iMechanica - electronic structure - Comments

Re:Comments on QCDFT/QC-OFDFT methods

Vikram Gavini — Fri, 19 Sep 2008 14:08:36 -0400

Dear Gang,

Many thanks for sharing your views and providing with additional information. As you rightly pointed out, in a scientific problem, esp. in its infancy, people have different perspectives, and it is great that these perspectives are coming out in this forum for the readers to appreciate the importance and the challenge this problem poses. I believe with time all the various methods proposed will continue to improve and the issues in these methods will get addressed systematically: like the need for rigorous justification of the quadrature rules proposed in Gavini et al. J. Mech. Phys. Solids. 55 697 2007, or the assumption of separation of length scales and issues concerning ghost forces in the formulations proposed in your recent articles.

I wish to clarify some of the points brought up in your comments:

(i) The choice of a local kinetic energy functional in QC-OFDFT method proposed in Gavini et al. J. Mech. Phys. Solids. 55 697 2007 was only for the prupose of demonstration of the method. Your comment was very accurate in pointing out that non-local kinetic energy functionals must be used for an accurate description of the system. However, the incorporation of these non-local kinetic energy functionals into QC-OFDFT is not difficult, and is in fact straightforward. The way to achieve this was already indicated in the appendix of Gavini et al. J. Mech. Phys. Solids. 55 669 2007. The numerical implementation of these functionals is work in progress.

(ii) As I pointed out in a previous comment, developing the QC techniques for KS-DFT is tricky because of the delocalized nature of the wave functions in terms of which the energy of the system is described. However, there are ways in which this issue can be circumvented without introducing any further approximations. It is too early for me to give a definitive answer on this as these are early days, and this is work in progress.

(iii) Regarding your comment that instead of considering wave-functions which are delocalized it may be better to address the problem in terms of electron-density which is local, I am not sure how one would achieve this with out resorting to an approximation on the kinetic energy functionals.

Re:A question about the QC method

Vikram Gavini — Fri, 19 Sep 2008 08:37:42 -0400

Dear Ajith,

The representative atoms are picked in such away that they account for every atom around the defect core, where as away from the defect core fewer representative atoms are picked. You have brought up an interesting question as to how one chooses these representative atoms? Often the displacement field governs the choice of these representative atoms. If the displacement field is rapidly varying or has large gradients (like near the defect core) more representative atoms are introduced. If the field is smooth then fewer representative atoms are introduced, as a smooth variation can be captured by few representative atoms.

Depending on whether one has a prior knowledge of the "nature" of the displacement field gives rise to two methods of choosing the rep atoms:

(i) Apriori mesh adaptation: Here the prior knowledge of the nature of the displacement field can be used to determine error estimates associated with the interpolation introduced through the rep atoms. These error estimates provide the optimal distribution of the rep atoms for a given problem.

(ii) Aposteriori mesh adaptation: Often the nature of the displacement field is not know. In such situation, one starts with a choice of the rep atoms and solves the problem. Then, they introduce a h-adaptation, i.e., they introduce an additional rep atom at various locations and see if the energy change is beyond a tolerance. If it is they accept the addition of the rep atom, if not, they reject it.

These adaptation schemes are analogous to mesh adaptation schemes in standard finite-elements but applied to a discrete lattice. For more information I suggest the following reference:

Knap, J., Ortiz, M., 2001. An analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899.

Further, in any QC calculation, convergence of the energy is always checked with respect to the rep atoms and once the convergence is attained the solution you are looking for has been captured. If there was some crucial information about the solution present in the sub-space not spanned by the constraint then one would not achieve the convergence with respect to the rep atoms.

Comments on QCDFT/QC-OFDFT methods

Gang Lu — Thu, 18 Sep 2008 18:18:38 -0400

Dear Vikram and others,

Thank you for your interesting posts. I'd like to offer a few comments here since our work has been mentioned in the discussions (Lu, et al., 2005): (1) In our original paper in 2005, we coupled Kohn-Sham DFT (KS-DFT) to the embedded atom method (EAM) for nonlocal QC simulations. In this approach, we viewed the empirical EAM model as an approximation to DFT because EAM was derived based on DFT and EAM potentials are often fitted from DFT data. In this sense, there are some physical connections between DFT and EAM. The main motivation of the work was to develop a simple and efficient method that can deal with extended defects, such as dislocations and cracks, with necessary quantum level accuracy maintained at the defect core. Anyone who is familiar with DFT and EAM simulations can implement the method easily and the computational overhead for the coupling is minimal. (2) Although the 2005 paper involved two disparate theories (DFT and EAM) in exchange of simplicity, one could develop a QC method that is entirely based on either OFDFT or KS-DFT. We have done so with OFDFT (Q. Peng, X. Zhang, L. Hung, E.A.Carter and G. Lu, "Quantum Simulation of Materials at Micron Scales and Beyond", PRB,78,054118 (2008) ) and the extension to KS-DFT is straightforward and is currently under way. The paper of X. Zhang and G. Lu, "Quantum mechanics/molecular mechanics methodology for metals based on orbital-free density functional theory", PRB 76,245111 (2007) points out the direction of achieving this goal. In our recent 2008 paper, we have performed QC-OFDFT calculations for nanoindentation of Al thin film with micron dimensions. Our method is based on a nonlocal formulation of OFDFT which is applicable to simple metals, like Al, Mg and Li. In the interest of a full disclosure, I should also mention a major difference between our QC-OFDFT method and the one developed by Vikram et al.: in its present form, Vikram's method takes a local approximation of the kinetic energy. The problems associated with the local approximation have been discussed in a recent paper by Emily Carter's group at Princeton (see P.11 of Ho et al. Phys. Chem. Chem. Phys. 2007, 9, 4951). It's not clear to me how the method of Vikram can be extended to nonlocal OFDFT or to KS-DFT. In Vikram's post, he has also acknowledged the difficulty in combining QC and KS-DFT within his approach. Having said that, it would be truly fantastic if someone figures out a way to do the coupling within the framework of Vikram et al. To their credit, Vikram's work has certainly laid down a solid foundation for future developments in this direction and I'm looking forward to its success. (3) Vikram mentioned at the end of the last post that "One of the major hurdles in developing the QC version for DFT is that the wavefunctions are delocalized and handing such a system with QC is tricky because QC relies on local perturbation of the system." I wish to comment that although wavefunctions are delocalized, electron density is not. Maybe one should think in terms of electron density in QCDFT, and that is actually what we did in our QCDFT method. Our method can be applied to semiconductor quantum dots or other systems that may be interested to this community. Overall, I agree with Vikram's posts; as in any scientific debates, people often have different perspectives on the same issues and I have just offered mine.

Re: Electrons to FEM (Cont'd)

N. Sukumar — Thu, 18 Sep 2008 05:12:18 -0400

Dear Vikram,

Thanks for the explanation, and for further clarifications on QC. The DFT everywhere and coarse-graining route does have its appeal over the approach of having different physics in different regions and then having to deal with `apt' matching conditions on the interface. In DFT calculations, systematic improvability and the ability to have strict control of the error is crucial, and hence the leaning to the former. In variational formulations, the basis-function viewpoint is often forgotten (in lieu of the common usage of shape functions, FE implementational details, etc.); your view of QC as a coarse-graining of basis functions is a valuable perspective.

A question about the QC method

Ajit R. Jadhav — Thu, 18 Sep 2008 02:51:07 -0400

Dear Vikram,

This journal club in general and your replies in particular make for an interesting reading. I have a few questions, even if they might be a bit subsidiary in nature.

In your above reply, you say:

The key idea of the QC method is to choose some special atoms called representative atoms and constrain the positions of other atoms with respect to these representative atoms (through the shape functions of the FE).This is tantamount to looking for a solution of the variational problem in a sub-space spanned by this constraint.

My questions are:

How does one pick up the representative atoms? What guiding principle is available in this respect? Roughly how many--what volume fraction--are they?

How does one know that the sub-space not spanned by this constraint does not carry something that is interesting (perhaps even critical) to the physical phenomena being modeled? What, possibly, could be the nature of such things (that possibly might get left out)?

BTW, please note that I am not challenging the very idea of having such an approach. I realize that there must be some great practical benefits following it, e.g., dramatically increasing the number of atoms being modeled or altering the kind of BCs that are being handled.... It is just that I want to make sure that I understand the approach conceptually and physically right before progressing further, that's all. (As you know already from our last year's discussions, I was, and still am, pretty much a novice to both DFT and QC.)

Thanks in advance.

Re: why it took so long for FEM to come to quantum mechanics?

Vikram Gavini — Wed, 17 Sep 2008 20:50:36 -0400

Pradeep,

Since this is a question on history of electronic structure calculations, what I am putting forth is the opinion I gathered through the literature and discussions with others working in this area. Firstly, I should mention that there were attempts in the mid 80's to use FE in electronic structure calculations. But the number of elements required to achieve chemical accuracy that chemists desire was far too many than the computational power at that time could handle. On the other hand plane-waves provided an ideal solution to atleast the problems involving perfect materials, and there were plenty of interesting problems in perfect materials to address given that the field was just beginning to build. Thus a lot of effort went in to developing commercial codes based on plane-wave basis, and, in my opinion, it became an accepted norm in the community. The issues started to surface when people wanted to address defects in materials using electronic structure calculation in the late 90's as Fourier space formulations could not account for defects at their realistic concentrations. This prompted the concurrent multi-scale schemes of the first kind that I discussed in my response to Sukumar's question---one where an electronic structure calculation is embedded in a less accurate model like empirical potentials or tight binding. But with the computational power available today and the fact that FE formulations can be implemented with ease in a parallel computing framework, it is worth revisiting the problem of using FE in electronic structure calculations. What the FE framework offers, which lacks in other real-space and Fourier space methods is the power of coarse-graining---being able to adapt the resolution of the basis set as necessary. I think this holds the key to be able to develop a seamless multi-scale scheme with DFT as the sole input and spanning across length scales from electronic structure to the continuum.

I have mentioned about the QC-OFDFT work in my repsonse to sukumar. One of the limitations of the orbital-free approach to DFT is that the kinetic energy functionals used are not good enough to describe covalent and ionic bonded systems (they are good only for metallic systems). Thus the QC formulation of the standard DFT needs to be developed to address problems in semicoductors and ferroelectrics where defects play an important role, like quantum dots in semiconductors and domain walls in ferroelectrics. One of the major hurdles in developing the QC version for DFT is that the wavefunctions are delocalized and handling such a system with QC is tricky because QC relies on local perturbations of the system. However, there may be ways to get around these issues, but this too early to give a definitive answer.

Re: Electrons to FEM

Vikram Gavini — Wed, 17 Sep 2008 20:11:50 -0400

Sukumar,

Many thanks for raising this question as this opens up discussion on more than one aspect of multi-scale methods, and also will provide an opportunity to put forth my view point of what the real QC method is.

In my opinion there are two kinds of concurrent methods. One where different physics is used to model different regions of the domain, and another where the same physics is used everywhere, but the range of multiple length scales are spanned by coarse-graing the numerical scheme. In my opinion the article Lu et al. falls in the former category. Here DFT is used to describe the physics right at the defect core, which is embedded in a region described by empirical potentials. Two very important question that comes up in such a formulation are (i) what happens on boundary of these heterogeneous theories---what are the conditions imposed? (ii) what errors do they introduce into the energetics of the system. Firstly, there is no right answer to the first question as the attempt is to stitch two heterogeneous theories; in Lu et al. this done by introducing an interaction energy term which is computed by using empirical potentials everywhere. Also, there is no way of quantifying the error associated with this approximation, even numerically for a particular problem (let alone rigorous bounds), because of the limitations on the size of DFT calculations. This turn out to be the major limitation of concurrent schemes which use heterogeneous mathematical theories to describe different regions of the domain.

Now coming to the second type, here one uses the same physics but use coarse-graining of numerical schemes to span across the multiple length scales. The quasi-continuum method was developed in this spirit. It relies on a variational principle: minimize the energy of the system with respect to the positions of atoms. The key idea of the QC method is to choose some special atoms called representative atoms and constrain the positions of other atoms with respect to these representative atoms (through the shape functions of the FE). This is tantamount to looking for a solution of the variational problem in a sub-space spanned by this constraint. Now this is nice because one can test for convergence (at least numerically) of the method as the same theory is being used every where and the subspace becomes increasingly richer with increasing the number of representative nodes. Further, I also like to clarify a common misconceptions about the QC method that the QC method uses only empirical potentials. The QC method is a method which coarse-grains the basis functions, it can be developed for any physics. It is most popular for empirical potentials, but there
are some efforts to develop the QC method for DFT too (by this I mean
DFT is the sole input physics, unlike what is done in Lu et al.). Below
is a reference where the QC method for orbital-free DFT is discussed.

Gavini V, Bhattacharya K, Ortiz M, Quasi-continuum orbital-free density-functional theory: A route to multi-million atom non-periodic DFT calculation, J. Mech. Phys. Solids 55 697 2007

Now coming to the heart of your question: "Is there a flow chat for concurrent multi-scale simulations". I would say if one follows the first approach there is none, as there is no systematic way to address the question of how to stitch heterogeneous mathematical theories. However, the QC method, in its true spirit, provides a systematic solution. Of course there are caveats to this statements and there are open questions, but, in my opinion, it provides a hope.

I hope I provided some inputs to the questions you have raised, atleast partially.

why it took so long for FEM to come to quantum mechanics?

Pradeep Sharma — Wed, 17 Sep 2008 15:37:39 -0400

Dear Vikram,

Thank for your interesting and thoughtful post. I have closely followed the literature on electronic structure calculations although more from the perspective of the "user" rather than the "developer" of computational methods. As you know, electronic structure calculations are central to the study of quantum dots----another area (like defects) where quantum mechanics meets head on with solid mechanics. I have a general question for you which struck me after reading your jclub issue: the finite element method has been around for a while.....why has it then taken the electronic structure community so long to get around to exploring its utility?

You did not mention your own work on orbital free DFT which I find to be quite interesting as well. Could you post a few lines describing that? In particular, I would be curious to hear what challenges one is likely to face if your work is to be extended for application towards semi-conductors (covalent solids) and ferroelectrics (highly polar dielectrics).

Electrons to FEM

N. Sukumar — Wed, 17 Sep 2008 03:04:15 -0400

Vikram,

Thanks for initiating this interesting theme, and for posting your overview on electronic-structure theories. The topic is timely, given the continued and growing interest of mechanicians in the development of multiscale computational methods. I am aware of the papers you have suggested for reading that pertain to the solution of the equations of DFT (Schrodinger and Poisson) via real-space methods, but am not well-versed with the details and intricacies on concurrent methods that attempt to bridge the scales. On taking a peek at Lu et al. (2005), I deciphered that they use the QC framework with the distinction that in the region in the vicinity of the defect, ab initio (first principles) calculations are done, unlike the original QC paper where classical potentials are used. Would it be possible to indicate via a step-by-step procedure what is entailed within a concurrent method, if one w'd like to include quantum-to-continuum simulations within a single umbrella, and to also briefly indicate some of the issues (e.g., interfacial conditions) that need to be addressed therein. I was trying to see if something along the lines of a flowchart for a FE analysis can be crafted for a concurrent multiscale simulation, so that one can better see and appreciate the various links. Possibly my question is a tad vague and overreaching, but I hope you get my drift? There are many experts on iMechanica who have contributed in this endeavor, and hence it would be beneficial to hear from others too about the strengths/weaknesses of competing methods, the current state-of-the-art if it exists, and the main challenges that remain.

Piezoelectric Effects

Zhenyu Zhang — Wed, 12 Sep 2007 11:41:13 -0400

Hi Harley et al.:

Perhaps another important class of mechanical-electronic/electrical coupling is the piezoelectric effects, and one recent significant work enabling the community to calculate such effects based on first principles was the formulation of David Vanderbilt and coworkers within the Berry phase approach. See, for example:

Spontaneous polarization and piezoelectric constants of III-V nitrides

Author(s): Bernardini F, Fiorentini V, Vanderbilt D

Source: PHYSICAL REVIEW B 56 (16): 10024-10027 OCT 15 1997

Abstract: The
spontaneous polarization, dynamical Born charges, and piezoelectric
constants of the III-V nitrides AlN, GaN, and InN are studied ab initio
using the Berry-phase approach to polarization in solids. The
piezoelectric constants are found to be up to ten times larger than in
conventional III-V and II-VI semiconductor compounds and comparable to
those of ZnO. Further properties at variance with those of conventional
III-V compounds are the sign of the piezoelectric constants (positive
as in II-VI compounds) and the very large spontaneous polarization.

Dynamic quantum behaviors?

Harley T. Johnson — Tue, 11 Sep 2007 09:47:37 -0400

Hi Henry,

Do these effects have to do with quantum mechanics of electrons? Or is there an analogy with electronic structure? What do you mean by "quantum"?

Dynamical quantum behaviours

Henry Tan — Mon, 10 Sep 2007 14:52:14 -0400

Researches on the quantum effcts of a passing shock wave are interesting. The quantum effcts may be used to control the behaviour of energetic materials under high speed impact.

Further on the nonlinear

Pradeep Sharma — Mon, 03 Sep 2007 15:40:18 -0400

Further on the nonlinear effects; as Harley has pointed out there are two issues: (i) small strain assumption and the (ii) validity of the typically adopted linear strain-band structure coupling. If I recall correctly there are some papers that have revisited the small strain assumption, one group from England (Faux and co-workers) and other from Poland (Majewski and co-workers).

As Harley alludes to, the second assumption of linear strain-electronic structure dependence is somewhat less studied. I collaborated on a work in which we studied, using ab initio methods, scaling of strain-energy gap coupling in Si clusters. We find that below roughly 5 nm, linear strain-energy gap relation is suspect (see Figure 4 in this paper). This paper is available from my website (# 30, Peng et. al., 2006). In a more recent collaborative work (with Harley), the nonlinear coupling between strain and electronic structure is even more evident (#36, Zhang et.al. 2007). However, I think we have barely scratched the surface...a thorough study of this issue would be quite interesting.

Interestingly, many years ago there was a paper by Zhang from NREL which re-examined the standard multiband kp model and suggested that (in presence of inhomogeneous strain) gradients of strain be included in the constitutive relation between strain and electronic structure. His derivation is quite rigorous and implies that this effect would be important for small quantum dots.

QM and SM: What I would like to have, eventually...

Ajit R. Jadhav — Mon, 03 Sep 2007 09:42:27 -0400

Dear Harley,

I make reference to the following passage in Pradeep's comment:

Pradeep said: Perhaps, during some point this month it would be a good idea to compile a supplementary list of "tutorial" papers and documents that can provide a clear and facile path for mechanicians to get involved in this interesting field. In due course, I will provide a few such references as well.

I very much like this idea. Further, I would like to add to the request.

At the end of the month, I would like it if someone could systematize all the comments and thoughts generated here, from the following two viewpoints: (i) The kind of modeling abstractions that have been used in the papers studied here (and other related papers), for each of QM and SM. (ii) The specific assumptions and techniques used in handling boundary conditions, and the lessons or insights these hold for further model-making, computational models included.

A systematization of the comments and discussions here, from the above two angles, would be interesting to have.

quantum effect on stability

Harley T. Johnson — Mon, 03 Sep 2007 09:26:13 -0400

I am familiar with this very interesting work, Zhenyu, and some of the subsequent work you did on this problem with Zhigang Suo. (Phys. Rev. B, 58, 5116-5120, 1998) One could refer to this generally as quantum mechanical coupling to surface energetics -- as opposed to coupling to deformation -- which would also connect to your idea about the quantum effect resulting in a frictional force. This is very fascinating.