Approximations integrals

Gutzwiller M C 1967 Phase-integral approximation in momentum space and the bound states of an atom J Math. Phys. 8 1979... 

From Table 13-31, a total of 394 time increments were necessary to distill all hut 22.08 Ih-mol of the initial charge of 99.74 Ih-mol following the establishment of total-reflux conditions. If this problem had to he solved by an explicit integrator, approximately 25,000 time increments would have been necessary. 

In this procedure, the value of the integrand can be determined numerically for every pair of elements and the double integral, approximately the sum of these values, then becomes ... 

Associated with the pole of the S-matrix is a Seigert state, I-Ves, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, // I res = z les,H being the system Hamiltonian.44 If a square integrable approximation to I res is constructed, then its time evolution, k . (/,), wiH exhibit pure exponential decay after a transient induction period. Of course any L2 state will show quadratic, and hence non-exponential, decay at short times since... 

Van Alsenoy, C. 1988. Ab Initio Calculations on Large Molecules The Multiplicative Integral Approximation. J. Comput. Chem. 9, 620-626. 

Poulsen, J. A. Nyman, G. Rossky, P. J., Quantum diffusion in liquid para-hydrogen an application of the Feynman-Kleinert linearized path integral approximation, J. Phys. Chem. B 2004,108, 19799-19808... 

There have been a number of direct solutions in the form of Equation (9.65) for convective combustion. These have been theoretical - exact or integral approximations to the boundary layer equations, or empirical - based on correlations to experimental data. Some examples are listed below ... 

Due to the integral approximations used in the MNDO model, closed-shell Pauli exchange repulsions are not represented in the Hamiltonian, but are only included indirectly, e.g., through the effective atom-pair correction terms to the core-core repulsions [12], To account for Pauli repulsions more properly, the NDDO-based OM1 and OM2 methods [23-25] incorporate orthogonalization terms into the one-center or the one- and two-center one-electron matrix elements, respectively. Similar correction terms have also been used at the INDO level [27-31] and probably contribute to the success of methods such as MSINDO [29-31],... 

Rearranging gives the computer algorithm for a first-order filter using the rectangular integration approximation. 

The line-integral approximation [ ]), Eq. (213) with (211) or (216), to the exact u,(r [n]) seems to be especially accurate. First of all, its asymptotic behavior for pure-state systems is the same as the behavior of the exact r, in Eq. (134), therefore guaranteeing the proper asymptotic form of KS orbitals. Next, it satisfies exactly the Levy-Perdew [33] identity... 

Computing the sensitivities is time consuming. Fortunately the direct integral approximation of the sensitivity matrix and its principal component analysis can offer almost the same information whenever the direct integral method of parameter estimation applies. 

Following the initial work of Pople, there appeared a number of integral approximation schemes, all combined with different sets of empirical parameters. One such approximation scheme that has been... 

O. Vahtras et al., Integral approximations for LCAO-SCF calculations. Chem. Phys. Lett. 213, 514-518 (1993)... 

For the first time, a systematic uniform ab initio study of endohedral complexes of C60, Si60 and Ge60 with monoatomic cations (Li+, Na+, K+ and Rb+) and anions (F-, Cl% Br and P) was performed. The 3-21 G level (3-21+G for the anions) was used. This could be achieved by application of the multiplicative integral approximation [86], as implemented in the program BRABO [87]. The sequence studied for the cages bears a close resemblance with isomorphic substitution in zeolites modifying its catalytic activity via bridging hydroxyls. 

Taylor series as functions of experimental conditions. This is exactly analogous to the analysis of In r described previously except that, by means of a tentative model, the primary reaction rate dependence on concentrations, temperature, and other experimental factors has been eliminated. This permits the rate equations to be Integrated approximately correctly. 

The overall strategy is to calculate Q for the system at hand, using sums, integrals, approximations, and so on, and then obtain measurable results for S, /(, and P from the appropriate logarithm or derivative of Q. 

We share the opinion expressed by Farrelly and Reinhardt (1983) that discrepancies between Stark effect results obtained by the use of the Carlini (JWKB) approximation and by accurate numerical calculations cannot be attributed to the break-down of the approximation, but are due to a failure to use the approximation in a correct way. An appropriate approach based on the phase-integral approximation of arbitrary order generated from an appropriately chosen base function is a still more efficient and often highly accurate method for the treatment of several problems, not only in quantum mechanics, but in various fields of theoretical physics. With... 

We shall first briefly describe the phase-integral approximation referred to in item (i). Then we collect connection formulas pertaining to a single transition point [first-order zero or first-order pole of Q2(z) and to a real potential barrier, which can be derived by... 

Phase-integral approximation generated from an unspecified base function... 

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