Tom, For a slightly more recent reference: I have been using this paper from 1989 recently which describes the evaluation of molecular integrals in an STO basis: J. Fernández Rico, R. López, and G. Ramírez, The Journal of Chemical Physics 91, 4204 (1989). https://doi.org/10.1007/BF00761325 I haven't read through the main paper in any detail as I have been mainly using the (useful) appendix on rotation matrices. From a brief glance at the paper body, it looks like the two centre overlap integrals are defined in Eqs. 27 & 28, with details on how to compute these in appendix D. These formulae seem to be more general (and more complicated) that what you are looking for, though I thought I should mention this paper in case it is useful to you. Regards, James On 12/08/18 09:27, Susi Lehtola susi.lehtola[]alumni.helsinki.fi wrote: > > Sent to CCL by: Susi Lehtola [susi.lehtola*alumni.helsinki.fi] > On 08/11/2018 12:29 AM, Thomas Manz thomasamanz . gmail.com wrote: >> Dear colleagues, >> >> I am trying to find an analytic formula and journal reference for the >> overlap integral of two simple exponential decay functions (different >> centers) in three-dimensional space. For example, consider the overlap >> integral of 1s Slater-type basis functions placed on each atom of a >> diatomic molecule. >> >> I have looked into the literature at a couple of sources. >> Frustratingly, I could not get some of the reported analytic formulas >> to work (i.e., some of the claimed analytic formulas in literature >> give wrong answers). Other formulas are horrendously complex involving >> all sorts of angular momentum and quantum number operators, almost too >> complicated to comprehend. >> >> I am trying to get an analytic overlap formula for the plain Slater >> s-type orbitals that are simple exponential decay functions. Does >> anybody know whether a working analytic formula is available for this? >> >> F.Y.I: I am aware of the formula given in Eq. 16 of Vandenbrande et >> al. J. Chem. Theory Comput. 13 (2017) 161-179. It is wrong and clearly >> doesn't match the numerical integration of the same integral (not even >> close as evidenced by comparing accurate numerical integration with >> the claimed analytic formula of the same integral). I am not trying to >> pick on this paper. I have tried other papers also, but many of them >> are so complicated that it is difficult to understand what is actually >> going on. > > This is exercise 5.1 in the purple bible [ > https://onlinelibrary.wiley.com/doi/book/10.1002/9781119019572 ]. The > overlap between two hydrogenic 1s STOs is > > S = (1 + R + 1/3 R^2) exp(-R) > > as given in eq 5.2.8. > > It's pretty straightforward to do the more general case where the > exponents differ from unity by using confocal elliptical coordinates as > advised by the book. The coordinates are > > mu = (r_A + r_B) / R > nu = (r_A - r_B) / R > > where mu = 1..infinity and nu=-1..1. r_A is the distance from nucleus A > and r_B is the distance from nucleus B, and R is the internuclear > distance. The third coordinate is phi = 0..2*pi. The volume element is > > dV = 1/8 R^3 (mu^2 - nu^2) dmu dnu dphi. > > The resulting expression is, however, a bit involved, and I don't have > the time to debug my Maple worksheet now. > > For a reference, you need to go pretty far back in the literature. This > is stuff that was done in the early days of quantum chemistry, when > Slater type orbitals were used as the basis and the molecules were small. > > I don't know if this it was the first one, but "A Study of Two-Center > Integrals Useful in Calculations on Molecular Structure. I" by C. C. J. > Roothaan in The Journal of Chemical Physics 19, 1445 (1951) presents the > necessary diatomic overlap integrals for the exponential type basis. > (The second part by Ruedenberg details the evaluation of two-electron > integrals for diatomics.) -- James C. Womack Email: jw5533]_[my.bristol.ac.uk Web: https://jcwomack.com
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