Different MO coeffients of the same molecule computed with different symmetry point groups in CASSCF

[葉哲和]

Dear Molpro users,

I recently came across a very weird situation in MULTI, where the CASSCF(2o,2e) calculation of the ground state of H2 molecule resulted in a very obsurd set of MO coefficients in the default D2h point group:

【  NATURAL ORBITALS

  ================

   Orb     Occ        Energy       Coefficients

                                                  1 1s      1 1s      1 2pz

   1.1  1.97401    -0.575420     0.984019 -0.248203  0.025124

   1.5  0.02599     0.584276     2.294243 -0.816536 -0.030234 】

However, if I switched to a lower point group, say, C2v, then the MO coefficients were reasonable by taking an equally weighted linear combination of the 1s orbitals from both hydrogen atoms (and of course 2s and 2pz orbitals by symmetry):

 【  NATURAL ORBITALS

  ================

   Orb     Occ        Energy       Coefficients

                                                   1 1s      1 1s      1 2pz     2 1s      2 1s      2 2pz

   1.1  1.97401    -0.575420     0.695807 -0.175506  0.017765  0.695807 -0.175506 -0.017765

   2.1  0.02599     0.584276     1.622275 -0.577378 -0.021379 -1.622275  0.577378 -0.021379 】

Very interestingly, in the D2h calculation, if I specify at the end of the input file "put,molden,<filename>", then the MO coefficients printed in the Molden file was again reasonable.

How can this happen, could someone explain? Molpro 2025.1 was used.

[Peter Knowles]

Molpro uses symmetry-adapted combinations of contracted atom-centred gaussians as basis functions. So there is nothing wrong with what you see in the print of orbital coefficients - it's 0.984019 times the sigma_g combination of atomic s functions, and 2.294243 times the sigma_u combination of atomic s.

The export to xml and molden resolves this symmetry adaptation, and gives coefficients of local atom-centred functions.

Peter

On 3 Jun 2025, at 09:10, 葉哲和 <chehoy...@nycu.edu.tw> wrote:

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<h_c2v.molden><h_d2h.out><h_d2h.molden><h_c2v.out>

[葉哲和]

Dear Prof. Knowles,

Thank you for the explanation.

I understand now the D2h point group utilizes the 𝜎g symmetry. Now may I understand the orbital 1.1 as 0.984019*𝜎g(from 1s bases) -0.248203*𝜎g(from 2s bases) + 0.025124*𝜎g(from 2pz bases)? (or more appropriately, "ag" instead of 𝜎g)

And so the symmetry was resolved in any subgroup of D2h, since then the s-bases will only combine in irrep number 1. Am I correct?

Best,

Che

Peter Knowles 在 2025年6月3日 星期二晚上11:51:50 [UTC+8] 的信中寫道：

[葉哲和]

Dear Prof. Knowles,

Thank you for your explanation. If I understand correctly, the D2h calculation then assigns orbital 1.1 as

0.984019*a_g(from 1s bases) - 0.248203*a_g(from 2s bases) + 0.025124*a_g(from 2pz bases),

and that the C2v calculation assigns orbital 1.1 as 

0.695807*a_1(from 1s basis of the first hydrogen) - 0.175506*a_1(from 1s basis of the first hydrogen) + 0.017765*a_1(from 1s basis of the first hydrogen) + 0.695807*a_1(from 1s basis of the second hydrogen) - 0.175506*a_1(from 1s basis of the second hydrogen) - 0.017765*a_1(from 1s basis of the second hydrogen).

C2v or lower point group give the same MO coefficients because they no longer differentiate a_g from b_{1u}, which is absorbed altogether in (projected to) the a_1 irrep. Am I correct? If so, then how is it that the MO coefficients in C2v calculations no longer produce some "good" combinations of the a_1 bases (like did in the case of D2h) to represent the overall molecular bases, but instead giving atom-centered local bases? Is the symmetry adaptation only applied to the D2h point group?

Best,

Che-Ho Yeh

Peter Knowles 在 2025年6月3日 星期二晚上11:51:50 [UTC+8] 的信中寫道：

Molpro uses symmetry-adapted combinations of contracted atom-centred gaussians as basis functions. So there is nothing wrong with what you see in the print of orbital coefficients - it's 0.984019 times the sigma_g combination of atomic s functions, and 2.294243 times the sigma_u combination of atomic s.

[Peter Knowles]

On 4 Jun 2025, at 08:38, 葉哲和 <chehoy...@nycu.edu.tw> wrote:

Dear Prof. Knowles,

Thank you for the explanation.

I understand now the D2h point group utilizes the 𝜎g symmetry. Now may I understand the orbital 1.1 as 0.984019*𝜎g(from 1s bases) -0.248203*𝜎g(from 2s bases) + 0.025124*𝜎g(from 2pz bases)? (or more appropriately, "ag" instead of 𝜎g)

Yes

And so the symmetry was resolved in any subgroup of D2h, since then the s-bases will only combine in irrep number 1. Am I correct?

I'm not sure what you are asking exactly, but in your example, the s functions appear also in irrep 5.  You can see the definition of the basis functions by putting

gprint,basis

early in the input file.

Peter

To view this discussion, visit https://groups.google.com/d/msgid/molpro-user/72f73de8-d3bc-447c-85a6-4e2ea3dcdd41n%40googlegroups.com.

[Peter Knowles]

On 5 Jun 2025, at 04:31, 葉哲和 <chehoy...@nycu.edu.tw> wrote:

Dear Prof. Knowles,

Thank you for your explanation. If I understand correctly, the D2h calculation then assigns orbital 1.1 as

0.984019*a_g(from 1s bases) - 0.248203*a_g(from 2s bases) + 0.025124*a_g(from 2pz bases),

and that the C2v calculation assigns orbital 1.1 as 

0.695807*a_1(from 1s basis of the first hydrogen) - 0.175506*a_1(from 1s basis of the first hydrogen) + 0.017765*a_1(from 1s basis of the first hydrogen) + 0.695807*a_1(from 1s basis of the second hydrogen) - 0.175506*a_1(from 1s basis of the second hydrogen) - 0.017765*a_1(from 1s basis of the second hydrogen).

Yes

C2v or lower point group give the same MO coefficients because they no longer differentiate a_g from b_{1u}, which is absorbed altogether in (projected to) the a_1 irrep. Am I correct?

Yes

If so, then how is it that the MO coefficients in C2v calculations no longer produce some "good" combinations of the a_1 bases (like did in the case of D2h) to represent the overall molecular bases, but instead giving atom-centered local bases? Is the symmetry adaptation only applied to the D2h point group?

The symmetry adaptation is always done, but in your example where you drop to the particular C2v subgroup (it's not the only one) that breaks the equivalence of the two nuclei, the adaptation does not result in any functions involving more than one atom centre.

Whether the MO coefficients show the symmetry in the lower symmetry basis depends entirely on how they are produced. Normally the Hartree-Fock code produces eigenfunctions of the Fock operator, which, except in special cases with additional degeneracy (eg two infinitely-separated atoms), will be bases for a single irrep.  But orbital localisation does something different.

Peter

To view this discussion, visit https://groups.google.com/d/msgid/molpro-user/1e94f09d-8926-43fa-819a-978668810ffbn%40googlegroups.com.

[葉哲和]

Dear Prof. Knowles,

Thanks again for the clarification.

Now I understand. If I did the same calculation is C2h point group then the a_g and a_u orbitals are again differentiated from each other and will produce orb 1.1 as 0.984019*a_g(from 1s bases) - 0.248203*a_g(from 2s bases) + 0.025124*a_g(from 2pz bases), so it really depends on how the molecule is oriented. Now the default style is to put the internuclear axis onto the z-axis, but if I orient it (on purpose) differently, say, over the y-axis, then choosing the active orbitals to be a_1 and b_2 will produce the same set of MO coefficients but with 2pz replaced with 2py functions.

Thank you so much. I think I have my doubts cleared.

Best,

Che-Ho Yeh

Peter Knowles 在 2025年6月5日 星期四下午1:54:03 [UTC+8] 的信中寫道：