1. Lecture 11. Quantum Mechanics
Lecture 11. Quantum Mechanics. Hartree-Fock Self-Consistent-Field Theory
Outline of today’s lecture
Postulates in quantum mechanics…
Schrödinger equation…
Simplify Schrödinger equation:
Atomic units, Born-Oppenheimer approximation
Solve Schrödinger equation with approximations:
Variation principle, Slater determinant, Hartree approximation,
Hartree-Fock, Self-Consistent-Field, LCAO-MO, Basis set

2. References
Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch. 1 & 8
Essentials of Computational Chemistry. Theories and Models, C. J. Cramer,
(2nd Ed. Wiley, 2004) Chapter 4
Molecular Modeling, A. R. Leach (2nd ed. Prentice Hall, 2001) Chapter 2
Introduction to Computational Chemistry, F. Jensen (1999) Chapter 3
A Brief Review of Elementary Quantum Chemistry
Molecular Electronic Structure Lecture
Wikipedia ( Search for Schrödinger equation, etc.

3. Potential Energy Surface & Quantum Mechanics
How do we obtain the potential energy E?
MM: Evaluate analytic functions (FF)
QM: Solve Schrödinger equation
3N (or 3N-6 or 3N-5) Dimension PES
for N-atom system x
Continuum
Modeling
Atomistic
Modeling
Geometry li θi
rij i
Quantum
Modeling
Charge
Force Field
For geometry optimization, evaluate
E, E’ (& E’’) at the input structure X
(x1,y1,z1,…,xi,yi,zi,…,xN,yN,zN) or {li,θi,i}.
Length
0.1 nm
1 nm
1m
1cm

4. The Schrödinger Equation
The ultimate goal of most quantum chemistry approach is
the solution of the time-independent Schrödinger equation. ?
(1-dim)
Hamiltonian operator  wavefunction
(solving a partial differential equation)

5. Postulate #1 of Quantum Mechanics
The state of a quantum mechanical system is completely specified by the wavefunction or state function that depends on the coordinates of the particle(s) and on time.
The probability to find the particle in the volume element located at r at time t is given by (Born interpretation)
The wavefunction must be single-valued, continuous, finite, and normalized (the probability of find it somewhere is 1).
= <|>
Probability
density

6. Born Interpretation of the Wavefunction: Probability Density

7. Probability Density: Examples
nodes

8. Postulate #2 of Quantum Mechanics
Once is known, all properties of the system can be obtained
by applying the corresponding operators to the wavefunction.
Observed in measurements are only the eigenvalues a which satisfy
the eigenvalue equation
eigenvalue
eigenfunction
(Operator)(function) = (constant number)(the same function)
(Operator corresponding to observable) = (value of observable)
Schrödinger equation: Hamiltonian operator  energy
with
(Hamiltonian operator)
(e.g. with )

9. Physical Observables & Their Corresponding Operators
with
(Hamiltonian operator)
(e.g. with )

10. Observables, Operators & Solving Eigenvalue Equations:
an Example
constant
number
the same function

11. The Uncertainty Principle
When momentum is known precisely, the position cannot be predicted precisely, and vice versa.
When the position is known precisely,
Location becomes
precise at the expense
of uncertainty in
the momentum

12. Postulate #3 of Quantum Mechanics
Although measurements must always yield an eigenvalue,
the state does not have to be an eigenstate.
An arbitrary state can be expanded in the complete set of
eigenvectors ( as where n  .
We know that the measurement will yield one of the values ai, but
we don't know which one. However,
we do know the probability that eigenvalue ai will occur ( ).
For a system in a state described by a normalized wavefunction ,
the average value of the observable corresponding to is given by
= <|A|>

13. The Schrödinger Equation for Atoms & Molecules

14. Atomic Units (a.u.)
Simplifies the Schrödinger equation (drops all the constants)
(energy) 1 a.u. = 1 hartree = eV = kcal/mol,
(length) 1 a.u. = 1 bohr = Å,
(mass) 1 a.u. = electron rest mass,
(charge) 1 a.u. = elementary charge, etc.
(before)
(after)

15. Born-Oppenheimer Approximation
Simplifies further the Schrödinger equation (separation of variables)
Nuclei are much heavier and slower than electrons.
Electrons can be treated as moving in the field of fixed nuclei.
A full Schrödinger equation can be separated into two:
Motion of electron around the nucleus
Atom as a whole through the space
Focus on the electronic Schrödinger equation

16. Born-Oppenheimer Approximation
(before)
(electronic)
(nuclear)
E =
(after)

17. Electronic Schrödinger Equation in Atomic Unit

18. Variational Principle
Nuclei positions/charges & number of electrons in the molecule
Set up the Hamiltonian operator
Solve the Schrödinger equation for wavefunction , but how?
Once  is known, properties are obtained by applying operators
No exact solution of the Schrödinger eq for atoms/molecules (>H)
Any guessed trial is an upper bound to the true ground state E.
Minimize the functional E[] by searching through all acceptable
N-electron wavefunctions =

19. Hartree Approximation (1928) Single-Particle Approach
Nobel lecture (Walter Kohn; 1998)
Electronic structure of matter
Impossible to search through
all acceptable N-electron
wavefunctions.
Let’s define a suitable subset.
N-electron wavefunction
is approximated by
a product of N one-electron
wavefunctions. (Hartree product)

20. Antisymmetry of Electrons and Pauli’s Exclusion Principle
Electrons are indistinguishable.  Probability doesn’t change.
Electrons are fermion (spin ½).  antisymmetric wavefunction
No two electrons can occupy the same state (space & spin).

21. Slater “determinants”
A determinant changes sign when two rows (or columns) are exchanged.
 Exchanging two electrons leads to a change in sign of the wavefunction.
A determinant with two identical rows (or columns) is equal to zero.
 No two electrons can occupy the same state. “Pauli’s exclusion principle”
“antisymmetric”
= 0
= 0

22. N-Electron Wavefunction: Slater Determinant
N-electron wavefunction aprroximated by a product of N one-electron
wavefunctions (hartree product).
It should be antisymmetrized ( ).
Hartree product is not antisymmetric!

23. Hartree-Fock (HF) Approximation
Restrict the search for the minimum E[] to a subset of , which
is all antisymmetric products of N spin orbitals (Slater determinant)
Use the variational principle to find the best Slater determinant
(which yields the lowest energy) by varying spin orbitals
= ij
(orthonormal)

24. Hartree-Fock (HF) Energy

25. Hartree-Fock (HF) Energy: Evaluation
Slater determinant
(spin orbital = spatial orbital * spin)
finite “basis set”
Molecular Orbitals as linear combinations of Atomic Orbitals (LCAO-MO)
where

26. Hartree-Fock (HF) Equation: Evaluation
No-electron contribution (nucleus-nucleus repulsion: just a constant)
One-electron operator h (depends only on the coordinates of one electron)
Two-electron contribution (depends on the coordinates of two electrons)
where

27. Potential energy due to nuclear-nuclear Coulombic repulsion (VNN)
*In some textbooks ESD doesn’t include VNN, which will be added later (Vtot = ESD + VNN).
Electronic kinetic energy (Te)
Potential energy due to nuclear-electronic Coulombic attraction (VNe)

28. Potential energy due to two-electron interactions (Vee)
Coulomb integral Jij (local)
Coulombic repulsion between electron 1 in orbital i and electron 2 in orbital j
Exchange integral Kij (non-local) only for electrons of like spins
No immediate classical interpretation; entirely due to antisymmetry of fermions
> 0, i.e., a destabilization

30. Hartree-Fock (HF) Energy: Integrals

31. Self-Interaction
Coulomb term J when i = j (Coulomb interaction with oneself)
Beautifully cancelled by exchange term K in HF scheme
 0
= 0

32. Constrained Minimization of EHF[SD]

33. Hartree-Fock (HF) Equation
Fock operator: “effective” one-electron operator
two-electron repulsion operator (1/rij) replaced by one-electron operator VHF(i)
by taking it into account in “average” way
and
Two-electron repulsion cannot be separated exactly into one-electron terms. By
imposing the separability, the Molecular Orbital Approximation inevitably involves
an incorrect treatment of the way in which the electrons interact with each other.

34. Self-Consistent Field (HF-SCF) Method
Fock operator depends on the solution.
HF is not a regular eigenvalue problem that can be solved in a closed form.
Start with a guessed set of orbitals;
Solve HF equation;
Use the resulting new set of orbitals
in the next iteration; and so on;
Until the input and output orbitals
differ by less than a preset threshold
(i.e. converged).

35. Koopman’s Theorem
As well as the total energy, one also obtains a set of orbital energies.
Remove an electron from occupied orbital a.
Orbital energy = Approximate ionization energy

36. Koopman’s Theorem: Examples

37. Restricted vs Unrestricted HF

38. Accuracy of Molecular Orbital (MO) Theory

39. EHF > E0 (the exact ground state energy)
Electron Correlation
A single Slater determinant never corresponds to the exact wavefunction.
EHF > E0 (the exact ground state energy)
Correlation energy: a measure of error introduced through the HF scheme
EC = E0 - EHF (< 0)
Dynamical correlation
Non-dynamical (static) correlation
Post-Hartree-Fock method
Møller-Plesset perturbation: MP2, MP4
Configuration interaction: CISD, QCISD, CCSD, QCISD(T)

40. Summary