CCL: looking for a parameter to characterize the degree of delocalization

Dear Youhao,
 There is a straighforward parameter defined within the context of QTAIM (quantum
 theory of atoms in molecules). It is called delocalization index. It measures
 the degree of electron sharing between two topological atoms. Many QTAIM
 programs can define delocalization index between atomic pairs but usually they
 do not partition it into orbital contributions. However, if you want you can
 partition DI into orbital contributions by having overlap matrix for each atom.
 Among AIM programs AIM2000 provides overlap matrix for atoms. I am not sure if
 Multiwfn can do it or not but you may ask its developer. Once you have overlap
 matrix for any atom, you can measure orbital contributions via a simple
 equation:
 DI(A,B) = - Sigma Sigma [Sij(A)Sij(B)]
 Where A and B are two atoms, by Sigma as you may guess I mean summation, and
 Sij(a and B) are overlap terms for each orbital. Please be careful that if you
 merely focus on pi-orbitals then a part of DI which is related to partial
 overlap between pi and sigma or delta... will be missing. For an ideally
 symmetric system this must be zero but this is not necessarily true for all
 systems. So, for non-planar/non-symmetric systems you may sum up contributions
 of pi, sigma, etc. together then compare it with total DI to have the
 contribution of mixing too. This type of computation is NOT conventional or
 straightforward and I cannot refer you to a particular paper but theoretically
 is possible. For more information regarding DI you may look at J. Phys. Chem. A,
 1999, 103, 304-314.
 On the other hand, some researchers believe that measuring current density in
 the presence of magnetic fields is also a measure of pi-delocalization. You may
 check some papers with the keyword current density but be aware that there is a
 fundamental difference between DI and current density. DI is a ground-state
 property but current density that is widely believed is a measure of pi electron
 delocalization is a "response" property. This means that DI is
 non-zero in the absence of external fields but current density is zero in the
 absence of an external magnetic field. 
 Good luck,Cina
 -------------------------------------------------------------------------Cina
 Foroutan-Nejad, PhD,CEITEC-Central European Institute of TechnologyMasaryk
 University, Brno,Czech Republichttps://muni.academia.edu/CinaForoutanNejad
      On Thursday, 19 March 2015, 3:28, Youzhao Lan lyzhao(_)aliyun.com
 <owner-chemistry.-$-.ccl.net> wrote:
 Sent to CCL by: "Youzhao  Lan" [lyzhao]_[aliyun.com]
 Dear all,
 Is there a calculated parameter or quantity to characterize
 the degree of delocalization of pi-electron?
 Any help will be appreciated.
 Best regards.
 Youzhao Lan
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 <html><body><div style="color:#000; background-color:#fff;
 font-family:bookman old style, new york, times,
 serif;font-size:16px"><div dir="ltr"><span>Dear
 Youhao,</span></div><div
 dir="ltr"><span><br></span></div><div
 dir="ltr" id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_7401">There is a straighforward
 parameter defined within the context of QTAIM (quantum theory of atoms in
 molecules). It is called delocalization index. It measures the degree of
 electron sharing between two topological atoms. Many QTAIM programs can define
 delocalization index between atomic pairs but usually they do not partition it
 into orbital contributions. However, if you want you can partition DI into
 orbital contributions by having overlap matrix for each atom. Among AIM programs
 AIM2000 provides overlap matrix for atoms. I am not sure if Multiwfn can do it
 or not but you may ask its developer. Once you have overlap matrix for any atom,
 you can measure orbital contributions via a simple
 equation:</span></div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span><br></span></div><div
 dir="ltr" id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_7763">DI(A,B) = - Sigma Sigma
 [Sij(A)Sij(B)]</span></div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span><br></span></div><div
 dir="ltr" id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_8105">Where A and B are two atoms,
 by Sigma as you may guess I mean summation, and Sij(a and B) are overlap terms
 for each orbital. Please be careful that if you merely focus on pi-orbitals then
 a part of DI which is related to partial overlap between pi and sigma or
 delta... will be missing. For an ideally symmetric system this must be zero but
 this is not necessarily true for all systems. So, for non-planar/non-symmetric
 systems you may sum up contributions of pi, sigma, etc. together then compare it
 with total DI to have the contribution of mixing too. This type of computation
 is NOT conventional or straightforward and I cannot refer you to a particular
 paper but theoretically is possible. For more information regarding DI you may
 look at J. Phys. Chem. A, 1999, 103, 304-314.</span></div><div
 dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span><br></span></div><div
 dir="ltr" id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_11442">On the other hand, some
 researchers believe that measuring current density in the presence of magnetic
 fields is also a measure of pi-delocalization. You may check some papers with
 the keyword current density but be aware that there is a fundamental difference
 between DI and current density. DI is a ground-state property but current
 density that is widely believed is a measure of pi electron delocalization is a
 "response" property. This means that DI is non-zero in the absence of
 external fields but current density is zero in the absence of an external
 magnetic field.&nbsp;</span></div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span><br></span></div><div
 dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span>Good
 luck,</span></div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span>Cina</span></div><div
 dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span><br></span></div><div
 dir="ltr" id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_12684">-------------------------------------------------------------------------</span></div><div
 dir="ltr" id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_12685">Cina Foroutan-Nejad,
 PhD,</span></div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_12686">CEITEC-Central European
 Institute of Technology</span></div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_12687">Masaryk
 Universit</span>y, Brno,</div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span>Czech
 Republic</span></div><div dir="ltr"
 id="yui_3_16_0_1_1426755704295_7400"><span
 id="yui_3_16_0_1_1426755704295_12885"><a href="https://muni.academia.edu/CinaForoutanNejad";
 id="yui_3_16_0_1_1426755704295_12884">https://muni.academia.edu/CinaForoutanNejad</a><br></span></div>
 <div dir="ltr" id="yui_3_16_0_1_1426755704295_7400"
 class="" style=""><span class=""
 style=""><br class=""
 style=""></span></div><br><div
 class="qtdSeparateBR"><br><br></div><div
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 On Thursday, 19 March 2015, 3:28, Youzhao Lan lyzhao(_)aliyun.com
 &lt;owner-chemistry.-$-.ccl.net&gt; wrote:<br> </font>
 </div>  <br><br> <div
 class="y_msg_container"><br>Sent to CCL by:
 "Youzhao&nbsp; Lan" [lyzhao]_[aliyun.com]<br>Dear
 all,<br>Is there a calculated parameter or quantity to
 characterize<br>the degree of delocalization of
 pi-electron?<br><br>Any help will be
 appreciated.<br><br>Best regards.<br>Youzhao
 Lan<br><br><br><br>-= This is automatically added to
 each message by the mailing script =-<br>To recover the email address of
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