Atomic basis

E.( ), possesses a complete set of eigenfiinctions, the matrix F whose dimension Mis equal to the number of atomic basis orbitals, has M eigenvalues e. and M eigenvectors whose elements are the. . Thus, there are... 

As larger atomic basis sets are employed, the size of the CSF list used to treat a dynamic correlation increases rapidly. For example, many of the above methods use singly- and doubly-excited CSFs for this purpose. For large basis sets, the number of such CSFs (N ) scales as the number of electrons squared uptimes the number... 

The localized nature of the atomic basis set makes it possible to implement a linear-scaling TB algoritluu, i.e. a TB method that scales linearly with the number of electrons simulated [17]. (For more infonnation on linear scaling methods, see section B3.2.3.3.)... 

Both HF and DFT calculations can be performed. Supported DFT functionals include LDA, gradient-corrected, and hybrid functionals. Spin-restricted, unrestricted, and restricted open-shell calculations can be performed. The basis functions used by Crystal are Bloch functions formed from GTO atomic basis functions. Both all-electron and core potential basis sets can be used. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

On an atomic basis, H/C ratios range from 1.5- -2.0. The range of elemental composition of cmde oil (3) may be given as follows ... 

Recommendations Plant measurements should be adjusted to close the constraints of the process. This adjustment shoiild be done on a component or subcomponent (e.g., atomic) basis. The adjustments should be done recognizing (at a minimum) the uncertainty in the measurements. While sophisticated routines have been developed for reconciliation, the vagaries of plant measurements may make them unsuitable in most applications. The routines are no substitute for accurate, precise measurements. They cannot compensate for the uncertainties and hmited information typically found in plant data. 

This model was later expanded upon by Lifshitz [33], who cast the problem of dispersive forces in terms of the generation of an electromagnetic wave by an instantaneous dipole in one material being absorbed by a neighboring material. In effect, Lifshitz gave the theory of van der Waals interactions an atomic basis. A detailed description of the Lifshitz model is given by Krupp [34]. 

The first (inconclusive) work bearing on the synthesis of element 104 was published by the Dubna group in 1964. However, the crucial Dubna evidence (1969-70) for the production of element 104 by bombardment of 94PU with loNe came after the development of a sophisticated method for rapid in situ chlorination of the product atoms followed by their gas-chromatographic separation on an atom-by-atom basis. This was a heroic enterprise which combined cyclotron nuclear physics and chemical separations. As we have seen, the actinide series of elements ends with 103 Lr. The next element should be in Group 4 of the transition elements, i.e. a heavier congenor of Ti, Zr and Hf. As such it would be expected to have a chloride... 

For H and He, the atomic basis set consists of a single Is orbital. For Li through Ne, the inner-shell electrons are treated as part of the nucleus and the basis functions used are atomic 2s, 2p c, 2py and 2p .. For Na through Al, the inner shell is treated as part of the nucleus and we consider only 3s, 3p, 3pj, and 3pj. orbitals. For Si through Cl we have to decide on whether or not to include the atomic 3d-orbitals in addition, and practice varies. Most authors include them. 

Gaussian Basis Functions for use in Molecular Calculations III Contraction of (10s, 6p) Atomic Basis Sets for the First-Row Atoms T. FI. Dunning, Jr... 

The KS orbitals can be determined by a numerical procedure, analogous to numerical HF methods. In practice such procedures are limited to small systems, and essentially all calculations employ an expansion of the KS orbitals in an atomic basis set. 

On an atomic basis, Li is much more soluble in K than vice versa (0.07, 0.43, 1.29 and 1.99 mol% Li compared with 0.007, 0.02, 0.05 and 0.07 mol% K). Although these solubilities are larger than those given by eqs. (a) and (b), both investigations point to a two-immiscible-liquid-type of phase diagram with a consolute T > 1000°C, and with the consolute composition being on the Li-rich side as in the Li-Na system. 

With this, a compact equation is found for the coefficients of the MSO expansions in the atomic basis, for each initial state /, as... 

Molecules and Clusters. The local nature of the effective Hamiltonian in the LDF equations makes it possible to solve the LDF equations for molecular systems by a numerical LCAO approach (16,17). In this approach (17), the atomic basis functions are constructed numerically for free atoms and ions and tabulated on a numerical grid. By construction, the molecular basis becomes exact as the system dissociates into its atoms. The effective potential is given on the same numerical grid as the basis functions. The matrix elements of the effective LDF Hamiltonian in the atomic basis are given by... 

The second part of this paper concerns the choice of the atomic basis set and especially the polarization functions for the calculation of the polarizability, o , and the hyperpo-larizabiliy, 7. We propose field-induced polarization functions (6) constructed from the first- and second-order perturbed hydrogenic wavefunctions respectively for a and 7. In these polarization functions the exponent ( is determined by optimization with the maximum polarizability criterion. These functions have been successfully applied to the calculation of the polarizabilities, a and 7, for the He, Be and Ne atoms and the molecule. 

Expanding the occupied 4>i over an orthonormal atomic basis set x of order m (which is assumed independent of the perturbation), one has... 

The atomic basis consists in a double-zeta set expanded with polarization functions (DZP) and augmented by diffuse functions (DZPR). Exponents and contraction coefficient are from McLean and Chandler 1980 [18] diffuse functions, centered on the heavy atoms with exponents of 0.023 for the s orbitals and 0.021 for the p orbitals are from Dunning and Hay 1977 [34]. Extension of the DZP basis set with two sets of diffuse s (0.0437, 0.0184) and p (0.0399, 0.0168) functions (DZPRR) has also been tested. 

In the case ofthe VB wave function, the natural orbitals and occupation numbers obtained by diagonalizing the matrix representation of the VB one-density function on the atomic basis set were used in Equation (14). 

EFP parameters for a particular fragment are generated for a given atomic basis set, they can be used in a variety of applications. The various components of the non-bonded interactions between molecules are evaluated using the EFP2 generated parameters. The procedure has been described in elsewhere [36] only the main points are summarized below. 

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