Chapter 3Electronic Structures

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Introduction• Wave function System• Schrödinger Equation Wave function

– Hydrogenic atom– Many-electron atom

• Effective nuclear charge• Hartree product• LCAO expansion (AO basis sets)

– Variational principle– Self consistent field theory– Hartree-Fock approximation

• Antisymmetric wave function• Slater determinants

– Molecular orbital• LCAO-MO

– Electron Correlation

Potential Energy Surfaces• Molecular mechanics uses empirical

functions for the interaction of atoms in molecules to approximately calculate potential energy surfaces, which are due to the behavior of the electrons and nuclei

• Electrons are too small and too light to be described by classical mechanics and need to be described by quantum mechanics

• Accurate potential energy surfaces for molecules can be calculated using modern electronic structure methods

Wave Function is the wavefunction (contains everything we

are allowed to know about the system)• ||2 is the probability distribution of the

particles• Analytic solutions of the wave function can be

obtained only for very simple systems, such as particle in a box, harmonic oscillator, hydrogen atom

• Approximations are required so that molecules can be treated

• The average value or expectation value of an operator can be calculated byO

d

d

*

*O

The Schrödinger Equation• Quantum mechanics explains how entities like electrons

have both particle-like and wave-like characteristics. • The Schrödinger equation describes the wavefunction of a

particle.• The energy and other properties can be obtained by

solving the Schrödinger equation.– The time-dependent Schrödinger equation

– If V is not a function of time, the Schrödinger equation can be simplified by using the separation of variables technique.

– The time-independent Schrödinger equation

t

tritrV

m

),(),(

22

2

/)()()(),( iEtettrtr

)()(H rEr

)(2

H 22

rVm

• The various solutions of Schrödinger equation correspond to different stationary states of the system. The one with the lowest energy is called the ground state.

)()(H 111 rEr )()(H 222 rEr

)()(H 333 rEr 321 EEE

1

Coordinate

En

erg

y

2 3

Wave Functions at Different States

PES’s of Different States

The Variational Principle• Any approximate wavefunction (a trial

wavefunction) will always yield an energy higher than the real ground state energy

– Parameters in an approximate wavefunction can be varied to minimize the Evar

– this yields a better estimate of the ground state energy and a better approximation to the wavefunction

– To solve the Schrödinger equation is to find the set of parameters that minimize the energy of the resultant wavefunction.

whenEE )(

whenEE )(

The Molecular Hamiltonian• For a molecular system

– The Hamiltonian is made up of kinetics and potential energy terms.

– Atomic Units• Length a0

• Charge e• Mass me

• Energy hartree

),( Rr

j jk jk

kj

r

ee

m 0

22

4

1

2H

I IJ IJ

JI

i ij ijI i Ii

I

R

eZZ

r

e

r

eZV

222

04

1

o

2

2

0 A 52917725.0em

ae

0

2

a

ehartree

The Born-Oppenheimer Approximation

• The nuclear and electronic motions can be approximately separated because the nuclei move very slowly with respect to the electrons.

• The Born-Oppenheimer (BO) approximation allows the two parts of the problem can be solved independently.– The Electronic Hamiltonian neglecting the kinetic energy

term for the nuclei.

– The Nuclear Hamiltonian is used for nuclear motion, describing the vibrational, rotational, and translational states of the nuclei.

i I IJ JI

JI

i ij jiI i iI

Ii

elec

RR

eZZ

rr

e

rR

eZ 2222

2

1H

),()(),(H RrRRr eleceffelecelec E

)()(H RR effnuclnucl ET

Nuclear motion on the BO surface

• Classical treatment of the nuclei (e,g. classical trajectories)

• Quantum treatment of the nuclei (e.g. molecular vibrations)

2

2

,,t

E nuc

nuc

R

aR

FmaF

)(2

ˆ

ˆ,

22

nuc

nuclei

A Anuc

nucnucnucnuceltotal

Em

RH

H

Solving the Schrödinger Equation

• An exact solution to the Schrödinger equation is not possible for most of the molecular systems.

• A number of simplifying assumptions and procedures do make an approximate solution possible for a large range of molecules.

Hartree Approximation• Assume that a many electron

wavefunction can be written as a product of one electron functions

– If we use the variational energy, solving the many electron Schrödinger equation is reduced to solving a series of one electron Schrödinger equations

– Each electron interacts with the average distribution of the other electrons

– No electron-electron interaction is accounted explicitly

)()()()( 2211 nn rrrr

Hartree-Fock Approximation• The Pauli principle requires that a wavefunction

for electrons must change sign when any two electrons are permuted (1,2) = - (2,1)

• The Hartree-product wavefunction must be anti-symmetrized can be done by writing the wave function as a determinant

• Slater Determinant or HF wave function

1 1 1

2 2 21 2

(1) (2) ( )

(1) (2) ( )1

!

(1) (1) ( )

n

n n n

n

n

n

n

Spin Orbitals• Each spin orbital i describes the distribution of

one electron (space and spin)• In a HF wavefunction, each electron must be in a

different spin orbital (or else the determinant is zero)

• Each spatial orbital can be combined with an alpha (, , spin up) or beta spin (, , spin down) component to form a spin orbital

• Slater Determinant with Spin Orbitals

)()()()()()()()()()(

)4()()4()()4()()4()()4()(

)3()()3()()3()()3()()3()(

)2()()2()()2()()2()()2()(

)1()()1()()1()()1()()1()(

!

1)(

2

2

2

2

2

2211

442424141

332323131

222222121

112121111

nrnrnrnrnr

rrrrr

rrrrr

rrrrr

rrrrr

nr

nnnnn n

n

n

n

n

Fock Equation• Take the Hartree-Fock wavefunction and

put it into the variational energy expression

• Minimize the energy with respect to changes in each orbital

yields the Fock equation

n 21

d

dE

*

*

var

H

iii F

0/var iE

Fock Equation• Fock equation is an 1-electron problem

– The Fock operator is an effective one electron Hamiltonian for an orbital

is the orbital energy

• Each orbital sees the average distribution of all the other electrons

• Finding a many electron wave function is reduced to finding a series of one electron orbitals

iii F

Fock Operator

• kinetic energy & nuclear-electron attraction operators

• Coulomb operator: electron-electron repulsion)

• Exchange operator: purely quantum mechanical - arises from the fact that the wave function must switch sign when a pair of electrons is switched

22

2ˆ

em

T

nuclei

A iA

Ane r

Ze2

V

KJVTF ˆˆˆˆˆ NE

ijij

j

electrons

ji d

r

e }{ˆ2

J

jiij

j

electrons

ji d

r

e }{ˆ2

K

Solving the Fock Equations

1. obtain an initial guess for all the orbitals i

2. use the current i to construct a new Fock operator

3. solve the Fock equations for a new set of i

4. if the new i are different from the old i, go back to step 2.

When the new i are as the old i, self-consistency has been achieved. Hence the method is also known as self-consistent field (SCF) method

iii F

Hartree-Fock Orbitals• For atoms, the Hartree-Fock orbitals

can be computed numerically• The ‘s resemble the shapes of the

hydrogen orbitals (s, p, d …)

• Radial part is somewhat different, because of interaction with the other electrons (e.g. electrostatic repulsion and exchange interaction with other electrons)

,)( YrRi

Hartree-Fock Orbitals• For homonuclear diatomic molecules, the

Hartree-Fock orbitals can also be computed numerically (but with much more difficulty)

• the ‘s resemble the shapes of the H2+ (1-

electron) orbitals , , bonding and anti-bonding orbitals

LCAO Approximation• Numerical solutions for the Hartree-

Fock orbitals only practical for atoms and diatomics

• Diatomic orbitals resemble linear combinations of atomic orbitals, e.g. sigma bond in H2

1sA + 1sB

• For polyatomic, approximate the molecular orbital by a linear combination of atomic orbitals (LCAO)

c

Basis Function• The molecular orbitals can be expressed

as linear combinations of a pre-defined set of one-electron functions know as a basis functions. An individual MO is defined as

: a normalized basis function

• ci : a molecular orbital expansion coefficients

• gp : a normalized Gaussian function;

• dp : a fixed constant within a given basis set

N

ii c1

1p

ppgd2

),( rlmn ezycxrg

N

pppii gdc

1

The Roothann-Hall Equations• The variational principle leads to a set of equations

describing the molecular orbital expansion coefficients, ci, derived by Roothann and Hall

– Roothann Hall equation in matrix form

• : the energy of a single electron in the field of the bare nuclei

• : the density matrix;

– The Roothann-Hall equation is nonlinear and must be solved iteratively by the procedure called the Self Consistent Field (SCF) method.

NcSFN

ii ,2,10)(1

SCFC

N N

core

1 121 ||PHF

coreH

P

..

1

*2Porbocc

iiicc

Roothaan-Hall Equations• Basis set expansion leads to a

matrix form of the Fock equations

F Fock matrix Ci column vector of the MO

coefficients i orbital energy S overlap matrix

iii SCFC

Solving the Roothaan-Hall Equations

1. choose a basis set2. calculate all the one and two electron

integrals3. obtain an initial guess for all the

molecular orbital coefficients Ci

4. use the current Ci to construct a new Fock matrix

5. solve for a new set of Ci

6. if the new Ci are different from the old Ci, go back to step 4.

iii SCFC

Solving the Roothaan-Hall Equations

• Known as the self consistent field (SCF) equations, since each orbital depends on all the other orbitals, and they are adjusted until they are all converged

• Calculating all two electron integrals is a major bottleneck, because they are difficult (6D integrals) and very numerous (formally N4)

• Iterative solution may be difficult to converge• Formation of the Fock matrix in each cycle is

costly, since it involves all N4 two electron integrals

The SCF Method• The general strategy of SCF method

– Evaluate the integrals (one- and two-electron integrals)

– Form an initial guess for the molecular orbital coefficients and construct the density matrix

– Form the Fock matrix– Solve for the density matrix– Test for convergence.

• If it fails, begin the next iteration. • If it succeeds, proceed on the next tasks.

Closed and Open Shell Methods

• Restricted HF method (closed shell)– Both and electrons are forced to be in the

same orbital

• Unrestricted HF method (open shell) and electrons are in different orbitals

(different set of ci )

N

iii c1

N

ii

N

ii

c

c

1

1

AO Basis Sets• Slater-type orbitals (STOs)

• Gaussian-type orbitals (GTOs)

• Contracted GTO (CGTOs or STO-nG)

• Satisfied basis sets• Yield predictable chemical

accuracy in the energies• Are cost effective• Are flexible enough to be

used for atoms in various bonding environments

rrYNr nmlmlnmln exp,,, 1

,,,,,,

2',,,,, exp,, rzyxNr cba

cbacba orbital radial size

,,,, ,,,,, rcr icbai

imln

Different types of Basis• The fundamental core & valence basis

– A minimal basis: # CGTO = # AO– A double zeta (DZ): # CGTO = 2 #AO– A triple zeta (TZ): # CGTO = 3 # AO

• Polarization Functions– Functions of one higher angular momentum

than appears in the valence orbital space

• Diffuse Functions– Functions with higher principle quantum

number than appears in the valence orbital space

Widely Used Basis Functions

• STO-3G: 3 primitive functions for each AO– Single : one CGTO function for each AO– Double : two CGTO functions for each AO– Triple : three CGTO functions for each AO

• Pople’s basis sets– 3-21G– 6-311G– 6-31G**– 6-311++G(d,p)

Polarization functions

Diffuse functions

core valence

Hartree Equation• The total energy of the atomic

orbital j

• The LCAO expansion

)'()('

2)'()(

2

22

2

rrrr

err

r

Ze

m kjk

kjjjjjj

,,

,

jjje

jjje

jj

CCh

h

C