| Incomplete Gamma functions were widely used in many fields,such as statistics, physics, finance and other fields,especially in the field of quantum chemistry. In recent years, ab initio has played a more and more important role in the chemistry and biology research. For lack of efficiency and accuracy of Ab initio, in this paper,we studies on algorithm improvement issues of the incomplete Gamma function which is very important in it. In the ab initio calculations based on molecular orbital, the amount of computation for two-electron integrals is enormous and time-consuming. And all of the two-electron integrals calculation which choose GTOs and STOs as basis functions can be transformed into a series of incomplete Gamma function Fm(T)(m=0…mmax)integrals calculation. So the computational efficiency of the incomplete Gamma function becomes the key factor of the all computation time. While the existing methods of the incomplete Gamma function can meet the computational accuracy demand for small molecular system, there is still huge room for improvement in terms of effciency. Also in the calculation of biological macromolecules, incomplete Gamma function is calculated faced with two problems: higher accuracy and higher efficiency. For the above issues, we worked as follows:(1)For the application scenarios of small molecular system, on the basis of studing on the advantages and disadvantages of existing methods, we presented an effective method based on combination of upwardwith downward recursive to calculate the incomplete Gamma function. Firstly, the incomplete Gamma function of continued fraction form Converged on the entire computation interval was deduced. And we analyzed mathematical properties of it. Followed by theoretical analysis and numerical experiments, we obtained suitable intervals of using upward and downward recursive method for the T and m with satisfying accuracy demand of the small molecular system. In addition, we proposed an improved method based on error function, achieving the multivariable computation transforming to univariate computation. But the efficiency is low. Next, the frist improved method were relatively compared with and traditional Taylor interpolation, Simpson quadrature formula and recursive computation methods in computation comparison or numerical precision. The results of Comparison showed that: our improved method is more efficiency than Taylor interpolation in the T <11, 11 ￡T <24.7 and T 324.7 three intervals; compared with Simpson quadrature formula, we avoided problem of the numerical precision of the method in certain areas which could not meet the accuracy needs of calculation and our computation far less than its computation; for downward recursive methods, we also have avoided in certain areas which could not meet the numerical accuracy of the calculation. Finally, the improved method was used in the calculation of quantum chemistry.(2)On the other hand, for biological macromolecular system, this paper proposeed a method based on fast interval estimation. Combined the convergence characteristics with numerical experiments, we could get an appropriate interval(Tmin, Tmax) of each formula. And on this interval, we could find a suitable Fm(Topt) to evaluate the expression instead of the full range of Fm(T) calculation expression. We got Fm(T) calculation expression about 10e10-=,16e10-= and20e10-= in(0, 1),(1, 8),(8, 20),(20, 30),(30, 40),(40, 60),(60, +￥). Finally, we performed numerical tests on each section, verifing correct of our theoretical analysis. Meanwhile, we gave a simple parallel test of two cores, achieving time-saving effect. For the macromolecular systems, this method have an exploring way of improved precision and the effective use of multicore, showing potential value. |
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