2. its.unc.edu 2
Outline
Introduction
Methods in Computational Chemistry
•Ab Initio
•Semi-Empirical
•Density Functional Theory
•New Developments (QM/MM)
Hands-on Exercises
The PPT format of this presentation is available here:
http://its2.unc.edu/divisions/rc/training/scientific/
/afs/isis/depts/its/public_html/divisions/rc/training/scientific/short_courses/
3. its.unc.edu 3
About Us
ITS – Information Technology Services
• http://its.unc.edu
• http://help.unc.edu
• Physical locations:
401 West Franklin St.
211 Manning Drive
• 10 Divisions/Departments
Information Security IT Infrastructure and Operations
Research Computing Center Teaching and Learning
User Support and Engagement Office of the CIO
Communication Technologies Communications
Enterprise Applications Finance and Administration
4. its.unc.edu 4
Research Computing
Where and who are we and what do we do?
• ITS Manning: 211 Manning Drive
• Website
http://its.unc.edu/research-computing.html
• Groups
Infrastructure -- Hardware
User Support -- Software
Engagement -- Collaboration
5. its.unc.edu 5
About Myself
Ph.D. from Chemistry, UNC-CH
Currently Senior Computational Scientist @ Research Computing Center, UNC-CH
Responsibilities:
• Support Computational Chemistry/Physics/Material Science software
• Support Programming (FORTRAN/C/C++) tools, code porting, parallel computing, etc.
• Offer short courses on scientific computing and computational chemistry
• Conduct research and engagement projects in Computational Chemistry
Development of DFT theory and concept tools
Applications in biological and material science systems
6. its.unc.edu 6
About You
Name, department, research interest?
Any experience before with high
performance computing?
Any experience before with
computational chemistry research?
Do you have any real problem to solve
with computational chemistry
approaches?
7. its.unc.edu 7
Think BIG!!!
What is not chemistry?
• From microscopic world, to nanotechnology, to daily life, to
environmental problems
• From life science, to human disease, to drug design
• Only our mind limits its boundary
What cannot computational chemistry deal with?
• From small molecules, to DNA/proteins, 3D crystals and
surfaces
• From species in vacuum, to those in solvent at room
temperature, and to those under extreme conditions (high
T/p)
• From structure, to properties, to spectra (UV, IR/Raman,
NMR, VCD), to dynamics, to reactivity
• All experiments done in labs can be done in silico
• Limited only by (super)computers not big/fast enough!
8. its.unc.edu 8
Central Theme of
Computational Chemistry
DYNAMICS
REACTIVITY
STRUCTURE CENTRAL DOGMA OF MOLECULAR BIOLOGY
SEQUENCE
STRUCTURE
DYNAMICS
FUNCTION
EVALUTION
10. its.unc.edu 10
What is Computational
Chemistry?
Application of computational methods and
algorithms in chemistry
• Quantum Mechanical
i.e., via Schrödinger Equation
also called Quantum Chemistry
• Molecular Mechanical
i.e., via Newton’s law F=ma
also Molecular Dynamics
• Empirical/Statistical
e.g., QSAR, etc., widely used in clinical and medicinal
chemistry
Focus Today
H
t
i ˆ
11. its.unc.edu 11
How Big Systems Can We
Deal with?
Assuming typical computing setup (number of CPUs,
memory, disk space, etc.)
Ab initio method: ~100 atoms
DFT method: ~1000 atoms
Semi-empirical method: ~10,000 atoms
MM/MD: ~100,000 atoms
12. its.unc.edu 12
i
j
n
1
i ij
n
1
i
N
1 i
2
i
2
r
1
r
Z
-
2m
h
-
H
n
i
j
n
1
i ij
n
1
i r
1
i
h
H
Starting Point: Time-Independent
Schrodinger Equation
E
H
H
t
i ˆ
13. its.unc.edu 13
Equation to Solve in
ab initio Theory
E
H
Known exactly:
3N spatial variables
(N # of electrons)
To be approximated:
1. variationally
2. perturbationally
14. its.unc.edu 14
Hamiltonian for a Molecule
kinetic energy of the electrons
kinetic energy of the nuclei
electrostatic interaction between the electrons and
the nuclei
electrostatic interaction between the electrons
electrostatic interaction between the nuclei
nuclei
B
A AB
B
A
electrons
j
i ij
nuclei
A iA
A
electrons
i
A
nuclei
A A
i
electrons
i e
R
Z
Z
e
r
e
r
Z
e
m
m
2
2
2
2
2
2
2
2
2
ˆ
H
15. its.unc.edu 15
Ab Initio Methods
Accurate treatment of the electronic distribution using the
full Schrödinger equation
Can be systematically improved to obtain chemical accuracy
Does not need to be parameterized or calibrated with
respect to experiment
Can describe structure, properties, energetics and reactivity
What does “ab intio” mean?
• Start from beginning, with first principle
Who invented the word of the “ab initio” method?
• Bob Parr of UNC-CH in 1950s; See Int. J. Quantum Chem.
37(4), 327(1990) for details.
16. its.unc.edu 16
Three Approximations
Born-Oppenheimer approximation
• Electrons act separately of nuclei, electron and
nuclear coordinates are independent of each other,
and thus simplifying the Schrödinger equation
Independent particle approximation
• Electrons experience the ‘field’ of all other
electrons as a group, not individually
• Give birth to the concept of “orbital”, e.g., AO,
MO, etc.
LCAO-MO approximation
• Molecular orbitals (MO) can be constructed as linear
combinations of atom orbitals, to form Slater
determinants
17. its.unc.edu 17
Born-Oppenheimer
Approximation
the nuclei are much heavier than the electrons and move more
slowly than the electrons
freeze the nuclear positions (nuclear kinetic energy is zero in the
electronic Hamiltonian)
calculate the electronic wave function and energy
E depends on the nuclear positions through the nuclear-electron
attraction and nuclear-nuclear repulsion terms
E = 0 corresponds to all particles at infinite separation
nuclei
B
A AB
B
A
electrons
j
i ij
nuclei
A iA
A
electrons
i
i
electrons
i e
el
r
Z
Z
e
r
e
r
Z
e
m
2
2
2
2
2
2
ˆ
H
d
d
E
E
el
el
el
el
el
el
el
el *
* ˆ
,
ˆ
H
H
18. its.unc.edu 18
Approximate Wavefunctions
Construction of one-electron functions (molecular orbitals,
MO’s) as linear combinations of one-electron atomic basis
functions (AOs) MO-LCAO approach.
Construction of N-electron wavefunction as linear
combination of anti-symmetrized products of MOs (these
anti-symmetrized products are denoted as Slater-
determinants).
down)
-
(spin
up)
-
(spin
;
1
i
i
u i
k
N
k
kl
i
l r
q
19. its.unc.edu 19
The Slater Determinant
z
c
b
a
z
c
b
a
z
z
z
z
c
c
c
c
b
b
b
b
a
a
a
a
n
z
c
b
a
z
c
b
a
n
z
c
b
a
n
n
n
n
n
n
n
n
3
2
1
3
2
1
3
2
1
3
2
1
3
2
1
3
1
2
3
2
1
3
2
1
Α̂
!
1
!
1
20. its.unc.edu 20
The Two Extreme Cases
One determinant: The Hartree–Fock method.
All possible determinants: The full CI method.
N
N
3
2
1 3
2
1
HF
There are N MOs and each MO is a linear combination of N AOs.
Thus, there are nN coefficients ukl, which are determined by
making stationary the functional:
The ij are Lagrangian multipliers.
N
l
k
ij
lj
kl
ki
N
j
i
ij u
S
u
H
E
1
,
*
1
,
HF
HF
HF
ˆ
21. its.unc.edu 21
The Full CI Method
The full configuration interaction (full CI) method
expands the wavefunction in terms of all possible Slater
determinants:
There are possible ways to choose n molecular
orbitals from a set of 2N AO basis functions.
The number of determinants gets easily much too large.
For example:
n
N
2
1
ˆ
;
2
1
,
CI
CI
CI
2
1
CI
c
S
c
H
E
c
n
N
*
n
N
9
10
10
40
Davidson’s method can be used to find one
or a few eigenvalues of a matrix of rank 109.
22. its.unc.edu 22
N
N
3
2
1 3
2
1
HF
N
l
k
ij
lj
kl
ki
N
j
i
ij u
S
u
H
E
1
,
*
1
,
HF
HF
HF
ˆ
N
i
li
ki
kl
N
l
k
kl
mn
N
n
m
mn u
u
P
nl
mk
P
h
P
E
H
1
*
1
,
2
1
1
,
nuc
HF
HF ;
ˆ
0
HF
E
uki
Hartree–Fock equations
The Hartree–Fock Method
23. its.unc.edu 23
|
S
Overlap integral
|
2
1
|
P
H
F
i
i
occ
i
c
c
2
P
Density Matrix
S
F
i
i
i c
c
The Hartree–Fock Method
24. its.unc.edu 24
1. Choose start coefficients for MO’s
2. Construct Fock Matrix with coefficients
3. Solve Hartree-Fock-Roothaan equations
4. Repeat 2 and 3 until ingoing and outgoing
coefficients are the same
Self-Consistent-Field (SCF)
S
F
i
i
i c
c
25. its.unc.edu 25
Semi-empirical methods
(MNDO, AM1, PM3, etc.)
Full CI
perturbational hierarchy
(CASPT2, CASPT3)
perturbational hierarchy
(MP2, MP3, MP4, …)
excitation hierarchy
(MR-CISD)
excitation hierarchy
(CIS,CISD,CISDT,...)
(CCS, CCSD, CCSDT,...)
Multiconfigurational HF
(MCSCF, CASSCF)
Hartree-Fock
(HF-SCF)
Ab Initio Methods
27. its.unc.edu 27
Size vs Accuracy
Number of atoms
0.1
1
10
1 10 100 1000
Accuracy
(kcal/mol)
Coupled-cluster,
Multireference
Nonlocal density functional,
Perturbation theory
Local density functional,
Hartree-Fock
Semiempirical Methods
Full CI
28. its.unc.edu 28
ROO,e= 291.2 pm
96.4 pm
95.7 pm 95.8 pm
symmetry: Cs
Equilibrium structure of (H2O)2
W.K., J.G.C.M. van Duijneveldt-van de Rijdt, and
F.B. van Duijneveldt, Phys. Chem. Chem. Phys. 2, 2227 (2000).
Experimental [J.A. Odutola and T.R. Dyke, J. Chem. Phys 72, 5062 (1980)]:
ROO
2 ½ = 297.6 ± 0.4 pm
SAPT-5s potential [E.M. Mas et al., J. Chem. Phys. 113, 6687 (2000)]:
ROO
2 ½ – ROO,e= 6.3 pm ROO,e(exptl.) = 291.3 pm
AN EXAMPLE
30. its.unc.edu 30
LCAO Basis Functions
’s, which are atomic orbitals, are called basis
functions
usually centered on atoms
can be more general and more flexible than atomic
orbital functions
larger number of well chosen basis functions yields
more accurate approximations to the molecular
orbitals
c
31. its.unc.edu 31
Basis Functions
Slaters (STO)
Gaussians (GTO)
Angular part *
Better behaved than Gaussians
2-electron integrals hard
2-electron integrals simpler
Wrong behavior at nucleus
Decrease too fast with r
r)
exp(
2
n
m
l
r
exp
*
z
y
x
32. its.unc.edu 32
Contracted Gaussian Basis Set
Minimal
STO-nG
Split Valence: 3-
21G,4-31G, 6-
31G
• Each atom optimized STO is fit with n
GTO’s
• Minimum number of AO’s needed
• Contracted GTO’s optimized per atom
• Doubling of the number of valence AO’s
33. its.unc.edu 33
Polarization /
Diffuse Functions
Polarization: Add AO with higher angular
momentum (L) to give more flexibility
Example: 3-21G*, 6-31G*, 6-31G**, etc.
Diffusion: Add AO with very small exponents for
systems with very diffuse electron densities
such as anions or excited states
Example: 6-31+G*, 6-311++G**
34. its.unc.edu 34
Correlation-Consistent
Basis Functions
a family of basis sets of increasing size
can be used to extrapolate to the basis set limit
cc-pVDZ – DZ with d’s on heavy atoms, p’s on H
cc-pVTZ – triple split valence, with 2 sets of d’s
and one set of f’s on heavy atoms, 2 sets of p’s
and 1 set of d’s on hydrogen
cc-pVQZ, cc-pV5Z, cc-pV6Z
can also be augmented with diffuse functions
(aug-cc-pVXZ)
35. its.unc.edu 35
Pseudopotentials,
Effective Core Potentials
core orbitals do not change much during chemical
interactions
valence orbitals feel the electrostatic potential of the
nuclei and of the core electrons
can construct a pseudopotential to replace the
electrostatic potential of the nuclei and of the core
electrons
reduces the size of the basis set needed to represent the
atom (but introduces additional approximations)
for heavy elements, pseudopotentials can also include of
relativistic effects that otherwise would be costly to treat
36. its.unc.edu 36
Correlation Energy
HF does not include correlations anti-parallel electrons
Eexact – EHF = Ecorrelation
Post HF Methods:
• Configuration Interaction (CI, MCSCF, CCSD)
• Møller-Plesset Perturbation series (MP2, MP4)
Density Functional Theory (DFT)
37. its.unc.edu 37
Configuration-Interaction (CI)
In Hartree-Fock theory, the n-electron wavefunction is approximated by one
single Slater-determinant, denoted as:
This determinant is built from n orthonormal spin-orbitals. The spin-orbitals that
form are said to be occupied. The other orthonormal spin-orbitals that
follow from the Hartree-Fock calculation in a given one-electron basis set of
atomic orbitals (AOs) are known as virtual orbitals. For simplicity, we assume that
all spin-orbitals are real.
In electron-correlation or post-Hartree-Fock methods, the wavefunction is
expanded in a many-electron basis set that consists of many determinants.
Sometimes, we only use a few determinants, and sometimes, we use millions of
them:
In this notation, is a Slater-
determinant that is obtained by
replacing a certain number of
occupied orbitals by virtual ones.
Three questions: 1. Which determinants should we include?
2. How do we determine the expansion coefficients?
3. How do we evaluate the energy (or other properties)?
HF
HF
c
HF
CI
38. its.unc.edu 38
Truncated configuration interaction:
CIS, CISD, CISDT, etc.
We start with a reference wavefunction, for example the Hartree-
Fock determinant.
We then select determinants for the wavefunction expansion by
substituting orbitals of the reference determinant by orbitals that
are not occupied in the reference state (virtual orbitals).
Singles (S) indicate that 1 orbital is replaced, doubles (D) indicate
2 replacements, triples (T) indicate 3 replacements, etc., leading
to CIS, CISD, CISDT, etc.
N
N
k
j
i
3
2
1
HF
etc.
,
3
2
1
,
3
2
1 N
N N
k
b
a
ab
ij
N
k
j
a
a
i
39. its.unc.edu 39
Truncated
Configuration Interaction
Level of
excitation
Number of
parameters
Example
CIS n (2N – n) 300
CISD … + [n (2N – n)] 2
78,600
CISDT …+ [n (2N – n)] 3
18106
… … …
Full CI
n
N
2
109
Number of linear variational parameters
in truncated CI for n = 10 and 2N = 40.
40. its.unc.edu 40
Multi-Configuration
Self-Consistent Field (MCSCF)
The MCSCF wavefunctions consists of a few selected determinants or CSFs. In the
MCSCF method, not only the linear weights of the determinants are variationally
optimized, but also the orbital coefficients.
One important selection is governed by the full CI space spanned by a number of
prescribed active orbitals (complete active space, CAS). This is the CASSCF method.
The CASSCF wavefunction contains all determinants that can be constructed from a
given set of orbitals with the constraint that some specified pairs of - and -spin-
orbitals must occur in all determinants (these are the inactive doubly occupied
spatial orbitals).
Multireference CI wavefunctions are obtained by applying the excitation operators to
the individual CSFs or determinants of the MCSCF (or CASSCF) reference wave
function.
k
C
C
c
k
k
k
k )
ˆ
ˆ
(
CISD
-
MR 2
1
k
k
k
k
k k
d
C
k
C
c 2
1
ˆ
)
ˆ
(
MRCI
-
IC
Internally-contracted MRCI:
41. its.unc.edu 41
Coupled-Cluster Theory
System of equations is solved iteratively (the convergence is
accelerated by utilizing Pulay’s method, “direct inversion in
the iterative subspace”, DIIS).
CCSDT model is very expensive in terms of computer
resources. Approximations are introduced for the triples:
CCSD(T), CCSD[T], CCSD-T.
Brueckner coupled-cluster (e.g., BCCD) methods use
Brueckner orbitals that are optimized such that singles don’t
contribute.
By omitting some of the CCSD terms, the quadratic CI method
(e.g., QCISD) is obtained.
42. its.unc.edu 42
Møller-Plesset
Perturbation Theory
The Hartree-Fock function is an eigenfunction of the
n-electron operator .
We apply perturbation theory as usual after decomposing the
Hamiltonian into two parts:
More complicated with more than one reference determinant
(e.g., MR-PT, CASPT2, CASPT3, …)
F̂
F
H
H
F
H
H
H
H
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
1
0
1
0
MP2, MP3, MP4, …etc.
number denotes order to which
energy is computed (2n+1 rule)
43. its.unc.edu 43
Semi-Empirical Methods
These methods are derived from the Hartee–Fock model, that is,
they are MO-LCAO methods.
They only consider the valence electrons.
A minimal basis set is used for the valence shell.
Integrals are restricted to one- and two-center integrals and
subsequently parametrized by adjusting the computed results to
experimental data.
Very efficient computational tools, which can yield fast
quantitative estimates for a number of properties. Can be used
for establishing trends in classes of related molecules, and for
scanning a computational poblem before proceeding with high-
level treatments.
A not of elements, especially transition metals, have not be
parametrized
44. its.unc.edu 44
Semi-Empirical Methods
Number 2-electron integrals (|) is n4/8, n = number of basis
functions
Treat only valence electrons explicit
Neglect large number of 2-electron integrals
Replace others by empirical parameters
Models:
• Complete Neglect of Differential Overlap (CNDO)
• Intermediate Neglect of Differential Overlap (INDO/MINDO)
• Neglect of Diatomic Differential Overlap (NDDO/MNDO, AM1, PM3)
45. its.unc.edu 45
A
B
AB
V
U
H
U from atomic spectra
VAB value per atom pair
0
H
, on the same atom
S
H AB
B
A
AB 2
1
One parameter per element
Approximations of 1-e
integrals
46. its.unc.edu 46
Popular DFT
Noble prize in Chemistry, 1998
In 1999, 3 of top 5 most cited
journal articles in chemistry (1st,
2nd, & 4th)
In 2000-2003, top 3 most cited
journal articles in chemistry
In 2004-2005, 4 of top 5 most
cited journal articles in chemistry:
• 1st, Becke’s hybrid exchange
functional (1993)
• 2nd, LYP correlation functional
(1988)
• 3rd, Becke’s exchange
functional (1988)
• 4th, PBE correlation functional
(1996)
http://www.cas.org/spotlight/bchem.html
Citations of DFT on JCP, JACS and PRL
47. its.unc.edu 47
Brief History of DFT
First speculated 1920’
•Thomas-Fermi (kinetic energy) and Dirac
(exchange energy) formulas
Officially born in 1964 with Hohenberg-
Kohn’s original proof
GEA/GGA formulas available later 1980’
Becoming popular later 1990’
Pinnacled in 1998 with a chemistry Nobel
prize
48. its.unc.edu 48
What could expect from DFT?
LDA, ~20 kcal/mol error in energy
GGA, ~3-5 kcal/mol error in energy
G2/G3 level, some systems, ~1kcal/mol
Good at structure, spectra, & other
properties predictions
Poor in H-containing systems, TS, spin,
excited states, etc.
49. its.unc.edu 49
Density Functional Theory
Two Hohenberg-Kohn theorems:
•“Given the external potential, we know the
ground-state energy of the molecule when
we know the electron density ”.
•The energy density functional is variational.
E
Ĥ
Energy
50. its.unc.edu 50
But what is E[]?
How do we compute the energy if the density is
known?
The Coulombic interactions are easy to compute:
But what about the kinetic energy TS[] and exchange-
correlation energy Exc[]?
,
]
[
,
]
[
,
]
[ 2
1
ext
ne
nn r
r
r
r
r
r
r
r
r
d
d
J
d
V
E
r
Z
Z
E
nuclei
B
A AB
B
A
E[] = TS[] + Vne[] + J[] + Vnn[] + Exc[]
52. its.unc.edu 52
All about Exchange-Correlation
Energy Density Functional
LDA – f(r) is a function of
(r) only
GGA – f(r) is a function of
(r) and |∇(r)|
Mega-GGA – f(r) is also a
function of ts(r), kinetic
energy density
Hybrid – f(r) is GGA
functional with extra
contribution from Hartree-
Fock exchange energy
r
r
r
r d
f
QXC
,
,
, 2
Jacob's ladder for the five generation of DFT functionals,
according to the vision of John Perdew with indication of
some of the most common DFT functionals within each rung.
53. its.unc.edu 53
LDA Functionals
Thomas-Fermi formula (Kinetic) – 1
parameter
Slater form (exchange) – 1 parameter
Wigner correlation – 2 parameters
3
/
2
2
3
/
5
3
10
3
,
F
F
TF C
d
C
T r
r
3
/
1
3
/
2
3
/
1
3
/
4
4
3
8
3
,
X
X
S
X C
d
C
E r
r
r
r
r
d
b
a
EW
C 3
/
1
1
54. its.unc.edu 54
Popular Functional: BLYP/B3LYP
Two most well-known functionals are the Becke exchange functional
Ex[] with 2 extra parameters &
The Lee-Yang-Parr correlation functional Ec[] with 4 parameters a-d
Together, they constitute the BLYP functional:
The B3LYP functional is augmented with 20% of Hartree-Fock
exchange:
r
r
r
r d
e
d
e
E
E
E c
x
c
x
xc
,
, LYP
B
LYP
B
BLYP
3
/
4
2
2
2
3
/
4
,
1
LDA
X
B
X E
E
r
d
e
t
t
C
b
d
a
E c
W
W
F
LYP
c
3
/
1
2
3
/
5
3
/
2
3
/
1
18
1
9
1
2
1
1
nl
km
P
P
b
E
E
a
E
N
l
k
kl
N
n
m
mn
c
x
xc
1
,
1
,
LYP
B
B3LYP
55. its.unc.edu 55
Density Functionals
LDA
local density
GGA
gradient corrected
Meta-GGA
kinetic energy density
included
Hybrid
“exact” HF exchange
component
Hybrid-meta-GGA
VWN5
BLYP
HCTH
BP86
TPSS
M06-L
B3LYP
B97/2
MPW1K
MPWB1K
M06
Better scaling with system
size
Allow density fitting for even
better scaling
Meta-GGA is “bleeding
edge” and therefore largely
untested (but better in
theory…)
Hybrid makes bigger
difference in cost and
accuracy
Look at literature if
somebody
has compared functionals
for
systems similar to yours!
Increasing
quality
and
computational
cost
56. its.unc.edu 56
Percentage of occurrences of the names of the several functionals indicated in Table 2, in
journal titles and abstracts, analyzed from the ISI Web of Science (2007).
S.F. Sousa, P.A. Fernandes and M.J. Ramos, J. Phys. Chem. A 10.1021/jp0734474 S1089-5639(07)03447-0
Density Functionals
57. its.unc.edu 57
Problems with DFT
ground-state theory only
universal functional still unknown
even hydrogen atom a problem: self-interaction
correction
no systematic way to improve approximations
like LDA, GGA, etc.
extension to excited states, spin multiplets,
etc., though proven exact in theory, is not
trivial in implementation and still far from
being generally accessible thus far
58. its.unc.edu 58
DFT Developments
Theoretical
• Extensions to excited states, etc.
• Better functionals (mega-GGA), etc
• Understanding functional properties, etc.
Conceptual
• More concepts proposed, like electrophilicity, philicity, spin-
philicity, surfaced-integrated Fukui fnc
• Dynamic behaviors, profiles, etc.
Computational
• Linear scaling methods
• QM/MM related issues
• Applications
60. its.unc.edu 60
Chemical Reactivity Theory
Chemical reactivity theory quantifies the reactive propensity of
isolated species through the introduction of a set of reactivity indices
or descriptors. Its roots go deep into the history of chemistry, as far
back as the introduction of such fundamental concepts as acid, base,
Lewis acid, Lewis base, etc. It pervades almost all of chemistry.
Molecular Orbital Theory
• Fukui’s Frontier Orbital (HOMO/LUMO) model
• Woodward-Hoffman rules
• Well developed: Nobel prize in Chemistry, 1981
• Problem: conceptual simplicity disappears as computational
accuracy increases because it’s based on the molecular orbital
description
Density Functional Theory (DFT)
• Conceptual DFT, also called Chemical DFT, DF Reactivity Theory
• Proposed by Robert G. Parr of UNC-CH, 1980s
• Still in development
-- Morrel H. Cohen, and Adam Wasserman, J. Phys. Chem. A 2007, 111,2229
61. its.unc.edu 61
DFT Reactivity Theory
General Consideration
• E E [N, (r)] E []
• Taylor Expansion: Perturbation resulted from an
external attacking agent leading to changes in N and
(r), N and (r),
'
'
2
!
,
,
2
2
2
2
r
r
r
r
r
r
r
r
2
1
r
r
r
r
r
r
2
d
d
E
d
N
E
N
N
N
E
d
E
N
N
E
N
E
N
N
E
E
N
N
N
Assumptions: existence and well-behavior of all above partial/functional derivatives
62. its.unc.edu 62
Conceptual DFT
Basic assumptions
•E E [N, (r)] E []
•Chemical processes, responses, and changes
expressible via Taylor expansion
•Existence, continuous, and well-behavedness
of the partial derivatives
63. its.unc.edu 63
DFT Reactivity Indices
Electronegativity (chemical potential)
Hardness / Softness
Maximum Hardness Principle (MHP)
/
1
,
2
2
1
2
2
S
N
E HOMO
LUMO
2
LUMO
HOMO
N
E
64. its.unc.edu 64
DFT Reactivity Indices
Fukui
function
N
f
r
r
– Nucleophilic attack
r
r
r N
N
f
1
– Electrophilic attack
r
r
r 1
N
N
f
– Free radical activity
2
r
r
r
f
f
f
65. its.unc.edu 65
Electrophilicity Index
Physical meaning: suppose an
electrophile is immersed in an electron
sea
The maximal electron flow and
accompanying energy decrease are
2
2
1
N
N
E
2
2
max
N
2
2
2
2
min
E Parr, Szentpaly, Liu, J. Am. Chem. Soc. 121, 1922(1999).
66. its.unc.edu 66
Experiment vs. Theory
Pérez, P. J. Org. Chem. 2003, 68, 5886. Pérez, P.; Aizman, A.; Contreras, R. J. Phys. Chem. A 2002, 106, 3964.
2
2
log
(k)
=
s(E+N)
67. its.unc.edu 67
Minimum Electrophilicity Principle
Analogous to the maximum hardness principle (MHP)
Separately proposed by Noorizadeh and Chattaraj
Concluded that “the natural direction of a chemical reaction is
toward a state of minimum electrophilicity.”
Noorizadeh, S. Chin. J. Chem. 2007, 25, 1439.
Noorizadeh, S. J. Phys. Org. Chem. 2007, 20, 514.
Chattaraj, P.K. Ind. J. Phys. Proc. Ind. Natl. Sci. Acad. Part A 2007, 81, 871.
non-
LA
1 2 3 4 5 6 7
Aa -0.091 -
0.085
-0.093 -0.093 -
0.088
-0.087 -0.083 -0.090
Bb -0.089 -
0.084
-0.088 -0.089 -
0.087
-0.087 -0.0842 -
0.0892
Aa -0.172 -
0.247
-0.230 -0.220 -
0.218
-0.226 -0.2518 -
0.2161
Bb -0.171 -
0.246
-0.247 -0.233 -
0.221
-0.226 -0.2506 -
0.2157
Yue Xia, Dulin Yin, Chunying Rong, Qiong Xu, Donghong Yin, and Shubin Liu, J. Phys. Chem. A, 2008, 112, 9970.
68. its.unc.edu 68
Nucleophilicity
Much harder to quantify, because it related to local
hardness, which is ambiguous in definition.
A nucleophile can be a good donor for one electrophile
but bad for another, leading to the difficulty to define a
universal scale of nucleophilicity for an nucleophile.
A
B
A
B
A
2
2
1
Jaramillo, P.; Perez, P.; Contreras, R.; Tiznado, W.; Fuentealba, P. J. Phys. Chem. A 2006, 110, 8181.
= -N - ½ S()2
Minimizing in Eq. (14) with respect to ,
one has
=-N and = - ½ N2.
Making use of the following relation
B
A
B
A
N
69. its.unc.edu 69
Philicity and Fugality
Philicity: defined as ·f(r)
• Chattaraj, Maiti, & Sarkar, J. Phys. Chem. A 107, 4973(2003)
• Still a very controversial concept, see JPCA 108, 4934(2004); Chattaraj,
et al. JPCA, in press.
Spin-Philicity: defined same as but in spin resolution
• Perez, Andres, Safont, Tapia, & Contreras. J. Phys. Chem. A 106,
5353(2002)
Nuclofugality & Electrofugality
2
)
( 2
A
En
2
)
( 2
I
Ee
Ayers, P.W.; Anderson, J.S.M.; Rodriguez, J.I.; Jawed, Z. Phys. Chem. Chem. Phys. 2005, 7, 1918.
Ayers, P.W.; Anderson, J S.M.; Bartolotti, L.J. Int. J. Quantum Chem. 2005, 101, 520.
70. its.unc.edu 70
Dual Descriptors
N
N
N
N
f
N
E
E
N
f
r
r
r
r
r
2
2
2
2
2
3rd-order cross-term derivatives
0
2
r
r d
f
r
r
r
f
f
f 2
r
r
r HOMO
LUMO
f
2
Recovering Woodward-Hoffman rules!
Ayers, P.W.; Morell, C., De Proft, D.; Geerlings, P. Chem. Eur. J., 2007, 13, 8240
Geerling, P. De Proft F. Phys. Chem. Chem. Phys., 2008, 10, 3028
71. its.unc.edu 71
Steric Effect
one of the most widely used
concepts in chemistry
originates from the space occupied
by atom in a molecule
previous work attributed to the
electron exchange correlation
Weisskopf thought of as “kinetic
energy pressure”
Weisskopf, V.F., Science 187, 605-612(1975).
72. its.unc.edu 72
Steric effect: a DFT description
Assume
since
we have
E[] ≡ Es[] + Ee[] + Eq[]
E[] = Ts[] + Vne[] + J[] + Vnn[] + Exc[]
Ee[] = Vne[] + J[] + Vnn[]
Eq[] = Exc[] + EPauli[] = Exc[] + Ts[] - Tw[]
Es[] ≡ E[] - Ee[] - Eq[] = Tw[]
r
r
r
d
TW
2
8
1
S.B. Liu, J. Chem. Phys. 2007, 126, 244103.
S.B. Liu and N. Govind, J. Phys. Chem. A 2008, 112, 6690.
S.B. Liu, N. Govind, and L.G. Pedersen, J. Chem. Phys. 2008, 129, 094104.
M. Torrent-Sucarrat, S.B. Liu and F. De Proft, J. Phys. Chem. A 2009, 113, 3698.
73. its.unc.edu 73
In 1956, Taft constructed a scale for the steric effect of different substituents,
based on rate constants for the acid-catalyzed hydrolysis of esters in aqueous
acetone. It was shown that log(k / k0) was insensitive to polar effects and thus,
in the absence of resonance interactions, this value can be considered as being
proportional to steric effects. Hydrogen is taken to have a reference value of
EsTaft= 0
Experiment vs. Theory
75. its.unc.edu 75
Glu 165 (the catalytic base), His 95 (the proton shuttle)
DHAP GAP
TIM 2-step 2-residue Mechanism
76. its.unc.edu 76
QM/MM: 1st Step of TIM
Mechanism
QM/MM size: 6051 atoms QM Size: 37 atoms
QM: Gaussian’98 Method: HF/3-21G
MM: Tinker Force field: AMBER all-atom
Number of Water: 591 Model for Water: TIP3P
MD details: 20x20x20 Å3 box, optimize until the RMS energy
gradient less than 1.0 kcal/mol/Å. 20 psec MD. Time step 2fs.
SHAKE, 300 K, short range cutoff 8 Å, long range cutoff 15 Å.
77. its.unc.edu 77
QM/MM: Transition State
=====================
Energy Barrier (kcal/mol)
-------------------------------------
QM/MM 21.9
Experiment 14.0
=====================
78. its.unc.edu 78
What’s New: Linear Scaling
O(N) Method
Numerical Bottlenecks:
• diagonalization ~N3
• orthonormalization ~N3
• matrix element evaluation ~N2-N4
Computational Complexity: N log N
Theoretical Basis: near-sightedness of density
matrix or orbitals
Strategy:
• sparsity of localized orbital or density
matrix
• direct minimization with conjugate
gradient
Models: divide-and-conquer and variational
methods
Applicability: ~10,000 atoms, dynamics
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600 700 800 900
Atoms
CPU
se
conds
pe
r
CG
ste
p
OLMO
NOLMO
Diagonalization
79. its.unc.edu 79
What Else … ?
Solvent effect
•Implicit model vs. explicit model
Relativity effect
Transition state
Excited states
Temperature and pressure
Solid states (periodic boundary condition)
Dynamics (time-dependent)
81. its.unc.edu 81
Popular QM codes
Gaussian (Ab Initio, Semi-empirical, DFT)
Gamess-US/UK (Ab Initio, DFT)
Spartan (Ab Initio, Semi-empirical, DFT)
NWChem (Ab Initio, DFT, MD, QM/MM)
MOPAC/2000 (Semi-Empirical)
DMol3/CASTEP (DFT)
Molpro (Ab initio)
ADF (DFT)
ORCA (DFT)
82. its.unc.edu 82
Reference Books
Computational Chemistry (Oxford Chemistry Primer) G. H.
Grant and W. G. Richards (Oxford University Press)
Molecular Modeling – Principles and Applications, A. R. Leach
(Addison Wesley Longman)
Introduction to Computational Chemistry, F. Jensen (Wiley)
Essentials of Computational Chemistry – Theories and Models,
C. J. Cramer (Wiley)
Exploring Chemistry with Electronic Structure Methods, J. B.
Foresman and A. Frisch (Gaussian Inc.)
83. its.unc.edu 83
Questions & Comments
Please direct comments/questions about research computing to
E-mail: research@unc.edu
Please direct comments/questions pertaining to this presentation to
E-Mail: shubin@email.unc.edu
The PPT format of this presentation is available here:
http://its2.unc.edu/divisions/rc/training/scientific/
/afs/isis/depts/its/public_html/divisions/rc/training/scientific/short_courses/
84. its.unc.edu 84
Hands-on: Part I
Purpose: to get to know the available ab
initio and semi-empirical methods in the
Gaussian 03 / GaussView package
• ab initio methods
Hartree-Fock
MP2
CCSD
• Semiempirical methods
AM1
The WORD .doc format of this hands-on exercises is available here:
http://its2.unc.edu/divisions/rc/training/scientific/
/afs/isis/depts/its/public_html/divisions/rc/training/scientific/short_courses/labDirections_compchem_2009.doc
85. its.unc.edu 85
Hands-on: Part II
Purpose: To use LDA and GGA DFT
methods to calculate IR/Raman spectra
in vacuum and in solvent. To build
QM/MM models and then use DFT
methods to calculate IR/Raman spectra
• DFT
LDA (SVWN)
GGA (B3LYP)
• QM/MM