From owner-chemistry %-% at %-% ccl.net Tue Jan 22 12:10:01 2013 From: "Sergio Manzetti sergio.manzetti!=!gmx.com" To: CCL Subject: CCL:G: AW: G: 6D, 5D and complex numbers Message-Id: <-48110-130122113521-27211-dhAt6IgAI0wdF+GEnmc81Q=server.ccl.net> X-Original-From: "Sergio Manzetti" Content-Type: multipart/alternative; boundary="========GMXBoundary259481358872509685835" Date: Tue, 22 Jan 2013 17:35:09 +0100 MIME-Version: 1.0 Sent to CCL by: "Sergio Manzetti" [sergio.manzetti[A]gmx.com] --========GMXBoundary259481358872509685835 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: 8bit Dear George, if the integrals between the orbitals are not complex, then the operation of the exchange coefficients should be a quadratic function, which would make them not complex. I am not sure I misinterpret the problem, Best wishes Sergio ----- Original Message ----- > From: Georg Lefkidis lefkidis:-:physik.uni-kl.de Sent: 01/22/13 04:58 PM To: Manzetti, Sergio Subject: CCL:G: AW: G: 6D, 5D and complex numbers Sent to CCL by: "Georg Lefkidis" [lefkidis]~[physik.uni-kl.de] Dear Susi, thank you for your reply, however, it does not really completely answer my question. If the coefficients refer to the spherical harmonics (which I also tend consider as the most probable) then the integrals between the atomic orbitals (which for instance Gaussian can print through IOP commands) should be complex as well. And this is not the case which is why I am perplexed... George -----Ursprüngliche Nachricht----- Von: owner-chemistry+lefkidis==physik.uni-kl.de]-[ccl.net [mailto:owner-chemistry+lefkidis==physik.uni-kl.de]-[ccl.net] Im Auftrag von Susi Lehtola susi.lehtola(a)alumni.helsinki.fi Gesendet: Dienstag, 22. Januar 2013 09:52 An: Lefkidis, Georg Betreff: CCL:G: 6D, 5D and complex numbers Sent to CCL by: Susi Lehtola [susi.lehtola-#-alumni.helsinki.fi] On Tue, 22 Jan 2013 02:42:54 -0500 "George Lefkidis lefkidis---physik.uni-kl.de" wrote: > Sent to CCL by: "George Lefkidis" [lefkidis#physik.uni-kl.de] Dear > all, > > I have a question about the implementation of basis sets in ORCA > (although my question pertains to other programs as well). I am > somewhat confused about the 5D and 6D (and 7F and 10F etc.). I > understand that 6D means Cartesian functions (xx, xy, etc), while (I > think) 5D means spherical harmonics. If this is true then 5D are > *complex* functions, and the matrix elements between them should be > complex as well, as I would expect the HF coefficients (LCAO expansion > coefficients). Since we're often dealing with real orbitals, it's handy to use the spherical harmonics in the real form as well. http://en.wikipedia.org/wiki/Spherical_harmonics#Real_form Note that you're not losing any degrees of freedom with this kind of a rotation of the basis set. You still are able to expand complex functions with this kind of a basis set, you will just need complex coefficients. >However quantum chemistry programs like ORCA, GAUSSIAN and GAMESS give >*real* numbers for everything. I know it is possible to create real >orbitals by combining the +m_l and -m_l. Gaussian, for example, cites >H. B. Schlegel and M. J. Frisch, Transformation between Cartesian and >Pure Spherical Harmonic Gaussians, Int. J. Quantum Chem., 54 (1995) >83-87, which claims to use Eq. (15). So, my question is, what do >actually these programs do and how exactly am I to understand the MO >coefficients? (I use my own codes on top of these software packages to >calculate nonlinear magnetooptics and spin dynamics > - up to now I always used 6D, now I got results from a collaborator >in 5D and I need to ensure the correct nomenclature of D0, D+1, D-1, > D+2 and D-2, etc.). The integrals are always evaluated with respect to cartesian functions. However, since spherical harmonics are just polynomials on the sphere, you can also write them in terms of cartesian functions. http://en.wikipedia.org/wiki/Solid_harmonics#Spherical_harmonics_in_Cartesian_form Eqn (15) in the Schlegel-Frisch paper gives you the coefficient for x^l y^m z^n in the complex form of the spherical harmonics, and in the following paragraph they say that they actually use the real form. Now, the idea is that now that you know what cartesian terms contribute to which component of the spherical harmonics function, you can obtain the integrals with respect to the spherical harmonics basis set by weighing the cartesian integrals correspondingly. Most GTO programs use the above method to use spherical harmonics as the basis set, and MO coefficients are reported in terms of these. The D0, D+1, etc coefficients of Gaussian refer to the Y_{20}, Y_{21} etc coefficients. -- -------------------------------------------------------- Mr. Susi Lehtola, M. Sc. Doctoral Student susi.lehtola^alumni.helsinki.fi Department of Physics http://www.helsinki.fi/~jzlehtol University of Helsinki Office phone: +358 9 191 50 632 Finland -------------------------------------------------------- Susi Lehtola, FM Tohtorikoulutettava susi.lehtola^alumni.helsinki.fi Fysiikan laitos http://www.helsinki.fi/~jzlehtol Helsingin Yliopisto Ty puhelin: (0)9 191 50 632 --------------------------------------------------------http://www.ccl.net/cgi-bin/ccl/send_ccl_messagehttp://www.ccl.net/chemistry/sub_unsub.shtmlhttp://www.ccl.net/spammers.txthttp://www.ccl.net/cgi-bin/ccl/send_ccl_messagehttp://www.ccl.net/chemistry/sub_unsub.shtmlhttp://www.ccl.net/spammers.txt--========GMXBoundary259481358872509685835 Content-Type: text/html; charset="utf-8" Content-Transfer-Encoding: quoted-printable Dear Geo= rge, if the integrals between the orbitals are not complex, then the operat= ion of the exchange coefficients should be a quadratic function, which woul= d make them not complex.
=20
=20 I am not sure I misinterpret the problem,
=20
=20 Best wishes
=20
=20 Sergio
=20
=20

=20 =C2=A0

=20
=20

=20 ----- = Original Message -----

=20

=20 From: = Georg Lefkidis lefkidis:-:physik.uni-kl.de

=20

=20 Sent: = 01/22/13 04:58 PM

=20

=20 To: Ma= nzetti, Sergio

=20

=20 Subjec= t: CCL:G: AW: G: 6D, 5D and complex numbers

=20
=20
=20
=20 
=20
Sent to CCL by: "Georg Lefkidis" [lefkidis]~[physik.uni-kl.de]=20
Dear Susi,=20

thank you for your reply, however, it does not really completely answer my =
question. If the coefficients refer to the spherical harmonics (which I als=
o tend consider as the most probable) then the integrals between the atomic=
 orbitals (which for instance Gaussian can print through IOP commands) shou=
ld be complex as well. And this is not the case which is why I am perplexed=
...=20

George=20


-----Urspr=C3=BCngliche Nachricht-----=20
Von: owner-chemistry+lefkidis=3D=3Dphysik.uni-kl.de]-[ccl.net [mailto:owner=
-chemistry+lefkidis=3D=3Dphysik.uni-kl.de]-[ccl.net] Im Auftrag von Susi Le=
htola susi.lehtola(a)alumni.helsinki.fi=20
Gesendet: Dienstag, 22. Januar 2013 09:52=20
An: Lefkidis, Georg=20
Betreff: CCL:G: 6D, 5D and complex numbers=20


Sent to CCL by: Susi Lehtola [susi.lehtola-#-alumni.helsinki.fi]=20
On Tue, 22 Jan 2013 02:42:54 -0500=20
"George Lefkidis lefkidis---physik.uni-kl.de" <owner-chemistry^ccl.net&g=
t;=20
wrote:=20
> Sent to CCL by: "George  Lefkidis" [lefkidis#physik.uni-kl.de] Dear=20
> all,=20
>=20
> I have a question about the implementation of basis sets in ORCA=20
> (although my question pertains to other programs as well). I am=20
> somewhat confused about the 5D and 6D (and 7F and 10F etc.). I=20
> understand that 6D means Cartesian functions (xx, xy, etc), while (I=
=20
> think) 5D means spherical harmonics. If this is true then 5D are=20
> *complex* functions, and the matrix elements between them should be=20
> complex as well, as I would expect the HF coefficients (LCAO expansion=
=20
> coefficients).=20

Since we're often dealing with real orbitals, it's handy to use the spheric=
al harmonics in the real form as well.=20
http://en.wikipedia.org/wiki/Spherical_harmonics#Real_form=20

Note that you're not losing any degrees of freedom with this kind of a rota=
tion of the basis set. You still are able to expand complex functions with =
this kind of a basis set, you will just need complex coefficients.=20

>However quantum chemistry programs like  ORCA, GAUSSIAN and GAMESS give=
=20
>*real* numbers for everything. I know  it is possible to create real=20
>orbitals by combining the +m_l and  -m_l. Gaussian, for example, cites=
=20
>H. B. Schlegel and M. J. Frisch,  Transformation between Cartesian and=
=20
>Pure Spherical Harmonic  Gaussians, Int. J. Quantum Chem., 54 (1995)=20
>83-87, which claims to  use Eq. (15). So, my question is, what do=20
>actually these programs do  and how exactly am I to understand the MO=
=20
>coefficients? (I use my own  codes on top of these software packages to=
=20
>calculate nonlinear  magnetooptics and spin dynamics=20
> - up to now I always used 6D, now I got results from a collaborator =
=20
>in 5D and I need to ensure the correct nomenclature of D0, D+1, D-1,=20
> D+2 and D-2, etc.).=20

The integrals are always evaluated with respect to cartesian functions.=20
However, since spherical harmonics are just polynomials on the sphere, you =
can also write them in terms of cartesian functions.=20
http://en.wikipedia.org/wiki/Solid_harmonics#Spherical_harmonics_in_Cartesi=
an_form=20

Eqn (15) in the Schlegel-Frisch paper gives you the coefficient for x^l y^m=
 z^n in the complex form of the spherical harmonics, and in the following p=
aragraph they say that they actually use the real form.=20

Now, the idea is that now that you know what cartesian terms contribute to =
which component of the spherical harmonics function, you can obtain the int=
egrals with respect to the spherical harmonics basis set by weighing the ca=
rtesian integrals correspondingly.=20

Most GTO programs use the above method to use spherical harmonics as the ba=
sis set, and MO coefficients are reported in terms of these. The D0, D+1, e=
tc coefficients of Gaussian refer to the Y_{20}, Y_{21} etc coefficients.=
=20
--=20
--------------------------------------------------------=20
Mr. Susi Lehtola, M. Sc.          Doctoral Student=20
susi.lehtola^alumni.helsinki.fi   Department of Physics=20
http://www.helsinki.fi/~jzlehtol  University of Helsinki=20
Office phone: +358 9 191 50 632   Finland=20
--------------------------------------------------------=20
Susi Lehtola, FM                  Tohtorikoulutettava=20
susi.lehtola^alumni.helsinki.fi   Fysiikan laitos=20
http://www.helsinki.fi/~jzlehtol  Helsingin Yliopisto=20
Ty puhelin: (0)9 191 50 632=20
--------------------------------------------------------http://www.ccl.net/=
cgi-bin/ccl/send_ccl_messagehttp://www.ccl.net/chemistry/sub_unsub.shtmlhtt=
p://www.ccl.net/spammers.txt=20


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