Introduction to Introduction to Computational Computational

Chemistry Chemistry

Introduction to Introduction to Computational Computational

Chemistry Chemistry Shubin Liu, Ph.D.

Research Computing Center

University of North Carolina at Chapel Hill

its.unc.edu 2

OutlineOutline

Introduction

Methods in Computational Chemistry

•Ab Initio

•Semi-Empirical

•Density Functional Theory

•New Developments (QM/MM)

Hands-on ExercisesThe 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 UsAbout 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 ComputingResearch 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 MyselfAbout 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 YouAbout 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!!!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 Computational Chemistry

Central Theme of Computational Chemistry

DYNAMICS

REACTIVITY

STRUCTURE CENTRAL DOGMA OF MOLECULAR BIOLOGY

SEQUENCE

STRUCTURE

DYNAMICS

FUNCTION

EVALUTION

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Multiscale Hierarchy of Modeling

Multiscale Hierarchy of Modeling

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What is Computational Chemistry?

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=ma

also Molecular Dynamics

• Empirical/Statisticale.g., QSAR, etc., widely used in clinical and medicinal chemistry

Focus TodayFocus Today

Ht

i ˆ

Ht

i ˆ

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How Big Systems Can We Deal with?

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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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

Starting Point: Time-Independent Schrodinger

Equation

EH

Ht

i ˆ

Ht

i ˆ

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Equation to Solve in ab initio Theory

Equation to Solve in ab initio Theory

EH

Known exactly:3N spatial variables

(N # of electrons)

To be approximated:1. variationally2. perturbationally

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Hamiltonian for a Molecule

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

mm

22

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

mm

22

22

22

2

22ˆ H

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Ab Initio Methods

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 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

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

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Approximate Wavefunctions

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

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The Slater DeterminantThe Slater Determinant

zcbazcba

zzzz

cccc

bbbb

aaaa

n

zcbazcban

zcba

n

n

n

n

n

nn

n

321

321

321

321

321

312321

321 Α

!1

!1

zcbazcba

zzzz

cccc

bbbb

aaaa

n

zcbazcban

zcba

n

n

n

n

n

nn

n

321

321

321

321

321

312321

321 Α

!1

!1

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The Two Extreme Cases

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

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The Full CI MethodThe 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

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.

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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 MethodThe Hartree–Fock Method

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|S Overlap integral

|

2

1|PHF

ii

occ

i

cc2PDensity Matrix

SF iii cc

The Hartree–Fock Method

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)

Self-Consistent-Field (SCF)

SF iii cc

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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 MethodsAb Initio Methods

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Who’s Who

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Size vs AccuracySize 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

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ROO,e= 291.2 pm 96.4 pm

95.7 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

AN EXAMPLE

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Experimental and Computed Enthalpy Changes He in

kJ/mol

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)

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LCAO Basis FunctionsLCAO 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

c

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Basis FunctionsBasis 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( r)exp(

2nml rexp*zyx 2nml rexp*zyx

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Contracted Gaussian Basis Set

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

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Polarization / Diffuse Functions

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: 6-31+G*, 6-311++G**

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Correlation-Consistent Basis Functions

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

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 EnergyCorrelation 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)

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Configuration-Interaction (CI)

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

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Truncated configuration interaction: CIS, CISD, CISDT, etc.

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.

NNkji 321HF NNkji 321HF

etc. ,321 ,321 NN NkbaabijNkja

ai etc. ,321 ,321 NN Nkba

abijNkja

ai

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Truncated Configuration Interaction

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.

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Multi-Configuration Self-Consistent Field (MCSCF)

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 k

kk

kk kdCkCc 21ˆ)ˆ(MRCI-IC

Internally-contracted MRCI:

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Coupled-Cluster Theory

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.

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Møller-Plesset Perturbation Theory

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, …)

F

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)

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Semi-Empirical MethodsSemi-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

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Semi-Empirical MethodsSemi-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)

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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

Approximations of 1-e integrals

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Popular DFTPopular 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

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Brief History of DFTBrief 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

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What could expect from DFT?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.

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Density Functional TheoryDensity 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

HEnergy

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But what is E[]?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

1extnenn rr

rr

rrrrr

ddJdVEr

ZZE

nuclei

BA AB

BA

E[] = TS[] + Vne[] + J[] + Vnn[] + Exc[]

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Kohn-Sham SchemeKohn-Sham Scheme

,|)(|)(

,)(

,||

)()(

,||

)(

,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

• 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[].

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All about Exchange-Correlation

Energy Density Functional

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

rrrr dfQXC ,,, 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.

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LDA FunctionalsLDA 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 3/11

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Popular Functional: BLYP/B3LYPPopular 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:

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

nlkmPPbEEaEN

lkkl

N

nmmncxxc

1,1,

LYPBB3LYP

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Density FunctionalsDensity Functionals

LDAlocal density

GGAgradient corrected

Meta-GGAkinetic energy density included

Hybrid“exact” HF exchange component

Hybrid-meta-GGA

VWN5

BLYPHCTHBP86

TPSSM06-L

B3LYPB97/2MPW1K

MPWB1KM06

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 somebodyhas compared functionals forsystems similar to yours!

Incr

easi

ng q

ualit

y a

nd c

om

puta

tional co

st

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 FunctionalsDensity Functionals

its.unc.edu 57

Problems with DFTProblems 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 DevelopmentsDFT 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

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Examples DFT vs. HFExamples DFT vs. HF

Hydrogen molecules - using the LSDA (LDA)

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Chemical Reactivity TheoryChemical 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

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DFT Reactivity TheoryDFT 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!

,,

22

2

2

rrrrr

rrr2

1

rrr

rrr

2

ddE

dNE

NN

N

E

dE

NN

E

NENNEE

NN

N

Assumptions: existence and well-behavior of all above partial/functional derivatives

its.unc.edu 62

Conceptual DFTConceptual 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

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DFT Reactivity IndicesDFT Reactivity Indices

Electronegativity (chemical potential)

Hardness / Softness

Maximum Hardness Principle (MHP)

HSAB (hard and Soft Acid and Base) Principle

/1,22

12

2

SN

E HOMOLUMO

2LUMOHOMO

N

E

its.unc.edu 64

DFT Reactivity IndicesDFT Reactivity Indices

Fukui function

N

fr

r

– Nucleophilic attack

rrr NNf

1

– Electrophilic attack

rrr 1 NNf

– Free radical activity

2

rrr

fff

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Electrophilicity IndexElectrophilicity 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

max N

2

2

2

2

minEParr, Szentpaly, Liu, J. Am. Chem. Soc. 121, 1922(1999).

its.unc.edu 66

Experiment vs. Theory

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 PrincipleMinimum 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

NucleophilicityNucleophilicity

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.

ABA

BA

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

BA

BAN

its.unc.edu 69

Philicity and FugalityPhilicity 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 AEn

2

)( 2 IEe

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.

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Dual DescriptorsDual Descriptors

NNN N

f

N

EE

Nf

r

r

rrr

2

2

2

22

3rd-order cross-term derivatives

02 rr df rrr fff 2 rrr HOMOLUMOf 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 EffectSteric 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).

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Steric effect: a DFT descriptionSteric 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

rdTW

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)

QM/MM Example: Triosephosphate Isomerase (TIM)

494 Residues, 4033 Atoms, PDB ID: 7TIM

Function: DHAP (dihydroxyacetone phosphate) GAP (glyceraldehyde 3-phosphate)

GAP

DHAPH2O

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Glu 165 (the catalytic base), His 95 (the proton shuttle)

DHAP GAP

TIM 2-step 2-residue Mechanism

TIM 2-step 2-residue Mechanism

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QM/MM: 1st Step of TIM Mechanism

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

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) MethodWhat’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

seco

nds

per C

G st

epOLMO

NOLMO

Diagonalization

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What Else … ? 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

Limitations and Strengths of ab initio

quantum chemistry

its.unc.edu 81

Popular QM codesPopular 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)

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Reference BooksReference 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 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

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/

its.unc.edu 84

Hands-on: Part IHands-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

its.unc.edu 85

Hands-on: Part IIHands-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