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# Quasi-continuum orbital-free density-functional theory : A route to multi-million atom electronic structure (DFT) calculation

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I would like to share the research work I have been pursuing over the past four years. I believe, through this forum, I will be able to reach researchers with various backgrounds and expertise. Suggestions and comments from members will be very useful. I am also attaching links to preprints of manuscripts describing this work. Please follow these links:

http://www-personal.umich.edu/~vikramg/academic/Preprints/QC-OFDFT.pdf

http://www-personal.umich.edu/~vikramg/academic/Preprints/OFDFT-FE.pdf

**Motivation**: Most materials exhibit varying features and undergo various processes across different length and time scales. Moreover, these features change quantitatively as well as qualitatively across different materials. Therefore, understanding all aspects of materials behavior in a cohesive picture calls for the bridging of length and time scales, which is a key issue in computational materials science. Multi-scale modelling is a

paradigm to address this key issue. The success of a multi-scale model depends on the accuracy and transferability of the theory used to model the materials as well as the scheme through which information is transferred across scales.

**Thesis work** The central theme of my doctoral thesis, working with Prof. Michael Ortiz and Prof. Kaushik Bhattacharya, focuses on developing a *seamless* multi-scale scheme with *density-functional theory* as its sole input. Density-functional theory of Hohenberg, Kohn and Sham (KS-DFT), which is derived from quantum mechanics, is widely accepted as a reliable computational tool to compute a wide range of material properties. In metallic systems, it is common to use an approximate orbital-free density-functional theory (OFDFT) where the kinetic energy is modelled and fitted to finer calculations. Below, I describe, in brief, the various steps that constitute my thesis work.

*Real-space formulation and analysis of OFDFT*: Owing to the computational complexity of density-functional theory, traditional implementations of

KS-DFT/OFDFT, have for the most part, been based on the use of a plane-wave basis and periodic boundary conditions on samples consisting of few atoms (around 200 atoms). I developed a real-space formulation of OFDFT to overcome the serious limitation of periodicity which is not an appropriate assumption for various problems of interest in materials science, especially defects. An important step in developing this formulation was to reformulate the electrostatic interactions that are extended in real-space as a local variational principle. This results in a saddle-point variational problem (*min-max* problem) with a local functional in real space. Further, I established that this problem is mathematically well-posed by proving existence of solutions using the direct method in calculus of variations.

*Finite-element discretization of OFDFT and Gamma-convergence*: The local and variational structure of the real-space formulation of OFDFT motivated the use of a finite-element basis to discretize and compute the formulation. The use of finite-element basis enables consideration of complex geometries, general boundary conditions and locally-adapted grids. I proved the convergence of the finite-element approximation, including numerical quadratures, using the mathematical technique of Gamma-convergence. Gamma-convergence, which is a variational form of convergence, states in spirit that the solutions of the sequence of approximate functionals generated by finer and finer finite-element approximations converge to the solution of the exact functional. The key ideas used in these proofs include Sobolev embeddings, inverse inequalities and a novel approach to treat the Gamma-convergence of *min-max* problems.

*Numerical implementation of OFDFT and examples:* Numerical implementation of OFDFT using the finite-element method requires care since the

electron-density and the electrostatic potential are localized near the atomic cores and are convected as the atomic positions change. Consequently, a fixed spatial mesh would be extremely inefficient as we alternate between relaxing these electronic fields and atomic positions. I overcame this hurdle by designing an approach which convects the finite-element mesh with the atomic positions. This approach uses a nested-mesh scheme, where the finite-element mesh which describes the electronic fields is constructed as a sub-grid of the triangulation of atomic positions.

I demonstrated the approach on a host of examples, which included atoms, molecules and clusters of aluminum, and validated it by comparison with other numerical simulations and experiments. I performed simulations on varying sizes of aluminum clusters, including those as large as 3730 atoms, and these demonstrate the efficacy and advantages of the approach. Being clusters, they possess no natural periodicity and thus are not amenable to plane-wave basis. Second, since the boundaries of the clusters satisfy physically meaningful boundary conditions, it is possible to extract information regarding the scaling of the ground state energy density with size.

Countours of electron-density on the mid-plane of an aluminum cluster with 5x5x5 fcc unit cells (666 atoms).

**Quasi-continuum orbital-free density-functional theory (QC-OFDFT): A route to multi-million atom non-periodic OFDFT calculation**

*Motivation* : The real-space formulation of OFDFT and its computation using a finite-element basis, though very effective in addressing non-periodic systems, is restricted to samples consisting of a few thousands of atoms. However, many properties of materials are influenced by defects -- vacancies, dopants, dislocations, cracks, free surfaces -- in small concentrations (parts per million). A complete description of such defects must include both the electronic structure of the core at the fine-scale (sub-nanometer) as well as the elastic, electrostatic and other interactions at the coarse-scale (micrometer and beyond). This in turn requires calculations involving millions of atoms which are well beyond the current capability. This was the main motivation for the development of QC-OFDFT, which is a seamless scheme for systematic and adaptive coarse-graining of OFDFT in a manner that enables consideration of multi-million atom systems at no significant loss of accuracy and without the introduction of spurious physics or assumptions.

*The key-idea: * The method is developed in the spirit of the theory of quasi-continuum (QC), which is a computational technique for seamlessly bridging atomistic and continuum scales by the judicious introduction of kinematic constraints on the atomic degrees of freedom. However, QC-OFDFT differs from the earlier QC approaches in several notable respects. Apart from the displacement field, QC-OFDFT requires additional representation of the electron-density and the electrostatic potential which exhibit sub-lattice structure as well as lattice scale modulation. These electronic fields are decomposed into a local, oscillating solution and a non-local correction. The local, oscillating component of the electronic fields is represented by a sub-lattice finite-element interpolation (fine-mesh) in the entire domain, whereas the non-local correction is effectively represented by a finite-element interpolation (coarse-mesh) which is sub-lattice close to defects and coarse-grains away from defects . The local solution is computed by performing a periodic calculation and the non-local correction is determined from a variational principle.

Key-Idea: (a) atomistic-mesh (triangulation of the lattice sites) coarse-grains away from the defect (red dot); (b) coarse-mesh, which describes the non-local corrections to electronic fields; (c) fine-mesh, which describes the local, oscillating component of electronic fields; atomistic and coarse mesh coarse-grain away from vacancy, whereas fine-mesh is a uniform triangulation.

In order to avoid computational complexities of the order of the fine-mesh, I exploit the conceptual framework of the theory of homogenization of periodic media to define quadrature rules of a complexity commensurate with that of the coarse-mesh. Convergence of the OFDFT solution with increasing number of nodes in the coarse-mesh was investigated using numerical tests. The reduction in computational effort afforded by QC-OFDFT, at no significant loss of accuracy with respect to a full-atom calculation, is quite staggering. For instance, I have analyzed million-atom samples with modest computational resources, giving access to cell-sizes that have never been analyzed using OFDFT.

*Some Results:*

Contour of electron-density around a vacancy in an million atom aluminum cluster

Contour of electron-density around a di-vacancy complex in an million atom aluminum cluster

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## Comments

## applications and details

Can you give an example: in what applications will this multiscale quasi-continuum orbital-free density-function calculation be applied to. Thanks.

Is the work published? I am very interested in reading some details.

## Details and Applications

Dear Henry Tan,

This work has not yet appeared in print, but is available online on the JMPS website as articles in press. For some reason my links in the post didn't appear. I have updated my post now. I am also giving the links to these papers here.

http://www.its.caltech.edu/~vikramg/academic/OFDFT-FE.pdf

http://www.its.caltech.edu/~vikramg/academic/DFT-FE-QC.jmps.pdf

Regarding applications of this approach, there are many. This methodological development seamlessly couples the quantum mechanical and continuum length scales with density-functional theory as its sole physics, which is the most reliable, computationally feasible and transferable theory to compute materials properties. So, broadly speaking, the applications are in studying materials with defects from a quantum mechanical perspective. Right now, I am looking at two applications: (1) Studying the problem of vacancy clustering and formation of prismatic loops in aluminum, which is important from the viewpoint of radiation damage in materials. We have observed some very interesting physics over here which have not been observed before (this work is not published, so I am unable to give you details; but will keep you posted) (2) Studying a single dislocation resolving all length-scales and computing the core-energy.

The other examples which I can list are : computing migration energies of these defects, studying surface reconstructions, phase transformations in materials. Once we extend this approach to the Kohn-Sham version of density-functional theory, one can study how optical, electronic and magnetic properties of complex materials (like ferro-electric, photonics materials) get influenced by defects like surfaces, domain walls and others.

There have been attempts to study these problems before, which broadly fall under the category of multi-scale modeling. But all these approaches have used heterogeneous theories to describe features at different length scales and various hypothesis are used to stitch these heterogeneous theories across the boundaries. This work differs from those approaches in an important way. We use the same physics everywhere in the domain (OFDFT) and use the power of coarse-graining to seamlessly transition from quantum mechanical length scale to continuum length scales. So we keep the same fundamental physics everywhere in the domain and play with the numerics.

I will be really happy to discuss more if you are interested.

## quite interesting

Vikram,

This is quite interesting. I did notice one of your papers that appears in the "Articles in Press" section of JMPS. The freedom from periodic boundary conditions is nice.

I have two simple questions: (1) Have you tried using/testing your methodology to study semiconductor clusters (quantum dots) from the viewpoint of electronic structure? (2) Did you implement your work within the quasicontinuum code available online?

## Re: quite interesting

Dear Pradeep,

Thanks! In response to your questions -

(1) We have developed the QC approach for orbital-free density-functional theory (OFDFT). OFDFT is a reliable computational tool for systems with electronic structure close to that of a free electron gas (like Al, simple metals ..), but can give erroneous results for systems which are covalently bonded. Hence, at this point we can't study semi-conductor devices. The most reliable computational tool for any material is the Kohn-Sham version of density-functional theory (KS-DFT). I am working on extending the present approach to achieve coarse-graining for the KS-DFT too. Once this is done, then we will have access to all materials and all properties (structural, optical, electronic, magentic ...)

(2) The standard quasi-continuum codes developed so far only handle the coarse-graining of the displacement field. But in DFT we need to coarse grain the electronic fields too, which is non-trivial. In our second manuscript, titled "Quasi-continuum orbital-free density-functional theory: A route to multi-million atom DFT calculation" (which is available as article under press on the JMPS website as well as a link is posted on my blog) we show how we achieve this. In short, we use a predictor-corrector approach and construct a numerical scheme which is supported by a theorem by Lions et. al (From Molecular models to continuum mechanics, ARMA 164 341, 2002). The non-periodic real-space approach and the finite-element discretization of OFDFT from our first manuscript ("Non-periodic finite-element formulation of OFDFT") are key to developing this scheme.

I will be really happy to discuss more with you.

## FEM and DFT

Read with interest your papers. Very nice and comprehensive-work that brings together tools and techniques from different fields along with high-performance computing in an attempt to `bridge the scales.' The essentials on OF-DFT are well-explained; am more familiar with KS-DFT using the pseudopotential approximation, and also some elements of Gamma/epi-convergence. Per what I have seen in the literature, most multiscale methods typically use empirical potentials and few start from first principles as you have proposed. I saw Pask's work on FE-DFT cited, and he is one among just a handful who are involved in the development and application of finite elements in self-consistent KS-DFT calculations. We're currently applying partition-of-unity finite elements for the Schrodinger equation (eigenproblem), and results are very encouraging (vis-a-vis other real-space methods and even planewaves). Will post/discuss the same when the papers are realized in the coming months. To create a link, you can highlight the URL (go into edit) and then click on the `chain' symbol in the menu and provide the URL there too.

## Re: FEM and DFT

Dear Sukumar,

Thankyou very much for your comments. I am looking forward to reading your papers.

## General questions on DFT itself

Hi Vikram and other mechanicians,

I am posting some questions of general nature on DFT here. Yet, I want to note right at the outset that DFT is not my area of special knowledge let alone of research. My interest in DFT (as of today) mostly is in the nature of a layman's curiosity. May be that's why I have a lot of naive (perhaps even dumb and stupid!) questions to ask about DFT.

I am posting these questions here, as a comment to Vikram's post, even though these aren't exactly comments on his two forthcoming papers. I am sure those papers is what he would like to have the discussion here mainly focused on. Yet, these questions of mine do pertain to DFT. So, I decided to post them here... If these questions become interesting enough, anyone may please feel free to create separate discussion threads and post them there... I hope Vikram won't mind thus starting out this discussion on the DFT-related points off his post.

(1.0) Why do people use the variational formulation of FEM for DFT? Two sub-questions...

(1.1) Why do they not use simple FDM (finite difference method)?

(1.2) Why not use Galerkin's method?

(2.0) The difference between DFT and ab initio calculations seems to be well noted in the literature. But what is the main difference between QFT and DFT?

(2.1) Why is DFT not used to analyze superconductivity?

All the following questions have this prefix: "Using the currently available DFT simulation technology..."

(3.0) Can one simulate ordinary liquids, say, water or molten tin or copper? Is there any salient difference between using DFT for solids vs. for liquids?

(3.1) Can one predict thermodynamic quantities like, say, heat of fusion?

(3.2) Can one model nucleation and growth in solidification?

(4.0) Can one model the motion of, say, a small cluster of water molecules against a backdrop of other millions of water molecules?

(4.1) Will it be possible to simulate some liquid passing through a nanotube? Can one think of any major obstacle against modeling this situation--something that would be very obvious to a DFT expert?

(4.2) Will it be possible to simulate a buckyball passing through a narrow slit? Will this simulate diffraction--as has been experimentally observed?

Thanks in advance for all your replies.

## Re: General questions on DFT

Dear Ajit,

I am answering your questions in brief below.

1. DFT itself is a variational formulation. It states that "The ground state properties of a system depend only on the electron-density". This is a variational statement. Now, FEM is better than other approaches like FD because it respects the variational structure of the problem. A discretized version of the variational formulation and a corresponding Gallerkin formulation both result in the same set of equations.

2,3,4 : Some of these applications are possible and are being attempted. However, there are two main obstacles which appear in most applications you mentioned.

(i) In most such applications one needs to break away from periodicity, which has been a serious restriction till date as most DFT codes are written using a plane-wave basis. This is where a real-space formulation is useful which can handel more general non-periodic systems.

(ii) Some applications you mention require simulations at non-zero temperature (higher temp) and this is still a holy grail problem in electronic structure calculations.

I hope these answers are useful.

## Further on the DFT-FEM questions

Hi Vikram,

Yes, the answers were helpful, but sometimes only in part.

1. I raised the question about FDM because: (i) I love its simplicity!; and (ii) the most important advantage of FEM--its ability to better handle more complicated BCs within a uniform framework--seems to have been absent here anyways (though I can't immediately see why).

Is it the case that DFT has not been formulated in terms of an equivalent differential equation at all? Is this the basic reason why FDM and the specifically Galerkin form of FE formulations have not been pursued thus far?

Not having a differential formulation of DFT is unexpected considering that (i) it is the variational form which is more restrictive, (ii) DFT work received a Nobel, and (iii) the way physics is usually done, whereby even a minor nuance of an alternative approache is not left unpursued. Possible, but unexpected.

2. A dumb question, again! What does periodicity basically mean in this context? The lattice-like regular arrangement (periodic in real space)? The particular form of BCs (their symmetry)? The basis in plane-waves (sort of like Fourier expansion)? Do the three have to be related necessarily?... It wouldn't seem to be so on the face of it... Some of my questions also probe these aspects... For instance, let me ask you one more question: You mention non-zero temp. OK. So, can DFT-FEM handle glass at 0 K (as a theoretical scenario)? How about a solid solution with randomly placed second component?

I did note Pradeep Sharma's comment about the freedom from periodicity that your work now allows. So your work does seem to be important. Yet, as an outsider, I can't quite see why the plane-wave basis would be a limitation... Just thinking aloud--you may leave this point alone.

3. Then, my questions also dealt with the aspect of *motion* within the system: whether and how DFT (and its FEM approximation) could handle motion. Yours seems to be a static model. Or is it the case that the inability to handle non-periodic BCs has also meant, so far, the inability to model motion of atoms/ionic cores?

4. Finally, my question on bucky-ball through a narrow slit also probed the power and suitability of the DFT approach to address (i) tunneling and (ii) wave-particle duality of matter. These are the routine problems for QFT folks to ponder on. How does DFT fare here? How well would a DFT-FEM simulation work out here?

In the present post, the question numbers 1 and 4 are more immediately interesting to me than the others.

## Re: Further questions on DFT

Dear Ajit,

1. The reasons we didnt use FD was because

(i) FD doesnot respect the variational structure of the problem

(ii) As you mentioned it cant handle arbitrary goemetries and boundaries

(iii) And, more importantly, it cant handle unstructured coarse-graining, which is the heart of our multi-scale scheme. Such a coarse-graining allows us to handel multi-million systems using OFDFT, which has not been possible to date. Such a scheme is important in simulating systems where the physics on the quantum scale as well as macroscopic scale are important, for ex. defects in materials as well as some applications you mentioned. Please refer to our second paper for more information.

Every variational formulation has an equivalent strong formulation (differential formulation), ofcourse assuming certain regularity in the system (but vice-versa is not true). The PDE associated with a variational formulation is the Euler-Lagrange equations. So DFT, in particular, has a differential formulation too.

I am familiar with one work recently (there have been some in the past too), where FD scheme is used to discretize DFT. Please refer to, "An Efficient Real Space Method for Orbital-Free Density-Functional Theory", C.J. García-Cervera,

Comm. Comp. Phys., 2 (2), pp. 334-357 (2007).Regarding Gallerkin formulation, as I mentioned before, a discrete variational formulation and Gallerkin formulation are equivalent (just like a variational formulation, weak-form are equivalent).

2. Yes, the three are related. We can discuss more on personal correspondence if this is not clear. Yes, DFT can handle glass at 0K.

3. Motion can be handled in DFT, though periodicity imposes serious restricions in modelling such systems. The physics of DFT is very fundamental, its the discretization and numerical schemes which pose the challenge.

4. DFT is derived from QM for ground-state properties and has all the features of QM. So, I am speculating it will be able to model the bucky-ball through a small slit though I havent come across this.

## How is the work on extending this approach to K-S DFT going on

Dear Vikram

Very interesting work. I want to know How the work on extending this approach to K-S DFT get along with and what the difficulties in the extending. To be honest, I am not familiar with the theory DFT, so my questions may be so simple.

I have read some literatures about the defect state of photonic crystals, unfortunately, we did not get much progress and gave up for some reasons. However, I am still interested in this area, so I want to konw the new ideas (especially those based on the computational/ soild mechanics).

I hope to hear your replies and thanks in advance .

## Ezio Bruno Deparment of

Ezio Bruno Deparment of Physics, University of Messina

Dear Vikram,

thank you very much for this interesting blog and congratulations for your very nice research work. I am really interested in. I'm also trying to go to million atoms following a very different path, though again the starting point is Density Functional theory. If you are interested in, here are the references to some papers of my group on the subject:

E. Bruno, F. Mammano, N. Fiorino and E.V. Morabito, "Coarse-grained density functional theories for metallic alloys: Generalized coherent-potential approximations and charge-excess functional theory", Phys. Rev. B

77, 155108 (2008) doi:10.1103/PhysRevB.77.155108E. Bruno, F. Mammano and B. Ginatempo, "Coarse grained Density Functional theory of order-disorder phase transitions in metallic alloys", arXiv:0810.5367v1 [cond-mat.mtrl-sci]