1. Hartree-Fock Theory Patrick Tamukong North Dakota State University
Department of Chemistry & Biochemistry
Fargo, ND U.S.A.
June 10, 2015

2. Julius Robert Oppenheimer
Electronic Structure Problem
Computer programs are written to solve the Schrödinger equation
Julius Robert Oppenheimer
(Berkeley- Los alamos, 1904 –1967)
Max Born
(German, )
Szabo, A. Ostlund, N. S. McGraw-Hill, New York, 1989, p. 40. 2

3. Born-Oppenheimer Approximation
The approximation used in solving the electronic structure problem
constant =0
Szabo, A. Ostlund, N. S. McGraw-Hill, New York, 1989, p. 40. 3

4. Atomic Units
Chosen for convenience such that (me=1, e=1, = h/2 = 1, ao=1, and the potential energy in the hydrogen atom (e2/ao = 1).
Other frequently used energy units:
1a.u. = eV = Kcal/mol = ·105 cm-1
1Kcal/mol = 4.184KJ/mol
Boltzmann’s constant: k = ·10-23J/K
Avogadro’s number: NA= ·1023mol-1
Rydberg constant: R∞= ·107m-1
Compton wavelength of electron: λC= ·10-12m
Stefan-Boltzmann constant: σ = ·108W/(m2K4) 4

5. Our Main Concern Static Electron Correlation
The need to include more than one electron configuration in the description of the total wave function
contributes some 80% to the total wave function at 2.80 Å but only 55% at 4.4 Å
Dynamic Electron Correlation
The need to account
for the coulomb hole 5

6. Electron Correlation in He
HF Approximation
Full Treatment
Full Treatment - HF
Helgaker et al. Molecular Electronic-Structure Theory, John Wiley & Sons Ltd, Chichester, England, 2000, p. 257. 6

7. Correlation Effect Cr2 Ground State MCSCF GVVPT2 7
Tamukong, P. K.; Theis, D.; Khait, Y. G.; Hoffmann, M. R. J. Phys. Chem. A 2012, 116, 4590. 7

8. Conceptual Picture Higher RMP2 MCSCF RHF 8
Hartree-Fock Theory lacks electron correlation and represents the total wave function as a single electron configuration
Higher
RMP2
MCSCF
RHF 8

9. Use of Molecular Orbitals
A MO is a wavefunction associated with a single electron. The use of the term "orbital" was first used by Mulliken in 1925.
MO theory was developed, in the years after valence bond theory (1927) had been established, primarily through the efforts of Friedrich Hund, Robert Mulliken, John C. Slater, and John Lennard-Jones. The word orbital was introduced by Mulliken in According to Hückel, the first quantitative use of MO theory was the 1929 paper of Lennard-Jones. The first accurate calculation of a molecular orbital wavefunction was that made by Charles Coulson in 1938 on the hydrogen molecule. By 1950, MO were completely defined as eigenfunctions of the self-consistent field
Friedrich Hund
Robert Sanderson Mulliken
Nobel 1966
John Clarke Slater
Sir John Lennard-Jones
Charles Alfred Coulson 9

10. Sir John Lennard-Jones
LCAO = MO
It was introduced in 1929 by Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Pauling for H2+.
Sir John Lennard-Jones
Linus Carl Pauling
Nobel 1962 10

11. The Variational Principle
Method of choice for approximate solutions to physical problems
The raison d'être for the LCAO
Define an approximate solution (with same boundary conditions as the eigenvectors of the Hamiltonian) to the Shrodinger equation as a LC of its unknown eigenvectors
then
The task then is to determine the optimal set of expansion coefficients which is accomplished by Lagrange’s method of undetermined multipliers 11

12. Lagrangian Method Example: One wishes to minimize subject to
In the Lagrangian method, a function (or functional) is
minimized (or maximized) subject to given equality constraints
Example: One wishes to minimize subject to
Construct the Lagrangian
Solve the equations
and y = ±1 OR
and x = 0 12

13. The Secular Equation HC = ESC Use the above philosophy
Consider small variations in
where
HC = ESC
All quantum chemistry methods solve this secular equation using different approximations. It is a matrix problem and reduces to that of matrix diagonalization. Often it is transformed into an eigenvalue problem 13

14. Matrix Eigenvalue Problem
Transform the secular equation into an eigenvalue problem by rewriting C as
HC = ESC
Hence
Eigenvalue problem
where
The above transformation is known as the symmetric (Löwdin) orthonormalization
The thus obtained matrix eigenvalue problem is the final problem solved in quantum chemistry 14

15. Hartree Approximation
The total Hamiltonian is approximated as a sum of one-electron operators and the wave function as a product of eigenvectors of those operatorsc
The variational principle then leads to
E=εi+εj+…+εn 15

16. Uncorrelated probabilities
The Problem
The form of ΨHP suggests the independence of Φi
Probability density given by ΨHP is equal to the product of monoelectronic probability densities
This is true only if each electron is completely independent of the other electrons
ΨHP - independent electron model
Φi – spin orbitals
A ♥ A♥
 PA=1/13 P♥=1/4 PA♥=1/52=PAP♥
PA is uncorrelated (independent) with P♥.
Uncorrelated probabilities
Correlated probabilities
In a n-electron system of electrons the motions of the electrons is correlated due to the Coulomb repulsion (electron-one will avoid regions of space occupied by electron two).
Electronic Hamiltonian can be rewritten:
vi is the monoelectronic term of the external potential:
Where:
E=εi+εj+…+εn
In HP, hi will act only on the wavefunction corresponding to the i-th electron. However, Vee depends on pairs of electrons so that we can not separate the variables in Schrödinger equation.
is the monoelectronic operator 16

17. Slater Determinants
The Hartree product ignores electron correlation completely and ignores Pauli’s exclusion principle
To fix this, the wave function is often written as a slater determinant or their linear combination
N is a normalization factor
In Hartree-Fock Theory, the wave function is a single slater determinant 17

18. Douglas Rayner Hartree English 1897-1958
Hartree-Fock Theory
Write the Hamiltonian as a sum of Fock operators
Douglas Rayner Hartree English
where the Hartree-Fock potential is defined in terms of coulomb and exchange operators
Vladimir Aleksandrovich Fock Russian 1898–1974
Coulomb integral
Exchange integral
Use the variational and langrangian methods to arrive at 18

19. Thank You !
Thank you for your attention.
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