The Energy Landscape Concept and Its Implications for Synthesis Planning

Pure Appl. Chem. 2014; 86(6): 883–898

Conference paper

Martin Jansen* The energy landscape concept and its implications for synthesis planning

Abstract: Synthesis of novel solids, in a chemical sense, is one of the spearheads of innovation in materials research. However, such an undertaking is substantially impaired by lack of control and predictability. We present a concept that points the way towards rational planning of syntheses in solid state and materials chemistry. The foundation of our approach is the representation of the whole material world, i.e., the known and not-yet-known chemical compounds, on an energy landscape, which implies information about the free energies of these configurations. From this it follows at once that all chemical compounds capable of existence (both thermodynamically stable and metastable ones) are already present in virtuo in this landscape. For the first step of synthesis planning, i.e., the identification of candidates that are capable of existence, we computationally search the respective potential energy landscapes for (meta)stable structure candidates. Recently we have extended our techniques to finite temperatures and pressures and calculated phase diagrams, including metastable manifestations of matter, without resorting to any experimental pre-information. The conception developed is physically consistent, and its feasibility has been proven. Applying appropriate experimental tools has enabled us to realize, e.g., elusive Na3N, including almost all of its predicted polymorphs, many years after the predictions were published.

Keywords: calculation of phase diagrams; global searches; inorganic materials synthesis; IUPAC Congress-44; local ergodicity; solid state chemistry; structure prediction; synthesis from atoms.

DOI 10.1515/pac-2014-0212

Introduction

The lines of action called “analysis” and “synthesis” continue to constitute the foundation of chemistry, providing it with genuine and distinct characteristics in a time of increasing overlap among the various dis- ciplines of natural sciences. Taken in its general sense, analysis goes far beyond determining chemical compositions, and includes investigating static structures, dynamic behaviors as well as all chemical or physical properties of matter, regardless of whether they might suggest applications or not. Synthesis, on the other hand, comprises all actions that lead to defined chemical compounds. Taken in their etymological senses, these terms appear to be in opposition to each other. In chemistry, however, analysis and synthesis go hand in hand, even in a synergetic manner, thus keeping chemistry on the track of steady and fascinating progress. Basically, two key factors are triggering innovation in chemistry: new methods in analysis, e.g., providing ever better spatial, temporal or energetic resolution in determining atomic or electronic structures, respectively, and in synthesis by providing new (classes of) chemical compounds. Synthesis and analysis often do

Article note: A collection of invited papers based on presentations at the 44th IUPAC Congress, Istanbul, Turkey, 11–16 August 2013.

*Corresponding author: Martin Jansen, Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany, e-mail: [email protected]

© 2014 IUPAC & De Gruyter 884 M. Jansen: The energy landscape concept not appear to be of equal weight for the chemist. The analytical tools rather serve synthesis, which is at the core of chemistry. Synthesis is in the focus for economic reasons, reflecting the huge share of world market volume as contributed by materials and drugs, for understanding and replicating our physical surroundings, be it biological or inorganic matter, and in particular for knowledge driven basic research, pleasing human curiosity and contributing to human culture. Therefore it is well understandable that control of synthesis has continued to be on the top of the chemical agenda. Although, in the chemist’s daily work, synthesis is meant to transform certain starting materials to the desired product compound, in an ultimately puristic sense, it starts from the elements. Thus the number of different elements available, and the multiplicity of bonding options they offer, stake out the territory for synthetic chemistry. That part of the universe accessible to human perception consists of one and the same set of chemical elements, out of which about 86 are stable and can be employed in chemical synthesis. From this number, one limiting factor results immediately, that is the maximal number of possible combinations of element types, which amounts to p = 286. From Fig. 1, and the inset, it follows that the resulting number of different chemical systems, even when considering only subsets, is beyond the power of our imagination [1]. It is interesting to note to what little extent these systems have been investigated, not to speak of being fully explored, so far. The answer to the more important question of how many different compounds can be made, using 86 stable elements, is most probably “enumerably infinite”. It is easy to give the reader a feeling for whether this statement comes close to the truth or not. For extended solids a special scenario develops, because of the variability of periodicity, which is in principle unlimited. As a consequence, e.g., binary SiC can form an infinite number of stacking variants (polytypes). We regard this inconceivable, yet breathtaking, plethora of possible chemical compounds to represent the true face of chemistry, virtually defining its identity [1]. From the numbers and examples given, it is obvious that only a minute portion of the compounds possible has been realized, so far. Admittedly, it is definitely hard to perceive these numbers as “limiting’’, they rather indicate superabundant opportunities for chemistry. Of course, one has to keep in mind the hypothetical character of the above combinatorial considerations. In exploring the chemical world physically, one will sooner or later run into practically insurmountable bar- riers. One quite obvious limitation results from the total mass of matter available in the universe. However, this aspect would only establish a restriction if one wanted to experimentally investigate all possible chemical systems simultaneously. Further arguments against accessibility of high component systems claim that it is impossible to force more than about seven different elements into one individual solid compound, via conventional all-solid-state reaction routes [2, 3]. The intricacies to be managed at running such reactions, in particular in a reproducible fashion, are well recognized [4]. As another approach for esti- mating the highest practically achievable number of constitutional components of a compound, it has been assumed that the limit is set by those observed in nature, or thus far synthetically realized [5, 6]. Again, this number is said to converge towards a limit of seven. However, by just looking into some pertinent databases

15 Unknown

10 log (combinations) 5 Known

10 20 30 40 50 60 70 80 Number of components

Fig. 1 “Mountain of materials’’ [1], number of element combinations calculated for a set of up to 86 different elements; the share of so far experimentally studied systems is marked in red color. M. Jansen: The energy landscape concept 885 we have already found compounds with up to 11 elements, see, e.g., references [7–9]. Furthermore, we do not see any justification for excluding solid solutions, when discussing limits on the number of elements present in a compound. These are regular manifestations of matter, playing an important role in earth sciences and in optimizing specific properties by extrinsic doping. Sometimes it has been questioned, whether we need to explore all the high component chemical systems, because relevant bonding scenarios and properties could be extrapolated from exemplary low component systems. If one considers the fact that a highly relevant material like the cuprate type superconductor with the highest critical temperature known, HgBa2Ca2Cu3O8+x [10], has been discovered in a five component system, the answer clearly is “yes, we need to”. But regardless of what the maximum possible number of constituent elements in a chemical compound precisely happens to be, our rough combinatorial estimates, even if ignoring practical restrictions, certainly give a proper impression of the general magnitude of the problem to be coped with in synthetic chemistry. From the numbers discussed, the overwhelming size of the tasks set of running syntheses in a rational and purposeful manner is quite obvious, and, at the same time they document what wide areas of chemistry have still remained unexplored. While in many areas of molecular chemistry a highly efficient tool box for synthesis planning has become available, the synthesis of new extended solids has remained virtually unplannable [1, 11, 12], and thus in this field of chemistry one still has to rely on explorative approaches. For overcoming this unsatisfactory, yet unacceptable, state of affairs the scientific imperative remains to continue developing and improving preparative solid-state chemistry until a goal-oriented rational approach becomes viable. Only then will it be possible to reach the, also economically crucial, efficiency in the development of new solid compounds that constitutes the first step in materials research. In this essay we report on our own efforts [1, 11–13] in this direction that have resulted in a consistent, universal and feasible concept for synthesis planning for solid materials.

The energy landscape concept

In an attempt to unify all classes of chemical compounds, one may look for a feature shared by all of them. As one such common feature we identify the basic preconditions for a compound to exist in a given equilibrium geometry. Quite obviously, there must be some gain in binding energy, as compared to the same ensemble of unbonded atoms, and moreover, the equilibrium geometry is typically associated with a minimum in binding energy. Since this holds true for each chemical compound capable of existence, and since one can move from one minimum to another by modifying the structures (polymorphs), or exchanging, adding or removing atoms, one easily arrives at the concept of a landscape of (binding)energy [1, 11–17] with the minima associated with stable configurations, cf. Fig. 2.

Fig. 2 Schematic visualization of a 3D section of the multiminima energy landscape of chemical matter (picture courtesy of Christoph Delago, University of Vienna). 886 M. Jansen: The energy landscape concept

A well defined and rather simple scenario results if one resorts to the hypothetic conditions of T = 0 K, and the zero-point vibrations suppressed. Then the relevant energy corresponds to the potential energy, and for each imaginable chemical configuration the energy can be calculated. The resulting continuous (hyper) surface of potential energy is directly related to the configuration space, and each minimum of the landscape is corresponding to a (meta)stable configuration, and vice versa [1, 11–17]. Admitting finite temperature and pressure, i.e., realistic conditions, all unstable configurations will decay, while the (meta)stable ones constitute locally ergodic minimum regions [1, 14], corresponding to a respective macroscopic thermodynamic state. Depending on the thermodynamic boundary conditions applied, one of these minimum regions repre- sents the thermodynamically stable state (ground state) of the system under consideration, while the (numerous) remaining minimum regions represent metastable ones, exhibiting a wide spread of life times [14, 18]. This way of projecting all known as well as not yet known chemical compounds onto an energy landscape is providing a sound concept for any attempt to analyze chemistry, in particular the issue of synthesis, on a universal foundation [1, 11–21]. As a particular strength of our concept, all classes of matter are included on an equal footing, thus artificial trenches are removed, e.g., between chemistry of molecular compounds and extended solids, or between natural and synthetic matter. Furthermore, valuable insights into many implications of chemical synthesis can be extracted without any effort. In the past, the question of whether a target compound would be stable was almost exclusively evalu- ated based on thermodynamic considerations [22, 23]. However, kinetic stability alone is already sufficient for a certain chemical structure to be experimentally accessible [1, 16]. Since composition and structure of a stable configuration (associated with a locally ergodic region on the energy landscape) are predetermined by natural laws, neither composition nor structure can be subject to any kind of arbitrary tuning or shaping by the chemist, and using the term “design” in the context of developing targets for chemical synthesis is definitely inappropriate [17]. As an even more substantial consequence of this view, chemistry can be approached in a deductive way by deriving the structure of the respective energy landscape from “first principles”. Finally, a clear definition of chemistry is provided with almost mathematical stringency: doing chemistry corresponds to exploring the energy landscape associated with chemical matter, thus accumulating the stockpile of all substances capable of existence.

Energy landscapes and chemical synthesis

As a highly satisfying feature, the Energy Landscape Concept applies universally and is encompassing all heuristic or inductive approaches to the issue of synthesis previously followed, the conceptual as well as the experimental ones, thus allowing for a holistic view at chemistry. Traditionally, chemical energy landscapes have been explored experimentally, while each successful synthesis of a new compound corresponds to the discovery of a new (locally) ergodic minimum region on the respective energy landscape. The underlying mental processes of directing syntheses have been extremely diverse, depending on the class of matter dealt with. Table 1 provides a survey of the most commonly followed strategies in conceiving syntheses, ranked according to the degree of control achieved.

Table 1 Approaches to chemical synthesis planning, a holistic view.

1. Experimental exploration by trial-and-error 2. Experimental exploration based on analogies and other heuristic concepts 3. Structure prediction based on crystal chemical rules 4. Local computational optimization of start configurations as derived by analogy and other empirical knowledge 5. Computational structure determination of already synthesized compounds, exploiting experimental input (e.g., lattice constants or composition) 6. Global optimization using non-physical cost functions 7. Structure prediction by mathematical enumeration and tiling 8. Global optimization with energy as the only cost function 9. Global optimization on ab initio level M. Jansen: The energy landscape concept 887

Although it is clear that energy is the pivotal factor determining the structure of the energy landscape of chemical compounds, and thus their capability to exist, rather non-physical tools have been employed for predicting not yet synthesized configurations. These latter approaches of anticipating structures commonly rely on topological features that are known to be stable from experience, and in principle, all concepts in use for classifying structures in chemistry also have some predictive potential, or are at least suited for validating predicted structures [1]. The classical crystal chemical rules as formulated by V.M. Goldschmidt [24] continue to provide a sound basis for anticipating extended inorganic structures. L. Pauling’s ideas on bond-length/ bond-strength relationships [25] have been further developed [26] and have been used as cost functions in computationally generating structures for prescribed compositions and lattice constants [27]. In a similar way, plausible structures can be derived by considering symmetry, space filling [28] and types of packings [29, 30] as well as the restrictions imposed by the intrinsic properties of 3D, e.g., reflected by the space groups [31] and Wyckhoff positions [32] available. In the area of intermetallic phases, the well established tools like structure field analysis [33], electron counts [34], and electronegativity (work function) balances [35] are providing more general than specific, or practically useful, structural information. All these heuristic approaches suffer from lack of generality and of predictive power because they inherently tend to extrapolate existing knowledge. They most probably will also fail in achieving completeness. In some domains of chemistry, the inductive approach [12, 36] has led to an admirable ability in predicting unknown configurations. In these domains, the technique applied to deriving structures is based on linking rigid structural fragments. Such strategies are farthest developed in organic chemistry, where the local stereochemistries of the atom types commonly involved are perfectly known. Among the earliest systematic applications at exhaustively and correctly predicting (enumerating) members of families of compounds is the work by Pòlya who, on the example of alkanes, even tackled this task by formulating it math- ematically [37]. In this context, one might also refer to that part of Leonhard Euler’s work dealing with convex polyhedra which has enabled scientists to immediately enumerate all possible topologies of fullerenes [38], when they were experimentally realized about two centuries later. The virtually infinite size of extended solids makes it much more intricate to deal with them, based on the concept of structural increments. Historically, the building block approach was first applied to solid state structures by deriving the familiar and rather consistent systematics of silicates [39–41]. Here again, the crite- ria of classification have been successfully employed for the purpose of prediction. For silicates, graph theoretical procedures [42] have been used in enumerating (part of) the plethora of connectivities possible, using the (SiO4 ) -tetrahedron as a building block. In the light of the considerations presented here, one notices the limitations of these systematics, since the nearly infinite ability to mix-and-match such building blocks in ever-increasing unit cells is virtually unlimited. Applying such approaches to coordination polymers has fur- nished an impressive breakthrough [43, 44]. Selecting directionally fixed connectivities between bidentate or tridentate linkers, of defined lengths, and a central metal, has allowed correct predictions of open framework structures, including the accessible porosities and the correct translational symmetries. In some instances, the configuration space accessible has been restricted successfully, hardly leaving any path to escape to non- desired configurations, during the experimental realization [45, 46]. In the category of non-energy based approaches for systematically generating crystal structures, mathematical enumeration techniques, like graph theory, are continuing to attract attention [42, 47–49]. A procedure for enumerating crystalline networks, based on mathematical tiling theory [50] has been suggested recently [51]. As a particular strength of this approach, the complete set of topologies possible is generated, at the boundary conditions given. Such a procedure has been discussed in the context of enumerating possible zeolites [52]. However, mathematical tilings of 3D do not necessarily correspond to stable chemical configurations, because the necessary definition of boundary conditions does not contain information on the chemical elements, which would be substantial because most of them show an appreciable variability with respect to chemical bonding. Nevertheless, (mathematical) enumeration of possible topologies is of significant momen- tousness in exploring special chemical landscapes. Its usefulness strongly depends on the respective system under consideration. For the alkanes, Pòlya’s approach exactly reproduces all possible configurations [37], including isomers, and each individual representative corresponds to a minimum on the energy landscape, at 888 M. Jansen: The energy landscape concept least at 0 K. However, with decreasing uniformity and rigidity of the stereochemistry of the chemical elements involved, the dependability and predictive power of purely mathematical enumerations will also decrease. When choosing the route (see Table 1) most efficient for the particular purpose, one has to carefully weigh up the advantages and disadvantages of the competing approaches. All heuristic approaches to organize chemical synthesis discussed above suffer from a common conceptual weakness: they rely upon geometrical reasoning, thus ignoring the physically correct cost function, which is the total energy of the system under consideration. Strategies for identifying stable configurations that are less fraught with prejudice and ambiguity require the proper cost function to be applied, see Table 1, number 9.

Computational exploration of chemical energy landscapes

In principle, the configurations and energies corresponding to the ground and excited states of a chemical system can be calculated by solving the respective Schrödinger equation. However, it is quite obvious that the number of particles to be considered in trying to cover the full compositional and structural diversity exhibited by an ensemble of atoms of realistic size makes such a straightforward approach intractable, at least by the currently available tools. Instead, for the exploration of landscapes of high complexities, like the energy landscapes of chemical matter under discussion, performing stochastic walks guided by appropriate cost functions have proved to be an efficient strategy. A variety of well-developed algorithms serving this purpose have become available, the most important approaches being based on Monte Carlo techniques, genetic algorithms or neural networks. Virtually each of them would, in principle, be suited for global searches of chemical landscapes. In our implementation we have chosen the “Metropolis Monte Carlo” variant of “simulated annealing” [11, 13, 19–21, 53, 54]. In a similar way, genetic algorithms have been employed at predicting inorganic crystal structures [55, 56]. The specific features of our implementation have been determined both by the basic objective of our efforts, i.e., predicting targets for chemical synthesis, and by technical feasibility, considering the performance of algorithms and computer hardware presently available. In marked contrast to the techniques previously employed in computational determinations of crystal structures of already synthesized solids [57], we allow the translational lattices as well as the compositions to be freely varied. These are indispensable requirements to make sure that the whole landscape under inspection is accessible and the full compositional and structural variability is explored. A flowchart of our implementation is displayed in Fig. 3. Regrettably, at the beginning, we had to resort to rather general two body potentials for the energy calculations, during the global Metropolis runs [11]. This was on one hand forced by the necessity to keep the

Fig. 3 Flowchart for the modular approach to predicting chemical compounds capable of existence. Upper part refers to the global search and preliminary ranking procedures, lower part to local ab initio optimization of particular structure candidates. M. Jansen: The energy landscape concept 889 computational effort needed for full explorations of systems of realistic sizes within bounds, and on the other to introduce as little bias as possible into the random walks. The price to be paid is a certain restriction of our approach to polar compounds. Furthermore, quantitative properties of the structures generated, such as lattice constants and total energies, are not very precise, on this level. Nevertheless, for many examples, we have been able to demonstrate that this approach is rather robust: in all systems explored, those compounds already known from experiment have been reproduced in our search runs, and all additional local minima discovered have stayed stable during subsequent local optimizations, using ab initio methods. However, it should be kept in mind that the shape of the energy landscape, as imaged by such a computational process, depends crucially on the quality and kind of the total energy calculations and addition- ally on the move classes applied, and will more or less deviate from the “true” landscape. As a particular drawback of the two-body potentials, consisting of a Coulombic and a Lennard-Jones term, they overweight ionic structures and are not suitable to address, for example, configurations containing homoatomic bonds. In particular, structures displaying electronically driven distortions would escape discovery at this level of total energy calculations. This weakness definitely challenges our ultimate objective of being able to predict stable configurations of any combination of the chemical elements. Therefore, at least for addressing systems in which homoatomic bonds are expected to occur, we have moved on to performing the total energy calculations on the ab initio level. Again one has to compromise between accuracy and computational costs of the global searches [20, 58–60]. As the result of the global explorations, either based on empirical potentials or ab initio methods, assum- ing T = 0 K and suppressing the zero-point vibrations, a continuous hypersurface of potential energy as a function of the underlying configurations is obtained (see Fig. 4a). If one excludes quantum mechanical effects such as tunneling, all minima identified would correspond to stable configurations. For extended solids, these energy landscapes already give a rather realistic idea of the most stable compounds to be encountered when investigating that system experimentally, since for solid matter temperature dependence of the enthalpies of formation as well as entropic contributions commonly are too small to let low lying minima disappear, when switching to nonzero and somewhat elevated temperatures. One should, however, keep in mind that entropically stabilized and nonperiodic configurations will be missed: the first due to focusing only on the minima of potential energy, the second due to using finite simulation cells. Beyond employing our implementation at identifying compounds capable of existence, our approach also offers the opportunity to explore the barrier structure [61] of the landscape of potential energy by applying “lid” or “threshold” techniques [62–64], or to spot entropically stabilized regions [65]. In order to validate the results of the global searches based on total energy calculations using empirical potentials or “cheap’’ ab initio methods, and to refine the structures and energies obtained, state of the art ab initio local optimizations are subsequently performed for the most promising structure candidates. Using public domain Hartree–Fock and DFT codes, e.g., CRYSTAL-, WIEN-, or VASP-programs, the lattice parameters and the positional parameters are optimized in an iterative process [66], and in order to get insights into

0 0 E/kJ E/kJ

X2/a.u. X2/a.u. X1/a.u. X1/a.u.

Fig. 4 Schematic presentation of the continuous landscape of potential energy, as a function of configuration space (left). Locally ergodic regions (indicated by blue basins), at finite temperature and with vibrations allowed, based on the same configuration space (right). 890 M. Jansen: The energy landscape concept the pressure dependence of the total energies of the candidate structures, these iterations are repeated for various fixed volumes [67]. The improvements in precision as achieved for the local post optimizations by switching to state of the art ab initio tools are significant; however, the overall structure of the energy landscape reflecting chemical configurations capable of existence remains virtually unchanged. In pursuing our objective of computationally exploring energy landscapes of chemical compounds, the combinatorial complexity, as pointed out above, has caught up with us. Global runs on a given system can easily end up with some tens of thousands of structure candidates, which have to be scrutinized for identi- cal ones and for those already known from experiment. Such a daunting and tedious task can hardly be done by hand. Therefore, we have developed tools for an automated processing of the results. In order to avoid arbitrariness or prejudice, neither with regard to translational or to rotational symmetry, no symmetry constraints are applied during the global exploration, and the thousands of candidates need to be analyzed regarding possible symmetries. Thus, in a first step, within given tolerances, a conventional unit cell and all symmetry elements present in the structures are elaborated [68], which is followed by an automatic determination of the space group [69]. Finally, using a pattern recognition algorithm which allows isostructures to be matched [70], even if they are on a grossly different scale, independently of the functions of the constituting atoms, e.g., anionic or cationic, the file of predicted configurations is searched for equivalent ones and the resulting unique set is compared to databases of already known structures.

Computational explorations of potential energy landscapes – illustrative results

The computational tools developed have allowed us to globally explore potential energy landscapes for selected chemical systems, to identify (local) minima, to survey its barrier structure and to analyse and clas- sify the (meta)stable atomic configurations encountered. As a quite pleasing general experience, for each system investigated those configurations already known from experiment have been retrieved, and many more new physically meaningful structure candidates have been identified, which has convincingly validated our approach. One of the first ionic systems whose energy landscape has been investigated [11] in detail using simulated annealing without recourse to experimental data is NaCl. A large number of local minima was found on the empirical energy landscape, and the global minimum of the landscape corresponded to the experimentally observed rocksalt structure. The structures of most of the energetically low-lying minima could be identified with typical AB-structure types like NiAs, PtS, CsCl or sphalerite. However, one deep-lying local minimum, denoted 5-5 structure type, exhibited an arrangement previously unknown in ionic systems. Here, Na and Cl are coordinating each other mutually in a trigonally bipyramidal fashion, resulting in a topology that resem- bles that of hexagonal BN. The energy barrier stabilizing this structure is only moderately high (0.01 eV per atom), suggesting that respective compounds might be difficult to synthesize with traditional solid state synthesis methods. Thus it came as a pleasant surprise when this new predicted structure type was found experimentally [71] to be the aristotype of newly synthesized Li4SeO5, where Li and Se occupy the Na positions and O the Cl positions in the 5-5 structure, respectively (see Fig. 5). By now, this structure type has been encountered also during the growth of ZnO films [72]. As mentioned above, the cost efficient two body potentials only apply to a subsection of the total energy landscape of chemical matter. Quite obvious, they will fail to correctly address systems relying on covalent bonding or electronically driven distortions. “Lone pair” electron configurations in combination with (partial) covalent bonding are known to induce conspicuously distorted coordination geometries. We have regarded it a meaningful touchstone and challenge to investigate whether our approach of global searches on the ab initio level is able to (re)produce correct and reasonable structures showing lone pair effects without recourse to any pre-information. For such a case study, we chose GeF2 [73]. Addressing the germanium(II) fluoride system computationally has posed particular challenges, like treating M. Jansen: The energy landscape concept 891

Fig. 5 The 5-5 structure proptotype as predicted (left); first experimental realization of the same connectivity as Li4SeO5 (right). the electron lone pair and the highly mutable bonding schemes involved, as there are strong covalent bonds between germanium and terminal fluorine atoms, significantly weaker ones to bridging F, and finally the van der Waals interactions effecting the bonding between the (GeF2)n chains present. Given this complexity, the results obtained are of impressive dependability (see Fig. 6): the experimentally known facts have been reli- ably reproduced and definitely meaningful new targets for synthesis have been suggested. In the meantime, one of the predicted structures, marked as (III) in Fig. 6, has been experimentally confirmed and identified as a high temperature polymorph [74].

[23] GeF2-II (paratellurite) GeF2-IV GeF2-I (experimental) GeF2-III P43212 (or P41212) Pbcm P212121 Pnma

GeF2-V (γ-SeO2) GeF2-VI (β-SeO2) GeF2-VII GeF2-VIII Pmc21 Pmc21 P21/c Pnma

GeF2-IX GeF2-X (α-SeO2) GeF2-XI GeF2-XII P421m P42/mbc Ima2 Pmc21

Fig. 6 Graphical representations of the twelve most stable structure candidates encountered for GeF2 during global search on the ab initio landscape. 892 M. Jansen: The energy landscape concept

-277.15 Structure types:

Li2CO3-Zabuyelite Li2CO3-type-HP α-Na CO -277.25 2 3 β-Na2CO3

γ-Na2CO3 β-K2CO3

γ-K2CO3 -277.35 β-Rb2CO3 γ-Rb2CO3 γ-Cs CO

tree) 2 3 Li2CO3-type-8 -277.45 Li2CO3-type-19

Li2CO3-type-55 Li2CO3-type-77 Energy (har -277.55

-277.65 b a -277.75 20 40 60 80 100 120 140 Volume (A3)

Fig. 7 E(V) curves for predicted polymorphs of Li2CO3; the modification stable at ambient pressure, and a predicted and a pos- teriori experimentally realized high-pressure polymorph, are marked by (a) and (b), respectively.

Iterating ab initio calculations for a given configuration at various fixed volumes has provided insights in the pressure dependence of the respective total energies, and thus to derive indications for the pressure conditions to be advantageously applied during attempts aiming at an experimental realization of a predicted structure candidate. Figure 7 displays the results of such a study [67] performed on Li2CO3 and is illustrating again the tremendous richness of the chemical energy landscapes: each of the E(V) curves shown is representing a local minimum structure, capable of existence. Remarkably, a predicted low volume candidate has been realized independently by experiment [75]. Beyond employing the implementation at identifying compounds capable of existence, our approach also offers the opportunity to explore the barrier structure [61] of the landscape of potential energy by applying “lid’’ or “threshold’’ techniques [62–64]. In a very much simplified representation of the results as a “tree graph” one can visualize the low energy structure candidates identified and the barrier structure relating them. As an example, in Fig. 8 the tree graph for Na3N [76], an elusive and long searched for basic inorganic solid, is displayed.

Relating configuration space and thermodynamic space

The energy landscape of chemical compounds as discussed thus far is based on atomic configurations, defined by the types and numbers of atoms involved, and their positional vectors [11, 14, 15, 20]. The behavior of real chemical systems, however, is controlled by the thermodynamic variables of state, i.e., by the pressure (volume) and temperature conditions applied, and the concentrations of the constituent elements. In the case of (global) thermodynamic equilibrium, setting these variables to certain values uniquely fixes the state of the system under consideration, i.e., its complete phase content. One should recall that structural information is neither needed for, nor extractable from, such a description. Furthermore, global equilibrium implies that we are considering essentially infinitely long timescales. Since it is our declared goal to provide generally applicable tools allowing for directed chemical syntheses, we cannot restrict our considerations to thermodynamically stable entities only, but have to include kinetically stable states of matter, that are thermodynamically metastable on finite timescales, as well. The equilibrium states of solid matter are commonly documented in phase diagrams that represent projections onto the space of the thermody- M. Jansen: The energy landscape concept 893

E/eV atom-1 0

-0.50 ReO -0.70 3

-0.90

-1.10 Li3N

-1.30

AuCu3 -1.50 Li3P -1.70 Al3Ti I-Na3N

Fig. 8 Results of a theoretical exploration of the energy landscape of Na3N. Structure candidates with low energy are shown, together with the barrier structure in form of a tree graph. namic variables of state of those parts of the (hyper)spaces of free enthalpy attributed to ergodic minimum regions of a given system that represent ground states, at respective p, T, xi conditions (cf. Fig. 9). Includ- ing metastable states requires us to consider the full (hyper)spaces of free enthalpy for all the phases in a chemical system that are kinetically stable. In principle, the latter can be derived from the configuration space [1] by determining all regions on the energy landscape that are locally ergodic on the timescale tobs, for temperature T [14]. In particular, this means that lattice vibrations including zero point vibrations need to be taken into account. The lattice vibrations will sample all configurations within the amplitudes of vibration, thus generating an ensemble of configurations among which the system fluctuates, depending on the conditions given. Raising the temperature would enable transport and thus structural transformations to take

Fig. 9 Three-component phase diagram at constant temperature and pressure, displaying surfaces of free enthalpies for three different locally ergodic regions. In blue: solid solution A/B/C, red: melt A/B/C, green: surface of free enthalpy for a metastable solid solution. 894 M. Jansen: The energy landscape concept place, and all configurations that are not kinetically stable will decay. As a result, we encounter discrete, locally ergodic minimum regions comprising a larger number of configurations constituting states that are all populated at the thermodynamic boundary conditions given (cf. Fig. 4). Such regions can encompass one or many local minima; they represent a macroscopic thermodynamic ensemble, the state functions of which can be calculated according to the prescriptions of statistical thermodynamics. If applied to crystalline solids, deriving the free enthalpy associated with a locally ergodic region, starting from the potential energy (Epot), requires only a few, nevertheless intriguing, steps. To the enthalpy, H = Epot + pV, the entropy terms have to be added. In this case, these are basically due to the phonon densities of states, and also to the configurational contributions, if many local minima contribute to the locally ergodic regions. By evalu- ating these quantities for various temperatures and pressures, the surface of free enthalpy can be scanned as a function of the thermodynamic variables for a given locally ergodic region of the energy landscape that can be associated with a prototype (ideal) structure [1]. A given chemical system will exhibit many different locally ergodic regions, one of them representing the global minimum of the free enthalpy. Two borderline types of behavior may be considered. For very long observation times, all locally ergodic regions will merge into a globally ergodic region, and thermodynamic equilibrium will be reached. At the opposite extreme, all configurations associated with local minima are assumed to survive for the time of observation, thus constituting metastable states. For the latter scenario, it is possible to determine the free enthalpies associated with locally ergodic regions R as a function of p, T, xi, and thus phase diagrams including metastable states of matter can be constructed. Of course, now the full function GR(p,T,xi) (i = 1 – n), restricted to such a region R, needs to be considered, and the elegant way of densely displaying the information on equilibrium states by projecting G onto the subspace of the variables p, T, xi is no longer viable. It should be noted that, as a highly welcome “spin-off”, all relevant thermodynamic entities like enthalpies, entropies, and specific heats are supplied. The thermodynamic data generated along this procedure provide a sound basis for predicting and assessing the (meta)stability of targets for chemical synthesis. Moreover, they are suited to complement and validate the conventional phase diagrams. Thus far, the various thermodynamic details known for a given system are fitted together using the elegant CALPHAD approach [77], which, however, only affords interpolation and consistency checks of the experimental data. A missing thermodynamically stable compound inevitably leads to a seriously wrong phase diagram. Here, implementing data from computational chemistry, accessible as described above, would improve the reliability significantly. In special cases, the construction of yet unexplored, or experimentally inaccessible, regions of the phase diagrams has turned out to be rather straightforward, employing our approach [78–80]. For most of the ternary combinations of the alkali halides, AX/BX, the high temperature regions of the phase diagrams are well investigated while the low temperature parts have proven to be experimentally inaccessible, due to slow kinetics. Here one might encounter miscibility gaps or stable and metastable, respectively, ordered crystalline compounds. Since the contributions from the phonon densities of state to the entropy of reaction between two very much related alkali metal halides, e.g., NaCl and LiCl, can be neglected to first order [78], the configurational entropy constitutes the essential contribution that determines whether a solid solution or an ordered compound is thermodynamically stable. The task has been tackled by first exploring the landscape of the ternary systems AX/BX globally. The resulting configurations are then examined and attributed to locally ergodic regions, according to their potential energies. Some of these minimum regions represent thermodynamically stable or metastable individual ternary structures, others constitute representative configurations of solid solutions. For the latter, the enthalpies of formation have been calculated, using various supercell descriptions. Finally, as the only contribution to the entropy of reaction, the configurational entropy is computed, and the free enthalpies of the various ordered compounds and solid solutions are obtained. As a demonstration of the feasibility, and moreover of the impressive correctness achievable, of the approach, Fig. 10 presents the quasi-binary phase diagram of NaCl/LiCl where the low temperature region has been calculated without using any experimental pre-information [78]. M. Jansen: The energy landscape concept 895

NaCl-LiCl 1100

900

700 T (k)

500

300 0 0.2 0.4 0.6 0.8 1 xLiCl

Fig. 10 Phase diagram for the quasi-binary system NaCl–LiCl, solidus/liquidus parts from experimental data, treated by the CALPHAD approach, sub-solidus region calculated without recourse to any experimental preinformation, red and blue dots mark experimental data.

Experimental verifications

As a crucial requirement in science, (theoretical) predictions need to be validated experimentally. Thus the next step of planning chemical syntheses consists of rationally developing an experimentally viable access to the desired configuration, predicted to be either kinetically or thermodynamically stable. This is a task of intriguing complexity, which includes monitoring the structural and compositional evolution of the system under consideration as a function of time. The reactions involved need to proceed spontaneously, and the system thus follows a descending trajectory on the hyperspace of free enthalpy. In many instances, such pathways would be a spinoff of the determination of the free enthalpy landscapes, addressed above. However, upon approaching the synthesis target many pathways leading to different modifications compete, and the final outcome is the result of a bifurcation in the population dynamics of sub- and super-critical nuclei. This final step in the synthesis of a specific solid is determined by the kind of nuclei that first reach critical size and start growing. Therefore, special measures need to be taken to direct the system into the minimum region corresponding to the desired configuration. Regrettably, neither the theoretical treatment nor the experimental control of such a process has yet reached a satisfactory level. In our attempts to experimentally realize metastable predicted compounds we have therefore restricted ourselves to creating synthesis conditions that would be particularly suited to preserve metastable configurations, i.e., employing but low thermal activa- tion. For this purpose, we have constructed a dedicated experimental setup (see Fig. 11) based on a high-vac- uum chamber [81]. This allows us to evaporate the elements constituting the desired compound as free atoms and to deposit them on a cooled substrate (at liquid-nitrogen or at liquid-helium temperature) in a random spatial distribution. The structural evolution of this solid reaction mixture, which is very much reminiscent of the starting configurations for the global computational exploration of the respective energy landscape, is monitored as a function of time and temperature. Amazingly, such mixtures undergo all solid-state reactions at temperatures far below room temperature, yielding well crystallized products [81]. Synthesizing crystalline solids at these low temperatures, i.e., at such extremely mild thermal conditions, is without precedent, and we have, indeed, been able to prepare several metastable compounds and metastable modifications along this novel approach. Some characteristics of our low-temperature atomic beam deposition (LT-ABD) have already become apparent over the past decade. Obviously, when crystalline nuclei form inside the amor- phous deposit, a shrinkage of volume occurs, thus generating effective negative pressures on the surfaces of these nuclei. Consequently, the first structures evolving are the metastable low-density ones, quite in line with our intention. Among the most impressive results is the synthesis of the elusive sodium nitride Na3N in an energetically high-lying structure, the ReO3 type [82]. Even more importantly, almost the full set of the 896 M. Jansen: The energy landscape concept

Fig. 11 UHV-set up for the synthesis of solids from atomically disperse educt mixtures (preparation chamber including hatch to the movable transfer system). 1) Sample holder with sapphire substrate, integrated heating, and feed for L-N2, 2) electron-beam vaporizer, 3) effusion cell, 4) MW-plasma source, 5) gate valve to the cryo-pumps, 6) transferbox with gate valve.

most stable predicted, and published nine years in advance, polymorphs of Na3N (see Fig. 8) have recently been realized [83]. This impressively documents the strength of our new rational way for solid-state synthesis, and this result has attracted a lot of interest, which in part arose because of generations of solid-state chemists having tried in vain to synthesize Na3N.

The quintessence, outlook

Representing the multitude of all known and still unknown chemical compounds on an energy landscape points the way to a deductive treatment of chemistry [12], quite in contrast to the inductive approach thus far preferably followed in this discipline. A rather simple scenario results if one resorts to the hypothetical conditions of T = 0 K, and the zero-point vibrations suppressed. Then for each imaginable configuration the energy can be calculated. The resulting continuous (hyper)surface of potential energy is directly related to the configuration space, and each minimum of the landscape corresponds to a stable configuration at T = 0 K, and vice versa. Admitting finite temperature and pressure, i.e., under realistic conditions, for a given observation time all unstable configurations will decay, while the (meta)stable ones constitute locally ergodic regions, corresponding to a particular macroscopic thermodynamic state. Depending on the thermodynamic boundary conditions applied, one of these regions corresponds to the thermodynamically stable state of the system under consideration, while the (numerous) remaining minimum regions represent metastable ones, exhibiting a wide spread of life times. Applying the procedures of statistical thermodynamics, free enthalpies for each locally ergodic region can, in principle, be derived, and phase diagrams even including metastable states can be constructed [18]. Such physically realistic energy landscapes, exhibiting numerous locally ergodic regions, offer a firm foundation for dealing with virtually all aspects of chemistry, on a rational basis. Since the sufficient and necessary precondition for any chemical compound to exist is that it belongs to a locally ergodic region, without any exception all manifestations of chemical matter are covered, and consequently the diverse fields of preparative chemistry are unified and can be dealt with on a comparable footing. According to the views presented, all chemical compounds capable of existence are already present on the energy landscape, and are just waiting to be discovered. Regarding against this background the fun- damental importance of solid matter in the form of useful materials for the welfare of mankind, we have proposed to start a systematic world-wide coordinated effort with the goal of tabulating all the (solid) compounds capable of existence, and we have compared it to the exemplary ™Human Genome Project, which has succeeded in fully deciphering the human genetic information [1]. The basic idea suggested as well as M. Jansen: The energy landscape concept 897 the lingual connotation of “materials” and “genome” seem to have provided the inspiration for the ambitious “Materials Genome Initiative” launched by the federal government of the USA in 2011. It is our conviction that the concept of the energy landscape of chemical matter, as presented here, will be a most sustainable approach to successfully address the issue of synthesis planning in solid state and materials chemistry.

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