ARMY RESEARCH OFFICE
Military University Research Initiative
Oct 16, 2003
Rodney J. Bartlett
Co-Workers
Mr. Andrew Taube Mr. Josh McClellan
Mr. Tom Hughes
Mr.Luis Galiano
Dr. Stefan Fau
Dr. DeCarlos Taylor (ARL)
Dr. Ariana Beste (ORNL)
Quantum Theory Project
Departments of Chemistry and Physics
University of Florida
Gainesville, Florida USA
OBJECTIVES
Identify and characterize the initial steps in nitramine and other detonation in the condensed
phase.
Progress requries NEW ab initio quantum mechancial techniques that have the accuracy
and appicability to provide reliable results for unimolecular and bimolecular reaction paths.
Study the series of molecules, nitramine (gas phase), methyl nitramine(liquid),
dimethylnitramine(solid) which have
(1) different reaction paths
(2) different condensed phase effects
Investigate their unimolecular, secondary, and bimolecular reaction mechanisms.
Obtain definitive results for the comparative activation barriers for different unimolecular
paths, particularly for RDX.
Study the nitromethane molecule and its various isomers as a prototype for nitroalkanes.
Generate ‘transfer Hamiltonians’ to enable direct dynamics simulations as a QM compliment
to classical potentials, and to be able to reliably describe many units of a condensed phase
explosive.
Provide high-level QM results to facilitate the development of classical PES for large scale
simulations.
University of Florida: Quantum Theory Project
Quantum Mechanics I (Isolated gas phase molecules, 0K)
Potential Energy Surface E(R)
Different Unimolecular Decomposition Paths
Activation Barriers
Spectroscopic signatures for intermediates and products
Quantum Mechanics II
Bi(tri...)molecular reactions
Long range (condensed phase, pressure) effects
Activation Barriers, Spectroscopy
Classical Mechanics-Representation E(R)
Large Molecule QM-Simplified Representation
of H(R) Transfer Hamiltonian
Electronic State Specific
Reactions of one Me2N-NO2
Me2N-NO2
 Me2N. + NO2.
 MeN.-NO2 + Me.
 H2C.-N(Me)-NO2 + H.
[ H2C=N+(Me)-N(O-)OH]  H2C=NMe + HONO
Reactions of two Me2N-NO2
Same reactions as before, causing a slight change in the interaction
energy with the second Me2N-NO2.
Additionally:
Me2N-NO2 + Me2N-NO2
 Me2N-ONO + Me2N. + NO2.
 Me2N-NMe2 + NO2. + NO2.
 Me2N-Me + Me-N.-NO2 + NO2.
 Me2N-N(Me)-NO2 + Me. + NO2.
 Me2N-H + H2C.-N(Me)-NO2 + NO2.
 Me2N-CH2-N(Me)-NO2 + H. + NO2.
products from CH3., H. (HONO, H-Me, …)
In reposne to Bob Shaw’s comment about
error bars in theoretical applications….
It is clear that if we want to know the right answer
for activation barriers competitive decompisiton paths..
We need a high-level of theory like CCSD(T), with a
large basis set.
Coupled Cluster Calculation of De’s
1.0
0.8
MP2
CCSD
+(T)
D
6.0
-8.3
-1.0
Dstd
7.5
4.5
0.5
0.6
r(De)
CCSD(T)
0.4
0.2
CCSD
0.0
-40.0
-30.0
-20.0
MP2
-10.0
0.0
De (kcal/mol)
From K. L. Bak et al., J. Chem. Phys. 112, 9229-9242 (2000)
10.0
20.0
30.0
40.0
Last year…
•Reported the detailed theory for the compressed (SVD) CC,
which ‘contracts’ the CC amplitudes in an optimum way to
make it possible to perform much higher level CC calculations
for large molecules.
Potenrial Energy Curve (1)
(HF, aug-cc-pVDZ, HF-bond stretcing)
0
-50
(E+100)*1000 (a.u.)
-100
r(eq)=1.733 bohr
a=0.25
-150
MRCI
CCSD
-200
CCSD(T)
CCSDT-1
COMP.SDT-1
-250
-300
0.5
1
1.5
2
2.5
r(HF)/r(eq)
3
3.5
4
Potential Energy Curve (2)
(H2O, aug-cc-pVDZ, OH-bonds
stretching)
100
(E+76)*1000 (a.u.)
0
r(eq)=1.809 bohr
a=0.25
-100
MRCI
CCSD
CCSD(T)
-200
COMP.SDT-1
CCSDT-1
-300
0.5
1
1.5
2
r(OH)/r(eq)
2.5
3
3.5
•Detailed a new extrapolation procedure for energies and forces that
has a mean error of nearly zero, and a maximum error of
0.75 kcal/mol for nitramne and its components.
•Reported on a series of CC studies of nitramine to assess its
decomposition paths.
This year….
OUTLINE
I.A NEW APPROACHE TO HIGH-LEVEL COUPLED-CLUSTER THEORY FOR
LARGER MOLECULES.
A. A natural orbital coupled-cluster method.
B. Comparisons to Compressed coupled-cluster theory.
II. NUMERICAL ILLUSTRATIONS FOR DMNA, DMNA DIMERS AND RDX.
Iii. CONCEPT OF A TRANSFER HAMILTONIAN AS A MEANS TO DESCRIBE
COMPLEX SYSTEMS WITH QUANTUM MECHANCIAL FORCES FOR MD
APPLICATIONS.
A. Illustration for nitromethane and its isomers.
University of Florida: Quantum Theory Project
How can we retain the accuracy of CCSD(T), ie an ~n7
method, but make it applicable to large molecules?
Dimer (bimolecular sytem) is 28 times as difficult as the
monomer, without modification.
I. Natural orbital coupled-cluster theory
II. Compressed coupled-cluster using
Singular Value Decomposition.
Frozen Natural Orbital
Coupled Cluster Theory
• Dependence on size of virtual sector of basis set
limits high-level CC to small molecules:
• CCSD ~ V4; CCSD(T) ~ V4; CCSDT ~ V5
• Natural Orbitals (NOs) are known to be the best
possible set of orbitals to truncate
• Too costly to get exact – use approximate
MBPT(2) NOs
• To maintain advantages of a HF reference, only
perform truncation in virtual space – leaving
occupied space alone – Frozen Natural
Orbitals (FNOs)
FNO Procedure
 SCF gives Molecular Orbitals U
 Construct MBPT(2) Density Matrix D in MO
Basis
 Solve DV=Vn
 Truncate V to V’ throwing out less occupied
virtuals, as measured by their occupation
numbers.
 Construct new Fock Matrix in FNO Virtual Space
 Diagonalize for new orbital energies
 Perform higher level (CC) calculation in
truncated virtual space
 For an estimate of truncation error, define:
∆MBPT(2) = MBPT(2) (Full) – MBPT(2)
(Truncated)
Note: On the following slides, the percentage
listed indicated the number of FNOs retained.
For example, 20% DZP means that there are
only 20% of the original number of DZP
virtual orbitals left.
Computational Details: Calculations were
performed on an IBM RS/6000 375 MHz
POWER3 processor with 3 GB RAM and 18
GB disk using the ACES II electronic structure
program.
Three Different Bases. Same Number of Orbitals.
FNO RHF CCSD(T) PES for Hydrogen Fluoride
-100.05
0.5
1
1.5
2
2.5
-100.1
Total Energy (Hartree)
-100.15
-100.2
-100.25
Reference
-100.3
-100.35
100% DZP
40% cc-pVTZ
-100.4
20% cc-pVQZ
100% cc-pVQZ
-100.45
Bond Length (Angstroms)
3
Average Timings for Determination of CCSD(T) PES for Six
Small Molecules* with FNO Truncation
3.5
LOG(Relative** Time)
3
2.5
20%
40%
60%
80%
100%
2
1.5
1
0.5
0
DZP
cc-pVTZ
cc-pVQZ
ANO-ADZP
Basis dissociation
Set
*F2, HF, CO, N2, NH3, H2O – PES of symmetric
*Relative to 20% DZP Basis Calculation
ANO-ATZP
Truncated Large Bases are Better than Similar-sized Small Bases
% Correlation Energy*
100
80
60
40
DZP
cc-pVTZ
cc-pVQZ
Points are in increments of 20% of basis set
20
0
0.5
1
1.5
2
LOG (Relative** Time)
2.5
3
* Total Correlation Energy Determined by 100% cc-pVQZ calculation
** Times Relative to 20% DZP Calculation
3.5
Dimethylnitramine
•
•
•
•
Model compound for RDX and HMX.
Two possible conformers:
C2v – X-ray crystallography / gas-phase e- diffraction1;
Cs – predicted by theory at SCF, MBPT(2), DFT, QCISD levels with
moderate basis sets2. It has been argued that theoretical
predictions are more accurate than experimental values.
• Dimer interactions are important to model dominant interaction in the
solid phase. Given X-ray structure, best to use C2v monomer.
Calculations have been done at SAPT level on fixed monomers3.
• DMNA can undergo decomposition via NO2 and HONO elimination.
HONO Elimination estimated to be exothermic by ~1-3 kcal/mol at
standard conditions4, therefore need high-level calculations, zeropoint energy corrections, etc.These have been investigated
theoretically locating transition states at the QCISD, MBPT(2) and
DFT levels2.
1) Stolevik, Rademacher, Acta Chem Scand 1969 23 672
2) Harris, Lammertsma, JPCA 1997 101 1370; Smith, et al., JPCB 1999 103 705;
Johnson, Truong, JPCA 1999 103 8840
3) Bukowski, Szalewicz, Chabalowski, JPCA 1999 103 7322
4) Shaw, Walker, JPC 1977 81 2572
DMNA Equilibrium Structure
C2v*
Cs**
Total Energies in
Hartree
CCSD(T) RHF Drop Core
C2v
Cs
C2v – Cs (kcal/mol)
DZP – 100%
-338.8157
-338.8302
9.09
DZP – 100% Time
0.39 hr
1.22 hr
DZP – 60%
-338.7549
-338.7671
DZP – 60% Time
0.06 hr
0.19 hr
DZP – 60% + ∆MBPT(2)
-338.8078
-338.8220
8.91
cc-pVTZ – 100%
-339.1252
-339.1399
9.24
cc-pVTZ – 100% Time
13.1 hr
39.5 hr
cc-pVTZ – 60%
-339.1032
-339.1180
cc-pVTZ – 60% Time
1.80 hr
7.00 hr
cc-pVTZ – 60% + ∆MBPT(2)
-339.1283
-339.1431
7.66
9.29
9.29
*Experimental geometry with HCH angles optimized at MBPT(2) Bukowski et.al. JPCA
1999 103 7322
**Theoretical prediction QCISD cc-pVDZ basis Johnson & Truong JPCA 1999 103 8840
DMNA Dimer Structures† - fixed monomer
geometry
M2
M1
M3
M4
† Monomer geometries are experimental methyl angles, optimized at
MBPT(2) level. Dimer structures are minima of SAPT method –
Bukowski, Szalewicz, Chabalowski JPCA 1999 103 7322
DMNA Dimer Interaction Energies
12
SAPT †
This work*
-Eint (kcal/mol)
10
8
6
4
2
0
M1
M2
M3
M4
SAPT Minima
* CCSD(T) FNO 60% DZP with core occupieds dropped and ∆MBPT(2) correction
† Bukowski, Szalewicz, Chabalowski JPCA 1999 103 7322
Background information for RDX
Decomposition Mechanisms:
(1), DSC closed pan(liquid-like) (2)
● NO Elimination: B3LYP/6-3311G**
2
(3)
● HONO Elimination
(4), DSC open pan(gas-like)(2)
● Triple Bond fission: IRMPD
(5), internal ring formation, -OH
● Other mechanisms: NO elimination (MALDI)
loss
Conformers of RDX:
(6)
● Solid α-RDX: AAE (Cs)
● Solid β-RDX: AAA
● Vapor phase(e diffraction): AAA(C v)
3
(7)
● Gas/liquid dynamically averaged structure
The energy difference between minima is on the order of 1 Kcal/mol (B3LYP/6311G**) with AAE being the most stable(1)
1)N. Harris, K. Lammertsma., J. Am. Chem. Soc. 119,6583 (1997)
2) G. Long, S. Vyazovkin, B. A. Brems, C.A. Wight, J. Phys. Chem. B., 104, 2570 (2000)
3) D. Chakraborty, R.P. Muller, S. Dasgupta, W. Goddard III, J. Phys. Chem. A., 104,2261 (2000)
4) X. Zhao, E. Hintsa, Y. Lee, J. Chem. Phys., 88, 2 ,801 (1988)
5) H.S. Im and E.R. Bernstein. J.Chem.Phys., 113, 18 ,7911 (2000)
6)B. Rice, C. Chabalowski, J. Phys. Chem. A., 101, 8720 (1997)
7) T. Vladimiroff, B. Rice J. Phys. Chem. A., 106, 10437 (2002)
RDX Minima*
AAA Chair
Total Energies in
Hartree
AAA
Chair
CCSD(T) RHF Drop Core
AAA Boat
AAA Boat
Chair-Boat
(kcal/mol)
DZP – 60%
-895.1732 -895.1728
-0.25
DZP – 60% + ∆MBPT(2)
-895.3311 -895.3289
-1.4
CC Time (hr)
30
145
Estimated Time for Full Basis (hr)
200
1000
*Conformations determined by B3LYP 6-31G(d) Calculations –
Chakraborty, et al JPCA 2004 104 2261
Triples at a Fraction of the Cost
Compressed Coupled Cluster &
FNO CC for DMNA
CCSDT-1 RHF DZ Basis Frozen Core:
Method
Comp. CC
60% FNO CC
CC Speed-up
Factor
8.6
6.3
5.9
CCSD(T) FNO Timings for
AAA Chair Conformer of RDX
Basis
Time
FNO* 60% DZP
30 hours
100% DZP
200 hours**
FNO* 60% cc-pVTZ
50 days**
100% cc-pVTZ
1 year**
FNO* 60% cc-pVQZ
2.5 years**
100% cc-pVQZ
20 years**
*FNO Speed-up ~ 8x faster with 50% truncation. For large numbers of occupied orbitals, FNO
speed-up determined by o3v3 term in CCSD equations – Not by o2v4 term
** Estimated.
Transfer Hamiltonian for large clusters of molecules.
Basic idea: Represent the CC Hamitonian in its
one-particle form by a low-rank operator, that permits
rapid generation of forces for MD, but can (hopefully)
retain the accuracy of CC theory, in the process.
TRANSFER HAMILTONIAN
In CC theory we have the equations…
exp(-T) Hexp(T) = Ĥ
Ĥ|0 = E|0 Where E is the exact correlated energy
m |Ĥ|0 =0 Where m| is a single, double, triple, etc excitation
which provides the equations for the coefficients in T, ie tia, tijab,
etc.
(R)E(R) = F(R) Provides the exact forces
r(x)= 0| exp(-T)(x-x’)exp(T) |0 gives the exact density
and m| Ĥ |n  Ĥ and ĤRk = kRk Gives the excitation
(ionization, electron attached) energies k and eigenvectors Rk
University of Florida: Quantum Theory Project
COMPARATIVE APPLICABILITY OF METHODS
10-10
CP
10-8
TB
10-6
COST
SE
TH
10-4
DFT
10-2
CC
1
ACCURACY
TRANSITION FROM MANY-PARTICLE HAMILTONIAN TO
EFFECTIVE ONE-PARTICLE HAMILTONIAN...
Wavefunction Approach
0|{i†a}Ĥ|0=0= a| Ĝ |i=0
Ĝ|i=i|i i
Parameterize Ĝ with a GA to satisfy E= 0|Ĥ|0,
E=F(R), r(r), (Fermi) = I
Density Functional Approach
Ĝ|i=i|i i
where Ĝ =t+E/r(x)
and E[r]=E, E=F(R),
r(r)= |i i|, (Fermi) = I
Future? Remove orbital dependence
and/or self-consistency?
RELATIONSHIP BETWEEN COUPLED-CLUSTER/DFT
HAMILTONIAN AND SIMPLIFIED THEORY
Second Quantized Ĝ
Ĝ =gpq{p†q} +ZAZB /RAB
Transition from orbital based to atom based- (hAA + AA)+ hAB(R) +  AB(R)
+ Z’AZ’B /RAB{akAexp[-bkA(RAB-ckA)2] +akBexp[-bkB(RABckB)2]}
hAB(R)=  ( + )KS(R)
AB(R) = [( RAB)2 +0.25(1/AA+1/BB)2]-1/2
University of Florida: Quantum Theory Project
Nitromethane Background
nitromethane (NMT)
methylnitrite (MNT)
aci-nitromethane
• Proper description of NMT unimolecular rearrangement is required for
adequate description of combustion, detonation and pollution chemistry.
• NMT is a model system for energetic materials, e.g. FOX-7, TNAZ.
• NMT→ CH3∙ +NO2∙ most energetically favored, ~63 kcal/mol*.
• NMT→MNT→ CH3O∙ +NO∙ second most energetically favored, ~69 kcal/mol*.
• Molecular beam experiments** demonstrate NMT → MNT
*CCSD(T)/cc-PVTZ, Nguyen et al., J. Phys. Chem. A 2003, 107, 4286
**Wodtke et al., J. Chem. Phys. 1986, 90, 3549
Unimolecular Decomposition
Pathways of NMT and MNT
CH3-NO2  CH3. + NO2.
 TS1  CH3ONO
Rearrangement
CH3ONO  CH3O . + NO.
 TS2  CH2O + HNO
Rearrangement
•G2MP2*
•CCSD/TZP
•CCSD(T)/cc-PVTZ**
Energies relative to NMT
* Hu et al., J. Phys. Chem. A 2002, 106, 7294
** Nguyen et al., J. Phys. Chem. A 2003, 107, 4286
Nitromethane HT
Force of C-N bond breaking
0.15
F (H/Bohr)
0.1
0.05
0
1.3
1.8
2.3
-0.05
-0.1
R (A)
2.8
3.3
CCSD/TZP
AM1
TH-CCSD
B3LYP/6-31G*
Nitromethane PES for C-N rupture
90
Energy(kcal/mol)
80
70
60
CCSD(UHF)/TZP
AM1(UHF)
TH(UHF)
B3LYP/6-31G*(UHF)
50
40
30
20
10
0
-10 0.9
1.9
2.9
R_C-N (A)
3.9
Reference Data
NMT Energy (UHF)
90
CCSD(UHF)
/TZP
40
-10
0.9
2.9
E-E_rel(kcal/mol)
E-E_rel(kcal/mol)
NMT Energy (RHF)
90
40
CCSD(RHF)
/TZP
-10
0.9
R_C-N (A)
R_C-N (A)
90
CCSD(RHF)
/DZP
-10
0.9
2.9
R_C-O (A)
aci-NMT Energy
E-E_rel(kcal/mol)
E-E_rel(kcal/mol)
MNT Energy
40
2.9
90
40
CCSD(RHF)
/TZP
-10
0.9
1.9
R_C-N (A)
Nitromethane Clusters
• Our nitromethane dimer and trimer calculations used local minima
found by Li, Zhao, and Jing in their application of BSSE corrected
DFT/B3LYP with 6-31++G** basis*. This reference also concludes
that a proper description of three body effects is needed for accurate
determination of potential energy surfaces for bulk nitromethane.
• The most stable dimer configuration was that in which two hydrogen
bonds of length 2.427Å are formed while trimer involved a ring
structure in which the three hydrogen bonds of lengths 2.329Å,
2.313Å, and 2.351Å. This configuration for the dimer minimum is
also supported by CP corrected SDQ-MBPT/DZP in which the
hydrogen bonding distance is found to be 2.25Å**.
• With TH-CCSD we observe the formation of methoxy radical in the
dimer and the formation of methylnitrite in the trimer, similar to
predicted unimolecular mechanisms found at the G2MP2/B3LYP/6311++G(2d,2p) level of theory ***.
* J. Li, F. Zhao, and F. Jing, JCC 24 (2003) 345.
** S. J. Cole, K. Szalewicz, G. D. Purvis III, and R. J. Bartlett, JCP 12 (1986) 6833.
S. J. Cole, K. Szalewicz, and R. J. Bartlett, IJQC (1986) 695.
*** W.F. Hu, T.J. He, D.M. Chen, and F.C. Liu, JPCA 106 (2002) 7294.
Breaking of H-Bonds in Nitromethane Dimers with
Frozen Monomers
SAPT equilibrium OH distance*
Nitromethane Decomposition in Dimer
H3CNO2* + H3CNO2  H3CNO + H3CO + NO2
UHF AM1 prediction
* Indicates bond rupture
UHF TH-CCSD predicts that
rearrangement
occurs when C-N bond is 2.42 Å
Nitromethane Rearrangement in Trimer
H3CNO2* + 2 H3CNO2  3H3CONO
UHF AM1 prediction
* Indicates bond rupture
UHF TH-CCSD predicts that
rearrangement
occurs when C-N bond is 1.94 Å
Equilibrium Geom of NMT
% error from CCSD/TZP
3
2.5
2
1.5
TH/CCSD
AM1
1
0.5
0
R_CH
R_CN
R_NO
Property
AHCN
AONC
TNAZ C-N Bond Rupture
Force of C-N bond breaking
0.3
0.25
B3LYP/6-31G
AM1
TH-CCSD
F (H/Bohr)
0.2
0.15
0.1
0.05
0
-0.05 1.3
1.8
2.3
-0.1
R (A)
*C-N bond breaking trans to N-NO2
2.8
SUMMARY
•Derived and illustrated the NO-CC methodwith applications to
DMNA and RDX. Savings is ~7 out of theoretical 16. With
rapid processors, will make state-of-the-art CC resutls possible in
a week, instead of a year.
•Awaiting analytical gradients (programmed) to enable geometry
and transition state searches.
•Illustrated tranfer Hamiltonian approach to retain the accuracy of
CC theory, but for much more complicated representations of the
condensed phase.
•Establsihed rigor of the theory. We are working on alternative,
and better realizations of the concept.
Nitramine: Dangers of DFT
Energy of N-N bond breaking
(E-Eeq) (kcal/mol)
100
80
60
40
0
-20
⇒
CCSD(T)/cc-PVTZ
B3LYP/6-31G*
20
1.1
1.6
2.1
2.6
R (A)
3.1
Forces from DFT are qualitatively and
quantitatively wrong at non-equilibrium geometries!