From owner-chemistry@ccl.net Mon Aug 13 10:44:01 2018 From: "Smith, Jack smith1106.:.marshall.edu" To: CCL Subject: CCL: overlap integral of two simple exponential decay functions (different centers) in three-dimensional space Message-Id: <-53433-180812095600-22473-2oUbksw4S4pITttNE5IrdQ-x-server.ccl.net> X-Original-From: "Smith, Jack" Content-Language: en-US Content-Type: multipart/alternative; boundary="_000_DCDCCB33DC5448E0BE2A76BA000E1444marshalledu_" Date: Sun, 12 Aug 2018 13:55:04 +0000 MIME-Version: 1.0 Sent to CCL by: "Smith, Jack" [smith1106+*+marshall.edu] --_000_DCDCCB33DC5448E0BE2A76BA000E1444marshalledu_ Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: base64 RnJhbmsgSGFycmlzIChodHRwczovL3d3dy5waHlzaWNzLnV0YWguZWR1L35oYXJyaXMvaG9tZS5o dG1sKSB3b3JrZWQgb3V0IGFsbCB0aGUgYW5hbHl0aWNhbCBleHByZXNzaW9ucyBmb3IgYWxsIDEt Y2VudGVyICgxLSBhbmQgMi1lbGVjdHJvbikgYW5kIDItY2VudGVyIDEtZWxlY3Ryb24gU1RPIGlu 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L3NwYW4+PGJyPg0KPHNwYW4+PC9zcGFuPjxicj4NCjxzcGFuPjwvc3Bhbj48YnI+DQo8L2Rpdj4N CjwvYmxvY2txdW90ZT4NCjwvZGl2Pg0KPC9ib2R5Pg0KPC9odG1sPg0K --_000_DCDCCB33DC5448E0BE2A76BA000E1444marshalledu_-- From owner-chemistry@ccl.net Mon Aug 13 21:31:01 2018 From: "Thomas Manz thomasamanz\a/gmail.com" To: CCL Subject: CCL: overlap integral of two simple exponential decay functions (different centers) in three-dimensional space Message-Id: <-53434-180813212954-28077-wgiN2EOCPtJV+a9DmpVfoQ=-=server.ccl.net> X-Original-From: Thomas Manz Content-Type: multipart/alternative; boundary="0000000000003361fc05735b2515" Date: Mon, 13 Aug 2018 19:29:46 -0600 MIME-Version: 1.0 Sent to CCL by: Thomas Manz [thomasamanz-,-gmail.com] --0000000000003361fc05735b2515 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable 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=E2=80=99t t= hink 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 (typo= s) > in his notes that I hand-corrected, and I=E2=80=99m not sure if I still h= ave them. > > You=E2=80=99re welcome to the PHATPSY code (in FORTRAN 66) if interested,= and I=E2=80=99ll > 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.f= i > 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-typ= e > 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 complicate= d > 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 =3D (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 =3D (r_A + r_B) / R > nu =3D (r_A - r_B) / R > > where mu =3D 1..infinity and nu=3D-1..1. r_A is the distance from nucleus= A > and r_B is the distance from nucleus B, and R is the internuclear distanc= e. > The third coordinate is phi =3D 0..2*pi. The volume element is > > dV =3D 1/8 R^3 (mu^2 - nu^2) dmu dnu dphi. > > The resulting expression is, however, a bit involved, and I don't have th= e > time to debug my Maple worksheet now. > > For a reference, you need to go pretty far back in the literature. This i= s > 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 integral= s > 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 > ------------------------------------------------------------------ > > > > -=3D This is automatically added to each message by the mailing script = =3D-http://www.ccl.net/chemistry/sub_unsub.shtmlConferences: http://server.ccl.net/ > chemistry/announcements/conferences/> > > --0000000000003361fc05735b2515 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
Hi,

Thanks to all of you that replied w= ith 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<= /a>> wrote:

You=E2=80=99re welcome to the PHATPSY code (in FORTRAN 66) if interest= ed, and I=E2=80=99ll look to see if I have those old notes stashed away som= ewhere, but you may want to dig around to see if Frank ever published those= notes.=C2=A0 Keep in mind, this was 40+ years ago.

BTW, most of the complexity is in the overlap of the spherical harmoni= cs with arbitrary orientation, not the exponential functions, so this may b= e overkill if all you want is the overlap of simple S functions.

- Jack=C2=A0

Jack A. Smith, PhD
Marshall University=C2=A0


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, con= sider 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 coup= le of sources. Frustratingly, I could not get some of the reported analytic= formulas to work (i.e., some of the claimed analytic formulas in literatur= e give wrong answers). Other formulas are horrendously complex involving all sorts of angular momentum and quant= um number operators, almost too complicated to comprehend.
I am trying to get an analytic overlap form= ula for the plain Slater s-type orbitals that are simple exponential decay = functions. Does anybody know whether a working analytic formula is availabl= e for this?
F.Y.I: I am aware of the formula given in E= q. 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 sam= e integral (not even close=C2=A0as evidenced by comparing accurate numerical integration with the claimed analytic form= ula of the same integral). I am not trying to pick on this paper. I have tr= ied other papers also, but many of them are so complicated that it is diffi= cult 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 =3D (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 ad= vised by the book. The coordinates are

mu =3D (r_A + r_B) / R
nu =3D (r_A - r_B) / R

where mu =3D 1..infinity and nu=3D-1..1. r_A is the distance from nuc= leus A and r_B is the distance from nucleus B, and R is the internuclear di= stance. The third coordinate is phi =3D 0..2*pi. The volume element is

dV =3D 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. Th= is is stuff that was done in the early days of quantum chemistry, when Slat= er 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 T= wo-Center Integrals Useful in Calculations on Molecular Structure. I" = by C. C. J. Roothaan in The Journal of Chemical Physics 19, 1445 (1951) pre= sents 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 =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2= =A0=C2=A0=C2=A0=C2=A0=C2=A0Junior Fellow, Adjunct Professor
susi.lehtola .. alumni.helsinki.fi =C2=A0=C2=A0University of Helsinki
http:/= /www.helsinki.fi/~jzlehtol =C2=A0Finland
------------------------------------------------------------------
Susi Lehtola, dosentti, FT =C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0=C2=A0= tutkijatohtori
susi.lehtola .. alumni.helsinki.fi =C2=A0=C2=A0Helsingin yliopisto
http:/= /www.helsinki.fi/~jzlehtol
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