Configuration Interaction

[amir11]

I am trying to calculate the excionic and biexcitonic energies in quantum dots using Configuration interaction method. I tried to calculate the two electron integrals for configuration interaction method. The integrals are of the form:
\int\int dr_1 dr_2 \Psi_1(r_1) \Psi_2(r_2)1/(r_1-r_2)\Psi_3(r_1)\Psi_4(r_2)

This integral for sure has singularities because of the 1/(r_1-r_2) term. I would be grateful if someone could help me understand how these singularities are treated.

Is there any code were I can find a routine for calculation of these integrals?

Many thanks for your time and attention.

 

[hess8]

Yes, these integrals have been around for a while. I'm not sure where to get code, though.

For example, see pg 26 of http://www.physics.byu.edu/Thesis/GetReport.aspx?id=133 , which shows how it's done for local orbitals. There are probably useful references in there, too.

 

[amir11]

any comment on the unbounded ness?
Unfortunately the PDF was not usefull for me. But any way many thxs.

 

[alxm]

I'm afraid this is one of these questions where you're asking how to do something you just shouldn't be trying to do. CI is not a suitable choice of method here, and is in fact used relatively little these days, since it does not result in the best accuracy/speed tradeoff. It's a method that scales factorially, so there's no way it would work for a system as large as even a small quantum dot. E.g. assume a model of 100 silicon atoms (which http://prl.aps.org/abstract/PRL/v87/i27/e276402" to be a model size that's been used), and conservatively, 10 spin-orbitals per atom. That means you have up to 2*(500)^4 or 125 billion integrals to evaluate per determinant. For the singlet and doublet states, you have {1000 \choose 1} + {1000 \choose 2} = 1 million determinants, giving you a grand total of 125 quadrillion integrals to evaluate, which is utterly unfeasible (to begin with, where are you going to find an exabyte of memory required to store all that?).

As for the actual question, you haven't actually said what integral you're trying to evaluate. I.e. what basis functions are you using? As was discussed in a https://www.physicsforums.com/showthread.php?t=440376", basis sets are often chosen specifically to make these integrals easier to evaluate.

You have to realize that there are entire fields (plural!) devoted do doing these calculations and developing the methods. Wave-based and density functional, molecular and solid-state. Being that I'm in the molecular field myself, I don't have enough expertise to say what method would be most appropriate here (except that it's surely not CI). Probably a DFT method. Nor can I really recommend a specific program, although there are many ab initio programs for solid state, both commercial and academic, many with source code licenses available.

 

[Cthugha]

This integral for sure has singularities because of the 1/(r_1-r_2) term. I would be grateful if someone could help me understand how these singularities are treated.

I am an experimentalist, but I am quite sure the theoreticians who do the calculations for us use the momentum space representation and use the Fourier transform of the 1/(r_1-r_2) term.

Our theoreticians usually do not use CI, but this paper on arxiv explains the Coulomb matrix elements in the appendix:

"A configuration interaction analysis of exchange in double quantum dots" by E. Nielsen and R. P. Muller, http://arxiv.org/abs/1006.2735

 

[amir11]

Well I'm using the single particle states as basis for the integration and am using the product of WF's instead of slater dets to make the comp cost low.
But any way I think there is something missing in the formula. I simply inserted the constant wave function (1/v) so the whole integration simplifies to
\int\int dr_1 dr_2 1/(r_1-r_2)
The integral is easy to evaluate numerically and see that it is not convergant.
I just saw the last post thanks I am reading it.

 

[alxm]

Well I'm using the single particle states as basis for the integration

Yes, the integral is over orbitals, i.e. single particle states. A basis function is what you describe your orbitals with. If you'd read (and understood) the thread I linked to, you'd have known that, as well as the fact that these integrals are not normally integrated numerically to begin with.

and am using the product of WF's instead of slater dets to make the comp cost low.

A Slater determinant is a product of wave-functions! If you were not using Slater determinants, you would not have the two-electron integral given in your first post, and you most certainly wouldn't be using the configuration interaction method, which is an expansion in Slater determinants. There is no way to change the scaling and hence the computational cost of the CI method. If you are not using an expansion in SDs over excitations then you are not using the CI method, period.

But any way I think there is something missing in the formula.[..] The integral is easy to evaluate numerically and see that it is not convergant.

Just because something like \int_0^\infty \frac{1}{x} dx diverges, hardly means that \int_0^\infty \frac{f(x)}{x}dx must do so. (Also, the correct numerator is |r_1 - r_2|)

So, in summary:
1) You want to calculate something using a method that isn't used and can't be used for that calculation. CI with 50 electrons is feasible, not with 150 atoms.
2) You want to write your own program to do the calculation, despite the fact that nobody who knew what they were doing ever writes such a program for the sake of a single calculation, or set of calculations, because:
3) It takes years to write such a program on the level of the ones in real use today, even if you were John Pople himself. There are hundreds if not thousands of people with PhDs in this area, but only a few dozen programs in widespread use.
4) You're not John Pople. Rather you seem to lack even the most basic knowledge about the method you supposedly want to implement. I teach an undergrad course where we expect the students to know what orbitals, basis sets and Slater determinants are, how the CI method works, and even something about how the integrals are evaluated. But I would never expect them to write a competent program if they so had a PhD in the subject. I have a PhD in the subject, and I've never written a complete program for the sake of doing a real calculation. On top of that, I'm now even doubting whether you're up to speed on the calculus involved.

It's not 'just' a matter of evaluating the integrals in question. Decades of research have gone into techniques for evaluating those integrals. Pople got his Nobel prize largely for it. The Helgaker book referenced in the other thread dedicates over 100 pages to describing modern techniques for Fock integral evaluation, and that's with Gaussian basis sets alone, and solid-state uses other ones as well, with entirely different integration techniques as a result. The way I see it, you have two options:

A) Take some course and/or read an intro textbook, educate yourself on the methods and models that are actually being used calculate what you want to calculate, get the programs implementing these methods, educate yourself on how to use them, and then do your calculations.
or
B) Take a course, continue to the graduate level, get a PhD, spend another number of years writing a program that's close enough to the state-of-the-art to be worth using, and then do your calculations using whatever method is most appropriate at that time. (Which, despite advances in computer technology, is still not going to be CI)

Other than that, I can't help you. Because what you're proposing right now is a fool's errand.

 

[Cthugha]

So, in summary:
1) You want to calculate something using a method that isn't used and can't be used for that calculation. CI with 50 electrons is feasible, not with 150 atoms.

I am afraid you misunderstand what CI is used for in QDs. You usually start by calculating the single particle electron and hole carrier energies by using 8-band k.p or similar approaches and then you use CI to address the direct and exchange energies between electrons and holes constituting the exciton or biexciton. Of course you do not use it to take all the atoms constituting the QD into account. That would indeed be a ridiculous attempt.
In fact CI is widely used for this purpose as one does not have to take into account too many electrons, holes and different configurations. However, how long it takes to write an adequate program to do this, I cannot judge.

I just saw the last post thanks I am reading it.

By the way I was wrong when I said that the theoreticians we work with do not use CI.
Maybe their following papers might help you, too (although the latter just includes a marginal discussion on the topic). Both should be freely available:

N. Baer et al.:"Coulomb effects in semiconductor quantum dots", Eur. Phys. J. B 42, 231–237 (2004)

S. Ritter et al., "Emission properties and photon statistics of a single quantum dot laser", Optics Express, 18, 9909-9921 (2010)

 

[amir11]

Yes, the integral is over orbitals, i.e. single particle states. A basis function is what you describe your orbitals with. If you'd read (and understood) the thread I linked to, you'd have known that, as well as the fact that these integrals are not normally integrated numerically to begin with.


A Slater determinant is a product of wave-functions! If you were not using Slater determinants, you would not have the two-electron integral given in your first post, and you most certainly wouldn't be using the configuration interaction method, which is an expansion in Slater determinants. There is no way to change the scaling and hence the computational cost of the CI method. If you are not using an expansion in SDs over excitations then you are not using the CI method, period.


Just because something like \int_0^\infty \frac{1}{x} dx diverges, hardly means that \int_0^\infty \frac{f(x)}{x}dx must do so. (Also, the correct numerator is |r_1 - r_2|)

So, in summary:
1) You want to calculate something using a method that isn't used and can't be used for that calculation. CI with 50 electrons is feasible, not with 150 atoms.
2) You want to write your own program to do the calculation, despite the fact that nobody who knew what they were doing ever writes such a program for the sake of a single calculation, or set of calculations, because:
3) It takes years to write such a program on the level of the ones in real use today, even if you were John Pople himself. There are hundreds if not thousands of people with PhDs in this area, but only a few dozen programs in widespread use.
4) You're not John Pople. Rather you seem to lack even the most basic knowledge about the method you supposedly want to implement. I teach an undergrad course where we expect the students to know what orbitals, basis sets and Slater determinants are, how the CI method works, and even something about how the integrals are evaluated. But I would never expect them to write a competent program if they so had a PhD in the subject. I have a PhD in the subject, and I've never written a complete program for the sake of doing a real calculation. On top of that, I'm now even doubting whether you're up to speed on the calculus involved.

It's not 'just' a matter of evaluating the integrals in question. Decades of research have gone into techniques for evaluating those integrals. Pople got his Nobel prize largely for it. The Helgaker book referenced in the other thread dedicates over 100 pages to describing modern techniques for Fock integral evaluation, and that's with Gaussian basis sets alone, and solid-state uses other ones as well, with entirely different integration techniques as a result. The way I see it, you have two options:

A) Take some course and/or read an intro textbook, educate yourself on the methods and models that are actually being used calculate what you want to calculate, get the programs implementing these methods, educate yourself on how to use them, and then do your calculations.
or
B) Take a course, continue to the graduate level, get a PhD, spend another number of years writing a program that's close enough to the state-of-the-art to be worth using, and then do your calculations using whatever method is most appropriate at that time. (Which, despite advances in computer technology, is still not going to be CI)

Other than that, I can't help you. Because what you're proposing right now is a fool's errand.

I wonder why you inseast that the method is not used and at all wether you have any background or knowldge in solid state calcs.
I should inform you that there are many publications where the CI is the method of choice e.g.:
Excitonic and biexcitonic properties of single GaN quantumdots modeledby8-band k·p theory
and configuration-interaction method PHYSICALREVIEWB 79,2453302009
there are many more.

1) one usully uses the single particle states calced by effective mass or k.p as bases and in my calcs I have not expanded them in terms of guissian states and I don't mean to.
2) the method converges usully with 6 electrons and 6 holes not 50 electrons for a simple quantum dot!
3) Had you ever written a code yourself, would have known how constructive and educative it is.(Much better than lisitning to a fool lecturer lecture)
4) In a system of disceret points the integral only diverges at a single point, that is when the two particles are at the same point. that is why I say the fact that the integral dosen't converge shows that there is another correction to the formual exists.
A) I sicerely encorage you not to replay to the threads you know nothing about. Your misleading the questioner.
B) I know many people that have written effective codes for calculation of states in dots using K.p method( which I believe you don't know what it is) in a month or a two.
C) Please don't answer the questions you don't have any knowldge, PLAESE

 

[amir11]

I am afraid you misunderstand what CI is used for in QDs. You usually start by calculating the single particle electron and hole carrier energies by using 8-band k.p or similar approaches and then you use CI to address the direct and exchange energies between electrons and holes constituting the exciton or biexciton. Of course you do not use it to take all the atoms constituting the QD into account. That would indeed be a ridiculous attempt.
In fact CI is widely used for this purpose as one does not have to take into account too many electrons, holes and different configurations. However, how long it takes to write an adequate program to do this, I cannot judge.



By the way I was wrong when I said that the theoreticians we work with do not use CI.
Maybe their following papers might help you, too (although the latter just includes a marginal discussion on the topic). Both should be freely available:

N. Baer et al.:"Coulomb effects in semiconductor quantum dots", Eur. Phys. J. B 42, 231–237 (2004)

S. Ritter et al., "Emission properties and photon statistics of a single quantum dot laser", Optics Express, 18, 9909-9921 (2010)

Thanks for your kind replay. I will check them.

 

[Cthugha]

Rereading your first post, the final hint I might be able to supply is the following website about CI applied to quantum dots made by Yasuhiro Tokura:

http://www.brl.ntt.co.jp/group/butsuden-g/tokura/ci/ci.html"

You mentioned that you are interested in sample code. While that link does not provide a complete routine, there is at least some small piece of sample code for calculating a part of the Coulomb matrix elements at the subheading "Bare Coulomb potential".
However, that webpage uses vertical QDs which have a rather high degree of symmetry compared to "real" QDs, so I do not know whether that helps you much.