[CP2K-user] [CP2K:18561] Re: Evaluation of basis functions over a grid and computation of overlap integrals

[Aleksandros Sobczyk]

Thank you very much for the feedback (especially over weekend).
It is very helpful, I think it is slowly clarifying.
I also found this reference in the 
meantime: https://onlinelibrary.wiley.com/doi/10.1002/qua.560540202
But the link you shared seems more readable.
I will post our findings (...or more questions!)

Best regards,
Aleksandros

On Saturday, March 18, 2023 at 6:19:31 PM UTC+1 Krack Matthias wrote:

> Hi Aleksandros
>
>  
>
> The basis set output in the top of the CP2K output is in Cartesian 
> representation (i.e. not spherical coordinates) which you can easily 
> recognize as soon as you have basis functions with l > 1 (d, f …), because 
> there are 6 Cartesian d orbitals printed. I just wanted to note that the 
> default output for integral matrices like the overlap matrix S, however, is 
> in the spherical orbital representation (5 d orbitals d(-2) … d(+2)). 
> Maybe, I confused you in that respect, since you are still trying to 
> reproduce the initial basis set normalization.
>
> I suggest to start exercising with a primitive basis set and a single He 
> atom, e.g.
>
> He
>
> 1 0 2 1 1 1 1
>
> 1.2  0.5  0.4  0.3
>
> and then try a contracted basis set with two primitive Gaussian functions 
> per l value
>
> He
>
> 1 0 2 2 1 1 1
>
> 1.2  0.5  0.4  0.3
>
> 0.8  0.4  0.2  0.1
>
> Here, I just selected exponents and coefficients randomly without testing 
> or calculating anything in an exemplary manner.
>
> Note, that Cartesian and spherical Gaussian functions have different normalization 
> factors 
> <https://iodata.readthedocs.io/en/latest/basis.html#cartesian-basis-functions>
> .
>
>  
>
> HTH
>
>  
>
> Matthias
>
>  
>
> *From: *cp... at googlegroups.com <cp... at googlegroups.com> on behalf of 
> Aleksandros Sobczyk <sobcz... at gmail.com>
> *Date: *Friday, 17 March 2023 at 18:17
> *To: *cp2k <cp... at googlegroups.com>
> *Subject: *Re: [CP2K:18555] Re: Evaluation of basis functions over a grid 
> and computation of overlap integrals
>
> Hi Matthias,
>
>  
>
> Thanks a lot for the response.
>
> I will try to explain better my understanding and my confusion.
>
> I am not sure what you mean by Cartesian representations and spherical 
> representations (I have little experience with contracted GTO).
>
>  
>
> 1) Let's take it a step back, to the cp2k definitions of the basis sets:
> https://www.cp2k.org/basis_sets
>
> The *φ_i *'s seem to be defined in spherical coordinates in that page, 
> and so are the contraction coefficients.
>
> I cannot see where the Cartesian representations come in.
>
>  
>
> 2) Assuming that we consider everything to be in spherical coordinates, as 
> defined in the basis_set page.
>
> We now want to transform the contraction coefficients of the input file, 
> to the "normalized" contraction coefficients that are reported in the 
> output of cp2k.
>
> Let's assume only the first orbital.
>
> From what I understand from your explanation, I need to perform the 
> following two steps:
>
> (Step 1): For all pairs of exponents a_i and coefficients c_i for this 
> orbital, we need to scale c_i such that the primitive Gaussian integral  
>
>         \int exp(-a_I*r^2) dr,
>
> will integrate to 1.
>
> (Step 2):  After this scaling, apply yet another scaling to the 
> coefficients c_i based on the procedure here 
>
>         
> https://github.com/cp2k/cp2k/blob/f26eaef31a9d3f80ca30d8d2f11790a2a072e370/src/aobasis/basis_set_types.F#L1075
>
>  
>
> After these two steps,  the coefficients that I will get out should match 
> the coefficients that are reported in the output of cp2k.
>
> Again, all of these are in spherical coordinates.
>
>  
>
> Thanks again for the response, let me know if I got something wrong.
>
>  
>
> Best,
>
> Aleksandros
>
>  
>
>  
>
>  
>
>  
>
> On Friday, March 17, 2023 at 5:39:59 PM UTC+1 Krack Matthias wrote:
>
> Hi
>
>  
>
> You want to normalize a contracted Gaussian function (cgf) basis set, 
> which is performed in two steps: (1) normalization of the primitive 
> Cartesian Gaussian functions and thereafter (2) normalization of the 
> contracted Gaussian functions as indicated in line 1075 
> <https://github.com/cp2k/cp2k/blob/f26eaef31a9d3f80ca30d8d2f11790a2a072e370/src/aobasis/basis_set_types.F#L1075>. 
> This is all in Cartesian representation. For the spherical orbital 
> representation, you will need a further Cartesian to spherical orbital 
> transformation step. CP2K calculates integrals internally in the Cartesian 
> representation, but the default printout, e.g. for the overlap integral 
> matrix, is in spherical orbitals.
>
> Firstly, I would try to reproduce the normalization for an uncontracted 
> basis set (one primitive function per l only). In the next step, you can 
> try two primitive functions. Normalization is required to keep for instance 
> the electron count (trace of PS). 
>
>  
>
> HTH
>
>  
>
> Matthias   
>
>  
>
> *From: *cp... at googlegroups.com <cp... at googlegroups.com> on behalf of 
> Aleksandros Sobczyk <sobcz... at gmail.com>
> *Date: *Friday, 17 March 2023 at 16:59
> *To: *cp2k <cp... at googlegroups.com>
> *Subject: *Re: [CP2K:18552] Re: Evaluation of basis functions over a grid 
> and computation of overlap integrals
>
> Below is the precise example we are using.
>
> These un-normalized orbitals:
>
>
>
>
>
>
>
>
>
> *Sr SZV-MOLOPT-SR-GTH SZV-MOLOPT-SR-GTH-q10 1 2 0 1 6 2 1     
>  7.290111894735  0.069364270475 -0.016182746349  0.035659445929     
>  2.536776771327 -0.571246927373  0.158928639982 -0.195822349727     
>  1.283099546928  0.167836311459 -0.041157757852  0.260320252229     
>  0.532449841650  0.904733330629 -0.431882417196  0.555386362294     
>  0.211628059408  0.250907816117 -0.073959284415  0.267635587013     
>  0.050841303698  0.007135721199  0.825600560103  0.001214128781*
>
> after the normalization (cp2k output) become as follows:
>
>
>
>
>
>
>
>
> *Sr SZV-MOLOPT-SR-GTH SZV-MOLOPT-SR-GTH-q10 1 2 0 1 6 2 1  7.290111894735 
>       0.221325      -0.071449       0.684627  2.536776771327      -0.825808 
>       0.317914      -1.004793  1.283099546928       0.145521      -0.049379 
>       0.569764  0.532449841650       0.405577      -0.267899       0.404860 
>  0.211628059408       0.056304      -0.022965       0.061570 
>  0.050841303698       0.000549       0.087969       0.000047*
>
>  
>
> None of the two aforementioned normalization procedures can reproduce this.
>
>  
>
> On Friday, March 17, 2023 at 4:54:28 PM UTC+1 Aleksandros Sobczyk wrote:
>
> Update on this:
>
> There appear to be two (maybe more) normalization procedures for orbital 
> coefficients:
>
> 1) 
> https://github.com/cp2k/cp2k/blob/f26eaef31a9d3f80ca30d8d2f11790a2a072e370/src/aobasis/basis_set_types.F#L1099
> 2) 
> https://github.com/cp2k/cp2k/blob/f26eaef31a9d3f80ca30d8d2f11790a2a072e370/src/atom_types.F#L2374
>
> The results of 1) seem to be closer to the output of cp2k (for 
> s-orbitals). For p-orbitals it starts to diverge.
>
>  
>
> It would be great if someone can confirm which normalization method is 
> used internally for contracted Gaussians.
>
> (what is the goal of the normalization? are the primitive Gaussians 
> normalized to integrate to 1? is it something more advanced?)
>
>  
>
> Best regards,
>
> Aleksandros Sobczyk
>
> On Monday, March 13, 2023 at 11:00:04 AM UTC+1 Aleksandros Sobczyk wrote:
>
> Dear prof. Hutter and CP2K developers,
>
>
> The example helped to figure out how to reproduce the overlap matrix for 
> our systems. We had to apply the following two steps:
>
> 1) For the basis sets, we had to use the normalized coefficients that are 
> found inside the output file of CP2K. The un-normalized coefficients from 
> the original BASIS_SET file did not work (as you suggested). For now, we 
> can directly use the CP2K output to derive these normalized coefficients 
> for our experiments, but it would be also helpful to know how to reproduce 
> them (i.e., what is the normalization procedure).
>
> 2) We also had to scale all the exponents of all the basis sets by a 
> factor of (1/0.529)^2 (to convert Angstrom to atomic units). This was an 
> arbitrary guess, but without it the overlaps do not match.
>
> Could you confirm that this is the correct way to compute the orbital 
> overlaps? (and if not, propose the correct way). 
> It would also be helpful if these details can be documented, e.g. in this 
> page: https://www.cp2k.org/basis_sets
>
> Best regards and thank you again for the feedback,
> Aleksandros Sobczyk
>
> On Friday, March 10, 2023 at 6:35:28 PM UTC+1 Aleksandros Sobczyk wrote:
>
> Dear Prof. Hutter,
>
>  
>
> Thank you very much for your reply and for the example.
>
> Let us investigate it with my colleagues and see if we can resolve our 
> problem.
>
>  
>
> Best regards,
>
> Aleksandros
>
> On Friday, March 10, 2023 at 3:18:28 PM UTC+1 Jürg Hutter wrote:
>
> Hi 
>
> are you assuming normalized or un-normalized Gaussians? 
> The basis set input in CP2K uses (like all QC codes) normalized Gaussians. 
> Internally, CP2K works with un-normalized Cartesian Gaussians, i.e. the 
> coefficients are adapted at the beginning of the calculation. 
>
> I have attached a simple example where you can play with the basis set 
> in the input and the overlap matrix is printed. 
>
> regards 
> JH 
>
> ________________________________________ 
> From: cp... at googlegroups.com <cp... at googlegroups.com> on behalf of 
> Aleksandros Sobczyk <sobcz... at gmail.com> 
> Sent: Friday, March 10, 2023 2:26 PM 
> To: cp2k 
> Subject: [CP2K:18530] Re: Evaluation of basis functions over a grid and 
> computation of overlap integrals 
>
> Update: I have also calculated several overlap integrals analytically (for 
> s-orbitals which are simpler), they still don't match the values of the S 
> matrix. 
> Any feedback would be greatly appreciated. 
>
> On Wednesday, March 8, 2023 at 12:57:02 PM UTC+1 Aleksandros Sobczyk 
> wrote: 
> Hello, 
>
> I have a set of atoms in real-space and the corresponding SZV basis sets. 
> I want to evaluate each basis function over a grid of points in the cell. 
> E.g., I have a grid of 3d points [r1, r2, ..., rk] and I want to evaluate 
> each 
> Φj(r1), Φj(r2), ... Φj(rk) 
> As a test, I tried to numerically integrate Φj * conj(Φj) over the grid 
> that it was evaluated, and compare the result with the corresponding entry 
> S[j, j] of the overlap matrix that 
> is returned by CP2K. 
> Unfortunately my integral differs substantially from the element S[j, j], 
> so I am doing something wrong. 
>
> Can we find somewhere more detailed documentation on the precise 
> mathematical formulation of the basis sets, and also on the specific 
> algorithms that are used by CP2K to compute the overlap integrals? 
> (So far I have followed as precisely as possible the following page: 
> https://www.cp2k.org/basis_sets 
> but it is still missing information, e.g. are the coefficients normalized? 
> do we assume that the spherical harmonics include the phase factor? etc.) 
>
> Thanks a lot in advance! 
> Aleksandros 
>
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