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# Journal Club for July 2018: Mechanics using Quantum Mechanics

Phanish Suryanarayana

Georgia Institute of Technology

**1. Introduction and Motivation**

Over the past few decades, Density Functional Theory (DFT) developed by Hohenberg, Kohn, and Sham [1,2] has been extensively used for understanding and predicting a wide array of material behavior, including their electronic, mechanical, thermal, and optical properties [3-6]. The tremendous popularity of DFT---free from any empirical parameters by virtue of its origins in the first principles of quantum mechanics---stems from its high accuracy to cost ratio when compared to other such ab-initio theories. However, the efficient and accurate solution of the DFT problem still remains a formidable task. In particular, the orthogonality constraint on the Kohn-Sham orbitals in combination with the substantial number of basis functions required per atom results in a cubic scaling with respect to the number of atoms that is accompanied by a large prefactor. Furthermore, the need for orthogonality gives rise to substantial amount of global communication in parallel computations, which hinders parallel scalability. Consequently, the size of physical systems accessible to DFT has been severely restricted to hundreds of atoms, which has limited the use of DFT in mechanics related applications.

In this journal club, I will outline efforts that enable the application of DFT based ab-initio calculations to three mechanics related problems: (i) study of materials under extreme conditions; (ii) study of the effect of mechanical deformations on the electronic properties of nanostructures, and their interaction with applied electrical and magnetic fields; and (iii) study of crystal defects, interactions betweem them, and interaction with macroscopic deformations. Note that this discussion will focus solely on Kohn-Sham DFT, and will try to provide a general overview, details of which (along with a more comprehensive review of literature) can be found in the cited references.

**2. SPARC: Simulation Package for Ab-initio Real-space Calculations**

Traditional methods for DFT utilize the plane-wave basis [7-9]. However, the non-locality of plane-waves makes them unsuitable for the development of approaches that scale O(N) with respect to the number of atoms, and makes parallelization over modern large-scale, distributed-memory computer architectures particularly challenging. Furthermore, they cannot be made compatible with non-traditional symmetries like cyclic and helical. Finally, the need for periodic boundary conditions limits their effectiveness in the study of non-periodic and localized systems such as defects. To overcome these limitations and therefore make DFT calculations amenable to the aforementioned mechanics related applications, we have developed a new real-space formulation and parallel implementation of DFT referred to as SPARC [10,11], which is able to outperform state-of-the-art implementations (typically developed by large teams of researchers over a couple of decades) by up to an order of magnitude or more, e.g., Fig. 1. In addition to mechanics, it is expected that SPARC will significantly impact a number of other fields such as materials science, physics, and chemistry.

Figure 1: Comparison of the performance of SPARC with other well-established plane-wave and real-space codes for representative aluminum systems with a vacancy [11].

**3. SQDFT: Ab-intio framework for materials under extreme conditons**

In order to overcome the critical cubic-scaling bottleneck with respect to system size, much research in the past two decades has been devoted to the development of linear-scaling solution strategies for DFT [12,13]. Rather than calculate the orthonormal Kohn-Sham orbitals, these techniques directly determine the quantities of interest with linear-scaling cost by exploiting the nearsightedness of matter. Though these efforts have yielded significant advances, there are a number of limitations. In particular, the accuracy and stability of linear-scaling methods remain ongoing concerns due to the need for additional computational parameters, subtleties in determining sufficient numbers and/or centers of localized orbitals, limitations of underlying basis sets, and calculation of accurate atomic forces, as required for structural relaxation and molecular dynamics simulations. In addition, efficient large-scale parallelization poses a significant challenge due to complex communications patterns and load balancing issues. Finally, and perhaps most importantly, the assumption of a band gap in the electronic structure makes these methods inapplicable to metallic systems.

High temperature calculations present additional challenges for DFT. These include the need for a significantly larger number of orbitals to be computed, as the number of partially occupied states increases, and need for more diffuse orbitals, as higher-energy states become less localized. Consequently, cubic-scaling methods as well as local-orbital based linear-scaling methods have very large prefactors, which makes them unsuitable for the study of materials under extreme conditions. In order to overcome these limitations, in the framework provided by the SPARC formulation and implementation, we have recently developed a linear-scaling DFT formulation and implementation referred to as SQDFT [14-16], whose cost actually decreases with increasing temperature. Furthermore, it can efficiently scale up to a hundred thousands computational processors (e.g., Fig. 2), and is therefore able to simulate systems whose sizes are two orders of magnitude larger than previously feasible. SQDFT is currently being utilized to study a variety of materials systems at extreme conditions, with applications in geomechanics.

Figure 2: Parallel scaling of SQDFT, with straight lines in the strong scaling representing ideal scaling [16].

**4. Symmetry-adapted DFT: Ab-initio framework for systems with non-traditional symmetries**

Nanostructures have tremendous number of applications, including energy harvesting, efficient power transmission, curing terminal diseases, and design of materials with high specific strength. Therefore, the development of techniques that enable the systematic design and discovery of novel nanostructures with tailored properties is of tremendous interest. Unfortunately, current experimental approaches are generally time consuming, expensive and typically rely on empirical insight. Further, accurate computational techniques like DFT are unable to characterize complex nanostructures and systematically traverse the enormous configurational space because of their large computational expense. This is mainly a consequence of their inability to exploit non-traditional symmetries that are typically present in nanostructures displaying exotic and novel properties. In order to overcome this, in the framework provided by SPARC, we are currently developing a novel DFT framework---based on the notion of objective structures [17]---that is compatible with all the symmetry groups, which will not only provide tremendous simplification in the characterization of nanostructures, but will also accelerate the design of new nanostructures by allowing the use of symmetry to parameterize the configurational space of nanostructures.

As first steps towards achieving this goal, we have developed Cyclic DFT [18] and Helical DFT [19] in the framework provided by the SPARC formulation, which can exploit the cyclic and helical symmetries present in the system to tremendously reduce the computational cost. Since uniform bending deformations can be associated with cyclic symmetry and uniform torsional deformations can be associated with helical symmetry, Cyclic and Helical DFT provide an elegant route to the ab-initio study of bending and torsion in nanostructures (e.g., Fig. 3). Ab-initio simulations of this nature are unprecedented and well outside the scope of any other systematic first principles method in existence. For example, Cyclic DFT was recently employed to study the properties of a 2 micron sized nanostructure, which is up to two orders of magnitude larger than state-of-the-art [20]. Cyclic and Helical DFT are currently being used to study the interaction of mechanical deformations with electric and magneic fields in nanostructures. Also, it is being used to study biological systems with helical symmetry.

Figure 3: Results for bending of a silicene nanoribbon [18]. (a) Electron density contour. (b) Cyclic band structure.

**5. Course-grained DFT: Ab-initio framework for the study of crystal defects**

Crystal defects, though present in relatively minute concentrations, play a significant role in determining material properties. This necessitates an accurate characterization of defects at physically relevant defect concentrations (parts per million), which represents a unique challenge since both the electronic structure of the defect core as well as the long range elastic field need to be resolved simultaneously. Since routine DFT calculations are limited to hundreds of atoms, this represents a truly challenging open problem. In order to solve this, we have developed a method to coarse-grain DFT (in the framework provided by the SPARC and SQDFT formulations) that is solely based on approximation theory, without the introduction of any new equations and resultant spurious physics [21,22]. This work has opened an avenue for the study of extended crystal defects using DFT, which represents a vital step towards understanding the deformation and failure mechanisms in solids. We are currently utilizing this framework to characterize dislocations, the interactions between them, and their interaction with macroscopic fields (e.g. strain). Such studies provide an avenue for the use of constitutive laws based on first principles in higher-scale simulations (e.g. dislocation dynamics).

Figure 4: Electron density contours on the mid and edge planes of sodium calculated using coarse-grained DFT [21]

**6. Concluding Remarks**

There is great potential and scope for the the use of quantum-mechanical methods like DFT in mechanics related applications. Unfortunately, many of these require system sizes that are well beyond the capabilities of traditional DFT formulations and implementations. Development of methods such as those described above have the potential to open new and exciting avenues for the routine use of ab-initio methods like DFT in mechanics.

**References**

[1] Hohenberg, P. and Kohn, W., 1964. Inhomogeneous electron gas. Physical review, 136(3B), p.B864.

[2] Kohn, W. and Sham, L.J., 1965. Self-consistent equations including exchange and correlation effects. Physical review, 140(4A), p.A1133.

[3] Jones, R.O. and Gunnarsson, O., 1989. The density functional formalism, its applications and prospects. Reviews of Modern Physics, 61(3), p.689.

[4] Baroni, S., De Gironcoli, S., Dal Corso, A. and Giannozzi, P., 2001. Phonons and related crystal properties from density-functional perturbation theory. Reviews of Modern Physics, 73(2), p.515.

[5] Marques, M.A. and Gross, E.K., 2004. Time-dependent density functional theory. Annu. Rev. Phys. Chem., 55, pp.427-455.

[6] Jones, R.O., 2015. Density functional theory: Its origins, rise to prominence, and future. Reviews of modern physics, 87(3), p.897.

[7] Gonze, X., Beuken, J.M., Caracas, R., Detraux, F., Fuchs, M., Rignanese, G.M., Sindic, L., Verstraete, M., Zerah, G., Jollet, F. and Torrent, M., 2002. First-principles computation of material properties: the ABINIT software project. Computational Materials Science, 25(3), pp.478-492.

[8] Kresse, G. and Furthmüller, J., 1996. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical review B, 54(16), p.11169.

[9] Giannozzi, P., Baroni, S., Bonini, N., Calandra, M., Car, R., Cavazzoni, C., Ceresoli, D., Chiarotti, G.L., Cococcioni, M., Dabo, I. and Dal Corso, A., 2009. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. Journal of physics: Condensed matter, 21(39), p.395502.

[10] Ghosh, S. and Suryanarayana, P., 2017. SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Isolated clusters. Computer Physics Communications, 212, pp.189-204.

[11] Ghosh, S. and Suryanarayana, P., 2017. SPARC: Accurate and efficient finite-difference formulation and parallel implementation of Density Functional Theory: Extended systems. Computer Physics Communications, 216, pp.109-125.

[12] [12] Goedecker, S., 1999. Linear scaling electronic structure methods. Reviews of Modern Physics, 71(4), p.1085.

[13] Bowler, D.R. and Miyazaki, T., 2012. Methods in electronic structure calculations. Reports on Progress in Physics, 75(3), p.036503.

[14] Suryanarayana, P., 2013. On spectral quadrature for linear-scaling density functional theory. Chemical Physics Letters, 584, pp.182-187.

[15] Pratapa, P.P., Suryanarayana, P. and Pask, J.E., 2016. Spectral Quadrature method for accurate O (N) electronic structure calculations of metals and insulators. Computer Physics Communications, 200, pp.96-107.

[16] Suryanarayana, P., Pratapa, P.P., Sharma, A. and Pask, J.E., 2018. SQDFT: Spectral Quadrature method for large-scale parallel O (N) Kohn–Sham calculations at high temperature. Computer Physics Communications, 224, pp.288-298.

[17] James, R.D., 2006. Objective structures. Journal of the Mechanics and Physics of Solids, 54(11), pp.2354-2390.

[18] Banerjee, A.S. and Suryanarayana, P., 2016. Cyclic Density Functional Theory: A route to the first principles simulation of bending in nanostructures. Journal of the Mechanics and Physics of Solids, 96, pp.605-631.

[19] Banerjee, A.S. and Suryanarayana, P., 2018. Ab initio framework for simulating systems with helical symmetry: formulation, implementation and applications to torsional deformations in nanostructures. In preparation.

[20] Ghosh, S., Banerjee, A.S. and Suryanarayana, P., 2018. Density Functional Theory in cylindrical coordinates: Ab-initio simulations of nanomaterials with uniform curvature. In preparation

[21] Suryanarayana, P., Bhattacharya, K. and Ortiz, M., 2013. Coarse-graining Kohn–Sham density functional theory. Journal of the Mechanics and Physics of Solids, 61(1), pp.38-60.

[22] Ponga, M., Bhattacharya, K. and Ortiz, M., 2016. A sublinear-scaling approach to density-functional-theory analysis of crystal defects. Journal of the Mechanics and Physics of Solids, 95, pp.530-556.

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## Comments

## DFT calculations for dislocations

Dear Phanish:

Thanks for the excellent discussion. You mention defects. What is the simplest crystal defect for DFT calculations? Does it have to be a periodic arrangement of defects or it is possible to analyze a single defect? Are you aware of any DFT calculations for dislocations? Thanks. Regards,Arash

## Re: DFT calculations for dislocations

"What is the simplest crystal defect for DFT calculations?"

In general, the simplest defect will be point defects, e.g., vacancy.

"Does it have to be a periodic arrangement of defects or it is possible to analyze a single defect?"

Traditionally, DFT approaches have employed the plane-wave basis (i.e., Fourier basis). Therefore, they are restricted to periodic boundary conditions, which translates to a periodic arrangement of defects. Using real-space methods like the one I described in my post (e.g. SPARC), it is possible to impose Dirichlet or periodic boundary conditions or a combination thereof. However, a number of questions remain, the main one being: what are the appropriate boundary conditions on the electronic structure quantities? (Recall that we are dealing with an eigenvalue problem). Indeed, an alternative to finding and applying such boundary conditions is to coarse-grain DFT [21,22].

"Are you aware of any DFT calculations for dislocations?"

Given their importance, there have been a number of efforts to study dislocations using DFT. Three broad classes of strategies that are adopted are: (i) Quadrupole or dipole method, e.g., [23,24], (ii) Development of new boundary conditions, e.g., [25,26], and (iii) Multiscale methods, e.g., [27]. Each of these approaches have their own limitations and strengths in terms of accuracy, efficiency, and the quantities they can calculate.

[23] Bigger, J.R.K., McInnes, D.A., Sutton, A.P., Payne, M.C., Stich, I., King-Smith, R.D., Bird, D.M. and Clarke, L.J., 1992. Atomic and electronic structures of the 90 partial dislocation in silicon.

Physical review letters,69(15), p.2224.[24] Dezerald, L., Proville, L., Ventelon, L., Willaime, F. and Rodney, D., 2015. First-principles prediction of kink-pair activation enthalpy on screw dislocations in bcc transition metals: V, Nb, Ta, Mo, W, and Fe.

Physical Review B,91(9), p.094105.[25] Woodward, C. and Rao, S.I., 2002. Flexible ab initio boundary conditions: Simulating isolated dislocations in bcc Mo and Ta.

Physical review letters,88(21), p.216402.[26] Yasi, J.A., Hector Jr, L.G. and Trinkle, D.R., 2010. First-principles data for solid-solution strengthening of magnesium: From geometry and chemistry to properties.

Acta Materialia,58(17), pp.5704-5713.[27] Lu, G., Tadmor, E.B. and Kaxiras, E., 2006. From electrons to finite elements: A concurrent multiscale approach for metals.

Physical Review B,73(2), p.024108.## Dear Prof. Suryanarayana,

Dear Prof. Suryanarayana,

Thanks for picking this topic.

I ,also, have a question regarding the boundary condition when DFT approaches has been employed in the plane-wave basis(aka Fourier basis).

As already mentioned by you this approach is restricted to periodic boundary conditions. I read few papers which are calculating surface energies (and interface energy) using plane wave basis approach by creating vacuum in one direction and using only one kpoint in that direction[1,2]. The argument is that the vacuum and single kpoint does not allow the interaction between the surfaces. Do you think this argument is correct in theory?

If yes, then using same argument can we create vacuum in all directions and restrict our kpoints in the vicinity of atoms rather than distributing them in supercell(which contains vacuum also). The only difference will be that we will be needing more than one kpoints in the directions of vacuum for kpoint convergence. The interaction between kpoints from nearby periodic cells can be restricted if kpoints are in vicinity of atoms. Kindly let me know your thoughts on this.

Also, I would like to know if the above mentioned codes are available on Github or Is there any plan, in future, to implement them in some open-source software packages?

Thank you,

[1] Miguel Fuentes-Cabrera, M. I. Baskes, Anatoli V. Melechko, and Michael L. Simpson, 2008. Bridge structure for the graphene/Ni(111) system: A first principles study. Physical review B 77, 035405.

[2] Zhiping Xu and Markus J Buehler. Interface structure and mechanics between graphene and metal substrates: a first-principles study. Journal of Physics Condensed Matter 22 485301.

## "Do you think this argument

"Do you think this argument is correct in theory? "

No, even when a single k-point is employed, a plane-wave code is employing periodic boundary conditions, i.e., there is interaction between a system and its replicas. The typical strategy is to utilize a large enough vacuum such that this interaction is negligible relative to the accuracy of interest. However, depending on the type of system (e.g. those which have dipole moment), the convergence with vacuum can be extremely slow.

"Also, I would like to know if the above mentioned codes are available on Github or Is there any plan, in future, to implement them in some open-source software packages?"

SPARC and SQDFT have already been released as open source codes. They can be found accompanying the paper. The others will be released as open source in the near future.

## Faster DFT calculations

Dear Phanish,

I had read with interest this great description which you have written. (Very good condensation.) A thought had struck me right on the first read. However, something else (including travel) came up in the meanwhile. ... Anyway, I am glad that there still is some time left to discuss it...

OK. Refer to your very first paragraph. You mention the cubic scaling with # of atoms, which implies the limitation of only a few hundred atoms. The first thing to strike me when I read it was this: how about multi-gridding?

So I was wondering whether your approach already makes use of it or what. (Sorry, no time to go through the papers you list... So, just asking...)

However, I also did a rapid Google search today and found, e.g., this: http://homepages.uni-paderborn.de/wgs/Dpubl/pss_217_685.pdf and this: https://repository.lib.ncsu.edu/bitstream/handle/1840.2/203/Bernholc_200... . Both are c. 2000 papers. There must have been more studies and developments since then.

Could you please share your thoughts on MG for DFT? Is it suitable? Does it work for the kind of (mechanics-related) problems you mention above? What are the constraints for using MG in DFT? ... Would like to know your views... Thanks in advance.

Best,

--Ajit

## Re: Faster DFT calculations

"You mention the cubic scaling with # of atoms, which implies the limitation of only a few hundred atoms. The first thing to strike me when I read it was this: how about multi-gridding?"

There have been efforts like the ones you referenced for developing multigrid preconditioning for the DFT problem, which can be used to reduce the prefactor associated with real-space DFT calculations. However, the use of multigrid does not change the cubic-scaling with respect to the number of atoms, which is a consequence of the underlying eigenproblem. Indeed, the scaling can be made linear by employing the nearsightedness principle, as discussed in my post. As is to be expected, the prefactor associated with linear-scaling approaches is significantly larger than the cubic-scaling approaches.

"So I was wondering whether your approach already makes use of it or what."

We do not utilize the multigrid preconditioning for our diagonalization based formulations and implementations (e.g. SPARC). Instead, we employ partial diagonalization based on CheFSI [28] whose performance is very weakly dependent on the spectral width of the matrix being diagonalized, therefore alleviating the need for preconditioning.

"Could you please share your thoughts on MG for DFT? Is it suitable? Does it work for the kind of (mechanics-related) problems you mention above? What are the constraints for using MG in DFT?"

Multigrid preconditioning can significantly reduce the prefactor associated with real-space DFT calculations. However, as mentioned above, it does not improve the scaling. Furthermore, methods like CheFSI alleviate the need for preconditioning, making them highly competitive. Finally, the complexity of multigrid in terms of formulation and implementation for eigenvalue problems (rather than the usual linear systems of equations) makes them less desirable.

[28] Zhou, Y., Saad, Y., Tiago, M.L. and Chelikowsky, J.R., 2006. Self-consistent-field calculations using Chebyshev-filtered subspace iteration.

Journal of Computational Physics,219(1), pp.172-184.## Re: Cubic scaling in DFT...

1. ``However, the use of multigrid does not change the cubic-scaling with respect to the number of atoms, which is a consequence of the underlying eigenproblem.''

Oh, I see.

2. ``[CheFSI's] performance is very weakly dependent on the spectral width of the matrix being diagonalized...''

Thanks for clarifying this part too, and indeed, for taking care to address all other issue I raised.

Best,

--Ajit