 

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 spinors further commute with the Kohn-Sham Hamiltonian and obey a commutative multiplication law, thereby making them an Abelian group isomorphic to the usual translation group [133]. But this means that they have the same irreducible representation, which is the Bloch theorem. So, we therefore have the generalized Bloch theorem ... 

If we now choose

[Pg.75]

In this section we will describe a second systematic method to construct density functionals, namely perturbation theory starting from the Kohn-Sham Hamiltonian [43,44]. The method is based on traditional perturbation theory and is comparable in computational cost. For this reason the method is less suited to the calculation of... 

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... 

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