Extended HÃ¼ckel Calculations on Solids Using the Avogadro

Technology Report Cite This: J. Chem. Educ. XXXX, XXX, XXX-XXX

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Extended Hü ckel Calculations on Solids Using the Avogadro Molecular Editor and Visualizer Patrick Avery, Herbert Ludowieg, Jochen Autschbach,* and Eva Zurek* Department of Chemistry, University at Buffalo, Buffalo, New York 14260-3000, United States S Supporting Information *

ABSTRACT: The “Yet Another extended Hückel Molecular Orbital Package” (YAeHMOP) has been merged with theAvogadro open-source molecular editor and visualizer. It is now possible to perform YAeHMOP calculations directly from the Avogadro graphical user interface for materials that are periodic in one, two, or three dimensions, and to visualize band structures, total and projected density of states, and crystal orbital overlap/ Hamilton populations (COOPs/COHPs). Calculations on graphite, silicon, sodium, and a one-dimensional hydrogen chain are provided to illustrate the functionality. Similar exercises have been carried out in an upper-level undergraduate quantum theory course.

KEYWORDS: Computer-Based Learning, Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Computational Chemistry, Crystals/Crystallography, Materials Science, MO Theory, Semiconductors, Theoretical Chemistry

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INTRODUCTION Solid-state materials have numerous applications in the modern world. Therefore, a basic understanding of the electronic structure of extended systems, i.e., their band structure, is of utmost importance to students of chemistry and related fields. Band structure, or crystal orbital (CO), theory is a straightforward extension of molecular orbital (MO) theory,1 but it is often omitted in the undergraduate curriculum. Nobel laureate Roald Hoffmann has put much effort into facilitating the communication between condensed-matter physicists, chemists, and materials scientists, by teaching chemists how to go between bonds and bands.2 Hoffmann has also pioneered extended Hückel theory (EHT) for molecules and crystals.3,4 EHT is a semiempirical quantum chemistry method that, unlike the original Hückel model, takes into account the atomic orbital (AO) overlap. With EHT, computations on real materials can be performed quickly on a personal computer. EHT results may not be quantitatively accurate, but they give a qualitatively correct and predictive chemical picture. EHT has been instrumental in developing the famous Woodward−Hoffmann rules of organic chemistry,5 and it has provided deep insight into the interactions of molecules with surfaces as well as bonding in solids.6−14 The aptly named Yet Another extended Hückel Molecular Orbital Package (YAeHMOP)15 is a numerical code developed by two Ph.D. students in Hoffmann’s group, Greg Landrum and Wingfield Glassey. YAeHMOP can carry out EHT calculations on molecules and systems that are periodic in one, two, or three dimensions. However, the visualization program accompanying YAeHMOP (called viewkel) does not © XXXX American Chemical Society and Division of Chemical Education, Inc.

run natively on non-Unix systems, and it requires the use of a terminal window, which is not familiar to students. The userfriendly multiplatform open-source molecular builder, editor, and visualizer Avogadro16,17 and its extensions18−21 have recently emerged as a popular choice for research and education. Avogadro’s ease of use and its advanced visualization capabilities render it an ideal graphical platform for YAeHMOP calculations. Consequently, we integrated YAeHMOP with Avogadro to create a computational engine for the electronic structure of crystalline materials. In this technology report, we illustrate with examples of archetypal solids how to calculate band structures, total and projected densities of states (DOS and PDOS), and crystal orbital overlap populations (COOPs) using the Avogadro−YAeHMOP interface. An article describing the Computer Aided Composition of Atomic Orbitals (CACAO) program, which uses EHT for MO calculations and presents the results graphically, was published in the Journal of Chemical Education in 1990.22 To date, this article has been cited over 830 times, highlighting the importance of EHT calculations and the visualization of the results in chemical education and research. We hope that the visualization of the results of EHT calculations for crystalline materials, facilitated by the software described herein, will have comparable impact. Received: September 11, 2017 Revised: October 31, 2017

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DOI: 10.1021/acs.jchemed.7b00698 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

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Figure 1. Dialog boxes for the YAeHMOP interface within Avogadro (accessed via “Extensions” → “Yaehmop” in the Avogadro menu). Extensive information about the options in each dialog box will show up in “tooltip” boxes when the mouse pointer hovers over the option. The input menus shown are for the (a) band structure, (b) total DOS, (c) PDOS, and (d) COOP.

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STRATEGIES TO INTRODUCE BAND STRUCTURE TO STUDENTS

material is covered in two review articles by Hoffmann that were specifically written for chemists23,24). Lecture notes with the example calculations shown herein were given to the students, and they repeated the calculations on their own in order to become familiar with the software and the underlying concepts. None of the students reported problems with the installation or execution of the software. The students then used the software for part of a homework assignment (see the Supporting Information) to compute and qualitatively assign the band structure of a hypothetical atomic solid of fluorine with a cubic grid, which was chosen as a three-dimensional analogue of the one-dimensional hydrogen grid discussed

The software and examples given in this technology report could potentially be used in a wide variety of courses. J.A. has used the software and the examples in a third-year undergraduate physical chemistry course for chemical engineering students (Fall 2016/17, 82/98 enrollment). Prior to the calculations, the course covered 4 weeks of basic concepts of quantum theory as they are typically taught in the undergraduate physical chemistry course sequence, 1 week of Hückel theory, and a 3 week introduction to band structure along the lines of the first 50 pages of Hoffmann’s book2 (the same B



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below. The fluorine example was chosen because it is easy to determine the bonding versus antibonding character of the valence bands “by hand”, sketching the crystal orbitals at selected points in the reciprocal unit cell and using only the criteria of (i) how much nearest-neighbor bonding versus antibonding overlap is present and (ii) that 2p orbitals have stronger σ than π overlap. These criteria then allow the students to assign the calculated bands to the 2s and the different 2p valence orbitals. Almost every student completed this part of the homework set successfully, and therefore, we plan to add more complex problems to the homework set in the future. We note, however, that students do not require a strong background in physical chemistry or quantum mechanics to be able to perform the calculations presented in this technology report and analyze the results. In Fall 2012, E.Z. cotaught a course entitled “Modern Materials: Theory, Applications, and Advanced Characterization Techniques” that was cross-listed for senior undergraduate students and beginning graduate students, with an enrollment of 12. Half of the course was devoted to theory and closely followed Hoffmann’s introduction to the electronic structure of solids and surfaces,2,23,24 where Hoffmann beautifully teaches chemists how to construct approximate band structure plots, and draw the corresponding DOS, PDOS, and COOP plots, without doing any calculations. The students only need to know approximate MO theory as taught in organic chemistry in order to understand Hoffmann’s texts. If the technology presented in this report had been available, E.Z. would have extensively used it for class assignments. Moreover, E.Z. recently presented the integration of YAeHMOP with Avogadro at the 254th ACS National Meeting and received positive feedback from faculty members who are very excited to use this technology. For example, one individual plans to incorporate it as a module in a Solid State Chemistry course, which is cross-listed for senior undergrads and graduate students. Typically the undergraduates are in their fourth year and have taken both Physical Chemistry and Inorganic Chemistry. Previously, the instructor carried out similar calculations with an old (and not very user-friendly) program package and found that the students could master the concepts and analyze the electronic structure calculated for a given crystal quite well. Another professor teaches elementary concepts of band structure theory in a senior-level undergraduate course on Materials Chemistry. The material in the course builds off symmetry concepts coupled with basic ideas of atomic structure that are taught in general chemistry. The book by P. A. Cox,25 which provides an excellent and not overly mathematical approach to the topic, is used as a textbook. Currently, the students employ Materials Studio, which is an eminently useful, but expensive, proprietary software package to calculate and analyze band structure diagrams in this course. The instructor was therefore very excited about the prospect of using the technology discussed herein because it is free, published under an open-source license, and can be run on a personal computer that employs either a Windows, Mac, or Linux operating system. Thus, elementary concepts concerning the electronic structure of materials are already being taught to senior-level undergraduates, and graduate students in physical, solid-state, and materials chemistry courses, and the instructors of these courses urgently need user-friendly technology that can be employed to carry out homework assignments, or a research project.

Below, we describe key elements of the software that are likely to be of interest to course instructors, and we provide a progression of example calculations that are designed to help the students develop an understanding of the electronic band structure and related properties of solid crystalline materials.

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FEATURES OF THE GRAPHICAL USER INTERFACE, AND CALCULATION EXAMPLES: SODIUM, SILICON, GRAPHITE, AND A HYDROGEN CHAIN The calculations are performed on crystal structures that are either stored in the extensive structure library included with Avogadro, or on systems whose coordinates can be easily input manually. Avogadro can also import any standard crystallographic information file (CIF) format. Figure 1 illustrates the dialog boxes in the graphical user interface (GUI) for the EHT band structure calculations. To maximize user-friendliness, the dialog boxes are prefilled with suitable default parameters. An extensive user manual and tutorial are provided as Supporting Information. Molecules have discrete bound MO energy levels, while the CO energy levels in solids are continuous and called bands. The bands are usually plotted along lines connecting high-symmetry points in the reciprocal space unit cell, called “special k-points” in condensed-matter theory jargon (“k-space” is often used to refer to the reciprocal space). The resulting band structure plot is often rightfully referred to as a “spaghetti diagram”. The coordinates of the special k-points are essentially quantum numbers that are specific to crystals and indicate how the COs change their phases between different direct-space unit cells in the crystal. For example, the Γ-point (denoted as “GM” in the GUI) is the origin of the reciprocal cell. At Γ, the COs in different unit cells all have the same phase. This means, for example, that a CO constructed from s AOs on adjacent atoms would have the lowest energy at Γ because all of the AOs would be in-phase and have totally bonding overlap throughout the crystal (examples of COs are shown as insets in Figure 5a). A learning outcome in our courses was that the students become acquainted with band structure plots and were able to tell, for instance, if they indicated metallic or semiconducting behavior. The energetic spread (called “dispersion”) of the bands, and the way they “run” up or down in energy between different k-values can be rationalized by sketching the bonding or antibonding overlap of the AOs giving rise to the COs in question.2,23,24,26 The DOS indicates the number of orbital energy levels per energy interval. In an MO diagram, a high DOS would correspond to a clustering of levels in a narrow energy range. Adding up the occupations of all MOs gives the number of electrons in a molecule. Likewise, integration of the DOS to the Fermi level, εF (which is, for our purposes, the energy of the highest occupied level), yields the total number of electrons per unit cell of the solid. A DOS plot is therefore the solid-state analogue of an MO level diagram. Usually, only the valence orbitals are calculated, such that the integrated DOS gives the number of valence electrons in the unit cell. The contributions from certain atoms or orbital types to the DOS can be plotted separately, too. In condensed-matter theory jargon, this is called “projected” DOS (PDOS). The PDOS indicates which atom types or AOs are interacting with each other to form the COs, and whether they are hybridizing. Avogadro can detect the symmetry of the crystal, to within a user-specified tolerance, and generate a list of special k-points associated with the lattice to be employed for the band structure plots. These k-points can be reordered in the dialog C



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Figure 2. Sodium metal. (a) Band structure and (b) density of states (DOS) in units of number of states eV−1/unit cell. Calculation for a primitive unit cell of sodium (1 atom in the unit cell) and εF set to 0. The blue line in part b is the integrated DOS. (c) The conventional body-centered cubic cell of sodium as displayed by Avogadro.

Figure 3. Silicon. (a) Band structure, (b) total DOS (the blue line is the integration), and (c) projected DOS (PDOS) and integrations (thin lines) for the primitive unit cell of silicon (2 atoms in the unit cell) and εF set to 0.

corners are shared among eight adjacent unit cells each, and therefore, the atom count per unit cell is 2. The YAeHMOP calculations were carried out on the primitive cell, and for this reason the half-filled band contains a total of one electron per unit cell, as seen from the DOS integration. A large DOS at εF shows that the system is metallic. Silicon crystallizes in the same structure as diamond, with a face-centered cubic lattice, wherein each atom is tetrahedrally coordinated. Silicon is the archetypal semiconductor and the namesake of Silicon Valley. It has a finite energy gap between the highest occupied valence bands and the lowest unoccupied “conduction” bands that is small enough so that electrons can become mobile in certain situations. The band structure calculated with YAeHMOP is provided in Figure 3, and it clearly shows the (indirect) band gap. The occupied bands are in very good agreement with those obtained from more accurate theoretical methods. The PDOS plots show that the 3s AOs contribute to the DOS primarily at lower energies, whereas the band structure closer to εF is primarily due to the 3p AOs. The DOS integration yields a total of 8 valence electrons, corresponding to the two Si atoms in the primitive cell. Graphite, the lowest-energy allotrope of carbon at ambient conditions, is formed from stacking planar honeycomb carbon

box if the user wants to compare the plot with one from the literature that has a different ordering. For the PDOS calculations, the dialog menu allows the user to partition the DOS into contributions from different atom types (e.g., carbon), or from individual atoms, or from different AO types (e.g., carbon s, px, py, pz). Checking the box “Display Band/DOS/COOP data?” will cause the numerical values of the calculated data to be displayed in a separate window at the end of the calculation, so that the user can analyze or plot them outside of the GUI. The calculation examples were chosen as representatives of a semiconductor or insulator, a metal, and a semimetal. It is important for students to understand why some materials do not conduct electricity while others are more or less good conductors at finite temperatures, and why metals would conduct electricity even at 0 K. Sodium was employed as an example of a metal. This soft, silvery, reactive element possesses a body-centered cubic structure with one valence 3s occupation per primitive unit cell. As a result, the band structure of sodium possesses a halffilled valence band, rendering the system metallic. In Figure 2 the conventional cell of sodium, which contains twice as many atoms as the primitive cell, is illustrated to show the full symmetry of the cubic lattice more clearly. The atoms in the D



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Figure 4. Graphite. (a) Band structure, (b) total DOS (the blue line is the integration), and (c) PDOS and integrations (thin lines) for the 4-atom primitive unit cell and εF set to 0. (d) A supercell of graphite.

Figure 5. Infinite one-dimensional hydrogen chain with a = 1 Å spacing and εF set to 0. (a) Band structure (with sketches of the crystal orbitals), (b) total DOS (the blue line is the integration), and (c) COOP. In panel c, H1,2 (blue) is the COOP for nearest neighbors, and H1,3 (green) is the COOP for second nearest neighbors.

of the sp2 bands, indicates the weaker interlayer orbital interactions. Because the pz bands cross εF but the DOS at εF is vanishingly small, graphite is a semimetal (The DOS plots show small but finite values around εF which is an artifact of the necessary smoothing. By applying different smoothing values, the students can see that the DOS really drops to zero). The 2s PDOS (blue) overlaps with the 2px,y PDOSs (green), which is indicative of hybridization, whereas the 2pz PDOS has minimal overlap with 2s. The PDOS integrated to εF yields 4.9 s, 7.0 px,y, and 4.1 pz occupations, yielding a total of 16 valence electrons in line with the four carbon atoms in the primitive unit cell. The

sheets called graphene, which are held together by a mix of dispersion and weak overlap interactions (The 2010 Nobel Prize in Physics was awarded to Geim and Novoselov for extracting graphene sheets from solid graphite and characterizing their properties). The primitive unit cell of graphite contains four carbon atoms. A supercell, shown in Figure 4, illustrates more clearly the layered graphite structure. The carbon atoms are three-coordinate. The filled sp2 hybrid bands have a large dispersion, indicating the strong covalent in-plane interactions. The pz AOs are perpendicular to the plane of the sheets. The weaker dispersion of the pz bands, compared to that E



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occupation ratio of px and py to s orbitals in the sp2 bands is 1.42, indicative of s-rich hybrids. A crystal orbital overlap population (COOP) plot provides information about the strength of the bonding or antibonding interactions between specific orbital- or atom-pairs in a solid. An MO-theory analogue would be the Mulliken overlap population. The COOP is also known as a density of states plot that has been weighted by the overlap population. For a one-dimensional hydrogen chain wherein each atom is separated from the next by a = 1 Å, as illustrated in Figure 5, the CO at Γ formed by the s AOs is completely in-phase, meaning that the level is totally bonding. This is reflected in a positive COOP between nearest neighbors at the bottom of the band. The CO at the Fermi level is nonbonding, so the COOP decreases to zero at εF. The CO at the top of the band is totally antibonding, which goes along with the most negative COOP value. Because the overlap is treated explicitly in EHT, the dispersion of the band relative to εF is not symmetric. This is similar to the familiar MO diagram for H2: When the overlap is included in the calculation, the bonding MO formed from the two 1s AOs is stabilized less, relative to the AOs, than the antibonding MO is destabilized. If the overlap is neglected, as in simple Hückel theory, the splitting would be symmetric. Figure 5c also provides the COOP between second-nearest-neighbor atoms in the chain. Its weaker magnitude compared to the nearest-neighbor COOP reflects the weaker overlap between second nearest neighbors, and its qualitative behavior is completely different. For example, the overlap in the CO at k = π/a is completely bonding between second nearest neighbors, which goes along with a positive COOP value.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS E.Z. acknowledges the NSF (DMR-1505817) for financial support. J.A. acknowledges support from the National Science Foundation, Grant CHE-1560881. We thank Greg Landrum for help with YAeHMOP and Daniel Fredrickson for providing us with and advising us on using his extended Hückel tuner. P.A.'s contributions to this work were supported by a Software Fellowship from the Molecular Sciences Software Institute, which is funded by the U.S. NSF (ACI-1547580).

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(1) Autschbach, J. Orbitals  Some fiction and some facts. J. Chem. Educ. 2012, 89 (8), 1032−1040. (2) Hoffmann, R. Solids and Surfaces: A Chemist’s View of Bonding in Extended Structures; VCH Publishers: New York, 1988. (3) Hoffmann, R. An Extended Hückel Theory. I. Hydrocarbons. J. Chem. Phys. 1963, 39 (6), 1397−1412. (4) Lowe, J. P. Quantum Chemistry, 2nd ed.; Academic Press: New York, 1993; pp 324−349. (5) Woodward, R. B.; Hoffmann, R. The Conservation of Orbital Symmetry. Angew. Chem., Int. Ed. Engl. 1969, 8 (11), 781−853. (6) Papoian, G.; Norskov, J.; Hoffmann, R. A Comparative Theoretical Study of the Hydrogen, Methyl and Ethyl Chemisorption on the Pt(111) Surface. J. Am. Chem. Soc. 2000, 122 (17), 4129−4144. (7) Zurek, E.; Hoffmann, R.; Ashcroft, N. W.; Oganov, A. R.; Lyakhov, A. O. A Little Bit of Lithium Does a Lot for Hydrogen. Proc. Natl. Acad. Sci. U. S. A. 2009, 106 (42), 17640−17643. (8) Nuspl, G.; Polborn, K.; Evers, J.; Landrum, G. A.; Hoffmann, R. The Four-Connected Net in the CeCu2 Structure and its Ternary Derivatives. Its Electronic and Structural Properties. Inorg. Chem. 1996, 35 (24), 6922−6932. (9) Tachibana, M.; Yoshizawa, K.; Ogawa, A.; Fujimoto, H.; Hoffmann, R. Sulfur-Gold Orbital Interactions that Determine the Structure of Alkenethiolate/Au(111) Self-Assembled Monolayer Systems. J. Phys. Chem. B 2002, 106 (49), 12727−12736. (10) Pancharatna, P. D.; Mendez-Rojas, M. A.; Merino, G.; Vela, A.; Hoffmann, R. Planar Tetracoordinate Carbon in Extended Systems. J. Am. Chem. Soc. 2004, 126 (46), 15309−15315. (11) Balakrishnarajan, M. M.; Pancharatna, P. D.; Hoffmann, R. Structure and Bonding in Boron Carbide: The Invincibility of Imperfections. New J. Chem. 2007, 31 (4), 473−485. (12) Fredrickson, D. C.; Lee, S.; Hoffmann, R. The Nowotny Chimney Ladder Phases: Whence the 14 Electron Rule? Inorg. Chem. 2004, 43 (20), 6159−6167. (13) Vajenine, G. V.; Hoffmann, R. Compounds Containing CopperSulfur Layers: Electronic Structure, Conductivity, and Stability. Inorg. Chem. 1996, 35 (2), 451−457. (14) Fredrickson, D. C. Electronic Packing Frustration in Complex Intermetallic Structures: The Role of Chemical Pressure in Ca2Ag7. J. Am. Chem. Soc. 2011, 133 (26), 10070−10073. (15) YAeHMOP home page. http://yaehmop.sourceforge.net (accessed Oct 2017). (16) Hanwell, M. D.; Curtis, D. E.; Lonie, D. C.; Vandermeersch, T.; Zurek, E.; Hutchison, G. R. Avogadro: an advanced semantic chemical editor, visualization, and analysis platform. J. Cheminf. 2012, 4 (1), 17. (17) Avogadro home page. https://avogadro.cc/ (accessed Oct 2017). (18) XtalOpt home page. https://xtalopt.github.io/ (accessed Oct 2017). (19) Lonie, D. C.; Zurek, E. XtalOpt: An Open-source Evolutionary Algorithm for Crystal Structure prediction. Comput. Phys. Commun. 2011, 182 (2), 372−387.

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CONCLUSION The incorporation of YAeHMOP into the open-source, userfriendly molecular builder and visualizer Avogadro enables fast calculations of the electronic structures of 1D, 2D, or 3D periodic materials on a personal computer. The graphical user interface makes it easy for experts and novices alike to carry out calculations and visualize the results. The software has been used by us for assignments in a third-year quantum theory and band structure course taught to chemical engineering undergraduate students. Binaries for different operating systems may be downloaded from the project’s web page.27

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.7b00698. Employed extended Hückel parameters and a detailed guide for installing the program and performing each type of calculation (PDF)

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REFERENCES

AUTHOR INFORMATION

Corresponding Authors

*E-mail: jochena@buffalo.edu. *E-mail: ezurek@buffalo.edu. ORCID

Patrick Avery: 0000-0003-2254-1345 Jochen Autschbach: 0000-0001-9392-877X Eva Zurek: 0000-0003-0738-867X F



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(20) Simpson, S.; Lonie, D. C.; Chen, J.; Zurek, E. A Computational Experiment on Single−Walled Carbon Nanotubes. J. Chem. Educ. 2013, 90 (5), 651−655. (21) Wach, A.; Chen, J.; Falls, Z.; Lonie, D.; Mojica, E. R.; Aga, D.; Autschbach, J.; Zurek, E. Determination of the Structures of Molecularly Imprinted Polymers and Xerogels Using an Automated Stochastic Approach. Anal. Chem. 2013, 85 (18), 8577−8584. (22) Mealli, C.; Proserpio, D. M. MO theory made visible. J. Chem. Educ. 1990, 67 (5), 399. (23) Hoffmann, R. How Chemistry and Physics Meet in the Solid State. Angew. Chem., Int. Ed. Engl. 1987, 26 (9), 846−878. (24) Hoffmann, R. A chemical and theoretical way to look at bonding on surfaces. Rev. Mod. Phys. 1988, 60 (3), 601−628. (25) Cox, P. A. The Electronic Structure and Chemistry of Solids; Oxford University Press: Oxford, 1987. (26) Canadell, E.; Whangbo, M. H. Conceptual aspects of structureproperty correlations and electronic instabilities, with applications to low-dimensional transition-metal oxides. Chem. Rev. 1991, 91 (5), 965−1034. (27) Avogadro with YAeHMOP download page. https://avogadroyaehmop.github.io/ (accessed Oct 2017).

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Extended HÃ¼ckel Calculations on Solids Using the Avogadro

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