ARMY RESEARCH OFFICE

Military University Research Initiative Oct 16, 2003

Quantum Theory ProjectDepartments of Chemistry and Physics

University of FloridaGainesville, Florida USA

Rodney J. Bartlett Co-Workers

Dr. Marshall CoryDr. Stefan Fau

Mr. Josh McClellan

I. INTRODUCTIONNature of problem and our objectives

II. NUMERICAL RESULTSDimethylnitramine and tests of quantum chemical methods to be used. (Stefan Fau)

III. PLAN AND PROGRESS FOR RDX (Stefan Fau, Marshall Cory)

IV. COMPRESSED COUPLED CLUSTER THEORY: A NEW APPROACH TO HIGH LEVEL CC FOR LARGE MOLECULES

V. SUMMARY OF PROGRESS AND FUTURE PLANS

University of Florida: Quantum Theory Project

OUTLINEOUTLINE

Identify and characterize the initial steps in nitramine detonation in the condensed phase.

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 including those for RDX.

Develop ‘response/dielectric function’ methods to incorporate the condensed phase effects into the quantum mechanical calculations.

Provide high-level QM results to facilitate the development of classical PES for large scale simulations.

Generate ‘transfer Hamiltonians’ to enable the direct dynamics simulations as a QM complement to classical potentials.

University of Florida: Quantum Theory Project

OBJECTIVES

Quantum Mechanics I (Isolated gas phase molecules, 0K)Potential Energy Surface E(R) Different Unimolecular Decomposition PathsActivation BarriersSpectroscopic signatures for intermediates and products

Quantum Mechanics IIBi(tri...)molecular reactionsLong range (condensed phase, pressure) effectsActivation Barriers, Spectroscopy

Classical Mechanics-- Representation E(R)

Large Molecule QM--Simplified Representationof H(R) Transfer HamiltonianElectronic State Specific

SEAMLESS WHOLE…

FROM QM [(CC) TO (DFT)TO (TH)] TO ADAPTIVE (CHARGE TRANSFER) POTENTIALS,TO CLASSICAL POTENTIALS (CP), AND BACK, ie

INSIST THAT E(R) LEADS BACK TO A H(R), THAT GIVESELECTRONIC DENSITY AND OTHERQM PROPERTIES

Reactions of one H2N-NO2

H2N-NO2 H2N. + NO2

.

H. + HN.-NO2

H2N-ONO

[ HN=N(O)OH] 3HN + 1HONO

( 1HN + 1HONO)

( 3HN + 3HONO)

Reactions of two H2N-NO2

Additionally:

H2N-NO2 + H2N-NO2 H2N-NH2 + NO2. + NO2

.

NH2. + NH2

. + O2N-NO2

H2NH + NO2. + HN.-NO2

H2N. + HONO + HN.-NO2

Same reactions as before, causing a slight change in the

interaction energy with the second H2N-NO2.

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, …)

Dimethylnitramine

Overview

• Immediate goals and methods

• Dimethylnitramine

• RDX

• Other Things We Can Do

Goals of Our Calculations Provide high quality energies (forces where feasible)

at points along various reaction coordinates for testing or fitting of faster methods.

Definitive answers for low energy dissociation reactions of dimethylnitramine (and more reliable ones for RDX).

Use nitramine as a test-case since better methods can be used. (More complete work if desired.)

Environmental effects by including second molecules.

How we do it

DFT: generally good minimum geometries less good for transition states, vdW, ... cheap

CCSD(T): good energies (and other properties) expensive.

Basis set extrapolation: necessary for high quality energies, ...

Single-point energies with basis set extrapolation on DFT/TZ minima and reaction paths.

Basis Set Extrapolations

DZTZ21

1

DZZ

1XZXZ

EEee

eEE

AeEE

CBSxf:

CBS3:

DZCBSxDZZ EEaEE

CBSxM: ECBSx-DZ from MBPT(2)

21X1XZXZ BeAeEE

CBS2:

(PWD)

Let’s introduce empirical parameters ...

A Broader Test of Basis Set Extrapolation Schemes

(H3C)2N-NO2,

H2C=N-CH3, cis HONO, 2A1 NO2, 2A2" CH3, H,

H2C=NH, HCN, NH3, 2B2 NH2, 3g- NH,

HNO, 2 NO, N2O,3B2 CH2, H2CO, CO, CO2,

H2O, 2 HO

Definition of CBS2Mf+E(CBS2Mf) = E(CCSD(T)-fc/cc-pVDZ)

+ 0.81 * EMBPT(2)-fc(CBS2 - cc-pVDZ)

For enthalpies of formation (riBP86/TZVP freqs.):

H(CBS2Mf+) = H(CBS2Mf) + 0.16 + nR * cR

c(H2) = -0.24

c(CH4) = -0.31

c(N2) = -0.17

c(O2) = -1.81

Determined by minimizing RMS of fHc-fHe.

Average error: 0.00 Standard deviation: 0.75 kcal/mol

Properties of the Extrapolation Scheme

• Standard deviation 0.75 kcal/mol.

• Using small reference molecules saves more expensive calculations.

• T2 diagnostic allows judgement of reliability for every

molecule! (Max. T2 < 0.15 is good. Calculations with

larger T2 may be unreliable.)

• Anions need diffuse basis sets in the gas-phase.

H2N-NO2 -> H2N. + NO2

. singlet & triplet

0

20

40

60

80

100

120

1 2 3 4 5 6 7

rNN

Ere

l

[kc

al/

mo

l]

t-MBPT(2)-fc/DZ

t-CCSD-fc/DZ

t-CCSD(T)-fc/DZ

t-MBPT(2)-fc/CBS2(XZ)

t-CCSD-fc/CBS2(XZ)

t-CCSD(T)-fc/CBS2(XZ)

100 * max. T2 (t)

A Difficult Reaction Path

A Difficult Reaction PathH2N-NO2 -> H2N

. + NO2. singlet & triplet

0

20

40

60

80

100

120

1 2 3 4 5 6 7

rNN

Ere

l

[kc

al/

mo

l]

MBPT(2)-fc/DZ

CCSD-fc/DZ

CCSD(T)-fc/DZ

CCSDT-1a-fc/DZ

MBPT(2)-fc/CBS2(XZ)

CCSD-fc/CBS2(XZ)

CCSD(T)-fc/CBS2(XZ)

100 * max. T2 (s)

t-MBPT(2)-fc/DZ

t-CCSD-fc/DZ

t-CCSD(T)-fc/DZ

t-MBPT(2)-fc/CBS2(XZ)

t-CCSD-fc/CBS2(XZ)

t-CCSD(T)-fc/CBS2(XZ)

100 * max. T2 (t)

rNN=

N-N bond breaks

riBP86/TZVP Energies of Some Primary Reaction Pathways in kcal/mol

DFT CBS2M

(H3C)2N-NO2 rE aE rH rH

(H3C)2N· + NO2

· 46 ~49 43. 49

H3C-N-NO2· + CH3

· 83 = 79 84

H2C·-N(CH3)-NO2 + H· 96 = 90 97

H2C=N-CH3···HONO -1 ~51 -3 -7

H3C-N=NO2CH3 13 ~51 12 11

H2C=N(CH3)-NO2H (w. H2O) (45) = -

DFT Energies of Secondary Reactions aE w.r.t.

[kcal/mol] rE aE rH DMNA

(H3C)2N 3 H3C-N + CH3

74.7 = 69.7 121

(H3C)2N H2C

-NH-CH3 -2.9 ~39 -2.2 85

H2C-NH-CH3 H2C=NH + CH3

25.1 ~29 20.7 72 (85)

H3C -N-NO2 H2C-NH-NO2 -7.2 ~39 ? 122

H2C-NH-NO2 H2C=NH + NO2

-7.9 ~4 ? 80 (122)

{H3C -N-NO2 H2C=NH + NO2 -15.1 ~39 -17.4} 122

H2C-NCH3-NO2 H2C=NCH3 + NO2

-9.1 ~3 -11.4 99

H2C-NCH3-NO2 H2C=NNO2 + CH3

33.1 ~38 28.8 134

A Small Summary The CBS2Mf+ extrapolation scheme gives enthalpies

of formation with an RMS error of 0.75 kcal/mol.

The T2 diagnostic indicates reliability of results.

While riBP86/TZVP is usually within 5-10 kcal/mol of the CBS2Mf results, the shape of the curves may be quite different.

Many reactions have been calculated at the CBS2Mf level, but this work is not yet complete.

Other Things We Can Do

Calculate triplet states (possibly important for strongly deformed geometries).

Use excited state methods (not quite fire and forget).

RDX

— Gas Phase Dynamics of RDX —

(unimolecular thermal decomposition)

• Purpose: Investigate/Reproduce the findings of Lee et-al1

with respect to the primary event

• Methodology: CCG2MP2/SCFFAF2 (and SCFSCF)

1) Zhao, Hintsa, Lee; JCP 88 801 (1988)

2) Runge, Cory, Bartlett; JCP 114 5141 (2001)

RDX

0.67

0.33

3 (H2C-N-NO2) - concerted

C3H6N5O4 + NO2 - simple bond rupture

Primary Event1

1) Zhao, Hintsa, Lee; JCP 88 801 (1988)

2nd-Order Reactant and Transition State

C3v C3

62.5 kcal/molUpper bound

Current

• Concerted - CCG2MP2 kinetic barrier and reaction swath

information generation

• SBR - Reactant and TS structure optimizations

Future

• Determine the theoretical reaction rates, k(T), and branching

ratios of the primary event

• The future direction of the dynamics work depends on what we

learn from the current effort

Compressed Coupled Cluster

Singular Value Decomposition Approach to

Coupled Cluster Calculations

Osamu Hino1, Tomoko Kinoshita2

and Rodney J. Bartlett1

Quantum Theory ProjectUniversity of Florida1

Graduate University for Advanced Studiesand Institute for Molecular Science, Japan2

Application of the coupled cluster method to larger systems　→　 several bottlenecks (CPU, Memory, Disk)

Background

Integral direct algorithmParallelization of programsLocal Correlation method

Important to exploit another approach

Use of Singular Value DecompositionCompressed CC method

Singular Value Decomposition (SVD) (1)

1 2

,

0

,

k

m

T Tk

T

A m n m n

U m

A USV

m

S m m s s s

V m n

× matrix,

orthogonal matrix, u

diagonal matrix,

row orthogonal matrix v

singular valuesleft singular vectorsright singular vectors

Singular Value Decomposition (SVD) (2)

1 1

2

1 1

,

,

m llT T

k k k k k kk k

m nl l

ij iji j

A s A s

d A A A A

u v u v

A(l) is the closest rank l matrix to A. SVD is a useful mathematical tool because of this remarkable property. If su (u>l) is nearly equal to zero, we can reconstruct the matrix A without losing much information.

CCD 1ab abij ijt t

Application of SVD to the Coupled Cluster Doubles (CCD) Amplitude (1)

First, we choose an approximate CCD amplitude. The simplest one is MBPT(2) amplitude. We assume the Hartree-Fock reference.

1 2

1

,

W

ab ij abij p p p

p

t s U V W O

The singular values which are less than the threshold are neglected.

SVD

Application of SVD to the Coupled Cluster Doubles (CCD) Amplitude (2)

We can define the following contracted two-electron creation and annihilation operators according to the SVD of the approximate amplitude.

† † † , ab ijA A a b I I j i

ab ij

C V a a A U a a Then we can define the approximate cluster operator.

†2

1

2

W WAI A I

A I

T t C A

Physical meaning of the procedures (1)

1 † †HF

1 † †HF

†HF

1

2

1vac

2

1

2

abij a b j i

abij

abij a b j i

abij

P P PP

t a a a a

t a a a a

s C A

exact HF MBPT(2):

Reduced density matrix for 1 1 1 1

, ,,ab ab ab cdij kl ij kl ab cd ij ij

ab ij

t t t t

Application of SVD to the Coupled Cluster Doubles (CCD) Amplitude (3)

The CCD equation becomes,

2 2HF HF HF, 0.T TA

Ic c

E He He

4 4

,

,

cd C ab cdij B A C

cd C abcd

ab cd t A C t A C V ab cd V

ab cd V A C O

Degrees of freedom of the equation2 2

2 2 2W O

V O V

Most expensive term in CCD calculation

Integral transformation is required only once.

Improvement of the quality of calculated results

Use better approximate amplitude (e.g. MBPT(3)…).

CCD 2 CCD 1, ,ab ab ab abij ij ij ijd t t d t t

Tighten the threshold.

†2

†2

1

2

1,

2

W WAI A I

A I

W WAI A I

A I

T t C A

T t C A W W

・ The CCSD model is one of the most reliable quantum chemical methods. However, it is often necessary to incorporate higher order cluster operators than connected doubles to achieve the chemical accuracy.

Background

・ CCSDT, CCSDTQ, and CCSDTQP are implemented and they produce highly accurate computational results. But they are too expensive to be performed routinely.・ Perturbative approach such as the CCSD(T) or CCSD(TQ) is one possible solution for this problem. But still there is a problem that the perturbative approaches are stable only in the vicinity of equilibrium molecular geometry.

Purpose of this study

To develop theoretical framework (1) including the connected triples (2) accurate(3) less expensive (4) stable under deformed molecular geometry

Compression of the connected triples

2

1

OVabc ai bjckijk X X X

X

t s Q R

(1) Apply SVD to the second order triples

† , 1,2, ,aiX X

ai

C Q a i X K (2) Create contracted mono-excitation operators

(3) Truncate the mono-excitation operator manifold

3 2 2 , 1K O V

(4) Compressed T3 cluster operator

3

1

6 XYZ X Y ZXYZ

T t C C C

Easy to manipulate T3 amplitude

Compressed CCSDT method

1 2 3 1 2 3

1 2 3 1 2 3

HF HF HF

HF HF

HF

e ,0 e

0 e ,0 e

T T T T T Tai

c c

T T T T T Tabij XYZ

c c

XYZ X Y Z

E H H

H H

C C C

(1) Equations

(2) Equations for connected triples

,

abc abc abcijk ijk ijk

XX X YZ YY XY Z ZZ XYZ XYZX X X

a ai aiXX i X X

ai

D t R

D t D t D t R

D D Q Q

Compressed CCSDT-1 method

Approximate treatment for the T3 amplitude

1 2 3 1 2

1 2 3

HF 3 HF

HF 2 HF

0 e 0 e

CCSDT-1a

no approximation for the T2 amplitude

CCSDT-1b

0 e 0 1

T T T T Tabij XYZ

c c

T T TXYZ XYZ cc

H H T

H H T

・ Easiest to implement・ Operation count for T3 amplitude scales as K2V2O・ Iterative counterparts of CCSD[T] and CCSD(T)

Potenrial Energy Curve (1) (HF, aug-cc-pVDZ, HF-bond stretcing)

-300

-250

-200

-150

-100

-50

0

0.5 1 1.5 2 2.5 3 3.5 4

MRCICCSDCCSD(T)CCSDT-1COMP.SDT-1

(E+

100)

*100

0 (a

.u.)

r(HF)/r(eq)

r(eq)=1.733 bohr=0.25

Potential Energy Curve (2) (H2O, aug-cc-pVDZ, OH-bonds stretching)

-300

-200

-100

0

100

0.5 1 1.5 2 2.5 3 3.5

MRCICCSDCCSD(T)COMP.SDT-1CCSDT-1

(E+

76)*

1000

(a.

u.)

r(OH)/r(eq)

r(eq)=1.809 bohr=0.25

SUMMARY OF PROGRESS

• Detailed study of nitramines to establish the accuracy of various quantum-mechanical results for application to uni- and bi-molecular reactions.

• Initial investigation of comparative reaction paths for DMNA with the goal of providing definitive results.

• Application to RDX to help resolve the nature of the initial step in its decomposition.

• Introduced compressed coupled-cluster theory as a new tool that can provide CC quality results at a fraction of the current cost.

PLANS FOR 2003-2004

• Resolve issue of comparative energetics among HONO elimination, loss of .NO2, and NO2 ONO in prototypical nitramines.

• Complete work on primary decomposition of dimethylnitramine and reactions of decomposition products with each other and new dimethylnitramine.

Apply the methods used with dimethylnitramine to RDX and compare to relative reaction rates from established technology.

Extend the compressed CC method to full triples and factorized quadruples.

Formulate analytical gradients for compressed CC.

Backup Slides

Enthalpies of formation for H2N-NO2 et al. (gas-phase)

cis HONO, NO2 (2A1), HN-NO2-, NH3, H2N (2B2),

HN (3g-)

average error [kcal/mol]B3L/6 CBS2M CBS2 CBS3 G2 CBS-Q

all 0.7 0.0 -0.4 -0.7 -2.2 -1.1no HN, HN-NO2

- 0.0 -0.2 -0.5 -0.6 -1.7 -0.6

H2N-NO2 -> H2N. + NO2

. BS dependence

0

20

40

60

80

100

1 2 3 4 5 6 7

rNN

Ere

l [

kca

l/mo

l]

MBPT(2)-fc/DZ

CCSD-fc/DZ

CCSD(T)-fc/DZ

MBPT(2)-fc/CBS2(XZ)

CCSD-fc/CBS2(XZ)

CCSD(T)-fc/CBS2(XZ)

100 * max. T2 (s)

MBPT(2)-fc/ADZ

CCSD-fc/ADZ

CCSD(T)-fc/ADZ

MBPT(2)-fc/CBS2(AXZ)

CCSD-fc/CBS2(AXZ)

CCSD(T)-fc/CBS2(AXZ)

100 * max. T2 (s)

Relative energy in kcal/mol along possible bimolecular reaction paths

C) 2 Me2N-NO2 MeN-NO2

. + Me2N-NO2Me

.

N N+

O-

OH

HH

HHH

NN+

O-

O

H

HH

HH

H

NN

+

O-

O

HH H

H

HH

NN

+

O-

O

CH HH

H

HH

r(CN) r(CO) riBP86   MBPT(2)-fc

1.460 3.450 0.0    

1.462 2.940 2.6    

1.468 2.440 7.2    

1.485 2.140 18.4    

1.568 1.840 46.8    

2.2813.247

1.640 64.157.0

   

  1.440      

The Very End