1. Chapter 3 Electronic Structures
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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
PES’s of Different States
Coordinate
Energy

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 orbital yields 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
choose a basis set
calculate all the one and two electron integrals
obtain an initial guess for all the molecular orbital coefficients Ci
use the current Ci to construct a new Fock matrix
solve for a new set of Ci
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
Pople’s basis sets
3-21G
6-311G
6-31G**
G(d,p)
valence
core
Diffuse functions
Polarization functions

32. Hartree Equation The total energy of the atomic orbital j
The LCAO expansion