Computational Materials Science 47 (2009) 237–253
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Computational Materials Science
journal homepage: www.elsevier.com/locate/commatsci
Density-functional tight-binding for beginners
Pekka Koskinen *, Ville Mäkinen
NanoScience Center, Department of Physics, University of Jyväskylä, 40014 Jyväskylä, Finland article info abstract
Article history: This article is a pedagogical introduction to density-functional tight-binding (DFTB) method. We derive it Received 1 June 2009 from the density-functional theory, give the details behind the tight-binding formalism, and give practi- Accepted 29 July 2009 cal recipes for parametrization: how to calculate pseudo-atomic orbitals and matrix elements, and espe- Available online 22 August 2009 cially how to systematically fit the short-range repulsions. Our scope is neither to provide a historical review nor to make performance comparisons, but to give beginner’s guide for this approximate, but PACS: in many ways invaluable, electronic structure simulation method—now freely available as an open- 31.10.+z source software package, hotbit. 71.15.Dx 2009 Elsevier B.V. All rights reserved. 71.15.Nc Ó 31.15.E
Keywords: DFTB Tight-binding Density-functional theory Electronic structure simulations
1. Introduction sizes were already possible. With DFT as a competitor, the DFTB community never grew large. If you were given only one method for electronic structure calcu- Despite being superseded by DFT, DFTB is still useful in many lations, which method would you choose? It certainly depends on ways: (i) In calculations of large systems [4,5]. Computational scal- your field of research, but, on average, the usual choice is probably ing of DFT limits system sizes, while better scaling is easier to achieve density-functional theory (DFT). It is not the best method for every- with DFTB. (ii) Accessing longer time scales. Systems that are limited thing, but its efficiency and accuracy are suitable for most purposes. for optimization in DFT can be used for extensive studies on dynam- You may disagree with this argument, but already DFT commu- ical properties in DFTB [6]. (iii) Structure search and general trends nity’s size is a convincing evidence of DFT’s importance—it is even [7]. Where DFT is limited to only a few systems, DFTB can be used among the few quantum mechanical methods used in industry. to gather statistics and trends from structural families. It can be used Things were different before. When computational resources also for pre-screening of systems for subsequent DFT calculations were modest and DFT functionals inaccurate, classical force fields, [8,9]. (iv) Method development. The formalism is akin to that of semiempirical tight-binding, and jellium DFT calculations were DFT, so methodology improvements, quick to test in DFTB, can be used. Today tight-binding is mostly familiar from solid-state text- easily exported and extended into DFT[10]. (v) Testing, playing books as a method for modeling band-structures, with one to sev- around, learning and teaching. DFTB can be used simply for playing eral fitted hopping parameters [1]. However, tight-binding could around, getting the feeling of the motion of atoms at a given temper- be used better than this more often even today—especially as a ature, and looking at the chemical bonding or realistic molecular method to calculate total energies. Particularly useful for total en- wavefunctions, even with real-time simulations in a classroom— ergy calculations is density-functional tight-binding, which is DFTB runs easily on a laptop computer. parametrized directly using DFT, and is hence rooted in first prin- DFTB is evidently not an ab initio method since it contains ciples deeper than other tight-binding flavors. It just happened that parameters, even though most of them have a theoretically solid density-functional tight-binding came into existence some ten basis. With parameters in the right place, however, computational years ago [2,3] when atomistic DFT calculations for realistic system effort can be reduced enormously while maintaining a reasonable accuracy. This is why DFTB compares well with full DFT with minimal basis, for instance. Semiempirical tight-binding can be * Corresponding author. Tel.: +358 14 260 4717. E-mail address: pekka.koskinen@iki.fi (P. Koskinen). accurately fitted for a given test set, but transferability is usually
0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.07.013 238 P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253
worse; for general purposes DFTB is a good choice among the dif- length (aB ¼ 0:5292 Å) and Hartree as the unit of energy (Ha ¼ ferent tight-binding flavors. 27:2114 eV). Selecting atomic mass unit (u ¼ 1:6605 10 27 kg) Despite having its origin in DFT, one has to bear in mind that DFTB the unit of mass, the unit of time becomes 1:0327 fs, appropriate is still a tight-binding method, and should not generally be consid- for molecular dynamicsp simulations.ffiffiffiffiffiffiffiffiffiffiffiffi Some useful fundamental 6 ered to have the accuracy of full DFT. Absolute transferability can constants are then h ¼ me=u ¼ 0:0234, kB ¼ 3:1668 10 , and never be achieved, as the fundamental starting point is tightly bound e0 ¼ 1=ð4pÞ, for instance. electrons, with interactions ultimately treated perturbatively. DFTB Electronic eigenstates are denoted by wa, and (pseudo-atomic) is hence ideally suited for covalent systems such as hydrocarbons basis states ul, occasionally adopting Dirac’s notation. Greek let- [2,11]. Nevertheless, it does perform surprisingly well describing ters l, m are indices for basis states, while capital Roman letters I, also metallic bonding with delocalized valence electrons [12,8]. J are indices for atoms; notation l 2 I stands for orbital l that be- This article is neither a historical review of different flavors of longs to atom I. Capital R denotes nuclear positions, with the posi- tight-binding, nor a review of even DFTB and its successes or failures. tion of atom I at RI, and displacements RIJ ¼jRIJj¼jRJ RIj. Unit b For these purposes there are several good reviews, see, for example vectors are denoted by R ¼ R=jRj. Refs. [13–16]. Some ideas were around before [17–19], but the DFTB In other parts our notation remains conventional; deviations are method in its present formulation, presented also in this article, was mentioned or made self-explanatory. developed in the mid-1990s [2,3,20–23]. The main architects behind the machinery were Eschrig, Seifert, Frauenheim, and their co-work- 2.2. Starting point: full DFT ers. We apologize for omitting other contributors—consult Ref. [13] for a more organized literature review on DFTB. The derivation of DFTB from DFT has been presented several Instead of reviewing, we intend to present DFTB in a pedagogi- times; see, for example Refs. [18,13,3]. We do not want to be cal fashion. By occasionally being more explicit than usual, we de- redundant, but for completeness we derive the equations briefly; rive the approximate formalism from DFT in a systematic manner, our emphasis is on the final expressions. with modest referencing to how the same approximations were We start from the total energy expression of interacting elec- done before. The approach is practical: we present how to turn tron system DFT into a working tight-binding scheme where parametrizations E ¼ T þ E þ E þ E ; ð1Þ are obtained from well-defined procedures, to yield actual num- ext ee II bers—without omitting ugly details hiding behind the scenes. Only where T is the kinetic energy, Eext the external interaction (including basic quantum mechanics and selected concepts from density- electron–ion interactions), Eee the electron–electron interaction, and functional theory are required as pre-requisite. EII ion–ion interaction energy. Here EII contains terms like The DFTB parametrization process is usually presented as v v v ZI ZJ =jRI RJj, where ZI is the valence of the atom I, and other contri- superficially easy, while actually it is difficult, especially regarding butions from the core electrons. In density-functional theory the en- the fitting of short-range repulsion (Section 4). In this article we ergy is a functional of the electron density nðrÞ, and for Kohn–Sham want to present a systematic scheme for fitting the repulsion, in or- system of non-interacting electrons the energy can be written as der to accurately document the way the parametrization is done. E n r T E E E E ; 2 We discuss the details behind tight-binding formalism, like Sla- ½ ð Þ ¼ s þ ext þ H þ xc þ II ð Þ ter–Koster integrals and transformations, largely unfamiliar for where Ts is the non-interacting kinetic energy, EH is the Hartree en- density-functional community; some readers may prefer to skip ergy, and Exc ¼ðT TsÞþðEee EHÞ is the exchange-correlation (xc) these detailed appendices. Because one of DFTB’s strengths is the energy, hiding all the difficult many-body effects. More explicitly, transparent electronic structure, in the end we also present se- Z ! X 3 lected analysis tools. 1 1 nðr0Þd r0 E n f w 2 V w E n E ; ½ ¼ ah aj r þ ext þ 0 j aiþ xc½ þ II We concentrate on ground-state DFTB, leaving time-dependent a 2 2 jr rj [24–27] or linear response [28] formalisms outside the discussion. ð3Þ We do not include spin in the formalism. Our philosophy lies in the limited benefits of improving upon spin-paired self-consistent - where fa 2½0; 2 is the occupation of a single-particle state wa with charge DFTB. Admittedly, one can adjust parametrizations for cer- energy ea, usually taken from the Fermi-function (with factor 2 tain systems, but the tight-binding formalism, especially the pres- for spin) ence of the repulsive potential, contains so many approximations 1 that the next level in accuracy, in our opinion, is full DFT. fa ¼ f ðeaÞ¼2 ½ expðea lÞ=kBT þ 1 ð4Þ P This philosophy underlies hotbit [29] software. It is an open- with chemical potential l chosen such that afa ¼ number of elec- source DFTB package, released under the terms of GNU general trons. The Hartree potential public license [30]. It has an interface with the atomic simulation Z 0 0 environment (ASE) [31], a python module for multi-purpose atom- nðr Þ V H½n ðrÞ¼ ð5Þ istic simulations. The ASE interface enables simulations with dif- jr0 rj ferent levels of theory, including many DFT codes or classical is a classical electrostatic potential from given nðrÞ; for brevity we potentials, with DFTB being the lowest-level quantum-mechanical R R R R will use the notation d3r ! , d3r0 ! 0, nðrÞ!n, and method. hotbit is built upon the theoretical basis described nðr0Þ!n0. With this notation the Kohn–Sham DFT energy is, once here—but we avoid technical issues related practical implementa- more, tions having no scientific relevance. Z X 1 E½n ¼ f hw j r2 þ V ðrÞnðrÞ jw i a a 2 ext a a ZZ 2. The origins of DFTB 1 0 nn0 þ þ E ½n þE : ð6Þ 2 jr r0j xc II 2.1. Warm-up So far everything is exact, but now we start approximating.
We begin by commenting on practical matters. The equations Consider a system with density n0ðrÞ that is composed of atomic 2 use h =me ¼ 4pe0 ¼ e ¼ 1. This gives Bohr radius as the unit of densities, as if atoms in the system were free and neutral. Hence P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 239 X IJ n0ðrÞ contains (artificially) no charge transfer. The density n0ðrÞ Erep ¼ V repðRIJÞ: ð14Þ does not minimize the functional E½nðrÞ , but neighbors the true I þ Exc½n0 þEII V xc½n0 ðrÞn0ðrÞ; ð7Þ 2.4. Charge fluctuation term while linear terms in dn vanish. The first line in Eq. (7) is the band- structure energy Let us first make a side road to recall some general concepts X from atomic physics. Generally, the atom energy can be expressed EBS½dn ¼ fahwajH½n0 jwai; ð8Þ as a function of Dq extra electrons as [32] a ! 2 0 @E 1 @ E where the Hamiltonian H ¼ H½n0 itself contains no charge transfer. 2 EðDqÞ E0 þ Dq þ Dq The second line in Eq. (7) is the energy from charge fluctuations, @Dq 2 @Dq2 being mainly Coulomb interaction but containing also xc- 1 ¼ E vDq þ UDq2: ð15Þ contributions 0 2 ZZ ! 0 2 1 d Exc½n0 1 The (negative) slope of EðDqÞ at Dq ¼ 0 is given by the (positive) E ½dn ¼ þ dndn0: ð9Þ coul 2 dndn0 jr r0j electronegativity, which is usually approximated as The last four terms in Eq. (7) are collectively called the repulsive v ðIE þ EAÞ=2; ð16Þ energy where IE the ionization energy and EA the electron affinity. The (up- Z Z 1 ward) curvature of EðDqÞ is given by the Hubbard U, which is Erep ¼ V H½n0 ðrÞn0ðrÞþExc½n0 þEII V xc½n0 ðrÞn0ðrÞ; 2 U IE EA ð17Þ ð10Þ and is twice the atom absolute hardness g (U ¼ 2g) [32]. Electro- because of the ion–ion repulsion term. Using this terminology the negativity comes mainly from orbital energies relative to the vac- energy is uum level, while curvature effects come mainly from Coulomb interactions. E½dn ¼EBS½dn þEcoul½dn þErep: ð11Þ Let us now return from our side road. The energy in Eq. (9) Before switching into tight-binding description, we discuss Erep and comes from Coulomb and xc-interactions due to fluctuations Ecoul separately and introduce the main approximations. dnðrÞ, and involves a double integrals over all space. Consider the space V divided into volumes VI related to atoms I, such that 2.3. Repulsive energy term X Z X Z VI ¼ V and ¼ : ð18Þ In Eq. (10) we lumped four terms together and referred them to I V I VI as repulsive interaction. It contains the ion–ion interaction so it is We never precisely define what these volumes V exactly are—they repulsive (at least at small atomic distances), but it contains also I are always used qualitatively, and the usage is case-specific. For xc-interactions, so it is a complicated object. At this point we adopt example, volumes can be used to calculate the extra electron popu- manners from DFT: we sweep the most difficult physics under the lation on atom I as carpet. You may consider Erep as practical equivalent to an xc-func- Z tional in DFT because it hides the cumbersome physics, while we 3 DqI dnðrÞd r: ð19Þ approximate it with simple functions. VI For example, consider the total volumes in the first term, the Hartree term By using these populations we can decompose dn into atomic ZZ contributions 1 n ðrÞn ðr0Þ X 0 0 d3rd3r0; ð12Þ 2 jr r0j dnðrÞ¼ DqIdnIðrÞ; ð20Þ I R divided into atomic volumes; the integral becomes a sum over atom 3 such that each dnIðrÞ is normalized, dnIðrÞd r ¼ 1. Note that Eqs. pairs with terms depending on atomic numbers alone, since n0ðrÞ VI (19) and (20) are internally consistent. Ultimately, this division is depends on them. We can hence approximate it as a sum of terms used to convert the double integral in Eq. (9) into sum over atom over atom pairs, where each term depends only on elements and R R pairs IJ, and integrations over volumes . their distance, because n0ðrÞ is spherically symmetric for free VI VJ First, terms with I ¼ J are atoms. Similarly ion–ion repulsions Z Z ! v v v v 0 2 Z Z Z Z 1 2 d Exc½n0 1 0 I J I J Dq þ dn dn : ð21Þ ¼ ; ð13Þ 2 I dndn0 jr r0j I I jRJ RIj RIJ VI VI v depend only on atomic numbers via their valence numbers ZI . Term depends quadratically on DqI and by comparing to Eq. (15) we Using similar reasoning for the remaining terms the repulsive en- can see that integral can be approximated by U. Hence, terms with 1 2 ergy can be approximated as I ¼ J become 2 UIDqI . 240 P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 Second, when I – Jxc-contributions will vanish for local xc- and FWHMI is the full width at half maximum for the profile. This functionals for which choice of profile is justified for a carbon atom in Fig. 1a. With these assumptions, the Coulomb energy of two spherically symmetric d2E xc d r r0 22 Gaussian charge distributions in Eq. (23) can be calculated analyti- 0 / ð ÞðÞ dnðrÞdnðr Þ cally to yield Z Z 0 0 and the interaction is only electrostatic, dnIdn erfðC R Þ J IJ IJ R ; 26 Z Z 0 ¼ cIJð IJÞ ð Þ 0 0 jr r j RIJ 1 dnIdn V V Dq Dq J ð23Þ 2 I J r r0 where VI VJ j j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ln2 between extra atomic populations DqI and DqJ. Strictly speaking, we CIJ ¼ : ð27Þ FWHM2 þ FWHM2 do not know what the functions dnIðrÞ are. However, assuming I J spherical symmetry, they tell how the density profile of a given In definition (26) we integrate over whole spaces because dnI’s are atom changes upon charging. By assuming functional form for the strongly localized. The function (26) is plotted in Fig. 1b, where we profiles dn ðrÞ, the integrals can be evaluated. We choose a Gaussian I see that when R FWHM we get point-like 1=R-interaction. Fur- profile [33], pffiffiffiffi thermore, as R ! 0, c ! C 2= p, which gives us a connection to 1 r2 on-site interactions: if I ¼ J dnIðrÞ¼ÀÁexp ; ð24Þ rffiffiffiffiffiffiffiffiffiffiffiffi 2 3=2 2r2 2prI I 8ln2 1 cIIðRII ¼ 0Þ¼ : ð28Þ where p FWHMI This is the on-site Coulomb energy of extra population on atom I. FWHMffiffiffiffiffiffiffiffiffiffiffiffiI rI ¼ p ð25Þ Comparing to the I ¼ J case above, we can approximate 8ln2 rffiffiffiffiffiffiffiffiffiffiffiffi 8ln2 1 1:329 FWHMI ¼ ¼ : ð29Þ p UI UI We interpret Eq. (28) as: narrower atomic charge distributions causes larger costs to add or remove electrons; for a point charge charging energy diverges, as it should. Hence, from absolute hardness, by assuming only Coulombic origin, we can estimate the sizes of the charge distributions, and these sizes can be used to estimate Coulomb interactions also be- tween the atoms. U and FWHM are coupled by Eq. (29), and hence for each element a single parameter UI, which can be found from standard tables, determines all charge transfer energetics. To conclude this subsection, the charge fluctuation interactions can be written as 1 X Ecoul ¼ cIJðRIJÞDqIDqJ; ð30Þ 2 IJ where ( UI; I ¼ J c ðR Þ¼ ð31Þ IJ IJ erfðCIJ RIJ Þ ; I – J: RIJ 2.5. TB formalism So far the discussion has been without references to tight-bind- ing description. Things like eigenstates jwai or populations DqI can be understood, but what are they exactly? As mentioned above, we consider only valence electrons; the repulsive energy contains all the core electron effects. Since tight-binding assumes tightly bound electrons, we use minimal lo- cal basis to expand X a waðrÞ¼ clulðrÞ: ð32Þ l Minimality means having only one radial function for each angular Fig. 1. (a) The change in density profile for C upon charging. Atom is slightly momentum state: one for s-states, three for p-states, five for charged ( 2=3 e and þ1 e), and we plot the averaged dnðrÞ¼jn ðrÞ n0ðrÞj (shadowed), where n ðrÞ is the radial electron density for charged atom, and n0ðrÞ is d-states, and so on. We use real spherical harmonics familiar the radial electron density for neutral atom. This is compared to the Gaussian from chemistry—for completeness they are listed in Table 2 in profile of Eq. (24) with FWHM ¼ 1:329=U where U is given by Eq. (17). The change Appendix A. in density near the core is irregular, but the behavior is smooth for up to 90 % of With this expansion the band-structure energy becomes the density change. (b) The interaction energy of two spherically symmetric X X Gaussian charge distributions with equal FWHMI ¼ FWHMJ ¼ 1:329=U with U ¼ 1, a a 0 EBS ¼ fa cl cmHlm; ð33Þ as given by Eq. (26). With RIJ FWHMI interaction is Coulomb-like, and approaches a lm U as RIJ ! 0. P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 241 where where 0 0 Hlm ¼huljH jumi: ð34Þ 1 1 h ¼ ð þ Þ; l 2 I m 2 J: ð42Þ lm 2 I J The tight-binding formalism is adopted by accepting the matrix ele- 0 This expression suggests a reasonable interpretation: charge fluctu- ments Hlm themselves as the principal parameters of the method. 0 ations shift the matrix element H according to the averaged elec- This means that in tight-binding spirit the matrix elements Hlm lm are just numbers. Calculation of these matrix elements is discussed trostatic potentials around orbitals l and m. As in Kohn–Sham in Section 3, with details left in Appendix C. equations in DFT, also Eqs. (39) and (40) have to be solved self-con- sistently: from a given initial guess for fDq g one obtains h1 and How about the atomic populations DqI? Using the localized I lm a basis, the total number of electrons on atom I is Hlm, then by solving Eq. (39) one obtains new fclg, and, finally, Z Z X X X new fDqIg, iterating until self-consistency is achieved. The number 2 3 a a 3 qI ¼ fa jwaðrÞj d r ¼ fa cl cm ulðrÞumðrÞd r: ð35Þ of iterations required for convergence is usually markedly less than a VI a lm VI in DFT, albeit similar convergence problems are shared. Atomic forces can be obtained directly by taking gradients of Eq. If neither l nor m belong to I, the integral is roughly zero, and if both (38) with respect to coordinates (parameters) RI. We get (with l and m belong to I, the integral is approximately dlm since orbitals rJ ¼ @=@RJ) on the same atom are orthonormal. If l belongs to I and m to some X X hi other atom J, the integral becomes F f ca ca H0 h1 S Z Z I ¼ a l m rI lm ðea lmÞrI lm a lm 1 1 X ulðrÞumðrÞ ulðrÞumðrÞ¼ Slm; ð36Þ Dq ðrIc ÞDq rIErep; ð43Þ VI 2 V 2 I IJ J J as suggested by Fig. 2, where Slm ¼huljumi is the overlap of orbitals l and m. Charge on atom I is where the gradients of cIJ are obtained analytically from Eq. (26), 0 and the gradients of Hlm and Slm are obtained numerically from an X X X 1 q ¼ f ca ca þ c:c: S ; ð37Þ interpolation, as discussed in Appendix C. I a 2 l m lm a l2I m 0 3. Matrix elements where c.c. stands for complex conjugate. Hence DqI ¼ qI qI , where q0 is the number of valence electrons for a neutral atom. This ap- I Now we discuss how to calculate the matrix elements H0 and proach is called the Mulliken population analysis [34]. lm S . In the main text we describe only main ideas; more detailed Now were are ready for the final energy expression, lm X X X X issues are left to Appendices Appendices A–C a a 0 1 IJ E ¼ fa cl cmHlm þ cIJðRIJÞDqIDqJ þ V repðRIJÞ; ð38Þ a lm 2 IJ I X1 2i V conf ðrÞ¼ v2ir ; ð46Þ i¼0 where the odd terms disappear because the potential has to be smooth at r ¼ 0. Since the first v0 term is just a constant shift, the 2 first non-trivial term is v2r . To first approximation we hence choose the confining potential to be of the form r 2 V conf ðrÞ¼ ; ð47Þ r0 where r0 is a parameter. The quadratic form for the confinement has Fig. 2. Integrating the overlap of local orbitals. The large shaded area denotes the appeared before [2], but also other forms have been analyzed [35]. volume of atom I, spheres represent schematically the spatial extent of orbitals, VI While different forms can be considered for practical reasons, they and the small hatched area denotes the overlap region. With m 2 J and l 2 I, the have only little effect on DFTB performance. The adjustment of the integration of ulðrÞ umðrÞ over atom I’s volume VI misses half of the overlap, since the other half is left approximately to VJ . parameter r0 is discussed in Section 5. 242 P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 The pseudo-atom is calculated with DFT only once for a given is the effective potential evaluated at the (artificial) neutral density confining potential. This way we get ul’s (more precisely, Rl’s), n0ðrÞ of the system. The density n0 is determined by the atoms in the localized basis functions, for later use in matrix element the system, and the above matrix element between basis states l calculations. and m, in principle, depends on the positions of all atoms. However, One technical detail we want to point out here concerns orbital since the integrand is a product of factors with three centers, two conventions. Namely, once the orbitals ul are calculated, their sign wave functions and one potential (and kinetic), all of which are and other conventions should never change. Fixed convention non-zero in small spatial regions only, reasonable approximations should be used in all simultaneously used Slater–Koster tables; can be made. using different conventions for same elements gives inconsistent First, for diagonal elements Hll one can make a one-center tables that are plain nonsense. The details of our conventions, approximation where the effective potential within volume VI is along with other technical details of the pseudo-atom calculations, V ½n ðrÞ V ½n ðrÞ; ð51Þ are discussed in Appendix A. s 0 s;I 0;I where l 2 I. This integral is approximately equal to the eigenener- 3.2. Overlap matrix elements gies el of free atom orbitals. This is only approximately correct since the orbitals ul are from the confined atom, but is a reasonable Using the orbitals from pseudo-atom calculations, we need to approximation that ensures the correct limit for free atoms. calculate the overlap matrix elements Second, for off-diagonal elements we make the two-center Z approximation: if l is localized around atom I and m is localized 3 around atom J, the integrand is large when the potential is local- Slm ¼ ulðrÞ umðrÞd r: ð48Þ ized either around I or J as well; we assume that the crystal field contribution from other atoms, when the integrand has three dif- Since orbitals are chosen real, the overlap matrix is real and ferent localized centers, is small. Using this approximation the symmetric. effective potential within volume VI þ VJ becomes The integral with ul at RI and um at RJ can be calculated also with u at the origin and u at RIJ. Overlap will hence depend on V s½n0 ðrÞ V s;I½n0;I ðrÞþV s;J½n0;J ðrÞ; ð52Þ l m b RIJ , or equivalently, on RIJ and RIJ separately. Fortunately, the b where V s;I½n0;I ðrÞ is the Kohn-Sham potential with the density of a dependence on R is fully governed by Slater–Koster transformation IJ neutral atom. The Hamiltonian matrix element is rules [36]. Only one to three Slater–Koster integrals, depending on Z the angular momenta of ul and um, are needed to calculate the 0 1 2 b Hlm ¼ ulðrÞ r þ V s;IðrÞþV s;JðrÞ umðrÞ; ð53Þ integral with any RIJ for fixed RIJ. These rules originate from the 2 properties of spherical harmonics. The procedure is hence the following: we integrate numerically where l 2 I and m 2 J. Prior to calculating the integral, we have to apply the Hamiltonian to u . But in other respects the calculation the required Slater–Koster integrals for a set of RIJ , and store them m in a table. This is done once for all orbital pairs. Then, for a given is similar to overlap matrix elements: Slater–Koster transforma- tions apply, and only a few integrals have to be calculated numeri- orbital pair, we interpolate this table for RIJ, and use the Slater– Koster rules to get the overlap with any geometry—fast and cally for each pair of orbitals, and stored in tables for future accurately. reference. See Appendix C.2 for details of numerical integration of Readers unfamiliar with the Slater–Koster transformations can the Hamiltonian matrix elements. read the detailed discussion in Appendix B. The numerical integra- tion of the integrals is discussed in Appendix C. 4. Fitting the repulsive potential Before concluding this subsection, we make few remarks about non-orthogonality. In DFTB it originates naturally and inevitably In this section we present a systematic approach to fit the repul- IJ from Eq. (48), because non-overlapping orbitals with diagonal sive functions V repðRIJÞ that appear in Eq. (38)—and a systematic overlap matrix would yield also diagonal Hamiltonian matrix, way to describe the fitting. But first we discuss some difficulties re- which would mean chemically non-interacting system. The lated to the fitting process. transferability of a tight-binding model is often attributed to The first and straightforward way of fitting is simple: calculate non-orthogonality, because it accounts for the spatial nature of dimer curve EDFT ðRÞ for the element pair with DFT, require the orbitals more realistically. EDFT ðRÞ¼EDFTBðRÞ, and solve Non-orthogonality requires solving a generalized eigenvalue V ðRÞ¼E ðRÞ ½ E ðRÞþE ðRÞ : ð54Þ problem, which is more demanding than normal eigenvalue prob- rep DFT BS coul lem. Non-orthogonality complicates, for instance, also gauge trans- We could use also other symmetric structures with N bonds having formations, because the phase from the transformations is not well equal RIJ ¼ R, and require defined for the orbitals due to overlap. The Peierls substitution [37], while gauge invariant in orthogonal tight-binding [38,39],is N V repðRÞ¼EDFT ðRÞ ½ EBSðRÞþEcoulðRÞ : ð55Þ not gauge invariant in non-orthogonal tight-binding (but affects In practice, unfortunately, it does not work out. The approximations only time-dependent formulation). made in DFTB are too crude, and hence a single system is insuffi- cient to provide a robust repulsion. As a result, fitting repulsive 3.3. Hamiltonian matrix elements potentials is difficult task, and forms the most laborous part of parametrizing in DFTB. From Eq. (34) the Hamiltonian matrix elements are Let us compare things with DFT. As mentioned earlier, we tried Z to dump most of the difficult physics into the repulsive potential, 0 1 2 H ¼ u ðrÞ r þ V s½n0 ðrÞ u ðrÞ; ð49Þ lm l 2 m and hence V rep in DFTB has practical similarity to Exc in DFT. In DFTB, however, we have to make a new repulsion for each pair where of atoms, so the testing and fitting labor compared to DFT function- als is multifold. Because xc-functionals in DFT are well docu- V s½n0 ðrÞ¼V extðrÞþV H½n0 ðrÞþV xc½n0 ðrÞð50Þ mented, DFT calculations of a reasonably documented article can P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 243 Âà 0 0 0 be reproduced, whereas reproducing DFTB calculations is usually E ðRABÞ E ðRABÞþE ðRABÞ V 0 ðR Þ¼ DFT BS coul ; ð57Þ harder. Even if the repulsive functions are published, it would be rep AB N a great advantage to be able to precisely describe the fitting pro- where the prime stands for a derivative with respect to R . The eas- cess; repulsions could be more easily improved upon. AB iest way is first to calculate the energy curve and use finite differ- Our starting point is a set of DFT structures, with geometries R, DFT ences for derivatives. In fact, systems treated this way can have energies EDFTðRÞ, and forces F (zero for optimized structures). A even different RAB’s if only the ones that are equal are chosen to vary natural approach would be to fit V rep so that energies EDFTBðRÞ and DFTB (e.g. a complex system with one appropriate AB bond on its surface). forces F are as close to DFT ones as possible. In other words, we DFT DFTB For each system, this gives a family of data points for the fitting; the want to minimize force differences F F and energy dif- number of points in the family does not affect fitting, as explained ferences jjEDFT EDFTB on average for the set of structures. There later. The dimer curve, with N 1, is clearly one system where this are also other properties such as basis set quality (large overlap ¼ method can be applied. For any equilibrium DFT structure things with DFT and DFTB wave functions), energy spectrum (similarity simplify into of DFT and DFTB density of states), or charge transfer to be com- pared with DFT, but these originate already from the electronic 0 0 0 0 EBS RAB Ecoul RAB part, and should be modified by adjusting V conf ðrÞ and Hubbard V 0 R0 ¼ ; ð58Þ rep AB N U. Repulsion fitting is always the last step in the parametrizing, and affects only energies and forces. 0 0 0 where R is the distance for which E R ¼ 0. In practice we shall minimize, however, only force differ- AB DFT AB ences—we fit repulsion derivative, not repulsion directly. The fit- 4.1.2. Homonuclear systems ting parameters, introduced shortly, can be adjusted to get energy differences qualitatively right, but only forces are used in the If a cluster or a solid has different bond lengths, the energy curve method above cannot be applied (unless a subset of bonds practical fitting algorithm. There are several reasons for this. First, forces are absolute, energies only relative. For instance, since we are selected). But if the system is homonuclear, the data points can be obtained the following way. The force on atom I is do not consider spin, it is ambiguous whether to fit to DFT dimer X curve with spin-polarized or spin-paired free atom energies. We 0 0 b FI ¼ FI þ V repðRIJÞRIJ ð59Þ could think that lower-level spin-paired DFT is the best DFTB – JXI can do, so we compare to spin-paired dimer curve—but we should 0 b fit to energetics of nature, not energetics in some flavors of DFT. ¼ FI þ IJ RIJ; ð60Þ J–I Second, for faithful dynamics it is necessary to have right forces and right geometries of local energy minima; it is more important 0 where FI is the force without repulsions. Then we minimize the first to get local properties right, and afterwards look how the sum global properties, such as energy ordering of different structural X 2 motifs, come out. Third, the energy in DFTB comes mostly from FDFT;I FI ð61Þ the band-structure part, not repulsion. This means that if already I the band-structure part describes energy wrong, the short-ranged with respect to , with ¼ 0 for pair distances larger than the cut- repulsions cannot make things right. For instance, if E ðRÞ and IJ IJ DFT off. The minimization gives optimum , which can be used directly, E ðRÞ for dimer deviates already with large R, short-range IJ DFTB together with their R ’s, as another family of data points in the repulsion cannot cure the energetics anymore. For transferability IJ fitting. repulsion has to be monotonic and smooth, and if repulsion is ad- justed too rapidly catch up with DFT energetics, the forces will go wrong. 4.1.3. Other algorithms For the set of DFT structures, we will hence minimize DFT and Fitting algorithms like the ones above are easy to construct, but DFTB force differences, using the recipes below. a few general guidelines are good to keep in mind. While pseudo-atomic orbitals are calculated with LDA-DFT, the systems to fit the repulsive potential should be state-of-the-art cal- 4.1. Collecting data culations; all structural tendencies—whether right or wrong—are directly inherited by DFTB. Even reliable experimental structures To fit the derivative of the repulsion for element pair AB,we can be used as fitting structures; there is no need to think DFTB need a set of data points fR ; V 0 ðR Þg. As mentioned before, fitting i rep i should be parametrized only from theory. DFTB will not become to dimer curve alone does not give a robust repulsion, because the any less density-functional by doing so. same curve is supposed to work in different chemical environ- As data points are calculated by stretching selected bonds (or ments. Therefore it is necessary to collect the data points from sev- calculating static forces), also other bonds may stretch (dimer is eral structures, to get a representative average over different types one exception). These other bonds should be large enough to ex- of chemical bonds. Here we present examples on how to acquire clude repulsive interactions; otherwise fitting a repulsion between data points. two elements may depend on repulsion between some other ele- ment pairs. While this is not illegal, the fitting process easily be- 4.1.1. Force curves and equilibrium systems comes complicated. Sometimes the stretching can affect chemical This method can be applied to any system where all the bond interactions between elements not involved in the fitting; this is lengths between the elements equal RAB or otherwise are beyond worth avoiding, but sometimes it may be inevitable. the selected cutoff radius Rcut. In other words, the only energy com- ponent missing from these systems is the repulsion from N bonds 4.2. Fitting the repulsive potential between elements A and B with matching bond lengths. Hence, e Transferability requires the repulsion to be short-ranged, and EDFTBðRABÞ¼EBSðRABÞþEcoulðRABÞþErep þ N V repðRABÞ; ð56Þ we choose a cutoff radius Rcut for which V repðRcutÞ¼0, and also e 0 where Erep is the repulsive energy independent of RAB. This setup V repðRcutÞ¼0 for continuous forces. Rcut is one of the main allows us to change RAB, and we will require parameters in the fitting process. Then, with given Rcut, after having 244 P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 0 collected enough data points fRi; V rep;ig, we can fit the function to be independent of the number of points in each family; a fit with 0 V repðRÞ. The repulsion itself is dimer force curve should yield the same result with 10 or 100 Z points in the curve. Hence for each system Rcut 0 pffiffiffiffiffiffi V repðRÞ¼ V repðrÞdr: ð62Þ R ri ¼ rs Ns; ð64Þ 0 Fitting of V rep using the recipe below provides a robust and where rs is the uncertainty given for system s, with Ns points in the unbiased fit to the given set of points, and the process is easy to family. This means that systems with the same rs’s have the same control. We choose a standard smoothing spline [40] for significance in the process, irregardless of the number of data points V 0 ðRÞ UðRÞ, i.e. we minimize the functional rep in each system. The effect of Eq. (64) is the same as putting a weight ! 0 2 Z 1=Ns for each data point in the family. Note that k has nothing to do XM V U R Rcut rep;i ð iÞ 00 2 with the number of data points, and is more universal parameter. SU½ ¼ðRÞ r þ k U ðRÞ dR ð63Þ i¼1 i The cutoff is set by adding a data point at UðRcutÞ¼0 with a tiny r. Fig. 3 shows an example of fitting carbon–hydrogen repulsion. 0 for total M data points fRi; V rep;ig, where UðRÞ is given by a cubic The parameters Rcut and k, as well as parameters ri, are in practice spline. Spline gives an unbiased representation for UðRÞ, and the chosen to yield visually satisfying fit; the way of fitting should not smoothness can be directly controlled by the parameter k. Large k affect the final result, and in this sense it is just a technical neces- means expensive curvature and results in linear UðRÞ (quadratic sity—the simple objective is to get a smooth curve going nicely V rep) going through the data points only approximately, while small through the data points. Visualization of the data points can be also k considers curvature cheap and may result in a wiggled UðRÞ pass- generally invaluable: deviation from a smooth behavior tells how ing through the data points exactly. The parameter k is the second well you should expect DFTB to perform. If data points lay nicely parameter in the fitting process. Other choices for UðRÞ can be used, along one curve, DFTB performs probably well, but scattered data such as low-order polynomials [2], but they sometimes behave sur- points suggest, for instance, the need for improvements in the elec- prisingly while continuously tuning Rcut. For transferability the tronic part. In the next section we discuss parameter adjustment behavior of the derivative should be as smooth as possible, prefer- further. ably also monotonous (the example in Fig. 3a is slightly non-monot- onous and should be improved upon). 5. Adjusting parameters The parameters ri are the data point uncertainties, and can be used to weight systems differently. With the dimension of force, In this section we summarize the parameters, give practical ri’s have also an intuitive meaning as force uncertainties, the lengths of force error bars. As described above, each system may instructions for their adjustment, and give a demonstration of their produce a family of data points. We would like, however, the fitting usage. The purpose is to give an overall picture of the selection of knobs to turn while adjusting parametrization. Occasionally one finds published comments about the perfor- mance of DFTB. While DFTB shares flaws and failures characteristic to the method itself, it should be noted that DFTB parametrizations are even more diverse than DFT functionals. A website in Ref. [41], maintained by the original developers of the method, contains var- ious sets of parametrizations. While these parametrizations are of good quality, they are not unique. Namely, there exists no auto- mated way of parametrizing, so that a straightforward process would give all parameters definite values. This is not necessarily a bad thing, since some handwork in parametrizing also gives feel- ing what is to be expected in the future, how well parametrizations are expected to perform. 5.1. Pseudo-atoms The basis functions, and, consequently, the matrix elements are determined by the confinement potentials V conf ðrÞ containing the parameters r0 for each element. The value r0 ¼ 2 rcov, where rcov is the covalent radius, can be used as a rule of thumb [2]. Since the covalent radius is a measure for binding range, it is plausible that the range for environmental confining potential should de- pend on this scale—the number 2 in this rule is empirical. With this rule of thumb as a starting point, the quality of the ba- sis functions can be inspected and r0 adjusted by looking at (i) band-structure (for solids), (ii) densities of states (dimer or other simple molecules), or (iii) amount of data point scatter in repulsion fit (see Section 4.2, especially Fig. 3). Systems with charge transfer should be avoided herein, since the properties listed above would depend on electrostatics as well, which complicates the process; adjustment of r0 should be independent of electrostatics. Fig. 3. (a) Fitting the derivative of repulsive potential. Families of points from The inspections above are easiest to make with homonuclear various structures, obtained by stretching C–H bonds and using Eq. (57). Here Rcut ¼ 1:8 Å, for other details see Section 5. (b) The repulsive potential V repðRÞ, which interactions, even though heteronuclear interactions are more is obtained from by integration of the curve in (a). important for some elements, such as for hydrogen. Different P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 245 chemical environments can affect the optimum value of r0, but this is why the parameter k is not given any guidelines here. Large usually it is fixed for all interactions of a given element. r’s can be used to give less weight for systems with (i) marginal importance for systems of interest, (ii) inter-dependence on other 5.2. Electrostatics parameters (dependence on other repulsions, on electrostatics, or on other chemical interactions), (iii) statistically peculiar stick- Electrostatic energetics, as described in Section 2.4, are deter- ing out from the other systems (reflecting situation that cannot mined by the Hubbard U parameter, having the default value be described by tight-binding or the urge to improve electronic U ¼ IE EA. Since U is a value for a free atom, while atoms in mol- part). ecules are not free, it is permissible to adjust U in order to improve (i) charge transfer, (ii) density of states, (iii) molecular ionization 5.4. Hydrocarbon parametrization energies and electron affinities, or (iv) excitation spectra for se- lected systems. Since U is an atomic property, one should beware To demonstrate the usage of the parameters, we present the when using several different elements in fitting—charge transfer hydrocarbon parametrization used in this article (this was first depends on U’s of other elements as well. shot fitting without extensive adjustment, but works reasonably Eq. (29) relates U and FWHM of given atom together. But since well). The Hubbard U is given by Eq. (17) and FWHM by Eq. (29), Eq. (21) contains, in principle, also xc-contributions, the relation both for hydrogen and carbon. The force curves have been calcu- can be relaxed, if necessary. Since FWHM affects only pair-interac- lated with GPAW [42,43] using the PBE xc-functional [44]. tions, it is better to adjust the interactions directly like Hydrogen: r0 ¼ 1:08, U ¼ 0:395. Carbon: r0 ¼ 2:67, U ¼ 0:376. Hydrogen–carbon repulsion: Rcut ¼ 3:40, k ¼ 35, and systems with CIJ ! CIJ=xIJ; ð65Þ force curve: CH , ethyne, methane, benzene; all with ri ¼ 1. Car- where xIJ (being close to one) effectively scales both FWHMI and bon–carbon repulsion: Rcut ¼ 3:80, k ¼ 200, and systems with 2 2 2 FWHMJ. If atomic FWHMs would be changed directly, it would af- force curve: C2,CC ,C3,C4 with ri ¼ 1, and C3 with ri ¼ 0:3. fect all interactions and complicate the adjusting process. Note that FWHM affects only nearest neighbor interactions (see Fig. 1) and al- 6. External fields ready next-nearest neighbors have (very closely) the pure 1=R interaction. Including external potentials to the formalism is straightfor- An important principle, general for all parameters but particu- ward: just add one more term in the Hamiltonian of Eq. (53).Ma- larly for electrostatics, is this: all parameters should be adjusted trix element with external (scalar) potential becomes, with within reasonable limits. This means that, since all parameters plausible approximations, have a physical meaning, if a parameter is adjusted beyond a rea- Z sonable and physically motivated limit, the parametrization will in H ! H þ u ðrÞV ðrÞu ðrÞd3r general not be transferable. If a good fit should require overly large lm lm l ext m Z Z parameter adjustments (precise ranges are hard to give), the origi- nal problem probably lies in the foundations of tight-binding. Hlm þ V extðRIÞ ulum þ V extðRJÞ ulum VI VJ 1 5.3. Repulsive potentials H þ V I þ V J S ð66Þ lm 2 ext ext lm The last step in the parametrization is to fit the repulsive poten- for a smoothly varying V ðrÞ. The electrostatic part in the Hamilto- tial. Any post-adjustment of other parameters calls for re-fitting of ext nian is the repulsive potential. The most decisive part in the fitting is choosing the set of struc- 1 1 I J hlm ¼ I þ J þ V ext þ V ext ð67Þ tures and bonds to fit. Parameters Rcut, r and k are necessary, but 2 they play only a limited part in the quality of the fit—the quality and naturally extends Eq. (42). and transferability is determined by band structure energy, elec- trostatic energy, and the chosen structures. In fact, the repulsion 7. van der Waals forces fitting was designed such that the user has as little space for adjust- ment as possible. Accurate DFT xc-functionals, which automatically yield the R 6 The set of structures should contain the fitted interaction in dif- long range attractive van der Waals interactions, are notoriously ferent circumstances, with (i) different bond lengths, (ii) varying hard to make [45]. Since DFT in other respects is accurate with coordination, and (iii) varying charge transfer. In particular, if short-range interactions, it would be wrong to add van der Waals charge transfer is important for the systems of interest, calculation interactions by hand—addition inevitably modifies short-range of charged molecules is recommended. parts as well. A reasonable initial guess for the cutoff radius is R ¼ 1:5 cut DFTB, on the other hand, is more approximate, and adding R , being half-way between nearest and next-nearest neighbors dimer physically motivated terms by hand is easier. In fact, van der Waals for homonuclear systems. It is then adjusted to yield a satisfying fit forces in DFTB can conceptually be thought of as modifications of for the derivative of the repulsion, as in Fig. 3, while remembering the repulsive potential. Since dispersion forces are due to xc-con- that it has to be short ranged (R ¼ 2 R , for instance, is too cut dimer tributions, one can see that for neutral systems, where dnðrÞ 0, large, lacks physical motivation, and makes fitting hard). Increasing dispersion has to come from Eq. (10). However, in practice it is bet- R will increase V ðRÞ at given R < R , which is an aspect that cut rep cut ter to leave V IJ ’s short-ranged and add the dispersive forces as can be used to adjust energies (but not much forces). Usually the rep additional terms parameters r are used in the sense of relative weights between systems, as often they cannot be determined in the sense of abso- X CIJ E ¼ f ðR Þ 6 ð68Þ lute force uncertainties. The absolute values do not even matter, vdW IJ IJ 6 I 8. Periodic boundary conditions can be calculated with standard methods, such as Ewald summation [51], and the repulsive part, 8.1. Bravais lattices X unitX cell X IJ 1 IJ V repðRIJÞ¼ V repðRIJ TÞð81Þ Calculation of isolated molecules with DFTB is straightforward, I We define also the energy-weighted density-matrix so that X ! e a a X X X qlm ¼ eafaclcm ; ð88Þ a faqI;a ¼ ðqIÞ¼Nel: ð100Þ and symmetrized density matrix I a I 1 Population on orbitals of atom I with angular momentum l is, q~ ¼ ðq þ q Þ; ð89Þ lm 2 lm lm similarly, X which is symmetric and real. Using q we obtain simple expres- l lm q ¼ ðq~SÞ : ð101Þ sions, for example, for I ll l2Iðll¼lÞ 0 EBS ¼ TrðqH Þð90Þ X X The partial Mulliken populations introduced above are simple, but Nel ¼ Trðq~SÞ¼ q~lmSml; ð91Þ enable surprisingly rich analysis of the electronic structure, as dem- Xl Xm onstrated below. qI ¼ TrIðq~SÞ¼ q~lmSml; ð92Þ l2I m 10.2. Analysis beyond Mulliken charges where EBS is the band-structure energy, Nel is the total number of At this point, after discussing Mulliken population analysis, we electrons, and qI is the Mulliken population on atom I.TrI is partial trace over orbitals of atom I alone. comment on the role of wave functions in DFTB. Namely, internally It is practical to define also matrices’ gradients. They do not di- DFTB formalism uses atom resolution for any quantity, and the rectly relate to density matrix formulation, but equally simplify tight-binding spirit means that the matrix elements Hlm and Slm notation, and are useful in practical implementations. We define are just parameters, nothing more. Nonetheless, the elements Hlm and S are obtained from genuine basis orbitals u ðrÞ using (with rJ ¼ @=@RJ) lm l well-defined procedure—these basis orbitals remain constantly dS S I; J lm ¼ rZ J lm ðl 2 m 2 Þ available for deeper analysis. The wave functions are X u ðr R Þr u ðr R Þ¼hu jr u ið93Þ a l I J m J l J m waðrÞ¼ clulðrÞð102Þ l and and the total electron density is dH ¼ H hu jHj u iðl 2 I; m 2 JÞð94Þ X X lm rJ lm l rJ m 2 nðrÞ¼ fajwðrÞj ¼ qlmumðrÞulðrÞ; ð103Þ a lm with the properties dSlm ¼ dSml and dHlm ¼ dHml. From these definitions we can calculate analytically, for instance, the time derivative of the overlap matrix for a system in motion awaiting for inspection with tools familiar from DFT. One should, however, use the wave functions only for analysis[58]. The formal- _ ; S ¼½dS V ð95Þ ism itself is better off with Mulliken charges. But for visualization _ and for gaining understanding this is a useful possibility. This dis- with commutator ½A; B and matrix Vlm ¼ dlmRI, l 2 I. Force from the band-energy part, the first line in Eq. (43), for atom I can be ex- tinguishes DFTB from semiempirical methods, which—in princi- pressed as ple—do not possess wave functions but only matrix elements (unless made ad hoc by hand). e FI ¼ TrIðqdH q dSÞþc:c:; ð96Þ which is, besides compact, useful in implementation. The density- 10.3. Densities of states matrix formulation introduced here is particularly useful in elec- tronic structure analysis, discussed in the following section. Mulliken populations provide intuitive tools to inspect elec- tronic structure. Let us first break down the energy spectrum into 10. Simplistic electronic structure analysis various components. The complete energy spectrum is given by the density of states (DOS), X One great benefit of tight-binding is the ease in analyzing the DOSðeÞ¼ drðe e Þ; ð104Þ electronic structure. In this section we present selected analysis a a tools, some old and renowned, others casual but intuitive. Other simple tools for chemical analysis of bonding can be found from where drðeÞ can be either the peaked Dirac delta-function, or some Ref. [57]. function—such as a Gaussian or a Lorentzian—with broadening parameter r. DOS carrying spatial information is the local density 10.1. Partial Mulliken populations of states, X r 2 The Mulliken population on atom I, LDOSðe; rÞ¼ d ðe eaÞjwaðrÞj ð105Þ a X X X R 3 qI ¼ TrIðq~SÞ¼ q~lmSml ¼ ðq~SÞll ð97Þ with integration over d r yielding DOSðeÞ. Sometimes it is defined l2I m l2I as 248 P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 X 0 2 LDOSðrÞ¼ fajwaðrÞj ; ð106Þ Let us start by discussing promotion energy. When free atoms a coalesce to form molecules, higher energy orbitals get occupied— electrons get promoted to higher orbitals. This leads to the natural where f 0 are weights chosen to select states with given energies, as a definition in scanning tunneling microscopy simulations[59]. Mulliken X charges, pertinent to DFTB, yield LDOS with atom resolution, free atom 0 X Eprom ¼ qðlÞ qðlÞ Hll: ð111Þ r l LDOSðe; IÞ¼ d ðe eaÞqI;a; ð107Þ a Promotion energy is the price atoms have to pay to prepare them- which can be used to project density for group of atoms R as selves for bonding. Noble gas atoms, for instance, cannot bind to X other atoms, because the promotion energy is too high due to the LDOSRðeÞ¼ LDOSðe; IÞ: ð108Þ large energy gap; any noble gas atom could in principle promote I2R electrons to closest s-state, but the gain from bonding compared For instance, if systems consists of surface and adsorbed molecule, to the cost in promotion is too small. we can plot LDOSmolðeÞ and LDOSsurf ðeÞ to see how states are distrib- Covalent bond energy, on the other hand, is the energy reduc- uted; naturally LDOSmolðeÞþLDOSsurf ðeÞ¼LDOSðeÞ. tion from bonding. We define covalent bond energy as [61] Similar recipes apply for projected density of states, where DOS E ¼ðE E Þ E : ð112Þ is broken into angular momentum components, cov BS free atoms prom X X This definition can be understood as follows. The term PDOSðe; lÞ¼ drðe e Þ ql ; ð109Þ a I;a E E is the total gain in band-structure energy as atoms a I ð BS free atomsÞ P coalesce; but atoms themselves have to pay Eprom, an on-site price such that, again lPDOSðe; lÞ¼ DOSðeÞ. that does not enhance binding itself. Subtraction gives the gain the system gets in bond energies as it binds together. More 10.4. Mayer bond-order explicitly, X Bond strengths between atoms are invaluable chemical infor- 0 Ecov ¼ EBS qðlÞHll ð113Þ mation. Bond order is a dimensionless number attached to the X l bond between two atoms, counting the differences of electron 0 ¼ qlmðHlm elmSlmÞ; ð114Þ pairs on bonding and antibonding orbitals; ideally it is one for sin- lm gle, two for double, and three for triple bonds. In principle, any bond strength measure is equally arbitrary; in practice, some mea- where sures are better than others. A measure suitable for many purposes in DFTB is Mayer bond-order [57], defined for bond IJ as 1 X e ¼ ðH0 þ H0 Þ: ð115Þ lm 2 ll mm MIJ ¼ ðq~SÞlmðq~SÞml: ð110Þ l2I;m2J This can be resolved with respect to orbital pairs and energy as The off-diagonal elements of q~S can be understood as Mulliken X r a 0 overlap populations, counting the number of electrons in the over- Ecov;lmðeÞ¼ d ðe eaÞqlmðHml emlSmlÞ: ð116Þ lap region—the bonding region. It is straightforward, if necessary, to a partition Eq. (110) into pieces, for inspecting angular momenta or Ecov;l;m can be viewed as the bond strength between orbitals l and eigenstate contributions in bonding. Look at Refs. [60,57] for further m—with strength directly measured in energy; negative energy details, and Table 1 for examples of usage. means bonding and positive energy antibonding contributions. Sum over atom orbitals yields bond strength information for atom 10.5. Covalent bond energy pairs X X Another useful bonding measure is the covalent bond energy, Ecov;IJðeÞ¼ ðEcov;lmðeÞþc:c:Þð117Þ which is not just a dimensionless number but measures bonding l2I m2J directly using energy [61]. and sum over angular momentum pairs Table 1 l l Xl¼ a lXm¼lb ÀÁ Simplistic electronic structure and bonding analysis for selected systems: C2H2 (triple lalb E ðeÞ¼ Ecov; ðeÞþ½c:c: ð118Þ CC bond), C2H4 (double CC bond), C2H6 (single CC bond), benzene, and graphene (C- cov lm point calculation with 64 atoms). We list most energy and bonding measures l m introduced in the main text (energies in eV). gives bonding between states with angular momenta la and lb (no Property C2H2 C2H4 C2H6 Benzene Graphene c.c. for la ¼ lb). For illustration, we plot covalent bonding contribu- q 0.85 0.94 0.96 0.95 H tions in graphene in Fig. 4. AH 1.23 0.45 0.26 0.32 ABH 2.75 3.27 3.34 3.39 Eprom;H 0.97 0.41 0.25 0.30 10.6. Absolute atom and bond energies qC 4.15 4.13 4.12 4.05 4.00 AC 5.86 5.86 5.78 6.48 6.94 While Mayer bond-order and Ecov are general tools for any tight- ABC 9.16 9.74 10.45 9.74 9.78 Eprom;C 5.62 5.69 5.64 6.46 6.94 binding flavor, neither of them take electrostatics or repulsions be- BCH 8.14 7.90 7.84 7.93 tween atoms into account. Hence, to conclude this section, we BCC 22.01 15.82 9.39 13.01 12.16 introduce a new analysis tool that takes also these contributions MCH 0.96 0.95 0.97 0.96 into account. M 2.96 2.02 1.01 1.42 1.25 CC The DFTB energy with subtracted free-atom energies, P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 249 11. Conclusions Here ends our journey with DFTB for now. The road up to this point may have been long, but the contents have made it worth the effort: we have given a detailed description of a method to do realistic electronic structure calculations. Especially the trans- parent chemistry and ease of analysis makes DFTB an appealing method to support DFT simulations. With these features tight- binding methods will certainly remain an invaluable supporting method for years to come. Acknowledgements One of us (PK) is greatly indebted for Michael Moseler, for intro- ducing molecular simulations in general, and DFTB in particular. Academy of Finland is acknowledged for funding though Projects 121701 and 118054. Matti Manninen, Hannu Häkkinen, and Lars Fig. 4. Covalent bonding energy contributions in graphene (C-point calculation Pastewka are greatly acknowledged for commenting and proof- with 64 atoms in unit cell). Both bonding and antibonding ss bonds are occupied, reading the manuscript. but sp and pp have only bonding contributions. At Fermi-level (zero-energy) the pp- bonding (the p-cloud above and below graphene) transforms into antibonding (p )—hence any addition or removal of electrons weakens the bonds. Appendix A. Calculating the DFT pseudo-atom The pseudo-atom, and also the free atom without the confine- ment, is calculated with LDA-DFT [62]. More recent xc-functionals 0 E ¼ E Efree atoms ð119Þ could be used, but they do not improve DFTB parametrizations, 1 X X X whereas LDA provides a fixed level of theory to build foundation. ¼ TrðqH0Þþ c Dq Dq þ V IJ qfree atomH ; ð120Þ 2 IJ I J rep ðlÞ ll However, better DFT functionals can—and should be—used in the IJ I Visualizing AI, BIJ, and ABI gives a thorough and intuitive measure Here a ¼ 1=137:036 is the fine structure constant, of energetics; see Table 1 for illustrative examples. Note that AI, Z LDA BIJ, and ABI come naturally from the exact DFTB energy expres- V ðrÞ¼ þ V ðrÞþV ðrÞþV ðrÞð129Þ s r H xc conf sion—they are not arbitrary definitions. 250 P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 with or without the confinement, and we defined a2 MðrÞ¼1 þ ½e V ðrÞ : ð130Þ 2 nl s The reminiscent of the lower two components of Dirac equation is hji, which is the quantum number j averaged over states with dif- ferent j, using the degeneracy weights jðj þ 1Þ; a straightforward calculation gives hji¼ 1. Summarizing shortly, for given l the potential in the radial equation is the weighted average of the potentials in full Dirac the- ory, with ignored spin–orbit interaction. This scalar-relativistic treatment is a standard trick, and is, for instance, used routinely for generating DFT pseudo-potentials[64]. The pseudo-atom calculation, as described, yields the localized basis orbitals we use to calculate the matrix elements. Our conven- e tions for the real angular part Y lðh; uÞ of the orbitals u ðrÞ¼ e l RlðrÞY lðh; uÞ are shown in Table 2. The sign of RlðrÞ is chosen, as usual, such that the first antinode is positive. Appendix B. Slater–Koster transformations Consider calculating the overlap integral Z 3 Sxx ¼ pxðrÞpxðr RÞd r ð131Þ for orbital px at origin and another px orbital at R, as shown in Fig. 5a. We rotate the coordinate system passively clockwise, such that the orbital previously at R shifts to R ¼ R^z in the new coordi- nate system. Both orbitals become linear combinations of p and x Fig. 5. Illustrating overlap integral calculation. (a) Originally one px orbital locates pz in the new coordinate system, px ! px sin a þ pz cos a, and the at origin, another px orbital at R. (b) Coordinate system is rotated so that another overlap integral becomes a sum of four terms orbital shifts to R^z; this causes the orbitals in the new coordinate system to become Z linear combinations of px and pz. Hence the overlap can be calculated as the sum of ^ 2 the so-called SðpppÞ-integral in (c), and SðpprÞ-integral in (d). The integrals in (e) pxðrÞpxðr RzÞ sin a ðFig:5cÞ and (f) are zero by symmetry. Z 2 þ pzðrÞpzðr R^zÞ cos a ðFig:5dÞ where x ¼ cos a is the direction cosine of R. For simplicity we Z ð132Þ assumed y ¼ 0, but the equation above applies also for y – 0 2 2 þ pxðrÞpzðr R^zÞ cos a sin a ðFig:5eÞ (using hindsight we wrote ð1 x Þ instead of z ). The integrals Z Z ^ 3 þ pzðrÞpxðr R^zÞ cos a sin a ðFig:5fÞ: SðpprÞ¼ pzðrÞpzðr RzÞd r Z ð134Þ 3 The last two integrals are zero by symmetry, but the first two terms SðpppÞ¼ pxðrÞpxðr R^zÞd r can be written as 2 2 are called Slater–Koster integrals and Eq. (133) is called the Sxx ¼ x SðpprÞþð1 x ÞSðpppÞ; ð133Þ Slater–Koster transformation rule for the given orbital pair (orbitals Table 2 R e e Spherical functions, normalized to unity ( Yðh; uÞ sin hdhdu ¼ 1) and obtained from spherical harmonics as Y / Y lm Ylm. Note that angular momentum l remains a good quantum number for all states, but magnetic quantum number m remains a good quantum number only for s-, pz-, and d3z2 r2 -states. e Visualization Symbol Yðh; uÞ sðh; uÞ p1ffiffiffiffiffi 4p qffiffiffiffiffi pxðh; uÞ 3 4p sin h cos u qffiffiffiffiffi pyðh; uÞ 3 4p sin h sin u qffiffiffiffiffi pzðh; uÞ 3 4p cos h qffiffiffiffiffiffiffi d3z2 r2 ðh; uÞ 5 2 16pð3 cos h 1Þ qffiffiffiffiffiffiffi dx2 y2 ðh; uÞ 15 2 16p sin h cos 2u qffiffiffiffiffiffiffi dxyðh; uÞ 15 2 16p sin h sin 2u qffiffiffiffiffiffiffi dyzðh; uÞ 15 16p sin 2h sin u qffiffiffiffiffiffiffi dzxðh; uÞ 15 16p sin 2h cos u P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 251 may have different radial parts; the notation SlmðpprÞ stands for ra- tions with an asterisk can be manipulated to yield another 16 dial functions RlðrÞ and RmðrÞ in the basis functions l and m). Similar transformations and the remaining ones can be obtained by inver- reasoning can be applied for other combinations of p-orbitals as sion, which effectively changes the order of the orbitals. This inver- well—they all reduce to Slater–Koster transformation rules involv- sion changes the sign of the integral according to orbitals’ angular ing SðpprÞ and SðpppÞ integrals alone. This means that only two momenta ll and lm, integrals with a fixed R is needed for all overlaps between any llþlm two p-orbitals from a given element pair. SlmðsÞ¼SmlðsÞ ð 1Þ ; ð136Þ Finally, it turns out that 10 Slater–Koster integrals, labeled ddr, because orbital parity is ð 1Þl. ddp, ddd, pdr, pdp, ppr, ppp, sdr, spr, and ssr, are needed to trans- The discussion here concentrated only on overlap, but the Sla- form all s-, p-, and d- matrix elements. The last symbol in the nota- ter–Koster transformations apply equally for Hamiltonian matrix tion, r, p,ord, refers to the angular momentum around the elements. symmetry axis, and is generalized from the atomic notation s, p, d for l ¼ 0; 1; 2. Table 3 shows how to select the angular parts for calculating Appendix C. Calculating the Slater–Koster integrals these 10 Slater–Koster integrals. The integrals can be obtained in many ways; here we used our setup with the first orbital at the ori- C.1. Overlap integrals gin and the second orbital at R^z. This means that we set the direc- tion cosines z ¼ 1 and x ¼ y ¼ 0inTable 4, and chose orbital pairs We want to calculate the Slater–Koster integral accordingly. Finally, Table 4 shows the rest of the Slater–Koster transforma- S ðsÞ¼hu ju ið137Þ lm ls1 ms2 tions. The overlap Slm between orbitals ul at Rl ¼ 0 and um at Rm ¼ R is the sum with X Slm ¼ csSlmðsÞ; ð135Þ e u ðrÞ¼R ðrÞY ðh; uÞ¼R ðrÞH ðhÞU ðuÞð138Þ s ls1 l s1 l s1 s1 for the pertinent Slater–Koster integrals s (at most three). The gra- and dients of the matrix elements come directly from the above expres- e sion by chain rule: Slater–Koster integrals SlmðsÞ depend only on R u ðrÞ¼RmðrÞY s ðh; uÞ¼RmðrÞHs ðhÞUs ðuÞ; ð139Þ b ms2 2 2 2 and the coefficients cs only on R. e For 9 orbitals (one s-, three p-, and five d-orbitals) 81 transfor- where RðrÞ is the radial function, and the angular parts Y si ðh; uÞ are mations are required, whereas only 29 are in Table 4. Transforma- chosen from Table 3 and depend on the Slater–Koster integral s in Table 3 ^ Calculation of the 10 Slater–Koster integrals s. The first orbital (with angular part s1) is at originR and the second orbital (with angular part s2)atRz. /sðh1; h2Þ is the function 2p e e resulting from azimuthal integration of the real spherical harmonics (of Table 2) / ðh ; h Þ¼ duY ðh ; uÞY ðh ; uÞ, and is used in Eq. (142). The images in the first row s 1 2 u¼0 s1 1 s2 2 visualize the setup: orbital (o) is centered at origin, and orbital (x) is centered at R^z; shown are wave function isosurfaces where the sign is color-coded, red (light grey) is positive and blue (dark grey) negative. e e s / ðh1; h2Þ Y s1 ðh1; uÞ Y s2 ðh2; uÞ s 5 2 2 dd d 2 2 d 2 2 r 3z r 3z r 8 ð3 cos h1 1Þð3 cos h2 1Þ dd d d 15 p zx zx 4 sin h1 cos h1 sin h2 cos h2 2 2 ddd dxy dxy 15 16 sin h1 sin h2 pffiffiffiffi pdr pz d3z2 r2 15 2 4 cos h1ð3 cos h2 1Þ pffiffiffiffi pdp px dzx 45 4 sin h1 sin h2 cos h2 ppr p p 3 z z 2 cos h1 cos h2 ppp p p 3 x x 4 sin h1 sin h2 pffiffi sdr s d3z3 r2 5 2 4 ð3 cos h2 1Þ pffiffi spr spz 3 2 cos h2 ssr ss1 2 252 P. Koskinen, V. Mäkinen / Computational Materials Science 47 (2009) 237–253 Table 4 0 HlmðsÞ¼huls jH jums ið143Þ Slater–Koster transformations for s-, p-, and d-orbitals, as first published in Ref. [36]. 1 2 To shorten the notation we used a ¼ x2 þ y2 and b ¼ x2 y2. Here x, y, and z are the is mostly similar to overlap. The potentials V s;I½n0;I ðrÞ in the b 2 2 2 direction cosines of R, with x þ y þ z ¼ 1. Missing transformations are obtained by Hamiltonian manipulating transformations having an asterisk ( ) in the third column, or by inversion. Here m is u ’s angular momentum and n is u ’s angular momentum. l m 0 1 2 H ¼ r þ V s;I½n0;I ðrÞþV s;J½n0;J ðrÞð144Þ l At 0 m At R cmnr cmnp cmnd 2 ss 1 with l 2 I and m 2 J, are approximated as * spx x * pffiffiffi conf s dxy 3xy V s;I½n0;I ðrÞ V ðrÞ V conf;IðrÞ; ð145Þ pffiffiffi s;I 1 s dx2 y2 3b 2 conf 2 1 where V r is the self-consistent effective potential from the con- s d 2 2 s;I ð Þ 3z r z 2 a * 2 2 fined pseudo-atom, but without the confining potential. The reason- px px x 1 x * px py xy xy ing behind this is that while V conf ðrÞ yields the pseudo-atom and the * 0 p p xz xz pseudo-atomic orbitals, the potential V ½n ðrÞ in H should be the x z pffiffiffi s 0 2 2 px dxy 3x y yð1 2x Þ pffiffiffi approximation to the true crystal potential, and should not be aug- px dyz 3xyz 2xyz pffiffiffi mented by confinements anymore. The Hamiltonian becomes 2 2 px dzx 3x z zð1 2x Þ pffiffiffi 1 1 px dx2 y2 3xb xð1 bÞ 0 2 conf conf 2 pffiffiffi H ¼ r þ V s;I ðrÞ V conf;IðrÞþV s;J ðrÞ V conf;JðrÞð146Þ 1 py dx2 y2 3yb yð1 þ bÞ 2 2 pffiffiffi 1 pz dx2 y2 3zb zb 2 ffiffiffi and the matrix element 2 1 p p d 2 2 2 x 3z r xðz 2 aÞ 3xz pffiffiffi conf conf 2 1 2 py d3z2 r2 yðz aÞ 3yz HlmðsÞ¼e SlmðsÞþhu jV V conf;I V conf;Jju ið147Þ 2 ffiffiffi l ls1 s;J ms2 2 1 p pz d3z2 r2 zðz aÞ 3za conf conf 2 ¼ e S ðsÞþhu jV V V ju i; ð148Þ * 2 2 2 2 2 2 2 m lm ls1 s;I conf;I conf;J ms2 dxy dxy 3x y a 4x y z þ x y * 2 2 2 dxy dyz 3xy zxzð1 4y Þ xzðy 1Þ 2 conf depending whether we operate left with r =2 þ V s;I or right * 2 2 2 dxy dzx 3x yz yzð1 4x Þ yzðx 1Þ with r2=2 þ V conf ; u ’s are eigenstates of the confined atoms with 3 1 s;J l dxy d 2 2 xyb 2xyb xyb x y 2 2 eigenvalues conf (including the confinement energy contribution 3 1 el d d 2 2 yzð1 þ 2bÞ yz x y 2 yzb yzð1 þ 2 bÞ 3 1 which is then subtracted). The form used in numerical integration is dzx dx2 y2 zxb zxð1 2bÞ xzð1 bÞ p2 ffiffiffi pffiffiffi pffiffiffi 2 ZZ 2 1 2 1 2 dxy d3z2 r2 3xyðz aÞ 2 3xyz 3xyð1 þ z Þ conf pffiffiffi 2 pffiffiffi 2 pffiffiffi H S dz d R r R r / h ; h 2 1 2 1 lmðsÞ¼el lmðsÞþ q q lð 1Þ mð 2Þ sð 1 2Þ dyz d3z2 r2 3yzðz aÞ 3yzða z Þ 3yza pffiffiffi 2 pffiffiffi 2 pffiffiffi hi 2 1 2 1 dzx d3z2 r2 3xzðz aÞ 3xzða z Þ 3xza 2 2 V conf r V r V r : 149 3 2 2 2 1 2 s;J ð 2Þ conf;Ið 1Þ conf;Jð 2Þ ð Þ dx2 y2 dx2 y2 b a b z þ b 4 pffiffiffi pffiffiffi pffiffiffi 4 1 2 1 2 1 2 dx2 y2 d3z2 r2 3bðz aÞ 3z b 3ð1 þ z Þb 2 2 4 As an internal consistency check, we can operate to um also directly 2 1 2 2 3 2 d3z2 r2 d3z2 r2 ðz aÞ 3z a a 2 2 2 2 4 with r , which in the end requires just d RmðrÞ=dr , but gives otherwise similar integration. Comparing the numerical results from this and the two versions of Eqs. (147) and (148) give way to estimate the accuracy of the numerical integration. question. 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