Gaussian approximation self-consistent calculations

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Self-consistency is achieved by a procedure in which a set of orbitals is assumed and the electron-electron repulsion is calculated. This energy is then used to calculate a new set of orbitals, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

Analytical self-consistent mean field theories were developed independently by Zhulina el al.29 30 and Milner et al.31,32 They are based on the assumption that for large stretchings of the grafted chains with respect to their Gaussian dimension, one can approximate the set of conformations of a stretched grafted chain by a set of most likely trajectories, and predict for such cases a parabolic density profile. In the calculations of the interactions between the two brushes, the interdigitation between the chains was ignored. 

The second approximation involving the ionic size can be improved upon even within the Gaussian approximation by including the structure of the ion in a self-consistent way [43], It would be worthwhile to extend our calculations to include this approach. 

The formal analysis of the mathematics required incorporating the linear combination of atomic orbitals molecular orbital approximation into the self-consistent field method was a major step in the development of modem Hartree-Fock-Slater theory. Independently, Hall (57) and Roothaan (58) worked out the appropriate equations in 1951. Then, Clement (8,9,63) applied the procedure to calculate the electronic structures of many of the atoms in the Periodic Table using linear combinations of Slater orbitals. Nowadays linear combinations of Gaussian functions are the standard approximations in modem LCAO-MO theory, but the Clement atomic calculations for helium are recognized to be very instructive examples to illustrate the fundamentals of this theory (67-69). 

ST. Tsuchiya, M. Abe, T. Nakajima, K. Hi-rao. Accurate rdativistic Gaussian basis sets for H through Lr determined by atomic self-consistent field calculations with the third-order Douglas-KroU approximation. /. Chem. Phys., 115(10) (2001) 4463 472. 

To avoid numerical problems in gaussian basis set due to the unbalance description of the basis set and the vanishing small eigenvalue of the density response matrix, the approximated KLI and LHF (or CEDA or ELP) effective exact-exchange methods can be used. These methods are still computationally much more elaborated than conventional local, semilocal or hybrid functionals, because the calculation of the Slater potential as well as the (self-consistent) correction term is required. A straightforward construction of the Slater potential... 

Fig. 6. The difference Compton profile for Ti and TiH experimental and theoretical for several models. The line A shows the experimental line profile difference normalised to two electrons before applying deconvolution for instrumental broadening and smoothing on row data of Ti and TiH. The solid line B shows the difference profile of Bandstructure calculations according to APW self-consistent approximation after a convolution with a Gaussian resolution function of a=0.30 a.u. The remaining lines C-F are the theoretical difference profiles of indicated model after convolution with the previous resolution function. 

The calculations were performed in the second-neighbor interaction approximation and, for the stacked chains, the geometry found in the mixed TCNQ-TTF crystaF (3.18 A interplane distance in the TCNQ stack and 3.47 A in the TTF stack) was applied. The GAUSSIAN 74 program generalized to infinite chains with periodic boundary conditions was used by applying the STO-3G< basis set. To attain self-consistency, about 30 iteration steps were needed to fulfill simultaneously the applied SCF criteria ... 

These expressions can be numerically implemented for a set of coefficients for the initial atomic orbitals in the system, as well as for other basis functions (e.g., of hydrogenic, Gaussian, or Slater type). An alternative method for computational implementation is to self-consistently solve the equations from the Hohenbeig-Kohn-Sham density functional theory, properly modified in order to include the extension of the spin characterization, wherefrom the molecular orbitals corresponding to the electronic distribution and of spin may directly result, hence, retaining only the HOMO and LUMO orbitals in the electronic frozen-core approximation with the help of which one can calculate and represent the contours of the frontier functions in any of the above (a) to (d) variants. 

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