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  1. Hartree-Fock Theory Erin Dahlke Department of Chemistry University of Minnesota VLab Tutorial May 25, 2006

  2. Elementary Quantum Mechanics • The Hamiltonian for a many-electron system can be written as : Potential energy Kinetic energy where the first two terms represent the kinetic energy of the electrons and nuclei, respectively, the third term is the nuclear-electron attraction, the fourth term is the electron-electron repulsion, and the fifth term is the nuclear-nuclear repulsion.

  3. Elementary Quantum Mechanics Atomic units: • The Hamiltonian for a many-electron system can be written in atomic units as : Kinetic energy Potential energy

  4. The Born-Oppenheimer Approximation To a good approximation one can assume that the electrons move in a field of fixed nuclei Kinetic energy Potential energy

  5. units of probability density What is a Wave Function? For every system there is a mathematical function of the coordinates of the system  the wave function () This function contains within it all of the information of the system. In general,for a given molecular system, there are many different wave functions that are eigenfunctions of the Hamiltonian operator, each with its own eigenvalue, E.

  6. Properties of a Wave Function The wave function must vanish at the boundaries of the system The wave function must be single-valued. The wave function must be continuous. • A description of the spatial coordinates of an electron is not enough. We must also take into account spin (). (Spin is a consequence of relativity.) • = spin up  = spin down • A many-electron wave function must be antisymmetric with respect to the interchange of coordinate (both space and spin) of any two electrons.  Pauli exclusion principle

  7. The probability of finding electron iin the volume element given by dr. Spin Orbitals and Spatial Orbitals orbital - a wave function for a single electron spatial orbital - a wave function that depends on the position of the electron and describes its spatial distribution spin orbital - a wave function that depends on both the position and spin of the electron

  8. Hartree Products Consider a system of N electrons The simplest approach in solving the electronic Schrödinger equation is to ignore the electron-electron correlation. Without this term the remaining terms are completely separable. Consider a system of N non-interacting electrons

  9. Hartree Products Each of the one-electron Hamiltonians will satisfy a one-electron Schrödinger equation. Because the Hamiltonian is separable the wave function for this system can be written as a product of the one-electron wave functions. This would result in a solution to the Schrödinger equation in which the total energy is simply a sum of the one-electron orbital energies

  10. Hartree Method Instead of completely ignoring the electron-electron interactions we consider each electron to be moving in a field created by all the other electrons From variation calculus you can show that: where Self Consistent Field (SCF) procedure 1. Guess the wave function for all the occupied orbitals Construct the one-electron operators Solve the Schrödinger equation to get a new guess at the wave function.

  11. Hartree Method The one-electron Hamiltonian h(i) includes the repulsion of electron i with electron j, but so does h(j). The electron-electron repulsion is being double counted. Sum of the one-electron orbital eigenvalues One half the electron-electron repulsion energy

  12. Hartree Products What are the short comings of the Hartree Product wave function? Electron motion is uncorrelated - the motion of any one electron is completely independent of the motion of the other N-1 electrons. Electrons are not indistinguishable Wave function is not antisymmetric with respect to the interchange of two particles

  13. Slater Determinants Consider a 2 electron system with two spin orbitals. There are two ways to put two electrons into two spin orbitals. Neither of these two wave functions are antisymmetric with respect to interchange of two particles, nor do they account for the fact that electrons are indistinguishable… Try taking a linear combination of these two possibilities

  14. Slater Determinants Try the linear combination with the addition. This wave function is not antisymmetric! What about the linear combination with the subtraction. This wave function is antisymmetric! An alternative way to write this wave function would be:

  15. Slater Determinants For an N electron system rows correspond to electrons, columns correspond to orbitals Slater Determinant  Electrons are not indistinguishable Wave function is not antisymmetric with respect to the interchange of two particles  ? Electron motion is uncorrelated

  16. Slater Determinants Is the electron motion correlated? Consider a Helium atom in the singlet state:

  17. Slater Determinants

  18. Slater Determinants What happens to the spin terms?

  19. Slater Determinants

  20. 1 0 0 1 Slater Determinants

  21. Slater Determinants After integrating out the spin coordinates, we’re left with only two terms in the integral:

  22. Coulomb Repulsion Classically the Coulomb repulsion between two point charges is given by: where q1 is the charge on particle one, q2 is the charge on particle 2, and r12 is the distance between the two point charges. An electron is not a point charge. Its position is delocalized, and described by its wave function. So to describe its position we need to use the wave function for the particle, and integrate its square modulus. (in atomic units the charge of an electron (e) is set equal to one. This is the expression for the Coulomb repulsion between an electron in orbital i with an electron in orbital j. Since is always positive, as is the Coulomb energy is always a positive term and causes destabilization.

  23. Slater Determinants Consider a Helium atom in the triplet state:

  24. Slater Determinants

  25. Slater Determinants

  26. 1 1 1 1 Slater Determinants

  27. Slater Determinants exchange integral

  28. Exchange Integral The exchange integral has no classical analog. The exchange integral gets its name from the fact that the two electrons exchange their positions as you go from the left to the right in the integrand. The exchange integrals are there to correct the Coulomb integrals, so that they take into account the antisymmetrization of the wave function. Electrons of like spin tend to avoid each other more than electrons of different spin, so the destabilization predicted by the Coulomb integrals is too high. The exchange integrals, which are always positive, lower the overall repulsion of the electron (~25%).

  29. Slater Determinants So no two electrons with the same spin can occupy the same point in space  the Pauli Exclusion principle

  30. Hartree–Fock Theory What if we apply Hartree theory to a Slater determinant wave function? one-electron Fock operator

  31. Hartree–Fock Theory Coulomb operator: Exchange operator:

  32. Self Interaction • The sums in these equations run over all values of j, including j = i. • Each of these terms contains a self-interaction term • - a Coulomb integral between an electron and itself • - an exchange integral between an electron and itself • Since both J and K contain the self-interaction term, and since we’re subtracting them from each other, the self-interaction cancels.

  33. units of probability density What is a Wave Function? For every system there is a mathematical function of the coordinates of the system  the wave function () This function contains within it all of the information of the system. In general,for a given molecular system, there are many different wave functions that are eigenfunctions of the Hamiltonian operator, each with its own eigenvalue, E.

  34. Hartree–Fock Theory For a molecular system we don’t know what the true wave function is. In general in order to approximate it we make the assumption that the true wave function is a linear combination of one-electron orbitals.

  35. Hartree–Fock Theory What happens to our Coulomb and exchange operators?? We call Pmnthe density matrix

  36. Hartree–Fock Theory One can write the one-electron Schrödinger equation as: Where we can define the following matrix elements: Fock matrix overlap matrix We can then rearrange the one-electron Schrödinger equation to get: Where we will have one such equation for each electron is our system.

  37. Hartree–Fock Theory In order find a non-trivial solution to this set of equations one can set up and solve the secular determinant: Solution of the secular determinant determines the coefficients cyi which can, in turn be used to solve for the one-electron energy eigenvalues, i.

  38. Choose a basis set Compute and store all overlap, one-electron, and two-electron integrals Choose a molecular geometry Guess initial density matrix Construct and solve the Hartree–Fock secular determinant Is new density matrix P(n) sufficiently similar to old density matrix P(n-1) Construct density matrix from occupied orbitals Replace P(n-1) with P(n) Output data for given geometry Flow chart of the implementation of Hartree–Fock Theory no yes

  39. Limitations of Hartree–Fock Theory - Energetics • Hartree–Fock theory ignores electron correlation • - cannot be used (accurately) in any process in which the • total number of paired electrons changes. • Even if the total number of paired electrons stays the same, if the nature of the bonds changes drastically HF theory can have serious problems. (i.e., isomerization reactions) • Does well for protonation/deprotonation reactions. • Can be used to compute ionization potentials and electron affinities. • Will not do well for describing systems in which there are dispersion interactions, as they are completely due to electron correlation effects, except by cancellation of errors. • HF charge distributions tend to be over polarized which give electrostatic interactions which are too large.

  40. Limitations of Hartree–Fock Theory - Geometries • HF theory tends to predict bonds to be too short, especially as you increase the basis set size. • Bad for transition state structures due to the large correlation associated with the making and breaking of partial bonds. • Nonbonded complexes tend to be far too loose, as HF theory does not account for dispersion interactions. Limitations of Hartree–Fock Theory - Charge Distributions • The magnitude of dipole moments is typically overestimated by 10–25% for medium sized basis sets. • Results are erratic with smaller basis sets.

  41. Summary • Hartree–Fock theory is an approximate solution to the electronic Schrödinger equation which assumes that each individual electron i, moves in a field created by all the other electrons. • Introduces the concept of exchange energy through the use of a Slater determinant wave function. • Ignores all other electron correlation. • Contains a self-interaction term which cancels itself out. • Often underbinds complexes. • Predicts bond lengths which are too short.

  42. References Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry. An Introduction to Advanced Electronic Structure Theory. Dover Publications: Mineola, NY; 1996. Pilar, F. L. Elementary Quantum Mechanics, 2nd Ed. Dover Publications: Mineola, NY; 2001. Cramer, C. J. Essentials of Computational Chemistry. Wiley: Chichester; 2002.