Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction
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Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction

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This presentation was given April 27, 2013 at Ibaraki University in Mito, Japan (Professor Seiji Mori's group). The presentation does not claim to give a complete overview of the complex field of ...

This presentation was given April 27, 2013 at Ibaraki University in Mito, Japan (Professor Seiji Mori's group). The presentation does not claim to give a complete overview of the complex field of DFTB parameterization, but rather focuses on the method's central approximations and discusses its performance in various applications.

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    Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction Presentation Transcript

    • All lectures online at: http://qc.chem.nagoya-u.ac.jp/presentations.html
    • 2 Density-Functional Tight-Binding (DFTB) as fast approximate DFT method Helmut Eschrig Gotthard Seifert Thomas Frauenheim Marcus Elstner Lecture II: Introduction to the Density-Functional Tight-Binding (DFTB) Method
    • 3 Density-Functional Tight-Binding 1. Tight-Binding 2. Density-Functional Tight-Binding (DFTB) 3. Bond Breaking in DFTB 4. Extensions 5. Performance and Applications
    • 4 Density-Functional Tight-Binding 1. Tight-Binding
    • 55 Lecture II 1. Tight-Binding Resources 1. http://www.dftb.org 2. DFTB Porezag, D., T. Frauenheim, T. Köhler, G. Seifert, and R. Kaschner, Construction of tight-binding-like potentials on the basis of density-functional theory: application to carbon. Phys. Rev. B, 1995. 51: p. 12947-12957. 3. DFTB Seifert, G., D. Porezag, and T. Frauenheim, Calculations of molecules, clusters, and solids with a simplified LCAO-DFT-LDA scheme. Int. J. Quantum Chem., 1996. 58: p. 185-192. 4. SCC-DFTB Elstner, M., D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Self-consistent-charge density-functional tight- binding method for simulations of complex materials properties. Phys. Rev. B, 1998. 58: p. 7260-7268. 5. SCC-DFTB-D Elstner, M., P. Hobza, T. Frauenheim, S. Suhai, and E. Kaxiras, Hydrogen bonding and stacking interactions of nucleic acid base pairs: A density-functional-theory based treatment. J. Chem. Phys,, 2001. 114: p. 5149-5155. 6. SDFTB Kohler, C., G. Seifert, U. Gerstmann, M. Elstner, H. Overhof, and T. Frauenheim, Approximate density-functional calculations of spin densities in large molecular systems and complex solids. Phys. Chem. Chem. Phys., 2001. 3: p. 5109- 5114. 7. DFTB3 Gaus, M.; Cui, C.; Elstner, M. DFTB3: Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method (SCC-DFTB). J. Chem. Theory Comput., 2011. 7: p. 931-948.
    • 6 Standalone fast and efficient DFTB implementation with several useful extensions of the original DFTB method. It is developed at the Bremen Center for Computational Materials Science (Prof. Frauenheim, Balint Aradi) and is the successor of the old Paderborn DFTB and Dylax codes. Free for non-commercial use. DFTB+ as part of Accelrys' Materials Studio package, providing a user friendly graphical interface and the possibility to combine DFTB with other higher or lower level methods. DFTB integrated in the ab initio DFT code deMon DFTB in the Gaussian code Amber is a package of molecular simulation programs distributed by UCSF, developed mainly for biomolecular simulations. The current version of Amber includes QM/MM support, whereby part of the system can be treated quantum mechanically, and DFTB is among the quantum mechanical methods available. Amber also has a stand-alone (pure QM) implementation. CHARMm (Chemistry at HARvard Macromolecular Mechanics) DFTB integrated in the Amsterdam Density Functional (ADF) program suite. DFTB+ DFTB+/Accelrys deMon GAUSSIAN G09 AMBER CHARMm ADF Implementations Lecture II 1. Tight-Binding
    • 7 • Tight binding (TB) approaches work on the principle of treating electronic wavefunction of a system as a superposition of atom-like wavefunction (known to chemists as LCAO approach) • Valence electrons are tightly bound to the cores (not allowed to delocalize beyond the confines of a minimal LCAO basis) • Semi-empirical tight-binding (SETB): Hamiltonian Matrix elements are approximated by analytical functions (no need to compute integrals) • TB energy for N electrons, M atoms system: • This separation of one-electron energies and interatomic distance- dependent potential vj,k constitutes the TB method Tight-Binding Lecture II 1. Tight-Binding
    • 8 are eigenvalues of a Schrodinger-like equation • solved variationally using atom-like (minimum, single-zeta) AO basis set, leading to a secular equation: where H and S are Hamiltonian and overlap matrices in the basis of the AO functions. In orthogonal TB, S = 1 (overlap between atoms is neglected) • H and S are constructed using nearest-neighbor relationships; typically only nearest-neighbor interactions are considered: Similarity to extended Hückel method Tight-Binding Lecture II 1. Tight-Binding
    • 9 • Based on approximation by W. Wolfsberg and L. J. Helmholz (1952) H Ci = i S Ci • H – Hamiltonian matrix constructed using nearest neighbor relationships • Ci – column vector of the i-th molecular orbital coefficients • i – orbital energy • S – overlap matrix • H - choose as a constant – valence shell ionization potentials • H = K S (H + H )/2 • K – Wolfsberg Helmholz constant, typically 1.75 Extended Huckel (EHT) Method Lecture II 1. Tight-Binding
    • Categories of TB approaches 10 Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
    • Slater-Koster (SK) Approximation (I) 11 Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
    • SK Approximation (II) 12 Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
    • SK Approximation (III) 13 Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
    • SK Approximation (IV) 14 Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
    • 15 SK Tables Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
    • 16 Density-Functional Tight-Binding 2. Density-Functional Tight-Binding (DFTB)
    • 17 DFTB MethodQuick review I Taken from Oliviera, Seifert, Heine, Duarte, J. Braz. Chem. Soc. 20, 1193-1205 (2009) ...open access Thomas Heine Helio Duarte
    • 18 DFTB MethodQuick review II Density Functional Theory (DFT) 2 3 1 3 3 , 1 1 '1 ' 2 ' '1 1 ' 2 ' 2 M i i ext i i N xc M i i rep i r E n v r d r r r Z Zr r E d rd r r r R R n E          at convergence: Various criteria for convergence possible: • Electron density • Potential • Orbitals • Energy • Combinations of above quantities Walter Kohn/John A. Pople 1998
    • 19 DFTB MethodQuick review III Phys. Rev. B, 39, 12520 (1989) Foulkes + Haydock Ansatz
    • 20 Self-consistent-charge density-functional tight-binding (SCC-DFTB) M. Elstner et al., Phys. Rev. B 58 7260 (1998) E r[ ] = ni fi ˆH r0[ ] fi i valence orbitals å 1    + ni fi ˆH r0[ ] fi i core orbitals å 2    + Exc r0[ ] 3  - 1 2 r0VH r0[ ] R3 ò 4    - - r0Vxc r0[ ] R3 ò 5    + Enucl 6 + 1 2 r1VH r1[ ] R3 ò 7    + 1 2 d2 Exc dr1 2 r0 r1 2 R3 òò 8    + o 2( ) Approximate density functional theory (DFT) method! Second order-expansion of DFT energy in terms of reference density 0 and charge fluctuation 1 ( 0 + 1) yields: Density-functional tight-binding (DFTB) method is derived from terms 1-6 Self-consistent-charge density-functional tight-binding (SCC-DFTB) method is derived from terms 1-8 o(3) Lecture II DFTB
    • 21 DFTB and SCC-DFTB methods  where  ni and i — occupation and orbital energy ot the ith Kohn-Sham eigenstate  Erep — distance-dependent diatomic repulsive potentials  qA — induced charge on atom A  AB — distance-dependent charge-charge interaction functional; obtained from chemical hardness (IP – EA) Lecture II DFTB
    • 22 DFTB method  Repulsive diatomic potentials replace usual nuclear repulsion energy  Reference density 0 is constructed from atomic densities  Kohn-Sham eigenstates i are expanded in Slater basis of valence pseudoatomic orbitals i  The DFTB energy is obtained by solving a generalized DFTB eigenvalue problem with H0 computed by atomic and diatomic DFT r0 = r0 A A atoms å fi = cmicm m AO å H0 C = SCe with Smn = cm cn Hmn 0 = cm ˆH r0 M ,r0 N [ ] cn Lecture II DFTB
    • 23 Traditional DFTB concept: Hamiltonian matrix elements are approximated to two-center terms. The same types of approximations are done to Erep. From Elstner et al., PRB 1998 0 0 (Density superposition) (Potential superposition) eff eff A B eff eff A eff B V V V V V A B D C A B D C Situation I Situation II Both approximations are justified by the screening argument: Far away, neutral atoms have no Coulomb contribution. Approximations in the DFTB Hamiltonian Lecture II DFTB
    • SCC-DFTB matrix elements 24 LCAO ansatz of wave function Rr i i c secular equations 0SHc i i variational principle pseudoatomic orbital Atom 1 – 4 are the same atom & have only s shell 1 4 2 3 r12 r23 r14 r34 r13 r24 How to construct? two-center approximation nearest neighbor off-diagonal elements only Hamiltonian Overlap pre-computed parameter •Reference Hamiltonian H0 •Overlap integral Sμν
    • SCC-DFTB matrix elements 25 LCAO ansatz of wave function Rr i i c secular equations 0SHc i i variational principle pseudoatomic orbital H11 H22 H33 H44 Atom 1 – 4 are the same atom & have only s shell Diagonal term Orbital energy of neutral free atom (DFT calculation) 1 4 2 3 r12 r23 r14 r34 r13 r24 Hamiltonian Overlap qH 2 1 Charge-charge interaction function Induced charge
    • SCC-DFTB matrix elements 26 LCAO ansatz of wave function Rr i i c secular equations 0SHc i i variational principle pseudoatomic orbital H11 H22 H33 H41 H44 Atom 1 – 4 are the same atom & have only s shell 1 4 2 3 r12 r23 r14 r34 r13 r24 r14 Two-center integral qSHH 2 10 Charge-charge interaction function Induced charge Hamiltonian Overlap Lookup tabulated H0 and S at distance r
    • r14 SCC-DFTB matrix elements 27 LCAO ansatz of wave function Rr i i c secular equations 0SHc i i variational principle pseudoatomic orbital H11 H22 H33 H41 H43 H44 Atom 1 – 4 are the same atom & have only s shell 1 4 2 3 r12 r23 r34 r13 r24 r34 Two-center integral qSHH 2 10 Charge-charge interaction function Induced charge Hamiltonian Overlap Repeat until building off-diagonal term Lookup tabulated H0 and S at distance r
    • 28 DFTB parameters Lecture II DFTB
    • DFTB repulsive potential Erep Which molecular systems to include? Development of (semi- )automatic fitting: •Knaup, J. et al., JPCA, 111, 56 37, (2007) •Gaus, M. et al., JPCA, 113, 11 866, (2009) •Bodrog Z. et al., JCTC, 7, 2654, (2011) 29 Lecture II DFTB
    • 30  Additional induced-charges term allows for a proper description of charge-transfer phenomena  Induced charge qA on atom A is determined from Mulliken population analysis  Kohn-Sham eigenenergies are obtained from a generalized, self-consistent SCC-DFTB eigenvalue problem SCC-DFTB method (I) Lecture II DFTB
    • 31 SCC-DFTB method (II) Basic assumptions: •Only transfer of net charge between atoms •Size and shape of atom (in molecule) unchanged Only second-order terms (terms 7-8 on slide 16): Lecture II DFTB
    • 32 SCC-DFTB method (III) Lecture II DFTB
    • 33 SCC-DFTB method (IV) Several possible formulations for : Mataga-Nishimoto < Klopmann-Ohno < DFTB Klopmann-Ohno: Lecture II DFTB
    • 34 Gradient for the DFTB methods The DFTB force formula The SCC-DFTB force formula computational effort: energy calculation 90% gradient calculation 10% Fa = - ni cmicni ¶Hmn 0 ¶a -ei ¶Smn ¶a é ë ê ù û ú mn AO å i MO å - ¶Erep ¶a Lecture II DFTB
    • 35 Spin-polarized DFTB (SDFTB) Lecture II DFTB  for systems with different and spin densities, we have  total density = +  magnetization density S = -  2nd-order expansion of DFT energy at ( 0,0) yields E r,rS [ ]= ni fi ˆH r0[ ] fi i valence orbitals å 1    + ni fi ˆH r0[ ] fi i core orbitals å 2    + Exc r0[ ] 3  - 1 2 r0VH r0[ ] R3 ò 4    - - r0Vxc r0[ ] R3 ò 5    + Enucl 6 + 1 2 r1VH r1[ ] R3 ò 7    + 1 2 d2 Exc dr1 2 r0 ,0( ) r1 2 R3 òò 8    + 1 2 d2 Exc drS ( ) 2 r0 ,0( ) rS ( ) 2 R3 òò 9    + o 2( ) The Spin-Polarized SCC-DFTB (SDFTB) method is derived from terms 1-9 o(3)
    • where pA l — spin population of shell l on atom A WA ll’ — spin-population interaction functional  Spin populations pA l and induced charges qA are obtained from Mulliken population analysis 36 Spin-polarized DFTB (SDFTB) Lecture II DFTB
    • 37 Spin-polarized DFTB (SDFTB) Lecture II DFTB  Kohn-Sham energies are obtained by solving generalized, self-consistent SDFTB eigenvalue problems where H- C- = SC- e- H¯ C¯ = SC¯ e¯ M,N,K: indexing specific atoms
    • 38 SCC-DFTB w/fractional orbital occupation numbers 1 2 2 tot i i rep i E f E q q 0vi i v c H S Fractional occupation numbers fi of Kohn-Sham eigenstates replace integer ni TB-eigenvalue equation Lecture II DFTB E 2fi 0 1 2 Finite temperature approach (Mermin free energy EMermin) 1 exp / 1 i i B e f k T 2 ln 1 ln 1e B i i i i i S k f f f f Te: electronic temperature Se: electronic entropy 0 1 2 N rep i i i i i EH H S F f c c q q SR R R R      0 1i f Atomic force M. Weinert, J. W. Davenport, Phys. Rev. B 45, 13709 (1992) EMermin = Etot - TeSe
    • 39 Fermi-Dirac distribution function: Energy derivative for Mermin Free Energy M. Weinert, J. W. Davenport, Phys. Rev. B 45, 13709 (1992) elect TS HF pulay charge TS i i i i i i i i i i i e i i i i F F F F F f x f T f x x f S f x x x     elect HF pulay charge i i i i i i i i i F F F F f f f x x x    Correction term arising from Fermi distribution function cancels out Lecture II DFTB
    • 0 1 2 3 4 5 0 20 40 60 80 40 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 Time[ps] Time[ps]Time[ps] Te= 0K Te= 1500K Te= 10kK Te=0 K always yields SCC convergence problemSCC iterations(time) Maximum iteration number is 70 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 2 4 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 (A) H10C60 Fe38 (B) Fe13C10 (C) Fe6C2 kbTe(10kK) ~0.87 eV ~half-width of 3d band in Fe38 DFTBLecture II
    • 41 Density-Functional Tight-Binding 3. Bond Breaking in DFTB
    • Bond breakingLecture II SCC-DFTB and SDFTB Dissociation of H2 +
    • Bond breakingLecture II SCC-DFTB and SDFTB Dissociation of H2 +
    • Bond breakingLecture II SCC-DFTB and SDFTB Dissociation of H2 + (correct) (wrong)
    • Bond breakingLecture II SCC-DFTB and SDFTB Dissociation of H2 M. Lundberg, Y. Nishimoto, SI, Int. J. Quant. Chem. 112, 1701 (2012)
    • 46 Density-Functional Tight-Binding 4. Extensions
    • 47 Extensions M=Sc, Ti, Fe, Co, Ni X=M,CHON d-elements with X partners: Geometries very good - “ballpark” energies only Recommendation: Use ONIOM(QM:QM), esp. ONIOM(DFT:DFTB) Lecture II
    • 48/25 New Confining Potentials Wa Conventional potential r0 Woods-Saxon potential k R r rV 0 )( R0 = 2.7, k=2 )}(exp{1 )( 0 rra W rV r0 = 3.0, a = 3.0, W = 3.0 Typically, electron density contracts under covalent bond formation. In standard ab initio methods, this problem can be remedied by including more basis functions. DFTB uses minimal valence basis set: the confining potential is adopted to mimic contraction • •+ • • 1s σ1s H H H2 Δρ = ρ – Σa ρa H2 difference density 1s Henryk Witek Electronic Parameters DFTB Parameterization 48
    • 2). DFTB band structure fitting •Optimization of parameter sets for Woods-Saxon confining potential (orbital and density) and unoccupied orbital energies •Fixed orbital energies for electron occupied orbitals •Valence orbitals : [1s] for 1st row [2s, 2p] for 2nd row [ns, np, md] for 3rd – 6th row (n ≥ 3, m = n-1 for group 1-12, m = n for group 13-18) •Fitting points : valence bands + conduction bands (depending on the system, at least including up to ~+5 eV with respect to Fermi level) Electronic Parameters DFTB Parameterization 1). DFT band structure calculations •VASP 4.6 •One atom per unit cell •PAW (projector augmented wave) method •32 x 32 x 32 Monkhorst-Pack k-point sampling •cutoff = 400 eV •Fermi level is shifted to 0 eV 49
    • Band structure for Se (FCC) Brillouin zone 50 Electronic Parameters DFTB Parameterization
    • Particle swarm optimization (PSO) Electronic Parameters DFTB Parameterization 51
    • 1) Particles (=candidate of a solution) are randomly placed initially in a target space. 2) – 3) Position and velocity of particles are updated based on the exchange of information between particles and particles try to find the best solution. 4) Particles converges to the place which gives the best solution after a number of iterations. • • • • • • • • •• • • •• •• • • •• • •• ••• • • • • •• •••••••• particle 1) 4) 2) 3) Particle Swarm Optimization DFTB Parameterization 52
    • Each particle has randomly generated parameter sets (r0, a, W) within some region Generating one-center quantities (atomic orbitals, densities, etc.) “onecent” Computing two-center overlap and Hamiltonian integrals for wide range of interatomic distances “twocent” “DFTB+” Calculating DFTB band structure Update the parameter sets of each particle Memorizing the best fitness value and parameter sets Evaluating “fitness value” (Difference DFTB – DFT band structure using specified fitness points) “VASP” DFTB Parameterization orbital a [2.5, 3.5] W [0.1, 0.5] r0 [3.5, 6.5] density a [2.5, 3.5] W [0.5, 2.0] r0 [6.0, 10.0] Particle Swarm Optimization 53
    • Example: Be, HCP crystal structure DFTB Parameterization Total density of states (left) and band structure (right) of Be (hcp) crystral structure 2.286 3.584 •Experimental lattice constants •Fermi energy is shifted to 0 eV 54 Electronic Parameters
    • 55 Band structure fitting for BCC crystal structures •space group No. 229 •1 lattice constant (a) Transferability checked (single point calculation) Reference system in PSO Experimental lattice constants available No POTCAR file for Z ≥ 84 in VASPa 55
    • 56 Band structure fitting for FCC crystal structures Reference system in PSO Experimental lattice constants available •space group No. 225 •1 lattice constant (a) a Transferability checked (single point calculation) 56
    • 57 Band structure fitting for SCL crystal structures Reference system in PSO Experimental lattice constants available •space group No. 221 •1 lattice constant (a) a Transferability checked (single point calculation) 57
    • 58 Band structure fitting for HCP crystal structures Reference system in PSO Experimental lattice constants available •space group No. 194 •2 lattice constants (a, c) c a Transferability checked (single point calculation) 58
    • 59 Band structure fitting for Diamond crystal structures Reference system in PSO Experimental lattice constants available •space group No. 227 •1 lattice constant (a) a Transferability checked (single point calculation) 59
    • DFTB ParameterizationTransferability of optimum parameter sets for different structures Artificial crystal structures can be reproduced well e.g. : Si, parameters were optimized with bcc only W (orb) 3.33938 a (orb) 4.52314 r (orb) 4.22512 W (dens) 1.68162 a (dens) 2.55174 r (dens) 9.96376 εs -0.39735 εp -0.14998 εd 0.21210 3s23p23d0 bcc 3.081 fcc 3.868 scl 2.532 diamond 5.431 Parameter sets: Lattice constants:bcc fcc scl diamond Expt. 60
    • Influence of virtual orbital energy (3d) to Al (fcc) band structure OPT The bands of upper part are shifted up constantly as orb (3d) becomes larger61
    • Influence of W(orb) to Al (fcc) band structure OPT The bands of upper part go lower as W(orb) becomes larger 62
    • Influence of a(orb) to Al (fcc) band structure OPT Too small a(orb) gives the worse band structure 63
    • Influence of r(orb) to Al (fcc) band structure OPT r(orb) strongly influences DFTB band structure 64
    • Correlation of r(orb) vs. atomic diameter Atomic Number Z Atomicdiameter[a.u.] Empirically measured radii (Slater, J. C., J. Chem. Phys., 41, 3199-3204, (1964).) Calculated radii with minimal- basis set SCF functions (Clementi, E. et al., J. Chem. Phys., 47, 1300-1307, (1967).) Expected value using relativistic Dirac-Fock calculations (Desclaux, J. P., Atomic Data and Nuclear Data Tables, 12, 311-406, (1973).) This work r(orb) In particular for main group elements, there seems to be a correlation between r(orb) and atomic diameter. 65 DFTB ParameterizationElectronic Parameters
    • Straightforward application to binary crystal structures Rocksalt (space group No. 225) •NaCl •MgO •MoC •AgCl … •CsCl •FeAl … B2 (space group No. 221) Zincblende (space group No. 216) •SiC •CuCl •ZnS •GaAs … Others •Wurtzite (BeO, AlO, ZnO, GaN, …) •Hexagonal (BN, WC) •Rhombohedral (ABCABC stacking sequence, BN)  more than 100 pairs tested 66
    • Selected examples for binary crystal structures element name Ga, As hyb-0-2 B, N matsci-0-2 Reference of previous work : •d7s1 is used in POTCAR (DFT) Further improvement can be performed for specific purpose but this preliminary sets will work as good starting points 67
    • 68 Extensions Gab DFTB = ¶2 Erep ¶a¶b + 2 ni Umi b cmmcni ¶Hmn 0 ¶a -ei ¶Smn ¶a é ë ê ù û ú mn AO å im MO å + + ni cmicni ¶2 Hmn 0 ¶a¶b -ei ¶2 Smn ¶a¶b - ¶ei ¶b ¶Smn ¶a é ë ê ù û ú mn AO å i MO å ( ) ( ) ( ) ååå å å å å å å å å å å D D+D D +DD+ +ú û ù ê ë é D +-D÷ ø ö ç è æ +-- - ú ú û ù ê ê ë é ÷ ø ö ç è æ -D+++ + ú ú û ù ê ê ë é ÷ ø ö ç è æ -D+++= atomsatomsatoms 2 MO AO atoms atoms MO AO 2atoms02 MO AO atoms0 rep 2 DFTB-SCC 2 1 2 1 2 1 2 1 2 K K A AK K K AAK K KA AK i K K K NKMKK NKMKi iii i i K KNKMKiii im i K KNKMKim b miiab b q q a q b q a qq ba a S b q q bbb ccn ba S q ba H ccn a S q a H ccUn ba E G ¶ ¶ ¶ ¶g ¶ ¶ ¶ ¶g ¶¶ g¶ ¶ ¶ ¶ ¶ gg ¶ ¶g ¶ ¶g ¶ ¶e ¶¶ ¶ egg ¶¶ ¶ ¶ ¶ egg ¶ ¶ ¶¶ ¶ mn mn nm mn mnmn nm mn mnmn nm Analytical Hessian for DFTB and SCC-DFTB H. A. Witek, S. Irle, K. Morokuma, J. Chem. Phys. 121, 5163 (2005) Lecture II
    • 69 Extensions  To determine response of molecular orbitals to nuclear perturbation, one has to solve a set of iterative coupled- perturbed SCC-DFTB equations Coupled Perturbed SCC-DFTB Equations Lecture II
    • 70 Frequency calculations performance test all-trans polyenes CnHn+2 Performance of DFTBLecture II
    • 71 How to treat interlayer interactions in graphite cheaply? Addition of empirical London dispersion term! • Ahlrichs et al., Chem. Phys. 19, 119 (1977): HFD (Hartree-Fock Dispersion), Ci parameters from expt. • Mooij et al., J. Phys. Chem. A 103, 9872 (1999): PES with London dispersion, C6 parameters from MP2 calculations • Elstner et al., J. Chem. Phys. 114, 5149 (2001): SCC-DFTB-D (dispersion- augmented SCC-DFTB), C6 parameters via Halgren,J. Am. Chem. Soc. 114, 7872 (1992) from expt. • Yang et al., J. Chem. Phys. 116, 515 (2002): DFT+vdW (DFT including van der Waals interactions), C6 parameters from expt. • Grimme, J. Comput. Chem. 25, 1463 (2004): DFT-D, C6 parameters from expt. • … several more plus many reviews and benchmarks Etot = EQM - f (R ) C6 /R6 (+1/R8 etc. terms) f (R ): damping function Fritz London 1/R6: 1922 ExtensionsLecture II
    • 72 Extensions London Dispersion Lecture II
    • 73 DFTB+Dispersion 4 7 , 66 3 3 , , , 2 2 , , 6 1 ( , ) * ( , ), , cubic mean rule 3 ( , 1 exp 3.0* ,damping funct ) ( ion ) * 2 , ij ij vd dispersio w tot DFTB dispersion ij i vdw j vdw ij vdw i vdw n i j j vdw i j E E E i j C i j r R R R E i j f r R R C f R i j a a a < æ öæ öæ ö ç ÷ç ÷= - - ç ÷ç ÷ç ÷ç ÷è = + = - + = èè = ø øø + å ( ) ( ) 1/ 21/ 2 / / , : atomic polarizability parameer , : Slater Kirkwood effective number of electrons i i j j i j i j N N N N a a a + Elstner et al., JCP 114, 5149 (2001) Lecture IX Extensions
    • 74 E ~ 1/R6 damping f(R ) = [1-exp(-3(rij/Rij,vdW )7)]3 Rij,vdW = 3.8 Å (for 1st row), 4.8 A (for 2nd row) DFTB+Dispersion Elstner et al., JCP 114, 5149 (2001) DFTB-D choice of C6 parameters: - generally hybridization dependent (i.e. not simply atomic values) - use “empirical” values for parameters to match BSSE-corrected MP2 interaction energies R [Å] E [ha] ExtensionsLecture II
    • 75 Application to Graphite Extensions DFTB-D Lecture II
    • Principles of GRRM ADD Anharmonic downward distortion S. Maeda and K. Ohno, J. Phys. Chem. A, 2005, 109, 5742. 76 UDC DDC One of the most powerful and reliable methods to find reaction pathways. Extensions GRRM with DFTB Lecture II
    • Example: Formaldehyde (RB3LYP/6-31G(d)) 4EQs 9TSs 3DDCs 9UDCs DDC at DFTB DFTB B3LYP Match EQ 4 4 4 TS 9 9 6 ~ 8 DDC 4 3 3 UDC 7 9 7 : Reproduced by DFTB : Found TS at DFTB 77 ExtensionsLecture II
    • 78 Density-Functional Tight-Binding 5. Performance and Applications
    • Performance for small organic molecules (mean absolut deviations) • Reaction energies: ~ 5 kcal/mole • Bond-lenghts: ~ 0.014 A° • Bond angles: ~ 2° •Vib. Frequencies: ~6-7 % 79 Performance SCC-DFTB: general comparison with experiment Lecture II
    • 80 Performance Accuracy of DFTB Geometries and Energies for Fullerene Isomers Fullerene Isomers Geometries and Energies vs B3LYP/6-31G(d) G. Zheng, SI, M. Elstner, K. Morokuma, Chem. Phys. Lett. 412, 210 (2005) RMS [Å] NCC-DFTB SCC-DFTB AM1 PM3 C20-C36 0.025 0.019 0.035 0.030 C60-C86 0.014 0.014 0.016 0.015 R2 (lin. reg.) NCC-DFTB SCC-DFTB AM1 PM3 C20-C36 0.88 0.93 0.77 0.73 C60-C86 0.97 0.98 0.86 0.84 Geometries Energetics •102 Fullerene Isomers • small cage non-IPR C20-C36 (35), large cage IPR C70-C86 (67) R2: Energy linear regression between E(Method) and E(B3LYP) Lecture II
    • What about non-cage carbon cluster structures? Some C28 isomers as example (from Scuseria, CPL 301, 98 (1999)) PerformanceLecture II
    • Performance What about non-cage carbon cluster structures? Some C28 isomers as example TACC’04 Symp. Proceedings AM1 and PM3 are performing very bad! DFTB includes effects of polarization functions through parameterization Wrong minimum structures! Lecture II
    • 83 Performance Accuracy of DFTB Geometries Table 3. The relative energies (in eV) of 11 different isomeric singlet structures of C28 calculated with B3LYP/6-31G(d), and DFTB. Structure a B3LYP/6-31G(d) DFTB B3LYP/6-31G(d) // DFTB buckyD2 0.00 0.00 0.00 ring 3.32 8.10 3.43 c24-6 3.17 3.56 3.66 2+2r14 5.08 9.66 5.22 2+2r16 6.01 10.25 6.13 c20-6o 5.41 5.52 5.96 c20-6 m 5.57 5.62 6.09 2 + 4 7.97 10.28 8.52 central7 5.86 6.07 6.47 8 + 8 7.43 9.43 7.41 4 + 4 9.91 14.27 10.20 R2 b 0.75 0.99 a Structures illustrated below, with the labels taken from “Portmann, S.; Galbraith, J. M.; Schaefer, H. F.; Scuseria, G. E.; Lüthi, H. P. Chem. Phys. Lett. 1999, 301, 98- 104.” b Squared correlation coefficients R2 in the linear regression analysis with B3LYP/6-31G(d) energies. Lecture II
    • 84 Performance Accuracy of DFTB frequencies 84 calculated frequencies scaled frequencies Testing set of 66 molecules, 1304 distinct vibrational modes Lecture II
    • 85 Performance Accuracy of DFTB frequencies Testing set of 66 molecules, 1304 distinct vibrational modes mean absolute deviation standard deviation maximal absolute deviation scaling factor SCC-DFTB 56 cm-1 82 cm-1 529 cm-1 0.9933 DFTB 60 cm-1 87 cm-1 536 cm-1 0.9917 AM1 69 cm-1 95 cm-1 670 cm-1 0.9566 PM3 74 cm-1 102 cm-1 918 cm-1 0.9762 HF/cc-pVDZ 30 cm-1 49 cm-1 348 cm-1 0.9102 BLYP/cc-pVDZ 34 cm-1 47 cm-1 235 cm-1 1.0043 B3LYP/cc-pVDZ 29 cm-1 42 cm-1 246 cm-1 0.9704 Lecture II
    • triphenylene Energy, frequencies, and Raman and IR intensities calculations DFT BLYP/cc-pVDZ Linux 2.4GHz machine 32 hours SCC-DFTB Linux 333MHz machine 24 seconds Performance Efficiency of DFTB frequencies Lecture II
    • Performance Improvement of DFTB ERep H. A. Witek, et al, J. Theor. Comp. Chem. 4, 639 (2005) and others Lecture II
    • 88 H. A. Witek, SI, G. Zheng, W. A. de Jong, K. Morokuma, J. Chem. Phys. 125, 214706 (2005) Performance C28 D2 Harmonic IR and Raman Spectra of C28 Lecture II
    • 89 H. A. Witek, SI, G. Zheng, W. A. de Jong, K. Morokuma, J. Chem. Phys. 125, 214706 (2005) Performance Harmonic IR and Raman Spectra of C60 C60 D2h~Ih Lecture II
    • 90 H. A. Witek, SI, G. Zheng, W. A. de Jong, K. Morokuma, J. Chem. Phys. 125, 214706 (2005) Performance Harmonic IR and Raman Spectra of C70 C70 D5h Lecture II
    • 91 Performance SCC-DFTB: Systematic comparison with other methods by Walter Thiel et al. Thiel and coworkers, J. Phys. Chem. A 111, 5751 (2007) (reference: exptl. H0) Lecture II
    • 92 Performance Thiel and coworkers, J. Phys. Chem. A 111, 5751 (2007) SCC-DFTB: Systematic comparison with other methods by Walter Thiel et al. Lecture II
    • 93 Performance But: NCC- and SCC-DFTB for radicals: own comparison with B3LYP/6-311G** Lecture II
    • 94 Performance NCC- and SCC-DFTB for radicals: own comparison NCC-DFTB B3LYP/6-311G(d,p) SCC-DFTB B3LYP/6-311G(d,p) Lecture II
    • 95 A B C DFT:PW91[1] -6.24 -5.63 -1.82 SCC-DFTB[2] -5.17 -4.68 -1.86 Adhesion energies (eV/atom) A B C PW91: An ultrasoft pseudopotential with a plane-wave cutoff of 290 eV for the single metal and the projector augmented wave method with a plane-wave cutoff of 400 eV for the metal cluster Fe-Fe and Fe-C DFTB parameters from: G. Zheng et al., J. Chem. Theor. Comput. 3, 1349 (2007) [1] Phys. Rev. B 75, 115419 (2007) [2] Fermi broadening=0.13 eV H10C60Fe H10C60Fe H10C60Fe55 Fe55 icosahedron H-terminated (5,5) armchair SWNT + Fe atom/cluster PerformanceLecture II
    • 96 Acknowledgements • Marcus Elstner • Jan Knaup • Alexey Krashenninikov • Yasuhito Ohta • Thomas Heine • Keiji Morokuma • Marcus Lundberg • Yoshio Nishimoto • Some others Acknowledgements Lecture II