By Abhay Ashtekar
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3 VECTORIAL FIRST-ORDER DIFFERENTIAL EQUATION AND LINE INTEGRAL 19 where (x, y) is some point in the interval x0 ≤ x ≤ x0 + x and y0 ≤ y ≤ y0 + y and x and y both → 0. It is readily noted that the value of f (x0 , y0 | ) depends solely on Curl τ . So, if τ fulfills Eq. 55), then the results as calculated along the two different infinitesimal contours are identical but differ if Curl τ = 0. Although and are infinitesimal contours, this does not retract from the generality of the derivation. In case we are interested in a regular-size region that contains one such point [which causes f (x f , y f | ) = 0], it has been shown (see Appendix C in Ref.
40, 1011 (1980). T. Pacher, L. S. Cederbaum, and H. K¨oppel, Adv. Chem. Phys. 84, 293 (1993). M. Baer, J. Phys. Chem. A, 104, 3181 (2000). 1) where Tn (s) is the nuclear kinetic energy and He (se |s) is the electronic Hamiltonian, which also contains the nuclear Coulomb interactions and depends parametrically on the nuclei coordinates. 2) where E is the total energy and | (se , s) is the complete wavefunction that describes the motion of both the electrons and the nuclei. 3) j=1 where the ψ j (s), j = 1, .
91). The two approaches are expected to yield the same results. In this section we discuss to some extent results, for closed contours. 66)] is the integral equation for a closed contour, and we have shown that the value of the integral is zero when the contour does not surround any of the points for which the tensorial field F is not defined. In case it surrounds one or more points, the integral becomes undefined. The same is expected for the explicit closed contour that follows from Eq. 93) we can state that the D matrix is the unit matrix I if the contour does not surround any of the pathological points.