Chapter 3 Electronic Structures

1
Chapter 3Electronic Structures

• 728345

2
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

3
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

4
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 by

5
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

6

• 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.

Wave Functions at Different States
PESs of Different States
7
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.

8
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

9
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.

10
Nuclear motion on the BO surface

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

11
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.

12
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

13
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

14
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

15
Fock Equation

• Take the Hartree-Fock wavefunction and put it
  into the variational energy expression
• Minimize the energy with respect to changes in
  each orbitalyields the Fock equation

16
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

17
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

18
Solving the Fock Equations

• obtain an initial guess for all the orbitals ?i
• use the current ?i to construct a new Fock
  operator
• solve the Fock equations for a new set of ?i
• 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

19
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)

20
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

21
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)

22
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
• c?i a molecular orbital expansion coefficients
• gp a normalized Gaussian function
• d?p a fixed constant within a given basis set

23
The Roothann-Hall Equations

• The variational principle leads to a set of
  equations describing the molecular orbital
  expansion coefficients, c?i, 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.

24
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

25
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 for a new set of
   Ci
6. if the new Ci are different from the old Ci, go
   back to step 4.

26
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

27
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.

28
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 c?i )

29
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

orbital radial size
30
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

31
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
• Poples basis sets
• 3-21G
• 6-311G
• 6-31G
• 6-311G(d,p)

valence
core
Diffuse functions
Polarization functions
32
Hartree Equation

• The total energy of the atomic orbital ??j
• The LCAO expansion