Calculating Molecular Binding Energies from Chemical Bonds to van der Waals Interactions Thom H. Dunning, Jr. Joint Institute for Computational Sciences.

Calculating Molecular Binding Energiesfrom Chemical Bonds to van der Waals Interactions

Thom H. Dunning, Jr.

Joint Institute for Computational SciencesUniversity of Tennessee – Oak Ridge National Laboratory

Oak Ridge, Tennessee

Joint Institute for Computational Sciences

Outline of Seminar

Statement of the Problem

Theoretical Methods

Basis Sets and Error Analysis Correlation Consistent Basis Sets Errors in Molecular Calculations

Errors in Molecular Binding Energies (for Four Classes) Intrinsic Errors of Methods Basis Set Convergence Errors Statistical Analysis of Errors in Binding Energies

Conclusions

Statement of Problem

Wide Range of Binding Energies

0.01 0.1 1.0 10.0 100.0 1000.0}

He2

De (kcal/mol)

Ne2 Ar2

Ar-HCl

N2-HFH-CO CO

H-CO-

H-C2H2

H-C2Hn

H-CHn

(HF)2

Ar-HF

N2

Statement of Problem

Importance of Electron Correlation

HF Expt’l

HF 100.3 141.6

N2 122.3 228.4

F2 -27.0 39.0

(HF)2 3.7 4.6

N2-HF 1.27 2.22

He2 – 0.0218

De (kcal/mol)

Chemical Bonds

Hydrogen Bonds

van der Waals “Bonds”

Electrostatic “Bonds”

Theoretical Methods

Configuration Interaction

e = 0 + Ciai

a +

Cijabij

ab + …

He C = Ee C

Long history in electronic structure theory

Very flexible, e.g., can describe both ground and excited states

Not size extensive/consistent

Perturbation Theory

He = H0 + H1

e = 0 + 1 + 22 + …

Ee = E0 + E1 + 2E2 + …

Most widely used technique for including electron correlation

Assumes that electron correlation is perturbation to the HF hamiltonian

Recent studies have revealed serious convergence problems

Theoretical Methods

Coupled Cluster Theory

e = eT0

T = t1 + t2 + t3 + …

t1 = tiaaa

+ai

t2 = tijabab

+aa+ajai

t3 =

Recent addition to electronic structure theory

Includes dominant higher-order terms as products of lower order terms

Rapid convergence if wavefunction is dominated by well localized electron pairs

Convergence problems if HF wave-function provides poor zero-order description of molecule

Basis Sets and Error Analysis

Correlation Consistent Basis Setscc-Sets based on detailed study of electron correlation in atoms

Correlation functions added in shells

Hartree-Fock orbitals cc-pVDZ: + (1s1p1d)

cc-pVTZ: + (2s2p2d1f)cc-pVQZ: + (3s3p3d2f1g)…

Augmented with diffuse functions for anions, long range interactions, etc.

Molecular properties often exhibit systematic dependence

Possible to extrapolate properties to complete basis set limit

De(CO)

n (cc-pVnZ)2 3 4 5 6

240.0

245.0

250.0

255.0

260.0

CBS Limit

HFOrbitals

+1s1p1d

+1s1p1d1f

+1s1p1d1f1g

cc-pVDZcc-pVTZ

cc-pVQZ

Basis Sets and Error Analysis

Definition of Errors (Method “M”)

Basis Set Convergence Error

QbsM(n) = Q(M,n) – Q (M,)

Intrinsic Error

QM = Q(M,) – Q(expt’l)

Calculational Error

Qcalc’dM(n) = Q(M,n) – Q (expt’l)

= QbsM(n) + QM

Basis Sets and Error Analysis

Convergence Types

n

Type II

Note:Qcalc’d

M 0

n

Type III

QM()

n

QbsM(n)

Type I

QM

Q(expt’l)

Qcalc’dM

Molecular Binding Energies:Chemical Bonds

T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000)

K. L. Bak, P. Jørgensen, J. Olsen, T. Helgaker, and W. Klopper, J. Chem. Phys. 112, 9229- 9242 (2000)

Intrinsic Errors in De

Chemical Bonds

CH HF CO N2

De (kcal/mol)a 83.9 141.6 258.6 227.4

MP2 -2.9 4.2 13.0 12.5

MP3 -1.4 -3.1 -8.5 -11.7

MP4 -0.6 1.1 5.7 4.7

MP5 -1.0

CCSD -1.0 -2.2 -8.1 -9.8

CCSD(T) -0.2 -0.1 0.0 -0.3

CCSDT -0.1 -0.2 -0.6 -1.1a Corrected for core-valence and relativistic effects.

Basis Set Convergence Errors in De

Chemical BondsD

ebs (

n)

(k

cal/

mol

)

n n2 3 4 5 6

-30.0

-25.0

-20.0

-15.0

-10.0

-5.0

0.0

2 3 4 5 6

cc-pVnZ sets

aug-cc-pVnZ Sets

CO

N2

HF

CH

Region of“false positives”

0

Statistical Analysis of Binding Energies

Q (Error in Q)

(Q

)

std = errorvariation

= average error

Intrinsic Errors in De

MP2 , CCSD, CCSD(T) Methods (6Z Set)

De (kcal/mol)

(D

e)

0.0

0.2

0.4

0.6

0.8

1.0

-40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0

MP2

CCSD(T)

CCSD

MP2 CCSD +(T)

6.0 -8.3 -1.0

std 7.5 4.5 0.5

0.0

0.2

0.4

0.6

0.8

1.0

-40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0

DZ TZ

QZ

5Z 6Z

Basis Set Convergence of De

CCSD(T) Method for Chemical Bonds

De (kcal/mol)

(D

e) = -1.0 kcal/molstd = 0.5 kcal/mol

Extrapolation of Binding Energies

Analysis of Electron Correlation in He

Principal ExpansionE = EHF(1s) + E2(2s2p)

corr + E3(3s3p3d)corr + …

Errors in He Atom

Klopper et al. [J. Phys. B. 32, R103 (1999)] showed error in truncating series after nth term in principal expansion is

Extrapolation Formula

€

ΔEncorr ∝

1

n3

€

En−1,n = EnHF +

n3Encorr − (n −1)3En−1

corr

n3 − (n −1)3

Extrapolation of De

CCSD(T) Method for Chemical Bonds

De (kcal/mol)

(D

e)

-40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.00.0

0.2

0.4

0.6

0.8

1.0

DZ-TZ

TZ-QZ

QZ-5Z

5Z-6Z

DT TQ Q5 56

-3.5 -0.1 0.0 -0.1

std 2.0 0.5 0.3 0.2

1.6

Molecular Binding Energies:Hydrogen Bonds

T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000)

A. Halkier, W. Klopper, T. Helgaker, P. Jørgensen, and P. R. Taylor, J. Chem. Phys. 111, 9157 (1999)

Intrinsic Errors in De

Hydrogen Bond in HF Dimer

(HF)2

De (kcal/mol) 4.56 0.05 a

MP2 -0.09

MP3 -0.03

MP4 -0.02

CCSD -0.16

CCSD(T) -0.02

a W. Klopper, M. Quack, and M. Suhm, J. Chem. Phys. 108, 10096 (1998).

Basis Set Convergence Errors in De

Hydrogen Bonds in (H2O)2 and Others

Deb

s (n

) (

kca

l/m

ol)

n n

-1.5

-1.0

-0.5

0.0

2 3 4 5

cc-pVnZ

aug-cc-pVnZ

d-aug-cc-pVnZ

(H2O)2

B

B

BB

J

J

J J

H

H

HH

P

P

PP

F

F

F

2 3 4 5

B HF-HF (4.53)

J HCl-HCl (1.95)

H HF-H2O (8.51)

P HF-HCN (7.31)

F H2O-H2O (4.96)

B

BB

B

J

J

JJ

2 3 4 5

uncorrected

corrected

Errors in De

Hydrogen Bonds

De(

n)

(k

cal/

mol

)

n

BS

SE

BS

CE

De(n=∞)

Extrapolation of De

Hydrogen Bond in HF Dimer

De(

n)

(k

cal/

mol

)

n

CalculatedExtrapolated

B

B

BB

J

J J

3.8

4.0

4.2

4.4

4.6

2 3 4 5

Molecular Binding Energies:Electrostatic Interactions

T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000)

Intrinsic Errors in De

Electrostatic Interactions

N2-HF Ar-HF Ar-FH Ar-HCl Ar-ClH

De (cm-1) 776 ± 30a 211 ± 4b 109 ± 10b 176 ± 5c 148 ± 10c

MP2 35 -10 -16 31 33

MP3 -36 -31 -31

MP4 38 7 -10 10 7

CCSD -52 -45 -36

CCSD(T) 17 0 -15 0 7

a R. J. Bemish, E. J. Bohac, M. Wu, and R. E. Miller, J. Chem. Phys. 101, 9457 (1994) and references therein. b J. M. Huston, J. Chem. Phys. 96, 6752 (1992) and references therein. c J. M. Huston, J. Chem. Phys. 89, 4550 (1988); J. M. Hutson, J. Chem. Phys. 96, 4237 (1992) and references therein.

Basis Set Convergence Errors in De

Electrostatic InteractionsD

ebs (

n)

(cm

-1)

n n2 3 4 5

-100.0

-80.0

-60.0

-40.0

-20.0

0.0

2 3 4 5

aug-cc-pVnZ

d-aug-cc-pVnZ

Ar-HF Ar-FH

Molecular Binding Energies:van der Waals Interactions

T. H. Dunning, Jr., J. Phys. Chem. A 104, 9062- 9080 (2000)

Intrinsic Errors in De

van der Waals Interactions

He2 Ne2 Ar2

De(cm-1) 7.59 a 29.4 b 99.6 c

MP2 -2.7 -10.5 13.4

MP3 -1.1 -7.1 -17.6

MP4 -0.5 -1.9 0.4

MP5 -0.2

CCSD -1.1 -6.8 -27.6

CCSD(T) -0.2 -1.0 -2.6

CCSDT -0.0a R. A. Aziz and M. J. Slaman, J. Chem. Phys. 94, 8047 (1991); R. A. Aziz, A. R. Janzen, and R. Moldover, Phys. Rev. Lett. 74, 1586 (1995). b R. A. Aziz, W. J. Meath, and A. R. Allnatt, Chem. Phys. 78, 295 (1983); R. A. Aziz and M. J. Slaman, Chem. Phys. 130, 187 (1989). c R. A. Aziz and M. J. Slaman, Mol. Phys. 58, 679 (1986); R. A. Aziz, J. Chem. Phys. 99, 4518 (1993).

Basis Set Convergence Errors in De

van der Waals InteractionsD

ebs (

n)

(cm

-1)

n n2 3 4 5 6

-20.0

-15.0

-10.0

-5.0

0.0

2 3 4 5 6

Ne2

aug-cc-pVnZ

d-aug-cc-pVnZ

t-aug-cc-pVnZ

Ne2

Ar2

Conclusions

Critical Assessment of Methods Coupled cluster method provides reliable means of

computing molecular properties for molecules well described by single configuration

Perturbation method is poorly convergent or even non-convergent; often does not achieve chemical accuracy for chemical bonds

Critical Assessment of Basis Sets Correlation consistent basis sets systematically

approach complete basis set limit, extrapolation possible Choice of cc-basis set family depends on molecular

system Chemically bound covalent molecules—standard sets

Conclusions (cont’d)

Chemically bound ionic, hydrogen-bonded, and electrostaticly bound molecules—singly augmented sets

van der Waals bound molecules—doubly augmented sets

Convergence with basis set is slow Difficult to describe coulomb hole using expansions in one-

electron functions Rate of convergence depends on molecular details

Single, double or triple bonds Chemically bound, hydrogen-bonded, electrostatically bound or van der

Waals bound

Extrapolation substantially improves convergence rate

Acknowledgements

It is a pleasure to acknowledge contributions of …Kirk Peterson, David Woon, David Feller, Ricky Kendall, Tanja van Mourik, and Angela Wilson to this work

It is also a pleasure to acknowledge work of …Poul Jørgensen, Trygve Helgaker, Wim Klopper, Jeppe Olsen and coworkers, whose work has also contributed greatly to calibrating the methods used for molecular calculations

Finally, I would like to thank …Division of Chemical Sciences, Office of Science, U.S. Department of Energy for their support of this work.

End of Presentation

Definitions

De and D0

E

De D0

A+B

ABZero Point Energy

Separated Atoms

Molecule

Higher Order Effects in He2V

(R)

(K

)

R (bohr)

0.0001

0.001

0.01

0.1

1.0

10.0

3 4 5 6 7 8 9 10 11 12

FCI-T

T-(T)

-0.32 K

-0.015 K