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”
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
...
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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)
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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)
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
Resolution of the N2 Problem
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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.
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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.
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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.
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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).
Using Basis Sets to Solve the Schrödinger Equation with Electron Correlation
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).