Ab-initio 4 beginers 2014 - EPFL

2/24/14

1

Chapter 1

Computa(onal Methods

The goal of this chapter: •  to give a brief and non-‐exhaus(ve overview of the commonly

used electronic structure methods. •  to deliver guidelines and help iden(fying which theore(cal

method to use for a given applica(on.

Soon, you will be able to understand the paragraph above.

2/24/14

2

or this one

3. Computational study DFT computations All (TD)-DFT computations were performed in Gaussian 091 using the ωB97X-D2 functional and def2-SVP3 basis set in combination with the SMD implicit solvent model4 with default parameters corresponding to water. The interaction and, if applicable, the excitation/fluorescence energies are given in Table S1. Note that the interaction energies are not corrected for basis set superposition, which is ill defined for an excimer.

Luisier, N.; Ruggi, A.; Steinmann, S.N.; Favre, L.; Gaeng, N.; Corminboeuf, C.; Severin, K. Org. Bio. Chem. 2012, 10, 7487.

Or, you will be able to interpret these figures.

Piemontesi, C.; Wang, Q.; Zhu, J. Org. Bio. Chem. 2013, 11, 1533.

2/24/14

3

Computa(onal Chemistry

Chapter 1.1

Molecular Mechanics

Quantum Mechanics

Approxima(ons to the Schrodinger equa(on. (HF, DFT, post-‐HF, semi-‐empirical methods).

Force Fields: Amber, CHARMM, MM2, TIP3P

Only nuclei, no electron: Molecules can be compared to balls and springs.

m2 m1

k broadly used

but not considered herein

HΨ = EΨ

H = Te + Vee + Vne + Vnn , E = total energy

Ψ (r1,r2 ,....,rN ) = N -particle wavefunction

Considered in this course

QM / MM

Computa(onal Chemistry

Chapter 1.1

2/24/14

4

Overview of Electronic Structure Methods

Chapter 1.1

Post-HF

DFT

Single-‐determinant

Hartree-Fock Mean-‐field theory, computa(onally “cheap”, not very accurate Ψ ρ

Mean field theory, computa(onally cheap, rela(vely accurate

LDA, BLYP, PBE, B3LYP, PBE0, M06-‐2X, ωB97x-‐D, CAM-‐B3LYP…

MP2, CCSD(T)

computa(onally demanding, highly accurate

CASSCF, CASPT2, MRCCSD

Mul(configura(onal methods

Jacob’s ladder

The electron density is the central object

The wavefunc(on is the central object

Chapter 1.1

Chapter 1.2

1.1 Ab ini2o methods

H EΨ = Ψ

So what’s the big deal

with Schrödinger anyway ?

Chapter 1.1

2/24/14

5

The Schrödinger Equa(on

Schrödinger equa(on:

Solu(on = complete informa(on on chemical systems !

Ĥ: Hamiltonian operator, contains all interac(ons present in the system

Ψ: wavefunc(on, contain coordinates of nuclei and electrons

no direct physical interpreta(on, Ψ2 related to probability of finding par(cles in space

Ĥ ac2ng on Ψ gives the total energy of the system E 2mes Ψ.

H EΨ = Ψ

Chapter 1.1

Hamiltonian for N electrons and M nuclei (in atomic units):

H = − 12∇i2

i

N

∑Kinetic energyof the electrons

− 1

2Mi

∇i2

i

M

∑Kinetic energyof the nuclei

−

Z jri − r jj

M

∑i

N

∑Electron-nucleiattraction

+ 1ri − r ji , j

N

∑Electron-electronrepulsion

+ZiZ jri − r ji , j

M

∑Nuclei-nucleirepulsion

Mi and Zi: mass and charge of the nuclei

: distance between two par(cles ri − r j

The Schrödinger Equa(on

Chapter 1.1

2/24/14

6

Born-‐Oppenheimer approxima(on

Born-‐Oppenheimer approxima(on: solving the Schrödinger equa(on for electrons only, fix nuclei. Vnucl is the constant nuclei-‐nuclei repulsion.

Ĥ becomes:

Jus(fica(on:

Electrons are much lighter than nuclei and can adapt instantaneously to their posi(on.

The BO approxima(on is generally valid for most problems addressed in organic chemistry.

2nucl

,

1 1ˆ2

N N M Nj

ii i j i ji j i j

ZH V= − ∇ − + +

− −∑ ∑∑ ∑r r r r

Chapter 1.1

Varia(onal Principle

Solve the Schrödinger equa(on: choose a trial wavefunc(on which depend on some varia(onal parameters

minimize with respect to those parameters

Exact energy E0 as a func(on of exact wavefunc(on Ψ0 and Ĥ:

Varia(onal principle:

For any trial wavefunc(on Ψ represen(ng the correct number of electrons:

The be6er the wavefunc?on, the lower the energy !

0 00

0 0

HE

Ψ Ψ=

Ψ Ψ

0

HE

Ψ Ψ≤

Ψ Ψ

Chapter 1.1

If we cannot find an analy(cal solu(on to the Schrödinger equa(on, a trick known as the varia(onal principle allows us to es(mate the energy of the ground state of a system.

2/24/14

7

Overview

J. P. Perdew, A. Ruzsinszky, L. A. Constan(n, J. Sun and G. B. I. Csonka, J. Chem. Theory Comput., 2009, 5, 902-‐908.

Chapter 1.1

Hartree-‐Fock: computa(onally cheap not very accurate

post-‐HF: Considerable computa(onal effort poten?ally very accurate

Physical Reason: HF: Electrons interact with an average poten(al

generated by the other electrons No instantaneous repulsion (no Coulomb correla(on)

post-‐HF: Electrons avoid each other (are correlated)

Linear combina(on of Slater determinants, many coefficients to op(mize

Hartree-‐Fock theory

Avoid the need for experimental data: ab ini?o theory

Hartree-‐Fock: wavefunc(on built as an an(symmetrized product of molecular orbitals

Ψ = ϕ1(1)ϕ2 (2)...ϕN (N ) =

ϕ1(1) ϕ2 (1) ϕN (1)

ϕ1(2) ϕ2 (2) ϕN (2)

ϕ1(N ) ϕ2 (N ) ϕN (N )Molecular orbital

index Electron coordinate

Wavefunc(on is a Slater determinant: ensures change of sign upon electron coordinate exchange (electrons are fermions !)

Ψ(ϕ1(1)ϕ2 (2)...,ϕi (i),ϕ j ( j),...ϕN (N )) = −Ψ(ϕ1(1)ϕ2 (2)...,ϕi ( j),ϕ j (i),...ϕN (N ))

Chapter 1.1

2/24/14

8

Hartree-‐Fock theory Molecular orbitals : linear combina(on of basis func(ons

ϕi = ckiχkk∑

Basis func(on (discussed later) Coefficient

Varia(onal principle: coefficients should minimize the energy

Hartree-‐Fock equa(ons ˆi i iFϕ ε ϕ=

: Fock operator, poten(al in which orbitals are op(mized. Only contains the averaged electron-‐electron repulsion!

Strong approxima(on ! Neglects instantaneous correla(ons between electrons (par(cle nature).

The missing frac(on of electron-‐electron repulsion is called electron correla(on.

F

Chapter 1.1

ϕi

Limits of Hartree-‐Fock method Hartree-‐Fock is an approxima(on: electron-‐electron repulsion is accounted for only in an averaged manner.

Ex: Reac(on energies F-‐

SN2 reac(on: F-‐ + CH3CN CH3F + CN-‐

Methods[1] Reaction Energy in kcal/mol Complete Basis Set (extrapolated) HF -8.92 CCSD(T)/aug-cc-pVTZ +1.29 Experiment +1.7± 2.3

[1] Gonzales, J. M.; Pak, C.; Cox, R. S.; Allen, W. D.; Schaefer III, H. F.; Csaszar, A. G.; Tarczay, G. Chem. Eur. J., 2003, 9, 2173

HF is here qualita(vely wrong because of the missing electronic correla(on effects.

2/24/14

9

Bond Separa(on Equa(ons (BSE) of linear alkanes:[2]

[2] Wodrich, M. D.; Corminboeuf, C.; Schleyer, P. v. R. Org. Lett., 2006, 8, 3631

CH3(CH2)mCH3 + m CH4 (m+1) C2H6

HF errors increase with size of the alkane !

MP2 and CCSD(T) are much closer to experimental values

Limits of Hartree-‐Fock method

Complexa(on energies:[3]

[3] Gonthier, J. F.; Steinmann, S. N.; Roch, L.; Ruggi, A.; Luisier, N.; Severin, K.; Corminboeuf, C. Chem. Commun., 2012, 48, 9239

Benzene parallel stacked 3.5 Å above phenanthrene in three loca(ons:

-‐  Above peripheral ring -‐  Above central ring -‐  Above indicated bond

HF interac(on energies

MP2 interac(on energies

HF predicts repulsive interac(on, whereas MP2 is ayrac(ve: HF lacks dispersion interac(ons due to the neglect of electron correla(on.

If correla(on effects or dispersion are important, post-‐HF methods can be used: MP2, CISD, CCSD(T)…

Limits of Hartree-‐Fock method

2/24/14

10

Basis sets

Molecular orbitals

Prac(cally: limited number of basis func(ons. Careful choice !

Natural choice: exact solu(ons for the hydrogen atom

Slater func(ons with spherical harmonics (s,p,d etc. func(ons)

Computa(onally, Gaussian func(ons are simpler and ozen used.

Scheme of Slater Type Orbital: exact solu(on for H atom, computa(ons difficult Scheme of Gaussian Type Orbital:

computa(ons easier but no cusp at nucleus and falls off too rapidly.

i ki kkcϕ χ=∑ infinite number of basis func(ons lead to exact

Hartree-‐Fock energy

Schemes from Wim Klopper ESQC 2011 course

To reduce the drawbacks of Gaussian func(ons: one uses a combina(on of Gaussian func(ons to mimick Slater-‐type func(ons.

Examples: STO-‐3G basis set uses 3 Gaussian to represent each Slater func(on.

STO-‐6G uses 6 Gaussian per Slater func(on…

STO-‐3G or STO-‐6G: minimal basis set, only one basis func(on per orbital.

Example: for H, STO-‐3G only has one s basis func(on ; for C only 2 s (one for core and one for valence) and 1 set of p func(ons (px, py and pz), etc…

Nota(on H (1s), C (2s1p)

The size of the basis sets may be gradually increased to converge toward an energy closer to the exact Hartree-‐Fock limit.

Basis sets

Chapter 1.1

2/24/14

11

Basis sets

More than one basis func2ons per orbital: Double-‐, triple-‐, quadruple-‐zeta basis sets: 2, 3 or 4 sets of func(ons per valence orbital

Example: 6-‐31G basis set for C contain 3 s (1 for core, 2 for valence) and 2 sets of p func(ons, etc…

Possible basis set improvements:

Example of a set of double zeta pz func(ons

Chapter 1.1

Add polariza2on basis func2ons: More flexibility to represent deforma(ons of the electronic density in molecules. Represented by * in Pople basis sets.

Examples: 6-‐31G* for C (3s, 2p,1d) or cc-‐pVDZ (double zeta polarized basis), cc-‐pVTZ (triple zeta polarized basis)…


Example of a d func(on polarizing a p func(on.

Polariza(on func(ons are always of higher angular momentum than the func(ons to be polarized.

Chapter 1.1

Basis sets

2/24/14

12

Add diffuse basis func2ons: Describe electronic density far from the nucleus. Ozen denoted by + or aug-‐.

Examples: 6-‐31+G* for C has 4 s (1 core, 2 valence and 1 diffuse), 3 p (2 valence and 1 diffuse) and one d basis func(on (polariza(on) or aug-‐cc-‐pVDZ, aug-‐cc-‐pVTZ…


Example of a set of double zeta px func(ons with a diffuse func(on added.

Chapter 1.1

Basis sets

Basis sets errors

Difference between exact Hartree-‐Fock energy and HF energy with finite basis:

basis set trunca(on error

check stability of results toward basis set increase

Problem for interac(on energies: E(int) = E(dimer) – E(monomerA) – E(monomerB)

Dimer basis set larger than monomer basis set: different basis set trunca(on errors on the monomer and dimer energies

Basis set superposition error on interaction energies

Possible remedy:

Compute energy of each monomer in dimer basis set: counterpoise correc(on

Basis func(ons of both monomers present but only electrons and nuclei from one monomer.

Use large basis sets: basis set trunca(on error and hence basis set superposi(on error tend to zero. Chapter 1.1

2/24/14

13

Restricted or unrestricted ?

Restricted Hartree-‐Fock (RHF=HF): one spin up and one spin down electron per orbital Same spa(al wavefunc(on for the two spins

Downside: Open-‐shell species cannot be treated: wrong dissocia(on limit for H2 HF (or RHF) is used for typical closed-‐shell singlet wavefunc(on Restricted open-‐shell Hartree-‐Fock (ROHF): only some electrons are unpaired Downside: more expensive computa(ons and complex implementa(ons Advantage: No spin contamina(on

ROHF is used when spin contamina(on is large using UHF Unrestricted Hartree-‐Fock (UHF): each spin treated separately

Simpler than ROHF, spin polariza(on on en(re molecule Downside: Wavefunc(on might be contaminated by higher spin states (i.e. singlet

may contain a bit of triplet state) Contamina(on may be eliminated by projec(on but addi(onal cost. One generally uses UHF for the treatment of open-‐shell systems.

Chapter 1.1

O2 is a triplet


OO+

O- OO

O

H

H1b1

3a1

The contribu2ng resonance structures for ozone.

Triplet methylene, linear or bended?

Chapter 1.1

2/24/14

14


Chapter 1.1

H2

Bond breaking/open-‐shell singlets

Beyond Hartree-‐Fock

Remark: Hartree-‐Fock computa(ons became rela(vely rare since the populariza(on of density func(onal theory in commercial computa(onal chemistry sozware (see Chapter 1.2).

Chapter 1.1

However, post-‐Hatree-‐Fock methods, which are s(ll widely used, are based on the Hartree-‐Fock reference wavefunc(on. They provide highly reliable benchmark values.

2/24/14

15

Configura(on Interac(on Hartree-‐Fock: electron-‐electron repulsion only as a mean field of other electrons

consequence of the wavefunc(on represented as a single Slater determinant

Improvement of the descrip(on by including more Slater determinants

1 1 2 2 1 2 1(1) (2)... ( ) (1) (2)... ( ) ...N NC N C Nϕ ϕ ϕ ϕ ϕ ϕ +Ψ = + +Hartree-‐Fock determinant Excited determinant: one or more virtual

orbital (here orbital N+1) replace one or more occupied orbital (here orbital N)

All determinants included: full CI, exact solu(on of the Schrödinger equa(on within a basis set.

At infinite basis set limit, full CI solves exactly the Schrödinger equa(on.

Very expensive method ! Applicable only on the smallest molecules ! Chapter 1.1

Advanced level

Selec(on of some determinants instead of all: truncated CI methods

Configura(on Interac(on Singles (CIS):

only determinants with a single excita(on (i.e. only one occupied replaced by one virtual orbital)

ground state energy unaffected, rough approxima(on to excited state energies

Configura(on Interac(on Singles and Doubles (CISD):

only determinants with single or double excita(on (one or two occupied replaced by virtual orbitals)

improves ground state and excited states energies.

Inclusion of triple (CISDT), quadruple (CISDTQ), etc… excita(on possible. Nowadays, Coupled Cluster methods are generally used instead of CI.

Configura(on Interac(on

Chapter 1.1

Advanced level

2/24/14

16

Configura(on Interac(on

Chapter 1.1

The number of excited determinants grows factorially with the size of the basis set. Full CI infeasible for all but the very smallest systems.

Advanced level

Size extensivity and consistency Essen(al proper(es of computa(onal methods are:

Size consistency

Energy of two molecules A and B infinitely separated in the same computa(on should be equal to the energy of A and B computed separately.

Size extensivity

The energy should scale linearly with the number of electrons in the system

Truncated CI methods are not size-‐consistent !

Ex: 2 HF molecules at infinite separa(on: ECISD=-‐200.559

Energy of one HF molecule mul(plied by 2 : 2 ECISD= -‐200.576

11 kcal/mol difference ! Error on interac(on energies, etc…

Numbers from Jürgen Gauss ESQC 2011 course,tzp basis set

Chapter 1.1

Advanced level

2/24/14

17

Perturba(on Theory Electronic correla(on energies < 1% of total energies

Perturba(on theory: gives solu(on of a slightly perturbed system (correlated) star(ng from unperturbed system (Hartree-‐Fock)

Hamiltonian rewriyen as

ˆ ˆ ˆH F Vλ= +Full Hamiltonian

Fock operator

Perturbation: ˆ ˆH F−

Parameter λ switches perturba(on on. λ=1 real system.

Electron correlation may be considered as a small perturbation

Chapter 1.1

Advanced level

(0) (1) 2 (2)

(0) (1) 2 (2)

......

E E E Eλ λλ λ

= + + +Ψ =Ψ + Ψ + Ψ +

Trunca(on of Taylor expansion at different powers of λ different orders of perturba(on theory.

MP1: first order, corresponds to Hartree-‐Fock

MP2: second order

MP3, MP4, MP5… are available in some quantum chemistry sozwares.

Energy and wavefunc(on wriyen as Taylor expansions:

MP2 much less expensive than CISD and size-‐consistent !

But non-‐varia(onal (usually lower than exact energy) and no guarantee that Taylor expansion converges

Perturba(on Theory

Chapter 1.1

Advanced level

2/24/14

18

Coupled Cluster Theory developed to be size-‐consistent. Wavefunc(on wriyen as exponen(al Ansatz:

is the cluster operator: generates all Slater determinant from the HF wavefunc(on.

CCSD: cluster operator truncated at double excita(ons

size-‐consistent thanks to exponen(al Ansatz !

Triple excita(ons expensive included through perturba(on theory

THFeΨ = Ψ

T

1 2ˆ ˆ ˆ ˆ... NT T T T= + + +

Single, double, …, N-triple excitations

CCSD(T) method Chapter 1.1

Advanced level

Coupled Cluster

CCSD(T): Golden standard of quantum chemistry

Ozen used to obtain reference values and validate other methods

Expensive ! Small and medium molecules

CC is not varia(onal but is more accurate than truncated CI.

Example: Difference to full CI energy for the CO molecule, cc-‐pVDZ basis set (in mH)

CI CC

SD 30.804 12.120

SD(T) - 1.47

SDT 21.718 1.009

SDTQ 1.775 0.061

Numbers from Jürgen Gauss ESQC 2011 course Chapter 1.1

Advanced level

2/24/14

19

Chapter 1.1

Classifica(on of Electron Correla(on •  Correla(on effects are normally par((oned into: 1.  Near-‐degeneracy effects (non-‐dynamic correla(on) 2.  Dynamic correla(on

•  Qualita(vely they differ in the way they separate the electrons. -‐ Non-‐dynamic correla(on: leads to a large separa(on in space of the two electrons in a pair (ex. on two different atoms in a dissocia(on process) -‐ Dynamic correla(on: deals with the interac(on between two electrons at short inter-‐electronic distance, the cusp region

-‐ CCSD(T) and MP2 deal with dynamic correla(on effects

All methods discussed so far are based on a Hartree-‐Fock reference wavefunc(on

Might be inappropriate if a single Slater determinant does not describe qualita(vely the system.

Mul(configura(onal character:

• Near degeneracy (virtual orbitals with low energy, different electronic configura(ons of similar energies.

•  Also a characteris(c of some organic intermediates (e.g. Bergman cycliza(on)

• Mutl(ple bonds in transi(on metal chemistry

• Transi(on states

• Strong configura(onal mixing in excited states


Chapter 1.1

see examples hereafter

2/24/14

20

I. Schapiro, M. N. Ryazantsev, L. M. Frutos, N. Ferre, R. Lindh, and M. Olivucci J. Am. Chem. Soc. 2011, 133, 3354.

The Ultrafast Photoisomeriza(ons of Rhodopsin and Bathorhodopsin are Modulated by Bond Length Alterna(on and HOOP Driven Electronic Effects

Examples: Conical Intersec(on

Examples: Transi(on States

D.H. Ess, A. E. Hayden, F.-‐G. Klarner, K. N. Houk J. Org. Chem. 2008, 73, 7586.

Transi(on States for the Dimeriza(on of 1,3-‐Cyclohexadiene: A DFT, CASPT2, and CBS-‐QB3 Quantum Mechanical Inves(ga(on

Chapter 1.1

2/24/14

21

Examples: Organic Chemistry

I. Interac2ng bis-‐allyl diradicals

W. T. Borden at al. "Through-‐Bond Interac(ons in the Diradical Intermediates Formed in the Rearrangements of Bicyclo[n.m.0] alkatetraenes", J. Am. Chem. Soc. 132, 14617, 2010.

II. Oxyallyl diradical

W. T. Borden at al. "The Lowest Singlet and Triplet States of the Oxyallyl Diradical," Angew. Chem. Int. Ed. 2009, 48, 8509.

Chapter 1.1

+

Example: singlet biradicals need at least two Slater determinants:

Chapter 1.1

Example: Bergman cycliza(on

Examples: Organic Chemistry

2/24/14

22

Wavefunc(on built from all necessary Slater determinants:

Difference with CI wavefunc(on:

many determinants may contribute equally

orbitals should be reop(mized Mul(-‐Configura(onal SCF (MCSCF)

MCSCF computa(ons usually require to choose an ac(ve space including all important orbitals needed for a qualita(ve descrip(on

1 1 2 2 1 2(1) (2)... ( ) (1) (2)... ( ) ...a N b NC N C Nϕ ϕ ϕ ϕ ϕ ϕΨ = + +Two degenerate orbitals


Chapter 1.1

In prac(ce, CASSCF or CASPT2 is used: not black-‐box approaches.

Advanced level


Electronic correla(on: difference between HF and full CI energies

Subdivision:

•  Sta(c correla(on arises from near-‐degeneracies and requires mul(configura(onal treatment

•  Dynamic correla(on arises rather from electrons instantaneously avoiding each other, and usually requires a big number of small contribu(ons from Slater determinants to be described

MCSCF (or CASSCF) lacks dynamic correla(on outside ac(ve space ! Can be added by CI (MR-‐CI methods) or perturba(on theory (CASPT2, GMCQDPT2, NEVPT2…)

Chapter 1.1

2/24/14

23

Composite methods Highly accurate computa(onal methods are expensive.

Idea: combine many methods to approximate highly accurate result.

Many flavors exist: G1-‐4, W1-‐4, CBS-‐QB3…

Composite methods

Example: summary of G4 method

1.  DFT (see next course) geometry and frequencies

2.  Hartree-‐Fock energy with very large basis sets, extrapolated to infinite basis

3.  MP4 computa(on of correla(on energy with small basis set

4.  Es(ma(on of effect of basis set increase on correla(on energies

5.  Es(ma(on of the difference between MP4 and CCSD(T)

6.  Higher level correc(ons (with empirical parameters)

Final energy should be close to CCSD(T) with large basis set.

Composite approaches are for non mul(reference systems. Chapter 1.1

Hückel theory 1930s, simple use of Schrödinger equa(on for aroma(c and conjugate systems

Wavefunc2on wriyen as a linear combina(on of atomic orbitals

k kkc ϕΨ =∑

Varia(onal principle: vary coefficients un(l the energy is minimal best Ψ

Op(mal coefficients leads to set of equa(ons represented by a secular determinant:

H11 − ES11 H12 − ES12H21 − ES21 H22 − ES22

= 0

ˆij i jH Hϕ ϕ= ij i jS ϕ ϕ=

0k

Ec∂ =∂

with and Chapter 1.1

2/24/14

24

Hückel theory

Hückel theory was developed in the 1930s and represents a simple use of Schrödinger equa(on that works reasonably for aroma(c and conjugate systems. Only molecules with π electrons, need parametriza(on.

Chapter 1.1

We will come back to Hückel theory when discussing aroma(city.

Hückel theory Hückel theory: π-‐systems in molecules

Only p orbitals on each atoms in the wavefunc(on expression

Further simplifica(ons:

• No overlap between func(ons:

• Only three possible values for H integrals:

0 if i j1 if i jij ijS δ

≠⎧= =⎨ =⎩

if i j if i and j are neighbors

0 otherwiseijH

αβ

=⎧⎪= ⎨⎪⎩

α and β are empirically determined, and depend on the type of atoms involved.

Chapter 1.1

2/24/14

25

Hückel theory: parameters Barrier for rota(on of ethylene can be used to es(mate β

Secular determinant equa(on for two p orbitals:

0E

Eα ββ α−

=−

2 solu(ons for E:

E1= α + β and E2 = α – β

β < 0 by conven(on, lowest energy is E1

2 electrons: barrier for ethene rota(on is 2β.

first es(ma(on for β: -‐30 kcal/mol from experimental data

Defini(ve value determined by average over different molecules

Chapter 1.1

Hückel theory: General Solu(on

Ek =α + 2β cos kπ(n +1)

Ek =α + 2β cos 2kπ(n)

acyclic chains

cyclic systems

n = number of atoms Ek= energy level k k = quantum number

iden?fying the MO

Etylene: E1= α + β

Benzene: E0 = α + 2β

E1 = α + β

Chapter 1.1

See more in the lecture on aroma2city

k = 1, 2, 3…

k = 0, ±1, ±2…n/2 for even n

2/24/14

26

Summary: ab ini?o methods

Chapter 1.1

Hartree-‐Fock (HF):

Pros Cons

Rela(vely cheap and simple Lack of correla(on-‐> various failures Monodeterminental

MP2 Widely implemented Frac(on of dynamic

correla(on

Not always accurate: e.g., Systema(c overes(ma(on of pi-‐pi stacking

CCSD(T) Widely implemented Gold Standard (very accurate)

size consistent

Large basis sets are needed Computa(onally very demanding: not applicable to large organic molecules

Mul2configura2onal

CASSCF

CASPT2 Implemented in several codes treat both dynamic and sta(c correla(on

Not a black-‐box approach, Computa(onal limita(ons (size of the ac(ve space)

Implemented in several codes treatment of sta(c correla(on

for systems with mul(configura(onal character.

Not a black-‐box approach, < 16 electrons in the ac(ve space

Mini Quiz 1

1.  Amongst the sets of theore(cal levels given below, which one (in each set) gives the lowest energy for H2O:

(a) HF/STO-‐3G and HF/6-‐31G*

(b) HF/cc-‐pVTZ or CCSD(T)/cc-‐pVTZ

2.  You need to compute very accurate reference energy values but the systems you are studying are slightly too large for a typical CCSD(T)/aug-‐cc-‐pVTZ computa(on (i.e., the gold standard). Which approaches will you use?

3.  How would you determine whether the rela(ve energy ordering for the xylene series is affected by dynamic correla(on ?

4.  How do you compute a rough geometry of a series of hydrocarbon radicals?

5.  How do you compute an accurate geometry for the propyl radical?

Chapter 1.1

Ab-initio 4 beginers 2014 - EPFL

Documents

Transcript of Ab-initio 4 beginers 2014 - EPFL