machine learning, model reduction and multiphysics...

Jiahao ChenEmail [email protected] (617) 721-5631Fax (617) 253-7030

Department of ChemistryMassachusetts Institute of Technology77 Massachusetts Avenue, Room 6–228Cambridge, Massachusetts 02139-4301

Machine learning, model reduction and multiphysics simulations ofmatter

The underlying physical laws necessary for the mathematical theory of a large part of physics and thewhole of chemistry are thus completely known, and the difficulty is only that the exact application ofthese laws leads to equations much too complicated to be soluble. It therefore becomes desirable thatapproximate practical methods of applying quantum mechanics should be developed, which can lead toan explanation of the main features of complex atomic systems without too much computation.— [1]

Figure 1: A simplified classification of models usedin theoretical chemistry, showing in principle thecoarse–graining relationships between them.

coar

sene

ss

ab initio theoriese.g. Hartree-Fock, density functional theory

empirical potentialse.g. force fields

continuum mechanicse.g. finite element models

hydrodynamicsheat transfer

electrochemistrymolecular biology

defect propagation

Examples of applications

photochemistry

exact quantum mechanics hydrogen atom

Method

mean field approximationslow energy limit

ignore subatomicstructure

ignore atomic granularitylong-wavelength limit

Quantum mechanics: the computationally intractable the-ory of matter Quantum mechanics allows the properties ofmatter to be predicted ab initio, i.e. axiomatically from firstprinciples without any empirical data. [2, 3] The extraordinaryinsight afforded by Schrödinger’s equation, [4,5] together withstatistical mechanics, have made possible the theoretical dis-ciplines of chemistry [6] and condensed matter physics. [7]However, quantum mechanics has exponential complexity, inthat adding a new particle to the system makes the wavefunc-tion exponentially harder to calculate, as every particle interactswith every other particle nonlocally.1 Therefore, it is difficult toapply quantum mechanics directly to macroscopic sized chunksof matter with ∼ 1023 atoms without suitable approximations.

Traditional computational studies have focussed on devel-oping approximations suitable for specific systems, as shown inFigure 1. Despite the success of these methods, more sophisti-cated methods are needed to study more complicated problemsin solution phase chemistry and into the realms of biology or

materials science. With coarse graining and multiphysics simulations, simulations of entire viruses [13], nanowires [14],or mechanical deformation of materials [15] are now possible, thus pointing the way toward efficient simulations ofunprecedented size. Understanding how to construct this chain of approximations is crucial to develop new simu-lations of large systems and verify their physical validity.

Figure 2: The multiscale physics and chemistry inbulk heterojunction solar cells across length scalesranging from the atomistic to the entire device; anexample of multiscale phenomena in real–world ap-plications From Ref. [16].

This proposal addresses the simulation of matter acrossmultiple length scales using both bottom–up and top–downapproaches. First, the applicability of quantum–mechanicalsimulations will be extended to much greater length scales thanis possible with a pure quantum mechanical treatment. Thereis now a well–established tradition of chemical simulationsthat employ a mix of quantum mechanics and classical em-pirical potentials, such as the quantum mechanical/molecularmechanical (QM/MM) method [17–19] and the “our own n-layered integrated molecular orbital and molecular mechanics”(ONIOM) [18] method. A reexamination of these methodswith a contemporary perspective on multiresolution and mul-tiphysics simulations should provide new insights into the ex-isting limitations of these methods and how they can be ame-liorated. Second, top–down approaches can be reformulatedwith the language and tools of random matrix theory, which will bring a fresh perspective to well–established physicalnotions like emergent universality and renormalization.

1Electronic structure theory lies in the Quantum Merlin Arthur (QMA)–hard complexity class, a superset of NP–hard. [8–12]

Research Proposal of 13



1 Multiscale quantum mechanical simulations of matterLocal scale ordering appears to be essential for high charge carrier mobility... There are no clear defininglength scale boundaries where one physical feature cannot occur.— [16]

The challenges of microscopic multiphysics simulations A multiphysics approach is necessary to simulate largechunks of matter, since full quantum mechanics calculations are impractical. Furthermore, the multiphysics perspectiveparallels the scientific understanding of natural phenomena across multiple length scales. For example, both chemistsand materials scientists are interested in condensed matter, but whereas chemists concern themselves primarily at thelevel of individual atoms and molecules, materials scientists are in general more concerned with larger features suchas microscopic defects up to the continuum bulk. However, multiphysics simulations below the submicron scaleposes unique consistency challenges. First, bulk descriptors such as temperature fields and dielectric constants failto become well–defined at smaller length scales, and their thermodynamics must be consistent with the statistics andmicroscopic granularity of individual atoms and molecules. Second, the classical Newtonian mechanics of continuumlength scales must be compatible with the underlying quantum mechanical behavior of atoms and electrons. This com-plicates the matching of boundary conditions and inductive responses across multiple physical levels. [15] Creating asuccessful multiphysics model for microscopic systems will entail a synthesis of physical ideas such as spatial renormal-ization, self–similarity and emergent universality [20] together with modern multiresolution techniques for large scalecomputations. [21]

Strategy 1. Exploit regularity of wavefunctions in multiresolution methods. One way to improve the quality ofmultiphysics simulations is to speed up more accurate models, allowing them to be applied to larger systems. Ab ini-tio theories, being the most expensive, have received much attention in this regard. First, the basis set of electronicstructure calculations can be improved; multiresolution techniques such as MADNESS [21] provide basis sets withwell–defined truncation errors. Second, electronic structure Hamiltonians can be replaced by low–rank approxima-tions. For example, the cost of coupled cluster theory can be reduced dramatically with a reduced rank Hamiltonianconstructed using higher order singular value decompositions. [22, 23] However, this is an a posteriori reduction tech-nique, in that the Hamiltonian must be constructed in part or fully before the simplification can be made. Reconcilingthese computational studies with the analysis of the regularity properties of wavefunctions [24,25] may provide insightinto how approximate Hamiltonians can be constructed that are consistent with the approximate wavefunctions.

Figure 3: Potential energy surface of benzene dimer (black)and benzene dimer cation (red) predicted using empirical fragmentcharges, showing excellent fits to ab initio data (crosses) using thesame set of parameters for the neutral dimer and the cation. [26]

Strategy 2. Enforcing consistency across multiple physicalmodels using forcematching. For two physical models to bemutually consistent, it is necessary that their predicted physicalquantities be as similar as possible. Theoretical chemists havehad success in developing fitting techniques based on reproduc-ing energies and forces in thermodynamically accessible regionsof the potential energy surfaces. We have implemented and de-veloped a semiautomatic parameterization tool to fit force fieldsto higher–level ab initio data using such techniques. [27] Ourresults, e.g. those in Figure 3, show that with such parameter-ization, empirical potentials can provide comparable or evenbetter accuracy than more expensive methods such as densityfunctional theory, for a fraction of the computational cost. [26]We expect that this computational tool can readily adapted tothe parameterization of the overlapping regions between twoarbitrary theoretical models. Iterating this process across allthe relevant physical models from the bottom up will allow usto study the usefulness of force– and energy– matching tech-niques for parameterizing multiscale simulations.




Strategy 3. Algorithmic model selection for empirical potential development. For many applications in materialsscience and molecular biology, force fields are currently the method of choice for providing a computationally tractableatomistic model, which has stimulated much work on improving their accuracy. One way is to improve the quality of theparameters, as discussed above. However, it is also important that the necessary terms to describe important physicaleffects are present in the functional form of these phenomenological models. New force fields can be developed to modelsuch as polarization or charge transfer, [28–31] or even transitions between multiple electronic states. [26] Statisticalmodel selection can be used to address questions about the necessity of various terms. By running simulations withdifferent empirical potentials and comparing their predictions, tools such as algorithmic information criteria can helpdetermine if specific physical effects are necessary by considering both accuracy and parsimony in the models. [32–34]

2 Post–simulation analyses with model reduction techniquesMultiphysics simulations will enable the simulation of very large physical systems. However, it will be easy to generateterabytes of raw trajectory data from saving the configuration of many atoms. How much data is enough? And howcan one extract physical and chemical insight from the data? Machine learning techniques proffer answers to thesequestions in lieu of labor–intensive analysis using expert intuition, and can be valuable when such intuition is absent.Furthermore, model reductions provided by machine learning can provide yet another bottom–up way to generatemultiphysics simulations. The focus in this section is on how model reductions can be performed using nonlineardimensionality reduction (NLDR) techniques to analyze simulation data from a detailed model with many degreesof freedom, which provide the flexibility to construct accurate Hamiltonians but may also confound the scientificinterpretation of the results.

Figure 4: Applying diffusion maps to summarize abinitio molecular dynamics simulation data, showing(a) the identification of the important subspace ofmolecular coordinates, (b) a projection of the molec-ular trajectories onto the data manifold coordinates,and (c) representative molecular snapshots collectedalong a manifold coordinate. From Ref. [35]

Previous work: finding reaction coordinates using nonlineardimensionality reduction (NLDR) We have found featureextraction using nonlinear dimensionality reduction (NLDR)techniques such as diffusion maps to be useful for automaticallysummarizing trajectory data from detailed ab initio molecu-lar dynamics simulations. [35] The specific problem consideredwas a barrierless relaxation of electronically excited molecules.The processes are highly non–equilibrium, so the conventionalpicture from transition state theory does not give a good de-scription of the photochemistry. 2 With no manual inter-vention, the algorithm found a single (reaction) coordinate onthe data manifold describing a fixed sequence of bond twistsand breaks, and furthermore classified the trajectories into twotypes, taking only positive and negative values of the manifoldcoordinate respectively.

Future work for model reduction Many questions remainabout the usefulness and limitations of NLDRs. Is there achemical interpretation of the quantitative value of the man-ifold coordinate? How can we interpolate along the data man-ifold to explore other regions of the potential energy surface?And can we find the characteristic time scale(s) associated withthe reaction coordinate? Several concrete investigations can bemade:

1. The coarse–grained dynamical trajectories can be further analyzed by symbolic regression techniques [37,38] toextract effective equations of motion along the reaction coordinate. These equations in turn can be analyzed toidentify any characteristic time scale(s) associated with the reduced dynamics, which cannot be identified using

2A reaction mechanism is equivalent to a one–dimensional mountain pass pathway in configuration space (superficially R3N ). [36]




just diffusion maps. These methods may greatly improve our ability to calculate the rate of chemical reactions,which are exponentially sensitive to the quality of the reaction coordinate that can be found, and also depend onthe curvature of the reaction coordinate in configuration space. [39]

2. The data manifold can be explored by splicing together short on–the–fly simulations using reverse integration.[40,41] Combining similar techniques with global sampling methods such as Markov state models [42–44] mayallow us to extend the diffusion map analysis [45–47] to studying processes on much larger dynamical systemssuch as proteins or spin glasses, which have rugged potential energy landscapes that are difficult to sample. Thesemay even be extensible to analyzing quantum dynamical data by considering quantum corrections to the classicalMarkov networks. [48]

3. Other technicalities remain open for investigation. For example, molecular configurations can be representedwith many coordinate systems. How sensitive are NLDRs to the choice of coordinate system? What are appro-priate distance metrics to judge the similarity of molecular configurations? How sensitive are the diffusion mapsto noise? And how universal are the diffusion maps across various systems, e.g. does the diffusion map for agas phase reaction look similar or different to that for the same reaction in solution with thousands of additionalsolvent molecules in the simulation box? Experience from electron transfer reactions shows that the effect of thesolvent can usually be described with a single collective coordinate; [49] it would be interesting to verify if NLDRmethods agree with this expectation. Investigating these questions will reveal the extent to which chemical andphysical insight can be extracted using NLDR techniques.

3 Random matrix theory and statistical mechanicsIn this section, we focus on how contemporary mathematics can improve the computational sampling of statisticalmechanical systems. In particular, random matrix theory offers new ways to construct quantum mechanical simulationswith intrinsic noise as well as suggest constructions to construct multiphysics models for disordered systems. Randommatrix theory may also aid in connecting bottom–up and top–down analyses by addressing the process of coarse grainingin a consistent framework.

Random matrix theory Random matrix theory has from its very beginning thrived and developed on the basis ofinterdisciplinary collaboration. From its twin roots in multivariate statistics and theoretical physics, random matrixtheory has since been applied a wide variety of problems including transport in disordered media, financial portfoliotheory, and the traffic patterns of public transport in Mexico. [50] Random matrix theory also has a long history ofapplications to physics, and has provided valuable insight into statistical fluctuations in disordered systems. [51, 52]However, applying random matrix theory to more accurate Hamiltonians, such as those constructed in the previoussections, requires new mathematical developments.

Figure 5: Cartoon of the computational procedure using ex-plicit statistical sampling of microstates (black arrows), showinghow random matrix theory (red arrow) can allow us to circumventthe additional cost of statistical sampling.

atomic coordinates electronic structure

dynamicsobservable

ensemble averaging

sampling in

phase space

...

rand

om m

atri

x th

eory

spatial disorder

spectral disorder

The sampling problem The detailed Hamiltonians con-structed in previous sections can be applied to systems withdisorder. However, the presence of noise greatly increases thecost of the simulations. As these methods require well–definedmolecular structures, explicit samples consistent with the ther-modynamic properties of the system must be generated beforeelectronic structure calculations can be performed. The desiredproperties can then be recovered by ensemble averaging over allthe explicit samples. In contrast, random matrix theory couldhelp circumvent the cost of explicit sampling by working withthe randomness directly in the electronic Hamiltonian. Byconsidering the mapping between the random (matrix) Hamil-tonian and the ensemble of corresponding wavefunctions, thisin principle will allow us to calculate the distribution of ex-perimental observables directly without explicit sampling. This




section outlines some initial concrete steps toward developingthis into a feasible computational technique.

Strategy 1. More efficient numerical free convolutions We have previously shown that free probability can producevery nice accurate approximations to problems in chemistry and condensed matter physics. If we can split the Hamil-tonian matrix into the sum of two matrices H = A+B, and if Q is a uniform Haar random matrix, then A+Q

′BQ

has density of states equal to the additive free convolution A� B. [53–55] However, calculating the free convolutionthis way does not actually reduce the cost of the calculation, but in fact destroys any possible sparsity structure in H .In contrast, we can calculate the free convolution using R–transforms, which is analogous to computing the (classical)convolution of two random variables using the Fourier transform. [56] While the latter can be computed efficientlyusing fast Fourier transforms, no such general purpose numerical method exists for computing the R–transform, de-spite some initial efforts. Developing the “fast R–transform” will enable much more rapid calculations on randommatrices, as this substitutes the problem of diagonalizing random matrices with integral convolutions of two one–dimensional probability distributions.

Strategy 2. Eigenvector information from free probability The calculation of experimental observables beyondthose of the density of states will require not just eigenvalues, but also eigenvectors. Much less is known about theeigenvectors of random matrices [57–59] and so these calculations will require additional collaborative work withmathematicians to develop the theory further.

Strategy 3. Universality and emergence The coarse graining of detailed models can be made more explicit by calcu-lating their universality classes. [60] This provides an alternative way of determining the relationship between detailedsimulations and model Hamiltonians, namely to identify the results of detailed simulations with the universality classesof well–studied simpler models.

4 Applications in chemistry and physicsFigure 6: Cartoon of a bulk heterojunction solar cell, showingincident sunlight (yellow arrow), a layer of intermixed donor andacceptor phases (blue and red), and charge separation at the in-terface followed by charge migration. Reproduced from Fig. 1 ofRef. [61].

This section explains how the mathematical techniques devel-oped in previous sections can be applied to important problemsin nanoscale chemistry and physics. Many important materi-als lack long–range order on the nanoscale and therefore haverugged potential energy surfaces with multiple thermodynami-cally accessible minima. [62–64] The intrinsic disorder in thesematerials also give rise to new physical and chemical effects,such as nonadiabatic phonon–assisted transitions, [65, 66] theStaebler–Wronski effect (in hydrogenated amorphous silicon),[67–69] dynamical localization, [70–72] anomalous diffusion,[73–75] and ergodicity breaking. [76] This raises the possibilityof engineering new devices that exploit these phenomena. De-tailed computer simulations can help reveal the underlyingmolecular mechanisms of energy collection, transport andstorage in nanoscale systems.

4.1 Solar energy collection and transport in bulkheterojunction organic semiconductors

“Local scale ordering appears to be essential for high charge carrier mobility... There are no clear defininglength scale boundaries where one physical feature cannot occur.” — Ref. [16]

Solar cells made with organic semiconductors have recently attracted interest as low–cost alternatives to conventionalcrystalline silicon solar cells. [77] Unlike the latter, the former have complicated donor—acceptor interfaces that are




not simple faces between ideal crystal surfaces. [16,78] This is particularly true of so–called bulk heterojunction devices,where the donor and acceptor phases are intermixed on the nanoscale as shown in Figure 6. Since the interface is wherecharge separation occurs, accurate knowledge of the shape and locationof interfaces is critical to understandinghowphotocurrent is generated. However, modeling phase boundaries and their associated phenomena such as surfacecharges or band bending [79] pose enormous challenges for existing simulation techniques, as accurate calculationsrequire explicit sampling as illustrated in Figure 5. This motivates the application of new simulation methods herein.

Figure 7: Left: Cross–correlation between the (eigenvalue) den-sity of states and the (eigenvector) delocalization lengths in metal–free phthalocyanine, showing an asymmetric correlation betweenthe excitation energy and the corresponding localization of theeigenstates, and in particular a blue–shifted subpopulation of lo-calized states. Right: A similar plot from a three–dimensional An-derson model. [80]

Excitation energy (eV)

RM

S le

ng

th (

no

rma

lize

d)

1.9 1.95 2 2.05 2.10

0.2

0.4

0.6

0.8

Advances in random matrix theoretic and machine learn-ing techniques automated will allow new computational stud-ies that reconcile the best aspects of simple theoretical mod-els and detailed atomistic simulations. We have already foundthat traditional descriptions of disordered electrons are insuffi-cient to describe the complicated band structure of excitonsin organic semiconductors. [80] These features cannot be re-produced with simple Anderson models [54,70,81] and moti-vates further development of new model Hamiltonians for ex-citons in organic semiconductors. These models will allow foraccurate insights into disorder–induced physics arising fromnanoscale morphological features like phase boundaries, ma-terial defects such as dislocations or grain boundaries, chemicalimpurities, and thermal fluctuations, and their effects on theelectrical and optical properties of disordered materials.

4.2 Energy transfer in photosynthetic systems“A high degree of order is clearly not essential fortransmitting excitation energy.” — Ref. [82]

The technologies that we will develop to study interfacial effects in disordered organic semiconductors can be adaptedreadily to study the naturally occurring analogues of solar cells, i.e. the various photosynthetic systems that can be foundin living organisms. [83] Biologists have uncovered a staggering complexity of naturally occurring light harvesting andenergy transducing photosynthetic complexes, [83] which also show structural features spanning multiple length scales,as shown in Figure 8. [82] Much controversy remains about how do these complexes work, and why they have suchcomplex structures with short–range orders over many length scales. [84–89]Figure 8: Atomic force micrographs of photosynthetic membranes inthe purple bacterium Rhodobacter sphaeroides. Scale bar: 100 nm. (a) Acluster of membrane patches. (b) Identifying two types of complexes ofsize 12 nm (red arrows) and 7 nm (green arrows). (c) Matching thesecomplexes to atomic models of photosynthetic protein complexes (inset).Reproduced from Fig. 1 of Ref. [82].

We expect that models with more accurate physicswill allow us to address these controversies definitively.For example, these models can have more accurateexciton—phonon interactions, [90–93] more accuratepigment—pigment interactions, [94, 95] or polarizationeffects. [96–98] Algorithmic model selection can helpdetermine which of these will turn out to be importantfor accurate modeling of photosynthetic systems. This inturn will allow studies of systems–level designs on greaterlength scales, like the reason for the diverse shapes ofphotosynthetic complexes [99, 100] and the interactionbetween different photosynthetic protein complexes onthe membrane level.

4.3 Energy storage in nanostructured su-percapacitorsElectricity often needs to be stored before use in devices

like capacitors. Capacitors have very high discharge rates and excellent cycle life, but their energy densities are too




low for many applications. [102] However, so–called supercapacitors, a.k.a. ultracapacitors or electric double layercapacitors, have thousands of times higher energy densities, [103–108] which opens up applications for energy storagein micron–sized devices. [109, 110] In particular, nanostructured carbon-based electrodes are believed to hold thegreatest potential for improving supercapacitor technology, due to the relatively low cost and extensively characterizedsurface chemistry. [104,108,111–113] However, themechanismfor energy storage in supercapacitors remainspoorlyunderstood.

Figure 9: (A) Plot of specific capacitance of the(CH3CH2)4N+·BF−

4 /CH3CN electrolyte interacting witha nanoporous carbon electrode. (B) Diagram of proposed solva-tion structures for pore size exceeding 2 nm, (C) between 1 nmand 2 nm, and (D) less than 1 nm. Reproduced from Fig. 3 ofRef. [101].

Conventional capacitors store charge on metal surfaces sep-arated by a bulk dielectric medium; this design dates back to thefirst Leyden jars of the 1740s. In contrast, supercapacitors arethought to store energy by charging up the electric double lay-ers that form between phase boundaries. [103] For nanoscaledevices, however, deviations from bulk behavior occur. [101]As shown in Figure 9, the capacitance became anomalouslylarge when the average pore size of the electrode fell below 1nm, i.e. when the pores became too small to admit fully sol-vated ions. [114] This cannot be explained by the electric dou-ble layer mechanism of capacitance.

New empirical potentials models treating polarization andcharge transfer effects [28–31] will prove to be essential to treatintermolecular charge transfer in large systems with many ions.This will provide the necessary atomistic description of the ionsolvation and transport properties, and redox chemistry thatoccur at the electrode surfaces. This will allow us to model theelectric double layer explicitly, [115] and we can then studythe belief that only the electric double layers which are pro-duced at interfacial layers contribute to the charge storage insupercapacitors. [103, 116] The simulations will also allow usto study how the local chemistry will be affected by electrodemorphology, [117] and the role of surface states and possible impurities. [112] These dynamical simulations will al-low us to understand the fundamental mechanisms for electrical energy storage on the nanoscale and provide insightinto how to create devices that harness anomalously high capacitances. [116]

References[1] Dirac, P. A. M. (1929) Quantum mechanics of many-electron systems. Proc. Roy. Soc. A 123, 714–733.

[2] Lieb, E. H & Loss, M. (2001) Analysis. (American Mathematical Soc., Providence, RI).

[3] Lieb, E. H & Seiringer, R. (2010) The stability of matter in quantum mechanics. (Cambridge University Press,Cambridge, UK; New York).

[4] Schrödinger, E. (1926) An Undulatory Theory of the Mechanics of Atoms and Molecules. Phys. Rev. 28,1049–1070.

[5] Schrödinger, E. (1926) Quantisierung als Eigenwertproblem. Ann. Phys. 384, 361–376.

[6] Szabo, A & Ostlund, N. S. (1996) Modern quantum chemistry: introduction to advanced electronic structure theory.(Dover, Mineola, NY).

[7] Ashcroft, N. W & Mermin, N. D. (1976) Solid state physics. (Holt, Rinehart and Winston, New York).

[8] Aaronson, S. (2009) Computational complexity: Why quantum chemistry is hard. Nature Physics 5, 707–708.

[9] Schuch, N & Verstraete, F. (2009) Computational complexity of interacting electrons and fundamental limita-tions of density functional theory. Nature Physics 5, 732–735.




[10] Jordan, S. P & Love, P. J. (2010) Quantum-Merlin-Arthur–complete problems for stoquastic Hamiltoniansand Markov matrices. Phys. Rev. A 81, 1–10.

[11] Brown, B, Flammia, S, & Schuch, N. (2011) Computational Difficulty of Computing the Density of States.Phys. Rev. Lett. 107, 1–4.

[12] Kassal, I, Whitfield, J. D, Perdomo-Ortiz, A, Yung, M.-H, & Aspuru-Guzik, A. (2011) Simulating chemistryusing quantum computers. Annu. Rev. Phys. Chem. 62, 185–207.

[13] Freddolino, P. L, Arkhipov, A. S, Larson, S. B, McPherson, A, & Schulten, K. (2006) Molecular dynamicssimulations of the complete satellite tobacco mosaic virus. Structure 14, 437–49.

[14] Wu, C, Malinin, S, Tretiak, S, & Chernyak, V. (2008) Multiscale Modeling of Electronic Excitations inBranched Conjugated Molecules Using an Exciton Scattering Approach. Phys. Rev. Lett. 100, 1–4.

[15] Walsh, P, Omeltchenko, A, Kalia, R. K, Nakano, A, Vashishta, P, & Saini, S. (2003) Nanoindentation of siliconnitride: A multimillion-atom molecular dynamics study. Appl. Phys. Lett. 82, 118–120.

[16] Rivnay, J, Mannsfeld, S. C. B, Miller, C. E, Salleo, A, & Toney, M. F. (2012) Quantitative determination oforganic semiconductor microstructure from the molecular to device scale. Chemical Reviews 112, 5488–519.

[17] Warshel, A & Levitt, M. (1976) Theoretical studies of enzymic reactions: dielectric, electrostatic and stericstabilization of the carbonium ion in the reaction of lysozyme. J. Mol. Biol. 103, 227–249.

[18] Svensson, M, Humbel, S, Froese, R. D. J, Matsubara, T, Sieber, S, & Morokuma, K. (1996) ONIOM: AMultilayered Integrated MO + MM Method for Geometry Optimizations and Single Point Energy Predic-tions. A Test for Diels−Alder Reactions and Pt(P( t -Bu) 3 ) 2 + H 2 Oxidative Addition. J. Phys. Chem. 100,19357–19363.

[19] Severo Pereira Gomes, A & Jacob, C. R. (2012) Quantum-chemical embedding methods for treating localelectronic excitations in complex chemical systems. Ann. Rep. Sec. C: Phys. Chem. pp. 222–277.

[20] Anderson, P. W. (1984) Basic notions of condensed matter physics, Frontiers in physics. (Benjamin-Cummings,Menlo Park, CA).

[21] Fann, G, Beylkin, G, Harrison, R. J, & Jordan, K. E. (2004) Singular operators in multiwavelet bases. IBM J.Research and Development 48, 161–171.

[22] Hohenstein, E. G, Parrish, R. M, & Martínez, T. J. (2012) Tensor hypercontraction density fitting. I. Quarticscaling second- and third-order Møller-Plesset perturbation theory. J. Chem. Phys. 137, 044103.

[23] Hohenstein, E. G, Parrish, R. M, Sherrill, C. D, & Martinez, T. J. (2012) Communication: Tensor hypercon-traction. III. Least-squares tensor hypercontraction for the determination of correlated wavefunctions. J. Chem.Phys. 137, 221101.

[24] Yserentant, H. (year?). Numer. Math., month = aug, number = 4, pages = 731–759, title = On the regularityof the electronic Schrödinger equation in Hilbert spaces of mixed derivatives, url = http://www.springerlink.com/in-dex/10.1007/s00211-003-0498-1, volume = 98, year = 2004.

[25] Yserentant, H. (2010) Regularity and Approximability of Electronic Wave Functions, Lecture Notes in Mathemat-ics. (Springer-Verlag Berlin Heidelberg).

[26] Chen, J & Van Voorhis, T. (2012) An empirical fragment charge model for multiple electronic states. submitted.

[27] Wang, L.-P, Chen, J, Martínez, T. J, & Van Voorhis, T. (2012) Systematic parameterization of polarizable forcefields from quantum chemistry data. submitted.

[28] Chen, J & Martínez, T. J. (2007) QTPIE: Charge transfer with polarization current equalization. A fluctuatingcharge model with correct asymptotics. Chem. Phys. Lett. 438, 315–320.




[29] Chen, J, Hundertmark, D, & Martínez, T. J. (2008) A unified theoretical framework for fluctuating-chargemodels in atom-space and in bond-space. J. Chem. Phys. 129, 214113.

[30] Chen, J & Martínez, T. J. (2009) The dissociation catastrophe in fluctuating-charge models and its implicationsfor the concept of atomic electronegativity. Prog. Theor. Chem. Phys. 19, 397–416.

[31] Chen, J & Martínez, T. J. (2009) Charge conservation in electronegativity equalization and its implications forthe electrostatic properties of fluctuating-charge models. J. Chem. Phys. 131, 044114.

[32] Akaike, H. (1974) A new look at the statistical model identification. IEEE Trans. Auto. Control 19, 716–723.

[33] Burnham, K. P, Anderson, D. R, & Burnham, K. P. (2002) Model selection and multimodel inference : a practicalinformation-theoretic approach. (Springer, New York).

[34] Trevor, H, Tibshirani, R, & Friedman, J. (2008) The Elements of Statistical Learning.

[35] Virshup, A. M, Chen, J, & Martínez, T. J. (2012) Nonlinear dimensionality reduction for nonadiabatic dynam-ics: The influence of conical intersection topography on population transfer rates. J. Chem. Phys. 137, 22A519.

[36] Lewis, A. S & Pang, C. H. J. (2011) Level set methods for finding critical points of mountain pass type.Nonlinear Anal. 74, 4058–4082.

[37] Bongard, J & Lipson, H. (2007) Automated reverse engineering of nonlinear dynamical systems. Proc. Natl.Acad. Sci. 104, 9943–8.

[38] Schmidt, M & Lipson, H. (2009) Distilling free-form natural laws from experimental data. Science 324, 81–5.

[39] Hänggi, P, Talkner, P, & Borkovec, M. (1990) Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys.62, 251–341.

[40] Frewen, T. A, Hummer, G, & Kevrekidis, I. G. (2009) Exploration of effective potential landscapes using coarsereverse integration. J. Chem. Phys. 131, 134104.

[41] Ferguson, A. L, Panagiotopoulos, A. Z, Kevrekidis, I. G, & Debenedetti, P. G. (2011) Nonlinear dimensionalityreduction in molecular simulation: The diffusion map approach. Chem. Phys. Lett. 509, 1–11.

[42] Bowman, G. R, Ensign, D. L, & Pande, V. S. (2010) Enhanced Modeling via Network Theory: AdaptiveSampling of Markov State Models. J. Chem. Theory Comput. 6, 787–794.

[43] Kasson, P. M & Pande, V. S. (2010) ”Cross-graining”: efficient multi-scale simulation via markov state models. Vol.268, pp. 260–268.

[44] Pande, V. S, Beauchamp, K, & Bowman, G. R. (2010) Everything you wanted to know about Markov StateModels but were afraid to ask. Methods 52, 99–105.

[45] Lafon, S & Lee, A. B. (2006) Diffusion maps and coarse-graining: A unified framework for dimensionalityreduction, graph partitioning, and data set parameterization. IEEE Trans. Pattern Anal. Machine Intelligence 28,1393–403.

[46] Nadler, B, Lafon, S, Coifman, R, & Kevrekidis, I. G. (2006) Diffusion maps, spectral clustering and reactioncoordinates of dynamical systems. Appl. Comput. Harm. Anal. 21, 113–127.

[47] Coifman, R. R, Kevrekidis, I. G, Lafon, S, Maggioni, M, & Nadler, B. (2008) Diffusion Maps, ReductionCoordinates, and Low Dimensional Representation of Stochastic Systems. Multiscale Modeling & Simulation7, 842.

[48] Cao, J. S & Silbey, R. J. (2009) Optimization of Exciton Trapping in Energy Transfer Processes. Journal ofPhysical Chemistry A 113, 13825–13838.




[49] Marcus, R. A. (1964) Chemical + Electrochemical Electron-Transfer Theory. Annu. Rev. Phys. Chem. 15,155–196.

[50] Akemann, G, Baik, J, & Di Francesco, P, eds. (2011) The Oxford Handbook of Random Matrix Theory, OxfordHandbooks in Mathematics. (Oxford University Press, Oxford).

[51] Beenakker, C. W. J. (1997) Random-matrix theory of quantum transport. Rev. Mod. Phys. 69, 731–808.

[52] Mehta, M. L. (2004) Random matrices, Pure and Applied Mathematics. (Elsevier/Academic Press, Amsterdam; San Diego, CA), 3rd edition, pp. xviii, 688 p.

[53] Voiculescu, D. (1991) Limit laws for Random matrices and free products. Inventiones Math. 104, 201–220.

[54] Chen, J, Hontz, E, Moix, J, Welborn, M, Van Voorhis, T, Suárez, A, Movassagh, R, & Edelman, A. (2012)Error analysis of free probability approximations to the density of states of disordered systems. Phys. Rev. Lett.109, 036403.

[55] Chen, J & Edelman, A. (2012) Partial freeness of random matrices. submitted.

[56] Voiculescu, D. (1986) Addition of Certain Non-commuting Random Variables. functional analysis 66, 323–346.

[57] Biane, P. (1998) Processes with free increments. Math. Z. 227, 143–174.

[58] Capitaine, M. (2011) Additive/multiplicative free subordination property and limiting eigenvectors of spikedadditive deformations of Wigner matrices and spiked sample covariance matrices.

[59] Belinschi, S. T, Speicher, R, & Treilhard, J. (year?) Operator-valued free multiplicative convolution: analyticsubordination theory and applications to random matrix theory. p. arXiv:1209.3508v1.

[60] Zirnbauer, M. R. (2010) in Oxford Handbook of Random Matrix Theory.

[61] Janssen, R. A. J, Hummelen, J. C, & Sariciftci, N. S. (2005) Polymer–Fullerene Bulk Heterojunction SolarCells. Mater. Res. Soc. Bull. 30, 33–36.

[62] Frauenfelder, H, Sligar, S. G, & Wolynes, P. G. (1991) The Energy Landscapes and Motions of Proteins. Science254, 1598–1603.

[63] Onuchic, J. N & Wolynes, P. G. (1993) Energy landscapes, glass transitions, and chemical reaction dynamicsin biomolecular or solvent environment. J. Chem. Phys. 98, 2218–2224.

[64] Wales, D. J, Doye, J. P. K, Miller, M. A, Mortenson, P. N, & Walsh, T. R. (2000) Energy landscapes: fromclusters to biomolecules. Adv. Chem. Phys. 115, 1–111.

[65] Fowler, W. B. (1968) Physics of color centers. (Academic Press), 1st edition.

[66] Townsend, P. D & Kelley, J. C. (1973) Colour centres and imperfections in insulators and semiconductors. (Crane,Russak).

[67] Staebler, D. L & Wronski, C. R. (1980) Optically induced conductivity changes in discharge-produced hydro-genated amorphous silicon. Journal of Applied Physics 51, 3262.

[68] Cody, G. D, Tiedje, T, Abeles, B, Brooks, B, & Goldstein, Y. (1981) Disorder and the Optical-AbsorptionEdge of Hydrogenated Amorphous Silicon. Phys. Rev. Lett. 47, 1480–1483.

[69] Fritzsche, H. (2003) Development In Understanding And Controlling The Staebler-Wronski Effect In A-Si:H.Annual Review of Materials Research 31, 47–79.

[70] Anderson, P. W. (1958) Absence of Diffusion in Certain Random Lattices. Phys. Rev. 109, 1492.




[71] Kramer, B & MacKinnon, A. (1993) Localization: theory and experiment. Rep. Prog. Phys. 56, 1469–1564.

[72] Evers, F & Mirlin, A. (2008) Anderson transitions. Rev. Mod. Phys. 80, 1355–1417.

[73] Bouchaud, J.-P & Georges, A. (1990) Anomalous diffusion in disordered media: Statistical mechanisms, modelsand physical applications. Phys. Rep. 195, 127–293.

[74] Shlesinger, M. F, Zaslavsky, G. M, & Klafter, J. (1993) Strange kinetics. Nature 363, 31–37.

[75] Mauro, J. C, Gupta, P. K, & Loucks, R. J. (2007) Continuously broken ergodicity. J. Chem. Phys. 126, 184511.

[76] Palmer, R. G. (1982) Broken ergodicity. Adv. Phys. 31, 669–735.

[77] Powell, D. M, Winkler, M. T, Choi, H. J, Simmons, C. B, Needleman, D. B, & Buonassisi, T. (2012) Crys-talline silicon photovoltaics: a cost analysis framework for determining technology pathways to reach baseloadelectricity costs. Energy Env. Sci. 5, 5874–5883.

[78] Pope, M & Swenberg, C. E. (1999) Electronic processes in organic crystals and polymers, Monographs on thephysics and chemistry of materials. (Clarendon, Oxford; New York), 2 edition.

[79] Zhang, Z & Yates, J. T. (2012) Band bending in semiconductors: chemical and physical consequences at surfacesand interfaces. Chem. Rev. 112, 5520–51.

[80] Chen, J, Welborn, M. G, Yost, S. R, & Van Voorhis, T. (2012) Excitonic bands in disordered phthalocynanines:a random matrix theoretic perspective. submitted.

[81] Welborn, M. G, Chen, J, & Van Voorhis, T. (2012) Accurate approximations to Anderson models in two andthree dimensions using free probability. submitted.

[82] Bahatyrova, S, Frese, R. N, Siebert, C. A, Olsen, J. D, Van Der Werf, K. O, Van Grondelle, R, Niederman,R. a, Bullough, P. a, Otto, C, & Hunter, C. N. (2004) The native architecture of a photosynthetic membrane.Nature 430, 1058–62.

[83] Kê, B. (2001) Photosynthesis: photobiochemistry and photobiophysics, Advances in Photosynthesis. (Kluwer Aca-demic, Dordrecht).

[84] Pullerits, T & Sundström, V. (1996) Photosynthetic Light-Harvesting Pigment−Protein Complexes: TowardUnderstanding How and Why. Acc. Chem. Res. 29, 381–389.

[85] van Amerongen, H & van Grondelle, R. (2001) Understanding the Energy Transfer Function of LHCII, theMajor Light-Harvesting Complex of Green Plants. J. Phys. Chem. B 105, 604–617.

[86] Scholes, G. D. (2003) Long-range resonance energy transfer in molecular systems. Annu. Rev. Phys. Chem. 54,57–87.

[87] Scholes, G. D & Fleming, G. R. (2006) Energy transfer and photosynthetic light harvesting. Adv. Chem. Phys.132, 57–129.

[88] Cheng, Y.-C & Fleming, G. R. (year?). Annu. Rev. Phys. Chem.

[89] Wong, C. Y, Alvey, R. M, Turner, D. B, Wilk, K. E, Bryant, D. a, Curmi, P. M. G, Silbey, R. J, & Scholes, G. D.(2012) Electronic coherence lineshapes reveal hidden excitonic correlations in photosynthetic light harvesting.Nature Chem. 4, 396–404.

[90] Damjanović, A, Kosztin, I, Kleinekathöfer, U, & Schulten, K. (2002) Excitons in a photosynthetic light-harvesting system: A combined molecular dynamics, quantum chemistry, and polaron model study. Phys. Rev.E 65, 1–24.




[91] Mohseni, M, Rebentrost, P, Lloyd, S, & Aspuru-Guzik, A. (2008) Environment-assisted quantum walks inphotosynthetic energy transfer. J. Chem. Phys. 129, 174106.

[92] Olbrich, C, Strümpfer, J, Schulten, K, & Kleinekathöfer, U. (2011) Theory and Simulation of the EnvironmentalEffects on FMO Electronic Transitions. J. Phys. Chem. Lett. 2011, 1771–1776.

[93] Olbrich, C, Jansen, T. L. C, Liebers, J, Aghtar, M, Strümpfer, J, Schulten, K, Knoester, J, & Kleinekathöfer,U. (2011) From atomistic modeling to excitation transfer and two-dimensional spectra of the FMO light-harvesting complex. J. Phys. Chem. B 115, 8609–21.

[94] Beljonne, D, Curutchet, C, Scholes, G. D, & Silbey, R. J. (2009) Beyond Förster resonance energy transfer inbiological and nanoscale systems. J. Phys. Chem. B 113, 6583–99.

[95] Renger, T & Schlodder, E. (2010) Primary photophysical processes in photosystem II: bridging the gap betweencrystal structure and optical spectra. Chemphyschem 11, 1141–53.

[96] Marini, A, Muñoz Losa, A, Biancardi, A, & Mennucci, B. (2010) What is solvatochromism? J. Phys. Chem. B114, 17128–17135.

[97] Huelga, S & Plenio, M. (2011) Quantum dynamics of bio-molecular systems in noisy environments. ProcediaChem, 3, 248–257.

[98] Shim, S, Rebentrost, P, Valleau, S, & Aspuru-Guzik, A. (2012) Atomistic study of the long-lived quantumcoherences in the Fenna-Matthews-Olson complex. Biophys. J. 102, 649–60.

[99] Koepke, J, Hu, X, Muenke, C, Schulten, K, & Michel, H. (1996) The crystal structure of the light-harvestingcomplex II (B800-850) from Rhodospirillum molischianum. Structure 4, 581–97.

[100] Liu, Z, Yan, H, Wang, K, Kuang, T, Zhang, J, Gui, L, An, X, & Chang, W. (2004) Crystal structure of spinachmajor light-harvesting complex at 2.72 A resolution. Nature 428, 287–92.

[101] Chmiola, J, Yushin, G, Gogotsi, Y, Portet, C, Simon, P, & Taberna, P. L. (2006) Anomalous increase in carboncapacitance at pore sizes less than 1 nanometer. Science 313, 1760–3.

[102] Pandolfo, A. G & Hollenkamp, A. F. (2006) Carbon properties and their role in supercapacitors. J. PowerSources 157, 11–27.

[103] Rightmire, R. A. (1966) U.S. Patent 3,288,641.

[104] Conway, B. E. (1999) Electrochemical Supercapacitors: Scientific Fundamentals and Technological Applications.(Kluwer Academic/Plenum: New York).

[105] Burke, A. (2000) Ultracapacitors: why, how, and where is the technology. J. Power Sources 91, 37–50.

[106] Kötz, R, Rötz, R, & Carlen, M. (2000) Principles and applications of electrochemical capacitors. Electrochim.Acta 45, 2483–2498.

[107] Winter, M & Brodd, R. J. (2004) What are batteries, fuel cells, and supercapacitors? Chem. Rev. 104, 4245–69.

[108] Inagaki, M, Konno, H, & Tanaike, O. (2010) Carbon materials for electrochemical capacitors. J. Power Sources195, 7880–7903.

[109] Lewis Jr., D. H, Waypa, J. J, Antonsson, E. K, & Lakeman, C. D. E. (2001) U.S. Patent 6,621,687.

[110] La O’, G. J, In, H. J, Crumlin, E, Barbastathis, G, & Shao-Horn, Y. (2007) Recent advances in microdevicesfor electrochemical energy conversion and storage. Int. J. Energy Res. 31, 548–575.

[111] Aricò, A. S, Bruce, P, Scrosati, B, Tarascon, J.-M, & van Schalkwijk, W. (2005) Nanostructured materials foradvanced energy conversion and storage devices. Nature Mater. 4, 366–77.




[112] Zhang, Y, Feng, H, Wu, X, Wang, L, Zhang, A, Xia, T, Dong, H, Li, X, & Zhang, L. (2009) Progress ofelectrochemical capacitor electrode materials: A review. Int. J. Hydrogen Energy 34, 4889–4899.

[113] Zhang, L. L & Zhao, X. S. (2009) Carbon-based materials as supercapacitor electrodes. Chem. Soc. Rev. 38,2520–31.

[114] Salitra, G, Soffer, A, Eliad, L, Cohen, Y, & Aurbach, D. (2000) Carbon Electrodes for Double-Layer CapacitorsI. Relations Between Ion and Pore Dimensions. J. Electrochem. Soc. 147, 2486–2493.

[115] Grahame, D. C. (1947) The electrical double layer and the theory of electrocapillarity. Chem. Rev. 41, 441–501.

[116] Daffos, B, Taberna, P.-L, Gogotsi, Y, & Simon, P. (2010) Recent Advances in Understanding the CapacitiveStorage in Microporous Carbons. Fuel Cells 10, 819–824.

[117] Zhang, H, Cao, G, Wang, W, Yuan, K, Xu, B, Zhang, W, Cheng, J, & Yang, Y. (2009) Influence of microstruc-ture on the capacitive performance of polyaniline/carbon nanotube array composite electrodes. Electrochim. Acta54, 1153–1159.


machine learning, model reduction and multiphysics...

Documents