AFRL-OSR-VA-TR-2015-0038

Exactly Embedded Wavefunction Methods for Characterizing Nitrogen

THOMAS MILLERCALIFORNIA INSTITUTE OF TECHNOLOGY

Final Report01/15/2015

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29-12-2014 2. REPORT TYPE

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Sep 2011 - Sep 2014

4. TITLE AND SUBTITLE

Exactly Embedded Wavefunction Methods for Characterizing Nitrogen Reduction

Catalysis

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FA9550-11-1-0288

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6. AUTHOR(S)

Thomas Francis Miller III

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7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

California Institute of Technology

1200 E. California Blvd.

Pasadena, CA 91125

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3-FINAL

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Air Force Defense Research Sciences Program

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SMALLMOLECULE

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14. ABSTRACT

The final year of research activity witnessed successful completion of the proposed research goals of developing and applying exactly embedded

density functional and wavefunction theory methods for the investigation of small-molecular activation catalysis, such as hydrogen and nitrogen

reduction. In a significant methodological advance from the past year, we developed an accurate and systematically improvable framework for

embedding a correlated wavefunction description of a subsystem in a density functional theory description of the surrounding molecular

environment (J. Chem. Phys., 140, 18A507 (2014).). Finally, we applied these newly developed embedding methods to the task of designing new

transition-metal catalysts for small-molecule activation. A central challenge in the development of inorganic hydrogen-evolution catalysts is to

avoid deleterious coupling of the energetics of metal-site reduction from the kinetics of metal-hydride formation. In work that we are currently

preparing for publication, we combined theoretical and experimental methods to investigate a new class of cobalt diimine-dioxime catalysts that

shows promise for achieving this aim by introducing an intramolecular proton-shuttle via a pyridyl pendant group.

15. SUBJECT TERMS

density functional theory, quantum embedding, nitrogen reduction, hydrogen evolution

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Thomas F. Miller III a. REPORT

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Miller, FA9550-11-1-0288 Final Report - 1

FINAL REPORT

Title: “Exactly Embedded Wavefunction Methods for Characterizing Nitrogen Reduction Cataly-sis”

Applicant/Institution: California Institute of TechnologyStreet Address/City/State/Zip: 1200 E. California Boulevard, Pasadena, CA 91125Principal Investigator (PI): Thomas F. Miller IIIPosition Title of PI: Professor of ChemistryMailing Address of PI: California Institute of Technology

1200 E. California Boulevard, MC 127-72Pasadena, CA 91125

Telephone Number of PI: 626-395-6588Email of PI: tfm@caltech.edu

With AFOSR support in the last funding period (Grant Number: FA9550-11-1-0288), theMiller group developed new electronic structure embedding methods for the investigation of small-molecular activation catalysis, such as hydrogen and nitrogen reduction. In particular, we made amajor advances by developing a robust, rigorous, and accurate density functional theory embed-ding method [1–5]. As is illustrated in Fig. 1, the method enables rigorous quantum embeddingcalculations in which a small part of a large molecular system is treated using an accurate (andcomputationally expensive) wavefunction theory (WFT) methods, such as CCSD(T), while the re-mainder of the system is treated using density functional theory (DFT). This method opens a widerange of application domains that are relevant to the catalysis of small-molecule activation, sincethe WFT-in-DFT embedding approach enables reactions involving transition-metal complexes tobe described using high-accuracy methods, but at only a small fraction of the computational cost.Using these new methods, we have worked in collaboration with experimentalist Jonas Peters (Cal-tech) to develop a new class of cobalt-based hydrogen evolution catalysts that successfully breakthe long-standing (and undesirable) correlation between the required overpotential and observedcatalytic rate in this important class of earth-abundant catalysts [5].

CCSD(T)  

DFT  WFT-­‐in-­‐DFT  Co  

FIG. 1: (Left) The WFT-in-DFT embedding approach is illustrated for cobalt-based hydrogen evolution catalyst isillustrated. Electronic density associated with the region containing the transition metal center is treated at the levelof wavefunction theory (CCSD(T), MRCI, CASPT2, etc.), while the remainder of the system is treated at the DFTlevel. This approach allows high accuracy to be employ in the physical regions that require it, allow for vastly reducedcomputational cost and systematically improvable accuracy.

Miller, FA9550-11-1-0288 Final Report - 2

Work performed during this function period (Grant Number: FA9550-11-1-0288) has led tothe publication of six papers in high-quality journals, including two in preparation:

• “A simple, exact density-functional-theory embedding scheme,” F. R. Manby, M. Stella, J.D. Goodpaster, and T. F. Miller III, J. Chem. Theory Comput., 8, 2564 (2012).

• “Density functional theory embedding for correlated wavefunctions: Improved methods foropen-shell systems and transition metal complexes,” J. D. Goodpaster, T. A. Barnes, F. R.Manby, and T. F. Miller III, J. Chem. Phys., 137, 224113 (2012).

• “Accurate basis set truncation for wavefunction embedding,” T. A. Barnes, J. D. Goodpaster,F. R. Manby, and T. F. Miller III, J. Chem. Phys., 139, 024103 (2013).

• “Accurate and systematically improvable density functional theory embedding for correlatedwavefunctions,” J. D. Goodpaster, T. A. Barnes, F. R. Manby, and T. F. Miller III, J. Chem.Phys., 140, 18A507 (2014).

• “Breaking the correlation between energy costs and kinetic barriers in hydrogen evolution:A new class of cobalt (pyridine-diimine-dioxime) catalysts,” P. Huo, C. Uyeda, J. D. Good-paster, J. C. Peters, T. F. Miller III, to be submitted.

• “Projection-based wavefunction-in-DFT embedding,” J. D. Goodpaster, T. A. Barnes, F. R.Manby, and T. F. Miller III, to be submitted. Invited Review Article for Annu. Rev. Phys.Chem.

In the following, we describe key aspects of the work from the previous funding period in greaterdetail.

A. Detailed Discussion

Methods that exploit the intrinsic locality of molecular interactions show significant promisein making tractable the electronic structure calculation of large-scale systems. In particular, em-bedded density functional theory (e-DFT) offers a formally exact approach to electronic structurecalculations in which complex condensed-phase chemical problems are decomposed into the self-consistent solution of individual smaller subsystems [6–10]. The objectives of the e-DFT approachare thus similar to those of more approximate partitioning and fragmentation schemes, but e-DFTavoids the uncontrolled approximations (such as link atoms) and errors associated with subsysteminterfaces that fundamentally limit other widely used methods.

However, in practice, previous e-DFT studies have employed substantial approximations in thedescription of subsystem interactions [9, 11, 12]. The subsystem embedding potentials that emergein the e-DFT framework include non-additive kinetic potential (NAKP) terms that enforce Pauliexclusion among the electrons of the various subsystems. Without knowledge of the exact func-tional for the non-interacting kinetic energy, previous e-DFT studies have employed approximateNAKP treatments that break down in cases for which the subsystem densities significantly over-lap (which include hydrogen-bonded or covalently bonded subsystems). Prior to our work in thisarea, e-DFT studies were thus limited to weakly interacting subsystems, a severe constraint that

Miller, FA9550-11-1-0288 Final Report - 3

excluded essentially all condensed-phase and reactive chemical applications. To overcome thisconstraint, my group has focused on the development of numerically exact methods for obtainingsubsystem embedding potentials in e-DFT.

Projection-based embedding. In the past AFOSR funding period, we made significant break-throughs in the efficiency and robustness of e-DFT calculations [1, 3, 4]. An exact scheme wasdeveloped that correctly accounts for NAKP contributions via a projection technique, while com-pletely avoiding any numerically demanding OEP calculations [1]. This projection-based embed-ding method introduces three simple innovations: (i) We replaced the iterated DFT-in-DFT with asingle conventional KS calculation. (ii) We completely avoided the issue of the NAKP through theuse of a level-shifting projection operator to keep the orbitals of one subsystem orthogonal to thoseof another (Fig. 2). And, (iii) we developed a simple but effective perturbation theory to eliminatepractically all dependence on the level-shift parameter. This approach provides the needed sim-plicity and robustness to enable broad utilization by the computational chemistry user communityin the near term, as is described in the next subsection.

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FIG. 2: Demonstration of the projectiontechnique that is employed in the new exactembedding scheme, using the example ofembedding the 10 electrons of the -OH moi-ety of ethanol in the environment producedby the ethyl subsystem. Following pro-jection, the subsystem orbitals are explic-itly orthogonalized, thus eliminating non-additive kinetic energy contributions [2].

Numerical tests of the new projection-based embed-ding method for a range of molecular systems, includ-ing those with conjugated and multiply-bonded subsystems,were shown to consistently yield DFT-in-DFT embeddingwith errors below 10−6 kcal/mol; these results mark adramatic improvement over the system-dependent 0.1 − 1kcal/mol (and sometimes larger) errors that can be expectedusing existing OEP-based approaches. Additionally, wedemonstrated that the method allows for accurate and ro-bust WFT-in-DFT embedding calculations and embeddedmany-body expansions [1]. Furthermore, we have devel-oped the method to rigorously and systematically controlWFT-in-DFT errors with respect to both basis set trunca-tion [3] and with respect to accuracy of the DFT embeddinginteractions [4], all while preserving the essential stability,efficiency, and simplicity of the method.

Among the most exciting aspects of the new project-based embedding method is that enables robust and efficientWFT-in-DFT calculations in complex chemical systems, en-abling practical applications studies to be performed in astraightforward manner. We have demonstrated this with amajor simulation study in which WFT-in-DFT embeddingwas employed to rationally design a promising new class ofcobalt-based catalysts for hydrogen evolution [5]. A cen-tral challenge in the development of inorganic hydrogen-evolution catalysts is to avoid deleterious coupling of theenergetics of metal-site reduction from the kinetics of metal-hydride formation. We worked in collaboration with theexperimental inorganic chemistry group of Jonas Peters atCaltech to discover a new class of cobalt diimine-dioxime catalysts that shows promise for achiev-ing this aim by introducing an intramolecular proton-shuttle via a pyridyl pendant group (Fig. 3).

Miller, FA9550-11-1-0288 Final Report - 4

Using over 200 projection-based CCSD(T)-in-DFT embedding calculations of the Co-based cata-lyst with a variety of pyridyl substituents, the energetic and kinetic barriers to hydrogen formationwere investigated, revealing nearly complete decoupling of the energetics of Co reduction and ki-netics of intramolecular Co-hydride formation. This work confirmed that WFT-in-DFT embeddingallows for local regions of a molecule to be described using high-accuracy wavefunction methods(such as CCSD(T)), thus overcoming significant errors in the DFT-level description of transition-metal complexes. This study provides a powerful demonstration of the utility of projection-basedWFT-in-DFT method for inorganic catalysis applications, as well as revealing a promising designprinciple for achieving uncoupled and locally tunable intramolecular charge-transfer events in thecontext of homogenous transition-metal catalysts.

feasibility of intramolecular P.T. reaction in 2.The main contributing factor to this low barrieris the short distance that proton need to travelto reach cobalt (∼0.7 Å).

Second, upon the changing of ortho substi-tution on pyridyl group, we calculate the re-duction potential of CoII/I couple and the reac-tion barrier of the P.T. reaction, and explic-itly show the decoupling between them. Asshown in 4, we can tune the barrier of the in-tramolecular P.T. reaction (red dots) by chang-ing the electron inductive effect (quantified ashammett constant) of the substituent, whilekeep the reduction potential unchanged (bluesquares) with these substitutions. The trendof the barrier upon substituent is correlatedwith the hammett constant of the substitutiongroups. Conversely, the CoII/I reduction poten-tial maintain unchanged with the same substi-tution. This is due to the relative large distancebetween the substituted pyridyl group and thecobalt microcycle complex, and the reductionis primary occurred on cobalt center and noton the ligand. The reason why the proposed

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Figure 4: To be finished with Jason’s WFT-in-DFT calculaitons Activation barrier ∆G‡ of in-tramolecular P.T. reaction and the reduction potentialof CoII/I (blue) with various substitutions at the orthoposition of the pyridyl group. Open square and circlesare the embedding-DFT results and the lines are thelinear fit. The reduction potential (right y axis) is inthe same scale as the free energy barrier (left x axis).

ortho substituted pyridyl complex 2 can decou-ple the anti-correlation is explained as follows:The previous cobalt complexes7,12–15 which ex-hibit “anti-correlation” all have the substituent

groups that influence the electron density oncobalt center. The influence of the electronicdensity on cobalt will impact reduction poten-tial and the hydride formation reaction energyat the same time, in different directions.7,11 Forcomplex 2, the substitution on pyridyl groupwill tune the proton transfer energy and bar-rier from pyridyl group to cobalt. However,it will not influence the electronic density oncobalt due to the large separation between thesubstituent and cobalt and the quickly dampedout electronic induction effects.

In conclusion, we find the ortho substitutedpyridyl group complex 2 is capable to decouplethe anti-correlation between reduction poten-tial and hydride formation barrier by regulatingthem separately. This provide the possibility todesign more efficient catalyst which evolve hy-drogen with a higher turnover frequency with-out compromise other novel properties such asrelatively positive reduction potential.

Proton binding sites and their in-fluence on CoII/I reduction potential

In this section, we demonstrate that upon pro-tonation of pyridyl group in complex 2, the re-duction potential of cobalt remains unchanged,which supports the fact that the complex 2 hasa clean spatial separation between substitutedproton donor and redox center, which is the keyfactor for decoupling the anti-correlation. Todo so, we synthesize complex 2 (with detailedprocedures and characterizations in SI), com-bine cyclic voltammetry (CV) measurementsand calculations to investigate the influence ofthe protonation on oxime and pyridyl groupsto the CoII/I reduction potential and the pKa

of these proton binding sites.These redox reactions denoted as reaction (A)

and (C) in 1 controls the primary energeticcosts to initiate the entire catalytic cycle. Com-bined experimental and theoretical results showthat the protonation on pyridien does not influ-ence the reduction potential; conversely, pro-tonation on oxime does strongly influence thecobalt reduction potential. Results for pKa

calculations on these proton binding sites re-veals that the pyridyl group is much easier to

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FIG. 3: WFT-in-DFT embedding has been used to dis-cover a new class of cobalt diimine-dioxime catalysts thatshows promise for decoupling the energetic (E◦) and ki-netic (∆G‡) barriers associated with proton- and electron-steps during hydrogen evolution [5].

Making projection-based embedding avail-able for wider use. We have already completedthe development of a user-friendly implemen-tation of the new projection-based approachto DFT-in-DFT and WFT-in-DFT embeddingin the Molpro quantum chemistry package,which is available to users in over 500 insti-tutions. We recognize that WFT-in-DFT em-bedding will have application domains that re-late to all aspects of simulation and modelingof complex chemical systems, and we antici-pate that a sizable user community will bene-fit from our methodological and software de-velopments. This work provides new tools forthe accurate description of catalytic and elec-tronically non-adiabatic processes in complexsystems, and it opens the door for widespreadadoption of rigorous quantum embedding ap-proaches in computational chemistry.

Indeed, our recent WFT-in-DFT implementation in the Molpro program is being tested andutilized by a range of theoretical chemistry groups, including those of: Prof. Donald Truhlar (U.Minnesota), Prof. Qiang Cui (U. Wisconsin), Prof. Berend Smit (UC Berkeley), Prof. AndresKohn (U. Stuttgart), Prof. Daniel Lambrecht (U. Pittsburgh), Prof. John Keith (U. Pittsburgh),Prof. Jeremy Harvey (U. Bristol), Dr. Betsy Rice (ARO), Prof. Fred Manby (U. Bristol), andKaito Miyamoto (U. Bristol / Toyota Co.).

Miller, FA9550-11-1-0288 Final Report - 5

[1] F. R. Manby, M. Stella, J. D. Goodpaster, and T. F. Miller III, A simple, exact density-functional theoryembedding scheme, J. Chem. Theory Comput., 8, 2564, (2012).

[2] J. D. Goodpaster, T. A. Barnes, F. R. Manby, and T. F. Miller III, Density functional theory embed-ding for correlated wavefunctions: Improved methods for open-shell systems and transition metalcomplexes, J. Chem. Phys., 137, 224113, (2012).

[3] T. A. Barnes, J. D. Goodpaster, F. R. Manby, and T. F. Miller III, Accurate basis-set truncation forwavefunction embedding, J. Chem. Phys., 139, 024103, (2013).

[4] J. D. Goodpaster, T. A. Barnes, F. R. Manby, and T. F. Miller III, Accurate and systematically improv-able density functional theory embedding for correlated wavefunctions, J. Chem. Phys., 140, 18A507,(2014).

[5] P. Huo, C. Uyeda, J. D. Goodpaster, J. C. Peters, and T. F. Miller III, Breaking the correlation betweenenergy costs and kinetic barriers in hydrogen evolution: A new class of cobalt (pyridine-diimine-dioxime) catalysts, in preparation.

[6] G. Senator and K. Subbaswamy, Density dependence of the dielectric constant of rare-gas crystals,Phys. Rev. B 34, 5754 (1986).

[7] P. Cortona, Self-consistently determined properties of solids without band-structure calculations, Phys.Rev. B 44, 8454 (1991).

[8] T.A. Wesolowski and A. Warshel, Frozen Density Functional Approach for ab-initio Calculations ofSolvated Molecules, J. Phys. Chem. 97, 8050 (1993).

[9] T.A. Wesolowski, in Computational Chemistry: Reviews of Current Trends - Vol. 10, pp. 1-82 (WorldScientific, Singapore, 2006).

[10] N. Govind, Y. A. Wang, A. J. R. da Silva, and E. A. Carter, Accurate Ab Initio Energetics of ExtendedSystems via Explicit Correlation Embedded in a Density Functional Environment, Chem. Phys. Lett.,295, 129 (1998).

[11] A. W. Gotz, S. M. Beyhan, and L. Visscher, Performance of Kinetic Energy Functionals for InteractionEnergies in a Subsystem Formulation of Density Functional Theory, J. Chem. Theo. Comput. 5, 3161(2009).

[12] M. Iannuzzi, B. Kirchner, and J. Hutter, Density functional embedding for molecular systems, Chem.Phys. Lett. 421, 16-20 (2006).