IntroductiontoCompChem_2009.pptbbbbbbbbbbb

Introduction to
Introduction to
Computational Chemistry
Shubin Liu, Ph.D.
Research Computing Center
University of North Carolina at Chapel Hill

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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/

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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

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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

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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

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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?

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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!

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Central Theme of
DYNAMICS
REACTIVITY
STRUCTURE CENTRAL DOGMA OF MOLECULAR BIOLOGY
SEQUENCE

STRUCTURE

DYNAMICS

FUNCTION

EVALUTION

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Multiscale Hierarchy of
Modeling

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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


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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

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 
   
 









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


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Equation to Solve in
ab initio Theory


 E
H
Known exactly:
3N spatial variables
(N # of electrons)
To be approximated:
1. variationally
2. perturbationally

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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

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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.

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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

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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

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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

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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

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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 nN 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 


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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
.

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       
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

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

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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

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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

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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

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ROO,e= 291.2 pm
96.4 pm
95.7 pm 95.8 pm
symmetry: Cs
Equilibrium structure of (H
Equilibrium structure of (H2
2O)
O)2
2
W.K., J.G.C.M. van Duijneveldt-van de Rijdt, and
W.K., J.G.C.M. van Duijneveldt-van de Rijdt, and
F.B. van Duijneveldt,
F.B. van Duijneveldt, Phys. Chem. Chem. Phys.
Phys. Chem. Chem. Phys. 2
2, 2227 (2000).
, 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

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Experimental and Computed
Enthalpy Changes He in kJ/mol
Exptl. CCSD(T) SCF G2 DFT
CH4
 CH2 + H2 544(2) 542 492 534 543
C2H4
 C2H2 + H2 203(2) 204 214 202 208
H2CO  CO + H2 21(1) 22  3 17 34
2 NH3
 N2 + 3 H2 164(1) 162 149 147 166
2 H2O  H2O2 + H2 365(2) 365 391 360 346
2 HF  F2 + H2 563(1) 562 619 564 540
Gaussian-2 (G2) method of Pople and co-workers is a combination of MP2 and QCISD(T)

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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

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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 


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

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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**

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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)

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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

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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)

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


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 






 
 




its.unc.edu 39
Truncated
Configuration Interaction
L
ev
elo
f
ex
citatio
n
N
u
m
b
ero
f
p
aram
eters
E
x
am
p
le
C
IS n
(2
N–n
) 3
0
0
C
IS
D …+[n
(2
N–n
)]2
7
8
,6
0
0
C
IS
D
T …
+[n
(2
N–n
)]3
1
8

1
0
6
… … …
F
u
llC
I 







n
N
2 
1
0
9
Number of linear variational parameters
in truncated CI for n = 10 and 2N = 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:

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.

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)

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

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)

its.unc.edu 45




A
B
AB
V
U
H 
 Ufrom atomic spectra
V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

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

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

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.

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

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[]

its.unc.edu 51
Kohn-Sham Scheme
,
|
)
(
|
)
(
,
)
(
,
|
|
)
(
)
(
,
|
|
)
(
,
2
1
and
)
(
)
(
)
(
ˆ
where
,
ˆ
2
3
2
























nk
nk
nk
xc
xc
ee
a a
a
ne
xc
ee
ne
nk
nk
nk
r
f
r
E
r
V
r
d
r
r
r
r
V
R
r
Z
r
V
K
r
V
r
V
r
V
K
H
H





The Only
Unknown
• Suppose, we know the
exact density.
• Then, we can formulate a
Slater determinant that
generates this exact density
(= Slater determinant of
system of N non-interacting
electrons with same density
).
• We know how to compute
the kinetic energy Ts
exactly from a Slater
determinant.
• Then, the only thing
unknown is to calculate
Exc[].

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.

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 



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

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

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

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

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

its.unc.edu 59
Examples DFT vs. HF
Hydrogen molecules - using the LSDA (LDA)

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

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

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

its.unc.edu 63
DFT Reactivity Indices
 Electronegativity (chemical potential)
 Hardness / Softness
 Maximum Hardness Principle (MHP)
 HSAB (hard and Soft Acid and Base) Principle





/
1
,
2
2
1
2
2













 S
N
E HOMO
LUMO
2
LUMO
HOMO
N
E 
















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

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).

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)

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.

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








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.

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

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).

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.

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

its.unc.edu 74
QM/MM Example:
Triosephosphate Isomerase (TIM)
494 Residues, 4033 Atoms, PDB ID: 7TIM
Function: DHAP (dihydroxyacetone phosphate) GAP (glyceraldehyde 3-phosphate)
GAP
DHAP
H2O

its.unc.edu 75
Glu 165 (the catalytic base), His 95 (the proton shuttle)
DHAP GAP
TIM 2-step 2-residue Mechanism

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.

its.unc.edu 77
QM/MM: Transition State
=====================
Energy Barrier (kcal/mol)
------------------------------------
-
QM/MM 21.9
Experiment 14.0
=====================

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
C
P
U
s
ec
o
n
d
s
p
e
r
C
G
s
tep
OLMO
NOLMO
Diagonalization

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)

its.unc.edu 80
Limitations and Strengths
of ab initio quantum
chemistry

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)

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.)

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:
/afs/isis/depts/its/public_html/divisions/rc/training/scientific/short_courses/

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:
/afs/isis/depts/its/public_html/divisions/rc/training/scientific/short_courses/labDirections_compchem_2009.doc

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

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