Basis functions, energy-dependent

In practical applications, the continuum is often approximated by a discrete spectrum. To this end, one conveniently introduces a potential wall at long internuclear separations and solves for the artifically bound states.171,172 Alternatively, basis set expansion techniques can be employed.195,196 In either case, the density of states depends on external conditions, that is, the size of the box or the number of basis functions. This dependence on external conditions has to be accounted for by the energy normalization. Instead of employing a single continuum wave function with proper energy E in Eq. [240], one samples over the discrete levels with energy E -... 

The use of basis functions which depend on the geometrical parameters shows itself in the final energy function as molecular integrals (overlap, one-electron and electron-repulsion) depending on these parameters so that, for example... 

The core contributions thus require the calculation of integrals that involve basis functions on up to two centres (depending upon whether 0, and 0 are centred on the same nucleus or not). Each element H)) can in turn be obtained as the sum of a kinetic energy Integra and a potential energy integral corresponding to the two terms in the one-electror HcUniltonian. 

Typically we fit up to the / = 3 components of the one center expansion. This gives a maximum of 16 components (some may be zero from the crystal symmetry). For the lowest symmetry structures we thus have 48 basis functions per atom. For silicon this number reduces to 6 per atom. The number of random points required depends upon the volume of the interstitial region. On average we require a few tens of points for each missing empty sphere. In order to get well localised SSW s we use a negative energy. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The orthogonal subspace used in Table I is spanned by the large basis 0=19sl4p8d6f4g3h2i, H=9s6p4d3f2g. The number of basis functions of the large basis that are (nearly) linearly dependent on the cc-pVnZ basis is drastically increased with the cc-pVnZ basis sets. We observe that the externally contracted MP2 calculations converge faster to the MP2 limit. As expected, the energies from the externally contracted MP2 method lie between the standard... 

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