Introduction to Computational Chemistry: Theory

Introduction to Computational Chemistry:Theory

Dr Andrew Gilbert

Rm 118, Craig Building, [email protected]

3023 Course Lectures - 2011

Introduction Hartree–Fock Theory Basis Sets

Lecture 1

1 IntroductionBackgroundThe Wave EquationComputing Chemistry

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

3 Basis SetsBasis functionsAdditional types of functionsComputational Aspects


Background

Computational chemistry

Computational Chemistry is the modeling of chemicalphenomenon using computers rather than chemicals.The models used vary in their sophistication:

CheminformaticsMolecular 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 ab initio quantum chemistry.


Background

Ab initio quantum chemistry

Ab initio means “from the beginning” or “from firstprinciples”, i.e. quantum mechanics.

“The fundamental laws necessary for the mathematical treatment of a large part ofphysics and the whole of chemistry are thus completely known, and the difficulty liesonly in the fact that application of these laws leads to equations that are too complex tobe solved.”

P.A.M. Dirac

Over the last four decades powerful molecular modellingtools have been developed which are capable of accuratelypredicting properties of molecules.These developments have come about largely due to:

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


Background

Advantages

Calculations aresafe: many experiments have an intrinsic dangerassociated with them.clean: there are no waste chemicals produced.cost effective: compared to performing experiments.easy: many experiments can be difficult to perform.

Calculations may also be to give greater insight into thechemistry by providing more information about the system.


Background

Disadvantages

Calculations can also beapplied uncritically: just because you can do somethingdoesn’t mean you should.wrong: there may be approximations in the models used,or bugs in the code.a black box: software packages insulate chemists from theunderlying theory.

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

Explain and rationalise known chemistryExplore new or unknown chemistry


The Wave Equation

Theoretical model

The theoretical foundation for quantum chemistry is thetime-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).Relativistic effects are usually small and will be ignored.


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

|RA − RB|

Vne is the nuclear-electron attraction term:

Vne = −∑iA

ZA

|RA − ri |

Vee is the electron-electron repulsion term:

Vee =∑i<j

1|ri − rj |


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 −12 hartree (exactly).

In more familiar 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

The ground-state wavefunction is simply a function of r .


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 parameter.Examples of external parameters include:

Geometric parameters (bond lengths, angles etc.).Applied electric fields (e.g. from a solvent)Magnetic field (in NMR experiments).

1st and 2nd derivatives are commonly available and used.Higher derivatives are required for some properties, butare expensive (and difficult!) to compute.


Computing Chemistry

Lecture 1 summary

Different models for chemistry:Cheminformatics (functional groups)Molecular mechanics (atoms)Semi-empirical models (approximate QM)Quantum chemistry (fundamental particles)

The Schrodinger wave equation: HΨ = EΨ

The Hamiltonian is made up of energy terms:

H = Tn + Te + Vnn + Vne + Vee

The wavefunction gives a complete description of thesystem.Chemical properties are obtained from derivatives of theenergy with respect to external parameters.


Lecture 2





The molecular orbital approximation

Hartree-Fock theory

HF theory is the simplest wavefunction-based method.It relies on the following approximations:

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

The Hartree-Fock model introduces an intrinsic error calledthe correlation energy.It forms the foundation for more elaborate electronicstructure methods.



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 means the nuclei are stationary w.r.t. the electrons.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.



The independent electron approximation

Consider the H2 molecule:

The total wavefunction involves 4x3 spatial 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 average field of all the otherelectrons 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.Antisymmetry is a consequence of the Pauli Principle.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 antisymmetrised Hartree 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 antisymmetric and written in termsof the one-electron molecular orbitals (MOs).What do the MOs look like?We write them as a linear combination of atomic orbitals:

ψi(ri) =∑µ

Cµiχµ(ri)

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 Ψ is normalised, the expectation value of the energy isgiven 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(ri) =∑µ

Cµiχµ(ri)

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. This is the variational method.


The self-consistent field

The self-consistent field method

Consider a 2-electron system with MOs ψ1(r1) and ψ2(r2).Electron 1 feels the nuclei and the field of ψ2(r2).Electron 2 feels the nuclei and the field of ψ1(r1).This creates a chicken and egg situation: we need ψ2 tosolve for ψ1, but we need ψ1 to solve for ψ2.

The SCF Process1 Guess a set of MOs, Cµi

2 Use MOs to compute Hi , Jij and Kij

3 Solve the HF equations to obtain a new set of 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:χαi (r, ω) = ψi(r)α(ω) and χβi (r, ω) = ψi(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 = φβi

The spatial part of the spinorbitals are different:

φαi 6= φβi



Pros and cons

Advantages of the UHF method:The UHF wavefunction has more flexiblity and can give alower energy (variationally better).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).



Lecture summary

The Born-Oppenheimer approximation clamps the nucleiand implies Tn = 0 and Vnn is constant.The P.E.S. is a consequence of the B.O.The independent electron (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(ri) =∑µ

Cµiχµ(ri)

The variational method and self-consistent field calculation


Lecture 3





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 they place fewer restrictions on thewavefunction.Larger basis sets attract a higher computational cost.Standard 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 χK is the degree of contraction of χDi are the contraction coefficients of χαi are the exponents of χThese types of basis functions are sometimes referred toas Gaussian type orbitals GTOs.


Basis functions

Minimal basis sets

The simplest possible atomic orbital representation iscalled a minimal basis set.Minimal basis sets contain one basis functions for eachoccupied atomic orbital.For example:H & He 1 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., e.g. a p shellconsists of three functions.


Basis functions

Minimal basis sets

The STO-3G basis set is a minimal basis set where eachatomic orbital is made up of 3 Gaussians.The STO-6G basis set is a minimal basis set where eachatomic orbital is made up of 6 Gaussians.Minimal basis sets are not well suited to model theanisotropic effects of bonding.Basis function exponents do not vary and therefore theorbitals have a fixed size and 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:

Double split valence basis sets: 3-21G, 6-31G and VDZTriple split valence basis sets: 6-311G and VTZ



Polarisation functions

Polarisation functions have higher angular momentum thanthe occupied AOs.They allow for anisotropic variations that occur in bonding

6-31G(d) or 6-31G* include d functions on the heavyatoms (non-hydrogen).6-31G(d ,p) or 6-31G** include d functions on heavy atomsand p functions on hydrogen atoms.



Diffuse functions

Diffuse basis functions are additional functions with smallexponents, and are therefore have large spatial extent.They 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 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 ,2pd).Dunning basis sets also exist, for example pVDZ and pVTZ(polarised double split valence and triple split valence,respectively).Some basis sets work better for HF and DFT calculations(e.g. Pople basis sets and Jensen’s pc-n bases)Others are best for correlated calculations (e.g. cc-pV*Z).Carefully designed sequences of basis sets can be used toextrapolate to the basis set limit. For example cc-pVDZ,cc-pVTZ, cc-pVQZ. . .



Effective core potentials

Effective Core Potentials replace core electrons with aneffective potential which is added to the Hamiltonian.ECPs have two main advantages:

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

Non-relativistic ECPs include HWMB (STO-3G) andLANL2DZ (6-31G).Relativistic ECPs include SRSC (6-311G*) and SRLC(6-31G).The size of the core can vary, for example:SRLC (large core): K = [Ar] + 3s, 2pSRSC (small core): K = [Ne] + 5s, 4pECPs are particularly useful for transition metals.


Computational Aspects

Counting basis functions

It is important to have an idea of how many basis functions arein your molecule as this will determine the cost of thecalculation.

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 3 summary

Gaussian basis functionsPrimitive functionsContracted basis functions

Minimal basis setsAdditional types of functions

Split valencePolarisation functionsDiffuse functions

Effective Core Potentials (ECPs)

The Correlation Energy Correlated Methods Density Functional Theory

Lecture 4

4 The Correlation EnergyThe correlation energyConfiguration expansion of the wavefunction

5 Correlated MethodsConfiguration expansion of the wavefunctionPost HF methods

6 Density Functional TheoryDensity functionalsThe Hohenberg–Kohn theoremsDFT models


The correlation energy

The story so far

The Hartree wavefunction is based on molecular orbitals:

ΨH(r1, r2) = ψ1(r1)ψ2(r2)

and models ET, EV and EJ, but is not antisymmetric.The Hartree-Fock wavefunction is antisymmetric and cantherefore model the exchange energy, EK

ΨHF(r1, r2) =1√2

[ψ1(r1)ψ2(r2)− ψ1(r2)ψ2(r1)

]Hartree-Fock theory has an intrinsic error. We call this thecorrelation energy and is defined by:

EC = E − EHF



Classification of methods

Post-HF

DFT HF

ΨEc

SCF

Hybrid DFT



Energy decomposition

The electronic Hamiltonian (energy operator) has severalterms:

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

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

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

The electron-electron repulsion term has been broken intothree terms: EJ + EK + EC



Electronic energy decomposition

EJ is the coulomb repulsion energy.This 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 energy.This 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 energy.This 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

The total electronic energy can be decomposed as follows:

E = ET + EV + EJ + EK + EC

For the Ne atom, the above energy terms are:ET = +129 EhEV = -312 EhEJ = +66 EhEK = -12 Eh 9.3%EC = -0.4 Eh 0.3%

The HF energy accounts for more than 99% of the energyIf 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 negative.There 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, . . . ψi , ψj , . . . ψN〉Ψa

i = |ψ1, ψ2, . . . ψa, ψj , . . . ψN〉Ψab

ij = |ψ1, ψ2, . . . ψa, ψb, . . . 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 theorem.



Lecture 4 summary

The origins of the electronic correlation energyThe importance of the electronic correlation energyThe 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 + . . .


Lecture 5






Configurations

To improve on the HF wavefunction, we need to considerexcited configurations:

These configurations can be mixed to give a betterapproximation to the wavefunction :

ΨCI = c0Ψ0 +∑

i

cai Ψa

i +∑

ij

cabij Ψab

ij + . . .



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 node in the σ∗ orbital allows the electrons to spend moretime apart, thus lowering the electron repulsion energy.



Orbital densities

The picture can be made clearer by considering density plots ofthe two orbitals:

!4 !2 2 4

0.1

0.2

0.3

0.4

0.5

0.6

This mixing is a compromise as ET and EV also change.This can be viewed as a dynamic correlation effect.



How does this help?

Adding in configurations also helps modelling stretchedbonds.

C0 ≈ 1 C0 ≈ 0C1 ≈ 0 C1 ≈ 1

This can be viewed as a static correlation effect.


Post HF methods


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 theexact 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:

CIS adds only single excitations (same as HF!)CID adds only double excitationsCISD adds single and double excitations, O(N6)CISDT adds single, double and triple excitations, O(N8)


Post HF methods

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 energies.Treats large systems poorly.

Coupled-cluster wavefunctions are like CI wavefunctions,but include terms to maintain size-consistency.

CCSD includes all single and double excitations, but alsoincludes some quadruple excitations.

Coupled-cluster wavefunctions are not variational.


Post HF methods

Size consistency

The CISD wavefunctions for two separate two-electronsystems includes double excitations on both:

When considered as a single system, however, these leadto quadruple excitations, which are not included in CISD.CCSD includes these types of excitation, making itsize-consistent.


Post HF methods

Møller-Plesset perturbation theory

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

H = H0 + λV

H0 is the Hartree-Fock hamiltonian.λV is a perturbation, which is assumed to be small.The wavefunction and energy are then expanded as apower series in λ (which is later set to 1):

Ψλ = Ψ0 + λΨ1 + λ2Ψ2 + . . .

Eλ = E0 + λE1 + λ2E2 + . . .

Ψ0 and E0 are the HF wavefunction and energy.


Post HF methods

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.


Post HF methods

Scaling

The number of basis functions N can be used as a measure ofthe size of the system. The cost of different methods scalesdifferently:

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

MPn scales as O(Nn+3)

CCSD and CISD are O(N6)

CCSD(T) scales as O(N7)

CCSDT scales as O(N8)


Post HF methods

An example

System tHF tMP2 tCCSD

Ala1 2.6 s 40 s 58 mAla2 47 s 7 mAla3 200 s 31 mAla4 8 m


Post HF methods

Summary of post-HF methods

Correlated wavefunction methods:

Theory Finite Expansion Variational Size-ConsistentCI 4 4 8

CC 4 8 4

MP 8 8 4

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 (< 1 kcal/mol error).


Lecture 6





Density functionals

Classification of methods

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) = N∫· · ·∫

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

ρ(r)dr gives a measure of the probability of finding anelectron in the volume 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 ) = df

dx D(x2 − 1) = 2xA 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?

The total energy can be decomposed into the following parts:

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 TheoremThe electron density ρ determines the external potential ν.

This theorem shows a one-to-one correspondencebetween ρ and ν and therefore (via the S.W.E.) Ψ.It also shows that there exists a universal and uniqueenergy functional of the density.

The Second Hohenberg-Kohn TheoremFor any valid trial density, ρ: Eν ≤ Eν [ρ]

The second HK theorem establishes a variational principlefor ground-state DFT.



Density functional theory

The HK theorems are non-constructive, so we don’t knowwhat the form of the universal functional is.Research in DFT largely focusses on the development ofapproximate 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 energydensity functional.Different DFT methods differ in the way they represent EXand 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 (VWN) based on the UEG, its form ismore complicated 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 costGood accuracy for structures and thermochemistryThe density is conceptually simpler than Ψ

Disadvantages of DFT methods include:Can fail spectacularly and unexpectedlyNo systematic way of improving the resultsIntegrals require numerical quadrature grids


DFT models

Lecture 6 summary

A functional takes in a function and returns a number.The density, ρ(r), contains all the information necessary, asshown by the Hohneberg-Kohn theorems.Density functionals can be used to compute the difficultexchange and correlation energies cheaply.LDA functionals, e.g. S-VWN, are based on the UEG, theyoverbind.GGA functionals, e.g. B-LYP, incorporate the reduceddensity gradient and are much more accurate.Hybrid functionals, e.g. B3LYP, incorporate Fock exchangeand are the most accurate.

Model Chemistries

Lecture 7

7 Model ChemistriesModel chemistriesPerformance

Model Chemistries

Model chemistries

Computable properties

Many molecular properties can be computed, these include

Bond energies and reaction energiesStructures of ground-, excited- and transition-statesAtomic charges and electrostatic potentialsDipole moments and multipole momentsVibrational frequencies (IR and Raman)Transition energies and intensities for UV and IR spectraNMR chemical shifts coupling constants and shieldingtensorsPolarisabilities and hyperpolarisabilitiesReaction pathways and mechanisms

Model Chemistries

Model chemistries

Levels of theory

Quantum chemistry abounds with many levels of theorythat represent a trade-off between cost and accuracy.

Small basis sets and treating correlation at a low-level oftheory gives low-cost methods.Using large basis sets and treating correlation at ahigh-level of theory gives high accuracy methods.

Both the cost and accuracy also depend on the propertybeing calculated:

Energies are cheapest to compute.Geometries are relatively cheap (1st derivatives).Harmonic frequencies more expensive (2nd derivatives).Anharmonic corrections are very expensive (3rd & 4thderivatives).

Computer memory may also be a limitation.

Model Chemistries

Model chemistries

The Pople Diagram

A minimal basis Hartree-Fock calculation forms ourbaseline, other levels of theory distinguish themselves bytheir treatment of the correlation energy (left to right) andthe size of the basis (top to bottom)

HF MP2 MP3 MP4 CCSD(T) · · · Full CIMinimal 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 values with experimental 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.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 along the diagonal towards the bottomright-hand (expensive) corner carrying out severalcalculations.When we see no significant improvement in the result, thenwe conclude 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 oftheoryThe notation for this is:energy-method / basis-set // geometry-method / basis-setThe // can be read as ‘optimised at’.For example:CCSD(T) / 6-311G(2d,p) // HF / 6-31G

Model Chemistries

Performance

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

Performance

Performance

Calculated electron affinity (eV) for Fluorine:

F + e− → F−

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

Model Chemistries

Performance

Performance

Convergence of MP methods relative to Full-CI using theSTO-3G basis.

Method HCN CN− CNMP2 -91.82033 -91.07143 -91.11411MP3 -91.82242 -91.06862 -91.12203MP4 -91.82846 -91.07603 -91.13538MP5 -91.83129 -91.07539 -91.14221MP6 -91.83233 -91.07694 -91.14855MP7 -91.83264 -91.07678 -91.15276MP8 -91.83289 -91.07699 -91.15666

Full-CI -91.83317 -91.07706 -91.17006∆E < 0.001 MP6 MP6 MP19

Model Chemistries

Performance

Performance

In general, the use of moderately large basis sets such as6-311G(d,p) combined with the MP2 treatment of electroncorrelation leads to calculated structures very close toexperiment.For difficult systems, the Coupled Cluster (CC) methodscorrespond to an electron correlation treatment better thanMP4 and thus greater accuracy is achieved, e.g. ozone:

Parameter MP2 CCSD CCSD(T) ExptO–O bond (A) 1.307 1.311 1.298 1.272

∠OOO (◦) 113.2 114.6 116.7 116.8

Documents

Introduction to Computational Chemistry: Theory