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
 -----UrsprÃngliche 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"
 <owner-chemistry^ccl.net>
 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
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