CMx Charges for SCC-DFTB and Some GaN

Vignettes

Christopher J. Cramer

University of Minnesota

DFTB Energy Functional

€

E ρ0 r( )[ ] = ψ i r( )hiKS ρ0 r( )[ ] ψ i r( )

i

occupied

∑ −1

2

ρ0 r1( )ρ0 r2( )

r1 − r2

dr1dr2∫∫

+ Exc ρ0 r( )[ ] − Vxc ρ0 r( )[ ]∫ ρ0 r( )dr + EN

€

μ hKS ν =εμ , μ = ν

μ T + veff ρ0, A r( ) + ρ0, B r( )[ ] ν , μ ∈ A, ν ∈ B

⎧ ⎨ ⎪

⎩ ⎪

€

Erep ρ0 r( )[ ] = Erep ρ0, A r( )[ ]A

atoms

∑ + Erep2( )

A<B

atoms

∑ ρ0, A r( ),ρ0, B r( )[ ]

SCC-DFTB Energy Functional

€

E ρ0 r( ) + δρ r( )[ ] = E ρ0 r( )[ ]

+1

2

1

r1 − r2

+δ 2Exc

δρ r1( )δρ r2( )ρ0

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟δρ r1( )δρ r2( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥∫∫ dr1dr2

€

δρ r( ) = ΔqAδ r − rA( )A

atoms

∑

€

1

2

1

r1 − r2

+δ 2Exc

δρ r1( )δρ r2( )ρ0

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟δρ r1( )δρ r2( )

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥∫∫ dr1dr2 =

1

2ΔqAΔqBγAB

A, B

atoms

∑

€

γAB =2ηA, A = B

aa bb( ), A ≠ B

⎧ ⎨ ⎪

⎩ ⎪

Class II Partial Charges (Population Analysis)

€

Pμν = 2 cμi

i

occ

∑ cνi

€

Sμν = φμφν∫ dr

€

N = tr PS( ) = tr SP( ) = tr S1/ 2PS1/ 2( )

€

N k = PS( )μμμ∈k

∑€

qk = Zk − N k

€

N k = S1/ 2PS1/ 2( )

μμμ∈k

∑

Mulliken Löwdin

Class IV Partial Charges (CM2 and CM3)

€

qkCMx = qk

(II) + Bk ′ k Ck ′ k Bk ′ k + Dk ′ k ( )′ k ≠k

∑

€

Bkk' = PS( )μλPS( )λν

λ

∑ν ∈k'

∑μ∈k

∑Mayerbondorder

Ckk ′=−Ck′k, Dkk′ =−Dk′k

empiricallinear and quadratic

parameters

x = 2, Li et al. J. Phys. Chem. A, 1998, 102, 1820.

x = 3, Winget et al. J. Phys. Chem. A 2002, 106, 10707Thompson et al. J. Comput. Chem. 2003, 24, 1291

Training Set and Error Functions

• Training set roughly 400 neutral and 25 ionic molecules

• Compare point-charge derived dipole moments to experimental values

• For ions, compare point-charge-derived moments to <|μ|> (MP2/cc-pVTZ, center of mass) and compare partial atomic charges to those determined from CHELPG fit to MP2/cc-pVTZ electrostatic potential

€

μ =± qk xk

k

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+ qk yk

k

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2

+ qkzk

k

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

1/ 2

Performance Example

O

OMe

H

O

HH

O

MeMe

OO

O

O

NH2H

O

NH2Me

S

HH

CH3CO2HH2O

MeOH MeNH2Me2O

NH3

NCNH2

HCN

MeCN

CH3F CH3SHCH3SiH3 H2S

CH3Cl

Performance Example 2

Mode l RMS Erro r (D)

CM2/ any level <0.20

MP2/6 -31G* < | | > 0.21

HF/ 6-31G* < | | > 0.31

HF/ 6-31G* CHELPG 0.33

PM3 < | | > 0.43

AM 1 < | | > 0.44

AM 1 Mullik en 0.89

HF/ 6-31G* Mul liken 0.93

PM3 Mullike n 1.00

HF/ 6-31G* NPA 1.05

Accurate, Density, and CM3 Dipole Moments

N NO

OH

HN N

H

H

O

ON N

HH

O

ON N

H

H

O

O

3.943.593.84

4.313.93 4.19

2.972.71 2.89

3.283.07 3.27

nitramide

MUE (density) = 0.30 debyes MUE (CM3) = 0.08 debyes

Accurate: mPW0/MG3S density dipole

MUE mean unsigned error:

Cs C2v Cs C2v

from mPW0/MIDI!

Approximate dipoles

Accurate, Density, and CM3 Dipole Moments

4.814.21 4.67

5.044.43 4.87

3.432.99 3.33

3.693.38 3.77

dimethylnitramine

MUE (density) = 0.49 debyes MUE (CM3) = 0.12 debyes

Accurate: mPW0/MG3S density dipole

N NO

OMe

MeN N

Me

Me

O

ON N

MeMe

O

ON N

Me

Me

O

O

MUE mean unsigned error:

Accurate, Density, and CM3 Dipole Moments

5.975.22 6.20

7.196.227.34

: RDX

MUE (density) = 0.86 debyes MUE (CM3) = 0.19 debyes

Accurate: mPW0/MG3S density dipole

N N

N

NO2

O2N NO2

MUE mean unsigned error;

Accurate, Density, and CM3 Dipole Moments

1.561.321.80

0.310.42 0.79

: HNIW; CL-20

MUE (density) = 0.32 debyes MUE (CM3) = 0.29 debyes

Accurate: mPW1PW91/MG3S density dipole

2.561.95 2.41

N N

N N

NNNO2O2N

NO2

NO2O2N

O2N

γ

MUE mean unsigned error:

[hexa-nitrohexaaza-iso-wurtzitane]

CM3 Delivers Consistent Partial Atomic Charges

Conformer CM3 ChElPG CM3 ChElPGγ-HNIW -12.6 -13.4 -12.4 -19.1-HNIW -13.2 -13.6 -13.0 -19.2-HNIW -13.7 -13.9 -13.7 -19.6

1 91/ !mPW PW MIDI / !HF MIDI

Polarization energies (in nitromethane) calculated using different charge schemes by wave function (kcal/mole):

MUD (CM3) = 0.1MUD (ChElPG) = 5.7

All 14 nitramines(0.2)(2.8)

MUD (Löwdin) = 5.9 (2.9)

MUD mean unsigned deviation:

GP =−12

−1ε

⎛ ⎝ ⎜

⎞ ⎠ ⎟ qkq ′ k γk ′ k k, ′ k ∑

electrostatic fitting

population analysis

SCC-DFTB Results — Before

Signed errors O(0.4 D), RMSE O(0.7 D)

Optimized Parameters (Mulliken mapping)

Linear (in B.O.)parameters

quadraticparameters

SCC-DFTB Results — After

CM3 Improvement

+ Mullikeno CM3

Gallium Nitride from Cyclotrigallazane

Kormos et al. JACS, 2005, 127, 1493

NH3

150° C[HGaNH]n GaN

substantial cubicform in addition

to wurtzite

What is Nature of [HGaNH]n?

Kormos et al. JPC A, 2006, 110, 494

XY

XY

XY

Y

XY

XY

XY

XY

XY

Y

XY

XY

XY

Y

XY

XY

X

Y

XYY

XXYY

XY

X

Y

Y

Y

X

X

XY

X

X

X

X

X

Y

X

Y

Y

Y

X

XYY

YX

X YYX

X Y

flat-chair (FC)

rolling-chair (RC) flat-boat (FB)

What is Nature of [HGaNH]n?

Kormos et al. JPC A, 2006, 110, 494

FC FB

RC

[HGaNH]n Is a Mixture of Nanorods

Kormos et al. JACS, 2005, 127, 1493

+etc.n GaN GeC

1 2.7 1.0

2 9.0 1.1

3 15.5 0.9

4 23.0 0.5

5 31.3 0.1

6 40.3 -0.4

7 49.7 -0.9

8 59.4 -1.5

9 69.3 -2.1

Dipole moment (D)

Error compared to DFT and MP2 • Data set included small molecules containing Ga, N, and H atoms

• B3LYP and MP2 with 6-311+G(2df, p) basis set on N and H and CEP-31G ECP and basis set on Ga

• Data set included six dimers for binding energies and intermolecular distances, seven reaction energies, and nine molecules for bond lengths and angles

mean unsigned error of SCC-DFTB

B3LYP MP2bond lengths 0.049 0.038

angles 3.86 4.03intermolecular distances 0.43 0.43

reaction energy 21.21binding energy 3.21 3.65

bond lengths in Åangles in degreesenergies in kcal/mol

[H2GaNH2]3 Binding Energy and Rod Growth

Dimer A

ΔEn(kcal/mol)n SCC-DFTB B3LYP2 63.8 2.23 57.6 -4.34 55.3 -3.85 53.8 -5.26 52.6 -6.27 51.8 -7.18 51.1 -7.69 50.6 -8.3

Binding Energies (kcal/mol)

Dimer MP2//RHF SCC-DFTBA -7.4 -3.08B -3.8 -1.41C -7.8 0.13D -4.1 -1.64

H3[(HGaNH)3]n–1H3 + H3[(HGaNH)3]H3

H3[(HGaNH)3]nH3 + 3H2

Future Plans

• Reparameterize SCC-DFTB to get better agreement with higher levels of theory– Hardness was not found to have sufficient

influence

– Reoptimize Erep to B3LYP data

• Add empirical dispersion term to get better binding energies and distances

Acknowledgments Biradicals, Diradicals, Ilk tRNA Dynamics Senior Collaborators

Dr. Benjamin Gherman Dr. Maria Nagan Prof. Dan Falvey (Maryland) Dr. Mark Seierstad Dr. Ed Sherer Prof. Laura Gagliardi (Genève)

Dr. William T. G. Johnson Stephanie Kerimo Prof. Wayne Gladfelter (Minn)

Dr. Youngshang Pak Prof. Shinobu Itoh (Osaka) Dr. Michael Sullivan Solvation Prof. Jaroslaw Kalinowski

Dr. Stefan Debbert Dr. Candee Chambers (Warsaw)

Dr. Bethany Kormos Dr. Jiabo Li Prof. Hilkka Kenttämaa (Purdue) Dr. Chris Kinsinger Dr. Tianhai Zhu Prof. Bogdan Lesyng (Warsaw)

John Lewin Dr. David Giesen Prof. Eric Patterson

David Heppner Dr. Gregory Hawkins (Truman State) Dr. Paul Winget Prof. Piotr Piecuch

Gallium Nitride Dr. James Xidos (Michigan St.)

Dr. Bethany Kormos Dr. Jason Thompson Prof. Bill Tolman (Minnesota) Joseph Scanlon Casey Kelly Prof. Don Truhlar (Minnesota)

Adam Chamberlin Dr. Eric Weber (US EPA)

Support from: US ARO, NSF, EPA, Minnesota Supercomputing Institute

AMSOL, SMxPAC, GAMESSPLUS, HONDOPLUS, OMNISOL, etc. available from various sources (see http://comp.chem.umn.edu/mccdir/software.htm)