Ab initio Methods

Ab initio Methods

Ab initio Methods

Lopez Sosa Luis & Tenorio Barajas A. Yair

Departamento de Qumica CINVESTAV

Computational ChemistryPhD. Andreas M. Kster

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Ab initio Methods

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IntroductionContent

Ab initio what means?Common uses

Hartree Fock MethodVariational principleSlater determinantLinear Combination of Atomic Orbitals

Post Hartree Fock MethodsElectronic CorrelationConfiguration InteractionPerturbation TheoryMller-Plesset Perturbation TheoryCoupled Cluster Method

Some results in calculationsAdvantages and Disadvantages

Conclusions

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The term Ab initio from its latin form means "from the beginning". This name is given to the computations that are derived directly from theoretical principles with no inclusion of experimental data.

An Ab initio calculation use the most fundamental physical quantities and only enters a value for a trial wave function.

Ab initio methods use the correct Hamiltonian

Introduction

Ab initio Methods

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IntroductionCommon Uses

Common Uses

Calculating:Molecular geometriesEnergiesVibrational frequenciesSpectra Ionization potentials andElectron affinities.

Ab initio Methods

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Hartree-Fock

Douglas Rayner Hartree Vladmir Aleksndrovich Fock William Hartree Bertha Swirles

IntroductionAuthors

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An Example He

The Hartree Fock Method for He

First we will write the two-electron wave function as product of orbitals, this is the Hartree product.

(r1, r2 )= (r1)(r2 )Functions and are equal, and we can write the potential energy due electron repulsion like:

(r1) (r2 )

V1eff (r1)=

*(r2 )1r12(r2 )dr2

The Hamiltonian over can be expressed like:(r1)

H!1eff

(r1)=12122r1+V1

eff (r1)And the effective Hamiltonian for He:

H!1eff

(r1)(r1)= i(r1)

(1)

(2)

(3)

(4)

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An Example He

Here depends from and this is a pseudo eigenvalue equation to solve it we need the method self consistent field SCF and is found in the variational principle

(r2 )H!1eff

(r1)

SCF Method follow the next steps

Guess a value for Evaluate potential energy for repulsion Solve Hartree Fock equation to obtain Evaluate again but using Solve HF equation to obtain again And so on until the difference between the value obtained and

the value introduced reach an acceptable value

(r2 )V1eff (r1)

(r1)V1eff (r1) (r1)

(r1)

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An Example He

One calculation for He with Hartree Fock is EHF =-2.862Eh,compared with the Eexact= -2.903Eh

The difference in values is due to the fact that both electronsare considered not interacting with each other, or an average interaction

Electrons are uncorrelated, and the correlation energy is:

CE=Eexact - EHF

For this case, CE= -110kJ/mol this is the energy for a chemical bond and thus is not enough precise

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Variational Principle


Schrdinger equation can be solved exactly only for H atom but we can solve it for other atoms approximately and a very good approach is performed by the variational principle

If is the ground state wave function of a system, then theapproximate function , is such that:0

E E 0

Consider a system with a fundamental state as the wave functionwith a solution with the Schrdinger equation E0 with eigenvalue

0

H!0 = E00

E0 =0*H!0 d0*0 d

we can write:

(5)

(6)

(7)

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Substituting other function instead of in equation (7)

E0 =0*H!0 d0*0 d

0

But according with the variational principle

E E0

So equality in energy only happen if = 0

If our trial wave function depends on parameters variational parameters the energy is going to depend on this parameters as well

,,, ...,

E(,,, ...) E0!And we can minimize the energy varying this parameters

dE(i)di= 0 with i = ,,, . . .


an obtain the best parameters for the trial wave function

(8)

(9)

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Slater Determinant

(1,2,...n)= 1n!

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

The wave function must be antisymmetric due the nature of theelectron so and be in agreement with Pauli exclusion principle

(i, j)=( j, i)

mathematically this can be done with a determinant

This Slater determinant is built with molecular orbitals whit i indexing the electron and j the orbital were the electron i is j(i)

! Exercice

Slater Determinant

(10)

John C. Slater

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LCAO

In molecular orbital theory orbitals inside Slater determinant are molecular orbitalsA systematic construction of this MO is achieved with the linear combination of atomic orbitals abbreviated LCAO

i(r )= Ci

(r )

This expansion will be used in calculation of the so called core Hamiltonian

The electronic Hamiltonian operator for a N electronic molecule with atomic nuclei A is

H! = 12i2

ZA

rA,i

+1ri ,ki>k

A,i

i

n

Defining the core Hamiltonian operator for kinetic and potential energy of one electron like:

h!(i)=12i2

ZA

rAiA

(11)

(12)

(13)

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Hartree Fock Method

The Coulomb operator involves electrostatic repulsion between electrons, this term is:

j!(a,b)= 1ra rb

Originally the core Hamiltonian acts over spin orbitals but this can be changed for spatial orbitals thanks to the model of independent particles

j! = 2 iijj

12ijji

j

occ

1

occ

Similarly the expectation value for Coulomb operator is:

j(i)

j(r ,

i)=

j(ri)j(i)

So we can write the expectation value of the core Hamiltonian with the 2 counting for two electrons in each orbital

h! = 2 ih!

ii

occ

(14)

(15)

(16)

(17)

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Hartree Fock Method

The Coulomb integral represents the repulsion experienced by electron 1 in orbital i from electron 2 in orbital j where it is distributed with probability density and there are 2 electrons in each orbital, so we can expect a total contribution of the form

J!(1,2) =i(1)

i(1)

j(2)

j*(2)

r1 r2 dr1dr2

K!(1,2) =i(1)

j*(1)

j(2)

i*(2)

r1 r2 dr1dr2

The exchange integral is a correction to the Coulomb integral due to the fact that exist an error when i=j

E

i

j

1

2

2

2

(18)

(19)

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Taking the order of the Hamiltonian elements H=Ke+Ven+Vee+Vnn core Hamiltonian + Coulomb operator + exchange operator

E= 2 ih!

ii

occ

+2 ii ! j j 12ij!

ij

j

occ

i

occ

E= 2 Hi ,i

i

n

+2 2 Ji , jKi , j j

n

i

n

or

This is the Hartree Fock closed-shell energy and only the value for the trial function is introduced, this is restricted to a single determinant and thus, correlation between electrons of oposite spin is neglected. This problem is treaty by post Hartree-Fock methods.

Hartree Fock Method

(20)

(21)

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Roothaan-Hall equations

The key to make MO calculations feasible was proposed by Roothaan and Hall in 1951, He express the MO as a linear combination of basis functions set of one electron LCAO.

i(r )= Ci

(r )

To represent correctly the MO the basis function must be form a complete set and this require an infinite number of basis functions, in practice if we chose a big enough number for mu, and correct functions, we can represent MO with a negligible error.

(11)

Clemens C. J. Roothaan George G. Hall

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If we use the LCAO and introducing this expansion in the core Hamiltonian we obtain

P = 2 C ,iCii

occ

h! = 2 C ,iCi , h!

i

occ

Here the core Hamiltonian matrix elements are defined over atomic orbitals mu and nu instead of molecular orbitals

h!

To simplify we introduce now the closed shell density matrix

And the above equation leads us:

h! = P , h!

(22)

(23)

(24)

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j! = 2 Ci , CiC jC j

12

,

i , j

occ

If we use the LCAO and introducing this expansion in the Coulomb operator in a similar way we obtain, and molecular orbital integralsare reduced to additions of integrals over atomic orbitals

Atomic orbitals form the basis for MO theory calculus and are called basis set

This basis set and geometry is the essential input for calculations in MO theory

Expansion coefficients are the MO coefficients, and are the values we are searching for and we will try to find it in our calculations.

Ci


(25)

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E= 2 C ,iCi , H

i

occ

+2 Ci , CiC jC j

12

,

i , j

occ

ij= CiSC j = i , j

,

Putting together the core Hamiltonian and the Coulomb operatorwith the core Hamiltonian matrix elements we have: h! =H

This is the expression for Hartree-Fock energy with the approximation of LCAO

The HF equations are derived from the premise that we need to find out the best coefficients in eq. (26) in order to minimize the E, and we need to take into account the restriction of orthonormalityof MO

S represent an element of the overlap matrix


(26)

(27)

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Is convenient to write eq (27) in matrix form. To work with quadraticmatrices of same dimensions all atomic and molecular orbitals aretaken into account

Fc= Sc

This is the Roothaan-Hall equation system. F is the Fock matrix, c isthe MO coefficient matrix, S the overlap matrix and lambda the Lagrange multiplier matrix.

(28)

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Simplifying by diagonalization of Lagrange multiplier matrix


with as diagonal matrix with eigenvalues of ,and U is an orthogonal unitary transformation matrix with properties:

where I is the unit matrix

inserting into and multiplying by U

if canonical MO coefficients are introduced where

the Roothaan-Hall equation can be written in its canonical form

U=U

UUt =UtU= I

=UUt Fc= Sc

FcU= ScU

c* = cU

Fc* = Sc*

(29)

(30)

(31)

(32)

(33)

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SCF Hartree-Fock,Roothaan-Hall flow path

Calculation of MO integrals

Calculation of start density

Calculation of HF matrix

Minimization of energyby diagonalization of HF matrix

Calculation of new density matrix

Calculation of SCF energy

Consistent energy? SCF

Total energyYesNo

General procedure

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Post-Hartree Fock Methods

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The interaction between electrons in a quantum system is known as electronic correlation.

We have seen approximation solutions to the real wave function. Hence, based on the variational principle, the energy computed will be higher than the ground state energy.

Electronic Correlation

Post HFElectronic Correlation

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The difference between these two energies is known as the correlation energy (Eq. 35).

(34)

(35)

Post HFElectronic Correlation

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1) Wave function-based methods: Configuration interaction (CI)

The Mller-Plesset perturbation theory (MP) Coupled cluster (CC)

2) Electron density based methods: Density functional theory (DFT)

Hence, the final step in improving MO calculations is to recover some of this correlation energy. These methods can be classified as:

Post HF

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Configuration Interaction (CI)

(36)

Post HFConfiguration Interaction

We start with a trial wave-function, which is written as a linear combination of determinants with expansion coefficients based on the variational principle.

Ab initio Methods

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a b c d

Energy

Ener

gy


The Configuration Interaction treatment for electron correlation is based on the idea that one can improve on the HF wave-function, and therefore the energy, adding terms into HF wave-function represents promotion of electrons from occupied to virtual MOs.

single double double

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With this idea, the total wave-function could be written as a linear combination of determinants (Eq. 37).

(37)


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(38)

Full CI calculations are possible only for very small molecules, because the promotion of electrons into virtual orbitals can generate a huge number of states unless we have only a few electrons and orbitals.

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To study big systems or use big basis, the expansion of configurations are truncated allowing only certain excitations. The easy way to do it is taking into account only simple and doble excitations, known as Configuration Interaction of Single and Doble states (CISD)

(39)

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The CI Method

1) Start with a basis set functions.2) Make a SCF calculation to obtain the OM SCF occupied

and virtual.3) Use those MO to form configuration functions.4) Write the wave function as a linear combination of

configuration functions.5) Use the variational principle to obtain determinat

coefficients.

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Perturbation Theory

(40)

The basic idea of perturbation theory is to expand the energy and wave-functions of the perturbed system in power series.

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(41)

(42)

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(43)

(44)

It is assumed that the correction factor is small compared to the initial Hamiltonian for this reason the perturbed wave function and energy can be expressed in the form of Taylor expansion in powers of the perturbation parameter.

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(45)

(46)

The perturbed Schrdinger equation (eq. 41) can be written as:

In general form this equation can be written as:

Ab initio Methods

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With the condition:

after several steps Eq. 46 give us a set of equations:

(47)

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Multiplying each of these equations (Eq. 47) by:

Using the orthonormality relation:

And the relation:

Ab initio Methods

(48)

We obtain the following expressions for the nth-order energies:

.

.

.

.

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Ab initio Methods

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The Mller-Plesset Perturbation

When the operator H0 is the Fock operator has the name of Mller-Plesset perturbation theory (MPPT), the more used.

If MPPT is a power series of second order is called MP2 and is used generally for geometry optimization. When the power serie is of fourth order is called MP4 and so on.

Perturbative methods are self-consistent in size in contrast withConfiguration Interaction methods then there is no problem if we truncate the power series.

Ab initio Methods

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The Coupled Cluster Method

The coupled cluster method was introduced by Coester and Kmmel in 1958. It is a numerical technique used for describing many electron systems.

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The wave-function of the CC theory is written as an exponential:

(49)

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(50)

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Depending on how many terms are actually included in the summation for T, one obtains coupled cluster doubles (CCD), coupled cluster singles and doubles (CCSD) or coupled cluster singles, doubles and triples (CCSDT) method and so on.

(51)

(52)

(53)T

Ab initio Methods

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Ab initio Methods

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Time of processor consumed to calculate the energy of a compoundrelated to its molecular formula and theory level used for calculations

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Hartree-Fock

STO-1G

MP2 CISD MP4(SDTQ) Full CI

STO-3G

3-21G

6-31G

Infinity base set Hartree-Fock limit

Exact solution of Schrdinger equation

Improvement of Correlation Treatment

Impr

ovem

ent o

f Bas

is S

et

Advantages

Useful for a wide range of systems. Able to calculate transition states and excited states. Rigorous accuracy. Predicted properties of novel compounds.

Disadvantages

Require large computer resources. Applicable efficiently to systems of tens of atoms.

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CONCLUSIONS

In general, ab initio calculations give very good qualitative results and can yield increasingly accurate quantitative results as the molecules become smaller. The advantage of ab initio methods is that they eventually converge to the exact solution, in general the relative accuracy of results is

HF

1. The Born-Oppenheimer approximation2. The use of an incomplete basis set3. Incomplete correlation4. Omission of relativistic effects

ab initio methods are expensive, they often take enormous amounts of computer CPU time, memory, and disk space. HF method scales N^4, with N referring to basis functions. Correlated calculations calculations scale more than this, in practice extreme accurate solutions are only feasible if the molecule contains dozenelectrons or less. However, results with an accuracy rivaling that of many experimental techniques can be obtained for moderate-sizeorganic molecules. Minimally correlated methods, such as MP2 and GVB, are often used when correlation is important to the description of large molecules.

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Ab initio Methods

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William Rowan Hamilton

Erwin Rudolf Josef Alexander Schrdinger

Sir John Pople

Wolfgang Ernst Pauli Joseph-Louis Lagrange Carl Friedrich Gauss Max Born J. Robert Oppenheimer

Christian Mller

Christian Mller Milton S. Plesset

Ab initio Methods

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