Analytic basis set

Schneider, B.I. (1975). A-matrix theory for electron-atom and electron-molecule collisions using analytic basis set expansions. Chem. Phys. Lett. 31, 237-241. 

Analytical basis set for higher-order calculations of transition amplitudes... 

B. I. Schneider, Tf-matrix theory for electron-atom and electron-molecule collisions using analytic basis set expansions, Chem. Phys. Lett. 37 237 (1975) B. I. Schneider, 77-matrix theory for electron-molecule collisions using analytic basis set expansions 2. Electron-//, scattering in static-exchange model. Phys. Rev. A 77 1957 (1975). 

There are methods that automate some of these steps. They are called composite methods because they combine results from several calculations to estimate the result that would be obtained from a more expensive calculation. The most popular families of composite methods are represented by Gaussian-3 (G3) theory [68,109] and CBS-APNO theory [110,111], where CBS stands for complete basis set. Both families of methods, which are considered reliable, include empirical parameters. The CBS theories incorporate an analytical basis set extrapolation based on perturbation theory, which is in contrast to the phenomenological extrapolation mentioned above. When the Gaussian software is used to perform these calculations, steps 2-, above, are performed automatically, with the result labeled G3 enthalpy (or the Hke) in the output file [20,99]. The user must still choose a reaction (step 1) and manipulate the molecular enthalpies (steps 5 and 6). The most precise composite methods are the Weizmann-n methods, which however are very intensive computationally [112]. 

H. M. Quiney, I. P. Grant, S. Wilson. On the accuracy of Dirac-Hartree-Fock calculations using analytic basis sets. /. Phys. B At Mol. Opt Phys., 22 (1989) L15-L19. 

In our analytic basis set approach for solving the APH surface function Hamiltonian q. (38) we use a linear combination of harmonic oscillators in each arrangement channel times Legendre polynomials. 

Preliminary results from the analytic basis set method indicate it to be a factor of 60 faster than the finite element method on the LiFH problem. 

The most time consuming part of the calculation, the determination of the surface functions, has been improved by almost two orders of magnitude. The DVR and analytic basis set methods are preferred over finite elements methods. We have designed an efficient numerical method for studying reactive scattering that can produce exact and approximate results for systems and energies which have heretofore been impossible to study. 

In Section 3, the performance of the software implementation of PS electronic methods, the PS-GVB suite of ab initio electronic structure programs, is discussed. Three issues are addressed numerical precision of the calculations as compared to conventional analytical basis set methods, computational efficiency as a function of system size for the various electronic structure methods in the program, and effectiveness in addressing real chemical problems. For the last of these, we focus on conformational energy differences, calculation of which we have been intensively studying over the past several years, and thermochemistry which we have just begun to study with GVB-MP2. 

In 1951, Hall [6] and, independently, Roothaan [7] put the Hartree-Fock equations - the ubiquitous independent particle model - in their matrix form. The Hartree-Fock equations describe the motion of each electron in the mean field of all the electrons in the system. Hall and Roothaan invoked the algebraic approximation in which, by expanding molecular orbitals in a finite analytic basis set, the integro-differential Hartree-Fock equations become a set of algebraic equations for the expansion coefficients which are well-suited to computer implementation. 

LG. Kaplan, Symmetry of Many-Electron Systems, translated by J. Gerratt, Academic Press, New York London 1975 (p. 269, the term algebraic approximation is employed to describe the use of finite analytic basis sets in molecular electronic structure calculations)... 

X-ray fluorescence (XRF) analysis is successfully used to determine chemical composition of various geological and ecological materials. It is known that XRF analysis has a high productivity, acceptable accuracy of results, developed theory and industrial analytical equipment sets. Therefore the complex methods of XRF analysis have to be constituent part of basis data used in ecological and geochemical investigations... 

To some other experts the meaning of the term ab initio is rather clear cut. Their response is that "ab initio" simply means that all atomic/molecular integrals are computed analytically, without recourse to empirical parametrization. They insist that it does not mean that the method is exact nor that the basis set contraction coefficients were obtained without recourse to parametrization. Yet others point out that even the integrals need not be evaluated exactly for a method to be called ab initio, given that, for instance, Gaussian employs several asymptotic and other cutoffs to approximate integral evaluation. 

The final geometry optimizations with the DZ+D basis set and the analytic calculation of the force fields and MP-2 corrections were done with the program GRADSCfI on a CRAY-IA. There are 128 basis functions for the difluoro-benzenes, 132 basis functions for the dihydroxybenzenes and 136 basis functions for the diaminobenzenes. The calculations on the difluorobenzenes require about 14 x 10 non-zero 2e integrals whereas the calculations on the diaminobenzenes require about 40 x 10 non-zero 2e integrals. The integral sort step required for the... 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

A completely different approach is taken in DMol. Recall that this program uses numerical basis sets rather than analytical functions. At the center of their implementation is Poisson s equation,... 

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