Introduction to Computational Quantum Chemistry: Theoryrsc.anu.edu.au/~agilbert/gilbertspace/uploads/Chem3108.pdf · Introduction to Computational Quantum Chemistry: Theory Dr Andrew

Introduction to Computational QuantumChemistry: Theory

Dr Andrew Gilbert

Rm 118, Craig Building, RSC

3108 Course Lectures 2007

Introduction Hartree–Fock Theory Configuration Interaction

Lectures

1 IntroductionBackgroundThe wave equationComputing chemistry

2 Hartree–Fock TheoryThe molecular orbital approximationThe self-consistent fieldRestricted and unrestricted HF theory

3 Configuration InteractionThe correlation energyConfiguration expansion of the wavefunction


Lectures

4 Correlated MethodsConfiguration interactionCoupled-cluster theoryPerturbation theoryComputational Cost

5 Basis SetsBasis functionsAdditional types of functions

6 Density Functional TheoryDensity functionalsThe Hohenberg–Kohn theoremsDFT models


Lectures

7 Model ChemistriesModel chemistries


Lecture





Background

Computational chemistry

Computational Chemistry can be described as chemistryperformed using computers rather than chemicals.It covers a broad range of topics including:

CheminformaticsStatistical mechanicsMolecular mechanicsSemi-empirical methodsAb initio quantum chemistry

All these methods, except the last, rely on empiricalinformation (parameters, energy levels etc.).

In this course we will focus on the last of these methods.


Background

Ab initio quantum chemistry

Ab initio means “from the beginning” or “from firstprinciples”.

Ab initio quantum chemistry distinguishes itself from othercomputational methods in that it is based solely onestablished laws of nature: quantum mechanics

Over the last two decades powerful molecular modellingtools have been developed which are capable of accuratelypredicting structures, energetics, reactivities and otherproperties of molecules.These developments have come about largely due to:

The dramatic increase in computer speed.The design of efficient quantum chemical algorithms.


Background

Nobel recognition

The 1998 Nobel Prize inChemistry was awarded toWalter Kohn “for hisdevelopment of the densityfunctional theory” andJohn Pople “for his developmentof computational methods inquantum chemistry”.


Background

Advantages

Calculations are easy to perform, whereas experimentsare often difficult.

Calculations are becoming less costly, whereasexperiments are becoming more expensive.

Calculations can be performed on any system, even thosethat don’t exist, whereas many experiments are limited torelatively stable molecules.

Calculations are safe, whereas many experiments have anintrinsic danger associated with them.


Background

Disadvantages

Calculations are too easy to perform, many black-boxprograms are available to the uninitiated.

Calculations can be very expensive in terms of the amountof time required.

Calculations can be performed on any system, even thosethat don’t exist!

Computational chemistry is not a replacement for experimentalstudies, but plays an important role in enabling chemists to:

Explain and rationalise known chemistry

Explore new or unknown chemistry


The wave equation

Theoretical model

The theoretical foundation for computational chemistry isthe time-independent Schrodinger wave equation:

HΨ = EΨ

Ψ is the wavefunction. It is a function of the positions of allthe fundamental particles (electrons and nuclei) in thesystem.

H is the Hamiltonian operator. It is the operator associatedwith the observable energy.

E is the total energy of the system. It is a scalar (number).

The wave equation is a postulate of quantum mechanics.


The wave equation

The Hamiltonian

The Hamiltonian, H, is an operator. It contains all the termsthat contribute to the energy of a system:

H = T + V

T is the kinetic energy operator:

T = Te + Tn

Te = −12

∑i

∇2i Tn = − 1

2MA

∑A

∇2A

∇2 is the Laplacian given by: ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2


The wave equation

The Hamiltonian

V is the potential energy operator:

V = Vnn + Vne + Vee

Vnn is the nuclear-nuclear repulsion term:

Vnn =∑A<B

ZAZB

RAB

Vne is the nuclear-electron attraction term:

Vne = −∑iA

ZA

RiA

Vee is the electron-electron repulsion term:

Vee =∑i<j

1rij


The wave equation

Atomic units

All quantum chemical calculations use a special system of unitswhich, while not part of the SI, are very natural and greatlysimplify expressions for various quantities.

The length unit is the bohr (a0 = 5.29× 10−11m)

The mass unit is the electron mass (me = 9.11× 10−31kg)

The charge unit is the electron charge (e = 1.60× 10−19C)

The energy unit is the hartree (Eh = 4.36× 10−18J)

For example, the energy of the H atom is -0.5 hartree. In morefamiliar units this is −1,313 kJ/mol


The wave equation

The hydrogen atom

We will use the nucleus as the centre of our coordinates.

The Hamiltonian is then given by:

H = Tn + Te + Vnn + Vne + Vee

= −12∇2

r −1r

And the wavefunction is simply a function of r: Ψ(r)


The wave equation

The Born-Oppenheimer approximation

Nuclei are much heavier than electrons (the mass of aproton ≈ 2000 times that of an electron) and thereforetravel much more slowly.

We assume the electrons can react instantaneously to anymotion of the nuclei (think of a fly around a rhinoceros).

This assumption allows us to factorise the wave equation:

Ψ(R, r) = Ψn(R)Ψe(r; R)

where the ‘;’ notation indicates a parametric dependence.

The potential energy surface is a direct consequence ofthe BO approximation.


Computing chemistry

The chemical connection

So far we have focused mainly on obtaining the totalenergy of our system.

Many chemical properties can be obtained from derivativesof the energy with respect to some external parameterExamples of external parameters include:

Geometric parameters (bond lengths, angles etc.)External electric field (for example from a solvent or othermolecule in the system)External magnetic field (NMR experiments)

1st and 2nd derivatives are commonly available and used.

Higher derivatives are required for some properties, butare expensive (and difficult!) to compute.

Some derivatives must be computed numerically.


Computing chemistry

Computable properties

Many molecular properties can be computed, these include

Bond energies and reaction energies

Structures of ground-, excited- and transition-states

Atomic charges and electrostatic potentials

Vibrational frequencies (IR and Raman)

Transition energies and intensities for UV and IR spectra

NMR chemical shifts

Dipole moments, polarisabilities and hyperpolarisabilities

Reaction pathways and mechanisms


Computing chemistry

Statement of the problem

The SWE is a second-order linear differential equation.Exact solutions exist for only a small number of systems:

The rigid rotorThe harmonic oscillatorA particle in a boxThe hydrogenic ions (H, He+, Li2+ . . .)

Approximations must be used:Hartree-Fock theory a wavefunction-based approach thatrelies on the mean-field approximation.Density Functional Theory whose methods obtain theenergy from the electron density rather than the (morecomplicated) wavefunction.

Relativity is usually ignored.


Computing chemistry

Classification of methods

Post-HF

DFT HF

ΨEc

SCF

Hybrid DFT


Lecture





Previously on 3108

The Schrodinger wave equation HΨ = EΨ

The Hamiltonian is made up of energy terms:

H = Tn + Te + Vnn + Vne + Vee

The Born-Oppenheimer approximation clamps the nucleiand implies Tn = 0 and Vnn is constant.

Atomic units (bohr, hartree, etc.)

Chemical properties are obtained from derivatives of theenergy with respect to external parameters, (e.g. firstderivatives of the energy w.r.t. nuclear coordinates can beused to find transition structures and equilibriumgeometries.).



Post-HF

DFT HF

ΨEc

SCF

Hybrid DFT


The molecular orbital approximation

Hartree-Fock theory

HF theory is the simplest wavefunction-based method.

It forms the foundation for more elaborate electronicstructure methods.

It is synonymous with the Molecular Orbital Approximation.It relies on the following approximations:

The Born-Oppenheimer approximationThe independent electron approximationThe linear combination of atomic orbitals approximation

It does not model the correlation energy, by definition.



Hartree-Fock theory

Consider the H2 molecule:

The total wavefunction involves 4 coordinates:Ψ = Ψ(R1,R2, r1, r2)

We invoke the Born-Oppenheimer approximation:Ψ = Ψn(R1,R2)Ψe(r1, r2).

How do we model Ψe(r1, r2)?



The Hartree wavefunction

We assume the wavefunction can be written as a Hartreeproduct: Ψ(r1, r2) = ψ1(r1)ψ2(r2)

The individual one-electron wavefunctions, ψi are calledmolecular orbitals.

This form of the wavefunction does not allow forinstantaneous interactions of the electrons.

Instead, the electrons feel the averaged field of all theother electrons in the system.

The Hartree form of the wavefunction is is sometimescalled the independent electron approximation.



The Pauli principle

One of the postulates of quantum mechanics is that thetotal wavefunction must be antisymmetric with respect tothe interchange of electron coordinates

The Pauli Principle is a consequence of antisymmetry.

The Hartree wavefunction is not antisymmetric:

Ψ(r2, r1) = ψ1(r2)ψ2(r1) 6= −Ψ(r1, r2)

We can make the wavefunction antisymmetric by adding allsigned permutations:

Ψ(r1, r2) =1√2

[ψ1(r1)ψ2(r2)− ψ1(r2)ψ2(r1)

]



The Hartree-Fock wavefunction

The antisymmetrized wavefunction is called theHartree-Fock wavefunction.

It can be written as a Slater determinant:

Ψ =1√N!

∣∣∣∣∣∣∣∣∣ψ1(r1) ψ2(r1) · · · ψN(r1)ψ1(r2) ψ2(r2) · · · ψN(r2)

......

. . ....

ψ1(rN) ψ2(rN) · · · ψN(rN)

∣∣∣∣∣∣∣∣∣This ensures the electrons are indistinguishable and aretherefore associated with every orbital!

A Slater determinant is often written as |ψ1, ψ2, . . . ψN〉



The LCAO approximation

The HF wavefunction is an antisymmetric wavefunctionwritten in terms of the one-electron MOs.

What do the MOs look like?

We write them as a linear combination of atomic orbitals:

ψi(r i) =∑

µ

Cµiχµ(r i)

The χµ are atomic orbitals or basis functions.

The Cµi are MO coefficients.



An example

The H2 molecule:

ψ1 = σ = 1√2

(χA

1s + χB1s

)ψ2 = σ∗ = 1√

2

(χA

1s − χB1s

)

For H2 the MO coefficients, Cµi , are ± 1√2



The HF energy

If the wavefunction is normalized, the expectation value ofthe energy is given by: E = 〈Ψ|H|Ψ〉For the HF wavefunction, this can be written:

EHF =∑

i

Hi +12

∑ij

(Jij − Kij)

Hi involves one-electron terms arising from the kineticenergy of the electrons and the nuclear attraction energy.

Jij involves two-electron terms associated with the coulombrepulsion between the electrons.

Kij involves two-electron terms associated with theexchange of electronic coordinates.



The HF energy

Remember that our wavefunction is given in terms of adeterminant: |ψ1, ψ2, . . . ψN〉And our MOs are written as a LCAO:

ψi(r i) =∑

µ

Cµiχµ(r i)

We can write the one-electron parts of the energy as:

Hi = 〈ψi |h|ψi〉=

∑µν

CµiCνi〈χµ|h|χν〉

The Jij and Kij matrices can also be written in terms of theMO coefficients, Cµi .


The self-consistent field

The variational principle

The MO coefficients, Cµi , can be determined using thevariational theorem.

Variational TheoremThe energy determined from any approximate wavefunction willalways be greater than the energy for the exact wavefunction.

The energy of the exact wavefunction serves as a lowerbound on the calculated energy and therefore the Cµi canbe simply adjusted until the total energy of the system isminimised.


The self-consistent field

The self-consistent field method

Thus, computing the HF energy implies computing the Cµi .

To compute the Cµi we must minimise the HF energyaccording to the variational principle.

Which comes first: the HF energy or the Cµi?

The SCF Process1 Guess a set of MOs, Cµi

2 Use MOs to compute Hi , Jij and Kij

3 Solve the HF equations for the energy and new MOs4 Are the new MOs different? Yes → (2) : No → (5)5 Self-consistent field converged


Restricted and unrestricted HF theory

Electron Spin

So far for simplicity we have ignored the spin variable, ω.

Each MO actually contains a spatial part and a spin part.

For each spatial orbital, there are two spin orbitals:χα

i (r, ω) = φi(r)α(ω) and χβi (r, ω) = φi(r)β(ω).

This is reasonable for closed-shell systems, but not foropen-shell systems.

H2 H−2




The spatial part of the spinorbitals are the same:

φαi = φβ

i

The spatial part of the spinorbitals are different:

φαi 6= φβ

i



Pros and cons

Advantages of the Unrestricted Hartree-Fock method:

Accounts for spin-polarisation, the process by whichunpaired electrons perturb paired electrons, and thereforegives realistic spin densities.

Provides a qualitatively correct description ofbond-breaking.

Provides a better model for systems with unpairedelectrons.

Disadvantages of the UHF method:

Calculations take slightly longer to perform than for RHF.

Can lead to spin-contamination which means thewavefunction is no longer a spin-eigenfunction (as it shouldbe).



Previously on 3108

The MO approximation and the Hartree wavefunction:

Ψ(r1, r2) = ψ1(r1)ψ2(r2)

Antisymmetry and the Hartree-Fock wavefunction:

Ψ(r1, r2) =1√2

[ψ1(r1)ψ2(r2)− ψ1(r2)ψ2(r1)

]The LCAO approximation

ψi(r i) =∑

µ

Cµiχµ(r i)

The variational method and self-consistent field calculation

Restricted and unrestricted Hartree-Fock theory




Post-HF

DFT HF

ΨEc

SCF

Hybrid DFT


Lecture





The correlation energy

Energy decomposition

The electronic Hamiltonian (energy operator) has severalterms:

Hel = Te(r) + Vne(r; R) + Vee(r)

This operator is linear, thus the electronic energy can alsobe written as a sum of several terms:

Eel = ET + EV︸︷︷︸ + EJ + EK + EC︸︷︷︸Te + Vne Vee

Where we have broken down the electron-electronrepulsion energy into three terms: EJ + EK + EC



Electronic energy decomposition

EJ is the coulomb repulsion energyThis energy arises from the classical electrostatic repulsionbetween the charge clouds of the electrons and is correctlyaccounted for in the Hartree wavefunction.

EK is the exchange energyThis energy directly arises from making the wavefunctionantisymmetric with respect to the interchange of electroniccoordinates, and is correctly accounted for in theHartree-Fock wavefunction.

EC is the correlation energyThis is the error associated with the mean-fieldapproximation which neglects the instantaneousinteractions of the electrons. So far we do not havewavefunction which models this part of the energy.



Electronic energy decomposition

E = ET + EV + EJ + EK + EC

For the Ne atom, the above energy terms are:

ET = +129 Eh

EV = -312 Eh

EJ = +66 Eh

EK = -12 Eh 9.3%EC = -0.4 Eh 0.3%

The HF energy accounts for more than 99% of the energy

If the correlation energy is so small, can we neglect it?



The importance of EC

Consider the atomisation energy of the water molecule:

Energy H2O 2 H + O ∆EEHF -76.057770 -75.811376 0.246393ECCSD -76.337522 -75.981555 0.355967

If we neglect the correlation energy in the atomisation of waterwe make a 30% error!



The electron correlation energy

The correlation energy is sensitive to changes in thenumber of electron pairs

The correlation energy is always negativeThere are two components to the correlation energy:

Dynamic correlation is the energy associated with thedance of the electrons as they try to avoid one another.This is important in bond breaking processes.Static correlation arises from deficiencies in the singledeterminant wavefunction and is important in systems withstretched bonds and low-lying excited states.

Electron correlation gives rise to the inter-electronic cusp

Computing the correlation energy is the single mostimportant problem in quantum chemistry



Modelling the correlation energy

There exists a plethora of methods to compute the correlationenergy, each with their own strengths and weaknesses:

Configuration interaction (CISD, CISD(T))

Møller-Plesset perturbation theory (MP2, MP3. . .)

Quadratic configuration interaction (QCISD)

Coupled-cluster theory (CCD, CCSD, CCSDT)

Multi-configuration self-consistent field theory (MCSCF)

Density functional theory (DFT)

In practice, none of these methods are exact, but they all(except for DFT) provide a well-defined route to exactitude.


Configuration expansion of the wavefunction

Configuration interaction

Recall the HF wavefunction is a single determinant madeup of the product of occupied molecular orbitals ψi :

Ψ0 = |ψ1, ψ2, . . . ψN〉

ψi =∑

µ

Cµiχµ

This is referred to as a single configuration treatment

If we have M atomic orbitals, the HF method gives us Mmolecular orbitals, but only the lowest N are occupied.

The remaining M − N orbitals are called virtual orbitals




We can create different configurations by “exciting” one ormore electrons from occupied to virtual orbitals:

Ψ0 = |ψ1, ψ2, . . . ψiψj . . . ψN〉Ψa

i = |ψ1, ψ2, . . . ψaψj . . . ψN〉Ψab

ij = |ψ1, ψ2, . . . ψaψb . . . ψN〉

These configurations can be mixed together to obtain abetter approximation to the wavefunction:

ΨCI = c0Ψ0 +∑

i

cai Ψa

i +∑

ij

cabij Ψab

ij + . . .

The CI coefficients, cai , c

abij . . . can be found via the

variational method



How does this help?

Consider a minimal H2 system with two MOs:

ψ1 = σ =1√2

(χA

1s + χB1s

)

ψ2 = σ∗ =1√2

(χA

1s − χB1s

)

The anti-bonding orbital, σ∗, has a node at the origin. Byallowing this orbital to mix in the electrons can spend more timeapart on average, thus lowering the repulsion energy.

Correlated Methods Basis Sets Density Functional Theory

Lecture





Previously on 3108

The MO approximation and the Hartree wavefunction:

Ψ(r1, r2) = ψ1(r1)ψ2(r2)

Antisymmetry and the Hartree-Fock wavefunction:

Ψ(r1, r2) =1√2

[ψ1(r1)ψ2(r2)− ψ1(r2)ψ2(r1)

]The LCAO approximation

ψi(r i) =∑

µ

Cµiχµ(r i)

The variational method and self-consistent field calculation

Restricted and unrestricted Hartree-Fock theory


Previously on 3108

The origins of the electronic correlation energy

The importance of the electronic correlation energy

The idea of a configuration (determinant):

Ψ0 = |ψ1, ψ2, . . . ψiψj . . . ψN〉Ψa

i = |ψ1, ψ2, . . . ψaψj . . . ψN〉Ψab

ij = |ψ1, ψ2, . . . ψaψb . . . ψN〉

The configuration interaction wavefunction

ΨCI = c0Ψ0 +occ∑i

vir∑a

cai Ψa

i +occ∑ij

vir∑ab

cabij Ψab

ij + . . .


Orbital densities

!4 !2 2 4

0.1

0.2

0.3

0.4

0.5

0.6



Post-HF

DFT HF

ΨEc

SCF

Hybrid DFT




If we allow all possible configurations to mix in then weobtain the Full-CI wavefunction. This is the most completetreatment possible for a given set of basis functions.

Complete-CI is Full-CI in an infinite basis set and yields theecact non-relativistic energy.

The cost of full-CI scales exponentially and is thereforeonly feasible for molecules with around 12 electrons andmodest basis sets.Truncated CI methods limit the types of excitations that canoccur:

CIS adds only single excitations (same as HF!)CID adds only double excitationsCISD adds single and double excitationsCISDT adds single, double and triple excitations



Size consistency

A method is size-consistent if it yields M times the energyof a single monomer when applied to M non-interactingmonomers.

HF and Full-CI theories are size consistent, but truncatedCI approaches are not.A method that is not size-consistent:

Yields poor dissociation energiesTreats large systems poorly because the correlation energyper monomer tends to zero as the number of monomersincreases.


Coupled-cluster theory

Coupled-cluster theories

The CID wavefunction can be written as ΨCID = (1 + T2)Ψ0

where:

T2 =14

occ∑i

occ∑j

virt∑a

virt∑b

cabij aaaba∗i a∗j

The a and a∗ are creation and annihilation operators.

The CCD wavefunction can then be written as :

ΨCCD = exp (T2)Ψ0

=

[1 + T2 +

T 22

2!+

T 32

3!+ . . .

]Ψ0

T 22 gives some quadruple excitations and leads to

size-consistency


Perturbation theory

The HF Hamiltonian

The Hartree-Fock wavefunction is not an eigenfunction ofthe Hamiltonian HΨ0 6= EHFΨ0

However, it can be considered as an eigenfunction of theHartree–Fock Hamiltonian:

H0 =∑

i

f (r i)

f (r i) = T(r i) + Vne(r i) + νHF(r i)

The difference between these operators, H− H0 gives thecorrelation energy


Perturbation theory

Møller-Plesset perturbation theory

In Møller-Plesset Perturbation Theory the Hamiltonian isdivided into two parts:

H = H0 + λV

The perturbation, λV, is assumed to be small

The wavefunction and energy are then expanded as apower series in λ (which is later set to unity)

Ψλ = Ψ0 + λΨ1 + λ2Ψ2 + . . .

Eλ = E0 + λE1 + λ2E2 + . . .

Ψ0 and E0 are the HF wavefunction and energy


Perturbation theory

Møller-Plesset perturbation theory

MPn is obtained by truncating the expansion at order λn

The MP1 energy is the same as the HF energy

The MP2 energy is given by:

EMP2 =occ∑i<j

vir∑a<b

|〈Ψ0|V|Ψabij 〉|

2

εi + εj − εa − εb

The cost of calculating the MP2 energy scales as O(N5)and typically recovers ∼80-90% of the correlation energy

The MPn energy is size-consistent but not variational

The MP series may diverge for large orders


Computational Cost

Scaling

HF formally scales asO(N4), practically as O(N2)

MPn scales as O(Nn+3)

CCSD and CISD scale asO(N6)

CCSD(T) scales as O(N7)

CCSDT scales as O(N8)


Computational Cost

An example

System tHF tMP2 tCCSD

Ala1 2.6 s 40 s 58 mAla2 47 s 6 m 30 sAla3 3m 20 s 30 m 50 sAla4 8m 20 s


Lecture





Previously on 3108

Correlated wavefunction methods:

Theory Ψ Variational Size-ConsistentCI (1 + T1 + T2 + . . .)Ψ0 ✓ ✗

CC exp(T1 + T2 + . . .)Ψ0 ✗ ✓

MP Ψ0 + λΨ1 + λ2Ψ2 + . . . ✗ ✓

Each of these methods gives a hierarchy to exactitude

Full-CI gives the exact energy (within the given basis set)

The concepts of variational and size-consistent methods

Coupled-cluster methods are currently the most accurategenerally applicable methods in quantum chemistry

CCSD(T) has been called the “gold standard” and iscapable of yielding chemical accuracy


Basis functions

Basis functions

The atom-centred functions used to describe the atomicorbitals are known as basis functions and collectively forma basis set

Larger basis sets give a better approximation to the atomicorbitals as the place fewer restrictions on the wavefunction

Larger basis sets attract a higher computational cost

Basis sets are carefully designed to give the bestdescription for the lowest cost


Basis functions

Gaussian basis representations

The 1s orbital of the H atom is an exponential e−α|r−A|

which gives rise to difficult integrals

Primitive Gaussians, e−β|r−A|2 yield easier integrals but donot have the correct behaviour

If we take fixed combinations of Gaussians {Die−βi |r−A|2}...we get the best of both worlds

∑i Die−βi |r−A|2


Basis functions








Basis functions








Basis functions








Basis functions

Gaussian basis functions

A Cartesian Gaussian basis function can be written:

χ(x , y , z) = xaybzcK∑

i=1

Die−αi (x2+y2+z2)

a + b + c is the angular momentum of χ

K is the degree of contraction of χ

Di are the contraction coefficients of χ

αi are the exponents of χ

These types of basis functions are sometimes referred toas GTOs


Basis functions

Minimal basis sets

The simplest possible atomic orbital representation iscalled a minimal basis set

Minimal basis sets contain the minimum number of basisfunctions to accommodate all of the electrons in the atom

For example:H & He a single function (1s)1st row 5 functions, (1s,2s,2px ,2py ,2pz)2nd row 9 functions, (1s,2s,2px ,2py ,2pz ,3s,3px ,3py ,3pz)

Functions are always added in shells


Basis functions

Minimal basis sets

The STO-3G basis set is a minimal basis set where eachatomic orbital is made up of 3 Gaussians. STO-nG alsoexist.

Minimal basis sets are not well suited to model theanisotropic effects of bonding

Because the exponents do not vary, the orbitals have afixed size and therefore cannot expand or contract


Additional types of functions

Split valence functions

Split-valence basis sets model each valence orbital by twoor more basis functions that have different exponents

They allow for size variations that occur in bonding

Examples include the double split valence basis sets,3-21G and 6-31G, and triple split valence basis sets suchas 6-311G



Polarisation functions

Polarisation functions have higher angular momentum

They allow for anisotropic variations that occur in bondingand help model the inter-electronic cusp

Examples include 6-31G(d) or 6-31G* which include dfunctions on the heavy atoms 6-31G(d ,p) or 6-31G**which include d functions on heavy atoms and p functionson hydrogen atoms



Diffuse functions

Diffuse basis functions are additional functions with smallexponents, and are therefore largeThey allow for accurate modelling of systems with weaklybound electrons, such as

AnionsExcited states

A set of diffuse functions usually includes a diffuse s orbitaland a set of diffuse p orbitals with the same exponent

Examples in include 6-31+G which has diffuse functions onthe heavy atoms and 6-31++G which has diffuse functionson hydrogen atoms as well.



Mix and match

Larger basis sets can be built up from these components,for example 6-311++G(2df ,2dp)

Dunning basis sets also exist, for example pVDZ and pVTZFor larger atoms Effective Core Potentials (ECPs) are oftenused. These replace the core electrons with an effectivepotential and have two main advantages:

They reduce the number of electrons (cheaper)They can be parameterized to take account of relativity

The valence electrons are still modelled using GTOs



Examples

Basis set Description No. functionsH C,O H2O C6H6

STO-3G Minimal 1 5 7 363-21G Double split-valence 2 9 13 66

6-31G(d) Double split-valence with polari-

sation

2 15 19 102

6-31G(d ,p) Ditto, with p functions on H 5 15 25 1206-311+G(d ,p) Triple split-valence with polarisa-

tion, p functions on H and diffuse

functions on heavy atoms

6 22 34 168



Accuracy

The accuracy of the computed properties is sensitive to thequality of the basis set. Consider the bond length anddissociation energy of the hydrogen fluoride molecule:

Basis set Bond Length (A) D0 (kJ/mol)6-31G(d) 0.9337 4916-31G(d ,p) 0.9213 5236-31+G(d) 0.9408 5156-311G(d) 0.9175 4846-311+G(d ,p) 0.9166 551Expt. 0.917 566

ZPVE = 25 kJ/mol MP2/6-311+G(d ,p)


Lecture





Previously on 3108

Gaussian basis functionsPrimitive functionsContracted basis functions

Minimal basis setsAdditional types of functions

Split valencePolarisation functionsDiffuse functions

Effective Core Potentials (ECPs)



Post-HF

DFT HF

ΨEc

SCF

Hybrid DFT


Density functionals

What is the density?

The electron density is a fundamental quantity in quantumchemistry

ρ(r1) =

∫· · ·

∫Ψ∗(r1, r2, . . . , rN)Ψ(r1, r2, . . . , rN)dr2 . . .drN

ρ(r)dr gives the probability of finding an electron in thevolume element dr

It is a function of three variables (x , y , z) and is therefore(relatively) easy to visualise


Density functionals

What is a functional?

A function takes a number and returns another number:

f (x) = x2 − 1 f (3) = 8

An operator takes a function and returns another function:

D(f ) = dfdx D(x2 − 1) = 2x

A functional takes a function and returns a number:

F [f ] =∫ 1

0 f (x)dx F [x2 − 1] = −2/3


Density functionals

What is a density functional?

A density functional takes the electron density and returnsa number, for example:

N[ρ] =

∫ρ(r)dr

simply gives the number of electrons in the molecule

Density functional theory (DFT) focusses on functionalsthat return the energy of the system


Density functionals

What is a density functional?

E = ET + EV + EJ + EK + EC

The classical potential energy terms of the total energy canbe expressed exactly in terms of the density:

EJ =12

∫ ∫ρ(r1)ρ(r2)

|r1 − r2|dr1dr2

EV = −∑

A

∫ZAρ(r)|RA − r|

dr

What about ET, EK and EC?


Density functionals

Orbital functionals

In Hartree-Fock theory, ET, EK and EC, are all orbitalfunctionals, eg:

ET = −12

∑i

∫ψi(r)∇2ψi(r)dr

No (known) exact expression for the kinetic energy interms of ρ exists

The exchange energy is non-classical, so should weexpect there to be an expression for the exchange energyin terms of the classical density?


The Hohenberg–Kohn theorems


The First Hohenberg-Kohn Theorem

The electron density ρ(r) determines the external potential

This theorem establishes the existence of a (universal andunique) energy functional of the density

The Second Hohenberg-Kohn Theorem

Any approximate density ρ which provides the external potentialν(r), determines its own wavefunction.

The second HK theorem can be used to establish avariational principle for DFT, although it restricts the theoryto ground states




The HK theorems are non-constructive, so we don’t knowwhat the form of the universal functional is

Research in the DFT largely falls down to the developmentof approximate functionals that model experimental data

Kinetic energy functionals are particularly problematic asET is so large and even a small relative error gives largeabsolute errors

Almost all DFT calculations rely on the Kohn-Shamapproximation, which avoids the need for a kinetic energyfunctional

Different DFT methods differ in the way they represent EX

and EC


DFT models

The uniform electron gas

The uniform electron gas is a model system with aconstant density of electrons

In 1930 Dirac showed that the exact exchange energy forthis system is given by:

EX = −Cx

∫ρ4/3(r)dr

Much later, Vosko, Wilk and Nusair parameterised acorrelation functional based on the UEG, its form is morecomplicated and it is inexact


DFT models

Local density approximation

Applying the UEG functionals to molecular system is calledthe local (spin) density approximation (LDA)

Combining the Dirac and VWN expressions gives theS-VWN functional

The LDA functional for EX underestimates the trueexchange energy by about 10% whereas the VWNfunctional overestimates EC by as much as 100%

Together they overbind molecular systems

The constant Cx is sometimes scaled to account for theover-binding, this gives Xα theory


DFT models

Gradient corrected functionals

Gradient corrected functions depend on ∇ρ as well as ρ

The gradient helps to account for deviations fromuniformity in molecular systems

The generalised gradient approximation exchangefunctionals have the form

EX =

∫ρ4/3(r)g(x)dr

where x is the reduced gradient

Different GGAs, such as Perdew ’86 and Becke ’88 aredefined by different g(x) functions


DFT models

GGA correlation functionals

There are also GGA correlation functionals such asLee-Yang-Parr (LYP) and Perdew ’86

EX and EC can be mixed and matched, although certaincombinations such as BLYP work particularly well

Combining a correlation functional with Hartree-Fockexchange does not work well, but hybrid functionals do:

EB3LYP = (1−c1)ED30X +c1EFock

K +c2EB88X +(1−c3)EVWN

C +c3ELYPC

B3LYP is the most popular density functional that is usedand yields very good structural and thermochemicalproperties


DFT models

Strengths and weaknesses

Advantages of DFT methods include:

Low computational cost

Good accuracy for structures and thermochemistry

The density is conceptually more simple than Ψ

Disdvantages of DFT methods include:

Can fail spectacularly and unexpectedly

No systematic way of improving the results

Integrals require numerical quadrature grids

Model Chemistries

Lecture

7 Model ChemistriesModel chemistries

Model Chemistries

Model chemistries

Levels of theory

Quantum chemistry abounds with many levels of theorythat represent a trade-off between computational cost andaccuracyA minimal basis Hartree-Fock calculation forms ourbaseline, other level of theory distinguish themselves bytheir treatment of the correlation energy and the size of thebasis

Cost favours small basis sets and a low-level treatment ofcorrelationAccuracy favours large basis sets and a high-leveltreatment of correlation

Model Chemistries

Model chemistries

The Pople Diagram

HF MP2 MP3 MP4 CCSD(T) · · · Full CI

Minimal Low-level · · · · · · · · · UnbalancedSplit-Valence . . . . . .PolarisedDiffuse . . .Polarised +Diffuse

... . . .. . .

Infinite Unbalanced Exact!

Model Chemistries

Model chemistries

Establishing the reliability of a method

Experimental data forms a valuable means of establishingthe reliability of a particular level of theory

Data sets such as the G2 and G3 sets are made up ofaccurate experimental values with uncertainties of lessthan 1 kcal/mol (chemical accuracy)

The G2 set consists of thermochemical data includingatomisation energies, ionisation potentials, electronaffinities and proton affinities for a range of smallmolecules (1 or 2 heavy atoms)

These data set can be used to benchmark a level of theory

What if we want to apply our method to an unknownsystem?

Model Chemistries

Model chemistries

Establishing the reliability of a method

If we wish to apply a level of theory to a system that has noexperimental data available, we need to converge the levelof theory to have confidence in our results

We start near the top left-hand (cheap) corner of the Poplediagram and move towards the bottom right-hand(expensive) corner carrying out several calculations

When we see no improvement in the result, then weconclude that we have the correct answer

Note that we cannot apply this approach to DFT methods(although we can converge the basis set)

Model Chemistries

Model chemistries

Specifying the level of theory

Geometric properties converge faster (with respect to thelevel of theory) than the energy (they are less sensitive tocorrelation)

It is common to optimise the geometry at a low-level oftheory, and then compute the energy at a higher level oftheory

The notation for this isenergy-method/basis-set//geometry-method/basis-set

For example:CCSD(T)/6-311G(2d,p)//HF/6-31G

Model Chemistries

Model chemistries

Performance

Average deviation from experiment for bond-lengths of 108main group molecules using 6-31G(d ,p)

Bond-length HF MP2 LDA GGA HybridDeviation A 0.021 0.014 0.016 0.017 0.011

Average deviation from experiment for atomisation energies of108 main group molecules using 6-31G(d ,p)

AE HF MP2 LDA GGA HybridDeviation kcal/mol 119.2 22.0 52.2 7.0 6.8

Model Chemistries

Model chemistries

Performance

Calculated electron affinity (eV) for Fluorine:

HF MP2 B3LYPSTO-3G -10.16 -10.16 -9.013-21G -1.98 -1.22 -0.866-31G(d) -0.39 +1.07 +1.056-311+G(2df ,p) +1.20 +3.44 +3.466-311+G(3df ,2p) +1.19 +3.54 +3.46Experiment +3.48

Documents

Introduction to Computational Quantum Chemistry: Theoryrsc.anu.edu.au/~agilbert/gilbertspace/uploads/Chem3108.pdf · Introduction to Computational Quantum Chemistry: Theory Dr Andrew