Gaussian expansions, for

Table 2.3 Coefficients and e-xponents for best-fit Gaussian expansions for the Is Slater type orbital [Hehre et al. 1969].

Comparison of the harmonic-osdllator expansion and Gaussian expansion for the two-body linear potential. The exact values are given by the first zero of the Airy function, which is 2.33811. 

Compact one- and two-Gaussian expansions for the components of the effective potentials of atoms in the first two rows are presented by Stevens-Basch-Krauss [485]. Later, the list of ECP was extended to the third-, fourth- and fifth-row atoms [486] and includes relativistic ECP (RECP). The pseudo-orbital basis-set expansions for the first two rows of atoms consist of four Gaussian primitives using a common set of exponents for the s and p functions. Analytic SBK RECP are generated in order to reproduce POs and eigenvalues as closely as possible. The semilocal SBK ECP are given by... 

The coefficients and the exponents are found by least-squares fitting, in which the overlap between the Slater type function and the Gaussian expansion is maximised. Thus, for the Is Slater type orbital we seek to maximise the following integral ... 

While being very attractive in view of their similarity to CLTST, on closer inspection (3.61)-(3.63) reveal their deficiency at low temperatures. When P -rcc, the characteristic length Ax from (3.60b) becomes large, and the expansion (3.58) as well as the gaussian approximation for the centroid density breaks down. In the test of ref. [Voth et al. 1989b], which has displayed the success of the centroid approximation for the Eckart barrier at T> T, the low-temperature limit has not been reached, so there is no ground to trust eq. (3.62) as an estimate for kc ... 

The starting point to obtain a PP and basis set for sulphur was an accurate double-zeta STO atomic calculation4. A 24 GTO and 16 GTO expansion for core s and p orbitals, respectively, was used. For the valence functions, the STO combination resulting from the atomic calculation was contracted and re-expanded to 3G. The radial PP representation was then calculated and fitted to six gaussians, serving both for s and p valence electrons, although in principle the two expansions should be different. Table 3 gives the numerical details of all these functions. 

The key feature is - both for the expansion of 1/r or e " in terms of even-tempered Gaussians - that, for large n, the cut-off error goes as exp(-anh) with h the step size and that the discretization errors goes as exp(—6/h), with a and b constants. While - for fixed n a small h is good for the discretization error, it is bad for the cut-off error and vice versa. The best compromise is that h 1/%/ni which implies that the overall error goes as exp(—Cy )-... 

It must be stressed that the use of GHOs in no way depends on this latter Gaussian expansion procedure since all molecular integrals reduce to standard STO forms. It has merely proved expedient to make use of the expansion method in calculations reported earlier. This ab initio technique provides the raw data with which to establish the patterns of behaviour of the GHOs. We can now address ourselves to one of our stated aims the development of approximation methods for a quantitative theory of valence. 

Expansion coefficients and Gaussian exponents for 3-2IG and 6-31G representations have been determined by Hartree-Fock energy minimization on atomic ground states. In the case of 6-3IIG representations, minimizations have been carried out at the MP2 level rather than at the Hartree-Fock level. 

Higher terms in the expansion for Z(p) can be straightforwardly calculated using the theory of Gaussian Markov processes. For example in the case of the Wiener process we obtain for the first term inside the summation of Eq. (35)... 

An explicit solution of Eq. (B.31) is possible for macromolecular rings which obey Gaussian statistics75. For open linear and branched molecules, only approximate solutions are known so far . One of these approximations is the so called cumulant expansion of S(q, t)77,78, which is a series expansion of the logarithmic TCF in powers of the delay time t... 

Now we write the same Fourier of expansion for the electric field and write everything according to the magnetic field intensity H = B, and we find with the case that (e/H)Aq co the amplitude fixed to the wavelength as is the case for some solitons, for Gaussian packets, we arrive at the same cubic Schrodinger equation ... 

An interesting mixed-basis-set method for use in SCF calculations has been described by Billingsley and Trindle,52 with application to LiC>2. One-centre and most two-centre integrals are evaluated analytically, whilst less tractable integrals are approximated by a gaussian expansion of the STO s. Examination of portions of the... 

Gaussian Expansions, AREP, and Spin-Orbit Operators for Pb ... 

Consistent Molecular-Orbital Methods. I. Use of Gaussian Expansions of Slater-Type Atomic Orbitals. See also D. Feller and E. R. Davidson, in Reviews in Computational Chemistry, K. B. Lipkowitz and D. B. Boyd, Eds., VCH Publishers, New York, 1990, Vol. 1, pp. 1-43. Basis Sets for Ab Initio Molecular Orbital Calculations and Intermolecular Interactions. 

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