Chemistry 6440 / 7440

Introduction to

Molecular Orbitals

Resources

• Grant and Richards, Chapter 2• Foresman and Frisch, Exploring Chemistry

with Electronic Structure Methods (Gaussian Inc., 1996)

• Cramer, Chapter 4• Jensen, Chapter 3• Leach, Chapter 2• Ostlund and Szabo, Modern Quantum

Chemistry (McGraw-Hill, 1982)

Potential Energy Surfaces

• molecular mechanics uses empirical functions for the interaction of atoms in molecules to calculate potential energy surfaces

• these interactions are due to the behavior of the electrons and nuclei

• electrons are too small and too light to be described by classical mechanics

• electrons need to be described by quantum mechanics

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

Schrödinger Equation

• H is the quantum mechanical Hamiltonian for the system (an operator containing derivatives)

• E is the energy of the system is the wavefunction (contains everything we

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

(as probability distribution, ||2 needs to be continuous, single valued and integrate to 1)

EH

Hamiltonian for a Molecule

• kinetic energy of the electrons• kinetic energy of the nuclei• electrostatic interaction between the electrons

and the nuclei• electrostatic interaction between the electrons• electrostatic interaction between the nuclei

nuclei

BA AB

BAelectrons

ji ij

nuclei

A iA

Aelectrons

iA

nuclei

A Ai

electrons

i e r

ZZe

r

e

r

Ze

mm

2222

22

2

22ˆ H

Solving the Schrödinger Equation

• analytic solutions can be obtained only for very simple systems

• particle in a box, harmonic oscillator, hydrogen atom can be solved exactly

• need to make approximations so that molecules can be treated

• approximations are a trade off between ease of computation and accuracy of the result

Expectation Values

• for every measurable property, we can construct an operator

• repeated measurements will give an average value of the operator

• the average value or expectation value of an operator can be calculated by:

Od

d

*

*O

Variational Theorem

• the expectation value of the Hamiltonian is the variational energy

• the variational energy is an upper bound to the lowest energy of the system

• any approximate wavefunction will yield an energy higher than the 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

exactEEd

d

var*

* ˆ

H

Born-Oppenheimer Approximation

• the nuclei are much heavier than the electrons and move more slowly than the electrons

• in the Born-Oppenheimer approximation, we freeze the nuclear positions, Rnuc, and calculate the electronic wavefunction, el(rel;Rnuc) and energy E(Rnuc)

• E(Rnuc) is the potential energy surface of the molecule (i.e. the energy as a function of the geometry)

• on this potential energy surface, we can treat the motion of the nuclei classically or quantum mechanically

Born-Oppenheimer Approximation

• freeze the nuclear positions (nuclear kinetic energy is zero in the electronic Hamiltonian)

• calculate the electronic wavefunction and energy

• E depends on the nuclear positions through the nuclear-electron attraction and nuclear-nuclear repulsion terms

• E = 0 corresponds to all particles at infinite separation

nuclei

BA AB

BAelectrons

ji ij

nuclei

A iA

Aelectrons

ii

electrons

i eel r

ZZe

r

e

r

Ze

m

2222

2

2ˆ H

d

dEE

elel

elelel

elelel *

* ˆ,ˆ

HH

Nuclear motion on the Born-Oppenheimer surface

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

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

22 /,/, tE nucnuc RaRFmaF

)(2

ˆ

ˆ,

22

nuc

nuclei

A Anuc

nucnucnucnuceltotal

Em

RH

H

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

)()()(),,,( 321321 rrrrrr

Hartree-Fock Approximation

• the Pauli principle requires that a wavefunction for electrons must change sign when any two electrons are permuted– since |(1,2)|2=|(2,1)|2, (1,2)=(2,1) (minus sign for fermions)

• the Hartree-product wavefunction must be antisymmetrized• can be done by writing the wavefunction as a determinant

– determinants change sign when any two columns are switched

n

nnn n

n

n

n

21222

111

)()1()1(

)()2()1(

)()2()1(

1

Spin Orbitals

• each spin orbital I describes the distribution of one electron

• in a Hartree-Fock wavefunction, each electron must be in a different spin orbital (or else the determinant is zero)

• an electron has both space and spin coordinates• an electron can be alpha spin (, , spin up) or beta spin

(, , spin down) • each spatial orbital can be combined with an alpha or

beta spin component to form a spin orbital• thus, at most two electrons can be in each spatial orbital

Fock Equation

• take the Hartree-Fock wavefunction

• put it into the variational energy expression

• minimize the energy with respect to changes in the orbitals while keeping the orbitals orthonormal

• yields the Fock equation

n 21

d

dE

*

*

var

H

iii F

*var / 0,i i j ijE d

Fock Equation

• 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 wavefunction is reduced

to finding a series of one electron orbitals

iii F

Fock Operator

• kinetic energy operator

• nuclear-electron attraction operator

22

2ˆ

em

T

nuclei

A iA

Ane r

Ze2V

KJVTF ˆˆˆˆˆ NE

Fock Operator

• Coulomb operator (electron-electron repulsion)

• exchange operator (purely quantum mechanical -arises from the fact that the wavefunction must switch sign when you exchange to electrons)

ijij

j

electrons

ji d

r

e }{ˆ2

J

jiij

j

electrons

ji d

r

e }{ˆ2

K

KJVTF ˆˆˆˆˆ NE

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.

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 orbitals

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

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+

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 polyatomics, approximate the molecular orbital by a linear combination of atomic orbitals (LCAO)

c

Basis Functions

’s are called basis functions• usually centered on atoms• can be more general and more flexible than

atomic orbitals• larger number of well chosen basis functions

yields more accurate approximations to the molecular orbitals

c

Roothaan-Hall Equations

• choose a suitable set of basis functions

• plug into the variational expression for the energy

• find the coefficients for each orbital that minimizes the variational energy

c

d

dE

*

*

var

H

Roothaan-Hall Equations

• basis set expansion leads to a matrix form of the Fock equations

F Ci = i S Ci

• F – Fock matrix

• Ci – column vector of the molecular orbital coefficients

I – orbital energy

• S – overlap matrix

Fock matrix and Overlap matrix

• Fock matrix

• overlap matrix

dF F

dS

Intergrals for the Fock matrix

• Fock matrix involves one electron integrals of kinetic and nuclear-electron attraction operators and two electron integrals of 1/r

• one electron integrals are fairly easy and few in number (only N2)

• two electron integrals are much harder and much more numerous (N4)

dh ne )ˆˆ( VT

2112

)2()2(1)1()1()|( ddr

Solving the Roothaan-Hall Equations

1. choose a basis set

2. calculate all the one and two electron integrals

3. obtain an initial guess for all the molecular orbital coefficients Ci

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

5. solve F Ci = i S Ci for a new set of Ci

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

Solving the Roothaan-Hall Equations

• also 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 (6 dimensional 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

Summary

• start with the Schrödinger equation• use the variational energy• Born-Oppenheimer approximation• Hartree-Fock approximation• LCAO approximation