Recent Developments and Applications of the DFTB Method Keiji Morokuma Cherry L. Emerson Center for Scientific Computation and Department of Chemistry, Emory University, Atlanta, GA (from 9/06, also Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto, Japan) A Symposium on DFTB: Theory and Applications 232st ACS National Meeting San Francisco, September 10-14, 2006 closed

Emory Group Dr. Guishan Zheng (Gaussian, analytical functions, TM, nano) Prof. Henryk Witek (Hessian,TM) Dr. Stephan Irle (nano,TM) Zhi Wang (nano) Benjamin Finck (nano) Dr. Petia Bobadova-Parvanova (TM) Dr. Djamaladdin G. Musaev (TM) Prof. Rajeev Prabhakar (TM) Gaussian, Inc. Dr. Michael J. Frisch Dr. Thom Vreven Paderborn Group Prof. Marcus Elstner (Braunschweig) Dr. Christof Köhler (Bremen) Prof. Thomas Frauenheim (Bremen ) Support National Science Foundation Air Force Office of Scientific Research (DURIP) Mitsubishi Chemical Corporation ACS, Petroleum Research Funds Gaussian, Inc. Pacific Northwest National Laboratory, EMSL Grand Challenge Acknowledgment

I. Analytical Hessian in DFTB Witek, Irle and KM, J. Chem. Phys. 121, 5163 (2004). Witek, KM and Stradomska, J. Chem. Phys., 121, 5171 (2004). Witek and KM, J. Comp. Chem. 25, 1858 (2004). Malolepsza, Witek and KM, Chem. Phys. Lett (2005). Witek, Stradomska and KM, J. Theo. Comp. Chem. 4, 639 (2005). Witek, Zheng, Irle, de Jong and KM, J. Chem. Phys. In revision.

Analytical Hessian

Accuracy of DFTB frequencies Testing set of 66 molecules, 1304 distinct vibrational modes (cm -1 ) mean absolute deviation standard deviation maximal absolute deviation scaling factor SCC-DFTB DFTB AM PM HF/cc-pVDZ BLYP/cc-pVDZ B3LYP/cc-pVDZ

IR and Raman Spectra of C 70

Dispersion Contribution SCC-DFTB-D (SCC-DFTB with dispersion) is based on a London-type dispersion energy between atoms : London dispersion interaction energy is only valid in case of non- interacting charge densities. Therefore, damping function f(R) has to be introduced for small interatomic distances: N=7, M=7, d=3.0, R 0 =3.8Å 1st row, 4.8Å 2nd row elements M. Elstner et al., JCP 114, 5167 (2001)) undamped damped We implemented dispersion contributions to Hessian

II. Development of analytical functions for DFTB parameters Guishan Zheng, COMP 342, Wednesday, 4.45pm

Motivation and Form  Original two-center parameters (overlap, Hamiltonian and repulsion) given on several hundred 1D grid points  Functions are smooth and accurate for high order energy derivative  Much smaller parameter data base  Functional form used:

An example of fitting: Overlap integral

An example of fitting: Core-core repulsion RMS and maximum deviation between fitted values and original grid values of repulsion curves. The unit used is Hartree.

Test optimization calculations for 264 molecules All 264 molecules consist of H, C, N and O atoms. Geometry optimization starting from the same point using the original parameter and the fitted function forms The optimized geometries are superposed in order to compare how close they are. The average geometry superposition deviation of 264 molecules between the original numerical grids and the new analytical functions are 0.007Å! Total energies are slightly different (should not mix two total energies), but energy differences (bond energies, etc.) are very close.

III. Development of DFTB parameters for first-row transition metal elements (Sc, Ti, Fe, Co, Ni) Guishan Zheng, Henryk Witek, Petia Bobadova-Parvanova, Stephan Irle, Djamaladdin G. Musaev, Rajeev Prabhakar, Keiji Morokuma, Marcus Elstner, Christof K ö hler and Thomas Frauenheim, J. Chem. Theo. Comp. to be submitted Guishan Zheng, COMP 342, Wednesday, 4:45pm Aslo Henryk Witek, COMP 339, Wednesday, 3:00pm

Motivation Extend the applicability of DFTB to problems containing transition metal elements. No reliable semiempirical method for TM. (Cu, Zn: Elstner, Cui, et al Au: Koskinen, Seifert, 2006). Candidate for low-level QM method in the ONIOM(QM:QM) ot (QM:QM:MM) method

Parameterization for First-Row Transition Metal Elements Only 3d, 4s and 4p orbitals are taken into account. Spin-polarized DFTB scheme (SDFTB) needs to be considered. Self-consistency of  and  spin densities on each shell of atoms is important. Analytical functions used for all two-center parameters Parameters determined for M-M and M-X with M = Sc, Ti, Fe, Co, and Ni, and X = H, C, N, and O.

Comparison between DFTB and B3LYP/SDD results Absolute mean bond length difference ( Å ) between DFTB and B3LYP/SDD results Absolute mean bond angle difference (degree) between DFTB and B3LYP/SDD results

Energy differences of different spin states

Conclusions 1.SDFTB analytical parameters for M-M and M-X (M=Sc, Ti, Fe, CO, Ni; X=H, C, N, O) have been determined. 2.SDFTB optimized geometries for M-containing compounds agree well with B3LYP results, except for very weak bonding cases. 3.SDFTB energetic orders qualitatively agree with B3LYP in most cases. Quantitatively cases exist with over- and underestimation of as large as 50 kcal/mol. Use with care. More tests needed. 4. A good candidate as a low-level QM method in ONIOM. Calibration in progress. 5. These analytical parameters will be available for download very soon.

IV. Implementation of the DFTB method into Gaussian03 COMP 342, Wednesday, 4.45pm: An efficient implementation of Density- Functional based Tight-Binding method (DFTB) in Gaussian 03 program: The calculation of energies, gradients, vibrational frequencies and IR spectrums Guishan Zheng, Michael Frisch and Keiji Morokuma

Motivation / Implementation Extend the applicability of DFTB to wider range of problems, e.g. TS. Take advantage of existing Gaussian functionalities, e.g. SCF convergence techniques, partial geometry optimization A smooth combination with ONIOM method Numerically efficient and stable implementation for dealing with large molecular systems, including Gradient, Hessian, IR and Raman intensities. SDFTB (both restricted and unrestricted) formalism with explicit on-site charge interaction and “analytical” parameter functions.

V. Applications to Nano Structure and Dynamics COMP 126, Monday 425pm: Quantum chemical molecular dynamics study of catalyst-free SWNT growth from SiC-derived carbon Zhi Wang, Stephan Irle, Guishan Zheng, K. Morokuma, Michiko Kusunoki COMP 159, Tuesday 230pm: DFTB-based QM/MD simulations of nanostructure formation processes far from thermodynamic equilibrium Stephan Irle, Zhi Wang, Guishan Zheng, Keiji Morokuma COMP 396, Thursday 1105am: The use of ONIOM in computational nanomaterials research Stephan Irle, Zhi Wang, Keiji Morokuma PHYS 269, Tuesday 120pm, QM/MD simulations of carbon nanotube and fullerene growth and dynamics Stephan Irle, Guishan Zheng, Zhi Wang, Keiji Morokuma

Va. Dynamics of formation of fullerenes and carbon nanotubes

0.0 ps0.1 ps1.6 ps 8.5 ps14.5 ps 40.2 ps56.8 ps81.1 ps 94.7 ps104.1 ps ps320.1 ps320.4 ps360.0 ps361.5 ps 2000 K 3000 K +10 C C C 2 “Shrinking Hot Giant” Road of Fullerene Formation J. Phys. Chem. B 110, 14531(2006).

Sc-entrapment in S3 trajectory at 2000 K (Finck. Irle, Morokuma: unpublished) 0 ps 0.7 ps 1.3 ps 12.1 ps 27.8 ps 36.5 ps

Nanotube growth from graphene sheets from the C face of SiC (JCP, 2006; unpublished) 42ps48ps: cap form 48ps: remove Si 96ps 42ps: Add one layer SiC 60ps 78ps 108ps

Vb. The Origin of Linear Relationship between CH 2 /NH/O-SWNT Reaction Energies and Sidewall Curvature: Armchair Nanotubes G. Zheng, Z. Wang, S. Irle, and K. Morokuma, J. Am. Chem. Soc. in press.

Linear relationship has long been recognized experimentally and computationally between the electrophile reaction energy and the sidewall curvature of SWCNT. What is the origin? DFTB: SCC-DFTB, geometry optimizations by Gaussian’s ‘external’ keyword/script DFT: B3LYP/6-31G(d) System: 15 Å (n,n) SWNT+O/CH 2 /NH DFTB: n=2~13 DFT: n=2~6 Studied Thermodynamic stabilities of exo- and endo-Adducts

Exo-addition Endo-addition Closed vs. Open Form (DFTB) open closed

DFTB Stabilization Energy : Exo/Endo 1/d Plots open closed

Relationship of energy components vs 1/d A:  E = DEF + INT for CH 2 adducts on the 15 Å SWNT with DFTB B: INT = ES + EX + ORB for CH 2 adducts on the 5 Å SWNT with HF/STO-3G. exo endo Closed only

Comparison of Orbital Interaction between Exo vs. Endo Adducts

Vc. Handedness by DFTB-D Vs. Suenaga et al. PRL’05Tashiro et al. JACS’06

K. Suenaga, H. Shinohara, S. Iijima: Determination of Handedness in Optical Active Chiral DWNTs, PRL 95, (2005) Models: tilted models simulated HRTEM of tilted models HRTEM of real tube “Fringe counting: this is a DWNT” DWNT with same LL/RR handedness are predominantly found! NCC-DFTB-D: 15 Å tubes (14,3)SWNT: C 252 H 34 (17,10)SWNT: C 374 H 54 GDVE02+ opt=loose external DFTBD full geometry optimizations Total stoichiometry: C 626 H 88 Absolutely no predominance found!

Still no predominance found! Our presumption holds: Chirality preference may be caused at DWNT nucleation stage What about defects? I-V defects are underway. Longer tube, different chirality:

Using SCC-DFTB-D w/Co instead of Rh +y -y 0.0 kcal/mol+2.9 kcal/mol Found two different isomers corresponding to preference for a certain chirality over another.

Conclusion DFTB is a very useful approximate method for geometries and energies of large molecular systems. Parameters for transition metal elements (Sc, Ti, Fe, Co and Ni) have been determined. More general method of determination of inter-element parameters is needed. An excellent candidate for low-level QM in ONIOM(QM:QM) and ONIM(QM:QM:MM) calculations. DFTB energy, gradient and Hessian will be available in the Gaussian code very soon.