ccl intro lecture

7/30/2019 CCL Intro Lecture

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An Introduction toComputational

Chemistry Laboratory

An Introduction toComputational

Chemistry Laboratory


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What is Computational Chemistry

Laboratory (CCL)?

CCL is a virtual chemistry laboratory (in many cases

substitutes a real laboratory.)

The aim: use of computers to aid chemical inquiry. Based on:

Physical background

theory (Classical Newtonian or

Quantum Physics) Mathematical numerical algorithms (optimization, linearalgebra, iteration procedures, numerical integration etc.) Computer software and hardware (HYPERCHEM 8.0,GAUSSIAN03 on Windows PC)

Chemical knowledge and intuition for understanding and

interpretation of the computational results


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Potential Energy Surface (PES) the main

chemistry inquiry

Chemistry is knowing the energy as a function of nuclear coordinates F. Jensen


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Potential energy surfaces (and

similar properties) calculation

Classical (Molecular) Mechanics

quick, simple; accuracy depends on parameterization; no consideration of electrons interaction)

Quantum Mechanics:1.

Molecular Wave Function Theory

Ab initio molecular orbital methods...much more demandingcomputationally, generally more accurate.

Semi-empirical molecular orbital methods ...computationallyless demanding than ab initio, possible on a pc for moderatesized molecules, but generally less accurate than ab initio,especially for energies.

2.

Density functional theory more efficient and often moreaccurate than Wave Function based approaches.


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Molecular Mechanics a theory

of molecules without electrons

Employs classical

(Newtonian) physics

Assumes Hookes Lawforces between atoms

(like a spring betweentwo masses)

EEstretchstretch

== kkss

(l(l --

lloo ))22

graph:graph: CC--CC;; C=OC=O

Force field = {kkss

}}


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Molecular Mechanics

More elaborate Force Fields (FF)


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Birth of quantum chemistry

Wave properties of matter

Birth of quantum chemistry

Wave properties of matter

Prince Louis de Broglie (1923):

=

h/mv= h/p

h= 6.62607550D-34 Js

p e

-i2x/

= e-i2px/h

(wave-particle duality

paradox)

-

probabilistic (statistic) wave.

Waves properties: interference, diffraction etc.

Possible explanation:The probabilistic behavior of elementary particles areconnected with structure of quantum vacuum.


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Basis of Quantum ChemistryBasis of Quantum Chemistry

Postulate I : A closed system is fully described by

Postulate II: Operator

for every physical quantity

(-ih/2)d/dx (e-i2px/h

p (e-i2px/h

(-ih/2)d/dx

(p

p

(p

Schrdinger equation (1926):

(can be solved exactly for the Hydrogen atom, but nothing larger)

P.A.M. Dirac, 1929: The underlying physical laws necessary for themathematical theory of a large part of physics and the whole ofchemistry are thus completely known.

di H E

dt


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One-dimensional Schrdinger wave

equation

2

2

2

22 2

2

2 2

2

2 2

2

Total energy kinetic potential

12 2

then

2

2

H E

pmv V V m

dp i p pp

dx

d d dp i i

dx dx dx

d

H Vm dx

dV E

m dx

Hamiltonian operator = operator of energy

SE = energy eigen-value equation

Extracts total energy, E

Many solutions

E

0

, E

1

,

E

n

x) wavefunction No direct physical meaning

x)|

Probability offinding particle withenergy E at point x

Single-valued, finite, continuous


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Molecular

Schrdinger equation (SE):

1MA 2A_ h282H = 2a_ h282melectrons

A

>

nuclei

a

ZArAa_e2nuclei electrons

aA

ZAZB

rABe2

nuclei

A 1

rabe2

electrons

baB >++

kinetic energy (nuc.) kinetic energy (elect.)

2 kinetic energy terms plus3 Coulombic energy terms:

(one attractive, 2 repulsive)

H = E

H = Hamiltonion operator


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Relativistic quantum mechanics

Dirac equation (1928) :


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Influence of Relativity on

Quantum World and vice-versa

(x)

(x)( )

(x)

(x)

L

L

S

S

x

REALITY IS RELATIVISTIC AND THUS IS QUANTUM (and vice-versa!)


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Diracs sea of electrons.

Quantum vacuum.


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The NR molecular wavefunction

physical meaning

The wavefunction, , is a key quantity in quantum chemistry.

depends on coordinates and spins. Spin of electron

relativistic

property, additional discrete

coordinate ; |ms1

|=1/2

In a three dimensional system ofn-electrons,

is the probability of simultaneouslyfinding electron 1 with spin ms1

in the volume dx1

dy1

dz1

at (x1

,y1

,z1

),

electron 2 with spin

ms2

in the volume dx2

dy2

dz2

at (x2

,y2

,z2

)

and so on

The wave function should be normalized, that is, the probability offinding all electrons somewhere in space equals 1.

2

1 1 1 1 1,..., , ,..., ....n s sn n n nx z m m dx dy dz dx dy dz

2

1 1 1 1 1 1

... , , ,..., , , ... 1n n n n n nall m

x y z x y z dx dy dz dx dy dz


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Wavefunctions general

properties

The wave function should be antisymmetric, that is, should

change sign when two electrons of the molecule interchange:

We can use the molecular wavefunction to calculate anyproperty of the molecular system. The average value, ,

of a

physical property of our molecular system is:

where, , is the quantum mechanical operator of the physicalproperty and

* C C d C

1 1 1

... ... n n nall m

dx dy dz dx dy dz d

1 1 1 ,..., 1

1 1 1 1

, , ,..., , , , , ,..., , , , ,...,

, , ,..., , , ,..., , , ,..., , , , ,...,

i i i j j j n n n s sn

j j j i i i n n n s sn

x y z x y z x y z x y z m m

x y z x y z x y z x y z m m


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Ab-initioWavefunction approachAb-initioWavefunction approach

Simplifying assumptions are employed to solve the Schrdinger equation approximately:

Born-Oppenheimer approximation

allows separate

treatment of nuclei and electrons Hartree-Fock independent electron approximationallows each electron to be considered as beingaffected by the sum (field) of all other electrons.

LCAO Approximation

Tools: Variational

Principle or Perturbation

Theory


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Born-Oppenheimer

Approximation -

1

MA2A

_ h282H

=

2a

_ h282m

electrons

A

>

nuclei

a ZArAa

_e2

nuclei electrons

aA

ZAZBrABe2nuclei

A 1rabe2

electrons

baB >

++

kinetic energy (nuc.) kinetic energy (elect.)

1 kinetic energy term plus2 Coulombic energy terms:(one attractive, 1 repulsive)

plus a constant for nuclei

constant


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Steps of solution of the Schrdinger

equation in the Born-Oppenheimer

approximation:

Htot

= (Tn

+ Vn

)

+ Te

+ Vne

+ Ve

= (Hn

)

+He

1.

Electronic SE:Hee

(r,R)=Ee

(R) e

(r,R)

2.

Nuclear SE: (T

n + V

n + E

e (R) )

n (R)=E

n

n (R)

Vn

+ Ee

(R) = potential energy surface (PES)TOTAL WF

: (r,R) =n

(R) e

(r,R)

In our laboratory we concentrate mainly onsolution of the electronic SE and working with PES(finding minimums, transition states etc.)


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Solving the Electronic SE:

Hartree-Fock (HF) approximation the physical background

Solving the Electronic SE:

Hartree-Fock (HF) approximation

the physical background

Multi-electronic SE:Hee

(r,R)=Ee

(R) e

(r,R)

is still very

complicated reduce it to the single-electronic equation

HF assumes that each electron experiences all the othersonly as a whole (field of charge) rather than individualelectron-electron interactions.

Instead of multielectronic Shrdinger equation introducesa one-electronic

Fock operatorF:

Fwhich is the sum of the kinetic energy of an electron, apotential that one electron would experience for a fixednucleus, and an average of the effects of the otherelectrons.


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Mathematical foundation of the HF (or

Self-consistent-field (SCF)) method

Molecular orbital theory approximates the molecular wave function

as a antisymmetrized

product of orthonormal

one-electron

functions (or molecular spin-orbitals)

where is the antisymmetrization operator and

where k=1/2; 1/2 =

; -1/2

=

.

The antisymmetrization

operator is defined as the operator that

antisymmetrizes a product of n one-electron functions andmultiplies them by normalization factor(n!)-1/2

1 2( .... )nf f f

( , , )i i i i i k f x y z


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Determinant of Slater

The antisymmetrized WF can berepresented as the Slaters determinant:

1 1 1 1 2 1 2 1 /2 1

1/2 1 2 1 2 2 2 2 2 /2 2

1 1 2 2 /2

( ) (1) ( ) (1) ( ) (1) ( ) (1) ... ( ) (1)( ) (2) ( ) (2) ( ) (2) ( ) (2) ... ( ) (2)

!... ...

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ... ( ) ( )

n

n

n n n n n n

x x x x xx x x x x

n

x n x n x n x n x n


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Variational Principle

The energyE

calculated from any

approximation

of the wavefunction

will be higher

than the true

energyE0

:

The better the wavefunction, the lower the energy(the more closely it approximates reality).

Changes (variation of parameters in ) are madesystematically to minimize the calculated energy.

At the energy minimum (which approximates thetrue energy of the system), dE/di

= 0.

*0

E H d E


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The Hartree-Fock energy

functional

We shall restrict ourselves to closed shell configurations, for

such cases, a single Slater determinant is sufficient to describe

the molecular wave function. Using the variational

principle

within this framework lead to the restricted HF theory. The

Hartree-Fock energy for molecules with only closed shells is/ 2 / 2 / 2

1 1 1

12 (2 )

2

n n ncore

HF i ij ij

i i j

E H J K

21 1

1(1) (1) (1) (1) / (1)2

core core

i i i i I I i

I

H H Z r

12 12(1) (2)1/ (1) (2) , (1) (2)1/ (1) (2)

ij i j i j ij i j j i

J r K r


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The Hartree-Fock equations

The Hartree-Fock equations are derived fromthe variational principle, which looks for thoseorbitals that minimizeEHF

.

For computational convenience the molecularorbitals are taken to be orthonormal:

The orthogonal Hartree-Fock molecular orbitals

satisfy the single-electronic equations:

(1) | (1)i j ij

(1) (1) (1)i i iF


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The (Hartree-) Fock operator

Single-electronic operator:

The Coulomb operatorJj

and the exchange

operatorKj

are defined by

where

f is an arbitrary function

/ 2

21 1

1

1 (1) / 2 (1) (1)2

n

I I j j

I j

F Z r J K

2

2

12

*

2

12

1 (1) (1) (1) (2)

(2) (2) (1) (1) (1)

j j

jj j

f f dvr

fK f dvr


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Next step:

MO-LCAO Approximation

Electron positions in molecular orbitals can be

approximated by a Linear Combination of AtomicOrbitals (LCAO).

This reduces the problem of finding the best

functional form for the molecular orbitals to themuch simpler one of optimizing a set of coefficients(cn

) in a linear equation:

= c1 c2 c3 c4 where

is the molecular orbital (MO) wavefunction

and

n

represent atomic orbital (AO) wavefunctions.


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One step more:

Basis sets (BS)

A basis set

is a set of analytical functions (k

) used to

represent the shapes of atomic orbitals n:

General

contracted BS: n

=k bk(n)

k(n)

Contraction coefficients are calculated in a separate

atomic HF calculation;if k=1 basis set is called uncontracted.

Basis sets in common use have a simple mathematicalform for representing the radial distribution of electron

density.

Most commonly used are Gaussian basis sets, whichapproximate the better, but more numericallycomplicated Slater-Type orbitals (STO).


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Hartree-Fock Self-ConsistentField (SCF) Method.

Hartree-Fock Self-Consistent

Field (SCF) Method.

Computational methodology (Jacobi iterations):

1. Guess the orbital occupation (position) of an electron(set of MO coefficients {cn

})

2. Calculate the potential each electron would experiencefrom all other electrons (Fock operatorF ({cn

}))

3.

Solve for Fock equations to generate a new, improved

guess at the positions of the electrons (new {cn })4. Repeat above two steps until the wavefunction for theelectrons is consistent with the field that it and the otherelectrons produce (SCF).


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Types of HF

Multiplicity (M) = 2*S+1

(S is the total spin of the system)

Electrons can have spin up or down

. Most

calculations are closed shell calculations (M=1),using doubly occupied orbitals, holding two

electrons of opposite spins. RHF restricted HFOpen shell systems (M>1)

are calculated by

1.

ROHF

restricted open shell HF

the same

spatial orbitals for different spin-orbitals from thevalence pair;2.

UHF

unrestricted HF

different spatial parts for

different spins from the same valence pair


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Illustrating an RHF singlet, and

ROHF and UHF doublet states


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Semi-empirical MO Calculations:Further Simplifications of HF

Semi-empirical MO Calculations:

Further Simplifications of HF

Neglect core (1s) electrons; replace integral for Hcore

by an

empirical or calculated parameter

Neglect various other interactions between electrons on

adjacent atoms: CNDO, INDO, MINDO/3, MNDO, etc.

Add parameters so as to make the simplified calculation giveresults in agreement with observables (atomic spectra ormolecular properties).

/ 2 / 2 / 2

1 1 1

12 (2 )

2

n n ncore

F ii ij ij

i i j

E H J K

21 1

1(1) (1) (1) (1) / (1)2

core core

ii i i i I I i

I

H H Z r

12 12

(1) (2)1/ (1) (2) , (1) (2)1/ (1) (2)ij i j i j ij i j j i

J r K r


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Beyond the SCF.

Correlated Methods (CM)

Include more explicit interaction of electrons than HF :

Ecorr

= E -

EHF , where E= H

Most CMs

begin with HF wavefunction, then incorporate

varying amounts of electron-electron interaction by mixing in

excited state determinants with ground state HF determinant

The limit of infinite basis set

& complete electron correlation

is

the exact solution of Schrdinger equation (which is still an

approximation)


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Beyond the SCF.

Correlation effects on properties.


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Two alternative ways of the

electron correlation treatmentHF (Hartree-Fock)

a singe determinant

theory

no correlation included!1. WF based multi-determinant

correlation methods:

1.

Configuration Interaction

(CI)

Variational; CISD, CSID(T)

2.

Many-body perturbation theory

(including coupled cluster)

Non-variational; MBPT2, MBPT3; CCSD; CCSD(T)2. Density functional theory (DFT)

correlation method not based on

wave-function, but rather on modification of the energy functional:

A single determinant theory including correlation!/ 2 / 2 / 2

1 1 1

2 ( )n n n

core Exch Corr

DFT i ij ij

i i j

E H J X


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Summary of Choices:

6-311++G**

6-311G**

6-311G*

3-21G*

3-21G

STO-3G

STO

HF CI QCISD QCISDT MP2 MP3 MP4 correlation

increasingsize of

basis set

increasing level of theory

Schrodinger

increasing accuracy,increasing cpu time

infinite basis set

full e

-


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The extra dimension:


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Summary: Levels ofQMTheory

H=E

Born-Oppenheimer approximation

Single determinant SCF

Semi-empirical methods

Correlation approaches:

1.Multi-determinantial (MCSCF, CI, CC, MBPT)

2. Single determinantial

(DFT)


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Some applications during your

work...

Calculation of reaction pathways (mechanisms)

Determination of reaction intermediates andtransition structures

Visualization of orbital interactions (formation of newbonds, breaking bonds as a reaction proceeds)

Shapes of molecules including their chargedistribution (electron density)

NMR chemical shift prediction.

IR spectra calculation and interpretation.


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Method Type Features Advantages Disadvantages Best for

Molecular

Mechanic

Uses classical

physics Relies on force-field

with embeddedempirical parameters

Computationallyleast intensive - fast

and useful withlimited computerresources

Good for:

Enthalpy ofFormation(sometimes)

Dipole Moment

Geometry (bondlengths, bond angles,

dihedral angles) oflowest energyconformation.

Particular force

field, applicable onlyfor a limited class ofmolecules

Does not calculateelectronic properties

Requires

experimental data(or data from ab

initio

calculations)

Large systems

(~1000 of atoms) Can be used for

molecules as large asenzymes

Systems or processeswith no breaking or

forming of bonds

Semi-Empirical Uses quantum

physics Uses experimentally

derived empiricalparameters

Uses manyapproximations

Less demandingcomputationally thanab

initio

methods

Capable ofcalculating transitionstates and excitedstates

Requiresexperimental data(or data from ab

initio) forparameters

Less rigorous thanab

initio) methods

Medium-sizedsystems (hundreds ofatoms)

Systems involvingelectronic transition

Ab

Initio Uses quantum

physics Mathematically

rigorous, noempirical parameters

Uses approximationextensively

Useful for a broadrange of systems

does not depend onexperimental data

Capable ofcalculating transitionstates and excitedstates

Computationallyexpensive

Small systems (tensof atoms)

Systems involvingelectronic transition

Molecules withoutavailableexperimental data

Systems requiringrigorous accuracy

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