Download Introduction to Computational Chemistry

Introduction to Computational Chemistry
WISOR XI
Bressanone, Italy, January 6-13, 2002
Peter R. Schreiner
Hans-Ullrich Siehl
Computational Center for
Molecular Structure and Design
University of Georgia, Department of Chemistry
Athens, GA 30602-2556
[email protected]
Abteilung Organische Chemie I
Universitaet Ulm
D-89069 Ulm, Germany
[email protected]
What is Computational Chemistry?
”Computational chemistry is to model all aspects of chemistry by
calculation rather than experiment."
Paul v. R. Schleyer (1987)
"Computational Chemistry has existed for half a century, growing
from the province of a small nucleus of theoretical work to a large,
significant component of scientific research. By virtue of the great
flexibility and power of electronic computers, basic principles of
classical and quantum mechanics are now implemented in a form
which can handle the many-body problems associated with the
structure and behavior of complex molecular systems."
John A. Pople (November 1997)
(Nobel prize for chemistry 1998, together with Walter Kohn )
Why do chemistry (an “experimental” science?) on computers?
• Improved chemical (physical) understanding
• Faster turnaround of ideas (not always)
• Reduced cost (and waste!)
• Safety
• Better accuracy (for very small systems)
Computational chemistry is far from black box!
Care must be taken in choice of method.
Elements of Computational Chemistry
• Decipher the “language” (many different approaches with even more
acronyms – learn)
• Technical aspects – how to transfer an idea (molecular structure or
property, chemical reaction, etc.) into the program and the computer (in
a reasonable time frame)
• Quality assessment: most important, most difficult! Required:
• Understand the basis of the underlying theoretical concepts of a
particular approach to solving the problem
• Understand the chemistry (above all!)
• Constantly compare experiment and theory (be critical of both)
Conceptual approach to computational chemistry
Prediction
Interpretation
Validation
"...give us insight, not numbers."
C. A. Coulson (Proc. Robert A. Welch Foundation Conf. XVI, S. 117)
Chemistry is knowing the energy as a function of the nuclear coordinates.
supplied
• graphically
• "by hand"
molecule
coordinates
• cartesian
• internal
different types
for different
purposes
human input:
choice!
difficult
program
molecular
properties
interpre
• structures
• energies
• molecular orbitals
• AMBER, CHARMM, GROMOS, Sybyl...
• IR, NMR, UV...
• AMPAC, MOPAC, VAMP...
• Gaussian, Gamess, MOLPRO, Jaguar, PSI...
many different ones:
(—> interpretation) (—> different strengths & purposes)
(—> different models)
Describing properties is knowing how the energy changes upon adding a perturbation.
Computational chemistry methods
• Molecular mechanics, force fields
easy to comprehend
quickly programmed
extremely fast
no electrons: limited interpretabilty
• ab initio methods
full quantum method
only expt. fundamental constants
very high accuracy
complete (all interactions are included)
very time consuming (“expensive”)
systematic improvement possible
• Semiempirical methods
quantum method
valence electrons only
fast
limited accuracy
• Density functional theory
quantum method
in principle “exact”
faster than trad. ab initio
variable accuracy
no systematic improvement
Force fields
Simple harmonic oscillator model
—> close relationship to potential energy surface
Typical accuracy (MM2 force field of Allinger et al.):
—
—
—
—
—
—
0.01 Å bond lengths
1˚ bond angles
few degrees for torsion angles
conformational energies: accurate to 1 kcal/mol (at best).
vibrational frequencies: 20 – 30 wavenumbers (at best)
configurational sampling (in MD): few kcal/mol
Semiempirical methods
• treat valence electrons only (core potential)
• minimal basis set (Slater (not Gaussian!) type orbitals, STO)
• neglect all two-electron integrals involving two-center charge distributions, i.e.,
all three-center and four-center two-electron integrals. Replace by parameters to
mimick experimental results (geometries and heats of formation).
• solve secular equations just like in self-consistent field (HF) theory
• typical methods: MNDO, MINDO, AM1, PM3 (order of development)
Caveat:
Semiempirical methods describe ground-states only and are geared towards
computing the heats of formation!
Semiempirical methods
• Faster than ab initio or DFT: x 1000 (N2 scaling) for energies
• N3 for gradient
• Harmonic force fields calculated from numerical finite differences of gradients
• Treat up to 10000 heavy (= non-hydrogen) atoms!
Accuracy and limitations (organic molecules):
• mean absolute deviations: 6.3 kcal/mol for heats of formation
• 0.014 Å for bond lengths
• 2.8° for bond angles
• 0.48 eV for first ionization potentials
Some qualitative deficiencies:
• Sterically crowded molecules are too unstable (e.g., neopentane vs. n-pentane).
• Four-membered rings are too stable (e.g. cyclobutane, cubane).
• Non-classical structures are normally too unstable.
• Rotational barriers are often underestimated (e.g. ethane)!
• Hydrogen bonds are much too weak and too long (e.g., water dimer)!
• Activation barriers are normally somewhat too high.
• Pericyclic reactions: biradicaloid mechanisms are artificially favored.
Hartee Fock (self-consistent field, SCF) theory: key steps
Schrödinger equation:
Space & time separation:
non-rel. Hamilton operator:
Born-Oppenheimer approximation:
Hψ = Eψ
ψ(r, t) = ψ(r) ψ(t) —> Hψ (r) = Eψ (r)
H = T + V = –(h2∇2/8pm) + V
ψ (r, R) ≈ ψ (r) ψ (R)
+∞
2
Normalization:
|cΨ| dτ = 1
-∞
Antisymmetrization:
Molecular Orbitals:
ψ (r1,...ri,rj,...rn) = – ψ (r1,...rj,ri,...rn)
ψ (r) = φ1 (r1) φ2 (r2) ...φn(rn)
Ν
LCAO:
φi =
∑ cµiχµ
µ=1
Basis functions:
gs(α, r) = Cxnymzlexp(-αr2)
Variation principle:
E (ψi) > E (ψ) with ψi ≠ ψ
HF/SCF quality assessment: general aspects
• Bond lengths and angles of "normal" organic molecules quite accurate (within 2%)
• Conformational energies accurate to 1–2 kcal/mol.
• Vibrational frequencies for most covalent bonds systematically too high by 10–12%
• Zero point vibrational energies: ~1-2 kcal/mol
• Isodesmic reaction energies accurate to 2–5 kcal/mol.
• Entropies accurate to 0.5 cal/K mol.
• Protonation/Deprotonation energies ~10 kcal/mol (gas phase)
• Atomization and homolytic bond-breaking reactions: (25–40 kcal/mol)
• Reaction barriers may (!) have large errors.
HF/SCF quality assessment: geometries and rotational barriers
Rotational barriers
Bond lengths and angles
Molecule
H3B-NH3
STO-3G
3-21G
6-31G**
Expt.
2.1
2.9
1.9
2.7
1.9
3.0
3.1
2.9
2.8
2.0
2.0
1.5
2.4
1.4
2.0
1.1
1.3
1.9
1.1
1.7
1.4
2.0
1.7
2.0
HO-OH cis
1.5
9.1
1.1
11.7
1.4
9.2
1.3
7.0
HS-SH cis
6.1
5.7
8.5
6.8
Basis
∆r (Å)
Angle (°)
STO-3G
3-21G
0.054
0.016
2.3
3.8
3-21G(*)
0.017
3.3
6-31G*
0.014
1.5
H3C-OH
H3C-SiH3
6-31G**
6-311G**
0.014
0.013
1.8
1.0
H3C-PH2
H3C-SH
H3C-CH3
H3C-NH2
HF/SCF quality assessment: isomerization energies
Formula
Reaction
HF/6-31G* Experiment
HCN
Hydrogen cyanide —> Hydrogen isocyanide
12.4
14.5
CH2O
Formaldehyde —> Hydroxymethylene
52.6
4.9
CH3NO
Formamide —> Nitrosomethane
65.3
62.4
C2H3N
Acetonitrile —> Methyl isocyanide
20.8
20.9
C2H4O
Acetaldehyde —> Oxacyclopropane
33.4
26.2
C3H6
Propene —> Cyclopropane
8.2
6.9
Classical failures of HF theory:
•FOOF: R.D. Amos, C.W. Murray, and N.C. Handy, Chem. Phys. Lett. 202, 489 (1993).
•Cr-Cr: G.E. Scuseria, J. Chem. Phys. 97, 7528 (1992).
•NO3: P.S. Monks, et al., J. Phys. Chem. 98, 10017 (1993).
Beyond HF/SCF: electron-correlation methods
HF/SCF:
electrons interact with each other only via the average electron density.
Reality:
instantaneous repulsion = correlation energy.
Most common methods to include electron correlation:
• Perturbation Theories (MPn)
• Configuration Interaction (CI)
• Coupled Cluster (CC)
• Density Functional Theory (DFT)
Correlation methods: Møller-Plesset perturbation theory
—> non-iterative corrections added to the Hartree-Fock energy
H = Ho + λH1
—> perturbation is small, H1 can be expressed in a series in terms of the parameter λ:
ψ = ψ(0) + λ1ψ(1) + λ2ψ(2) + λ2ψ(2) + ...
E = E(0) + λ1E(1) + λ2E(2) + λ3E(3) + ...
Problems:
• overcorrection possible
• non-convergent series
Correlation methods: configuration interaction
—> wavefunction expressed as a linear combination of all possible HF-determinants:
n
ψ = b0 ψ0 + ∑ bsψs
s>0
HF
S
S
D
D
Problems:
• very time-consuming
• not size-extensive
• slow convergence
• strongly basis set-dependent
T
Q
Correlation methods: coupled-cluster theory
—> tries to include all corrections (S, D, T, ...) to infinite order
—> operator T performs changes of the electron distribution, as expressed through
the molecular orbitals φi to compute the correlation energy:
ψ = eTφo
H eTΨ0 = E eTΨ0
—> single, double…etc substitutions (CCD, CCSD, CCSD(T)…)
—> CCSD(T) is most common and probably the best choice as the triple
contributions are included perturbatively through an additional term arising from
5th order MP theory.
—> QCISD and QCISD(T) are similar to CCSD and CCSD(T), respectively, but
some of the terms have been omitted. As there is virtually no computation
advantage of the CI methods in this context, QCI should not be used!
Correlation improves dissociation energies dramatically
Reaction
H2 -> H•+H•
HF/6-31G** MP2/6-31G** MP3/6-31G** MP4/6-31G**
Expt.
85
45
52
101
48
52
105
49
49
106
58
47
109
32
50,56
FH -> F•+H•
3CH2 -> CH•+H•
1CH2 -> CH•+H•
93
101
131
109
127
108
128
107
141
105,106
70
89
90
91
98
OH2 -> OH•+H•
BH3 -> BH2•+H•
86
90
83
119
106
110
115
108
108
116
109
109
126
57-107
116
87
109
110
110
113
LiH -> Li•+H•
BeH -> Be•+H•
NH3 -> NH2•+H•
CH4 -> CH3•+H•
Correlation methods: scaling
Method
Ave. Error (kcal/mol) vs. FCI
approx. time factor
HF
DFT
MP2
5-30
2–10
17.4
ON2-3
ON3
ON4
MP3
MP4
MP5
14
4
3.2
ON5
CISD
CCSD
CCSD(T)
13.8
4.4
CCSDT
CCSDTQ
0.5
0.0
ON6
O2N7
O2N>7
O2N>>7
Correlation methods: systematic quality improvement
Basis set
minimum
HF
DZ
DZP
TZP
QTZ
...
infinite
HF limit
Method
CID
CISD
CISDT
CISDTQ
FCI
"exact"
Density functional theory (DFT)
Hohenberg-Kohn theorem: energy and all other electronic properties of the ground
state are uniquely determined by its charge density ρ(r):
EV [ρ ]= FHK + ∫ ρ(r )V (r )dr
• What is the form of FHK (functional)?
• How can the exact charge density ρ(r) be determined?
One particle eigenvalue: effective one electron Hamilton (UHF!), HKS, which
incorporates both, FHK[ρ] and V:
 1
ZA
H φ (r1 ) =  − ∆ + ∑
+
2
R
−
r
A

A
1
γ
K S iσ
∫

ρ γ (r2 )
+ VXC (r ) φ iσ (r1 ) = ε iσ φ iσ (r1 )
r1 − r2

Density functional theory (DFT)
• Self consistent solutions for φiσ resemble those of HF theory, but DFT
orbitals have no physical significance other than constituting the charge
density (still being discussed).
• DFT wavefunction is not a Slater determinant of spin orbitals! In a strict
sense there is no N-electron wave function available in DFT!
• KS equations give exact solution of N-electron problem, provided the
correct functional dependence of exchange correlation EXC energy with
respect to the charge density ρ(r) is available.
Density functional theory performance
F• + H2 —> FH + H•
Re
barrier
Cr2 properties
De
vib.
HF
14.2
1.465
–19.4
1151
CCSD
CCSD(T)
3.3
2.3
1.560
1.621
–2.9
0.5
880
705
DFT
3.6
1.590
1.5
597
Expt.
2.0
1.679
1.4
470
Wave function interpretation
Why can the wave function not be interpreted directly?
ψ = φ 1φ 2... φ n
—>
with
φ i = ∑ cir χr
r
and
χr = ∑ arn ξn
n
results in a list of numbers without obvious chemical meaning.
The electron density ρ does have physical meaning:
N/2
ρ(r) =
ψ(r) 2
with
ρ(r) = 2∑ φ(r)
i
N/2
2
and
dr ρ(r) = 2∑
Problem:
ρ is four-dimensional (number of electrons plus x,y,z coordinates).
Presentation and interpretation is difficult.
"Atomic" information is desireable: Charges, bonds, dipoles, etc.
i
dr φ(r) 2= N
Wave function interpretation
Filter (translate into chemical lingo) out partial information
Solution:
a)
Density methods (do something with the density so it can be interpreted):
•
•
•
b)
Bader’s “Atom in Molecules” (density gradients)
Electrostatic potentials (molecular polarization)
Density surface (spatial extension, shape analysis)
Molecular orbital methods (do something with the orbitals so they can be interpreted):
•
•
•
Mulliken population analysis
Löwdin population analysis
Natural population analysis, natural bond orbital analysis
Mulliken population analysis
Distribution of electrons based on occupations of atomic orbitals (AOs), φi.
Principle:
N=
MO's
AO's
∑
∑r
i
N(i)
AO's
MO's
cir2
+2
Atomic part
∑
i
N(i)
∑
r>s
cir cis Srs
overlap contributions
But:
Atomic population n(k) does not include electrons from the overlap population N(k,l)!
Mulliken:
Divide overlap population exactly in the middle, regardless of type, EN, etc.:
MO's
N(k) =
∑
i
N(i) ∑ cir k cir k + ∑ cis lSr ksl
rk
sl≠k
Mulliken, R. S. J. Chem. Phys. 1955, 23, 1833, 1841, and 2338.
with
qk = Zk – N(k)
Mulliken population analysis
Advantages:
• easy to program
• fast
• commonplace (easy for comparisons)
Drawbacks:
• differences of atom types are not properly accounted for
• strong basis set dependence
• N(k) may be larger than 2: violation of Pauli-principle
Remedy:
Lödwin analysis
• Symmetric transformation of Mulliken-analysis in orthogonal coordinate system so that
0 < N(k) < 2
Löwdin, P.-O. Adv. Quantum Chem. 1970, 5, 185.
Natural population analysis / natural bond orbital methods (Weinhold et al.)
Principle:
Uses orbitals which diagonalize the one-particle density matrix (natural orbitals, NO).
Diagonalisation of the two-center subdeterminants gives the "natural bond orbitals."
Advantages:
• very detailed analysis possible
• little basis set dependent
• atomic properties fully included
Drawbacks:
• more time consuming than simpler methods (unimportant, though)
• sometimes numerically unstable
• highly delocalized are troublesome
Reed, A. E.; Weinstock, R. B.; Weinhold J. Chem. Phys. 1985, 83, 735.
Examples: LiH charges with Mulliken and NPA analyses
Level
Li (LiH)
Li (LiF)
HF/STO-3G
-0.016
+0.229
HF/6-31G*
+0.170
+0.666
HF/6-311+G**
+0.358
+0.718
HF/STO-3G
+0.425
+0.493
HF/6-31G*
+0.731
+0.917
HF/6-311+G**
+0.815
+0.977
Mulliken
NPA
Examples: Diborane (B2H6)
HF/STO-3G
±0.00 (Mulliken)
–0.01 (NPA)
HF/3-21G
±0.02 (Mulliken)
+0.08 (NPA)
HF/STO-3G
+0.08 (Mulliken)
–0.20 (NPA)
HF/3-21G
–0.09 (Mulliken)
–0.04 (NPA)
NBO Bond order:
0.59 (HF/STO-3G) !
0.61 (HF/6-31G*) !
HF/STO-3G
–0.04 (Mulliken)
–0.09 (NPA)
HF/3-21G
±0.02 (Mulliken)
+0.02 (NPA)
Enols and enolates: electron density, potential and charges
–0.27
–0.04
–0.28
enol
AM1 optimized structures; HF/6-31G* densities; NBO charges
–0.60
+0.19
–0.66
enolate
Charges in acetone and protonated acetone (or methylated)
–0.29
–0.10
+0.22
+0.32
remember: "S N 2-like"
Nu –
Not the case!
AM1 optimized structures; HF/6-31G* NBO charges
O
Charges and dipoles on typical “Hammett-systems”
–0.36
+0.35
–0.35
–0.35
+0.56
+0.35
–0.35
+0.23
–0.38
–0.39
+0.36
+0.23
–0.33
–0.31
µ = 2.4 D
–0.32
µ = 3.4 D
–0.59
+0.36
–0.55
–0.40
+0.36
+0.57
µ = 12.5 D
AM1 optimized structures; HF/6-31G* NBO charges
µ = 4.5 D
µ = 6.0 D
–0.57
+0.20
–0.44
+0.36
µ = 13.7 D
Computational limitations (as of 2002…)
Good methods
Difficult
Not yet realized
available
(well under development)
(often attempted)
Geometries
Reaction rates
Crystal structures
(prediction)
Energies (small
Circular dichroism (optical,
Melting points
systems)
magnetic, vibrational)
IR
Spin-orbit couplings
Protein folding
NMR
Full relativistics
Full reaction dynamics
ESR
Excited states (vertical)
Molecular dynamics
MW
Solvent effects
Excited states (adiabatic)
Multipole moments
Density matrix methods/geminals
Solvent dynamics
PE-spectra
Linear scaling
Systematic improvement
of DFT
Local correlation methods
r12-methods
Accurate enzyme-substrate
interactions
Case study - “classical” norbornyl cation controversy
≠
δ−
X
δ+
X
H
H
exo
nonclassical
+ ROS
≠
+X
H
X
endo
OS
+
-
H
H
X
classical
exo/endo (X = OBrosylate) rate ratio= 350 (AcOH)
i.e. energy differences in transition states 2-5 kcal/mol
H
Case study - “classical” norbornyl cation controversy
DFT (B3LYP/6-31G*) optimized structures of the cations:
C
C
C
1.711ÅC
2.087Å
C
C
1.892Å
C
1.414Å
C
C
1.395Å
0.0
C
C
C
C
C
12–15 kcal/mol
Case study - “classical” norbornyl cation controversy
X
X
Me
-X-
-X-
C
C
50
1.543Å
C
55,000
C
C
1.437Å
C
-X-
Me
2.246Å
C
-X-
1.471Å
C
Me
Me
X
0.00035
X
60
Solvation — Application of MC-Methods: Example
∆∆G (kcal mol-1)
0.8
0.6
0.4
∆∆G (kcal/mol)
0.2
0
0
0.25
0.5
0.75
1
1.25
structural parameter λ
J. Am. Chem. Soc. 1995, 117, 2663
How to treat a solvolysis reaction computationally?
E
R+ + Xca. 200 kcal/mol
R• + X•
R-X
reaction coordinate
Problem: homolytic bond cleavage in the gas phase!
How to treat a solvolysis reaction computationally?
Trick: protonation leads to cation smoothly!
E
R+ + HX
R-XH+
reaction coordinate
Case study - “classical” norbornyl cation controversy
C
C
1.581Å
C
C
C
C
C
1.449Å
2.447Å
C
C
1.522Å
O
2.396Å
C
C
1.791Å
mol-1
3.7 kcal
including ground state
energy difference
forward
reaction:
∆H ‡ = 2.8
kcal mol-1
C
C
1.550Å
C
C
3.104Å
1.399Å
1.997Å
C
O
reverse
association:
∆H‡ = 5.6
kcal mol-1
TSexo
C
1.514Å
2.432Å
C
C
1.623Å
C
C
reverse
association:
∆H‡ = 1.9
kcal mol-1
C
C
C
C
C
C
1.519Å
2.527Å
C
1.615Å
1.874Å
C
C
C
1.395Å
O
C
O
O
1 (X=OH2+)
(0.0 kcal mol-1)
forward
reaction:
∆H‡ = 5.3
kcal mol-1
C
1.560Å
C
C
TSendo
2.356Å
6
{0.9
kcal mol-1}
2 (X=OH2+)
{1.2
kcal mol-1}
Case study - “classical” norbornyl cation controversy