DFTB Symposium

Looking at DFTB from a Semiempirical Perspective

Walter Thiel

Max-Planck-Institut für Kohlenforschung, Mülheim, Germany

ACS National Meeting at San Francisco, 11 September 2006

DFTB: Original tight-binding approach

● LCAO-MOs from solution of secular equations with overlap. In usual matrix notation: H0C=SCE.

● Hamiltonian matrix elements calculated using the kinetic energy operator and an effective Kohn-Sham potential which is approximated as the sum of the Kohn-Sham potentials of associate neutral atoms (A, B).

● Basis orbitals and potentials V0 taken from DFT calculations on atoms.

● Only two-center terms computed

● Non-iterative tight-binding treatment.

● Total energy as sum of orbital energies and repulsive two-center correction terms determined by fitting the differences between reference DFT and tight-binding DFTB potential curves in suitable reference molecules.

T̂

G. Seifert, H. Eschrig, and W. Bieger, Z. Phys. Chem. (Leipzig) 267, 529 (1986).G. Seifert, D. Porezag, and T. Frauenheim, Int. J. Quantum Chem. 58, 185 (1996).

bν

BAAμμν VVTH ΦˆΦ 00

0

).,( 0μνμν SH

A B

AB

occ

ijirep

occ

ii UψHψEεE 00

ˆ

),( 00BA VV

μΦ

iεABU

SCC-DFTB: Self-consistent-charge tight-binding approach

● Improve DFTB by allowing for charge fluctuations.

● Second-order expansion of the DFT total energy with respect to the charge density variation relative to a chosen reference density.

● Charge density variation represented by sum of atomic contributions qA which are approximated by Mulliken charges:

● Damped Coulomb interaction between Mulliken charges with correct asymptotic behavior for large distances (→ 1/RAB) and for small distances (→ one-center-term: chemical hardness computed from PBE).

● Working equations:

● Iterative SCF treatment.

● Parametrization of repulsive two-center terms Erep (part of E0).

ABA B

BA

CBCACC

qqEE

qSHH

SCEHC

γ

γγμνμνμν

ΔΔ

Δ

2

12

1

0

0

M. Elstner, D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Phys. Rev. B 58, 7260 (1998).

0AAA qqq Δ

AAγ

ABγ

SCC-DFTB vs. MNDO-type methods: Basic Features

Valence-electron SCF-MO treatment

Minimal basis set of atomic orbitals

Only one-center and two-center terms

Type of integral approximation

Overlap included in secular equations

One-center integrals derived from

Damped two-center two-electron integrals

- with correct limits (R = 0, R = )

Electrostatic balance (attraction/repulsion)

Two-center one-electron integrals

Repulsive atom-pair correction terms

SCC-DFTB

+

+

+

CNDO

+

DFT

+

+

+

DFT-TB

+

MNDO-type

+

+

+

NDDO

-

Exp

+

+

+

empirical

+

SCC-DFTB vs. MNDO-type methods: Practical issues

Parametrization against

Reference data in parametrization

Number of parameters

Computational scaling

Estimated relative cpu time

Analytic gradient

Analytic Hessian

SCC-DFTB

DFT

Energies

10 per atom pair

N3

1.5

+

+

MNDO-type

Exp

Many

5-18 per atom

N3 1.0

+

+

SCC-DFTB validation by the Elstner group

● Reaction energies for 28 reactions involving 22 small molecules (CHNO):

Mean absolute deviation of 4.3 kcal/mol relative to G2 reference data,

compared with BLYP deviations of 5.1 (3.6) kcal/mol for cc-pVDZ(cc-pVTZ)

basis.

● Harmonic frequencies for 196 normal modes of these 22 molecules

(CHNO): Mean absolute deviation of 75 cm-1 from B3LYP/cc-pVTZ

reference data.

● Bond lengths and bond angles of these molecules: Excellent agreement

with MP2/6-31G* and BLYP/cc-pVTZ reference data, with mean absolute

deviations of 0.017 (0.016) Å and 1.6° (1.4°) vs. MP2 (BLYP).

● Occasional failures reported (H2O2 planar, CO energetics, N2H4 frequencies).

T. Krüger, M. Elstner, P. Schiffels, and T. Frauenheim, J. Chem. Phys. 122, 114110 (2005).

Heats of formation: General considerations

● MNDO-type methods: Evaluation from computed atomization energies

and experimental heats of formations of the atoms.

● Implies that zero-point vibrational and thermal corrections are incorporated

into through the parametrization.

● Not done in the SCC-DFTB parametrization.

● First option in SCC-DFTB:

Include zero-point vibrational and thermal corrections explicitly.

● Second option in SCC-DFTB:

Add empirical atomic increments for converting the computed total energies

into heats of formations, or equivalently, treat as an adjustable

parameter rather than computing it.

AfHΔ

M. R. Ibrahim and P. v. R. Schleyer, J. Comput. Chem. 6, 157 (1985).W. Thiel, Tetrahedron 44, 7393 (1988).

A

Af

A

Ael

moltotf HEEH ΔΔ

moltotE

AelE

SCC-DFTB heats of formation: Explicit calculation

● Standard CHNO test set with 140 mostly organic molecules:

Strong overbinding in 139 cases (exception: H2), mean absolute deviation

of 54.5 kcal/mol from experiment.

● Errors (kcal/mol) for selected molecules:

H2 + 28.0, methane -14.1, ethane -27.7, ethylene -18.8, acetylene -17.4,

n-hexane -75.1, benzene -56.7, N2 -32.4, ammonia -18.2, HCN -34.2,

water -13.3, dimethylether – 38.1, CO -31.8, CO2 -37.5, formaldehyde

-25.7, acetic acid -40.9, nitric acid -130.8.

● Errors increase with molecular size.

● Errors particularly large for triple bonds (N2, HCN, CO) and NO bonds.

● Typical overbinding per bond (kcal/mol): C–H 3-4, N–H and O–H 6-7, C–C and C=C 4-7, C≡C ca. 10, C≡N ca. 25.

● Direct calculation with explicit inclusion of zero-point vibrational and thermal corrrections leads to inaccurate heats of formation in SCC-DFTB.

● Potential problems with dissociation reactions.

SCC-DFTB heats of formation: Increment approach

● Applied by the Jorgensen group.

● Electronic energies of the atoms optimized by fitting against the experimental heats of formation of the PDDG/PM3 training set with 134 reference molecules.

● Results (in eV):

● Use of the optimized values removes systematic errors from the SCC- DFTB atomization energies.

● Reaction energies are not affected.

● All subsequent results for SCC-DFTB heats of formation are based on these fitted values.

elAE

Element

Optimized

Computed

H

-7.7196

-6.4923

C

-39.7799

-38.0530

N

-59.9909

-56.1120

O

-85.9785

-83.9753

elAE

elAE

K. W. Sattelmeyer, J. Tirado-Rives, and W. L. Jorgensen, J. Phys. Chem. A 110, 13551 (2006).

SCC-DFTB validation by the Jorgensen group: Energetics

● All values in kcal/mol, N comparisons.

● Experimental reference data, except for H-bond energies from CCSD(T).

● SCC-DFTB problems: Molecules with NO bonds, three-membered rings, some small molecules (H2, H2C=CH2).

Heats of formation

- neutral hydrocarbons

- neutral CHNO Molecules

- ions and radicals

Conformational energies

Isomerization energies

Hydrogen bond energies

N

254

622

30

15

34

12

AM1

5.6

6.8

7.0

1.4

6.6

3.2

PM3

3.6

4.4

9.8

1.8

4.3

4.5

PDDG/PM3

2.6

3.2

10.0

1.8

2.4

4.1

SCC-DFTB

4.8

5.8

13.9

1.2

5.0

1.9

B3LYP/6-31G(d)

(3.4)

0.4

3.1

K. W. Sattelmeyer, J. Tirado-Rives, and W. L. Jorgensen, J. Phys. Chem. A 110, 13551 (2006).

SCC-DFTB validation by the Jorgensen group: Other properties

● N comparisons for neutral CHNO molecules

● Reference geometries from MP2/cc-pVTZ

● Reference dipole moments from gas-phase experiments

Bond lengths (Å)

Bond angles

Dihedral angles (deg)

Dipole moments (D)

N

218

126

30

47

AM1

0.017

1.5

2.8

0.23

PM3

0.012

1.7

3.2

0.25

PDDG/PM3

0.013

1.9

3.7

0.23

SCC-DFTB

0.012

1.0

2.9

0.39

K. W. Sattelmeyer, J. Tirado-Rives, and W. L. Jorgensen, J. Phys. Chem. A 110, 13551 (2006).

Own validation: Standard CHNO molecules

Mean absolute errors ( N comparisons) for a standard validation set of mostly organic compounds (C, H, N, O).

Propertya N MNDO AM1 PM3 OM1 OM2 OM3 DFTB

ΔHf (kcal/mol) 140 6.3 5.5 4.2 3.5 3.1 2.9 7.7

R (pm) 242 1.4 1.7 1.1 1.2 1.6 2.0 1.5

(degree) 101 2.6 1.9 2.1 1.8 2.2 1.8 1.3

IP (eV) 52 0.46 0.35 0.42 0.32 0.26 0.45 3.82

μ (D) 53 0.35 0.26 0.27 0.25 0.28 0.27 0.37

ν (cm-1) 112 241 172 151 189 155 120 90

a) Heats of formation ΔHf, bond lengths R, bond angles , vertical ionization

potentials IP, dipole moments μ, vibrational wavenumbers ν.

Own validation: Heats of formation

Mean absolute errors (kcal/mol) for N comparisons

Neutral CHNO molecules

- Hydrocarbons

- CHN compounds

- CHO compounds

- XNO compounds

Anions

Cations

Radicals

N

140

57

32

39

8

24

33

42

MNDO

6.3

5.9

6.2

4.8

16.3

14.4

11.5

11.9

AM1

5.5

4.9

4.6

5.5

11.4

11.3

9.8

10.6

OM2

3.1

1.7

3.9

4.5

2.9

8.4

7.2

5.0

DFTB

7.7

6.3

6.1

2.7

43.9

12.7

14.5

17.0

Own validation: G2 and G3 sets

Mean absolute errors (N comparisons) for heats of formation (kcal/mol)Reference data from G2 and G3 studies a,b

Compounds N G3 B3LYP MNDO AM1 PM3 OM1 OM2 OM3 DFTBe

G2 CHNO 81 0.69 2.35 7.72 7.37 6.77 4.39 3.36 3.82 9.19

G3 CHNO 47 0.94 7.12 7.13 6.27 4.43 4.36 3.15 3.62 4.50

G2 IPs 32 1.13 5.15 12.55 12.22 11.93 10.57 7.13 6.91 9.70

G2 EAs 29 0.97 3.57 15.44 11.80 9.22 13.70 8.60 8.39 12.30

Alkanes C1-C16 16 0.49c 15.44d 1.81 10.94d 2.24 1.54 2.03 0.44 5.99d

a) L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 112,

7374 (2000).

b) P. C. Redfern, P. Zapol, L. A. Curtiss, and K. Raghavachari, J. Phys. Chem. A

104, 5850 (2000).

c) G3 data only up to C8H18.

d) Error increases with molecular size, e.g., up to 30 kcal/mol for C16H34 in B3LYP.

e) N=78, 26, 22 in rows 1, 3, 4 (triplets excluded).

Own validation: Pericyclic reactions

● Reference data for barriers taken from experiment and published

calculations at the CCSD(T), MP4, MP2, and B3LYP level.

● Reference data for transition structure mostly from B3LYP/6-31G*, partly

also from MP2 and CCSD(T) calculations.

● Reaction studied: Diels-Alder reaction, electrocyclic ring opening, Cope

and Claisen rearrangement, dipolar cycloaddition, ene reaction.

● Mean absolute deviations (N comparisons):

Barriers (kcal/mol)

X · · · Y bond lengths (Å)

N

15

24

AM1

4.7

0.20

OM2

4.3

0.22

DFTB

10.4

0.11

Own validation: Peptides

● Reference data taken from RHF, B3LYP, and MP2 calculations:

Geometries generally from RHF/6-31G* or RHF/6-31G**, relative energies

generally from MP2/6-31G*//RHF or LMP2/cc-pVTZ(-f)//RHF and sometimes

from B3LYP/6-31G*

● Reference systems for geometries: N-methylacetamide complexes (3),

Ac-Ala-NHMe dipeptides (7), Ac-(Gly)2-NHMe turns (4), Ac-(Gly)3-NHMe

turns (5), Ac-(Ala)3-NHMe tetrapeptides (10), Ac-(Ala)n-NHMe (n=2-6) helix

and C7eq conformers (10).

● Reference systems for relative energies: All except first and last group above.

● Mean absolute deviations (N comparisons):

Relative energies (kcal/mol)

Backbone H-bond lengths (Å)

Backbone dihedral angles (deg)

N

22

67

190

AM1

2.0

0.22

17.0

OM2

1.7

0.34

12.0

DFTB

1.1

0.26

9.0

K. Möhle, H. J. Hofmann, and W. Thiel, J. Comput. Chem. 22, 509 (2001).

Own validation: Hydrogen bond energies

● Reference geometries from B3LYP/aug-cc-pVTZ optimizations.

● Reference energies from single-point counterpoise-corrected MP2//B3LYP energies using the aug-cc-pVDZ and aug-cc-pVTZ basis sets and subsequent complete basis set extrapolation: E (MP2/CBS).

● Higher-order correlation effects estimated from the difference between single- point CCSD(T) and MP2 calculations with the aug-cc-pVDZ basis: Ecorr.

● Reference binding energy: E0 = E(MP2/CBS) + Ecorr.

● Reference systems: All 57 CHNO complexes from the MMFF94 data base, see T. A. Halgren, J. Comput. Chem. 17, 520 (1996).

● Mean absolute deviations of computed H-bond energies (kcal/mol):

Geometry

Optimized

N

57

AM1

2.8

OM2

1.5

DFTB

2.7

Own validation: Hydrogen bond geometries

● Reference geometries from B3LYP/aug-cc-pVTZ optimizations.

● Reference systems: All 57 CHNO complexes from the MMFF94 data base, see T. A. Halgren, J. Comput. Chem. 17, 520 (1996).

● Mean deviations (N comparisons): N AM1 OM2 DFTB

X· · ·H · · ·Y bond lengths (Å) 148 0.12 -0.14 -0.03 X· · ·H · · ·Y bond angles (deg) 74 -32.1 -10.3 0.1

● Mean absolute deviations (N comparisons): N AM1 OM2 DFTB

X· · ·H · · ·Y bond lengths (Å) 148 0.25 0.20 0.08 X· · ·H · · ·Y bond angles (deg) 74 33.7 12.1 6.2

Treatment of excited states in large molecules

Ref. 1: Excited state surfaces within TDDFT response theory● Standard TDDFT and TD-DFTB share similar limitations in applicability and accuracy (for conjugated organic molecules).● TDDFT with GGA/hybrid functionals should be applied to photochemical problems with great care.● Issues: Long-range charge transfer or polarization, multiconfigurational ground state.

Ref. 2: Calculating absorption shifts for retinal proteins● Comparison of different methods● Recommendation: Use SCC-DFTB for ground-state optimization or MD and OM2-GUGACI or ab initio SORCI for excitation energies.

Ref. 3: Color tuning in rhodopsins● Successful application of this approach to analyze spectral shifts between rhodopsins.

[1] M. Wanko, M. Elstner et al, J. Chem. Phys. 120, 1674 (2004).

[2] M. Wanko, W. Thiel, F. Neese, M. Elstner et al, J. Phys. Chem. B 109, 3606 (2005).

[3] M. Hofmann, K. Schulten, W. Thiel, M. Elstner et al, J. Am. Chem. Soc. 128, 10818 (2006).

SCC-DFTB validation: Assessment

● Viable alternative to established semiempirical methods

● Comparable overall accuracy

● Accuracy ranking dependent on systems and properties considered

● Present evidence suggest overall: AM1 < SCC-DFTB < OM2, PDDG/PM3

● SCC-DFTB excellent for geometries

● SCC-DFTB performs well for biological systems

● SCC-DFTB may show large errors in “unusual” systems

● SCC-DFTB less suitable for excited states

● More elements need to be parametrized

N. Otte, M. Scholten, and W. Thiel, J. Phys. Chem. A 111, 5751 (2007).

Acknowledgements

Marco Bocola Marcus Elstner

Axel Koslowski Bill Jorgensen

Nikolaj Otte Paul Strodel