CCL: overlap integral of two simple exponential decay functions (different centers) in three-dimensional space

Hi,
 Thanks to all of you that replied with information about the Slater
 s-orbital overlap integrals. I am looking into the resources you suggested.
 Sincerely,
 Tom
 On Sun, Aug 12, 2018 at 7:55 AM, Smith, Jack smith1106.:.marshall.edu <
 owner-chemistry ~~ ccl.net> wrote:
 > Frank Harris (https://www.physics.utah.edu/~harris/home.html) worked out
 > all the analytical expressions for all 1-center (1- and 2-electron) and
 > 2-center 1-electron STO integrals back in the early 70s.  I used them in a
 > program I wrote called PHATPSY.  I further generalized the diatomic
 > integrals for arbitrary origin and orientation (and to exponentially
 > shielded nuclear attraction integrals).  Unfortunately, I don’t
 think Frank
 > ever formally published those notes.  I used a preprint copy from the QTP
 > library at the University of Florida. There were a few minor errors (typos)
 > in his notes that I hand-corrected, and I’m not sure if I still have
 them.
 >
 > You’re welcome to the PHATPSY code (in FORTRAN 66) if interested,
 and I’ll
 > look to see if I have those old notes stashed away somewhere, but you may
 > want to dig around to see if Frank ever published those notes.  Keep in
 > mind, this was 40+ years ago.
 >
 > BTW, most of the complexity is in the overlap of the spherical harmonics
 > with arbitrary orientation, not the exponential functions, so this may be
 > overkill if all you want is the overlap of simple S functions.
 >
 > - Jack
 >
 > Jack A. Smith, PhD
 > Marshall University
 > smith1106 ~~ marshall.edu
 >
 >
 > On Aug 12, 2018, at 3:27 AM, Susi Lehtola susi.lehtola[]alumni.helsinki.fi
 > <owner-chemistry ~~ ccl.net> 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.)
 > --
 > ------------------------------------------------------------------
 > Mr. Susi Lehtola, PhD             Junior Fellow, Adjunct Professor
 > susi.lehtola .. alumni.helsinki.fi   University of Helsinki
 > http://www.helsinki.fi/~jzlehtol  Finland
 > ------------------------------------------------------------------
 > Susi Lehtola, dosentti, FT        tutkijatohtori
 > susi.lehtola .. alumni.helsinki.fi   Helsingin yliopisto
 > http://www.helsinki.fi/~jzlehtol
 > ------------------------------------------------------------------
 >
 >
 >
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