using basis sets to solve the electronic schrödinger ...1. “gaussian basis sets for use in...

Solution of theElectronic Schrödinger Equation

Using Basis Sets to Solve the Electronic

Schrödinger Equation with Electron

Correlation

Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation

Errors in HF Predictions: Binding Energies

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 NB 0.0218

De (kcal/mol)

Chemical Bonds

Hydrogen Bonds

van der Waals “Bonds”

Electrostatic “Bonds”


Mathematical Models for Electron Correlation

Configuration Interaction

e = 0 + Cia

ia +

Cijab

ijab + …

He C = Ee C

R Long history in electronic structure

theory

RVery flexible, e.g., can describe both

ground and excited states

RNumber of configurations grows

rapidly with excitation level

R Truncated CI not size extensive

Perturbation Theory

He = H0 + H1

e = 0 + 1 + 22 + …

Ee = E0 + E1 + 2E2 + …

RMost widely used technique for

including electron correlation

RH0 usually taken to be the HF

Hamiltonian

R Recent studies have revealed serious

convergence problems


Models for Electron Correlation (cont’d)

Coupled Cluster Theory

e = eT 0

T = t1 + t2 + t3 + …

t1 = tiaaa

+ai

t2 = tijabab

+aa+ajai

t3 = ...

R Recent addition to electronic structure

theory

R Includes dominant higher-order terms

as products of lower order terms

R Rapid convergence if wavefunction is

dominated by well localized electron

pairs, e.g., CCSD is exact if electron

pairs are completely separate

R Convergence problems if HF wave-

function provides a very poor zero-

order description of molecule


Notation for Correlated Calculations

Perturbation Theory Methods

He = H0 + H1

= 0 + 1 + 22 + ...

Coupled Cluster Methods

= eT 0

Variational Methods

= 0 + Caiai + Cab

ijabij + ...

T = T1 + T2 + T3 + ...

{ai} {ab}{ij}

MP2

MP3

MP4

...

CCD

CCSD

CCSDT

...

SDCI

SDTCI

...

MRCI

T2

T1+T2

T1+T2+T3

...


SDCI Calculations on the Oxygen Atom

-0.1

-1.0

-10.0

-100.0

1 2 3 4

(nsnp)

(nd)(nf)

(ng)

(1h)

En

,n+

1 (m

Eh)

En,n-1 = Ecorr(n, l) - Ecorr(n-1, l)


Contributions to Correlation Energy (SDCI)

-0.1

-1.0

-10.0

-100.0

1 2 3 4

cc-pVDZ

cc-pVTZ

cc-pVQZ

cc-pV5Z

Nbf

Basis Function Groupings

Contributions of basis functions to the

correlation energy for the first row atoms

fall into distinct groups with

E1,0(sp) E1,0(d)

E2,1(sp) E2,1(d) E1,0(f)

E3,2(sp) E3,2(d) E2,1(f) E1,0(g)

These grouping form the foundation for

the construction of correlation consistent

basis sets:

cc-pVDZ: HF Orbitals + (1s1p1d)

cc-pVTZ: HF Orbitals + (2s2p2d1f)

cc-pVQZ: HF Orbitals + (3s3p3d2f1g)

where to balance the errors

cc-pVDZ: HF Set = (9s4p)

cc-pVTZ: HF Set = (10s5p)

cc-pVQZ: HF Set = (12s6p)


Atomic Calculations with cc-Sets

B

J

H

Ecorr

(m

Eh)

BB B B

J

J J JH

HH HP

PP

P

F

F

FF F

R

R

RR R

2 3 4 5 6

C

N

O

F

Ne

-50.0

-100.0

-150.0

-200.0

-250.0

-300.0

-350.0

BB

H

BJ

PP

Exponential Convergence

Ecorr(n) = Ecorr( ) + Ecorr(2)e- (n-2)

Inverse Powers of lmax (=n)

Ecorr(n) = Ecorr( ) + A/n3


Errors in Molecular Calculations

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


Illustration of Types of Errors in Calculations

n

Type II

Note:

Qcalc’dM 0

n

Type III

QM( )

n

Qbs

M(n)

Type I

QM

Q(expt’l)

Qcalc’d

M


Confusion: Convergence of De(N2) with MPn

200.0

210.0

220.0

230.0

240.0

MP2 MP3 MP4

228.4 kcal/mol

De

(k

cal/

mol)

Basis set:

cc-pVTZ


Resolution of the N2 Problem


Binding Energies: Chemically Bound Molecules

De(expt’l)a 83.9 141.6 228.4 259.3

De(core-valence) -0.2 -0.2 -0.8 -0.9

De(valence-only) 83.7 141.4 227.6 258.4

CCSD -0.8 -2.0 -9.9 -7.5

CCSD(T) 0.0 0.1 -0.3 0.1

CCSDT 0.1 0.0 -0.9 -0.3

MP2 -2.7 4.4 12.4 13.6

MP3 -1.2 -3.3 -11.8 -7.9

MP4 -0.4 1.3 4.2 5.9

MP5 -0.8

CH HF N2 CO

a Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules; Van

Nostrand,; Princeton, 1979.


Binding Energies: Hydrogen-bonded Molecules

De(expt’l)a, kcal/mol 4.56

CCSD -0.16

CCSD(T) -0.02

MP2 -0.09

MP3 -0.03

MP4 -0.02

(HF)2

a Cayton, D. C.; Jucks, K. W.; Miller, R. E. J. Chem. Phys.

1989, 90, 2631; Klopper, W.; Quack, M.; Suhm, M. A. J.

Chem. Phys. 1998, 108, 10096.


Binding Energies: Weakly Bound Molecules

De(expt’l), cm-1 776.±30 211.±4 109.±10 176.±5 148.±10

CCSD -52 -45 -36 — —

CCSD(T) 17 0 -15 0 -1

MP2 35 -10 -16 31 33

MP3 -36 -31 -31 — —

MP4 38 7 -10 10 7

N2-HFa Ar-HFb Ar-FHb Ar-HClc Ar-ClHc

a Lovejoy, C. M.; Nesbitt, D. J. J. Chem. Phys. 1987, 86, 3151; Nesbitt, D. J.; Child, M. S. J. Chem. Phys. 1993, 98, 478; Nesbitt,

D. J.; Lindeman, T. G.; Farrell, J. T., Jr.; Lovejoy, C. M. J. Chem. Phys. 1994, 100, 775; Bemish, R. J.; Bohac, E. J.; Wu, M.;

Miller, R. E. J. Chem. Phys. 1994, 101, 9457; Farrell, J. T.; Sneh, O.; Nesbitt, D. J. J. Phys. Chem. 1994, 98, 6068; Tang, S. N.;

Chang, H-C.; Klemperer, W. J. Phys. Chem. 1994, 98, 7313.b Hutson, J. M. J. Chem. Phys. 1992, 96, 6752 and references therein.c Hutson, J. M. J. Chem. Phys. 1988, 89, 4550; Hutson, J. M. J. Phys. Chem. 1992, 96, 4237; and references therein.


Binding Energies: Very Weakly Bound Molecules

De(expt’l), cm-1 7.59 29.4 99.6De(core-valence) +0.05 -0.8

De(valence-only) 29.4 98.8

CCSD -1.1 -6.8 -26.8CCSD(T) -0.2 -1.0 -1.8CCSDT 0.0

MP2 -2.7 -10.5 13.2MP3 -1.1 -7.1 -16.8MP4 -0.5 -1.9 1.2MP5 -0.2

He2a Ne2

b Ar2c

a Aziz, R. A.; Slaman, M. J. J. Chem. Phys. 1991, 94, 8047. Aziz, R. A.; Janzen, A. R.; Moldover, R. Phys.

Rev. Lett. 1995, 74, 1586.b Aziz, R. A.; Meath, W. J.; Allnatt, A. R. Chem. Phys. 1983, 78, 295. Aziz, R. A.; Slaman, M. J. Chem.

Phys. 1989, 130, 187.c Aziz, R. A.; Slaman, M. J. Mol. Phys. 1986, 58, 679. Aziz, R. A. J. Chem. Phys. 1993, 99, 4518


References

1. “Gaussian basis sets for use in correlated molecular calculations. I. The atoms

boron through neon and hydrogen,” T. H. Dunning, Jr., J. Chem. Phys. 90, 1007-

1023 (1989).

2. “Electron affinities of the first-row atoms revised. Systematic basis sets and wave

functions,” R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys.

96, 6796-6806 (1992).

3. “Gaussian basis sets for use in correlated molecular calculations. III. The second

row atoms, Al-Ar,” D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 980,1358-

1371 (1993).

4. “Gaussian basis sets for use in correlated molecular calculations. IV. Calculation of

static electrical response properties,” D. E. Woon and T. H. Dunning, Jr., J. Chem.

Phys. 100, 2975-2988 (1994).

5. “Gaussian basis sets for use in correlated molecular calculations. V. Core-valence

basis sets for boron through neon,” D. E. Woon and T. H. Dunning, Jr., J. Chem.

Phys. 103, 4572-4585 (1995).


References (cont’d)

6. “Gaussian basis sets for use in correlated molecular calculations. VI. Sextuple-zeta

correlation-consistent sets for boron through neon,” A. K. Wilson, T. van Mourik,

and T. H. Dunning, Jr., J. Molec. Struct. (Theochem) 388, 339-349 (1996).

7. “Gaussian basis sets for use in correlated molecular calculations. VII. The atoms

aluminum through argon revisted,” T. H. Dunning, Jr., K. A. Peterson, and A. K.

Wilson, J. Chem. Phys. 114, 9244-9253 (2001).

8. “Gaussian basis sets for use in correlated molecular calculations. VIII. Standard

and augmented sextuple zeta correlation consistent basis sets for aluminum through

argon,” T. van Mourik and T. H. Dunning, Jr., Intern. J. Quant. Chem. 76, 205-221

(2000).

9. “Gaussian basis sets for use in correlated molecular calculations. IX. Correlation

consistent sets for the atoms gallium through krypton,” T. H. Dunning, Jr., J.

Chem. Phys. 110, 7667-7676 (1999).

10. “Accurate correlation consistent basis sets for molecular core-valence effects: The

second row atoms, Al-Ar, and the first row atoms B-Ne revisited,” K. A. Peterson

and T. H. Dunning, Jr., J. Chem. Phys. 117, 10548-10560 (2002).

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