In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence properties are analyzed to show that the approach presented in this work is a valuable contribution to the existing literature on molecular integral calculations as well as on spherical Bessel integral calculations. [ABSTRACT FROM AUTHOR].

Plane trigonometry mostly deals with the relationships among the three sides and three angles of a triangle that lies in a plane. |

A ray is that portion of a line that starts at a point on the line and extends indefinitely in one direction. The starting point of a ray is called its vertex. If two rays are drawn with a common vertex, they form an angle. One of the rays of an angle is called the initial side, and the other ray is the terminal side. The angle that is formed is identified by showing the direction and amount of rotation from the initial side to the terminal side. If the rotation is in the counterclockwise direction, the angle is positive; if the rotation is clockwise, the angle is negative. |

The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, written 360°. Thus, one degree, 1°, is 1/360 of a revolution. One-sixtieth of a degree is called a minute, written 1′. |

By using a circle of radius r, an angle can be constructed whose vertex is at the center of this circle and whose rays subtend an arc on the circle whose length equals r. Such an angle measures 1 radian. For a circle of radius r, a central angle of θ radians subtends an arc whose length s is given by Ep. (1) | (1) | |

Because a central angle of 1 revolution (360°) subtends an arc equal to the circumference of the circle (2πr), it follows that an angle of 1 revolution equals 2π radians; that is, 2π radians = 360°. See also: Plane geometry; Radian measure. |