From owner-chemistry@ccl.net Sun Aug 12 04:29:00 2018 From: "Susi Lehtola susi.lehtola[]alumni.helsinki.fi" To: CCL Subject: CCL: overlap integral of two simple exponential decay functions (different centers) in three-dimensional space Message-Id: <-53431-180812042753-11634-BPDopk7a6glXKkus3NwKAg-x-server.ccl.net> X-Original-From: Susi Lehtola Content-Language: en-US Content-Transfer-Encoding: 8bit Content-Type: text/plain; charset=utf-8; format=flowed Date: Sun, 12 Aug 2018 11:27:39 +0300 MIME-Version: 1.0 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 ------------------------------------------------------------------ From owner-chemistry@ccl.net Sun Aug 12 10:08:00 2018 From: "J C Womack jw5533%%my.bristol.ac.uk" To: CCL Subject: CCL: overlap integral of two simple exponential decay functions (different centers) in three-dimensional space Message-Id: <-53432-180812100048-22680-Fb6F9LPm7gfPrelA5BzhAA!A!server.ccl.net> X-Original-From: J C Womack Content-Type: multipart/signed; micalg=pgp-sha512; protocol="application/pgp-signature"; boundary="eVtXI7RRMPuZ9pliKRrkiOqcYc4JwIFCf" Date: Sun, 12 Aug 2018 15:00:35 +0100 MIME-Version: 1.0 Sent to CCL by: J C Womack [jw5533 ~ my.bristol.ac.uk] This is an OpenPGP/MIME signed message (RFC 4880 and 3156) --eVtXI7RRMPuZ9pliKRrkiOqcYc4JwIFCf Content-Type: multipart/mixed; boundary="d0M70XpURu5hCrcmkQRil5iEXyCGjlr7Q"; protected-headers="v1" > From: J C Womack To: CCL Subscribers Message-ID: <7d4dbf05-397c-6bd7-e188-51908244bc36]_[my.bristol.ac.uk> Subject: Re: CCL: overlap integral of two simple exponential decay functions (different centers) in three-dimensional space References: <53431-180812042753-11634-Ov8LtAKf/spQZAglm5IBCA]_[server.ccl.net> In-Reply-To: <53431-180812042753-11634-Ov8LtAKf/spQZAglm5IBCA]_[server.ccl.net> --d0M70XpURu5hCrcmkQRil5iEXyCGjlr7Q Content-Type: text/plain; charset=utf-8 Content-Language: en-US Content-Transfer-Encoding: quoted-printable 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. Fern=C3=A1ndez Rico, R. L=C3=B3pez, and G. Ram=C3=ADrez, The Journal o= f 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: >=20 > 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=C2=A0as evidenced by comparing accurate numerical integration wi= th >> 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. >=20 > 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 >=20 > S =3D (1 + R + 1/3 R^2) exp(-R) >=20 > as given in eq 5.2.8. >=20 > 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 >=20 > mu =3D (r_A + r_B) / R > nu =3D (r_A - r_B) / R >=20 > where mu =3D 1..infinity and nu=3D-1..1. r_A is the distance from nucle= us A > and r_B is the distance from nucleus B, and R is the internuclear > distance. The third coordinate is phi =3D 0..2*pi. The volume element i= s >=20 > dV =3D 1/8 R^3 (mu^2 - nu^2) dmu dnu dphi. >=20 > The resulting expression is, however, a bit involved, and I don't have > the time to debug my Maple worksheet now. >=20 > 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 smal= l. >=20 > 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 th= e > necessary diatomic overlap integrals for the exponential type basis. > (The second part by Ruedenberg details the evaluation of two-electron > integrals for diatomics.) --=20 James C. Womack Email: jw5533]_[my.bristol.ac.uk Web: https://jcwomack.com --d0M70XpURu5hCrcmkQRil5iEXyCGjlr7Q-- --eVtXI7RRMPuZ9pliKRrkiOqcYc4JwIFCf Content-Type: application/pgp-signature; name="signature.asc" Content-Description: OpenPGP digital signature Content-Disposition: attachment; filename="signature.asc" -----BEGIN PGP SIGNATURE----- Version: GnuPG v2 iQIcBAEBCgAGBQJbcD2HAAoJEECnhgp8lzVjSOoP+gN0DJkn4Y62G9TrT7eADhPl PPmnONriDy/Fy2uGzxam0N828rYACoqLqSyYdkOmlS/iMGjIXGVoRMqAE9pJEKWI m9uKDY8IgLubVN54WyakSdoNIcWXXlGiIqrKDBpqSjwpFHbCgK/gWjNUWEWmxEJ2 AIHw5vPKIjBMEetksCDTEzhCODUK4BGaN//BH3qi4ykvu4eAignjNrzEekU6zQ1H NQtFR4YDquoSIjstwPSBtycFklm/s14KulvtSZy79jADT4mXtGftt5w+HCBgK6Pm sSPrmZ18q5QUF6z7h5ujnV4EsAS7qX4Q2zSxeMeJ3Fihnrr9I4T23GMt5bvjUnCV xGO7ssvtpRQcsEToqAnj7E/4v4xdzLLQid4LEeXmYFsjiS+ry1LUT9wo6GXCRMze 5lZEkx2AupGxHRyh2Be2rF6RwTFuNlzrVrcLkAVHeVhwHv6eBz+AwMhfo975Y3ic hxNpfjC7XvHtCfIELW2EuJHBRC/+cqykd9deA0T4IQGyFbhWuVujXfGeyZ1rS6dC tT2phSC89PgKNkuB3WJ8LClEkPSdcN4uMYQV0IHNvmMFkXfzQMtrBEK82/vVvCg2 YhMq0l3ePXmIvMb+GIcHtCmtZ3KPOGSF36rrvnYf9XYPvKsHTdeaaOkaooV0PLGI tXrVlnmIYKwLtlaH6Tso =dpCg -----END PGP SIGNATURE----- --eVtXI7RRMPuZ9pliKRrkiOqcYc4JwIFCf--