AB INITIO MOLECULAR ORBITAL

DefinitionThe term ab initio means from first

principles. It does not mean that we are solving the Schrödinger equation exactly.

It means that we are selecting a method that in principle can lead to a reasonable approximation to the solution of the Schrödinger equation and then selecting a basis set that will implement that method in a reasonable way.

Hartree-Fock method

The essential idea of the Hartree-Fock or molecular orbital method is that, for a closed shell system, the electrons are assigned two at a time to a set of molecular orbitals.

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Here each molecular orbital i is now expanded as a linear combination of basis functions, µ Our aim is to find the value of the coefficients Cµi that gives the best molecular orbitals. The sum is over n basis functions. n is the number of basis functions chosen for the system.

n

iC1

We expand each molecular orbital in terms of a set of basis functions which are normallycentred on the atoms in the molecule.

Basis of M.O. Theory...Basis of M.O. Theory...Three Simplifying assumptions are employed to ‘solve’ the

Schrödinger equation approximately:– Born-Oppenheimer approximation allows separate

treatment of nuclei and electrons– Hartree-Fock independent electron approximation allows

each electron to be considered as being affected by the sum (field) of all other electrons.

– LCAO Approximation

Variational Principle

Schrödinger equation

The energy of an isolated molecule can be obtained by the solution of the Schrödinger equation. In its time-independent form

is the Hamiltonian operator, is the wave function, and E is the energy of the system

H

EH ˆ

Hamiltonian OperatorThe Hamiltonian contains one- and two-electron terms. The two-electron terms (summed over i and j) are just the repulsion potential energies between all pairs of electrons.

1/rij is the repulsion between a pair of electrons (distance rij apart).

i j iji

i rhH

1ˆ

The one-electron terms (summed over i) are more varied. For each electron, there is a kinetic energy term and a sum of attractive potential energy terms for each nucleus in the molecule.

-1/2 i2 is the kinetic energy term

ZA/rAi is the coulombic attraction between electron i and nucleus A.

ZA is the nuclear charge (atomic number) of atom A and rAi is the distance between electron i and nucleus A.

2

12

2

2

2

2

222

iii zyxi

A Ai

Aii r

Zh

Born-Oppenheimer Approx.States that electron motion is independent of nuclear motion,

thus the energies of the two are uncoupled and can be calculated separately.

Derives from the large difference in the mass of nuclei and electrons, and the assumption that the motion of nuclei can

be ignored because they move very slowly compared to electrons

Htot a (Tn) + Te + Vne + Vn + Ve

Kinetic energy Potential energy

(Tn is omitted; this ignores relativistic effects, yielding the electronic Schrödinger equation.)

Hartree-Fock ApproximationHartree-Fock ApproximationAssumes that each electron experiences all the others only as

a whole (field of charge) rather than individual electron-electron interactions.

Introduces a Fock operator F:

F

which is the sum of the kinetic energy of an electron, a potential that one electron would experience for a fixed

nucleus, and an average of the effects of the other electrons.

Born-Oppenheimer approximation The masses of the nuclei are much larger and their velocities much smaller than those of the electrons. The Schrödinger equation by separating it into two parts, one describing the motions of the electrons in a field of fixed nuclei and the other describing the motions of the nuclei. The electronic Schrödinger equation:

elelelel EH ˆ

The orbital approximation

The orbital approximation assume that each electron is associated with a separate one-electron wavefunction or spin orbital

Hartree proposed that the wavefunction could be expressed simply as a product of spin orbitals, one for each electron:

=  1(1)  2(2) ....  n(n)

The LCAO approximation

Each spin orbital is actually a product of a spatial function,  i(x,y,z), and a spin function, a or  b . The spatial molecular orbitals,  i, are usually expressed as linear combinations of a finite set of known one-electron functions. This expansion is called a linear combination of atomic orbitals (LCAO)

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LCAO ApproximationElectron positions in molecular orbitals can be approximated by a

Linear Combination of Atomic Orbitals.

This reduces the problem of finding the best functional form for the molecular orbitals to the much simpler one of optimizing a set of

coefficients (cn) in a linear equation:

= c1 f1 + c2 f2 + c3 f3 + c4 f4 + …

where is the molecular orbital wavefunction and fn represent atomic orbital wavefunctions.

Variational Principle

The energy calculated from any approximation of the wavefunction will be higher than the true energy.

The better the wavefunction, the lower the energy (the more closely it approximates reality).

Changes are made systematically to minimize the calculated energy.

At the energy minimum (which approximates the true energy of the system), dEcalc-real ~ 0.

Basis sets

A basis set is a set of mathematical equations used to represent the shapes of spaces (orbitals) occupied by the electrons and their energies.

Basis sets in common use have a simple mathematical form for representing the radial distribution of electron density.

Most commonly used are Gaussian type basis sets, which approximate the better, but more complicated Slater-Type orbitals (STO).

Slater-type orbitals (STO)Slater-type orbitals describe the

electron distribution quite well, but they are not simple

enough to manipulate mathematically.

Several Gaussian-type orbitals can be added together to

approximate the STO. Here 4 GTO’s mimic 1 STO fairly

well.

Basis SetsSTO-3G (Slater-type orbitals approximated by 3 Gaussian functions)… a minimal basis set, commonly used in Semi-

Empirical MO calculations.

Derivation of the 2-electron secular equation

The energy is calculated by taking into account the interaction of each electron with the average field of all the other electrons. For each electron i, the Schrödinger equation may be written as:

iiif f is the one-electron Fock operator

which includes kinetic and potential energy terms of one electron in the average field of the others.

LCAO approximation for i:

SCFC

CC

iii

iii**

Multiplying both sides by  * and integrating gives

F is called the Fock matrix and S is called the overlap matrix

• the i (the eigenvalues) represent the energies of the orbitals  i, • the Ci (the eigenvectors) represent the molecular orbitals coefficients of  i.

n

ii C

N

ii

N

i CCf11

iii SCFC

Self-Consistent-Field (SCF) theoryFock matrix itself depends on the

coefficients Ci and therefore it is necessary to use an iterative procedure when solving the equation.

We guess values for the Ci initially and the variational principal (which states that Eguess is always > Etrue) allows us to optimise the Ci until we reach self consistency

Electron configurationClose-shell system-all electrons are pairedOpen-shell system - to pair up as many

electrons as possible – A system containing unpaired electrons (eg

radicals, biradicals, etc). •ROHF: Restricted Open Shell

Hartree-Fock method •UHF: Unrestricted Hartree-Fock The alpha and beta molecular orbitals

are not constrained to have the same molecular orbital coefficients.

multiplicity = 2S + 1S = the number of unpaired electrons x 1/2

Outline of a calculation Read input & calculate a geometry Assign basis set Calculate nuclear repulsion energy Calculate integrals Assign electronic configuration Generate initial guess Perform self-consistent field iterations (i.e.

calculate the electronic energy) Calculate total energy = nuclear repulsion

+ electronic Perform electron density analysis Carry out further steps...

Assumptions of ab initio quantum chemistry Assume that the Born-Oppenheimer

approximation holds (ie. that the nuclei remain fixed on the scale of electron movement). This assumption can become invalid when two electronic states lie very close together.

Assume that basis sets adequately represent molecular orbitals.

Assume that electron correlation is adequately included within a certain level of theory

Capabilities of ab initio quantum chemistry Can calculate the geometries and energies of

equilibrium structures, transition structures, intermediates, and neutral and charged species

Can calculate ground and excited states Can handle any electron configuration Can handle any element Can calculate wavefunctions and detailed

descriptions of molecular orbitals Can calculate atomic charges, dipole moments,

multipole moments, polarisabilities, etc. Can calculate vibrational frequencies, IR and

Raman intensities, NMR chemical shifts Can calculate ionisation energies and electron

affinities Can include the electrostatic effects on solvation

Limitations, strengths & reliability of ab initio quantum chemistryLimitations Requires more cpu time than empirical or semi-empirical

methods Can treat smaller molecules than empirical and semi-

empirical methods Calculations are more complex Have to worry about electronic configuration

Strengths No experimental bias Can improve a calculation in a logical manner (basis sets,

level of theory) Provides information on intermediate species, including

spectroscopic data Can calculate novel structures (no experimental data is

required) Can calculate any electronic state