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Introduction to Computational Introduction to Computational Chemistry Chemistry
Shubin Liu, Ph.D.Renaissance Computing Institute
University of North Carolina at Chapel Hill
10/1/2007 Introduction to Computational Chemistry 2
Outline• Introduction
• Methods in Computational Chemistry – Ab Initio– Semi-Empirical – Density Functional Theory
– New Developments (QM/MM)
• Hands-on Exercises
10/1/2007 Introduction to Computational Chemistry 3
Goals of Course
• To get familiar with computational chemistry methods available
• To serve as the starting point for further reading and applications
• Hands-on experiments via G03/GaussView
10/1/2007 Introduction to Computational Chemistry 4
Prerequisites
• UNIX & LSF basics– Basic kernel commands (e.g., ls, cd, more, vi, rm, …, bsub, bjobs, …)
• Introduction to Scientific Computing• Introduction to Gaussian/GaussView• An account on Emerald cluster with
csh/tcsh Shell (type “echo $SHELL”)
10/1/2007 Introduction to Computational Chemistry 5
About Us• ITS
– http://its.unc.edu– Physical locations: 401 West Franklin Street; 211 Manning Drive– 12 Divisions
IT Infrastructure and Operations Research Computing Teaching and Learning Technology Planning and Special Projects Telecommunications User Support and Engagement Office of the Vice Chancellor Communications Enterprise Applications Enterprise Data Management Financial Planning and Human Resources Information Security
• RENCI– http://www.renci.org/– Anchor Site: 100 Europa Drive, suite 540, Chapel Hill – A number of virtual sites on the campuses of Duke, NCSU and UNC-Chapel Hill, and
regional facilities across the state – Mission: to foster multidisciplinary collaborations; to enable advancements in science,
industry, education, the humanities and the arts; to provide the technical leadership and expertise; to work hand-in-hand with businesses and communities to utilize advanced technologies
10/1/2007 Introduction to Computational Chemistry 6
About Us
• Where/Who are we and do we do?– ITS Manning: 211 Manning Drive– Website
http://www.renci.org/unc/computing/– Groups
• Infrastructure • Engagement • User Support
10/1/2007 Introduction to Computational Chemistry 7
About Myself• Ph.D. from Chemistry, UNC-CH• Currently Senior Computational Scientist Renaissance Computing Institute at UNC-CH• Responsibilities:
– Support Comp Chem/Phys/Material Science software, Support Programming (FORTRAN/C/C++) tools, code porting, parallel computing, etc.
– Engagement projects with faculty members on campus– Conduct own research on Comp Chem
• DFT theory and concept• Systems in biological and material science
10/1/2007 Introduction to Computational Chemistry 8
About You
• Name, department, group, research interest?
• Do you have any real problem that is intended to be studied by computational chemistry approaches?
• If yes, what is it?
10/1/2007 Introduction to Computational Chemistry 9
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 do?– 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!
10/1/2007 Introduction to Computational Chemistry 10
Central Theme of Computational Chemistry
DYNAMICS
REACTIVITY
STRUCTURE CENTRAL DOGMA OF MOLECULAR BIOLOGY
SEQUENCE
STRUCTURE
DYNAMICS
FUNCTION
EVALUTION
10/1/2007 Introduction to Computational Chemistry 11
Multiscale Hierarchy of Modeling
10/1/2007 Introduction to Computational Chemistry 12
What is Computational Chemistry?
Application of computational methods and algorithms in chemistry
– Quantum Mechanicali.e., via Schrödinger Equation
also called Quantum Chemistry– Molecular Mechanical
i.e., via Newton’s law F=maalso Molecular Dynamics
– Empirical/Statisticale.g., QSAR, etc., widely used in clinical and medicinal chemistry
Focus TodayFocus Today
Ht
i ˆ
Ht
i ˆ
10/1/2007 Introduction to Computational Chemistry 13
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
10/1/2007 Introduction to Computational Chemistry 14
ij
n
1i ij
n
1i
N
1 i
2i
2
r
1
r
Z-
2m
h- H
n
ij
n
1i ij
n
1i r
1ih H
Starting Point: Time-Independent Schrodinger Equation
EH
Ht
i ˆ
Ht
i ˆ
10/1/2007 Introduction to Computational Chemistry 15
Equation to Solve in ab initio Theory
EH
Known exactly:3N spatial variables
(N # of electrons)
To be approximated:1. variationally2. perturbationally
10/1/2007 Introduction to Computational Chemistry 16
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
BA AB
BAelectrons
ji ij
nuclei
A iA
Aelectrons
iA
nuclei
A Ai
electrons
i e
r
ZZe
r
e
r
Ze
mm22
22
22
2
22ˆ H
nuclei
BA AB
BAelectrons
ji ij
nuclei
A iA
Aelectrons
iA
nuclei
A Ai
electrons
i e
r
ZZe
r
e
r
Ze
mm22
22
22
2
22ˆ H
10/1/2007 Introduction to Computational Chemistry 17
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.
10/1/2007 Introduction to Computational Chemistry 18
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
10/1/2007 Introduction to Computational Chemistry 19
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
BA AB
BAelectrons
ji ij
nuclei
A iA
Aelectrons
ii
electrons
i eel r
ZZe
r
e
r
Ze
m
2222
2
2ˆ H
nuclei
BA AB
BAelectrons
ji ij
nuclei
A iA
Aelectrons
ii
electrons
i eel r
ZZe
r
e
r
Ze
m
2222
2
2ˆ H
d
dEE
elel
elelel
elelel *
* ˆ,ˆ
HH
d
dEE
elel
elelel
elelel *
* ˆ,ˆ
HH
10/1/2007 Introduction to Computational Chemistry 20
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
iiu ik
N
kklil rq
down)-(spin
up)-(spin ;
1
iiu ik
N
kklil rq
10/1/2007 Introduction to Computational Chemistry 21
The Slater Determinant
zcbazcba
zzzz
cccc
bbbb
aaaa
n
zcbazcban
zcba
n
n
n
n
n
nn
n
321
321
321
321
321
312321
321 Α̂
!1
!1
zcbazcba
zzzz
cccc
bbbb
aaaa
n
zcbazcban
zcba
n
n
n
n
n
nn
n
321
321
321
321
321
312321
321 Α̂
!1
!1
10/1/2007 Introduction to Computational Chemistry 22
The Two Extreme Cases
One determinant: The Hartree–Fock method.
All possible determinants: The full CI method.
NN 321 321HF NN 321 321HF
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
lkijljklki
N
jiij uSuHE
1,
*
1,HFHFHF ˆ
N
lkijljklki
N
jiij uSuHE
1,
*
1,HFHFHF ˆ
10/1/2007 Introduction to Computational Chemistry 23
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 basis functions.
• The number of determinants gets easily much too large. For example:
n
N2
1ˆ ;
2
1,CICICI
2
1CI
cScHEc
n
N
*n
N
1ˆ ;
2
1,CICICI
2
1CI
cScHEc
n
N
*n
N
91010
40
91010
40
Davidson’s method can be used to find one or a few eigenvalues of a matrix of rank 109.
10/1/2007 Introduction to Computational Chemistry 24
NN 321 321HF NN 321 321HF
N
lkijljklki
N
jiij uSuHE
1,
*
1,HFHFHF ˆ
N
lkijljklki
N
jiij uSuHE
1,
*
1,HFHFHF ˆ
N
ilikikl
N
lkklmn
N
nmmn uuPnlmkPhPEH
1
*
1,21
1,nucHFHF ; ˆ
N
ilikikl
N
lkklmn
N
nmmn uuPnlmkPhPEH
1
*
1,21
1,nucHFHF ; ˆ
0HF
Euki
0HF
Euki
Hartree–Fock equations
The Hartree–Fock Method
10/1/2007 Introduction to Computational Chemistry 25
|S Overlap integral
|
2
1|PHF
ii
occ
i
cc2PDensity Matrix
SF iii cc
The Hartree–Fock Method
10/1/2007 Introduction to Computational Chemistry 26
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)
10/1/2007 Introduction to Computational Chemistry 27
Semi-empirical methods(MNDO, AM1, PM3, etc.)
Semi-empirical methods(MNDO, AM1, PM3, etc.)
Full CIFull CI
perturbational hierarchy(CASPT2, CASPT3)
perturbational hierarchy(CASPT2, CASPT3)
perturbational hierarchy(MP2, MP3, MP4, …)
perturbational hierarchy(MP2, MP3, MP4, …)
excitation hierarchy(MR-CISD)
excitation hierarchy(MR-CISD)
excitation hierarchy(CIS,CISD,CISDT,...)
(CCS, CCSD, CCSDT,...)
excitation hierarchy(CIS,CISD,CISDT,...)
(CCS, CCSD, CCSDT,...)
Multiconfigurational HF(MCSCF, CASSCF)
Multiconfigurational HF(MCSCF, CASSCF)
Hartree-Fock(HF-SCF)
Hartree-Fock(HF-SCF)
Ab Initio Methods
10/1/2007 Introduction to Computational Chemistry 28
Who’s Who
10/1/2007 Introduction to Computational Chemistry 29
Size vs Accuracy
Number of atoms
0.1
1
10
1 10 100 1000
Acc
urac
y (k
cal/m
ol) Coupled-cluster,
Multireference
Nonlocal density functional,Perturbation theory
Local density functional,Hartree-Fock
Semiempirical Methods
Full CI
10/1/2007 Introduction to Computational Chemistry 30
ROO,e= 291.2 pm
96.4 pm95.7 pm 95.8 pm
symmetry: Cs
Equilibrium structure of (HEquilibrium structure of (H22O)O)22
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. 22, 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
10/1/2007 Introduction to Computational Chemistry 31
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
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)
10/1/2007 Introduction to Computational Chemistry 32
LCAO Basis Functions
’s are called basis functions• usually centered on atoms• can be more general and more flexible than
atomic orbitals• larger number of well chosen basis functions
yields more accurate approximations to the molecular orbitals
c
c
10/1/2007 Introduction to Computational Chemistry 33
Basis Functions
• Slaters (STO)
• Gaussians (GTO)
• Angular part *• Better basis than Gaussians• 2-electron integrals hard
• 2-electron integrals simpler• Wrong behavior at nucleus• Decrease to fast with r
r)exp( r)exp(
2nml rexp*zyx 2nml rexp*zyx
10/1/2007 Introduction to Computational Chemistry 34
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
• 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
• Contracted GTO’s optimized per atom• Doubling of the number of valence AO’s
10/1/2007 Introduction to Computational Chemistry 35
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 statesExample: 6-311++G**
10/1/2007 Introduction to Computational Chemistry 36
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)
10/1/2007 Introduction to Computational Chemistry 37
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
10/1/2007 Introduction to Computational Chemistry 38
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)
10/1/2007 Introduction to Computational Chemistry 39
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
cHFCI
10/1/2007 Introduction to Computational Chemistry 40
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.
NNkji 321HF NNkji 321HF
etc. ,321 ,321 NN NkbaabijNkja
ai etc. ,321 ,321 NN Nkba
abijNkja
ai
10/1/2007 Introduction to Computational Chemistry 41
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
N2 109
Number of linear variational parametersin truncated CI for n = 10 and 2N = 40.
10/1/2007 Introduction to Computational Chemistry 42
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.
kCCck
kkk )ˆˆ(CISD-MR 21
kk
kkk kdCkCc 21
ˆ)ˆ(MRCI-IC
Internally-contracted MRCI:
10/1/2007 Introduction to Computational Chemistry 43
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.
10/1/2007 Introduction to Computational Chemistry 44
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
FHH
FH
HHH
ˆˆˆ
ˆˆ
ˆˆ
1
0
10
FHH
FH
HHH
ˆˆˆ
ˆˆ
ˆˆ
1
0
10
MP2, MP3, MP4, …etc.number denotes order to which energy is computed (2n+1 rule)
10/1/2007 Introduction to Computational Chemistry 45
Semi-empirical molecular orbital methods
• Approximate description of valence electrons• Obtained by solving a simplified form of the
Schrödinger equation• Many integrals approximated using empirical
expressions with various parameters• Semi-quantitative description of electronic
distribution, molecular structure, properties and relative energies
• Cheaper than ab initio electronic structure methods, but not as accurate
10/1/2007 Introduction to Computational Chemistry 46
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
10/1/2007 Introduction to Computational Chemistry 47
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)
10/1/2007 Introduction to Computational Chemistry 48
AB
ABVUH
AB
ABVUH Ufrom atomic spectraVvalue per atom pair
0H 0H on the same atom
SH AB SH AB BAAB 21 BAAB 21
One parameter per element
Approximations of 1-e integrals
10/1/2007 Introduction to Computational Chemistry 49
Popular DFT
• Noble prize in Chemistry, 1998• In 1999, 3 of top 5 most cited journal articles in chemistry
(1st, 2nd, & 4th)• In 2000-2004, top 3 most cited journal articles in chemistry • In 2005, 4 of top 5 most cited journal articles in chemistry
– 1st, Becke’s hybrid exchange functional (1993)– 2nd, Lee-Yang-Parr correlation functional (1988)– 3rd, Becke’s exchange functional (1988)– 5th, PBE correlation functional (1996)
http://www.cas.org/spotlight/bchem.html
10/1/2007 Introduction to Computational Chemistry 50
Advantageous DFT
• Computationally efficient
Hartree-Fock-like computationally (~N3) , but included electron correlation effects
• Theoretically rigorous
Two Hohenberg-Kohn theorems guarantee an exact theory in ground state
• Conceptually insightful
Provides basis to understand chemical reactivity and other chemical properties
10/1/2007 Introduction to Computational Chemistry 51
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
10/1/2007 Introduction to Computational Chemistry 52
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.
10/1/2007 Introduction to Computational Chemistry 53
Density Functional Theory• 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.
EEnergy
EEnergy
00 ifEE
10/1/2007 Introduction to Computational Chemistry 54
Can we work with 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[]?
• How do we determine the density variationally? We must make sure that the density is derived from a proper N-electron wavefunction (N-representability problem) and a given external potential vext (v-representability problem).
, , , 2
1Coulombextextnuc rr
rr
rrrrr
ddEdVEr
ZZE
nuclei
BA AB
BA
10/1/2007 Introduction to Computational Chemistry 55
The Kohn-Sham (KS) Scheme• 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 from a Slater determinant.
• The N-representability problem will then be solved (density is obtained from an anti-symmetric N-electron function).
• Then, the only thing unknown is to calculate Exc[].
mn
N
nmmn
n
iiin tPtTEdddn
1,1
kin32
2
1 ˆ ˆ , rrrr
10/1/2007 Introduction to Computational Chemistry 56
Kohn-Sham Equations
,|)(|)(
,)(
,||
)()(
,||
)(
,2
1
and
)()()(ˆ
where
,ˆ
2
3
2
nknknk
xcxc
ee
a a
ane
xceene
nknknk
rfr
ErV
rdrr
rrV
Rr
ZrV
K
rVrVrVKH
H
The Only Unknown
10/1/2007 Introduction to Computational Chemistry 57
All about Exchange-Correlation Energy Density Functional
• LDA – f is a function of (r) only• GGA – f is a function of (r) and ∇(r)
• Mega-GGA – f is also a function of ts(r), kinetic energy density
• Hybrid – f is GGA functional with extra contribution from Hartree-Fock exchange energy
rrrr dfQXC ,,, 2
10/1/2007 Introduction to Computational Chemistry 58
LDA Functionals
• Thomas-Fermi formula (Kinetic) – 1 parameter
• Slater form (exchange) – 1 parameter
• Wigner correlation – 2 parameters
3/223/5 310
3, FFTF CdCT rr
3/13/23/13/4 438
3, XX
SX CdCE rr
rr
r
db
aEWC 3/1
3/2
1
10/1/2007 Introduction to Computational Chemistry 59
GGA Functional: BLYP
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:
rrrr dedeEEE cxcxxc , , LYPBLYPBBLYP
3/4
2
2
23/4 ,1
LDA
XBX EE
rdettCbd
aE cWWF
LYPc
3/123/53/23/1 18
1
9
12
1
1
10/1/2007 Introduction to Computational Chemistry 60
Hybrid Functional: B3LYP
FxB and Fc
LYP have been fitted against ab initio data (one could call this computational approach a “semi-ab-initio method”).
In a very popular variant, denoted B3LYP, the functional is augmented with a little of Hartree-Fock-type exchange:
nlkmPPbEEaEN
lkkl
N
nmmncxxc
1,1,
LYPBB3LYP
10/1/2007 Introduction to Computational Chemistry 61
Other Popular Functionals
• LDA– SVWN
• GGA– PBE– PW91– HCTH– Mega-GGA
• Hybrid functionals
10/1/2007 Introduction to Computational Chemistry 62
Disadvantageous DFT
• ground-state theory only
• universal functional unknown
• no systematic way to improve approximations like LDA, GGA, etc.
10/1/2007 Introduction to Computational Chemistry 63
Examples DFT vs. HF
Hydrogen molecules - using the LSDA (LDA)
10/1/2007 Introduction to Computational Chemistry 64
DFT Reactivity Indices
• Electronegativity (chemical potential)
• Hardness / Softness
• HSAB Principle and Maximum Hardness Principle
2LUMOHOMO
N
E
2LUMOHOMO
N
E
/1,22
2
SN
E HOMOLUMO
/1,22
2
SN
E HOMOLUMO
FOR MORE INFO...
Parr & Yang, Density Functional Theory of Atoms and Molecules (Oxford Univ. Press, New York, 1989).
10/1/2007 Introduction to Computational Chemistry 65
DFT Concept: Fukui Function
rrr 1 NNf rrr 1 NNf
• Fukui function
N
fr
r
N
fr
r
Nucleophilic attack
rrr NNf
1 rrr NNf
1
Electrophilic attack
Free radical activity
2
rrr
fff
2
rrr
fff
10/1/2007 Introduction to Computational Chemistry 66
Fukui Function: An Example
10/1/2007 Introduction to Computational Chemistry 67
Fukui Function: Another Example
10/1/2007 Introduction to Computational Chemistry 68
New Development: Electrophilicity Index
• Physical meaning: suppose an electrophile is immersed in an electron sea
The maximal electron flow and accompanying energy decrease are
2
2
1NNE
2
2
2
2
max N
2
2
minE
Parr, Szentpaly, Liu, J. Am. Chem. Soc. 121, 1922(1999).
10/1/2007 Introduction to Computational Chemistry 69
New Development:Philicty and Spin-Philicity
• 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)
10/1/2007 Introduction to Computational Chemistry 70
New Development: Steric Effect
r
r
rdEs
2
8
1
r
r
r
r
rr
22
4
1
8
1
s
s
E
S.B. Liu, J. Chem. Phys. 126, 244103(2007).
10/1/2007 Introduction to Computational Chemistry 71
BLACK CIRCLE: Total Energy Difference; RED SQUARE: Electrostatic; GREEN DIMOND: Quantum; BLUE TRIANGLE: Steric
New Development: Steric Effect
S.B. Liu and N. Govind, to be published
10/1/2007 Introduction to Computational Chemistry 72
What’s New: QM/MM
• Focus: Enzyme catalytic reactions• Strategy: QM for active site and MM for the rest• Main Issue: boundary between QM and MM.• Models: Link-atom, pseudo-orbital, pseudo-bond,
etc.• Limitation: active site should be small;
– long-range charge transfer– conformation change (protein folding)
10/1/2007 Introduction to Computational Chemistry 73
QM/MM Example: Triosephosphate Isomerase (TIM)
494 Residues, 4033 Atoms, PDB ID: 7TIM
Function: DHAP (dihydroxyacetone phosphate) GAP (glyceraldehyde 3-phosphate)
GAP
DHAPH2O
10/1/2007 Introduction to Computational Chemistry 74
Glu 165 (the catalytic base), His 95 (the proton shuttle)
DHAP GAP
TIM 2-step 2-residue Mechanism
10/1/2007 Introduction to Computational Chemistry 75
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.
10/1/2007 Introduction to Computational Chemistry 76
QM/MM: Transition State
=====================
Energy Barrier (kcal/mol)
-------------------------------------
QM/MM 21.9
Experiment 14.0
=====================
10/1/2007 Introduction to Computational Chemistry 77
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
10/1/2007 Introduction to Computational Chemistry 78
0
10
20
30
40
50
60
70
80
90
100
0 100 200 300 400 500 600 700 800 900
Atoms
CP
U s
ec
on
ds
pe
r C
G s
tep
OLMONOLMO
Diagonalization
O(N) Method: An Example
10/1/2007 Introduction to Computational Chemistry 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)
10/1/2007 Introduction to Computational Chemistry 80
Limitations and Strengths of ab initio quantum chemistry
10/1/2007 Introduction to Computational Chemistry 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)
10/1/2007 Introduction to Computational Chemistry 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.)
10/1/2007 Introduction to Computational Chemistry 83
QUESTIONS & COMMENTS?
Please direct comments/questions about Comp Chem to
E-mail: [email protected]
Please direct comments/questions pertaining to this presentation to
E-Mail: [email protected]
Please direct comments/questions about Comp Chem to
E-mail: [email protected]
Please direct comments/questions pertaining to this presentation to
E-Mail: [email protected]
10/1/2007 Introduction to Computational Chemistry 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
10/1/2007 Introduction to Computational Chemistry 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