Ab Initio Theoretical Ch i l Ki tiChemical Kinetics

CEFRC Summer SchoolPrinceton

June, 2010

Stephen J. Klippenstein

Ab Initio TheoreticalChemical Kinetics

Stephen J. KlippensteinJames A. Miller

Lawrence B. HardingYuri Georgievskii

Outline1. Overview of Ab Initio Transition State Theory

a) Transition State Theory (TST)b) Introduction to Electronic Structure Theoryc) Multireference Electronic Structure Theory

2. Ab Initio TST for Different Classes of Reactiona) Tight TST for Reactions with Barriersb) TST for Radical-Radical Reactionsc) Multiple Transition States and Dynamics

3. Pressure Dependent Kineticsa) Single Well Reactionsb) Multiple Well Time Dependent Master Equationc) Examples

Review ArticlesThe Current Status of Transition State TheoryDonald G. Truhlar, Bruce C. Garrett, and Stephen J. KlippensteinJ. Phys. Chem. (Centennial Issue) 100, 12771-12800 (1996).RRKM Theory and Its ImplementationStephen J. Klippenstein, in Comprehensive Chemical Kinetics, 39, 55, (2003).Modeling the Kinetics of Bimolecular ReactionsAntonio Fernandez Ramos, James A. Miller, Stephen J. Klippenstein, and Donald G.Truhlar, Chem. Rev., 106, 4518-4584 (2006).Predictive Theory for Hydrogen Atom-Hydrocarbon Radical AssociationKineticsLawrence B. Harding, Yuri Georgievskii, and Stephen J. KlippensteinJ. Phys. Chem. A, feature article, 109, 4646-4656 (2005).Predictive Theory for the Association Kinetics of Two Alkyl RadicalsStephen J. Klippenstein, Yuri Georgievskii, and Lawrence B. HardingPhys. Chem. Chem. Phys., invited article, 8, 1133-1147 (2006).Master Equation Methods in Gas Phase Chemical KineticsJames A. Miller and Stephen J. KlippensteinJ. Phys. Chem. A, feature article, 110, 10528-10544 (2006).Ab Initio Methods for Reactive Potential SurfacesLawrence B. Harding, Stephen J. Klippenstein, and Ahren W. JasperPhys. Chem. Chem. Phys., 9, 4055-4070 (2007).

Transition State Theory

1. Motivation2. Dynamical Derivation and the Variational

Principle3. Partition Functions, Numbers of States,

and Density of States4. Conventional Transition State Theory5. Accuracy

MotivationChemical Models Need Rate Coefficients for 1000’s of ReactionsExperiments• Too time consuming• Often can’t perform experiments at T,P of interestTheory• Empirical Models

– Correlations and extensions of related experiments– Greater accuracy desirable

• Classical Trajectories– Need global potential energy surface– No quantum effects– Timescale ~ 1 ns or shorter

• Quantum Dynamics– Too computationally intensive– 4 Atom Systems

• Transition State Theory– Compromise of Efficiency and Accuracy

Ab Initio Transition State Theory

How accurate is transition state theory?• Depends on how it is implementedHow efficient is it?• Depends on how it is implementedGenerally can get better than a factor of two accuracy with ~ a day of

effortImplementation requires an evaluation of the partition function for

both the reactants and the transition stateDetermine these from information on structure, energy, and

rovibrational properties of reactants and transition stateUse ab initio electronic structure theory to determine these properties

=> Ab Initio Transition State Theory

k(T) = kBTh

Q±

Qreac

exp(−E ± /kBT)

Theory of Unimolecular and Recombination Reactions; R. G. Gilbert and S. C. Smith, Blackwell Scientific

k = dq∫ dpP(q,p)δ s− s±( )χ q,p( ) psm

Trajectory Expressionfor the Rate ConstantP(q,p)probability of specific q,pStatisticalAssumptionCanonical

Microcanonical

P(q,p) =exp −βH(q,p)[ ]dqdpexp −βH( )∫

P(q,p) =δ E − H( )

dqdpδ E − H( )∫Microcanonical J-Resolved

P(q,p) =δ E − H( )δ J − JT( )

dqdpδ E − H( )δ J − JT( )∫

s=s± defines dividing surface ps/m = velocity through s±

χ= reactivity function•1 if trajectory is reactive•0 otherwise

Transition State Assumptionand the Variational Principle

Transition State Assumption• Assume all trajectories with positive velocity

through dividing surface are reactive• Replace χ with Θ (ps/m)

In Reality Θ (ps/m) ≥ χ• k TST ≥ k Trajectory

⇒Variational PrincipleVariational Transition State Theory

• Consider family of TS dividing surfaces• Best Dividing Surface is one that provides

minimum estimate for k

Transition State Theory (TST)

Canonical

k(T) = kBTh

Q±

Qreac

exp(−E ± /kBT)

Microcanonicalk(E) = N ±(E)

hρreac (E)

QQuan = exp −βEi( )i

∑

Qclass =dqdphn

exp −βH( )∫

NQuan (E) = Θ E − Ei( )i

∑

QclassαC(T,mi) dqexp −βV( )∫

Nclass E( ) = dqdpΘ E − H( )∫ /hn

Nclass E( )αB(mi) dq E −V( )∫n / 2

ρ(E) = ∂N /∂E

Microcanonical - J Resolved k(E,J) = N ± (E,J) / hρreac (E,J)[ ]

Thermodynamic Formulation of TSTEquilibrium Constant = ratio of Partition Functions

Keq =Qproducts

Qreactants

= exp −ΔGkBT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

k(T) = kBTh

Q±

Qreac

exp(−E ± /kBT)

k(T) =kBT

hexp −

ΔG±

kBT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

k(T) =kBT

hexp −

ΔH ±

kBT

⎛ ⎝ ⎜

⎞ ⎠ ⎟ exp ΔS±( )

Variational =>Maximize ΔG±

Maximize ΔH±

Minimize ΔS±

Conventional Transition State Theory• Fixed Transition State Location

– Saddle point on potential Energy Surface• Rigid Fragments

– Erot (J,K) = B J(J+1) + (A-B) K2

– A,B are rotational constants– J,K are rotational quantum numbers

• Harmonic Vibrations– V(qi) = kqi

2/2– Evib (vi) = hνi (vi+ 1/2)– νi is the harmonic vibrational frequency– vi is the quantum number for vibrational mode i

Rigid Rotor Harmonic Oscillator (RRHO)

Statistical Mechanics, D. A. McQuarrie, Harper & Row

Canonical Partition Functions

Q = QtransQrotQvibQelec

Qtrans =2πMkBTh2

⎛ ⎝

⎞ ⎠

32

V

Qrot =π12

σ8π 2IakBT

h2⎛ ⎝ ⎜

⎞ ⎠ ⎟

12 8π 2IbkBT

h2⎛ ⎝ ⎜

⎞ ⎠ ⎟

12 8π 2IckBT

h2⎛ ⎝ ⎜

⎞ ⎠ ⎟

12

Qvib

quan = Πi=1

n exp(−hν i /2kBT)

1− exp(−hν i /kBT)Q

vib

class = Πi=1

n kBT

hν i

Qelec = gi exp −Ei

kBT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

i=1∑

Number of States

N(E,J) = Θ E − Evibi− Erot J ,K( )

i,K∑

N(E) = dε∫ Nvib E − ε( )ρrot ε( )

N(E) = N(E,J) 2J +1( )J∑

Beyer-Swinehart for Nvibωj = vibrational frequency

for mode j1. Nvib(i) = 1, for all i2. For j = 1,n3. For i = ωj,m4. Nvib (i) = Nvib (i) + Nvib

(i- ωj)5. End i loop6. End j loopρrot (E) =

∂Nrot E( )∂E

=2

σE

B2C⎛ ⎝

⎞ ⎠

1/ 2

Theory of Unimolecular and Recombination Reactions; R. G. Gilbertand S. C. Smith, Blackwell Scientific

Reaction Path Degeneracies

Multiply partition function bym

σm = number of optical isomers σ = rotational degeneracy of given configurationDo for both reactants and transition stateYields correction to rate constant that is equal to the reaction pathdegeneracy

m±σmσ ±

Beyond Conventional RRHO TST AnharmonicitiesSeparable - With Beyer SwinehartSums and Densities of Fully Coupled Anharmonic Vibrational States: A Comparison

of Three Practical MethodsT. L. Nguyen, J. R. Barker, J. Phys. Chem. A, 114, 3718-3730 (2010).

Spectroscopic Perturbation TheoryRelate power series expansion of E in quantum numbers vi to higher orderderivatives of potentialProblems due to resonances and due to turnovers in E(vi)

Monte CarloQ - ∫ dq1 .. dqn exp[-βV(q1,…dqn)]

Pitzer-GwinnQ quantum,HR = Q quantum,HO * Q class,HR / Q class,HO

Path IntegralsV. A. Lynch, S. L. Mielke, D. G. Truhlar, J. Chem. Phys. 121, 5148 (2004)

VibSCF; VibCI - MULTIMODE - Bowmanhttp://www.chemistry.emory.edu/faculty/bowman/multimode/

Beyond Conventional RRHO TST TorsionsMoments of Inertia - Coupling of external and internal momenta

A. L. L. East, L. Radom, J. Chem. Phys. 106, 6655 (1997).Torsional Modes - Hindered Rotors

Q ~ φ exp[-V(φ)/kBT)]Pitzer-GwinnNonseparability of multiple torsional modesIntramolecular Hydrogen Migration in Alkylperoxy and Hydroperoxyalkylperoxy

Radicals:Accurate Treatment of Hindered Rotors

S. Sharma, S. Raman, and W. H. Green, J. Phys. Chem. A, 114, 5689-5701 (2010).

An extended hindered rotor model with incorporation of Coriolis and vibration-rotational coupling for calculating partition functions and derived quantities

P. Vansteenkiste, D. Van Neck, V. Van Speybroeck, and M. Waroquier, J. Chem.Phys. 124, 044314 (2006).

How should we calculate multi-dimensional potential energy surfaces for an accuratereproduction of partition functions?P. Vansteenkiste, V. Van Speybroeck, E. Pauwels, and M. Waroquier, Chem.Phys. Lett. 314 109-117, (2006).

Beyond Conventional RRHO TSTVariational Transition State Theory

Follow reaction path - Imaginary mode at saddle pointW. L. HaseD. G. TruhlarW. H. MillerProject vibrational frequencies along reaction pathEvaluate TS partition function along reaction path;Find minimum

• Canonical - k(T) = min {∫ dE dJ P(E,J) k(E,J,R); R}• Microcanonical - k(T) = ∫ dE P(E) min {∫dJ P(J) k(E,J,R);R}• E, J Resolved - k(T) = ∫ dE dJ P(E,J) min { k(E,J,R);R}

Beyond Conventional RRHO TSTTunneling Corrections (Γ)

Most Important - Calculate Imaginary Frequency AccuratelyOne Dimensional

Wigner - Perturbative ExpansionΓ = 1 - (h ν / kB T)2 /24

Eckart - Asymmetric form parametrized by reactant, product andsaddle point energies

Analytic form for ΓMultidimensional and Corner Cutting

Small Curvature (SCT)Large Curvature (LCT)Optimized Multidimensional Tunneling (OMT)

Polyrate - Freewarehttp://comp.chem.umn.edu/polyrate/

Variational transition state theory with multi-dimensional tunneling[1]

AB + C

A + BC

Saddle point

R(A-B)

R(B

-C)

Optimizedgeneralizedtransition state

[1] For example, see reviews: Truhlar, Garrett, and Klippenstein J. Phys. Chem. 1996, 100, 12771. Fernandez-Ramos, Ellingson, Garrett, and Truhlar, Rev. Comp. Chem. 2007, 23, 125.

One-dimensionalWigner tunnelingpath

Minimum-energypath

Multi-dimensional Tunneling path

SCT ≡ small-curvature tunnelingLCT ≡ Large-curvature tunnelingOMT ≡ optimized multidimensional tunneling

≡ max SCT LCT

Accurate Electronic Structure – Move along diagonalBasis Set

MethodAM1

HF MP2 CCSD(T)

CASPT2 Full CICCSDTQ

CASSCF CISD MRCI Act. ↑

DFT

cc-pVTZ

aug-cc-pVDZ

cc-pVDZ6-31G*

Sto-3G

3-21G

cc-pVQZ

aug-cc-pVQZ

aug-cc-pVTZ

6-311++G(d,p)

6-311++G(3df,2pd)

Quantum Monte Carlo

cc-pCVDZ

Complete Basis Set (CBS) Full CI with CBS

Chemical AccuracyChemical Accuracy ~ 1.0 kcal/mol

CCSD(T) and big enough basis setCheat with Additivity approximations --

e.g., MP2/Big +CCSD(T)/small - MP2/smallDFT Geometry OptimizationsGreat for equilibriumNeed to be much more careful for transition states

Subchemical Accuracy ~0.1 kcal/molMany factors that affect energies at this levelFocal Point AnalysisCCSD(T) geometries; MP2-R12; CCSDTQCore Correlation; Diagonal Born Oppenheimer CorrectionsSpin-Orbit Coupling; Vibrational Anharmonicities

Ab Initio Transition State Theory– Errors from Electronic Structure and State Counting Should be Comparable

Electronic Structure

KineticMethodologyRRK

Classical Trajectories

QRRK

Quantum Dynamics

VRC-TST

Semiclassical Dynamics

Path IntegralsMultiple TS’s

ConventionalRRHO-TST

cc-pVDZ

cc-pVQZ

cc-pVTZ

Quantum Scattering Theory Full CI / CBS Potential

VTSTTunnelingAnharmonic

6-31G*

Sto-3G

3-21G

CBS

MP2

HF

Full-CI

CCSDTQ

CCSD(T)

MRCI

CASPT2

DFT

Kinetic Accuracy

Kinetic Accuracy ~ Factor of 2Energy to chemical accuracyTransition State TheoryRRHO Energy LevelsEckart Tunneling

Subkinetic Accuracy ~ 20% AccuracyMany factorsMultidimensional Tunneling; Variational EffectsAnharmonicities; Transition State RecrossingEnergy Transfer Probabilities; 2D Master Equation

Empirical Normalization - Adjust some parameter to reproduceexperiment hope for good interpolation/extrapolation

Introduction toElectronic Structure Theory

1. Hartree-Fock (HF)2. Second-Order Moller Plesset Perturbation

Theory (MP2)3. Configuration Interaction (CISD)4. Coupled Cluster Theory [CCSD(T);

QCISD(T)]5. Basis Sets6. High Level Schemes7. Density Functional Theory

Introduction to Computational Chemistry, Frank Jensen, Wiley

Hartree-Fock (HF)Born Oppenheimer - electrons onlyProvides potential V for nuclear motions

φk = Spin-OrbitalsΨk = Configuration State Functions (CSF)

Self Consistent Field (SCF)Mean FieldIterative

Fock Operatorh = one electron; J = CoulombK = exchangeHartree-Fock Equations

Spin-symmetryUnrestricted or Restricted

Helec = −1

2∇ j2 +

Za

rjaa∑

⎛

⎝ ⎜ ⎞

⎠ ⎟ j∑ +

1

rjkj>k∑

Ψ0 = φ1 r1( )LφN rN( )

hi +Vmean r1( )( )φ j r1( ) = ε jφ j r1( )

Vmean r1( ) ~ drk∫ φk* rk( ) 1

r1kφk rk( )

k≠1∑

Fφi = hφi + J j − K j( )j occupied∑ φi

Fφi = ε i, jφ j

Self-Consistent determination of Vmean and φi

HelecΨelec = VΨelec

LCAO-MO ExpansionSpin-Orbitals φi are Molecular OrbitalsExpand them in Atomic Orbitals χμ

φi = Cμ,iχμμ∑

Expand Atomic Orbitals in Gaussian Basis Sets

χμ r,θ,φ( ) = Na,b,c xa ybzc dμi exp −α ir

2( )i∑

a, b, c => s, p, d, …αi => 1s, 2s, 3s, …Polarization FunctionsDiffuse Functions

Beyond HF and Mean Field

Make wavefunction more general - more CSFs

Ψ1s = φs r1( )φ2 r2( )LφN rN( )

Ψ12sr = φs r1( )φr r2( )LφN rN( )

Singly Excited

Doubly Excited

Ψ = Ψ0 + cirΨi

r

i,r∑ + cij,rsΨij

rs

i,r∑ +L

…Correlation Energy = Difference From HFPerturbation TheoryConfiguration InteractionCoupled-Cluster TheorySingle Reference vs. Multiple Reference

Second Order Perturbation Theory - MP2Rayleigh Schrodinger Perturbation TheoryE = E0 + V00 + ∑V0iVi0/(E0-Ei) + …Consider sum of Fock operators as reference H

Single ReferenceEffect of Single or Double excitations from ground state CSFSize consistent - Energy of (A+B) = Energy of A + Energy of BNonvariationalGround State onlyMPn = oscillatory then divergesMP3 - considers up to triple excitationsMP4 - considers up to quadruple excitations

E = EHF +Ψ0H'Ψij

rs ΨijrsH'Ψ0

EHF − Eijrs

r<s∑

i< j∑

Singles and Doubles Configuration Interaction

Consider single or double excitations from HF groundstate (CISD)

Not PerturbativeExcited StatesVariationalNot size consistent - Energy of (A+B) ≠ Energy of A +

Energy of BConverge to exact answer with triples, quadruples,

etc. => Full CI - not practical

Ψ = Ψ0 + cirΨir

i,r∑ + cij,rsΨij

rs

i,r∑

Coupled ClusterCCSD(T) - Gold standard - Accuracy <0.5 kcal/molΨCC = exp(T)Ψ0

T = T1 + T2 + T3 +L

T2Ψ0 = tijrs

i> j;r<s∑ Ψij

rs

CCSD, CCSDT, CCSDTQ, … Truncation of T ExpansionCCSD(T) = CCSD + Perturbative TriplesQCISD(T) ~ CCSD(T) with slightly fewer terms

– conceptually worse but empirically betterSize consistent; Not VariationalT1 diagnostic - Measure of Multireference Effects<0.02 Closed Shell; <0.03 Radicals; >0.06 Catastrophe

Include excitations to all orders viaexponentialTi creates excitations of order i from Ψ0

Pople Basis SetsSto-3G too small• minimal basis set - 1 function for each electron• 3 => each atomic orbital is a fit to 3 Gaussians3-21G still too small• Valence double zeta - two basis functions for each valence orbital• 3 => number of Gaussians for core orbitals• 2,1 => implies number of Gaussians for 1st and 2nd valence orbitals6-31G* Smallest basis set to provide semiquantitative accuracy• Valence double zeta• * => polarization functions, but only on heavy atoms (C, O, N; not on H)6-311++G**• valence triple zeta, polarization & diffuse (+) functions on all atoms6-311++G(3df,2pd) , etc.Polarization functions strongly effect geometry and energyDiffuse very important for negative ions and for hydrogen bondingHigher level schemes employ 6-31G* and larger

Dunning Basis Setscc-pVNZ; N=D,T,Q,5, …

Correlation-consistent polarized valence n zetasystematic convergence to complete basis set (CBS) limitextrapolate from T,Q to CBS usually accurate enough

aug-cc-pVNZ - add diffuse functionscc-cpVNZ - to treat core correlationcc-pV(N+d)Z - for third row atoms, e.g., Scc-pVNZ-PP and cc-pVNZ-DK

Effective Core Potentials - Treat Core Electrons in effectivemannerTransition Metals, etc

Frequencies - aTZ, TZ, or aDZ for quantitative accuracycc-pVNZ-F12 - explicitly correlated basis and methods

treats cusp at r12 = 0get CBS results with TZ basis

Basis SetsBasis Set Superposition Error• More basis sets for monomer when calculating E in

presence of other monomer• For long-range interactions find augmented functions

actually give worse interaction energies• CBS schemes treat BSSE

DFT less sensitive to Basis sets• not trying to converge so no real point in large basis set• some functionals designed for specific basis sets• some sensitivity for low frequencies

High Level Schemes EfficientG3, G3MP2, G3B3, G3X, G4 - Pople

QCISD(T)/small; MP2 and MP4 large; Pople basessmall = 6-31G*

CBS-QB3, CBS-APNO - Peterssonlike GnQB3 small ~ 6-31+G*; APNO small = (14s9p4d2f)/[6s6p3d2f]

BAC - Melius Bond Additivity Correction

corrections related to known errors for specific bond types

Multicoefficient methods - TruhlarOptimize coefficients in sets of calculations with different methods andbasis setsFor both Gn-like methods and for DFT based methods

Approaches valuable for more than 7 heavy atoms

High Level Schemes AccurateFocal Point - AllenHeat - StantonW1, W2, W3, W4 - Martin

Increasingly more accurate from W1 to W4geom CCSD(T)/cc-pV(Q+d)ZCCSD/CBS from AV5Z, AV6ZCCSD(T)/CBS from AVQZ, AV5ZCCSDT/CBS from DZ,TZCCSDTQ/DZ and CCSDTQ5/DZcore correlation, spin-orbit, scalar relativistic, DBOC, Exptl E00.1 kcal/mol accuracy

My approach - Intermediate Accuracyopt and freq from B3LYP/6-311++G(d,p) or CASPT2QCISD(T)/CBS from TZ, QZ~0.5 kcal/mol for equilibrium~1 kcal/mol for TS

Zheng, Zhao, Truhlar, J. Chem. Theory Comput. 5, 808 (2009)

Zheng, Zhao, Truhlar, J. Chem. Theory Comput. 5, 808 (2009)

Density Functional TheoryGreat for geometry searchesGreat for vibrational frequenciesEnergies are more troublesomeNo way to converge to correct answerLots of functionals with something appropriate for every

situationBut have to choose correct oneTruhlar’s M08 functionals appear useful for energies;

especially for larger moleculesB3LYP is not the best functional but it is still widely used

because it is one of the more reliable onesNo dispersion in B3LYP

Commerical Software PackagesGAUSSIAN - Pople

good at lots of things, probably not best at anything exceptas a black box

I use for DFT geometry optimizationsMOLPRO - Werner and Knowles

my favoritegreat for CCSD(T); CASPT2; MRCI;DFT needs analytical second derivatives

QCHEM - Head-GordonJaguar - Schaefer

Freeware Electronic Structure PackagesACES - Bartlett,

Coupled cluster theory with analytical second derivativesCFOUR - Stanton, Gauss

Alternative/Successor to ACESMRCC - Kallay

Advanced coupled cluster theoryAdd on to MOLPRO or to CFOUR

PSI - SchaeferMultireference coupled cluster

COLUMBUS - Shepard (Argonne)Multireference CI

GAMESS - GordonGeneral purpose, DFT, CASSCF, CR-CCSD(T), IRC

NWCHEM - PNNLGeneral purpose, Massively parallel

Multi-ReferenceElectronic Structure Theory

1. Motivation2. Complete Active Space (CAS) Wavefunction3. Second Order Perturbation Theory with CAS

Reference (CASPT2)4. Multi-Reference Configuration Interaction

(MRCI)5. Multi-Reference Coupled Cluster Theory [MR-

CCSD(T)]

CH4 Bond Dissociation Curves - Single Reference

SolidRestricted

DashedUnrestricted

Blue - HF

Orange - MP2

Green - QCISD(T)

Red - CCSD(T)

Black - Full CI

T1 Diagnostic

MulticonfigurationSelf-Consistent Field (MCSCF)

MCSCF - CI with optimization of both coefficients andmolecular orbitals

Ψ = Ψ0 + cirΨir

i,r∑ + cij,rsΨij

rs

i,r∑

Optimize both Cμi and cir, …Don’t consider all single and double excitations

too timeconsuming and not necessaryChoose limited set of excitations based on chemistry

φi = Cμ,iχμμ∑

Ψ0 = φ1 r1( )LφN rN( )

Complete Active Space CASSCFCASSCF

Partition electrons/orbitals into active and inactive spacesInclude all excitations within active space and none within inactivespace

Occupation numbers tell which orbitals need multireference descriptionChoose on basis of deviation of occupation numbers from 0 or 2HOMOs and LUMOs (Highest Occupied & Lowest Unoccupied)Bond Breaking - sigma,sigma* or two radical orbitals - (2e,2o)More generally - include all bonds whose chemistry changesPi bonds should almost always be included

State Averaged - to treat degenerate statesNot Black Box - Calculation will often choose different orbitals in

different regions of spaceOrbital Rotations often helpfulStretch bonds to get correct active orbitals then shrink back down to

correct geometryFull Valence Maximum ~ 14e,14o

Second Order Perturbation Theory withCAS Reference (CASPT2)

CASSCF has no dispersion energyDo perturbation theory from CAS referenceEssentially size consistentProblems with degeneraciesUse shiftVery useful for weakly interacting systems and for geometries

and vibrational frequenciesNot very accurate energies for complete chemical changesMOLPRO - two forms

RS2 - has analytic derivatives but no contractionRS2C - contracted but no analytic derivativescontraction very useful for larger active spacesmakes little difference for small active spacesParallel molpro implies RS2C frequency analysis can be rapid

Energies for multireference configurationsTraditional:

Find active space that treats chemistry of reactants, transitionstate, and productsOften hard to do and energies not especially accurate

Alternative:Find active space for particular multireference geometryEvaluate singlet-triplet splitting with CASPT2 or CAS+1+2+QCEvaluate energy of triplet state relative to reactants andproducts with QCISD(T)

Multi-Reference Configuration Interactionfrom CAS Reference (MRCI)

CAS+1+2 - Include all single and double excitations from CASreference space

CAS+1+2+QC - Davidson correction for higher order excitations

MRCI is not size consistentComparison of CASPT2 and MRCI usefulWhen similar then they are probably both reliableWhen different there is something to think aboutCan try to bring into agreement by expanding active spaceOften method that predicts lower barrier appears to be correct

CH4 Bond Dissociation Curves - Multi-Reference

Blue - CASSCF

Purple - CASPT2

Red - CAS+1+2+QC

Black - Full CIT1 Diagnostic

CH4 Bond Dissociation Curves - DFT and Basis Set

Blue - MPW1K

Purple - B3LYP

Gray - CASPT2

Black - Full CI

Solid - 6-31G* Dotted - adzDashed - atzDot-dash - aqz

C2H6 Bond Dissociation Curves

Blue - MPW1K

Purple - B3LYP

Orange - MP2

Green - CASSCF

Red Solid - CCSD(T)Dashed - QCISD(T)

Gray Solid - CASPT2Dashed - CAS+1+2+QC

Spin-Flip Coupled Cluster Theory Krylov

Renormalized Coupled Cluster TheoryCR-CCSD(T) Piecuch

Multi-Reference Coupled Cluster Theory[MR-CCSD(T)]

MRCI and CASPT2 not accurate enough for some problemsToo much multireference character for CCSD(T) to be

accurateNeed more than single and doubles excitationsMR-CCSD(T) combines the best of both MRCI and CCSD(T)Active research topicFormalism is complicatedState-Specific MR-CCSD(T) limited to a few configurations but

quite appealingTriple excitations in state-specific multireference coupled

cluster theory: Application of Mk-MRCCSDT and Mk-MRCCSDT-n methods to model systemsEvangelista FA, Simmonett AC, Allen WD, Schaefer HF,Gauss J, J. Chem. Phys. 128, 124104, (2008).

Higher OrderCoupled Cluster Theory

CCSDT, CCSDTQ, CCSDTQP, …Great for when multireference effects are significant but

not overwhelmingApproximate treatment of higher excitations in

coupled-cluster theory. II. Extension to generalsingle-determinant reference functions andimproved approaches for the canonical Hartree-Fock case

Kallay M, Gauss J, J. Chem. Phys. 129, 144101(2008).

MRCC program by Kallay

Multi-Reference SummaryMust use multi-reference approach for any singlet diradical or other

low spin polyradicalCASPT2 is a great approach for geometry optimization and

frequency analysisMOLPRO has a very nice set of CASPT2 routinesCASPT2 is a great approach for weakly interacting species - e.g.,

long-range radical-radical interactionsMRCI+Q is sometimes but not always better than CASPT2Use singlet triplet splitting trick when you canCommon mistakes:

1. Use of CCSD(T) for singlet diradicals when T1 diagnostic islarge

2. Use of CAS for geometry optimizations - it’s just notaccurate enough

3. Assumption that MRCI+Q energies are highly accurate