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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
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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