Hartree-Fock Theory Patrick Tamukong North Dakota State University Department of Chemistry & Biochemistry Fargo, ND 58108-6050 U.S.A. June 10, 2015

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, 1882-1970) Szabo, A. Ostlund, N. S. McGraw-Hill, New York, 1989, p. 40. 2

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

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. = 27.212 eV = 627.51 Kcal/mol = 2.1947·105 cm-1 1Kcal/mol = 4.184KJ/mol Boltzmann’s constant: k = 1.38066·10-23J/K Avogadro’s number: NA= 6.02205·1023mol-1 Rydberg constant: R∞= 1.097373·107m-1 Compton wavelength of electron: λC= 2.426309·10-12m Stefan-Boltzmann constant: σ = 5.67032·108W/(m2K4) 4

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

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

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

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

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 1932. 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 1896-1997 Robert Sanderson Mulliken 1996-1986 Nobel 1966 John Clarke Slater 1900-1976 Sir John Lennard-Jones 1894-1954 Charles Alfred Coulson 1910-1974 9

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 1894-1954 Linus Carl Pauling 1901-1994 Nobel 1962 10

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

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

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

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

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

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

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

Douglas Rayner Hartree English 1897-1958 Hartree-Fock Theory Write the Hamiltonian as a sum of Fock operators Douglas Rayner Hartree English 1897-1958 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

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