sfsu electronicstructure lect 1

7/29/2019 SFSU ElectronicStructure Lect 1

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Introduction to Computational Chemistry

NSF Computational Nanotechnology and Molecular Engineering

Pan-American Advanced Studies Institutes (PASI) WorkshopJanuary 5-16, 2004

California Institute of Technology, Pasadena, CA

Andrew S. Ichimura


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For the Beginner

There are three main problems:

1. Deciphering the language.

2. Technical implementation.

3. Quality assessment.


3/31

Focus on

Calculating molecular structures and relative

energies.

1. Hartree-Fock (Self-Consistent Field)

2. Electron Correlation

3. Basis sets and performance


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Ab initio electronic structure theory

Hartree-Fock (HF)Electron Correlation (MP2, CI, CC, etc.)

Molecular

properties

Geometryprediction

Benchmarks for

parameterization

Transition States

Reaction coords.

Spectroscopicobservables

Prodding

Experimentalists

Goal: Insight into chemical phenomena.


5/31

Setting up the problem

What is a molecule?A molecule is composed of atoms, or, more generally as a collection of charged

particles, positive nuclei and negative electrons.

The interaction between charged particles is described by;

Coulomb Potential

Coulomb interaction between these charged particles is the only important

physical force necessary to describe chemical phenomena.

Vij V(rij ) qiqj

40rijqiqj

rijrij

qi

qj


6/31

But, electrons and nuclei are in constant motion

In Classical Mechanics, the dynamics of a system (i.e. how the systemevolves in time) is described by Newtons 2nd Law:

F maF = force

a = acceleration

r= position vector

m = particle mass

dV

dr m

d2r

dt2

In Quantum Mechanics, particle behavior is described in terms of a wavefunction, .

HY i

Y

t

Hamiltonian OperatorH

Time-dependent Schrdinger Equation

i 1; h 2


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Time-Independent Schrdinger Equation

IfH is time-independent, the time-

dependence ofY may be separated out as a

simple phase factor.

H(r,t) H(r)

Y(r,t) Y(r)eiEt/

H(r)Y(r) EY(r) Time-Independent Schrdinger Equation

Describes the particle-wave duality of electrons.


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Hamiltonian for a system with N-particles

Sum of kinetic (T) and potential (V) energy

HT

V

T Ti 2

2mii1

N

i1

N

i2 2

2mii1

N

2

xi2

2

yi2

2

zi2

i2

2

xi2

2

yi2

2

zi2

Laplacian operator

Kinetic energy

V Vij

j1

N

i1

N

qiqj

rijj1

N

i1

N

Potential energy

When these expressions are used in the time-independent Schrodinger Equation,

the dynamics of all electrons and nuclei in a molecule or atom are taken into

account.


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Born-Oppenheimer Approximation

Since nuclei are much heavier than electrons, their velocities are much

smaller. To a good approximation, the Schrdinger equation can beseparated into two parts:

One part describes the electronic wavefunction for a fixed nucleargeometry.

The second describes the nuclear wavefunction, where the electronicenergy plays the role of a potential energy.

So far, the Hamiltonian contains the following terms:H Tn Te Vne Vee Vnn

Tn Te

Vne Vee Vnn


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Born-Oppenheimer Approx. cont.

In other words, the kinetic energy of the nuclei can be treated separately. This

is theBorn-Oppenheimer approximation. As a result, the electronicwavefunction depends only on thepositions of the nuclei.

Physically, this implies that the nuclei move on a potential energy surface

(PES), which are solutions to the electronic Schrdinger equation. Under the

BO approx., the PES is independent of the nuclear masses; that is, it is the

same for isotopic molecules.

Solution of the nuclear wavefunction leads to physically meaningful

quantities such as molecular vibrations and rotations.

0

E

H H

H. + H.


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Limitations of the Born-Oppenheimer approximation

The total wavefunction is limited to one electronic surface, i.e. a particular

electronic state.

The BO approx. is usually very good, but breaks down when two (or more)

electronic states are close in energy at particular nuclear geometries. In such

situations, a non-adiabatic wavefunction - a product of nuclear and

electronic wavefunctions - must be used.

In writing the Hamiltonian as a sum of electron kinetic and potential energyterms, relativistic effects have been ignored. These are normally negligible

for lighter elements (Z


12/31

Self-consistent Field (SCF) Theory

GOAL: Solve the electronic Schrdinger equation, HeY=EY.

PROBLEM: Exact solutions can only be found for one-electron systems,e.g., H2

+.

SOLUTION: Use the variational principle to generate approximate

solutions.

Variational principle - If an approximate wavefunction is used inHeY=EY, then the energy must be greater than or equal to the exactenergy. The equality holds when Y is the exact wavefunction.

In practice: Generate the best trial function that has a number of

adjustable parameters. The energy is minimized as a function of theseparameters.


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SCF cont.

Antisymmetric wavefunctions can be written as

Slater determinants.

Since electrons are fermions, S=1/2,the total electronic wavefunction must beantisymmetric (change sign) with respect to the interchange of any two electron

coordinates. (Pauli principle - no two electrons can have the same set of quantum

numbers.)

Consider a two electron system, e.g. He or H2

. A suitable antisymmetric

wavefunction to describe the ground state is:

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

Each electron resides in a spin-orbital, a product of spatial and spin functions.

(Spin functions are orthonormal: | = | =1; | | 0)

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

Interchange the coordinates of the two electrons,

(He: 1 =2 = 1s)

(H2: 1 = 2 = bonding MO)


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A more general way to represent antisymmetric electronic wavefunctions is in the

form of a determinant. For the two-electron case,

1,2 1(1) 2(1)

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

For an N-electron N-spinorbital wavefunction,

SD

1 1 2(1) N(1)

1 2 2(2) N(2)

1 N 2(N) N(N)

, i |j ij

ASlater Determinant (SD) satisfies the antisymmetry requirement.

Columns are one-electron wavefunctions, molecular orbitals.Rows contain the electron coordinates.

One more approximation: The trial wavefunction will consist of a single SD.

Now the variational principle is used to derive the Hartree-Fock equations...

SCF cont.


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Hartree-Fock Equations

(1) Reformulate the Slater Determinant as,

A 1(1)2(2) N(N) A is the diagonal producA the antisymmetrizer

A 1

N!

(1)p Pp 0

N1

1

N!

1 Pij Pijkijk

ij

Pis the permutation operator. Pij permutes two electron coordinates .

(2) He Te

Vne Vee

Vnn

Te 12

i2

i

N

Vne

ZaRa ria

i

N

Vee 1ri rjji

N

i

N

Vnn ZaZbRa Rbb a

a

One electron

termsDepends on

two electrons


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hi 1

2i

2 Za

Ra ria

gij 1

ri rj

He hi

i1

N

gijji

N

i

N

Vnn

One-electron operator - describes electron

i, moving in the field of the nuclei.

Two-electron operator - interelectron

repulsion.

Hamiltonian

Expectation value over

Slater Determinant

Ee | He |

Ee

A |

He |

A (1)p

p 0

N1

|

He |

P

(3) Calculation of the energy.

Examine specific integrals:

| Vnn | VnnNuclear repulsion does not depend

on electron coordinates.


18/31

The one-electron operator acts only on electron 1 and yields

an energy, h1, that depends only on the kinetic energy and

attraction to all nuclei.

For coordinate 1,

| h1 | 1(1)2(2) N(N) | h1 | 1(1)2(2) N(N) 1(1) |

h1 |1(1) 2(2) |2 (2) N(N) |N(N) h1

| g12 | 1(1)2(2) N(N) | g12 | 1(1)2(2) N(N) 1(1)2 (2)| g12 |1(1)2 (2) 3 (3) |3 (3) N(N) |N(N)

= 1(1)2 (2)| g12 |1(1)2 (2) J12

Coulomb integral,J12: represents the classical repulsion

between two charge distributions 12(1) and 2

2(2).

| g12 |P12 1(1)2(2) N(N) | g12 | 2(1)1(2) N(N)

1(1)2 (2)| g12 |2 (1)1(2) 3 (3) |3 (3) N(N) |N(N)

= 1(1)2 (2)| g12 |2 (1)1(2) K12

Exchange integral,K12: no classical analogue. Responsible for

chemical bonds.


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The expression for the energy can now be written as:

Sum of one-electron, Coulomb,

and exchange integrals, and Vnn.

To apply the variational principle, the Coulomb and Exchange integrals are

written as operators,

Ee i |

hi |ii1

N

1

2 j |

Ji |j j |

Ki |j

jN

iN

Vnn

Ji |j(2) i(1) | g12 |i(1) j (2)

Ki |j (2) i(1) | g12 |j(1) i(2)

Ee

hii1

N

1

2 (JijKij )j

N

iN

Vnn

The objective now is to find the best orbitals (i, MOs) that minimize theenergy (or at least remain stationary with respect to further changes in i),

while maintaining orthonormality ofi.


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Employ the method of Langrange Multipliers:

f(x1,x2, xN)

g(x1,x2, xN) 0L(x1,x2, xN,) f(x1,x2, xN) g(x1,x2, xN)

OptimizeL such thatLxi

0,Li

0

Function to optimize.

Rewrite in terms of another function.

Define Lagrange

function.

Constrained optimization ofL.

L E ij i |j ij ij

N

L E ij i |j i |j ij

N

0

In terms of molecular orbitals, the Langrange function is:

Change inL with respect to small

changes in i should be zero.

E i |hi |i i |

hi |i i1

N

i | Jj Kj |i i | Jj Kj |iij

N

Change in the energy with respect changes in i.


21/31

Define the Fock Operator, Fi

Fi hi Jj Kj j

N

Effective one-electron operator, associatedwith the variation in the energy.

E i |Fi |i i |

Fi |ii1

N

Change in energy in termsof the Fock operator.

L i | Fi |i i | Fi |i i1N

ij i |j i |j ijN

0

According to the variational principle, the best orbitals, i, will make L=0.

Fii ijjj

N

After some algebra, the final expression becomes:

Hartree-Fock Equations


22/31

After a unitary transformation, ij0 and iii.

Fii ' ii ' HF equations in terms ofCanonical MOs anddiagonal Lagrange multipliers.

i i '| Fi |i ' Lagrange multipliers can be interpreted asMO energies.

Note:

1. The HF equations cast in this way, form a set of pseudo-eigenvalue

equations.

2. A specific Fock orbital can only be determined once all the other

occupied orbitals are known.

3. The HF equations are solved iteratively. Guess, calculate the

energy, improve the guess, recalculate, etc.4. A set of orbitals that is a solution to the HF equations are called

Self-consistent Field (SCF) orbitals.

5. The Canonical MOs are a convenient set of functions to use in the

variational procedure, but they are not unique from the standpoint

of calculating the energy.


23/31

Koopmans Theorem

The ionization energy is well approximated by the orbital energy, i.

* Calculated according to Koopmans theorem.


24/31

Basis Set Approximation

For atoms and diatomic molecules, numerical HF methods are available.

In most molecular calculations, the unknown MOs are expressed in terms of aknown set of functions - a basis set.

Two criteria for selecting basis functions.

I) They should be physically meaningful.

ii) computation of the integrals should be tractable.

It is common practice to use a linear expansion of Gaussian functions in the MObasis because they are easy to handle computationally.

Each MO is expanded in a set of basis functions centered at the nuclei and arecommonly called Atomic Orbitals.

(Molecular orbital = Linear Combination of Atomic Orbitals - LCAO).


25/31

MO Expansion

i ci

M

Fi ci

M

i ci

M

FC SC

F |F |

S

|

LCAO - MO representation

Coefficients are variational parameters

HF equations in the AO basis

Matrix representation of HF eqns.

Roothaan-Hall equations (closed shell)

F - element of the Fock matrix

S - overlap of two AOs

Roothaan-Hall equations generate M molecular orbitals from M basis functions.

N-occupied MOs

M-N virtual or unoccupied MOs (no physical interpretation)


26/31

Total Energy in MO basis

One-electron integrals, M2 Two-electron integrals, M4

Computed at the start; do not change

Products of AO coeff form Density Matrix, D

E i |hi |i

i1

N

12

ij | g|ij ij | g|ji j

N

i

N

Vnn

E cici

M

| hi | i1

N

12

cicjcicj | g| | g| Vnn

M

ij

N

Total Energy in AO basis

D cjcjj

occ.MO

; D cicii

occ.MO


27/31

General SCF Procedure

Obtain initial guess

for coeff., c i,form

the initial D

Form the Fock matrix

Diagonalize the Fock Matrix

Form new Density Matrix

Two-electronintegrals

Iterate


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Computational Effort

Accuracy

As the number of functions increases, the accuracy of the Molecular Orbitals

improves.

As M, the complete basis set limit is reached Hartree-Fock limit.

Result: The best single determinant wavefunction that can be obtained.

(This is not the exact solution to the Schrodinger equation.)

Practical Limitation In practice, a finite basis set is used; the HF limit is never reached.

The term Hartree-Fock is often used to describe SCF calculations with

incomplete basis sets.

Formally, the SCF procedure scales as M4 (the number of basis

functions to the 4th power).


29/31

Restricted and Unrestricted Hartree-Fock

1

2

3

4

5

RHF

Singlet

ROHF

Doublet

UHF

Doublet

Energy

Restricted Hartree-Fock (RHF)

For even electron, closed-shell singlet states, electrons in a given MO

with and spin are constrained to have the same spatial dependence.

Restricted Open-shell Hartree-Fock (ROHF)

The spatial part of the doubly occupied orbitals are restricted to be the same.

Unrestricted Hartree-fock (UHF)

and spinorbitals have different spatial parts.

Spinorbitalsis(n)


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Comparison of RHF and UHF

R(O)HF

and spins have same spatialpart

Wavefunction, , is an

eigenfunction of S2 operator.

For open-shell systems, theunpaired electron () interacts

differently with and spins.

The optimum spatial orbitals are

different. Restricted

formalism is not suitable for spin

dependent properties.

Starting point for more advanced

calculations that include electron

correlation.

UHF

and spins have differentspatial parts

Wavefunction is notan

eigenfunction of S2. may be

contaminated with states of

higher multiplicity (2S+1).

EUHF ER(O)HF

Yields qualitatively correct

spin densities.

Starting point for more

advanced calculations that

include electron correlation.


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Ab Initio(latin, from the beginning) Quantum Chemistry

Summary of approximations Born-Oppenheimer Approx.

Non-relativistic Hamiltonian Use of trial functions, MOs, in the variational procedure

Single Slater determinant

Basis set, LCAO-MO approx.

RHF, ROHF, UHF

Consequence of using a single Slater determinant and

the Self-consistent Field equations:

Electron-electron repulsion is included as an average effect. The electron

repulsion felt by one electron is an average potential field of all the others,

assuming that their spatial distribution is represented by orbitals. This is

sometimes referred to as theMean Field Approximation.

Electron correlation has been neglected!!!

sfsu electronicstructure lect 1

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