Kohn-Sham orbital expansion

C., the coefficient of zth basis function in the expansion ofy th Kohn-Sham orbital... 

It is our purpose to briefly review expansion (1) through the adiabatic perturbation theory of Gorling and Levy [11], which arrives at the formal expression for the second-order energy, Ec(2)[n], in terms of Kohn-Sham orbitals. 

Car and Parrinello in their celebrated 1985 paper [2] proposed an alternative route for molecular simulations of electrons and nuclei altogether, in the framework of density functional theory. Their idea was to reintroduce the expansion coefficients Cj(G) of the Kohn-Sham orbitals in the plane wave basis set, with respect to which the Kohn-Sham energy functional should be minimized, as degrees of freedom of the system. They then proposed an extended Car-Parrinello Lagrangian for the system, which has dependance on the fictitious degrees of freedom Cj(G) and their time derivative Cj (G) ... 

N (Kohn-Sham) orbitals can be expanded into atomic orbitals according to (4). Furthermore, the expansion coefficients C, can be determined by requiring that they optimize the total (Kohn-Sham) energy. This results in the (Kohn-Sham) matrix equation similar to (5)... 

If the basis set expansion for the Kohn-Sham orbitals in Equation (3.57) is substituted into the Kohn-Sham equations then it is possible to express them in a matrix form, identical in form to the Roothaan-Hall equations ... 

Using the LCGTO expansions for the Kohn-Sham orbitals (Equation (2)) and the electronic density (Equation (3)), the Kohn-Sham SCF energy expression can be expressed as ... 

In Equation (11), S represents the overlap matrix, c the molecular orbital coefficient matrix, and e the Kohn-Sham orbital energies. The expansion coefficients of the approximate density, necessary for the construction of the Kohn-Sham matrix, are calculated by the minimization of... 

The (Kohn-Sham) orbitals, electron density, p, can be determined numerically or variationally as an expansion of basis functions. 

In this section we will present some simple calculations using a plane-wave expansion of the Kohn-Sham orbitals [4]. The plane-wave basis set is or-... 

In O Eq. 7.67, the summation is over aU reciprocal lattice vectors G which fulfill the condition G T = 27tM, M being an integer number. In practice, this plane-wave expansion of the Kohn-Sham orbitals is truncated such that the individual terms all yield kinetic energies lower than a specified cutoff value, Ecut,... 

While plane waves are a good representation of delocalized Kohn-Sham orbitals in metals, a huge number of them would be required in the expansion (O Eq. 7.67) to obtain a good approximation of atomic orbitals, in particular near the nucleus where they oscillate rapidly. Therefore, in order to reduce the size of the basis set, only the valence electrons are treated explicitly, while the core electrons (i.e., the inner shells) are taken into account implicitly through pseudopotentials combining their effect on the valence electrons with the nuclear Coulomb potential. This frozen core approximation is justified as typically only the valence electrons participate in chemical interactions. To minimize the number of basis functions the pseudopotentials are constructed in such a way as to produce nodeless atomic valence wavefunctions. Beyond a specified cutoff distance from the nucleus, Ecut the nodeless pseudo-wavefunctions are required to be identical to the reference all-electron wavefunctions. 

Using the above expansions for the Kohn-Sham orbitals O Eq. 16.15 and the density Eq. 16.16, the Kohn-Sham energy expression O Eq. 16.7 can be rewritten in terms of atomic orbitals. 

The latter can be evaluated using, for instance, plane wave and multipole expansions as it was employed in (L)APW - - lo - - LO method/ In the solid state codes the Kohn-Sham orbitals are usually normalized over the volume of a single unit cell (not over the volume of the Born-von Karman cluster) so that... 

The Kohn-Sham equations are solved in a self-consistent field fashion. Initially a charge density is needed so that Exc can be computed. To obtain the charge density, an initial guess to the Kohn-Sham orbitals is needed. This initial guess can be obtained from a set of basis functions whereby the coefficients of expansion of the basis functions can be optimized just like in the HF method. From the function of Exc in terms of the density, the term xc is computed. The Kohn-Sham equations (Equation 9-50) are then solved to obtain an improved set of Kohn-Sham orbitals. The improved set of Kohn-Sham orbitals is then used to calculate a better density. This iterative process is repeated until the exchange-correlation energy and the density converge to within some tolerance. 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

Approximate solutions of the time-dependent Schrodinger equation can be obtained by using Frenkel variational principle within the PCM theoretical framework [17]. The restriction to a one-determinant wavefunction with orbital expansion over a finite atomic basis set leads to the following time-dependent Hartree-Fock or Kohn-Sham equation ... 

Similarly, expanding the KS potential in an LCAO expansion makes molecular density-functional calculations practical [9]. For metals and similar crystalline solids, it is best to expand the Kohn-Sham potential in momentum space via Fourier coefficients. For molecular solids various real-space method are under investigation. For molecules studied with the big, well-chosen Gaussian basis sets of quantum chemistry, it is undoubtedly best to expand the KS potential in linear-combination-of-Gaussian-type-orbital (LCGTO) form [10]. 

We used the discrete variational (DV-Xa) method which uses a linear combination of atomic orbitals (LCAO) expansion of molecular orbitals to calcidate the silicate cluster electronic state. (19, 20) In this method the exchange-correlation potentials are approximated by the simple Kohn-Sham-Slater form... 

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