Introduction to Computational Chemistry. 10/1/2007Introduction to Computational Chemistry2 Outline...

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


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


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”)


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

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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

http://www.renci.org/unc/computing/


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


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?


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!


Central Theme of Computational Chemistry

DYNAMICS

REACTIVITY

STRUCTURE CENTRAL DOGMA OF MOLECULAR BIOLOGY

SEQUENCE

STRUCTURE

DYNAMICS

FUNCTION

EVALUTION


Multiscale Hierarchy of Modeling


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


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


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


Equation to Solve in ab initio Theory

EH

Known exactly:3N spatial variables

(N # of electrons)

To be approximated:1. variationally2. perturbationally


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


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.


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


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


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


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


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


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.


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


|S Overlap integral

|

2

1|PHF

ii

occ

i

cc2PDensity Matrix

SF iii cc

The Hartree–Fock Method


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)


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


Who’s Who


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


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


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)


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


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


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


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**


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)


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


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)


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


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


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.


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:


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.


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)


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


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


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)


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


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


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


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


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.


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


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


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


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


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


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


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


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


Other Popular Functionals

• LDA– SVWN

• GGA– PBE– PW91– HCTH– Mega-GGA

• Hybrid functionals


Disadvantageous DFT

• ground-state theory only

• universal functional unknown

• no systematic way to improve approximations like LDA, GGA, etc.


Examples DFT vs. HF

Hydrogen molecules - using the LSDA (LDA)


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).


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


Fukui Function: An Example


Fukui Function: Another Example


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).


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)


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).


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


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)


QM/MM Example: Triosephosphate Isomerase (TIM)

494 Residues, 4033 Atoms, PDB ID: 7TIM

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

GAP

DHAPH2O


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

DHAP GAP

TIM 2-step 2-residue 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.


QM/MM: Transition State

=====================

Energy Barrier (kcal/mol)

-------------------------------------

QM/MM 21.9

Experiment 14.0

=====================


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

CP

U s

ec

on

ds

pe

r C

G s

tep

OLMONOLMO

Diagonalization

O(N) Method: An Example


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)


Limitations and Strengths of ab initio quantum chemistry


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)


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.)


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]


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


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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